QlnrnpU Intersttg IGtbrarg BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF ftenrg HI. Sage 1891 B. ab'^5 a.1 aV^li 1J06 Cornell University Library TK 147.B41 Direct and alternating current manual; wi 3 1924 004 406 199 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004406199 By the same aatbor Alternating Cuirents : An Analytical and Graphical Treatment for Students and Engineers By Frederick Bedell and Albert C. Crehorb Translated into German by Alfred H. Bucherer Theorie der Wechselstrome: in analytischer und graphisher Darstellung Translated into French by J. Bertkon Etude Analytique & Giaphique des Courants Alternatifs The Principles of the Transformer By Frederick Bedell A Laboratory Manual of Physics and Applied Electricity Arranged and Edited by Edward L. Nichols VOL. H— PART II Experiments with Alternating Currents By Frederick Bedell DIRECT AND ALTERNATING CURRENT MANUAL WITH DIRECTIONS FOR TESTING AND A DISCUSSION OF THE THEORY OF ELECTRICAL APPARATUS BY FREDERICK BEDELL, Ph.D. PROFESSOR OP APPLIED ELECTRICITY IN CORNELL UNIVERSITY ASSISTED BY CLARENCE A. PIERCE, Ph.D. SECOND EDITION, ENLARGED AND REVISED NEW YORK D. VAN NOSTRAND COMPANY 25 Park Place LONDON: CONSTABLE & CO., Ltd. 1915 E-V. Copyright, 1909 and 191 1 By Frederick Bedell All rights reserved First Edition, September, 1909 ; November, 1909 Second Edition (enlarged), July, 1911 ; (revised), December, igii August, 19x2; November. 19x2; Maxch, 1914, August, 1915 Note. — The first edition was issued under the title " Direct and Alternating Current Testing." PRESS OP The new era printing cohpany Lancaster, pa. PREFACE. This manual consists of a series of tests on direct and alter- nating current apparatus, selected with reference to their practical usefulness and instructive value. While the book has been pre- pared primarily for students, it is hoped that it may prove helpful to others. The presentation is in the form of a laboratory manual; the author, however, has not restricted himself to a mere statement of instructions for conducting tests but has directed the reader's attention to the principles that underlie the various experiments and to the significance of the results. Ex- perience has shown that theory is more readily grasped when it is thus combined with its application and that the application is more intelligently made when its broader bearings are understood. The material has been systematically arranged and it is believed that the book may be found useful for reference or as a text, aside from its use in testing. From the text proper are excluded specialized tests and those that are of limited application or require unusual testing facili- ties, such tests being described in the appendices to the several experiments. These appendices thus permit a fuller discussion of some of the details of the tests and various modifications than could be included to advantage in the text proper. The tests in general are those that can be performed in any college laboratory. No attempt has been made to make the work exhaustive or complete ; on the contrary every effort has been made to eliminate matter of secondary importance and that which is of questionable technical or pedagogical value. The aim has been to arrange an introductory series of experi- ments of a comprehensive nature, so that in a reasonable time and with a reasonable amount of effort the student may acquire VI PREFACE. the power to proceed to problems requiring a continually increas- ing initiative and originality. Although standardized tests afford the quickest way for obtaining certain desired results and, in the case of a student, for obtaining a knowledge of testing methods, the ability to conduct such tests with full instructions given is soon acquired. Beyond this point the exclusive use of standard- ized tests should be avoided. Standards in electricity serve best aS new points of departure. The student who is to become more than the " ordinary slide-rule engineer " or " mental mechanic " will have sufficient intellectual curiosity to desire more than any standardized tests can give him and should be encouraged in every way to seek new results and to devise ways and means for obtain- ing them with the facilities at hand. To attempt to formulate such work would at once deprive it of its freshness. The student may well be referred to the current technical press and to the transactions of the engineering societies for suggestions as to subject matter for further study and also as to methods to be adopted. With reference to prepared blanks and forms, the writer be- lieves that their use can be, and often is, carried too far, leading perhaps to good technical but not to good pedagogical results. In a certain sense the one who prepares the forms and lays out the work is the one who really performs the experiment, the tabu- lators of data being assistants who, for commercial work, require only a common school education. Progress undoubtedly results from the development of indi- vidualism and if room for such development is to be given in a college course — specifically in a college laboratory course — ^the more or less standardized instruction must needs be reduced so as not to fill the entire available time. The natural tendency has been quite the reverse. Two decades ago, the study of electrical engineering meant, practically, the study of direct currents, there being little else. Laboratory courses were developed in which the whole available time was well filled with test after test upon PREFACE. vii direct current generators and motors. The transformer and alternator were then added, with extensive time-consuming tests, with the apparent assumption that the full development of alter- nating currents was reached. In succeeding years came the gen- eral development of polyphase currents, the rotary converter, in- duction motor, etc., these subjects being added to a crowded course by a process of compression rather than judicious elimina- tion. The student was given more than he could possibly assimi- late. As types of machines have multiplied, it would take years to perform all the permutations and combinations of tests on all the diflferent types. But is this necessary for a student? Why not develop a student's powers by a few typical experiments on a few typical kinds of apparatus? With this end in view the writer has made selection from ma- terial which has long been collecting in the form of typewritten outlines. These have been in a process of continual evolution, frequently rewritten and used by many classes. By a process of elimination and survival, experiments consisting of a large amount of mechanical data-taking and tabulation and a relatively small amount of technical content have been dropped in favor of those experiments which have proved most effective in student de- velopment. Various demands upon the writer's time prevented his preparing for the press a book on testing a number of years ago and the present appearance of the book is due in no small way to the valuable assistance of Dr. Pierce. Meanwhile several admirable manuals have appeared, which differ, however, in aims or scope from the present work. The author hopes to find leisure, in the near future, to make good some of the omissions of the present volume and to include in a later edition additional chapters on alternating current motors and converters. The present work is self-contained and requires only such pre- liminary courses in physical and electrical measurements as are usually given in colleges. The book may be used to advantage in conjunction with standard texts on electrical engineering, as those viii PREFACE. by Franklin and Esty, S. P. Thompson, and Samuel Sheldon, or with an introductory text such as that by H. H. Norris. The experiments given in the book may be supplemented by others of an elementary, intermediate or advanced nature, as circumstances may require. The division of experiments into parts and sections will be found to add materially to the flexibility of the book. The author desires to express his appreciation of the initial instruction and inspiration of Professor H. J. Ryan and of the continuous cooperation for many years of Professor G. S. Moler. He wishes also to express his indebtedness to many who have been associated with him in laboratory instruction, in particular to Dr. A. S. McAllister, as many references in the present text bear evi- dence. He likewise desires to express his appreciation of the spirit of cooperation shown by Professors H. H. Norris and V. Karapetoff and other engineering colleagues. The author is in- debted, furthermore, to various professors and students, who have used and corrected this book in proof during the last year and to a number of engineers who have looked over the proof sheets and have made valuable suggestions. For all shortcomings the author alone is responsible. Ithaca, N. Y., July i, 1909. PREFACE TO THE SECOND EDITION. The preparation of this edition for the press has given the author an opportunity to make good certain omissions in the first edition and to include discussions of the induction motor, the induction generator, frequency changers, the synchronous motor, the synchronous converter, wave analysis and a selection of special problems. In a restricted sense the book is now complete. It is not exhaustive and has many short-comings, but it is believed that the reader who has become familiar with the principles of testing herein contained can proceed to conduct such further tests as he desires and to manipulate and use new types of apparatus without special instruction. In order to present the discussion of the circle diagram for an induction motor with the utmost conciseness and clearness, the graphical constructions for determining slip, efficiency and power factor — which are not essential to the understanding of the circle diagram piroper — ^have been omitted from the main discussion and placed in an appendix. For the same purpose, in the dis- cussion of the theory of the synchronous motor (Experiment lO-B), particular emphasis has been laid upon the essential principles which govern the operation of the motor and matters of secondary importance have been sub-ordinated. Although much has been written on the subject of wave' analysis, common experience has been that it is indeed a laborious task to familiarize one's self with the usual methods and to use them for accurately analyzing a wave. In an endeavor to elimi- nate all unnecessary labor, the author has given explicit instruc- tions for wave analysis that occupy only two pages (pp. 335-6), a numerical example for determining the old harmonics up to the seventeenth occupying two pages more. It is believed that this ix X PREFACE. method, which is based upon the work of Runge, will be found generally useful. While the problems in Chapter XII. may prove useful, they should be looked upon as chiefly suggestive; if they inspire the reader to undertake other than standardized tests, they will have served their purpose. Ithaca, N. Y., May ij 1911. CONTENTS. CHAPTER I. Direct Current Generators. Page. Experiment i-A. Generator Study and Characteristics of a Series Generator i Experiment i-B. Characteristics of a Compound Generator. . 13 CHAPTER 11. Direct Current Motors. Experiment 2-A. Operation and Speed Characteristics of a Direct Current Motor (Shunt, Compound and Differential). 27 Experiment 2-B. EfiSciency of a Direct Current Motor (or Generator) by the Measurement of Losses 41 CHAPTER III. Synchronous Alternators. Experiment 3-A. Alternator Characteristics 62 Experiment 3-B. Predetermination of Alternator Character- istics 73 CHAPTER IV. Single-Phase Currents. Experiment 4-A. Study of Series and Parallel Circuits Con- taining Resistance and Reactance 102 Experiment 4-B. Circle Diagram for a Circuit with Resistance and Reactance 123 xii CONTENTS. Page. CHAPTER V. Transformers. Experiment S-A. Preliminary Study and Operation of a Transformer 128 Experiment 5-B. Transformer Test by the Method of Losses. 150 Experiment S-C. Circle Diagram for a Constant Potential Transformer 179 CHAPTER VI. Polyphase Currents. Experiment 6-A. A General Study of Polyphase Currents 196 Experiment 6-B. Measurement of Power and Power Factor in Polyphase Circuits 222 CHAPTER VII. Phase Changers, Potential Regulators, Etc. Experiment 7-A. Polyphase Transformation 241 Experiment 7-B. Induction Regulators 250 CHAPTER VIII. Induction Motors. Experiment 8-A. Preliminary Study of an Induction Motor and the Determination of its Performance by Loading 257 Experiment 8-B. Predetermination of the Performance of an Induction Motor by Means of the Circle Diagram 278 CHAPTER IX. Induction Machines: Frequency Changers and Induction Generators. Experiment 9-A. Operation. and Test of a Frequency Changer (Secondary Generator) . , 291 CONTENTS. xui Page. Experiment 9-B. Operation and Test of an Induction Gener- ator (Primary Generator) 295 CHAPTER X. Synchronous Machines. Experiment io-A. Study and Operation of a Synchronous Motor 304 Experiment io-B. Special Study of a Synchronous Motor... 316 Experiment io-C. Study of a Synchronous Converter 321 CHAPTER XI. Wave Analysis. Experiment ii-A. Analysis of a Complex Wave by the Method of 18 Ordinates 331 CHAPTER XII. Problems 345 CHAPTER I. DIRECT CURRENT GENERATORS. Experiment i-A. Generator Study and Characteristics of a Series* Generator. PART I. GENERATOR STUDY. § I. Faraday discovered (1831) that when a conductor cuts lines of force an electromotive force is generated in the conductor proportional to the rate at which lines are cut, and all dynamos (or generators as they are now commonly called) operate on this principle. To generate an electromotive force, it is essential there- fore to have a conductor (or several conductors combined by various winding schemes) forming the armature as one member; and to have lines of force or magnetic flux set up by field mag- nets which form the second member. For operation it is neces- sary also to have a source of mechanical powerf by which either one of these members can be given a motion with respect to the other. A generator may have| a stationary field and revolving armature ; or, a revolving field and stationary armature, desig- nated as the revolving field type. Although the latter is useful for large alternators, serious objections to it have been found in direct current machines, for with the commutator stationary the brushes must revolve, which leads to difficulties in construc- tion and operation. It is the custom, therefore, to build all * Where a series generator is not available, this study may be taken without experimental work or in connection with Part I. of Exp. i-B. t Power is required to overcome friction and other losses, and to over- come a counter torque (§§ 1-3, Exp. 2-A) which varies with the load. t Outside of this classification is the inductor alternator which has a stationary armature, a stationary field winding and a revolving inductor of iron; its study should be taken up later under alternators. 2 DIRECT CURRENT GENERATORS. [Exp. direct current generators and motors with a stationary field and revolving armature. § 2. The student should consult any of many excellent treatises for a detailed discussion of different types of generators and if possible should note one or two machines in the laboratory or elsewhere which are examples of each important type. Machines noted should illustrate the following terms, some of which are briefly explained in later paragraphs : Stationary field, revolving field, bipolar, multipolar, separately-excited, self-excited, series wound, shunt wound, compound wound, magneto-generator, open and closed coil armature, drum armature, Gramme {or ring) armature. Only the general structure of the various machines need be noted. Observe particularly the magnetic circuit of each machine and the disposition of the field winding. Keep in mind that magnetic flux is proportional to magnetomotive force (field ampere-turns) divided by the reluctance of the complete magnetic circuit, i. e., the sum of the reluctance of each part (air gap, core, etc.). As the reluctance of any part of a magnetic circuit is equal to the length divided by the product of the cross-section and per- meability, it is obvious that an unnecessarily long magnetic circuit should be avoided, a fact neglected in some early machines. § 3. In a bipolar generator, one pole is north and the other south; in a multipolar generator (with 4, 6, 8, etc., poles) the poles are alternately north and south. Each armature conductor accordingly passes first underneath a north and then underneath a south pole and has induced in it an electromotive force first in one direction* and then in the *(§3a). An exception is the so-called unipolar, homopolar or acyclic dynamo, which has a unidirectional electromotive force generated in the armature conductor; it accordingly delivers direct current to the line with- out commutation. Faraday's disk dynamo (one of the earliest dynamos) was of this type. For years it was the dream of zealous electricians to make this type of machine practicable, but it was considered only as an in- teresting freak, for at ordinary speeds the voltage generated is too low 1-A] SERIES GENERATOR. 3 other, i. e., an alternating electromotive force. The simplest form of generator is therefore the alternator, the current being taken from the armature to the line without any commutation. If the armature is stationary, the alternating current from the armature is taken directly to the line; if the armature is revolving, the armature windings are connected to collector rings (or slip rings) from which the current is taken to the line by means of brushes. In a direct-current generator the armature windings are con- nected* to the several segments or bars of a commutator, from which the current is taken by brushes to the line. The alter- nating electromotive force generated in each coil is thus com- mutated, or reversed in its connection to the line, at or near the time of zero value of the electromotive force of the coil. The electromotive force in each coil increases from zero to a maximum and back to zero, and at any instant the electromotive forces in the various individual coils have different values rang- ing from zero to a maximum, according to the positions of the coils. The sum of these coil-voltages, as impressed upon the line as terminal voltage, is however practically constant. for most purposes. But changed conditions have made it a practical and important machine (i) driven at high speed by the steam turbine, or (2) driven at moderate speed to generate large currents at low voltage for elec- trochemical work. Dynamos of this class are not included in this study. For further information, see " Acyclic Homopolar Dynamos," by Noeg- gerath, A. I. E. E., Jan., 1905; also, Standard Handbook, or Franklin and Esty's Electrical Engineering. For description of some structural im- provements, see pp. 560 and 574, Electrical World, Sept. 12, igo8. * (§3b). The details of armature windings will not be here discussed; they are amply treated in many text and handbooks. In almost all machines a closed coil winding is used. (The Brush and T-H arc dynamos and a few special machines use open coil winding.) In a closed winding, the armature coils are connected in series and the ends closed. There are two ways of connecting the coils in series : wave winding and lap winding. In the wave or series winding there are always two brushes and two paths for the current from brush to brush, irrespective of the number of poles. In the lap or parallel winding, generally used in large generators, there are as many paths (and brushes) as poles. The two schemes are essentially the same in a bipolar machine. 4 DIRECT CURRENT GENERATORS. [Exp. § 4. Field magnets are usually* energized by direct current passed through the field windings ; permanent magnets being used only in small machines, called magneto-generators, used for bell- ringers, etc. A generator is separately-excited or self-excited according to whether the current for the field is supplied by an outside source or by the machine itself. Alternators are sepa- rately excited; direct current generators are usually self-excited. § 5. A direct current machine (either generator or motor) may be : ( I ) Series wound, with the field winding of coarse wire in series with the armature and carrying the whole armature current ; (2) Shunt wound, with a field winding of fine wire in shunt with the armature and carrying only a small part of the whole current ; (3) Compound wound, with two field windings, the principal one in shunt and an auxiliary one in series with the armature. The compound generator is in most general use, being best suited for all kinds of constant potential service, both power and lighting; the shunt generator performs similar service but not so well. The characteristics of these machines will be studied fully in Exp. i-B. The series generator is of interest because: (i) It is one of the earliest types and of historical importance; (2) It is the simplest type and illustrates in a simple manner the principles which underlie all dynamo-electric machinery, both generators and motors; (3) In a compound wound generator or motor, the series winding is an important factor in the regulation of poten- tial or speed. In itself the series generator is of relatively small importance, because neither current or -voltage stay constant; it is used only in some forms of arc light machines with regulating devices for constant current. In direct current motors, all three types of winding are em- ployed : series wound motors for variable speed service in traction, crane work, etc.; shunt and compound wound (including differ- *The induction generator, to be studied in a later experiment, does not come under this classification. i-A] SERIES GENERATOR. S ential wound) motors for more or less constant speed service (Exp. 2-A). PART II. CHARACTERISTICS OF A SERIES GENERATOR. ' The characteristic curves to be obtained are: the magnetiza- tion curve, with the machine separately excited; the external series characteristic, with the machine self excited; and the total characteristic, which is computed. § 6. Magnetization Curve. — This curve shows the armature voltage (on open circuit) corresponding to different field currents when the generator is separately excited from an outside source, as in Fig. i. No load is put upon the machine. Means for vary- ing the field current must be provided ; see Appendix, § 14. § 7. Data. — Readings are taken of field current, ar- _^_ 'CA/ mature voltage and speed; current from the first reading is taken outside source^ with field current zero, showing the voltage due to . , , , . rn, Fig. 1. Connections for magnetization residual magnetism, ihe ,. , ., , 1^-^ " t, curve, — separately excited. field current is then in- creased by steps from zero to the maximum* rating of the machine, the readings taken at each step giving the " ascending " curve. The descending curve is then obtained by decreasing the field current by steps again to zero. In Fig. 3, only the ascend- ing curve is shown ; see also Fig. 2, of Exp. i-B. To obtain a smooth curve, the field current must be increased or decreased continuously; there will be a break in the curve if a step is taken backwards or if the field circuit is broken during *(§7a). Current Density.— Vor field windings an allowable current density is 800-1,000 amp. per sq. in. (1,600-1,275 circ. mils per amp.) ; for armatures, 2,000-3,000 arap. per sq. in. (640-425 c. mils per amp.). For a short time these limits can be much exceeded. The sectional area of a wire in circular mils is the square of its diameter in thousandths of an inch. ^-aM^W 6 DIRECT CURRENT GENERATORS. [Exp. a run. This is true of all characteristics or other curves involv- ing the saturation of iron. § 8. Brush Position. — During the run the brushes are kept in one position; if for any reason they are changed, the amount should be noted. For a generator the best position is the position of least sparking and of maximum voltage, which locates the brushes on the " diameter " or " line " of commutation. Under load this line is shifted forward from its position at no load, on account of field distortion caused by armature reaction,* and the brushes are accordingly advanced a little to avoid sparking. As it is desirable to keep the brushes in the same position in taking all the curves, with load or without load, it is well to give the brushes at no load a little lead, but not enough to cause much sparking. § 9. Speed Correction. — If the speed varied during the run, the values of voltage as read are to be corrected to the values they would be at some assumed constant speed. Since, for any given field current, voltage variesf directly with speed, this correction is simply made by direct proportion ; each voltmeter reading is *(§8a). Armature Reactions. — Armature current has a demagnetizing effect and a cross-magnetizing effect, the two effects together being called armature reaction, as discussed in various text books. The demagnetizing effect due to back ampere-turns weakens the field; the cross-magnetizing effect due to cross ampere-turns distorts the field (weakening it on one side and strengthening it on the other) and shifts forward the line of commutation. In many early machines this made it necessary to shift the brushes forward or back with change of load to avoid sparking; in modern machines the armature reactions are not sufficient to make this necessary and the brushes are kept in one position at all loads. If a very accurate determination of the neutral position of the brushes is desired, it can be found by a voltmeter connected to two sliding points which are the exact width of a commutator bar apart. The neutral posi- tion is the position of zero voltage between adjacent commutator bars, and this is shown by the voltmeter. t (§9a). If the speed can be varied at will, this can be verified for one field excitation. A peripheral speed of 3,000 feet per minute is permissible with the ordinary drum or ring armature. x-A] SERIES GENERATOR. multiplied by the assumed constant speed and divided by the observed speed. § lo. Curve. — After the speed correction is applied, the magne- tization curve is plotted as in Fig. 3. The abscissae of this curve, field amperes, are proportional to field ampere-turns or magneto- motive force; the ordinates, volts generated at constant speed, are (by Faraday's principle, §1) proportional to magnetic flux. The curve, therefore, is a magnetization curve (shovi^ing the relation between magnetic flux and magnetomotive force) for the magnetic circuit of the generator, which is an iron circuit with an air gap. The bend in the curve indicates the saturation of the iron. § II. External* Series Characteristic. — This characteristic, which is the operating or load characteristic of the machine, shows the variation in terminal voltage for different ' cur- rents, when the machine is self excited and the exter- nal resistance is varied. The armature, field and external circuit are in series, as in Fig. 2; read- ings are taken of current. Fig. 2. Connections for series character- istic, — self excited. The voltage and speed, for an ascending curve as in Fig. 3. descending curve may be taken if desired. For any point on the curve, the resistance of the external cir- cuit is i? = £ -^ /, or the tangent of the angle between the /-axis and a line drawn from the point to the origin. Below the knee of the curve, it will be seen that a small change in the external resistance will make a large change in current and voltage. * (§ iia.) In any characteristic the term "external" indicates that the values of current and voltage external to the machine are plotted ; the term " total" indicates that the total generated armature current and voltage are the quantities used. 8 DIRECT CURRENT GENERATORS. [Exp. If the speed varied during the run, the external characteristic should be corrected* for speed as before (§9). The watts output, for any point on the external characteristic is given by the product of current and voltage, and may be plotted as a curve. § 12. If the field coil is connected so that the current from the armature flows through it in the wrong direction, so as to demagnetize instead of building up the residual magnetism, the machine will not " pick up." For each direction of rotation, the proper connection of the field will be found to be independent of the direction of the residual magnetism. Note the effect of pre- vious magnetization (from an outside source) first in one and then in the other direction, and the effect in each case of revers- ing the field connections. § 13. Total Series Characteristic. — The total characteristic is derived from the series characteristic, so as to show the total generated electromotive force instead of the terminal brush voltage. Resistance Data. — The only additional data needed are the volt- age drops through the field and armature for different currents ; this is plotted as a curve (Fig. 3) which is practically a straightt line. With the armature stationary, current from an outside source is passed through the field or armature (separately) ; the current is measured and the difference of potential at the termi- nals. The ratio E-^ I gives the resistance. This is called meas- * (§ lib). This correction is applied to the external characteristic and not to the total characteristic for convenience. Inasmuch as it is the generated electromotive force which is proportional to speed, to be accurate the correction should be applied to the total and not to the external char- acteristic. t (§ 13a)- This would be a straight line if the resistance were constant. The resistance varies with temperature ; see Appendix, § 15. The armature resistance also varies with current since it includes the resistance of brushes and brush contact, which depends upon current density. The hot resistance is to be measured after the machine has run awhile, and is to be considered constant. i-A] SERIES GENERATOR. uring resistance by "fall of potential" method; see Appendix, §17- Curve. — By adding to the external characteristic the RI drop for field and armature, we have the generated voltage or total characteristic Fig. 3. Interpretation. — The total characteristic falls below the magne- tization curve on account of armature reaction, that is, the de- magnetizing effect of the armature current which weakens the field and hence reduces the gen- erated voltage; for, in taking the magnetization curve, there was no armature current and hence no armature reac- tion. The external char- acteristic falls below the total series characteris- tic, on account of resist- ance drop. The magnetization curve would be higher than the total character- istic for all currents, if in taking it the brushes were given no lead, that is were in the position of maximum volt- age. Giving the brushes a lead lowers the magnetization curve so that for small values of the current it may fall below the total characteristic. Fig. 3. AMPERES Characteristics of a series generator. lO DIRECT' CURRENT GENERATORS. [Exp. APPENDIX I. MISCELLANEOUS NOTES. § 14. Current and Voltage Adjustment. — For currents of small values, when a wide range of adjustment is desired, a series resist- ance (Fig. i) is frequently inadequate and it is better to shunt off current from a resistance R, as in Fig. 4. Fig. 4. Fig. 5. Methods for adjusting voltage or current. By adjusting the slider p, the voltage delivered to the apparatus under test can be given any desired value from zero up to the value of the supply voltage. A modification which is sometimes conveni- ent employs two resistances, B and C, Fig. 5. The adjustment is made by short circuiting or cutting out more or less of one resistance or the other, but not of both. The full amount of one resistance should always be in circuit. § 15. Temperature Corrections. — The conductivity of copper varies with temperature, according to the law given below. Resistance values to be significant should therefore be for some specified tem- perature; known for one temperature they can be computed for any other. Temperature rise can be computed from increase in resist- ance. In all cases where accuracy of numerical results is important, as 'in commercial tests for efficiency, regulation, etc., definite tempera- ture conditions should be obtained ; for this the detailed recommenda- tions of the A. I. E. E. Standardization Rules should be consulted. To meet standard requirements, a run of several hours is commonly required. In practice work this is not necessary, it being usually sufficient to specify resistances as cold when taken at the beginning and hot when taken at the close of the test. Let Rf be the resistance of a copper conductor at a temperature i-A] SERIES GENERATOR. 1 1 t° C. At a higher temperature the resistance will be greater and experiment shows that the increase in resistance will be in direct proportion to the temperature rise. At a temperature {t + 6) °C. the resistance is accordingly The temperature coefficient u, (per degree C.) depends upon the initial temperature t (degrees C), or the temperature for which the resistance is taken as loo per cent, and has for copper the following values :* — t 0° 6° 12° i8° 25° 32° 40° 48° a ,0042 .0041 .0040 .0039 .0038 .0037 .0036 .0035 From the formula given above, if the resistance is known for one temperature, the resistance can be computed for any other tempera- ture or for any temperature rise. § 16. From this formula we can also compute the temperature rise 6, above the initial temperature t, corresponding to a known increase in resistance. By transformation the formula becomes The temperature rise above an initial temperature t is accordingly equal to the per cent, increase in resistance divided by a. § 17. Fall of Potential Method for Measuring Resistance. — This method is based upon the fact that the fall of potential through a resistance R carrying a current / is E = RI (Ohm's Law). The resistance R which is to be determined may be the resistance of any conductor whatever (transformer coil, iield winding, armature, etc.) which will carry a measurable current without undue heating and is not itself a source of electromotive force. An armature, there- fore, must be stationary while its resistance is being measured by this method. Connect the unknown resistance to a source of direct current through a regulating resistance. Fig. 6 (see also § 14), so that the current will not unduly heat the resistance or exceed the range of instruments. Take readings of the two instruments simultaneously, * A. I. E. E. Standardization Rules ; also, A. E. Kennelly, Electrical World, June 30, 1906. 13 • DIRECT CURRENT GENERATORS. [Exp. and without delay so as to minimize the effect of heating. The re- sistance R is equal to E-~I. Fig. 6 shows the usual arrangement of apparatus, in case the volt- meter current is but a small part of the total current. The voltmeter leads should be connected di- Adjueting Resistance + ■AA/^ rectly to the resistance to be measured (not including un- necessary connectors, etc.) or should be pressed firmly against its terminals. The resistance of an armature winding is taken by pressing Fig. 6. Measurement of resistance by , , , , fall-of-potential method. ^he voltmeter leads agamst the proper bars i8o° or 90° apart; if resistance of brushes and connections is to be included, the voltmeter is connected outside of these connections. In case the ammeter current is very small, so that the voltmeter current is a considerable part of the total current, the voltmeter should be connected outside the ammeter so as to measure the combined drop of potential through the ammeter and unknown resistance. With the voltmeter connected either way, an error is introduced which may often be neglected but can be corrected for when par- ticular accuracy is desired. § 18. The voltmeter should always be disconnected before the cir- cuit is made or broken, or any sudden change is made in the current, to avoid damage to the instrument. If the resistance being measured is highly inductive, not only the instrument but also the insulation of the apparatus under test may be damaged by suddenly breaking the current through it on account of the high electromotive force induced by the sudden collapse of the magnetic field. This may be avoided by gradually reducing the current before breaking the circuit. § 19. The value of an unknown resistance can be found in terms* of a known resistance placed in series with it by comparing the drops in potential around the two resistances, the current in each having the same value. i-B] COMPOUND GENERATOR. 13 Experiment i-B. Characteristics of a Compound* Generator. § I. Introductory. — A compotind generator is made for the purpose of delivering current at constant potential either at the terminals of the machine or at some distant receiving point on the line. In the former case the machine is f,at compounded, the ideal being the same terminal voltage at full load as at no load, giving a practically horizontal voltage characteristic. In the latter case the machine is over compounded, giving a terminal voltage which rises from no load to full load to compensate for line drop, so that at the receiving end of the line the voltage will be constant at all loads. Constant potential' service is used both for power and for lighting. Constant delivered voltage is essential in lighting for steadiness of illumination and in power for constant speed. § 2. For such service, the series generator is not at all adapted, its voltage being exceedingly low at no load and, for a certain range, increasing greatly with load. § 3. A shunt generator almost meets the conditions, generating a voltage which is nearly constant but decreasing slightly with load (Figs. 4 and 6). Obviously by increasing the field excita- tion (field ampere-turns) when the machine is loaded, the voltage can be increased to the desired value ; this is true, however, only in case the iron is not saturated and it is accordingly possible for the increase in field ampere-turns to produce a corresponding in- crease in the magnetic flux. (Compare Fig. 2.) In a shunt machine this increase in field excitation can be obtained by an increase in field current produced either by an attendant who adjusts the field rheostat or by an automaticf regulator. * (§ia). This experiment can be applied to a Shunt generator by omit- ting §§20-25. t (§ 3a) • Tirrell Regulator. — Many older forms of regulators, which oper- ated by varying field resistance, are superseded by the Tirrell Regulator. This regulator operates through a relay as follows: (i) When the volt- age is too low, it momentarily short circuits the field rheostat, causing the H DIRECT CURRENT GENERATORS. [Exp. § 4. In a compound generator, the necessary increase of field excitation with load is simply and effectively obtained by means of an auxiliary series winding. Since the current in the series winding is the load current, the magnetizing action of the series winding (that is its ampere-turns or magnetomotive force) in- creases in direct proportion to the load. This increases the mag- netic flux and hence the generated voltage by an amount depend- ent upon the degree of saturation of the iron. Looked at in another way, a shunt winding (which alone gives a falling characteristic) and a series winding (which alone gives a rising characteristic) are combined so as to give the desired flat compounding or a certain degree of over-compounding. As the shunt winding alone gives very nearly the desired charac- teristic, the shunt is the principal winding, the series winding being supplementary and of relatively few ampere-turns. The characteristic curves for a shunt or compound generator may be classed as no-load characteristics, and load characteristics. PART I. NO-LOAD CHARACTERISTIC. § 5- There is one no-load characteristic, the saturation curve, which shows the saturation of the iron for different field exci- tations ; for this the generator is usually self-excited but may be separately excited when so desired. §6. (a) No-load Saturation Curve.*— This curve shows the terminal voltage for different values of field current. § 7. Data. — The machine is connected as a self-excited shunt voltage to rise; (2) when the voltage is too high, it momentarily removes the short circuit, causing the voltage to fall. The voltage would be much too high or too low, if the short circuit were permanently made or broken. The short circuit is, however, rapidly made and broken, and of a varying duration, a nearly constant voltage being thus secured. It may be applied directly to a generator (D.C. or A.C.) or to its exciter. It may be used advantageously in connection with a compound winding, and may be arranged so as to cause the voltage to rise with load in the same manner as in an over compounded generator. * Also called excitation characteristic, or internal shunt characteristic. I-B] COMPOUND GENERATOR. 15 RHEOSTAT Fig. I. Connections for no- load saturation curve. generator, Fig. i, and is driven without load at constant speed. Readings are taken of field current, terminal voltage and speed. The field current is varied by adjust- ing the field rheostat by steps from its position of maximum to minimum re- sistance. This gives the ascending curve ; the resistance is then increased again to its maximum for the descend- ing curve. If the rheostat, with re- sistance all in, does not sufficiently reduce the field current, a second rheostat may be placed in series with it. The machine " builds up " from its residual magnetism as does the series generator ; if the field winding is connected to the armature in the wrong direction, the machine will not pick up but will tend to be- come demagnetized. Should the direction of rotation be reversed, the field connection should be reversed. § 8. Curves. — Voltage read- ings are corrected by propor- tion for any variation in speed (§9, Exp. i-A), and the curves plotted as in Fig. 2. §9. Interpretation of Curves. — The curves in Fig. 2 show the saturation of the iron and are much the same as the characteristic of a series dynamo. The current through the armature is small, being only a few per cent, of full-load current; the resistance drop through the armature may accordingly be neglected and the meas- ured terminal voltage be taken as (practically) equal to the total Fig. 2. FIELD AMPERES No-load saturation curve. i6 DIRECT CURRENT GENERATORS. fExp. generated voltage. Likewise, the armature current is so small that armature reactions are negligible, and the curve is practically the same as a separately-excited magnetization curve. There is no necessity, therefore, for taking curves both self- and sepa- rately-excited. By separately exciting a generator, it is pos- sible to obtain a higher magnetization and consequently a higher generated voltage than can be obtained by self-excitation. In design work and in manufacturing tests, the saturation curve is commonly plotted with field ampere-turns, instead of amperes, as abscissae. However plotted, the abscissae are proportional to magneto-motive force and the ordinates to magnetic flux.* § lo. Saturation Factor and Percentage of Saturation. — There are two ways for expressingf numerically the amount of satura- tion for any point P on the working part of the curve. ( i ) The saturation factor, f, is the ratio of a small percentage increase in field excitation to the corresponding percentage increase in volt- age thereby produced. (2) The percentage of saturation, p, is the ratio OA -f- OB, when in Fig. 2 a tangent to the curve at P is extended to A. Compute these two for some one point on the curve, corre- *(§9a)- Magnetic Units. — For electrical quantities there are three sys- teins of units in use — the C.G.S. electromagnetic, the C.G.S. electro- static and the practical or volt-ohm-ampere system. For magnetic quanti- ties there is only one system of units in use, the C.G.S. electromagnetic system; mag;netic units of the practical system would be of inconvenient size, they have no names and are never used. The unit of magnetic flux is the maxwell, which is one C.G.S. line of force. The unit of flux density is the gauss, which is one maxwell per square centimeter. The unit of magnetomotive force is the gilbert, which is (io-^4t) ampere-turn. The unit of reluctance is the oersted, which is a reluctance through which a magnetomotive force of one gilbert pro- duces a flux of one maxwell. The maxwell and the gauss are author- ized by International Electrical Congress, but not the gilbert and the oersted. Analogous to Ohm's Law (current = electromotive force-;- resistance), we have the corresponding law for the magnetic circuit: flux (maxwells) = magnetomotive force (gilberts) -H reluctance (oersteds). t A. I. E. E. Standardization Rules, 57, 58. i-B] COMPOUND GENERATOR. 17 spending say to normal voltage, and check by the relation /> = I — I//. These terms are useful because they make possible an exact numerical statement of the degree of saturation of a machine, under working conditions, without the reproduction of a satura- tion curve. For a more complete study, compute p and / for different points and plot. PART II. LOAD CHARACTERISTICS. § II. The usual load characteristics are the shunt, compound and armature characteristics. In takiiig the shunt and compound characteristics, the machine is left to itself with the field rheostat in one position during the run, the curve showing the variation in terminal voltage with load.. In taking the armature characteristic the field rheostat is con- stantly adjusted ; the curve shows the variation in excitation- necessary to maintain a constant terminal voltage at different loads. The differential and series characteristics are not commercial characteristics but are included to show more fully the operation of the series winding. (For full-load saturation curve, see § 33.) § 12. (&) Shunt Character- istic. — This is the working characteristic of the machine when operated at normal speed as a shunt-wound gen- erator and shows the varia- tion in terminal voltage with load (Curve A, Fig. 4). § 13. Data. — The connec- tions are shown in Fig. 3. Readings are taken of terminal voltage, field current, line current and speed. No speed correction is made, there being none which is simple and accurate. The field rheostat is set in one position and no change is made in it during the run. 3 Fig. 3. Connections for shunt chacac- teristic. iS DIRECT CURRENT GENERATORS. [Exp. § 14. The setting of the rheostat for commercial testing (§ 21a) is made for normal voltage at full load. For the purposes of this experiment, it is usually preferable to set the rheostat for normal voltage (or for any selected value of voltage) at no load; in this case the shunt, compound and differential curves. Fig. 6, all start from the same no load voltage. The load current is then increased from no load up to about 25 per cent, overload and then decreased, if so desired, back to no load. The return curve will fall a little below, on account of hysteresis. Data are also to be taken for a characteristic starting at no load with a voltage below normal (§ 18). Armature resistance is measured by the fall-of-potential method, (§ 17, Exp. i-A). 100 120 AMPERES 140 160 Fig. 4. Shunt characteristics. § 15. Curves. — The armature RI drop is plotted as Curve D in Fig. 4. For the external shunt characteristic (Curve A, Fig. 4), plot observed line current as abscissae and observed terminal voltage as ordinates. i-B] COMPOUND GENERATOR. 19' For the total shunt characteristic (Curve B), plot total arma- ture current* (line current plus field current) as abscissae, and total generated voltage (terminal voltage plus armature RI drop) as ordinates. § 16. Interpretation {Armature Reactions and Regulation). — An ideal characteristic would be the straight horizontal line, Curve C, indicating a constant voltage at all loads. As a matter of fact the terminal voltage (Curve A) decreases with load. There are, at constant speed,t three causes for this : ( i ) armature resistance drop, (2) armature reactions which reduce the magnetic flux and (3) decreased field excitation as the voltage decreases. The difference between Curves A and B shows the effect of (i) resistance drop; the difference between B and C shows the effect of (2) armature reaction and (3) decreased excitation, and of (4) if speed varies. The difference between B and C will show the effect of arma- ture reactions (2) alone J if a run is made at constant excitation and constant speed, thus eliminating (3) and (4). This is the practical method for determining armature reactions. The ma- chine may be self excited or (preferably) separately excited. § 17. The regulation of the generator is shown by the drop in Curve A. To express regulation numerically as a per cent., the rated voltage at full load is taken as 100 per cent. In a commercial test, therefore, the curve is taken by beginning at full load at rated voltage (100 per cent.) and proceeding to open circuit. The regulation]] is the per cent, variation from normal * The difference between line and armature currents is so small that for many practical purposes the distinction between them can be neglected. t (§i6a). Should the generator slow down under load, as when driven by an induction motor, this would constitute a fourth cause (4). t (§ i6b). Included, as a part of armature reaction, is the effect of local self-induction of the armature conductors, when traversed by the arma- ture current which (in any one conductor) is rapidly reversing in direc- tion. II A. I. E. E. Standardization Rules 187, et seq. 20 DIRECT' CURRENT GENERATORS. [Exp. full-load voltage (in this case the per cent, increase) in going from full load to no load. § i8. Characteristics taken, with Low Field Excitation. — On short circuit a shunt generator has no field excitation and the short-circuit current (depending on residual magnetism) is com- monly less than normal full-load current. The current, however, is much greater before short circuit is reached. On account of this excessive current, the complete characteristic curve cannot be obtained with the field rheostat in its normal setting. To show the form of the complete shunt characteristic, set the field rheostat for a no-load voltage much below normal, and take Curve E (Fig. 4) from open circuit to short circuit, and Curve F returning from short circuit to open circuit. The form of these curves should be interpreted. § 19. With a weak field, armature reactions cause the terminal voltage to fall off with load more rapidly than with a strong field. This is seen by com- paring Curves E and F with Curve A. The effect of armature reactions is least when the iron is highly saturated, for then any decrease in magneto- motive force (due to arma- ture ampere-turns) does not cause a corresponding decrease in magnetic flux. (Compare Fig. 2.) It follows, therefore, that a shunt generator gives the best regulation when worked above the knee of the saturation curve. It will be found (§22) that this is not so for a compound generator. § 20. (c) Compound Characteristic. — The connections for taking the compound characteristic, Fig. 5, are the same as for the shunt characteristic. Fig. 3, with the addition of the series SERIES FIELD Fig. s. Connections for compound char- acteristic. i-B] COMPOUND GENERATOR. 21 field winding which is in series* with the armature. The same readings of terminal voltage, field current, line current and speed are taken as for the shunt characteristic and no speedf correc- tions are made. ded) Fig. 6. Series, shunt, compound and differential characteristics. §21. In Fig. 6 are plotted shunt, compound and differential characteristics, beginning with the same no-load voltage.:]: The compound characteristic cannot be made a perfectly straight line from no load to full load. What can be done is to have the terminal voltage at full load the same as the no-load *(§2oa). Short Shunt and Long Shunt. — The connection shown in Fig. 5 is short shunt; it would be long shunt if the shunt field were con- nected to the line terminals ac, instead of to the armature terminals ah. Both methods of connection are used commercially, the difference between them being slight. t (§2ob). A generator is compounded for the particular speed at which it is to operate. When it is to be driven by an induction motor, it may be compounded so as to take into account the slip of the motor, i. e., its slowing down under load. t (§2ia). In commercial testing, the compound and shunt characteristics would be taken with the same normal voltage at full load (§14). The differential characteristic would not be taken. 22 DIRECT CURRENT GENERATORS. [Exp. voltage (flat compounding) or a definite percentage higher (over compounding). In either case the regulation is the maximum percentage deviation from the ideal straight line at any part of the characteristic, rated full-load voltage being taken as loo per cent. (See § 17, and Standardization Rules.) § 22. If the field magnets of a compound generator are highly saturated, the increase in field ampere-turns with load due to the series winding cannot cause a corresponding increase in the mag- netic flux and there will be considerable deviation from the ideal straight line characteristic. A compound generator accordingly gives better regulation when the iron is below saturation, which is opposite to the conclusion reached for the shunt generator (§19)- In a compound generator there is less cause for sparking and shifting of brushes than in a shunt generator, on account of the strengthening of the field by the series winding under load. For fluctuating loads, as railway service, the compound generator is accordingly superior and generally used. Obviously, on account of the series winding, it is much worse to overload or short circuit a compound than a shunt generator. § 23. Shunt for Adjusting Compounding. — If the characteristic of a compound generator rises more than is desired, there are too many series ampere-turns. These can be reduced without changing the number of turns by reducing the current which flows through them. This is done by a shunt resistance in paral- lel with the series winding. A generator i? usually given more series turns than are necessary, the desired amount of compound- ing being obtained by adjusting the shunt resistance. This is much easier than changing the number of series turns and makes it possible to change the amount of compounding at any time, even after the machine is in use. § 24. {d) Differential Characteristic. — The connections for this are the same as for the compound characteristic (Fig. 5) except that the series field winding is reversed so as to be in i-B] COMPOUND GENERATOR. 23 opposition to the shunt winding. The eilect of the series wind- ing is now to decrease (instead of increase) the magnetization of the iron, as the armature current increases, causing the volt- age to fall off with load more rapidly than with the shunt field alone. As there is no demand for this, generators with differ- ential winding are not used. (In a motor, Exp. 2-A, a differ- ential winding is useful in giving constant speed). § 25. (e) Series Characteristic. — This characteristic shows the effect of the series winding alone, with the shunt winding not connected. The procedure is the same as in testing a series generator, the connections being as in Fig. 2, Exp. i-A. 10000 8000 ■ 10,100 4000 2000 - CONSTANT SPEED CONSTANT TERMINAL VOLTAGE -7,900 40 80 120 160 ARMATURE AMPERES 200 Fig. 7. Armature characteristic or field compounding curve, showing that at full load 2,200 more ampere-turns are needed than at no load for constant terminal voltage. » § 26. (/) Armature Characteristic. — This curve is used in de- termining the proper number of series turns for compounding a generator; and is therefore frequently called a field compounding curve.* It shows, Fig. 7, the variation in field excitation * This has also been called an " excitation characteristic," a name which is ambiguous since it may be taken to mean the saturation curve, §6. 24 DIRECT CURRENT GENERATORS. [Exp. (amperes or ampere-turns*) necessary at different loads to main- tain a constant voltage at the terminals of a shunt generator driven at constant speed.f The connections are shown in Fig. 3, readings being taken of field current, line current, terminal voltage and speed. Separate excitation may be used when a higher excitation is wanted than can be obtained by self -excita- tion. The load current is increased from no load to about 25 per cent, overload. At each load, before readings are taken, the voltage is brought to the desired constant^ value by adjusting the field rheostat. § 27. The rise in the armature characteristic shows the in- crease in ampere-turns of excitation needed to compensate for loss in voltage due to resistance drop, armature reactions, etc. (§16). If in service the machine is to be operated as a shunt gene- rator, this increase in excitation can be obtained by adjusting the field rheostat as was done in obtaining this curve. If, however, the machine is to be operated as a compound gen- erator, this increase in excitation is to be obtained by the ampere- turns of the series winding. § 28. Determination of Proper Number of Series Turns. — We know from the armature characteristic the additional ampere- turns of excitation which must be provided at full load to pro- duce the desired terminal voltage. We know also the amperes' (load current) which will flow through these turns at full load. The necessary number of turns is accordingly readily found by dividing ampere-turns by amperes. Thus in Fig. 7, we note that * To plot in ampere-turns, the number of turns in the shunt field must be known; sA Appendix I. The number of turns multiplied by field current gives the number of field ampere-turns. t (§26a). In case the generator is to be normally driven by an induc- tion motor, with speed decreasing with load, it should be so operated in taking the armature characteristic. (See §§ i6a, 20b.) t (§26b). The curve may be taken for a voltage which increases with load; such a curve would show the series ampere-turns to be added for" over-compounding. i-B] COMPOUND GENERATOR. 25 for full load (200 amperes) there are needed 2200 more ampere- turns excitation than at no load. The series winding will be traversed by the current of 200 amperes, and must accordingly have II turns in order to make the required 2,200 ampere-turns. If the armature characteristic were a straight line, the series turns calculated as above would be the same for all loads and the generator could be compounded so as to have perfect regulation and give an exactly constant voltage at all loads. But the armature characteristic always curves, bending more after saturation is reached. The series turns are, therefore, calcu- lated for one definite load (full load) ; for other loads the com- pounding will be only approximately correct (§21). The armature characteristic and hence the proper number of series turns for correct compounding, will differ for different speeds and terminal voltage, — an interesting subject for further investigation. APPENDIX I. MISCELLANEOUS NOTES. § 29. Determining the Number of Shunt Turns. — The number of shunt turns on a generator can be more or less accurately deter- mined, if the machine has a series winding or a temporary auxiliary winding with a known number of turns. With the machine separately excited, take an ascending no-load saturation curve, using the shunt field winding of unknown turns; take a second similar curve, using the series or auxiliary field wind- ing of known turns. A comparison of the two curves shows that the shunt winding requires a much smaller current than does the auxiliary winding to give the same generated armature voltage. De- termine this ratio of currents for equal terminal voltage (found for several voltages and averaged) and suppose it to be i : 40. The shunt turns are then 40 times as many as the auxiliary turns, the ampere-turns for equal terminal voltage being the same. If for example the auxiliary turns are 10, the shunt turns are accordingly 400. § 30. The number of turns in two field windings can be compared 26 DIRECT CURRENT GENERATORS. [Exp. by the use of a ballistic galvanometer (or voltmeter or ammeter used as a ballistic galvanometer) ; the chief advantage of the method is that it does not require facilities for running the machine. With the armature stationary and the galvanometer connected to the ter- minals of one field, break a certain armature current and note the throw of the galvanometer. Repeat, breaking the same armature current with the galvanometer connected to the other field. The ratio of galvanometer throws gives the desired ratio of field turns. It is best to take a series of readings and average the results. § 31. An estimate of the number of turns in a coil can be made from its measured resistance, size of wire and mean length of turn. This can be used as a check, but the method is commonly only approximate on account of the uncertainty of the data. § 32. To Compound a Generator by Testing with Added Turns. — The proper number of series turns required to compound a generator can be ascertained by trial by means of temporary auxiliary turns. With the generator running at full load, pass current from an inde- pendent source through these auxiliary turns and adjust this current until the terminal voltage of the generator has the desired full-load voltage. This current (say 220 amperes), multiplied by the number of auxiliary turns (say 10) through which it flows, shows that 2,200 extra ampere-turns are needed at full load. If the full-load current is 200 amperes, the generator would accordingly require 11 series turns. § 33. Full-load Saturation Curve. — For obtaining this curve, the field excitation is varied and the load adjusted at each reading, so that the external current remains constant at its full-load value. Field currents are plotted as abscissae and terminal voltages as ordi- nates. Such a curve is to be taken later (Exp. 3-A) on an alter- nator; it may accordingly be omitted, in the present experiment, if time is limited. CHAPTER II. DIRECT CURRENT MOTORS. Experiment 2-A. Operation and Speed Characteristics of a Direct Current Motor, (Shunt, Compound and Differential). PART I. INTRODUCTORY. § I. Generators and Motors Compared. — Structurally a direct current generator and a direct current motor are alike,* the essential elements being the field and the armature. The same machine may accordingly be operated either as a generator or as a motor. Operating as a generator, the machine is supplied with me- chanical power which causes the armature to rotate against a counterf or opposing torque ; this rotation of the armature gen- erates an electromotive force which causes current to flow and electrical power to be deHvered to the receiving circuit. Operating as a motor, the machine is supplied with electrical power which causes current to flow in the armature against a counterf or opposing electromotive force ; this current creates a torque which causes the armature to rotate and mechanical power to be delivered to the shaft or pulley. *(§ia). Since generators are built in much larger sizes than motors, one generator being capable of supplying power for many motors, there may be a difference in design due to size. Moderate size machines, gene- rators or motors, are built with few poles, — four being" common in small motors. On the other hand, very large machines — that is generators — are built with many poles. In all direct current machines, — generators or motors — it is, common practice to use a stationary field and a revolving armature (§ i, Exp. i-A). t (§ib). There is no counter torque in a generator until current flows in the armature; there is no counter electromotive force in a motor until there is rotation of the armature. 27 28 DIRECT CURRENT MOTORS. [Exp. It is seen that the operation, either as a generator or as a motor, involves (i) the generation of an electromotive force and (2) the creation of a torque, both of which depend upon funda- mental laws of electromagnetism. § 2. Generation of Electromotive Force. — An electromotive force is generated in a generator or in a motor due to the cutting of lines of force, this electromotive force being proportional to the rate at which the lines of force or flux are cut, as already discussed in § i, Exp. i-A. In a generator this electromotive force causes (or tends to cause) a current to flow; in a motor, it is a counter electromotive force and opposes the flow of current. § 3. Creation of Torque. — A torque is created in a generator or motor due to the forces acting upon a conductor carrying current in a magnetic field. In a motor this torque causes (or tends to cause) a rotation of the armature with respect to the field ; in a generator, it is a counter torque and opposes the rota- tion of the armature. The creation of torque depends upon the following funda- mental principle : — When a conductor carrying current is located in a magnetic field, it is acted upon by a force that tends to move the conductor in a direction at right angles to itself and to the magnetic flux', the force being proportional* to the current and to the flux density. This force creates, a torque, — that is a turning moment or couple — equalf to the product of the force and the length of the * (§3a). In C.G.S. units this force is equal to the product of the cur- rent, flux density, length of conductor and sine of the angle between the conductor and direction of flux. This sine is unity when the conductor and flux are at right angles, as in most electrical machinery. When there are a number of conductors, each conductor is subject to this force; the total torque of a motor is therefore proportional to the total number of armature conductors. t (§3^). Torque may be expressed as pounds at one foot radius, pound- feet, kilogram-meters, etc. Power is proportional to the product of torque 2-A] SPEED CHARACTERISTICS. 29 radius or lever arm to which the force is applied. It accordingly follows that : torque is proportional to armature current and to the flux density of the field; this is irrespective of whether the armature is rotating* or not. A reversal of either the current or the flux alone reverses the direction of the torque. Of the total torque, part is used in overcoming friction, wind- age and core loss ; the remainder is useful torque and is available at the pulley. § 4. Automatic Increase of Current with Load. — The counter- electromotive force E' is always a few per cent, less than the sup- ply voltage E. The difference is due to the resistance drop in the motor armature, — including brushes, brush contact and con- nections, and the series field (if any) ; that is E' = E — RI. (i) and speed; thus, if R.P.M. is revolutions per minute and T is torque in pound-feet jjp ^ 27rX R.P.M. y, 33,000 If power is known, torque may be found by dividing power by speed. In pound-feet, torque is „ 3'?,oco , , H.P. 2K ^ R.P.M. ■ When power is in watts, it is frequently convenient to express torque in " synchronous watts " ; thus, _ Watts r.p."m:" (One synchronous watt ^7.04 pound-feet.) (One pound-foot = 0.142 synchronous watt.) Torque is also expressed in "watts at 1,000 R.P.M." ; thus, „ Watts 7-=i,oooXr:pm:- *(§3c). Torque with the armature at rest {static torque) can be de- termined for various field currents and for various armature currents by means of a lever arm attached to the armature and a spring balance or platform scales. 3° DIRECT CURRENT MOTORS. [Exp. Within the usual range of operation, this RI drop for a com- mercial motor is only a few per cent, of the total line voltage. Good design does not permit more, inasmuch as the output and efficiency are decreased by the same percentage. The current which flows in the armature is seen to be If under running conditions the current / is not sufficient to give the motor enough torque (which is proportional to current and flux) to do its work at the speed at which it is running, the motor will begin to slow down, thus decreasing the counter- electromotive force E' (which is proportional to speed and flux) . As £' decreases / increases. Until the torque is sufficient to meet the demands upon the motor. The current accordingly increases automatically with the load, and this increase can be continued until the safe* limit, determined by heating, is reached. On the other hand, if the current / is more than is needed to give the torque required for the load at a certain running speed, the surplus torque will cause the armature to accelerate, thus increasing £' and decreasing / to a value which gives the proper torque for the load and speed. It will be seen that a small change in £' is sufficient to cause a large change in / and therefore in the torque. As an example, suppose £' = loo, £=:i04; if an increase in speed causes E' to increase 2 per cent., that is to 102, the current / will be reduced 50 per cent. § 5. Relations between Speed, Flux and Counter-electromo- tive Force. — Counter-electromotive force is proportional to speed (5") and flux ((^) ; that is £'oc<^5-. (3) Hence, speed varies directly as the counter-electromotive force and inversely with the flux ; that is *A motor is usually rated so that it can be run for several hours at 25 per cent, over its rated load. 2-AJ SPEED CHARACTERISTICS. 31 E' or, ^"^-f' (4) E-Rl ^^—^- (5) This is the speed equation for a motor. It is seen that if <^ is reduced the speed will increase. The equation shows the defi- nite numerical relations of the quantities involved. Hoiv an increase in speed is brought about by a decrease in flux is made more clear in § 7. § 6. Speed of a Shunt Motor. — A shunt motor with constant supply voltage has a constant field current and therefore a con- stant fiux. It accordingly follows that the speed is nearly con- stant. The RI drop causes it to decrease with load (compare equation 5) ; this is partially offset, however, by the effect of armature reactions, as seen later (§8). §7. It is seen from equation (5) that the speed may be in- creased or decreased by weakening or strengthening the field. The process is explained as follows : — When the field is weakened the counter-electromotive force is reduced; this permits more current to flow in the armature, thus giving greater torque* and speed. The speed accordingly in- creases until E' has increased so as to limit the current (and hence the torque) to a value which will give no further accel- eration. The cause for increase of speed is surplus torque . § 8. Armature Reactions and Brush Position. — If the brushes are given a backward lead (which is usual in motors running in one direction, in order to obtain better commutation) the field is * (§7a)- As an example, suppose the field is weakened so that the flnx is reduced 2 per cent, and E' the same amount; and suppose the armature current increases 50 per cent. Torque is proportional to flux and armature current and in this assumed case is increased 47 per cent. ; for .98 x 1.50 = 1.47. This increase is only temporary, for the armature current and the torque decrease as higher speeds are reached. 32 DIRECT CURRENT MOTORS. [Exp. weakened by the demagnetizing effect of armature reactions (§8a, Exp. i-A). This causes the flux to decrease with load, so that the speed does not decrease as much as it would with the brushes in the neutral position. On account of armature reac- tions, therefore, the speed regulation of a motor is better; the voltage regulation of a generator is worse (§ i6, Exp. i-B). The proper brush position for best commutation is the posi- tion which gives minimum speed. § 9. If the backward lead of the brushes is increased, the speed of the motor under load can be increased tmtil it equals or ex- ceeds the speed at no load. Such a control of speed by brush adjustment is not practicable, however, on account of bad com- mutation and destructive sparking; the brushes should be given the position of best commutation. A small variation of speed can be made, if desired, by shifting the brushes, provided it is not enough to cause much sparking. § ID. Speed Control. — From equation (5) it is seen that the speed of a motor can be varied : by changing the impressed volt- age, E; by varying resistance, R (series controller) ; or, by vary- ing flux, . Each of these methods is in use for operating variable speed motors. (a) Varying line voltage. Several line voltages can be ob- tained by using a number of line wires. Such a system is called a multiple-voltage system. (b) Varying resistance. The series controller is in common use with series motors; it is used occasionally with shunt motors of small size. (c) Varying flux. This can be accomplished either by a change in excitation (magnetomotive force) or a change in reluctance; for flux = magnetomotive force -^ relvictance. ( I ) Speed control by varying excitation is obtained in a shunt motor by a rheostat in series with the field (§ 7) ; in a series motor, by an adjustable resistance in parallel with the field. 2-A] SPEED CHARACTERISTICS. 33 (The possible method of control by brush-shifting, § 8, is not used.) (2) Speed control by varying reluctance is obtained in certain shunt motors by varying the air-gap. A limit to speed control by a variation in flux (by varying either excitation or reluctance) is reached on account of arma- ture reactions ; a considerable reduction in flux causes bad com- mutation. For varying the speed through a wide range, there- fore, these methods can only be used if the effects of 'armature reactions are overcome. This was first SatisfaC- Constant Potential supply torily accomplished by the compensated winding of Prof. H. J. Ryan, which was placed in slots in the pole faces. This compensa- tion is now generally ac- complished by the more easily constructed inter- poles or commutating poles of the interpole motor. PART II. OPERATION. § II. Shunt Motor.— If the motor is compound, cut the series coil out of the circuit. Connect the sup- ply lines to the main terminals of the motor and complete the connections, as in Fig. i. Note the queries, § 15. To start the motor, have all the starting box resistance in cir- cuit and all the field rheostat out of circuit; make sure that the field circuit is complete. The circuits should be so arranged that closing the supply circuit will excite the field (which takes 4 STARTING BOX /^ "\. ^ \ Fig. I. Connections for operating shunt motor. 34 DIRECT CURRENT MOTORS. [Exp. an appreciable time) before* the armature circuit is closed. The armature circuit is then closed and this is commonly done by the starting box lever. Bring the motor, unloaded, up to speed by cutting the starting box resistance slowlyf out of circuit until the whole resistance is cut out. Note the ammeter during the process and the in- crease of speed as indicated by the hum of the motor. The starting box should be kept in circuit only during starting, for (except in special- cases) it is not designed for continuous operation. If the motor does not now run at normal speed, the speed can be increased by gradually varying the field current by means of the field rheostat. Do not reduce the field current too much, nor under any circumstances break the field circuit,^ or the motor will run at a dangerous speed. Note the speed at no load for several excitations; also, when facilities permit, for several supply voltages. (For example, operate a no- volt motor with 55 volts on armature and on field; with 55 volts on armature and no volts on field; but not with no volts on armature and 55 volts on field.) § 12. Stopping. — Motors are commonly stopped by opening the supply switch and not by first opening the armature circuit. * (§ iia). If the starting box were made with suflSciently high resistance, so as to properly limit the current irrespective of counter-electromotive force, the armature circuit could be closed simultaneously with the field. This, however, is not usual practice. t (§ lib). Starting boxes are sometimes made so that it is impossible to manipulate them too rapidly. The "' multiple-switch " motor starter, used particularly in starting large motors, has a number of switches, thrown successively by hand; these give good contact for large currents and re- quire time for cutting out the successive sections of the resistance. t (§ lie). Automatic Release. — This danger is commonly guarded against by a solenoid on the starting box which releases the lever and allows it to spring back to the starting position when there is no current in the field circuit. This also acts as a " no-voltage " release, giving protection against damage which might occur were the current supply cut off and put on again with the starting box resistance all out. 2-A] SPEED CHARACTERISTICS. 35 There is then no sudden discharge of field magnetism and con- sequent liability to damage; for, as the armature slows down it generates a gradually decreasing electromotive force which main- tains the field excitation so that it too decreases gradually. (If there is an automatic release on the starting box, it opens the armature and field circuits after the field excitation has decreased to a low value.) The effects of induced electromotive force caused by sudden field discharge can be reduced by absorbing its energy in a high resistance shunt in parallel with the field circuit, or in a short- circuited secondary circuit around the field core. A brass field- spool will act in this way. Throwing power suddenly off the line, by opening the supply switch, may cause fluctuations in line voltage, — particularly in case of large motors under load. To avoid this, before the sup- ply switch is opened, the starting resistance may first be gradu- ally introduced into the armature circuit, which, however, is not to be opened; then the supply switch is opened. § 13. Compound Motor. — In a compound* motor, the series winding strengthens the field as the armature current increases. On starting or under heavy load («'. e., at times when the arma- ture current is large) the motor is accordingly given a very strong field and therefore has — for a given armature current — a greater torque than it would have with the shuntf winding only. * (§ 13a). To tell whether a series winding is connected "compound" or •'differentially," throw off the belt and start the motor (for a moment) with the series coil only. If the motor tends to start in the same direction as it does with the shunt coil, the winding is " compound " or "'cumula- tive;" if in the reverse direction, the winding is "differential." t(§i3b)- This means a greater torque than it would have with the same shunt winding only. The motor could be given a different shunt winding which would give as strong a field and as great a torque as is obtained by means of the compound winding. Such a shunt winding, however, would give the strong field at all times ; whereas the compound winding gives the strong field only at particular times, — i. e., at starting and under load. 36 DIRECT CURRENT MOTORS. [Exp. Under load the compound winding, by strengthening the field, causes the motor to slow down. For certain kinds of service — as in operating rolling mills, cranes, elevators, etc. — this is desir- able in that the motor can work at great overload without the excessive demand for power which would be made by a constant speed motor. As compared with a shunt motor, it works under load at greater torque and less speed, and can stand a greater overload. In this respect it is similar to the series motor (see § 18). It differs from the series motor in that at light load there is still a certain strength of field due to the shunt winding, and the speed, therefore, cannot exceed a certain value, whereas a series motor will attain a dangerous speed if the load is thrown off. Under some operating conditions the compound motor can accordingly be used where neither the shunt nor the series motor would be suitable. If slowing down with load is not wanted and a constant speed is desired at all loads, together with a large torque at starting, the series winding is used during starting only and is then cut out or short-circuited. § 14. Differential Motor. — Since a differential winding weak- ens the field as the load increases, such a winding makes possible a speed which increases with load. This is practically not desir- able. In some cases, however, it is desirable to have the same speed at full load as at no load and to use a series winding just sufficient to overcome the tendency which a shunt motor has to slow down with load. If the series turns are too many for this, their effect can be cut down by a shunt of proper resistance con- nected in parallel with the series winding. The starting torque of a differential motor is poor, particularly under load, inasmuch as the large starting current in the differ- ential winding greatly weakens the field. For this reason, when a differential winding is used, it is usually cut out of circuit or short-circuited during starting. w-A] SPEED CHARACTERISTICS. 37 If there are many series turns and no shunt is used, the cur- rent taken by a differential motor may become excessive as the load increases, thus weakening the field so that the motor races, or even reversing the field so that the motor suddenly reverses. § IS. Queties. — For increasing the speed, is the field current increased or decreased? Why? What is the use of the starting box? In starting, why do you not close the field and armature circuits simultaneously? Why is the starting box connected in series with the armature and not in series with the line? Why is a strong field needed for starting? Does this become of more or of less importance when starting under load? Would an added series winding be an advantage or a disadvantage in starting? Why would it be dangerous to break the field circuit? What is the effect of shifting the brushes? What is the proper position for the brushes? What is effect of interchanging positive and negative supply lines? What changes in connections are necessary to reverse the direction of rotation of the armature? (Be careful not to run more than a moment in the reverse direction, if the brushes would thus be damaged.) PART III. SPEED CHARACTERISTICS. § i6. Shunt, Compound and Differential Motor. — It is the pur- pose of the experiment to determine the variation of speed with load for the same motor connected in three ways, — shunt, com- pound and differential ; the line voltage is constant throughout the three runs. The brushes should be in one position during all the runs (§8), or the amount of any change noted. With the motor connected as a shunt machine. Fig. t, adjust the field current by means of the field rheostat so that the motor runs, on no load, at the speed for which it is designed, and keep the field current constant at this value during the run. For. the other two runs, compound and differential, adjust the field cur- rent for this same no-load speed* and keep the field current constant during each run. * (§ i6a). Starting with the same no-load speed, and making runs from no load to full load, gives the three speed characteristics of Fig. 2 coincid- ing at no load ; this is the best procedure for instruction purposes. In commercial testing, the field should be adjusted so that the motor runs at rated speed at full load. The curve is then taken from full load to no load; the maximum per cent, variation in speed from its full load value is the per cent, speed regulation. (Standardization Rules, 195.) 38 DIRECT CURRENT MOTORS. [Exp. Vary* the load on the motor by steps between no load and 25 per cent, overload, reading line voltage, field current, armaturef current (or else line current) and speed, for each step. Make runs with the motor connected shunt, compound and differential. With current as abscissae (either line current or armature current) and speed as ordinates, plot speed characteristics for the three runs as in Fig. 2. 1100 - 1000 900 - ■ 1 , "^ IT 1 800 . 1 700 - ^^^^<£is:ou^ 1 06OO - """t**-^-^ iij ,lu500 - 1 » 8i ■n 1 400 .3 gl 300 2 ^1 —1 800 - £{ 100 r 8 12 36 iO Fig. 2. 16 20 21 23 3 ARMATURE CURRENT Speed characteristics of a motor, — shunt, compound and differential. § 17. It is instructive to take runs as a differential motor with different resistances in shunt with the series coil; also, to take the various runs (shunt, compound and differential) with the field excitation above and below saturation. * (§i6b). This may be done by means of a brake, a blower, a belted generator or other convenient load; if a generator is used, its output may be absorbed in resistance or pumped back into the line (§26, Exp. 2-B). t(§i6c). If the armature current is measured, the field current is added to give the line current; if the line current is measured, the field current is subtracted to give the armature current. 2-A] SPEED CHARACTERISTICS. 39 APPENDIX I. SERIES MOTOH. § i8. Operation. — A series* motor is distinctly a variable speed motor. Its characteristics are shown in Fig. 3. The speed increases rapidly as the load is decreased, becoming dangerouslyf great if the The series motor, therefore. p I t I I / Maximumv / Safe SpeeA \/ - % ^„.^^ENCY - ^r^- / \ r^X c /I / 1 1 V / /^---gJPEEO If \ / 1 1 X ■■ / ^ « ^r 3 Is , Lrri . u. . . , J 1 1 , L AMPERES Fig. 3. Characteristics of a series operated at constant voltage. motor, load is removed or reduced too much cannot be run at no load and normal voltage; it can be run at no load with a series resistance in circuit. The series motor, be- sides being used for trac- tion,J is used for hoists, etc. For such service it is well adapted. The im- portant characteristic is that by slowing down under heavy load, it can increase its torque with- out requiring a corresponding increase in power; for torque = power -^ speed (§ 3b). If the speed did not decrease with load, it is seen that the power would have to be greatly increased to give the same torque. This would require a much larger motor. *. (§ i8a). For the purpose of comparison with the shunt, compound and differential motor, the characteristics of the series motor are here described, although its test is not usually to be included as a part of the present experiment. When the test is made, it is well to combine it with efficiency measurements, § 33, Exp. 2-B. t (§i8b). In the laboratory, be prepared to shut down quickly if ex- cessive speed is reached. With a belted load, there is danger of the belt flying off; with a brake, there is danger of an unintentional sudden de- crease in load. J(§i8c). In traction, the controller is usually so arranged that two motors can be connected in series or in parallel with each other for speed control, thus giving each motor half or full voltage. The series resistance is likewise used for control and for starting. In starting, the resistance and both motors are all in series. 4° DIRECT CURRENT MOTORS. [Exp. § 19. Torque. — Since torque varies as flux X current, the torque would vary as P, if flux were proportional to current. For small currents — below saturation — this is more or less true. For large currents — after saturation — the flux is practically constant and the torque increases directly as /. The torque curve. Fig. 3, is there- fore at first more or less parabolic and then becomes a straight line. § 20. Speed. — From equation (5) it is seen that speed varies in- versely with flux. For small currents, if we consider RI negligible and flux proportional to current, speed varies as i/I ; the speed curve (Fig. 3) would then be an hyperbola. For larger currents satura- tion is reached, the flux becomes practically constant and the speed more nearly constant. On account of Rl drop, speed continues to gradually decrease as current increases, even after saturation is reached. Series' motors are sometimes overwound, that is, wound so that saturation (and hence more constant speed) is soon reached. § 21. Test. — The load is varied between an overload (determined by maximum safe current) and an underload (determined by maxi- mum safe speed). The line voltage is constant; a series resistance is used for starting and may be used for adjusting voltage. Any method for loading can be used. If a shunt generator is used as a load, its output may be absorbed in resistance or pumped back into the line. (See §26, Exp. 2-B.) The pumping back method has been modified by A. S. McAllister, so as to form a convenient method for determining the torque of any kind of motor, direct or alternating {Standard Handbook, 3-239 and 8^151 ; McAllister's Alternating Current Motors, p. 185). § 22. Power. — Power is equal to EI and, when E is constant, power is directly proportional to /. In Fig. 3, power would be represented by a straight line passing through the origin. It will be seen, there- fore, that the power required does not increase as rapidly as does the torque. 2-B] EFFICIENCY. 4^ Experiment 2-B. Efficiency of a Direct Current Motor* (or Generator) by the Measurement of Losses. § I. Introductory. — Efficiency is the ratio of output to input. The obvious and direct method for determining the efficiency of a motor is, therefore, to measure the outputf and the input and take their ratio. An indirect method, known as the method of losses or stray power method, avoids the measurement of output. In this method the losses are measured and the output obtained by subtracting the losses from the input ; the efficiency is then determined. This method of losses possesses several advantages over meth- ods that involve the measurement of output. The motor output is in some cases a troublesome quantity to measure, especially if accuracy is essential ; but, even with the same degree of accuracy in the measurement of output and of losses, the efficiency cannot be as accurately determined^ from the former as from the latter. *With the appendices, this experiment covers the main features of the usual methods for determining the efficiency of any machine, direct or alternating. The main experiment is expHcit for determining the efficiency of a shunt motor, and, it is suggested that the student, without reference to the Appendices, first performs this main experiment. The Appendices should then be read and, if desired, a second experiment made (either now or later) under some of the special conditions which are there treated. t (§ la). Direct Measurement of Output. — The output of a motor can be determined directly by electrical measurement (using for a load a calibrated generator, §24), or by mechanical measurement (measuring .torque by means of a Prony brake, Brackett cradle dynamometer, etc.). Power can be readily computed when torque and speed are known (§3b, Exp. 2-A). There are various forms of absorption and transmission dynamometers conveniently arranged for the direct measurement of power. For description of Prony brake, see Flather's Dynamometers and the Measurement of Power and the usual hand and text books; also Electric Journal, I., 419. For the cradle dynamometer, see Nichols' Laboratory Manual, Vol. II., and elsewhere. t(§ib). Let us suppose that the error in measuring the input, output or losses is one per cent, due to inaccuracies in the instruments or in 42 DIRECT CURRENT MOTORS. [Exp. A further advantage of this method is that a load run is not an essential, as will be seen later, and hence may be omitted. Conditions often arise, as in testing large machines, when a load test is impossible and this advantage then becomes important. It is always best, however, to make the load run when this can be done. The method of losses is general and can be applied for deter- mining the losses, and hence the efficiency, of a shunt, compound, differential or series wound motor or generator. In the follow- ing paragraphs the directions are full and explicit for testing- a shunt-wound motor. Modifications are outlined in the Ap- pendices for applying the method to other types of motors and generators. § 2. For testing any machine two runs are made : a load run to ascertain working conditions, and a no-load run (or runs) to determine losses under these same conditions. In making the no-load run for losses the machine can be driven electrically as a motor or mechanically as a generator. The for- mer method is used in this experiment (§ 7) ; the latter method is described in § 21 of Appendix I. The resistance* of the armature is to be found by the fall of potential method both before and after the load run, in order that it may be determined both cold and hot (see § 17, Exp. i-A). Since this includes the resistance of the brushes and of brush contact, which varies with current, to be exact it would be necessary to measure the armature resistance for each load their reading. Assume the true output to be 95 when the true input is 100. The output, as measured, might vary from 94.05 to 95.95 and the input, from 99 to loi ; hence the efficiency, determined from output, might vary from 93.1 to 96.9 per cent. On the other hand with the same percentage error in their determination, the measured losses might vary from 4.95 to 5.05 and the measured input from 99 to loi ; hence the efficiency, determined from losses, could only vary from 94.9 per cent, to 95.1 per cent. *In measuring armature resistance the voltmeter is to be connected to the same points as in the load run. 2-B] EFFICIENCY. 43 current. No account will be taken of a possible difference be- tween the contact resistance with machine running and that measured with armature stationary. § 3. Load Run (Shunt Motor). — This run is made* to ascer- tain the working conditions for which the losses are to be deter- mined, that is, to ascertain the load current and hot resistances for calculating copper losses and to ascertain the normal speed and excitation for which the iron and friction losses are to be determined in the no-load run. (The load run is a repetition of the run made in Exp. 2-A for obtaining speed characteristics.) § 4. Connect the motor to the supply lines, the voltage of which should remain practically constant during the run. (See Fig. I of Exp. 2-A.) Adjust the field current by means of the field rheostat so that the motor runs at its rated full-loadf speed (or the speed for which its efficiency is desired) and keep the field current constant at this value during the run. Care in keeping the field current constant will increase the accuracy of the results ; it is not sufficient to leave the rheostat in one posi- tion and assume the field current constant because it is very nearly so. * (§3a). Omission of Load Run. — It will be seen that the load run is not essential and that the method may be employed even when the load run is impossible. Whenever it is possible, however, the load run should be taken, since it serves to get the machine " down to its bearings," that is, down to its working condition of friction as well as of temperature. When the load run is omitted, cold resistances are measured and hot resistances determined by suitable temperature corrections or assumptions. Values of field current and speed are determined for no load; values are assumed for full load which it is believed will most nearly represent the operating conditions for which the efficieiicy is to be obtained. In a motor, for example, we may assume a constant excitation and a constant speed, or a speed which is say 5 per cent, lower at full load, etc. In a generator we may assume a constant speed and a constant excitation, or an excitation which is a certain amount lower (shunt generator) or higher (compound generator) at full load. t(§4a). For commercial testing the speed should be adjusted to its rated value at full load; in laboratory practice the adjustment, when de- sired, may be made at no load. 44 DIRECT CURRENT MOTORS. [Exp. § 5. Beginning at about 25 per cent, overload, as estimated from the input, vary the load by steps from overload to no load or vice versa; at each step measure the line voltage, armature cur- rent,* field current and speed. § 6. The motor may be loaded in any manner that is convenient. A brake may be used for this, but it is frequently more con- venient to load with a generator and to absorbf the output of the generator by suitable resistances. § 7. No-load Run (Shunt Motor) ; Machine Driven Electric- ally 4 — For a shunt machine one|| no-load run is made; the machine is operated as a motor at the same constant excitation as in the load run. The object is to determine the losses for different speeds at this constant excitation. Before taking read- ings the motor should be run awhile so as to attain its normal working condition of lubrication, temperature, etc. With the motor running unloaded, adjust the field current to the same value as during the load run and hold constant at this value during the no-load run. By varying the electromotive force impressed on the armature terminals, vary the speed of the motor by steps so as to cover as wide a range of speed as pos- sible ; this will give more accurate results than if only the speed range of the load run is covered. At each step measure the * See § i6c, Exp. 2-A. t If a direct current generator of suitable voltage is used, the current from the generator may be "pumped back" into the motor supply line (§26). t (§ 7a) . This run can be made with the machine driven mechanically (§21) instead of electrically. II (§7b). Although a run at only one excitation is necessary for de- termining the efficiency of a shunt motor, runs at other excitations are recommended. These additional runs may be taken by the two voltage method (§7d). They are necessary if hysteresis loss is to be separated (Appendix I.) or if flux density is variable (Appendix III.). If a run is wanted at a very high saturation, a higher voltage may be supplied to the field than the rated voltage supplied to the armature. 2-B] EFFICIENCY. 45 -^ 'f) r^WM? eHUNT FIELD W5 electromotive force impressed on the armature terminals, arma- ture current,* field current and speed. By using two resistances, B and C, arranged as in Fig. i, the electromotive force impressed on the armature may be varied '^""'''"^ ''°'^"*''' ^"•'p'^' by short circuiting more or less of B or of C. A single series resistance B may suffice, but the adjustment in many cases can be better made with two. An independent genera- tor can be used as a supply to obtain variable voltages for the armature circuit, or the two voltagesf of a three- wire system. § 8. Results.— The losses of the motor include: (i) Copper losses of field and armature ; (2) Iron losses of armature; (3) Friction and Windage, or air resistance. Losses (2) and (3) are rotation losses and are independent of load. * (§7c). For the no load run the armature current is small; if a low reading ammeter is used, it should be short-circuited at starting to avoid damage by the initial rush of starting current. t(§7d). Two-voltage Method. — For instruction purposes a complete series of armature voltages and corresponding speeds is desirable. Where two supply voltages (as no and 220 volts on a 3- wire system) are avail- able, accurate results may be obtained by a two-voltage method, by taking 8 or 10 readings and averaging first with say 220 and then with no volts impressed on the armature of a 220 volt motor. These points, accurately determined, are sufficient for working up results by the straight line method of Fig. 2, in which they are represented by black dots p and q. By this method the trouble of adjusting armature voltage is avoided. Fig. I. ■'ConnecHqn for no-load run as a shunt motor for determining losses. 46 DIRECT CURRENT MOTORS. [Exp. § 9. Copper Losses. — The copper losses for any circuit can be computed, if the current and resistance through which it flows are known, being equal to RP where R is resistance and / is cur- rent. The armature copper loss is thus computed ; it is a vari- able loss, changing with load. The field copper loss is a constant loss and does not vary with load. It also can be computed by the formula RP, or more con- veniently from the formula EI, the product of current in the field circuit and voltage supplied at its terminals. (The formula EI cannot be thus used unless copper loss is the only expenditure of energy; it cannot be used for determining copper loss of an armature or other circuit in which there is a back electromotive force.) In a self-excited machine, in which a field rheostat is used under normal operation, the loss in the rheostat is to be included in the field circuit loss. § 10. Iron Losses. — The iron losses are losses due to hysteresis and eddy currents ;* they are independent of load, but vary with the speed and with the flux density in the armature. At con- stant speed, hysteresis loss (within the usual working range) varies approximately as the 1.6 power of the flux density; eddy currents as the square of the flux density. At constant flux density, hysteresis loss varies directly with the speed and eddy currents with the square of the speed. If the field current of the motor is held constant, the flux density in the armature will be practically constant for all loads. It will be modified under loadf to a small extent by armature reaction, the effect of which will be neglected. Hence in a shunt motor run with constant *This includes eddy currents in the pole pieces and in armature copper as well as in armature iron. t (§ loa). Loa ^| ellipse.* At one end of the "-mdN^ '^ RHEOSTAT _i characteristic (near open cir- Fig. 3. Connections for loading an cuit), an alternator tends to. regulate for constant voltage; at the present day, this is the usual working part of the characteristic. At the other end (near short circuit), an alternator tends to regulate for constant current. The earliest alter- nators were constructed for such operation. Constant current alternators are used (less now than formerly) for series arc lighting. For this service an alternator should have high armature reaction so as to limit the current on short circuit to the desired value; a reac- tance external to the arma- ture will serve equally well. §9. Full-load Saturation Curve. — The machine is run 40 60 80 100 ARMATURE AMEERES Fig. 4. External characteristic of an alternator at unity power factor. (The dotted parts of these curves were cal- culated according to Exp. 3-B.) at constant speed so as to give its normal full-loadf current at different field excitations. The connections are as in Fig. 3. To obtain the curve for unity power factor, a non-inductive resistance * See discussion of Fig. 7, Exp. 3-B. t(§9a)- Curves taken at intermediate loads (one fourth, one half and three fourths full load) would lie between the no-load and full-load 68 SYNCHRONOUS ALTERNATORS. [EXF, is used as load; with constant armature current, readings are taken of terminal voltage for different field currents, and plotted as in Fig. 2. For the first reading, adjust the field rheostat to its maximum resistance;* with field circuit open, reduce the load resistance to zero {i. e., short-circuit the armature through the ammeter) ; close the field circuit and adjust the field rheostat until the de- sired value of armature current is obtained. For each succeed- ing reading, increase the load resistance by a small step and re- adjust the field rheostat until. the desired value of armature current is again obtained, taking care that the increase or de- crease in excitation is continuous. § 10. In Fig. 2, the excitation data are as follows : Excitation. Volts. Amperes. Ampere Turns. Ko Load. Full I.o2d. 6.66 7-33 3090 340« 575 627 575 A comparison of the no-load and full-load saturation curves. Fig. 2, shows the following : At constant excitation, the difference in the ordinates of the two curves (their distance apart vertically) shows the difference in terminal voltage of the alternator at no load and at full load. At constant terminal voltage, the difference in the abscissae of the two curves (their distance apart horizontally) shows the difference in excitation (magnetomotive force) required at no load and full load in order to maintain the voltage constant. At constant excitation, a voltage of 575 at full load increases to 627 when the load is thrown off, giving a regulation of 9 per curves of Fig. 2. To take these is unnecessary, unless some special object is in view. For inductive load, the full-load saturation curve will be lower than with non-inductive load (as shown in Fig. i, Exp. 3-B, for zero power factor). For different power factors, see § 13, and take data as in § 14. * This resistance should be sufficient to reduce the field current to but a small fraction of its normal value. 3-A]' CHARACTERISTICS. 69 10 F ^ < " ■* LU D- ^^__^^,,J>l.=-}-=^A25 Amp. cc - 3 Li.6 Ampj H _ S 6 UJ < " DC a -J UJ - a. - A. < " TERMINALj VOLTAGE, 575 _J - "Ol LU « 31 li-l ' 1 1 i-l — 1 1 I I U. . cent., the same as already obtained from the external character- istic. Fig. 4. For constant terminal voltage of 575, the excitation must be increased from 6.66 amperes at no load to 7.33 amperes, at full load. This will be found to check approximately with the arma- ture characteristic. Fig. 5 ; an exact check can not be expected. Fig. 2 shows that, as we go above saturation, there is less difference between the no-load and full-load voltages, '" i. e., the regulation is better (§7). §11. Armature Character- istic or Field Compounding Curve. — This curve i^ taken for an alternator* in the same way as for a direct current generator (§26, Exp. i-B). The curve in Fig. 5, taken for a constant terminal volt- age of 575 at unity power factor, shows that in going from no load to full load (43.4 amperes) the excitation is increased from 6.6 to 7.25 amperes. This checks with the increase 6.66 to 7.33 amperes in Fig. 2. Armature characteristics for lower power factors than unity will rise more rapidly (§ 13). * (§ iia). Composite Winding. — Although an alternator can not be com- pounded by a series winding carrying the line or armature current, as in the case of a direct current generator (since the field winding requires a direct current and the line or armature current is alternating), the result can be accomplished by rectifying part of the alternating current and passing it through what is called an auxiliary field winding. Such an alternator is said to be composite wound. The alternating current to be rectified is commonly derived from the secondary of a transformer, through the primary of which flows the line or armature current; for the core of this transformer the arniature frame or spider is used. The 4000 3000 2000 1000 20 40 60 80 J^RMATURE AMPERES Fig. 5. Armature characteristic, or field compounding curve ; unity power factor ; speed constant. 70 SYNCHRONOUS ALTERNATORS. [Exp. APPENDIX I. MISCELLANEOUS NOTES. § 12. Tests on Polyphase Generators. — The tests described above may be made on polyphase generators in the same manner as on single-phase machines. The polyphase generator when loaded should ordinarily be given a balanced load, i. e., one that is divided equally between the several circuits. Tests may also be made by loading down one phase only and taking measurements on the unloaded as well as the loaded phases. In plotting curves, plot voltage and current per phase (the more usual way) ; or, line voltage and equivalent single-phase current. See Exp. 6-A, particularly §§ 28-30. §13. Power Factors Less than Unity. — The characteristics of an alternator under load vary with the power factor of the load. With a power factor less than unity and current, lagging, the regulation will be poorer, the full-load saturation curve will be lower, the exter- nal characteristic lower and the armature characteristic higher than at unity power factor. The reverse is true when the current is lead- ing (instead of lagging), as it may be when there is capacity in the line or in the load, or when the load consists in part of over-excited synchronous motors or converters. These facts may be fully shown by calculation (Exp. 3-B), or by a complete series of runs made with loads of different* power factors. If such runs are to be made, it will be more profitable to make them after Exp. 3-B. At present, it will suffice to illustrate these facts by a few readings only, as in the next paragraph. § 14. Tests to Compare Effects of Inductive and Non-inductive Loads. — The difference between inductive and non-inductive loads composite winding is not, however, being extensively used, for it can not give constant voltage under all conditions — e. g., varying power factor — and the rectifying commutator is liable to spark. The Tirrell regulator (§33, Exp. i-B), applied to the exciter of an alternator, can maintain constant voltage under all conditions of load. * (§ 13a). This will require special facilities for adjusting power factor; for an inductive load, this can be done by means of an adjustable resist- ance and adjustable reactance in parallel. Runs should be made at one high power factor, one medium, and one as low as can be obtained. 3-A] CHARACTERISTICS. 7^ can be illustrated by the following tests, or by modifications which may be devised by "the experimenter. 1. Load the alternator on inductive load, using for this any one particular load which can be conveniently obtained. An induction motor can be used for a load, as in commercial practice ; but a choke coil will serve fully as well. With the same speed and excitation as were used in taking the external characteristic on non-inductive load. Fig. 4, take readings* of load current and terminal voltage with the inductive load. These readings are plotted,! in Fig. 4, as the point p, which is one point on a characteristic for low power factor. (For more complete curves, see Fig. 7, Exp. 3-B.) Throw off the load and ( at the same speed and excitation) read the no-load voltage ; the per cent, increase in voltage when the load is thrown off gives the per cent, regulation. 2. With the same speed and excitation, repeat with a non-inductive load, so adjusted as to obtain the same load current as in i. Note the terminal voltage under load, the no-load voltage when the load is thrown off, calculate the regulation and compare with the regulation in i. 3. With the same speed and terminal voltage as were used for obtaining the armature characteristic on non-inductive load. Fig. 5, note the increase in field current required with inductive load to maintain constant terminal voltage and plot the point q, Fig. 5. 4. Repeat with a non-inductive load (adjusted for the same load current) and compare results. § 15. Efficiency.— If the alternator is driven by a direct current motor, the friction and core loss are conveniently determined by the method of § 21, Exp. 2-B. If the driving motor is alternating, a wattmeter is used to measure its input, the increase in motor input * (§ 14a). If a wattmeter reading is also taken, the power factor can be found by dividing the reading of the wattmeter by the product of current and voltage. t (§ 14b). Since the same value of exciting current may at different times give different amounts of magnetization (as in the case of the ascending and descending curves), the point p thus located — and the point q as located later — may not be exact in their positions, as compared with the characteristics previously taken. They will, however, serve to illus- trate the effects in question. 72 SYNCHRONOUS ALTERNATORS. [Exp. giving the friction and core loss of the alternator — any changes in motor losses being corrected for, if necessary. The copper losses of field and armature are calculated from resist- ance measurements, and the efificiency so determined. If the armature has large, solid conductors, the loss in them will, be greater with alternating than with direct current, this additional loss being a load loss. Load losses are losses which occur under load in addition to the losses already accounted for, i. e., in addition to core loss, RP, friction and windage. There is no simple and accurate method for determining load losses in alternators. The A. I. E. E. Standardization Rules (116-7) give a method for estimating these losses by assuming them to be — in the absence of more accurate infor- mation — equal to one third of the short-circuit core loss. 3-Bl PREDETERMINATION. 73 ExPERiHBNT 3-B. Predetermination of Alternator Charac- teristics.* § I. Introductory. — It is desirable to be able to predetermine the performance of any machine without loading, and this is particularly true of alternators ; for, in the case of large ma- chines, the regulation can not be conveniently found in any other way. There are two simple methods for predetermining the per-- formance of an alternator approximately, — the electromotive force method and the magnetomotive force method. Although other more complex methods are proposed for the more exact determination, no one method has been found which is generally accepted and gives correct results in all cases. It is well to first thoroughly study the electromotive force method, on account of the insight it gives into the general performance of the alterna- tor and into other methods of dealing, with the subject. The magnetomotive force method should then follow; after which, other methods (essentially modifications of these two) can be made a special study by those who desire to pursue the subject further. (See Appendices I. and II.) § 2. There are primarily two causes for the change in termi- nal voltage of an alternator with load : 1. The effect of armature resistance, which is small and defi- nite; this causes a drop in electromotive force which is in phase with the armature current and is equal to R I. 2. The efifect of the flux set up by the armature current, a much larger and less definite efifect, discussed in the next para- graph. § 3. All the flux set up by the armature current encircles the * To be preceded by Exp. 4-A. See § 9 for a statement of data to be taken. For a short experiment, take §§ 1-18 and 26-30, plotting curves for unity power factor only. The curves used to illustrate this experi- ment and Exp. 3-A all relate to the same machine. 74 SYNCHRONOUS ALTERNATORS. [Exp. armature conductors. There are, however, different paths which the flux may follow, causing different inductive effects. § 4. (a) True Armature Reaction. — By one path, flux set up by the armature conductors passes into the pole pieces and through the magnetic circuit of the field magnets (Fig. 10), linking with the windings of the field coils. This flux has a demagnetizing effect, weakening* the field by a certain mag- netomotive force produced by the ampere-turns of the armature. This flux through the field magnets is maintained by successive armature conductors; in a single-phase alternator it is pulsating, but in a polyphase alternator, due to the combined effect of the armature currents in the different phases, it is constant both in position and in magnitude. § 5. (&) Local Armature Reactance. — By a different path, flux set up by the armature current encircles the armature con- ductors without entering the pole-pieces ; this flux (the fine lines in Fig. 9) is entirely in the armature, or partly in the armature and partly in the air gap. The flux surrounding any particular conductor varies periodically and produces a reactance electro- motive force or reactance drop, XI, in quadrature with the arma- ture current and proportional to it, — as in any alternating cur- rent circuit. § 6. By another and somewhat similar path, flux encircles the armature conductors by entering into and returning from the poles without linking with the windings of the field circuit; this flux is shown by heavy lines. Fig. 9. This is cross-magnetizing flux and distorts the field; it does not weaken the field except incidentally to a small extent by saturating the pole pieces. This cross-magnetization is alternating with respect to the armature conductor, as in (&) ; with respect to the pole pieces, it is con- stant in a polyphase and pulsating in a single-phase alternator, as in (o). It may be treated separately; or with (a) or (&). * The field is weakened by a lagging current, but strengthened by a leading current, §§ 46-8. 3-B] PREDETERMINATION. 75 § 7. It is thus seen that there are two somewhat different effects produced by the armature current: the first (a) is a magnetomotive force, which reduces tlie field flux and so reduces the generated voltage; the second (6) is an electromotive force, which is subtracted from the generated electromotive force (in the proper phase) so as to give a lower terminal voltage. These two eft'ects operate simultaneously to lower the terminal voltage, the relative amounts of the two varying according to details of design, — saturations, air-gap, shape of slots, etc. To take full and accurate account of the two effects — treating one as a magnetomotive force and the other as an electromotive force — is difficult* and will not be undertaken here. § 8. We may, however, instead of treating the two effects separately, treat them combined, following either one of two methods : (a) The magnetomotive force or ampere-turn method, which assumes that all the effect is magnetomotive force; or, (&) The electromotive force or reactance method, which as- sumes that all the effect is electromotive force. If the saturation curve were a straight line, the two methods would be identical ;f for, magnetomotive force would produce a proportional electromotive force. With the saturation curve, however, not a straight line, a given increase or decrease in mag- netomotive force will cause a less than proportional change in electromotive force. Hence, if we consider that all the effect of armature flux is a magnetomotive force, we will have a less drop in terminal voltage than if we consider that all the effect is an electromotive force. The magnetomotive force method is, accordingly, opti- mistic (Behrend) and gives the generator a better regulation * See Appendix II. fThis would be true if tlie details of the two methods were in all respects the same. Differences in the details of the two methods, as usually applied, cause differences in the results, even though the saturation curve is straight. 76 SYNCHRONOUS ALTERNATORS. [Exp. than it actually has ; the electromotive force method, on the other hand, is pessimistic, giving the generator a poorer regula- tion than the actual. The two methods, therefore, give the limits between which is the true performance of the machine. § 9. Data. — For either method, the data required are obtained from the following two* runs, which are made without loading the generator: 1. An open-circuit run, giving the open-circuit voltage £0, for different field currents, — i. e., the no-load saturation curve, obtained as in §5, Exp. 3-A. See Curve (i). Fig. i. To save labor in the many subsequent calculations, it is customary to use only the ascending curve. 2. A short-circuit run, giving the short-circuit current Is, for different field currents, — called also a synchronous impedance test, — as described in the next paragraph. See Curve (2), Fig. i. These data enable us to ascertain the synchronous impedance of the armature and hence to compute the volts impedance drop for the electromotive force method ; they also enable us to ascer- tain the magnetomotive force required to overcome the mag- netizing effect of the armature, for the magnetomotive force method. The hot armature resistancef is to be found by the fall-of-po- tential method. § 10. Test for Short-circuit Current and Synchronous Impe- dance. — With the armature short-circuited through an ammeter,^ * (§9a)- Two such runs are common in testing many kinds of appa- ratus; note, for example, the open-circuit and short-circuit tests for trans- formers, Exp. 5-B. t (§9h). On account of eddy currents, the resistance will be greater for alternating currents than the value found by direct current. T'his is of importance as affecting efficiency (§ 15, Exp. 3-A), but is of little con- sequence so far as regulation is concerned, for RI drop has only a small effect at high power factors and is negligible at low power factors, as will be seen later. J (§ loa). The ammeter leads should be short and heavy; for, by the 3-B] PREDETERMINATION. 77 10 11 12 13 14 FIELD AMPERES 15 16 Fig. I. No-load saturation curve (i) and short-circuit current (2 and 3) far different field excitations. Also full-load saturation curves (4, 5, and 6) for zero power factor, current lagging. the short-circuit current is found for diflferent values of field current. The ammeter should have a range of about three times full-load current. The speed should be normal, but special care in maintaining constant speed is not necessary.* methods of computations used later, any drop in them is included in the impedance drop of the armature. * (§ lob). If facilities for varying the speed are provided, with constant excitation vary the speed through wide range and note the practical absence of change in the short-circuit current. Note, however, that the open-circuit voltage is proportional to speed. How are these facts explained? 78 SYNCHRONOUS ALTERNATORS. [Exp. Beginning with the field weakly excited, increase the field cur- rent by steps so that the short-circuit armature current (/s) is increased from, say, i normal to i^ or 2 times* normal full-load current. At each step read field and armature currents and plot as in Curves 2 and 3 of Fig. i. In the short-circuit test, we may have either the field or the armature under normal full-load working conditions, but not both at the same time. § II. The curve for short-circuit current, will (as in Fig. i) be a straight line through a wide working range, and may be extended as a straight linef beyond the observed data. The ultimate bending of the curve depends upon the relative satura- tions of various parts of the magnetic circuit, — armature, teeth, poles, etc. Fig. I shows that normal excitation, OH = 7.33 amperes, gives a short-circuit current of 116 amperes. (Normal excita- tion is the excitation giving rated voltage, 575, at full load, unity power factor; for this machine— see Figs. 6 and 7 — the corre- sponding no-load voltage is found to be 627.) * (§ loc). By taking the run quickly, even higher values of current can be reached. Running an alternator on short circuit, as described, affords the best means for drying armature insulation. An alternator in shipment may have been unduly exposed to weather or have been allowed to stand in a damp place. The insulation readily takes up moisture and is much impaired thereby. In such a case, as soon as the alternator is installed it should be run for one day with the armature short-circuited, the field excitation being so low that the normal armature current flows; there is no high voltage to break down the insulation. The armature is thus baked and the insulation restored. This precaution, particularly in the case of high voltage machines, may avoid a break-down of insulation upon starting up. t (§ iia). Extrapolation as a straight line (2) gives (after saturation is reached) a diminishing value for synchronous impedances Z = £0 -=- Is, as used later. It thus favors the machine by giving a smaller impedance drop; in the electromotive force method this is justifiable because it par- tially offsets the pessimistic tendency of that method. This justification is empirical. Curve (3) has been extrapolated by assuming £o-^/s to be constant. 3-B] PREDETERMINATION. 79 An excitation, OG = 2.6 amperes, is required to cause normal full-load current (43.4 amp.) on short circuit. The corresponding impedance voltage is £2 = 234, for on short circuit the whole generated voltage is used in overcoming the internal or armature impedance. § 12 Synchronous Impedance. — On short circuit, the whole generated voltage is equal to the internal impedance drop in the armature. Impedance is equal to impedance drop divided by current; hence, the synchronous impedance of the armature — i. e., its impedance when running at synchronous speed — is equal 600 627 Volts-Jj'' 575 Volts-Jj''d. °y/ E // < ^,500 II i c g O400 /W 3 5 z Ksoo fi^ ■e g 4^34 Volts=40.7i< 1 j2200 X\ f- 100 "/ 1 Resistance Drop: 32/ 1 _ 20 140 Fig. 40 80 80 100 120 AMPERES ON SHORT CIRCUIT: /s Impedance, reactance, and resistance drop. (All the curves in Exps. 3-A and 3-B relate to the same machine.) to the generated voltage £o, divided by the short-circuit current h. For any field current, the values of JSo and 7s are obtained from curves (i) and (2), Fig. i ; the corresponding synchronous impedance, Z = £o -^- /s, should be plotted as a curve (not shown). It will be found nearly constant for a wide range, — diminishing, however, for high values of field current. § 13. In Fig. 2, the curve marked impedance drop is plotted by 8o SYNCHRONOUS ALTERNATORS. [Exp. taking, from Fig. i, corresponding values for Eo and Is- Eventually there is a tendency for the curve to bend, although in this instance there is none within the range for which Fig. 2 is drawn. The ratio of any ordinate to the corresponding abscissa gives the value of the synchronous impedance; thus, in Fig. 2, the impedance drop is 234 volts for a full-load current of 43.4 amperes, and the impedance is, therefore, 234 -f- 43.4 = 5.4 ohms. The normal full-load voltage of this machine is 575 ; the impedance drop is, accordingly, 40.7 per cent. This is called* the impedance ratio. An open-circuit voltage of 627 is seen to give a short-circuit current of 116 amperes, as already seen in Fig. I. § 14. Resistance drop is plotted as a straight line. Fig. 2. The resistance, found by the fall-of-potential method, is 0.17 ohms; the resistance drop, for 43.4 amperes, is 0.17 X 43-4 = 7-4 volts. §15. The reactance drop is £x = V-Ez^ — £r^; or, for 43.4 amperes, reactance drop = "v'234^ — 7-4^ == 233.9 volts. Usu- ally, as in this case, resistance is small so that there is little differ- ence between the values of synchronous impedance and synchro- nous reactance. It is common, therefore, not to calculate the value of reactance drop, but to use the value of impedance drop in its place. Synchronous reactance is proportional to speed; hence, syn- chronous impedance is practically proportional to speed. Synchronous impedance and synchronous reactance are ficti- tious quantities, comprising not only the real impedance and re- actance of the armature, but also including the effect of arma- ture reactions. It is instructive to compare the curves of Fig. 2 with similar curves for a transformer ; see Fig. 7, Exp. S-B. § 16. Electromotive Force Method. — Aside from its usefulness in predetermining the performance of alternators, this method serves as an excellent illustration of the use of vector diagrams ♦Standardization Rule, 208. 3-B] PREDETERMINATION. 8l in solving alternating current problems ; it is a practical applica- tion* of the elementary principles discussed in detail in Exps. 4-A and 4-B. The electromotive force method is general, apply- ing to all classes of alternating current problems, — transmission lines (§ 56), transformers (Exp. S-C), etc. For this reason the method will be treated in considerable de- tail. S 17. Unity Power Fac- .0 i=43.4 ' £t=575 B1:'C tor. — ^With a non-inductive V load, the power factor of "^ the load is unity; the cur- ^"'- 3- Electromotive force diagram, at unity power factor ; current in phase with rent which flows is, accord- terminal voltage, ingly, in phase with the ter- minal voltage. This is shown in Fig. 3, in which the terminal voltage Et, is in phase with the current /. The armature resis- tance drop, Er^RIj is in the direction of — in phase with — the current / ; the reactance drop. Ex = XI,- is in quadrature with I. The total generated electromotive force £0, is accordingly the vector sum of the following three electromotive forces: £t de- livered to the load ; RI to overcomef armature resistance and XI to overcome armature reactance. * (§ i6a) . This application illustrate.s the way that general principles can be put to practical purposes; the application was first made indepen- dently, and more or less simultaneously, by various engineers. The writer used the method in numerical problems to illustrate the elementary principles of Bedell and Crehore's Alternating Currents in the early nine- ties soon after the issue of that book, and applied it a little later to laboratory data. The data and some of the curves here given are taken from a laboratory outline prepared by the writer for student use and printed in the Sibley Journal, 1897-8, p. 215. t (§i7a). The arrows show the direction of the vectors in the sense that BC and CA are electromotive forces to overcome resistance and reactance, respectively; in the reverse sense, CB and AC are the electro- motive forces produced by resistance and reactance. 7 82 SYNCHRONOUS ALTERNATORS. [Exp. § i8. Knowing the values of resistance drop RI, and reactance drop XI, we may have either of two problems to solve : (a) Given the terminal voltage Et, to determine the open- circuit voltage Eq; or, (&) Given the open-circuit voltage Eq, to determine the termi- nal voltage Et. The following examples will make clear the solution of either problem. (a) Given £t=575; RI = 'J4; Z7 = 233.9. Required to find Eo. Lay off to scale the values of £t, RI and XI, as in Fig. 3 ; by construction Eq is found to be 627. Designating the total in- phase voltage by £p, and the quadrature voltage by Eq; we have, by computation, Eq ^ V £p^ + -fcp' = V C-fcx + ^^)' + C^-^)^ = V(57S + 74)" + 233-9' = 627. The regulation is 9 per cent., Eo being 9 per cent, greater than Et. (b) Given £0^627; 72/^7.4; Z7^ 233.9. Required to find Et. Lay off RI and XI to scale, as in Fig. 3. From A as a. center and radius £0 = 627, strike an arc cutting at the line OB, drawn as a continuation of BC. By this construction, Et is found to be 575; by computation £x=V£o'-(^/)=-i?/=V627''- 233.9^ — 7.4=575. At unity power factor, it is seen that the terminal voltage is always less than the generated or no-load voltage. § 19. Power Factor Less than Unity, Current Lagging. — With an inductive load, the power factor of the load is less than unity and the current, accordingly, lags behind the terminal electro- motive force. This is shown in Fig. 4 in which the current I lags behind the terminal electromotive force £t by an angle 6 = 30°, the power factor of the load, in this case, being cos 30° = 0.866. 3-B] PREDETERMINATION. 83 Fig. 4 is drawn by first constructing to scale the triangle ABC, with two sides equal to RI and XI, respectively, and then laying Ccse = lr cose=o Fig. 4. Electromotive force diagram, at power factor 0.866 ; current lagging 30° behind terminal voltage. off OB at an angle with BC, so that cos 9 equals the power factor of the load. (a) Given £t = S7S, we find by construction £0 = 726; or, by computation £0 = V-Ep' + Eq' = VI^ cos e + Riy + (£t sin 6 + xiy = V(S75 X .866 + 7.4)^ + (575 X -5 + 233-9)' = 726. The regulation is 26.3 per cent. With inductive load, the regulation is always poorer than with non-inductive load. The dotted quadrant indicates the locus of the point for different power factors. {b) Given £0 and power factor; required the terminal voltage £t. Lay off a line in the direction BO making the proper angle 6. 84 SYNCHRONOUS ALTERNATORS. [Exp. Strike an arc from ^ as a center, with a radius Eo, cutting the line OB at 0, thus giving* OB = Er. § 20. Power Factor Less than Unity, Current in Advance. — This case is shown in Fig. 5. The triangle ABC is drawn as be- fore, and OB is laid off making an angle 6 with BC, so that cos B equals the power factor of the load. The current /, for this case, is 30° in advance of Et. (a) Given £t = S75, we find by construction Eo=^^o%; or, by computation, £0 = V EPITE^, = V(£t cos e + Riy+ (£t sin 6 — Xiy, ^ V(57S X .866 + 7.4)' + (575 X .5 — 233-9)' = 5o8. The regulation is- -12 per cent. Co?0 = O.5,<'' (b) Given Eq; the terminal voltage Ei is found, as before, by striking an arc from A as a center, with a ra- dius Eo, cutting the line OB at 0. For a leading current, the terminal voltage is always greater than for a lagging current or for unity power factor, and ^ may even be equal to or Fig. 5. Electromotive force diagram, at greater than the no-load power factor 0.866 ; current 30° in advance voltage, of terminal voltage. o o . • , ^ .. § 21. bpecial Case of Zero Power Factor. — At zero power factor, cos 6 = 0, sin 6=1. * (§ 19a) . The graphical construction for this case will usually be pre- ferred; an analytical expression for Ej, derived from the figure, is El = VE6'— {XI cos e—RI sin ey— (RI cos e + XI sin 9). 3-B] PREDETERMINATION. 85 From JFigs. 4 and 5 it is seen, that the RI drop becomes ineflfec- tive, being at right angles to £t, and can be neglected. Hence, practically, Et:=:Eq — XI, for lagging current; £t = £0 + XI, for leading current. For this case, the various voltages are combined algebraically. Practically, XI ^ZI^Ez, and these expressions become £t = Eq ± £z. This expression, approximate for 5^90°, would be exact for a value of 6 a little less than 90° ; so that, in Fig. 4, OB A forms a straight line and tan 6 = XI -^ RI. § 22. Given the Terminal Voltage at One Power Factor, to Determine it at Any Other Power Factor. — Given £t at any power factor, £0 is found by method (a) of the preceding para- graphs. With £0 thus known, the value of £t is readily found for any desired power factor by method (&). In conducting tests, it is often difficult or impossible to deter- mine £t at unity or high power factors, on account of the power required. The value of £t can, however, be found by test at a low power factor (§52) and then determined by calculation for any desired high power factor. Usually £0 is found by test and resistance drop is known; the reactance drop is not known. In this case the procedure is as follows : In Fig. 4, lay off resistance drop BC; at right angles draw the indefinite line CA, — the value of reactance drop being unknown. At an angle 6 with BC, lay oiiE BO equal to the value of £t found by test at power factor cos 6. Draw OA = £0, as found by test, cutting CA at A. The point A being located and £0 known, values of £t at any power factor are determined by method (b) above. In this manner, if the regulation is known for one power fac- tor, it can be calculated for any power factor. At constant terminal voltage, the locus of the point O will be the arc of a 86 SYNCHRONOUS ALTERNATORS. [Exp. circle with 5 as a center; at constant excitation, £o is constant and the locus of is the arc of a circle with ^ as a center. §23. Application of Electromotive Force Method. — Knowing the armature resistance and synchronous reactance* — obtained from the short-circuit test, — the electromotive force method can be used for predetermining the regulation, the external charac- teristic and the full-load saturation curve for any power factor. § 24. Predetermination of Regulation at Different Power Factors. — By method (a) of §§ 17-20, determine the open-circuit voltage Eq, corresponding to rated full-load voltage at rated full- load current, for different power factors. The values of arma- ture RI drop and XI drop corresponding to full-load current will be constant in all the computations, R and X being taken as con- stant.f Plot the values of £0, thus obtained, with power factor (or B) as abscissse, as in Fig. 6. This is to be done for lagging and for leading currents. Arrange, also, a scale — as on the right of Fig. 6 — to show the values of £0 as per cent, of full-load voltage. § 25. The curves show the increase (or decrease) in voltage when full-load current is thrown off at different power factors; in per cent., this gives the regulation. At power factor i.o, the *(§23a). Synchronous reactance .is practically equal to synchronous impedance. In Figs, i and 2, synchronous impedance is Z = £0 -=- 7s, and is more or less constant ; it can be computed for the value of £0 or for the value of Is corresponding to working conditions. Thus, for normal field excitation, corresponding to £0 = 627, we obtain 2 = 627-^116 = 5.4 ohms; the armature current 116 amp. is, however, far above normal. For normal full-load current, 43.4 amp., we obtain Z = 234 -4- 43.4 =: 5.4 ohms; in this case the field excitation is far below normal. It is thus seen that Z can be computed from the short-circuit test either for normal field current or for normal armature current; but field and armature currents can not simultaneously be normal. When Z is constant, the two computations give identical results. When Z is not constant, the two computations give different results; either may be used, but it is justifiable to use the method which gives the smallest value for Z as being least pessimistic. (See §§ iia and 33.) t See § 26a. 3-B] PREBETERMINATION. 87 regulation is 9 per cent.; at power factor 0.5 (lagging current), it is 37 per cent. ; at power factor 0.0, it is 40 per cent. At high power factors, it is seen that a small change in power factor causes a marked change in regulation ; while at lower power fac- tors the regulation is nearly constant. The reason for this will appear from a consideration of the construction in Figs. 4 and 5. This fact is made use of in § 52. Mn-toaci voltage, nagging current) q—e Full-load voltage, 57S -L-h I 1 L. ■100-j-O a/ SO-j-lO = 20" 30 £ 10 lift 0.9 0.8 0.7 0.0 0.5 0.4 0.3 0.2 0.1 POWER. FAtTOR Fig. 6. Curves showing no-load voltage corresponding to a constant full-load voltage (575) for full-load current (43.4 amperes) at different power factors. § 26. Predetermination of External Characteristics. — For a definite open-circuit voltage £0 and various power factors, com- pute (by method {b) of §§ 17-20) the terminal voltage £t, for different load currents. Armature XI drop and RI drop are to be taken as proportional to current ; i. e., X and R are taken as constant.* Data are thus obtained for plotting the complete ex- ternal characteristic, from open circuit to short circuit, for differ- ent power factors. * (§26a). In §§24, 26 and 32, the same constant values of X and Z are to be used. In § 26 it is proper that X and Z be considered constant for the reason that field excitation is constant. In § 24 the armature current is constant, but not the field, and strictly speaking X and Z might not remain constant, although for simplicity and for ease in comparison they are so taken. In the case of § 32, X and Z should only be taken as con- stant for a certain range, and for very high saturations should be taken as variable as in § 33. 88 SYNCHRONOUS ALTERNATORS. [Exp. § 27. Fig. 4, Exp. 3-A, shows the characteristic for unity power factor. Power is zero on open circuit and on short cir- cuit. Maximum power is, in this case, obtained at about twice full-load current ; at short circuit, the current is about 2j- times full-load current. A small short-circuit current* is an element ' Koor 10 3a 30 10 120 130 50 60 70 80 90 100 110 AMPERES ARMATURE . Fig. 7. External characteristics at different power factors. of safety, obtained, however, by large impedance drop and poor regulation. Compare § 24a, Exp. 5-C. § 28. External characteristics for different power factors, with current lagging and leading, should be plotted as in Fig. 7. The lowest possible characteristic is a straight line; it is obtained for a power factor (cos $) of such a value that tan 6= (armature reactance-drop) -^ (armature resistance-drop). See §21. The *. (§273). This is the working part of the characteristic for constant current operation, see § 8, Exp. 3-A. The armature should have a high reactance for constant current and low reactance for constant potential. 3-B] PREDETERMINATION. 89 characteristic for zero power factor is a little higher than the straight line for the limiting case ; the difference, however, is in- appreciable. When the scale used is such that the ordinate on open circuit is equal to the abscissa on short circuit, the characteristics are ellipses with a 45° line as axis (Steinmetz, Alternating Current Phenomena, 3d ed., p. 304). In any alternator, armature resistance is small and armature reactance, relatively large, so that the armature impedance is practically all reactance; this gives curves as in Fig. 7. If the conditions were reversed, resistance being large and reactance negligible, the curves for cos 6=1 and cos 6^0 would have to be interchanged. Unity power factor would give the poorest regulation and the straight line characteristic now obtained for zero power factor; for, with reactance zero, Et.= Eq — RI, in place of Et^Eo — XI, as in §21. § 29. Predetermination of Full-load Saturation Curve from No-load Saturation Curve. — By method {b) of §§ 17-20, com- pute the terminal voltage Ei corresponding to the different open- circuit voltages of the no-load saturation curve; this is to be done* for full-load current at unity power factor and at zero power factor, current lagging. In this manner, full-load saturar, tion curves are plotted for unity power factor (Fig. 2, Exp. 3-A) and for zero power factor (Fig. i of this experiment). § 30. The interpretation of the full-load saturation curve for unity power factor is given in §§ 10, Exp. 3-A. The curve for zero power factor is capable of similar interpretation. It is seen that, for the same terminal voltage, the excitation must be much greater at zero than at unity power factor; or, for the same excitation, the terminal voltage is much lower. § 31. In determining the full-load saturation curves for any power factor, X and Z can be taken as they are (somewhat vari- *It is unnecessary to construct intermediate curves for part load and for other power factors, unless a special study is to be made. 9° SYNCHRONOUS ALTERNATORS. [Exp. able, § 33) or they can be assumed constant,* § 32. The com- putations can be readily made by either method ; it is only above saturation that the results differ. This will be discussed in greater detail in the case of zero power factor. §32. For zero power factor, the terminal voltage (§21) is Et^Eo — Ez', that is, the impedance drop, £z is subtracted arithmetically from E. In Fig. I, if impedance drop Ez is taken as constant, we obtain Curve (4) differing from Curve (i) by a constant distance (jEz) vertically.f This is satisfactory below saturation, but above saturation is too pessimistic. § 33. If we wish to extend the curve above saturation, it is better to take a variable value, Z^Eo-^h, computed from Curves (i) and (2), Fig. i, for each value of Eo, — that is, for each excitation. This gives a decreasing value for Z and results in Curve (5) instead of (4). Instead of subtracting from Curve ( I ) a constant Ez, we now subtract Ez = ZI = j~Eo. Vs Here / is full-load current (43.4 amp.) ; Eo is taken from Curve ( I ) and Is is the corresponding short-circuit current from Curve (2). The formula can be interpreted thus: if a current Is uses up in the armature a voltage Eo, a current / will use up a proportional voltage, £z= (/-^/s)£o. * See § 26a. fBy the magnetomotive force method (Appendix I.), Curve (6) differs from Curve (i) by a constant distance (Mz) horizontally; at high satu- rations this is too optimistic. 3-B] PREDETERMINATION. 9 1 APPENDIX I. MAGNETOMOTIVE FORCE METHOD.* § 34. In the magnetomotive force method, instead of combining vectorially various electromotive forces — as was done in the electro- motive force method, Figs. 3, 4 and 5 — the corresponding magneto- motive forces are so combined. § 35. The magnetomotive force corresponding to any electromotive force is found by reference to the no-load saturation curve, and is commonly expressed in ampere-turns. For a given machine, with constant number of field turns, field amperes are proportional to field ampere-turns and may be used as a measure of magnetomotive force. In Fig. i of this experiment and Fig. 2 of Exp. 3-A, it is seen, for example, that 627 volts corresponds to a field excitation of 7.33 field amperes, or 3,401 field ampere-turns, either of which may be taken as a numerical measure of magnetomotive force. § 36. It is readily seen that a straight saturation curve gives mag- netomotive forces proportional to electromotive forces, so that the same results will be obtained from the use of either, if the procedure is otherwise identical. On the other hand, a saturation curve which is not straight gives values of magnetomotive forces not proportional to electromotive forces, so that different results will be obtained according to whether magnetomotive forces or electromotive forces are used. § 37. Method. t- — The three magnetomotive forces Mo, Mz and Mt are combined vectorially, as in Fig. 8 ; cos 6 is the power factor of the load. These three quantities Mo, Mz and Mj may be interpreted by their correspondence:]: to the three electromotive forces £0, £z and Et, * No additional data are required ; see § 43 for the particular application of the method to be made. t (§37a). This is the common interpretation of the method (see Rush- more, p. 740, Vol. I., St. Louis Elect. Congress, 1904). In Franklin & Esty's Electrical Engineering, Mo is obtained as the resultant of two mag- netomotive forces which correspond not to £t and Ez, but to £p and £q (the in-phase and quadrature components of £0). t(§37b). If the saturation curve were a straight line and magneto- motive forces were proportional to electromotive forces, the triangles for magnetomotive forces and electromotive forces would be similar and each side of one triangle would be perpendicular to the corresponding side of the other. 92 SYNCHRONOUS ALTERNATORS. [Exp, respectively. A magnetomotive force Mt is required for a terminal voltage Et, corresponding values being taken from the saturation curve; at no load no other magnetomotive force is required. Under load, an additional magnetomotive force BA = Mz is required to overcome the magnetizing effect of the armature. In terms of mag- netomotive force, Mz is equal to the ampere-turns of the armature; Cosff Fig. 8. cose=o Magnetomotive force method. in terms of its corresponding electromotive force, it is a magneto- motive force which will produce an electromotive force equal to the armature impedance drop, £z. The total magnetomotive force which the field must provide is the vector sum. Mo. In this sense. Mo is the resultant of Mr and Mz {^ BA ) , in the same way that £o is the resultant of Et and £z. Interpreting these quantities further as magnetomotive forces: Mo is the magnetomotive force produced by the field ; Mz (= AB, in the direction of armature current, 7) is the magnetomotive force produced by the armature ; Mt is the combined magnetomotive force and produces the electromotive force Et. In this sense. Mi is the resultant of Mo and Mz ( =AB). On open circuit the field ampere- 3-B] PREDETERMINATION. 93 turns (or amperes) give us the value of the magnetomotive force Mt; for, in this case, Mz = 0. On short circuit, the field ampere-turns (or amperes) give us the value of Mz; for, in this case, Mt = 0. That is, on short circuit the field and armature ampere-turns are (practically) equal and oppo- site (compare §21). In Fig. I it is seen that, on short circuit, full-load current (43.4 amp.) is given by a magnetomotive force Mz = OG = 121 ampere- turns (2.6 amperes) ; the corresponding impedance voltage, as used in the electromotive force method, is £z = GF = 234. § 38. Procedure; Any Power Factor. — The value of Mz is known, as in the preceding paragraph ; also the power factor, cos 6, of the load. Given £t to find £0. Construct the triangle OBA, Fig. 8, from the known values of Mz and cos 6, and the value of Mr corresponding to Et; the value of Mo and the corresponding value of £0 is thus determined. Given £0, the converse procedure is followed to obtain £t. The most important cases are for unity and zero power factors. § 39. Unity Power Factor. — For this case, cos ^= i, and OBA (Fig. 8) becomes a right triangle. The same procedure is followed as in the preceding paragraph. § 40. The following procedure, known as the Institute* Method (proposed by a committee but not adopted) differs from the fore- going by taking special account of the armature RI drop. Armature RI drop is significant at unity power factor; it becomes less so as the power factor decreases and becomes negligible at zero power factor. The Institute Rule is : " When in synchronous machines the regulation is computed from the terminal voltage and impedance voltage, the exciting ampere-turns corre- sponding to terminal voltage plus armature resistance-drop, and the ampere-turns at short-circuit corresponding to the armature impedance- drop, should be combined vectorially to obtain the resultant ampere-turns, and the corresponding internal e.m.f. should be taken from the saturation curve." By the reverse procedure £t is determined when £0 is known. * Rule 71, p. 1087, Vol. XIX. 94 SYNCHRONOUS ALTERNATORS. [Exp. §41. Zero Power Factor.— When cos 6 = 0, it is seen that, by the construction of Fig. 8, Mz and Mr are in one straight line; hence Mt = Mo — Mz; or, Mo = Mt + Mz. At no load-JWo = Mt. Under load, if Mt (and £t) is to have the same value as at no load, the field excitation Mo is to be increased by an amount Mz added in this case arithmetically* § 42. Determination of Full-load Saturation Curve. — Given the no- load saturation curve. Fig. i ; the full-load saturation curve for zero power factor is found by adding the constant magnetomotive force Mz = OG. The two curves (i) and (6) are accordingly a constant distance apart, measured horizontally. § 43. Application. — To illustrate the use of the magnetomotive force method, it will suffice to apply the method, using observed data, to the following typical cases : 1. Using the Institute Method, § 40, obtain Eo, corresponding to rated voltage, Ei, at full load, unity power factor. Plot this as the point p, Fig. 6. Note that this point is a little lower than Eo obtained by the electromotive force method, i. e., the regulation is better. 2. Also, locate p by the method of § 39. 3. By the method of §§ 38 and 41, locate the point q, Fig. 6, that is £0 corresponding to rated £t at full load, zero power factor. Note that this is considerably lower than £0 obtained by the electromotive force method. 4. Construct a full-load saturation curve (§42) for zero power factor. ■ § 44. Justification of the Magnetomotive Force Method. — The con- struction of Fig. 8 shows that the armature ampere-turns are com- bined with the field ampere-turns in such a way as to have the great- est efifect for power factor zero, cos = o ; the least effect for cos 6=1; and intermediate effects for intermediate values of cos Q. This will be shown to be qualitatively correct, although quantitatively it is only correct approximately or under certain assumptions. § 45. Fig. 9 shows two conductors of an armature coil, one midway under a north pole, the other midway under a south pole. In this position the electromotive force induced in the armature conductors * (§4ia). The corresponding electromotive forces at zero power factor are likewise added arithmetically; £o = £t-|-£z. (See § 21.) 3-B] PREDETERMINATION. 95 is a maximum. The armature current will likewise be a maximum, if it is in phase with this electromotive force. In this position, the flux set up by the armature current has a cross-magnetizing effect; the flux passes transversely through the pole piece but does not pass through or link with the field winding and so does not directly oppose the field ampere-turns. Fig. 10 shows the armature conductors midway between poles; the coil, to which these conductors may be assumed to belong, is exactly opposite a pole. In this position the electromotive force induced in Fig. g. Distortion of field by transverse magnetization, or cross- magnetizing effect of armature cur- rent ; produced by an in-phase cur- rent, or component of current. Fig. 10. Weakening of field by de- magnetizing effect of armature current ; produced by a wattless or quadrature current, or component of current. the armature conductors is zero ; at zero power factor the armature current — lagging 90° behind the electromotive force — is a maximum. It will be seen from the figure that in this position the armature has the greatest demagnetizing effect, the flux produced by the armature passing through the field winding and directly opposing the field ampere-turns. § 46. It is seen that when the armature current is in phase with the generated electromotive force it produces distortion and cross-mag- netization ; when the armature current is in quadrature it produces demagnetization without distortion, the armature ampere-turns being in direct opposition to the field ampere-turns. When the current has a phase displacement, with respect to the induced electromotive force, between 0° and 90°, it may be considered as composed of two components, an in-phase component producing cross-magnetization and a quadrature component producing demag- netization. § 47. On short circuit, the current in the armature lags 90° (or 96 SYNCHRONOUS ALTERNATORS. [Ekp. nearly so, on account of high armature reactance). The armature and field ampere-turns on short circuit are, therefore, practically equal and opposite. If they were exactly equal and opposite, there would be no electromotive force generated; as a matter of fact, there is a very small electromotive force equal to the armature RI drop. That the armature ampere-turns due to a current lagging 90° opposes or weakens (and does not aid or strengthen) the field is verified by this short-circuit test, and its resultant small electromotive force. § 48. A leading current, on the other hand, directly aids and strengthens the field. § 49. In the foregoing discussion of Figs. 9 and 10, the reaction of the armature has been considered for the particular moment and position when the armature current is a maximum. In reality, the armature assumes successively all positions and the current takes all values; in intermediate positions, demagnetization and cross-magneti- zation are both present in varying amounts dependent upon the posi- tion of the armature and the armature current at any instant. The general nature of the reaction, however, may be considered as defined by its character when the current is a maximum. The real effect is a summation of the effects at each instant through a cycle. A more complete discussion would involve some knowledge or assumption as to flux distribution in the pole pieces, and other design factors. As a matter of fact, a sinusoidal flux distribution has been assumed in order to make it possible to treat Mo as a vector in Fig. 8; the assumption tacitly made is that the field flux passing through an arma- ture coil varies as a sine function of time, so that the generated elec- tromotive force (e = — dcjy-i-dt) is also a sine function differing in phase by 90°. This assumption justifies the treatment of Mo and £0 as vectors at 90°. But distortion, by its very nature, disturbs the flux distribution and makes the assumption necessarily an impossible one. No diagram using plane vectors can exactly represent all the quantities. The justification of the magnetomotive force method is, therefore, partly empirical. It is found to give fairly good result on many modern alternators in which armature reaction is large as compared with armature reactance and in which too high saturation is not reached; it is least accurate in alternators with high saturation and relatively large armature reactance. 3-Bl PREDETERMINATION. 97 APPENDIX II. OTHER METHODS. § 50. There are a number of methods for determining the regula- tion and characteristics of alternators which are essentially modifi- cations of the electromotive force and magnetomotive force methods, or a combination of the two ; these methods are based on test data alone (obtained from open-circuit and short-circuit tests, §9), on design data alone, or partly on design and partly on test data. Methods based on design data are of particular interest to the design- ing engineer but cannot be taken up here; they include methods for calculating armature reaction and reactance and for predetermining the behavior of a machine before its construction. (For further dis- cussion, see references, § 55.) In all methods use is made of the fundamental principles brought out in the electromotive force and magnetomotive force methods, which should therefore be carefully studied before other methods are undertaken. For those whose object is a general understanding of the behavior of alternators, a study of these two methods is sufficient ; but those who desire to pursue the subject further should consult the references in § 55. It has been pointed out that, so far as results are concerned, these two methods give the pessimistic and optimistic limits. Other methods give intermediate, and in some cases more correct results; there is, however, no one absolutely correct method. In reference to this, Mr. Behrend says : " It appears wise to admit the existing dilemma. The question of accu- rately determining the regulation of alternators can not be solved. . . . It seems to the speaker far more dignified and more in accordance with the science that we are working in, to say that this case is so complex, so intricate, there are so many factors to be taken into account, that it can no more be solved than you can state to one thousandth of an inch the dis- tance between two chalk marks drawn on the floor." (A. I. E. E., Vol. XXIIL, p. 326.) § 51. Test Methods. — The aim in various methods is to test the alternator under real or equivalent load conditions with only a small expenditure of power. The machine may be actually loaded and the power returned by some opposition method (§§27, 27a, Exp. 2-B), or it may be tested without any load by simulating working load 8 98 SYNCHRONOUS ALTERNATORS. [Exp. conditions. In the preceding pages this was done by two tests, the open-circuit test at normal voltage and zero current, and the short- circuit test at normal current and zero voltage, in each test the power output being zero. But, inasmuch as power output is the product of current, voltage and power factor, E and I may simultaneously have normal full-load values without involving expenditure of power if the power factor is zero. This leads to the low power factor tests (§52) and split field tests (§53), concerning which only a brief state- ment will be made ; for fuller information consult references. These tests are used in heat runs and efficiency tests, as well as in test for the determination of regulation. § 52. Tests at Low Power Factor. — When operated at low power factor, an alternator may have full-load current and normal voltage with only a small expenditure of energy. If Eo and £t are thus determined for one power factor, their values and the regulation can be calculated (§22) for unity or any other power factor. This cal- culation is usually made either for the same terminal voltage or for the same excitation (same £0). The load may consist of react- ances, unloaded induction motors or a synchronous motor with low or no-field excitation. The power factor is known from readings of ammeter, voltmeter and wattmeter. Any power factor less than 0.20 or 0.25 may be considered as zero, for between these limits (see Fig. 6) there is practically no change in regulation. .When a synchronous motor is used, the generator voltage is adjusted by the field rheostat of the generator; the armature current by the field rheostat of the motor. In this way a full-load saturation curve for low power factor can be obtained (Fig. i) and compared with the no-load curve; or points can be plotted for an external charac- teristic, as in Fig. 7. § 53. Split Field Method. — When an alternator is operated at low power factor with a synchronous motor load, as in the preceding paragraph, electric energy is given out by the alternator to the motor one quarter-cycle and is practically all returned the next quarter- cycle; power circulates between the two machines. Circulation of power in one machine was first proposed by Mordey*; this was accomplished by dividing the armature coils in two parts, one opposed to the other. In this way part of the armature acted as a generator and part as a motor. This, however, proved open to objection. * W. M. Mordey, Journal Brit. Inst, of Elect. Eng'rs, Vol. II., 1893. 3-B] PREDETERMINATION. 99 Behrend (see his St. Louis paper, § 55) has developed a method for circulating power in one machine by dividing not the armature but the field and reversing the excitation of one part of the field. The armature acts as a generator with respect to one part of the field and as a synchronous motor with respect to the other part. Each part of the field has its own rheostat, one controlling the generator and the other the motor action. Tests are made in much the same way as though two machines were used, § 52. For a later modifica- tion of this method, see paper by S. P. Smith, § 55. § 54. Arguments for and Against Specifying Regulation at Zero Power Factor. — The opinion is growing among engineers that regula- tion should be specified at zero power factor. Tests at unity power factor are objectionable, not only on account of the use of much power which may be prohibitive, but also on account of errors in the results. In Fig. 7 it is seen that the difference in regulation for a small change in power factor is very small near zero power factor, but is considerable near unity power factor. At unity power factor, therefore, any inductance or capacity in the load introduces a large error. The use of a water rheostat as a load causes an error for this reason, for it possesses a capacity which, though small, is suflScient to give an alternator a better regulation than it would have if the power factor were unity. Tests at zero power factor, on the other hand, have the advantage that such errors are insignificant; furthermore, the tests are less diffi- cult to make on account of the small amount of power required. They can often be made when tests at unity power factor are not possible. For these reasons, specification of regulation at zero power factor (rather than unity power factor) has been advocated; such specifica- tion can be checked by experiment and, furthermore, it gives the regulation under the worst conditions. On the other hand, this is objected to because, by itself, the regulation at zero power factor is no positive indication of the behavior of the machine at unity power factor ; two machines with the same regulation at zero power factor may have very different regulations at unity power factor. This is largely due to resistance drop, which is of importance at unity power factor, but has practically no effect at zero power factor. Specifica- tion of regulation at zero power factor is, therefore, insufficient — loo SYNCHRONOUS ALTERNATORS. [Exp. unless, in addition, the resistance drop is separately stated. Tests at zero power factor are also objected to because such tests are made when the distorting influence of cross-magnetization is absent. (See Vol. I., p. 761, Int. Elec. Cong., 1904.) § 55. References. — References are given below to a few leading articles on the subject of alternator regulation. A complete list would be a long one, but the references here given are the best ones to consult first; they contain references to practically all that has been written on the subject. Rushmore's paper, with twenty- four references, sum- marizes the work of others and is one of the best papers to read first, particularly in connection with variations of the magnetomotive force and electromotive force methods. The discussion, found at the dose of some of these papers, will be found very valuable. Transactions International Elect. Congress, St. Louis, 1904: The Regulation of Alternators, by D. B. Rushmore, Vol. I., p. 729; The Testing of Alternating Current Generators, by B. A. Behrend, Vol. L, p. 528; Methods of Testing Alternators According to the Theory of Two Reac- tions, by A. Blondel, Vol. L, p. 620 ; Methods of Calculation of Armature Reactions of Alternators, by A. Blondel, Vol. L, p. 635. Transactions American Inst, of Electrical Engineers: The Determination of Alternator Characteristics, by L. A. Herdt, Vol. XIX., p. 1093, 1902 ; The Experimental Basis for the Theory of the Regulation of Alter- nators, by B. A. Behrend, Vol. XXL, p. 497, 1903 ; A Contribution to the Theory of the Regulation of Alternators, by Hobart and Punga, Vol. XXIIL, p. 291, 1904. Journal British Inst, of Electrical Engineers: Henderson and Nicholson, p. 465, 1905 ; S. P. Smith, paper read November 12, 1908 (also Lond. Electrician, November 13). See also Guilbert, Elect. World, 1902-3; Torda-Heymann, Lond. Elec- trician, Vol. LHL, p. 6, 1904 ; C. A. Adams, Harvard Eng. Journal, 1902-3. Parts of the subject will be found treated in various text-books: S. P. Thompson's Dynamo Electric Machinery, Karapetoff's Exp. Elect. Eng., Franklin and Esty's Elect. Eng., etc 3-B] PREDETERMINATION. lol APPENDIX III. MISCELLANEOUS NOTES. § 56. Transmission Line Regulation. — In the electromotive force method, §§ 16-22, a complete treatment is given of the effect upon delivered voltage of resistance drop and reactance drop in the arma- ture of an alternator. The treatment, however, is general and is not limited to alternators. The same treatment will apply to any resist- ance and reactance drop, wherever located, and may accordingly be applied to the case of a transmission line. In the geometrical treat- ment of any problem, resistance drop is always in phase with the current, reactance drop in quadrature. Example i. — Given a transmission line in which RI drop = 7.4 ; XI drop = 233.9. What must be the voltage £0 applied at the sending end of the line to maintain a voltage of 575 at the receiver for a load of 43.4 amperes, at unity power factor, at power factor 0.866 (current lagging 30°), and at power factor 0.866 (current leading 30°)? Figs. 3, 4 and 5 show that 627, 726 and 508 volts, respectively, are required at the sending end in the three cases, the corresponding line regulation being g, 26.3 and — 12 per cent. In this example the same numerical values have been used for a transmission line as were used in Figs. 3, 4 and S for an alter- nator. Practical values for a transmission line would give a relatively greater resistance drop and smaller reactance drop, as in example 2. Example 2. — A transmission line gives 1,000 volts at the receiver. The resistance drop is 100 volts, reactance drop is 200 volts ; what is the regula- tion for different power factors? Curves as shown in Figs. 6 and 7 can be drawn for a transmission line. These curves have been discussed for an alternator ; the discus- sion can, however, be applied to a transmission line. In calculating the regulation of a transmission line, the values of resistance and reactance can be taken from tables in various hand- books and elsewhere. In testing a transmission line, the reactance drop can be found by an open-circuit test and a short-circuit test, as in the case of alterna- tors. With a low voltage, short-circuit the line and measure h ; open- circuit the line and measure £0. The line impedance is Z = £o-^/s; the line reactance is X^ \/Z' — R'. In the laboratory a line with resistance and reactance can be tested in this way as a transmission line; the regulation for loads of dif- ferent power factors can be predetermined (Figs. 6 and 7) and com- pared with actual load tests. CHAPTER IV. SINGLE-PHASE CURRENTS. Experiment 4-A. Study of Series and Parallel Circuits Con- taining Resistance and Reactance. § I. Introductory. — The object of this experiment is to acquaint one with the fundamental relations between currents and electro- motive forces in alternating current circuits. These relations will be brought out by a study of series and parallel circuits contain- ing resistance and inductance, the clear understanding of which is essential for one undertaking any study of alternating currents. Practically every problem in alternating currents involves — or can be reduced to — a problem of series and parallel circuits. A study of alternator characteristics (see Figs. 3-5, Exp. 3-B) is a study of series circuits; the transformer. (see Figs. 6-9, Exp. S-C) can be reduced to equivalent series and parallel circuits, and so, too, the induction motor. This is true of nearly all types and kinds of alternating current apparatus. It will be found that the study of series and parallel circuits brings out the general principles that are common to all alternating current problems. Such cir- cuits are studied, therefore, not merely as leading up to the sub- ject proper, but as actually being the subject matter of all alter- nating current testing. Part I. contains an outline of the underlying principles of the subject, which will be found discussed in detail in Bedell and Crehore's Alternating Currents and in other treatises. Part II. describes the tests to be made and Part III. describes the results derived from them. For the convenience of the reader, some paragraphs on theory are included in Part III. 4-A] SERIES AND PARALLEL CIRCUITS. 103 PAKT I. ELEMENTARY PRINCIPLES. §2. Defining Relations.^In a direct current circuit, the cur- rent which flows is I^E^^R, irrespective of whether the cir- cuit is inductive or not; the power expended is the product of electromotive force and current. § 3. In a non-inductive* alternating current circuit, this is also true ; the current is determined by the resistance, as in a direct current circuit, and the power is the product of electromotive force and current ; thus, I = E-^R; W = EI. The impedance, defined below, consists in this case of the resistance R only. § 4. In an inductive alternating current circuit, the current is less than E^r-R and the power is less than EI; thus, I^E^Z ; W = EI X power factor. The impedance Z, defined as the volts per ampere, is greater than the resistance R on account of the reactance X ; thus, Z = E^I=yR^ + X''=^yR^-\-L'oy'. The reactance (defined in §40) for an inductive circuit has a value X^L(o, where L is the inductance, or coefficient of self- induction of the circuit, and « is 2ir X frequency in cycles per second (§1, Exp. 3-A). Impedance and reactance are expressed in ohms. It is seen that inductive reactancef depends not only * (§33). A circuit is inductive when a current in it sets up a magnetic field (§ 14) ; it is non-inductive when a current in it produces no magnetic field. A circuit is never entirely non-inductive, but may be made nearly so. This is practically accomplished when the outgoing and return con- ductors are placed so close together that the magnetic effects of the currents in the two conductors neutralize each other. Iij a solenoid this is accomplished by using a double winding, the currents in the two halves of which flow in opposite directions. t (§4a). In a circuit with capacity C, the reactance is i/Cw. When L and C are both present, the total reactance is the difference between the capacity reactance and inductive reactance; Xt=Lw — i/Cw. See §57. I04 SINGLE-PHASE CURRENTS. [Exp. upon L, which is a constant of the circuit depending upon its form and dimensions, but also upon the frequency of the alter- nating current supply. § 5. The preceding equations can be written The admittance Y of an alternating current circuit, defined as the amperes per volt, is the reciprocal of impedance; F = /-^£. The unit of admittance is commonly called the mho. § 6. Power factor, defined as the ratio of true power W to apparent power or volt-amperes El, is always less than (or equal to) unity. Power factor^ cos ^, where 6 is the phase difference between E and I; see Figs. 2 and 7 discussed later. In a circuit with resistance R and reactance X, tane = X^R. The subject will be most readily understood by considering: first, circuits with R, only; second, circuits with X, only; and finally circuits with both R and X. §7. Series Circuit with Resistance Only. — In an alternating current circuit containing only a resistance R, the electromotive force required to make flow a current /, is E& = RI, as in a direct current circuit. The current is in phase with the electromotive force. As the electromotive force rises from zero to a maximum and falls again to zero, the current i at each instant is proportional to the electro- motive force e at that instant; e = Ri. The current is zero when the electromotive force is zero, and is a maximum when the electromotive force is a maximum. § 8. If E- is repre"sented as a vector. Fig. 5, the current / is represented as a vector in the same direction or phase as E; 4-A] SERIES AND PARALLEL CIRCUITS. 105 that is, to cause a current / to flow through a resistance R, an in-phase electromotive force equal to RI is required. § 9. Significance of Vectors. — In developing the theory of vec- tor diagrams for alternating current quantities, the vectors rep- resent the maximum values of quantities which vary according to a sine law. In applying these diagrams, however, the vectors are usually drawn to represent the effective (or virtual) values, as measured by ammeter and voltmeter, — the effective value of a sine wave being ^V^ times its maximum value.* Furthermore, vectors are used for currents and electromotive forces which do not vary exactly as a sine law, although the results in these cases are not, in general, theoretically correct.f In drawing vector diagrams, it is implied, therefore, that the currents and electro- motive forces have wave forms which are sine waves or may be represented by equivalent sine waves of the same effective values. The phase difference 6, between equivalent sine waves for cur- rent and electromotive force, is determined by the relation: cos 6 =: power factor =:W -^ EI. §10. Direction of Rotation. — Counter-clockwise rotation is usually taken as the direction of rotation of alternating current vector diagrams, and this convention will be here followed. By considering a diagram as making one complete revolution (360°) in one cycle, the projections, from instant to instant, of the various lines of the diagram upon any fixed line of reference will be proportional to the instantaneous values of the quantities represented by those lines. By reversing all diagrams as in a mirror, the corresponding diagrams for clock-wise rotation will be obtained. §11. Electrical Degrees. — In alternating current vector dia- grams, "angle" is a measure of time, 360° indicating the time *See Bedell and Crehore's Alternating Currents, p. 38, and other text- books. t (§9a). Compare §§60-64; for further discussion, see references given in § gb, Exp. 5-C. io6 SINGLE-PHASE CURRENTS. [Exp. of one complete period or cycle, 90° indicating ^ period, etc. A degree is, therefore, a unit of time, being sometimes designated a " time-degree " or " electrical degree." This designation is, how- ever, unnecessary except in discussions where " space-degrees " are also used. § 12. Series Circuit with Reactance Only. — In an alternating current circuit containing only a reactance oi X ohms, the electro- motive force required to make flow a current /, is Ex=^XI; and I = Ex^X, as shown in §§ 14-17. When the reactance X, is due to inductance, the electromotive force to overcome reactance is Ex = XI^Lu>I. Reactance is the same as resistance in that an electromotive force proportional to it is required to cause a current to flow, the electromotive force being XI for reactance and RI for resistance. Reactance is, however, different from resistance in that it consumes no energy; when the current is increasing, energy is stored* in the magnetic field (as in a fly-wheel), this energy being returned to the circuit when the current is decreasing. In a reactance, the current and electromotive force are not in phase but are in quadrature with each other, i. e., the current and electro- motive force differ in phase by a quarter of a cycle or 90°, and when one is a maximum the other is zero. *(§i2a). The energy of the magnetic field is equal to ^/iLP, cor- responding to the energy of a moving body, l/^MV. d *. 3 1^ 6^ A , I ' Fig. I. Vector diagram for circuit with inductive re- actance 4-A] SERIES AND PARALLEL CIRCUITS. 107 §13. For inductive reactance,* the electromotive force to over- come reactance is in advance of the current by 90°, as in Fig. i, and is not in phase as in Fig. 5. The current lags behind the electromotive force by 90°, that is, the current reaches a positive maximum | cycle later than the electromotive force reaches its positive maximum. When R^o, tan^ = Z-^i?= 00 ; ^=90" ; power = EI cos 6 = 0. A current and electromotive force in quadrature represent no power and are said to be "wattless.'' § 14. Theory. — When a current flows in an inductive circuit, the current sets up magnetic flux which is linked with the circuit. When the current changes, this flux changes and a counter- electromotive force is induced in the circuit tending to oppose any change in the current, — the current seemingly possessing inertia. The electromotive force produced by self-induction depends upon the rate of change of current,! and is e oc — di/dt ; or, e = — L di/dt. The negative sign indicates that the electromotive force is counter to the impressed electromotive force. The equal and opposite impressed electromotive force to over- come self-induction is e=^L di/dt. * (i 13a). For capacity reactance, the electromotive force to overcome reactance X=i/C(i) is XI z=lI -^Ca and is 90° behind the current; the current is 90° in advance of the electromotive force ; see § 55. t (§ 14a). The electromotive force produced by self-induction, expressed in terms of rate of change of flux, is e = — S d/dt. (Compare §§33, 33a, Exp. S-A.) In the absence of iron, i and 0,are proportional to each other and L is constant. In this case Li = S, and L = S0 -i- i ; or, the inductance of a coil is equal to the flux-linkages or flux-turns S for unit current. Since 4>xSi, it follows that LocS', other things (including dimensions of coil and leakage) being equal; the inductance of a coil is approximately proportional to the square of the number of turns. In the presence of iron, i and are not proportional, and L is not constant but varies with saturation. io8 SINGLE-PHASE CURRENTS. [Exp. § 15. The inductance L of a circuit is defined by the foregoing equations. When e is in volts and % is in amperes, L is in henries. A circuit has an inductance of one henry when a change of cur- rent at the rate of one ampere per second induces an electro- motive force of one volt. § 16. When the current varies according to a sine law, 2 = /max sin (of. The impressed electromotive force is, accordingly, e = L di/dt = Lw/max cos at = Lial^^j. sin {at -\-go°). The impressed electromotive force to overcome self-induction is, therefore, 90° in advance of the current; the current, on the other hand, lags 90° behind the electromotive force. § 17. The maximum value of this electromotive force is seen to be La times the maximum value of the current; hence, the effective value of this electromotive force is La times the effective value of the current, that is, Ex = La>I = XL Fig. i and the statements in §§12, 13 are thus established. § 18. Series Circuit with Resistance and Inductive React- ance. — In a circuit with both R and X, the electromotive force required to cause a current / to flow consists of two components, which have been separately discussed in the preceding paragraphs : RI, in phase with /, to overcome resistance; XI, 90" ahead of /, to overcome reactance. Thus in Fig. 2, if OD is current, OC is the electromotive force to overcome resistance and CA is the electromotive force to over- come* reactance, OA being the total impressed electromotive force. These electromotive force relations are fundamental and * (§ i8a). These electromotive forces, CA and OC are components of the impressed electromotive force. In the opposite sense, as counter- electromotive forces, we have the counter-electromotive force AC, lagging 90° behind the current, produced by inductive reactance ; and, the counter- electromotive force CO, opposite in phase to the current, produced by resistance. Compare §15, Exp. 6-A. 4-A] SERIES AND PARALLEL CIRCUITS. 109 are shown by the electromotive force triangle, Fig. 2, and by the following equations : E = Ve^ + E^ = VRf + Lmf = iV R^ + noy\ The impedance triangle, Fig. 3, is derived by dividing the elec- tromotive forces. Fig. 2, by /. § 19. It is seen that the electromotive forces XI and RI are added as vectors. If, instead of a single X and R, there were o I "o RI c Fig. 2. Electromotive force triangle. Fig. 3. Impedance triangle. several, the same procedure could be followed: RJ, RJ, RJ, etc., would be laid off in phase with I ; and XJ, XJ, XJ, etc., in quadrature with /. Electromotive forces in a series circuit are added as vectors. Impedances, resistances and reactances in a series circuit are added as vectors. § 20. The total drop in phase with / is tRI ; the total drop in quadrature with / is SX7. Hence, for any series circuit, E=y{%Riy + {%Xry, and Z=V(5i?)^+(5Z)^ The total resistance of a series circuit is seen to be the arith- metical sum of the separate resistances; the total reactance is the arithmetical sum of the separate reactances. For further discussion of series circuits, see §§38-50; for par- allel circuits see §§ 51-53. no SINGLE-PHASE CURRENTS. [Exp. PART II. MEASUHEMENTS. §21. The following tests require a resistance, which is non- inductive and is designated R^ ; and a coil, which is inductive and is designated RzL^. It is desirable to have the resistance and the coil take currents which are comparable in value with each other, for the frequency at which the tests are made; thus, if at no volts, 60 cycles, the coil takes a current of 10 amperes, the resist- ance should be so selected that at no volts it takes a current of, say, from 5 to 20 amperes. Except for § 28, the coil should not have an iron core, so that there are no losses except RP. For the tests of § 26a (which may precede the main tests), the windings of the coil should be divided in two equal parts, which can be connected in series and in parallel. § 22. The instruments required consist of a voltmeter, capable of reading the supply voltage and lower voltages; an ammeter capable of measuring the combined currents of the coil and resist- ance; and a wattmeter having a voltage range corresponding to the range of the voltmeter and a current range corresponding to the range of the ammeter. A voltmeter switch will be found convenient for the series tests ( §§ 29-31 ) and an ammeter switch for the parallel tests ( §§ 32-34) . On all tests the frequency should be known. § 23. (a) Resistance Alone, — With an adjusting resistance in series, as in Fig. 4, connect the resistance R^ to the supply circuit (say no volts, 60 cycles) and measure the current /, the voltage E at the terminals of R^, and the watts W consumed by R^. The current coil of the wattmeter is connected in series as an ammeter and the potential coil in shunt as a voltmeter, the arrangement* of instruments being shown in Fig. i, Exp. 5-B. * (§23a). In these tests no account is ordinarily to be taken of the fact that the instruments themselves consume a certain small amount of power, as fully discussed in Appendix III., Exp. S-A; this fact, however, should not be neglected in accurate testing, as for example in the accurate determination of L by the impedance method, § 47. 4-A] SERIES AND PARALLEL CIRCUITS. 1 1 1 § 24. Vary the adjusting resistance,* and in this way take sev- eral sets of readings. If there is any question as to the accuracy of the instruments, assume the ammeter and voltmeter to be correct and determine a correction for the wattmeter, so that in (a) the watts as read by the wattmeter are equal to the product of volts and amperes, as read by the voltmeter and ammeter. This serves as a calibration of the wattmeter, to be used in this and subsequent tests. § 25. Take readings, in a like manner, at a second frequency. §26. (&) Coil Alone. t — Repeat (a) using the coil R^L^ alone, as in Fig. 6, instead of the resistance i?i. § 27. Take readings at a second frequency. § 28. Effect of Iron. — Gradually introduce an iron core and watch the ammeter ; or, introduce iron wires, a few at a time, thus gradually increasing the amount of iron. At present, only the general effect of iron is to be noted and explained; a more com- plete study of iron in the form of a closed magnetic circuit is made in the subsequent experiments on the transformer. §29. (c) Resistance and Coil in Series. — Connect the resist- ance Ri and the coil i?2-^2 i" series, and, together with an adjust- ing resistance, connect to the supply, as in Fig. 8. For a certain current, take readings of the voltage drop, the current and the watts consumed as follows : first, for the resistance ; second, for * (§24a). This adjustment should be so made that the readings of the various instruments are taken at open parts of the scales. t(§26a). Series and Parallel Connections. — It is instructive to use a coil with two equal windings. In this case, the regular tests should be made with the two windings either in parallel or in series and additive, — 4. e., setting up magnetic flux in the same direction. If one winding is reversed, it will oppose the other so that the resultant flux (and hence the impedance) is small. A few volts may cause a very large current. Preliminary Test.— With, the resistance Ri in series as a safeguard, to avoid excessive current, measure the current and voltage and determine the impedance, of each winding alone and of the two windings connected in series and in parallel, additively and differentially. The additive wind- ing is inductive; the differential winding is non-inductive, — except so far as there is magnetic leakage. 112 SINGLE-PHASE CURRENTS. [Exp. the coil; and third, for the resistance and coil combined. Vary the current, by means of the adjusting resistance, and take several sets of readings, the current being kept constant for each set; see §24a. § 30. The ammeter and current coil of the wattmeter are in series with the circuit for all readings and their location is un- changed. The voltmeter and the voltage coil of the wattmeter are in parallel with each other and are connected : first, across the terminals of the resistance; second, across the terminals of the coil; and third, across the terminals of the resistance and coil combined. These changes can be most readily made by means of a voltmeter switch, the current being maintained constant during one set of readings by means of the. adjusting resistance. Some error is here introduced on account of the power consumed in the instruments. § 31. Repeat at a second frequency. §32. (d) Resistance and Coil in Parallel. — Connect the resist- ance i?i and the coil R^L^ in parallel, and, together with the adjusting resistance, connect to the supply as in Fig. 10. For a certain constant voltage E, take readings of current, voltage and watts: first, for the resistance alone; second, for the coil alone; and third, for the resistance and coil together in parallel. Vary the voltage by means of the adjusting resistance, and take several sets of readings, the voltage being kept constant for each set; see §24a. § 33. The voltmeter and potential coil of the wattmeter are not changed during the readings. The ammeter and the current coil of the wattmeter are shifted from one circuit to another, being: first, in series with the resistance ; second, in series with the coil ; and third, in the main circuit. Since, during one set of readings, the voltage is maintained constant, the readings thus obtained* ♦This would be true if the instruments themselves took no power; §23a. 4-A] SERIES AND PARALLEL CIRCUITS. 113 will be the same as readings obtained simultaneously with three ammeters and three wattmeters. § 34. Repeat at a second frequency. § 35. (e) Measurement of Resistance. — Measure the resist- ances i?i and i?2 by direct current, § 17, Exp. i-A. PART III. RESULTS. §36. In each test, (a), (b), (c), and (d), select say two sets of readings at each frequency and construct vector diagrams showing the magnitude and relative phase positions of the various currents and voltages. Compute for the various circuits, and parts of circuits, the power factor and the phase difference between current and voltage. The prime object is to obtain a clear understanding of the relations between the various quanti- ties, rather than to obtain exact numerical values. Besistance — vywvj s Fig. 4. Circuit containing resistance i?i. §37. (a) Resistance Alone. — For this case, the current and electromotive force are in phase, and true power is equal to the product, volts X amperes. Power factor=H^-^£/^i; cose=i; ^ = 0. See Fig. 5. §38. (b) Coil Alone. — The current / lags behind the electro- motive force E by an angle 6, as in Fig. 7. The true power W, indicated by the wattmeter, is less than the volt-amperes or apparent power, EI; thus 9 114 Hence SINGLE-PHASE CURRENTS. W = EI X power factor = EI cos 6. cos 5^ power f actor ^ PF -e- £/. [Exp. The angle 6 is thus computed from the readings of the wattmeter, voltmeter and ammeter. In constructing Fig. 7, lay off OA = E ; then lay off OD = I, at an angle 6 determined as above, and construct the right tri- angle of electromotive forces, OCA. Adjusting Resistance — wy^ r Fig. 6. Circuit containing coil R2L2. Compute the components of electromotive force and current, and verify the various relations discussed in the following ■ paragraphs. §39. Components of Electromotive Force. — In the manner just described, the electromotive force is resolved into the power com- ponent, Ef=OC, in phase with 7, and the wattless component, Eq = CA, in quadrature with I. These components are E-p^E cos 9^E X power factor ; £q=£ sinS = £ X reactive factor.* The vector sum of these two components gives the total im- pressed electromotive force. £=V£p^-f £q^ * (§39a). Designating power factor by p and reactive factor by q, it is seen that p^ + q'=i. Compare Standardization Rule 56. 4-A] SERIES AND PARALLEL CIRCUITS. "5 § 40. From these electromotive forces, we have the definitions : Impedance is total electromotive force divided by current; Resistance is power or in-phase component of electromotive force divided by current; R=EcosO-^I. (In general, when motors, transformers, etc., are included in the circuit, this gives apparent resistance.) Reactance* is the wattless or quadrature component of electro- motive force divided by current ; X = Es{nd-^I. §41. Components of Current.'\ — In a similar manner, the cur- rent may be resolved into a power component, If = I cos 6, in phase with E, and a wattless component /q = / sin 6, in quadra- ture with E; the total current is I=^yiT^ -{- Iq^. § 42. From these currents, we have the definitions : Admittance Y is total current divided by electromotive force; Y^I^E. Conductance g is the power or in-phase component of current divided by electromotive force; g^I cos 6^- E. Susceptance b is the wattless or quadrature component of cur- rent divided by electromotive force; b^I sin 6^^ E. We have, then, the following relations; Total current = / = £ X Y- Power current ^/ cos 5^ £ X 5'- Wattless current ^I sin 9^ EX b. g=Y cos 6. b^Y sine. Admittance = y/g^ -}- b^. *This is the general definition, La, l/Cai, etc., being merely particular values; see paper on Reactance, by Steinmetz and Bedell, p. 640, Vol. XL, Transactions A. I. E. E., 1894. t (§4ia). As an illustration of the resolution of current, see Fig. 2 and other figures in Exp. S-C. It is usual to resolve electromotive force into components for series circuits and current into components for parallel circuits. ii6 SINGLE-PHASE CURRENTS. [Exp, Admittance is the reciprocal of impedance ; but conductance is not the reciprocal of resistance (as with direct currents), nor is susceptance the reciprocal of reactance. §43. Power. — It is seen that the expression for true power, EI cos 6, may be written in two ways : PF = £cos6X I (resolving electromotive force); or, W^= IcosO'XE (resolving current). §44. Resolving the electromotive force into components, we have: True power is equal to the product of current (I) and the component of electromotive force (E cos 6) which is in phase with the current. § 45. Resolving the current into components, we have : True power is equal to the product of the impressed electromotive force (E) and the component of current {I cos 6) which is in phase with the electromotive force. §46. Calculation of L and X by Wattmeter Method. — React- ance is by definition (§40) equal to the quadrature electromotive force, £x, divided by current. Referring to Fig. 7, the reactance and inductance of the coil R^L^ are computed as follows: LjO) = X2 = CA -^ I, ohms ; L^^X^-^a^X^-i- 2Trn, henries. By this method, X^ and L^ are determined by measurements of E, I and W, and are independent of the measured value of R^. (See §§47 and 49.) Note also that R^^OC-^I^W -^I^, and that tani9 = Z2-^i?2. § 47. Calculation of L and X by Impedance Method. — By the impedance method, L^ depends upon E, I and the measured value of i?2, and is independent of the wattmeter reading. The cal- culations are made as follows: Impedance (ohms) ==^2 ^£-f-7. Reactance (ohms) = Xj = ^/Z^ — R^ 4-A] SERIES AND PARALLEL CIRCUITS. 117 Here R^ is the resistance of the coil, as measured by direct current. The inductance, in henries, is L2 = X2^h2irn. For the accurate determination of L by either of these methods, the wave form of electromotive force should be sinusoidal and the losses in instruments should be taken into consideration, § 23a. §48. (c) Resistance and Coil in Series. — In a series circuit there is one current which is the same in all parts of the circuit; electromotive forces are added vectorially, i. e., the voltage drops around the separate parts of the circuit, when added as vectors, give the total impressed electromotive force of the circuit. Fig. 8. O / D Resistance and coil in series, B.,1 B Rfl Fig. 9. The three readings of the voltmeter, E, E^ and E^, are, accord- ingly, drawn to scale so as to form the triangle OAB, in Fig. 9. The current / is laid off in phase with E^, since the current and electromotive force in the non-inductive resistance are in the same phase. OCA is then drawn as a right triangle. We have then the in-phase electromotive forces, OB = RJ to overcome the resistance R-^, and BC = RJ to overcome the resist- ance i?2 ; and the quadrature electromotive force, CA = L^ml = XJ, to overcome the reactance X^. It will be seen that Fig. 9 is the same as Figs. 5 and 7 combined in one diagram so drawn that the current in both parts of the circuit is the same in magni- tude and in phase. §49. Three-voltmeter Method. — The foregoing construction, known as the three-voltmeter method, enables us to calculate L^ ii8 SINGLE-PHASE CURRENTS. [Exp. and X2, the results being dependent upon three voltmeter read- ings and current, and not dependent upon the wattmeter (as in the wattmeter method, § 46), nor upon the measurement of resist- ance (as in the impedance method, §47). Referring to Fig. 9, we have hence and L2 = X2 -T- =C'A-~I. § 53. For any number of parallel circuits, the total current in phase with E is 5/ cos 6 ; the total quadrature current is S/ sin 0. Hence /=V (5/ cos (9) ^H- (5/ sin e)^ Dividing by E, we have The total conductance of a number of parallel circuits is the arithmetical sum of the separate conductances; the total suscept- ance is the arithmetical sum of the separate susceptances. (Com- pare with § 20 for series circuits.) APPENDIX I. CIRCUITS WITH CAPACITY. § 54. It is not intended in this experiment that tests with capacity be included, the following summarized statements concerning capacity being made for reference and for comparison with the relations already discussed concerning inductance. § 55. Circuits with Resistance and Capacity. — In theory, circuits containing capacity (C) can be treated exactly the same as circuits containing inductance, if the following differences are noted : Inductive reactance = L — i/C is flux. The instantaneous value of the voltage, and hence the effective or virtual value, is accordingly proportional to the number of turns, and the ratio of voltages in any two coils is the ratio of the number of turns in the coils. (See Appendix I.) * In connecting together transformers of different makes, care must be taken, for their polarities may be indicated by different systems. t For current ratio and tests on commercial transformers, see Appendix I. 134 TRANSFORMERS. [Exp. If any combination of coils gives a voltage which is beyond the range of the voltmeter, these tests can be made by using a lower supply voltage; it may be found convenient to connect the high potential side of the transformer to the line, thos step- ping the voltage down to a lower voltage in the secondary. § lo. Prove that the voltage of the secondary is either in phase, or i8o° out of phase,f with the primary voltage. To do this, join together one terminal of the primary and one terminal of the secondary, so that the two windings are in series ; the supply voltage is connected to the terminals of the primary. Measure the voltage across the primary, the voltage across the secondary, and the voltage across the two, measured between the terminal of the primary and the terminal of the secondary which are not joined together. Either the sum or the difference of the first two readings will equal the third reading; whether it is the sum or the difference will depend on which terminals of the two windings are connected together. If the two voltages were of different phase, the total would be found to be not the algebraic sum but the vector sum. §11. Use as an Auto-transformer. — As ordinarily used, a transformer has two independent circuits, a primary and a sec- ondary, and any particular winding is used as part of one of t (§ loa). The secondary eletromotive force is in the same phase as the primary counter electromotive force, being induced by (substantially) the same flux; hence it is opposite in phase. to the primary impressed or line electromotive force. It follows that the secondary current, when the transformer is loaded, is nearly opposite in phase to the primary current, this being discussed more fully in Exp. S-C. This opposition of currents is verified by the auto-transformer test, § ii. That primary and secondary currents are opposite to each other in phase may be further illustrated by the following experiment. Take a straight upright core surrounded by a primary circuit. Place around it (loosely) a closed ring forming a secondary circuit. Connect the primary to an alternating current supply. When the primary circuit is closed, the secondary will be thrown off violently, showing that the currents in the two circuits are in opposite directions. The secondary ring may be held down by threads, so as to float as a halo. 110 Volt Supply -R, S-A] STUDY AND OPERATION. I35 these only. In the auto-transformer,* or single-coil transformer, part of the windings is common to both primary and secondary. Connect the transformer coils as an auto-transformer, and verify the different values of voltage transformation. To do this connect all coils in series and consider any one or more of the coils, as may be desired, to be primary or secondary. Some of the coils will at the same time form part of both primary and a a' B b'c c d d' secondary; these coils will carry, ^Jl!lfiiI/^JiafliLkQfilHLT^^ therefore, both the primary and ^ the secondary currents, which are Load opposite in phase (§ loa) and so p^^_^_ g^^^.^^^^ J^^l give a resultant current approxi- former, using coils A B C D as mately equal Jto the arithmetical P""ary; coil D is also used as secondary. difference of the two. §12. Connect the coils as a step-down auto-transformer (Fig. 2) and as a step-up auto-transformer or "booster" (Fig. 3). Using suitable resistances as a load, determine the currentf in each coil, in the resistance and in the supply line and explain their relative values. The currents and voltages for other com- binations of coils can be computed and compared, or determined experimentally. Suppose a 3 : 2 ratio is desired ; with A, B, C as primary, how would the use of C, D as secondary compare with the use oi B, CI § 13. Advantages of the Auto-transformer. — It will be found that the auto-transformer requires less copper than a transformer with separate primary and secondary coils; it has, therefore, not only lower first cost but less copper loss and copper drop, giving better efficiency and regulation. The saving in space on account * Also called " balance coil " or " compensator " ; the term auto-converter should be discarded. t In making measurement of current, it will be found convenient to use one ammeter and a 3-way ammeter switch. 136 TRANSFORMERS. [Exp. of less copper makes it possible to reduce also the iron and iron loss. This advantage of an auto-transformer will be seen to be greater the nearer the ratio of transformation is 1:1. For a comparison of output of transformers and auto-transformers, see §§8, 9, Exp. y-B. An auto-transformer cannot be used when complete insulation of the primary from the secondary is necessary, as in house lighting from high potential lines. no Volt Supply % — __pl ^MMAAMMMM/VWV-' Load Resistance The step-down auto-transformer of Fig. 2 is in common use as a starting device for induction mo- FiG. 3. Step-up auto-trans- ^o^s, giving a lower voltage than former or booster, using coils full line voltage while the motor ^BC as primary; coils ABCD j^ ^^^j ^^ ^ ^^ ^ . ^^^ p; g^ are used as secondary. Exp. 7-A. A common use of the step-up arrangement of Fig. 3 is as a booster to raise the voltage on remote parts of a distribution system, say from 2,000 to 2,200 volts. For this a standard 2,000/200 volt transformer can be used, with the low-potential coil in series with the primary to boost the voltage, as in Fig.2, Exp. 7-B. This becomes a "negative booster" if the connec- tions of the low-potential coil (coil D in Fig. 3) are reversed. (If a standard transformer is to be tried in the laboratory, a lOO-volt circuit may be boosted to no volts, or reduced to 90 volts.) § 14. Constant Potential Operation. — Transformers are usually operated from a constant potential circuit, so as to transform — either step-up or step-down — from a constant primary potential to a constant secondary potential. §15. Open Circuit. — Connect a no-volt alternating current supply circuit across two of the transformer coils in series, as S-A] STUDY AND OPERATION. I37 a primary. Measure the no-load primary current, I,,, called the exciting current. Predict, and then measure, the value of !„ when the two primary coils are in parallel and connected to a SS-volt supply — i. e., half the preceding voltage. Compare the relative values, for the two cases, of primary turns, ampere turns, volts, volts per turn and flux density. Measure I^ when the two primary coils are in series, and con- nected to a 55-volt supply; and interpret the results (see Fig, 2, Exp. s-B). Commercial transformers are commonly built with two pri- maries for connection in series (for, say, 2,200 volts) or parallel (for 1,100 volts) ; and two secondaries for connection in series (for, say, 220 volts) or parallel (for no volts). § 16. Operation Under Load* — Join twof of the coils in series to form a primary and join the other two coils in series to form a secondary — or make such other arrangement of coils as may, be desired. Connect the primary with an alternating current supply — say no volts, 60 cycles — appropriate to the arrange- ment of coils adopted. A voltmeter, ammeter and wattmeter are connected J in the primary, as in Fig. i. § 17. With the secondary on open circuit, measure the primary voltage, the primary current (in this case, the no-load current, /(,) and the primary power (in this case, the no-load or core losses, Wf,). * Time should not be spent in an attempt to get very accurate results in this test, particularly if it is to be followed by the more accurate test by the method of losses, Exp. 5-B. t (§ i6a). Where there is a choice of coils, select an arrangement which avoids great magnetic leakage. If each coil forms one layer or section, to take the first two for primary and the other two for secondary would not be a good arrangement. In a commercial transformer, the primary and secondary windings are so placed as to reduce magnetic leakage ; to secure this end, however, all the windings should be used, that is, no coil should be left idle. An arrangement of coils commonly used is as follows : low, high, high, low, potential. % With instruments arranged as in Fig. i, no corrections need be made. (See Appendix III.) 13S TRANSFORMERS. [Exp. § 18. Load the secondary by means of suitable non-inductive resistance. Change this resistance by steps so as to vary the secondary current between no load and 25 per cent, overload. At each step measure the primary voltage £1, current I^, and power, Wii also the secondary* voltage E^, and secondary cur- rent I^. The product of the secondary voltage and current will give the secondary power W,, the secondary load being non- inductive. In practice, a load of incandescent lamps is non- inductive, but not so a motor load. § 19. Measure the resistance of primary and secondary. (See § IS, Exp. 5-B.) §20. For each load, compute the power factor (W^-i-EJ,^) ; also the angle 6 by which the primary current lags behind the electromotive force. (Power factor = cos 6.) Plot /i, Wi, power factor, 6, E^ andW^ for different values of I2, as in Fig. 4. Plot, also, the copper loss for primary (i?i/i^) and for secondary {RJ^) and the core loss Wf, |*the value of W^ on open circuit) which is constant at all loads, as in Fig. 8, Exp. s-B. Note 'that E^ decreases with load. Determine the per cent, regulation — the per cent, increase in E^ in going from full load to no load. Note the current ratio for different loads. It will be seen that as the transformer becomes loaded (by decreasing resistance in the secondary) the secondary current becomes more nearly equal to the primary current (multiplied hy S^-^S^). In a loaded transformer, primary and secondary ampere-turns are practically (but not exactly) equal. It is seen that in a transformer there is a loss in volts, a loss . in amperes and a loss in watts, this last determining the efficiency. While best for illustrating the operation of a transformer, the * By means of suitable transfer switches one voltmeter and one ammeter may be used for both primary and secondary. S-A] STUDY AND OPERATION. 139 loading method is not so good for the accurate determination of efficiency and regulation. These can be computed much more _0£e7i^ircuit_yoltage 10 15 SECONDARY CURRENT; AMPERES 20 25 Fig. 4, Curves for 2,000/100 volt, 2 K.W. transformer ; see also Fig. 8, Exp. S— B. accurately from the losses, determined without loading, as in Exp. S-B. §21. Load the transformer with an inductive load and take one reading of the instruments. It will be seen that the sec- ondary voltage is somewhat less than it was with non-inductive load — that is, the regulation is poorer.* This happens when induction motors are operated from transformers. In this case the secondary current is lagging. If the secondary current were leading, the secondary voltage in some cases would increase, instead of decrease, with the load. The results are similar to those obtained for an alternator ; see Exp. 3-B, particularly Fig. 7. §22. Design Data and Computation of Flux Density. — Note the construction and essential dimensions of the transformer, * (§2ia). If the leakage reactance of a transformer is small, compared with its resistance, the regulation may be better at low than at high power factor ; compare § 28, Exp. 3-B. 14° TRANSFORMERS. [Exp. including the cross section of the magnetic circuit and size of wire, but do not remove parts, destroy insulation or damage the transformer in any way in seeking this information. Data fur- nished by the maker can be used for this purpose. § 23. Compute the current density in amperes per square inch and in circular mils per ampere, for the primary and the sec- ondary windings. Current densities from 1,000 to 2,000 circular mils per ampere are common, but less copper was often allowed in early transformers. §24. Compute the maximum value of the total flux in C.G.S. lines or maxwells (see §9a, Exp. i-B) ; thus Flux = ^X5„...= AXi2L where E is the voltage and 5" the number of turns for any coil, and n is frequency. The quantity E-^S is the volts per turn. For proof of formulae, see Appendix II. Compute the maximum value of the flux density in gausses (flux per sq. cm.) ; thus r-1 ^ ■. R £Xio^ Flux density = S_. = ^^^— -^, where A is the cross section* of the core in sq. cms. If A is in square inches, 5max. is the flux density in lines per square inch. If A, in square inches, is multipHed by 6.45, the formula gives Bma.%. in gausses — for, unfortunately, this mixture of C.G.S. and English units is in common use. §25. The computations for B should be made for standard frequency (60 cycles), and two other frequencies (30 and 120) with the same value of E. If values of A and 5" are not obtain- able, assumed values may be assigned for practice computations. If the cross section of the core is not uniform, B will have dif- * (§24a). The net cross section is, say, 15 per cent, less than the gross cross section on account of lamination. S-A] STUDY AND OPERATION. H' ferent values for different parts of the magnetic circuit. From these computations it can be seen whether B will be more or less, if a transformer is operated at a higher or lower frequency than rated and at the same voltage. (Note in what manner E should be changed to maintain B the same at different frequencies.) Practically, transformers are run at different frequencies without changing E, if the frequency is not too far below the frequency for which the transformer is designed. For a discussion of the effect of frequency upon core loss, see §§ 8-14, Exp. 5-B. In transformer design,* B is given a wide range (4,000-14,000 gausses at 60 cycles), being sometimes greater in small than in large transformers and greater in transformers designed for low than in those for high frequency. In design, E and n being given, B may be assumed and the product A XS determined. This product being fixed, the designer may adjust the values of A and 5" to suit his purpose, increasing A and decreasing 5" to use more iron and less copper, or zdce versa. § 26. From the formula for flux density, it will be seen that the electromotive force of any coil of a transformer is propor- tional to the number of turns in the coil, a fact already noted. The volts-per-turn should be computed as a constant for the transformer. For small transformers this may be one third or one half, being greater for large transformers, perhaps 2 to 4 for transformers above 30 K.W. The reciprocal gives the turns- per-volt. The volts-per-turn, when known for a certain type and size of transformer, may be used as a design constant. §27. Other data of interest to the designer (which may be de- termined when worth while) are the weight of copper and of iron, total and per K.W. This may range from 5 to 25 lbs. per K.W. for either copper or iron. The space factor for copper is the * (§2Sa). For more complete design data, see handbooks, etc. As mag- netic material is improved, higher magnetic densities are possible for the same loss. While densities of 4,000-8,000 were used with ordinary grades of iron, densities of 6,000-12,000 are now common with alloy steel. 142 TRANSFORMERS. [Exp. ratio of the cross section of copper to the total cross section of the windings, i. e., to the cross section of copper plus insulation and air space. Similarly the space factor for the iron is the ratio of its net to gross section. APPENDIX I. POLARITY AND RATIO OF COMMERCIAL TRANSFORMER. §28. Polarity; Alternating Current Method. — The coils are con- nected in series, two at a time, and notice is taken whether the voltage around the two is the sum or the difference of the separate voltages. There are several ways in which this can be carried out. As an example, let us take a transformer with two primaries for 1,000 volts each and two secondaries for 50 volts each. Connect the two 1,000-volt primaries in series and con- 50 Volt -^1 T nect the terminals of one* of the primaries -i-i|)0 to a low potential supply circuit, say 50 Supply x^ « 1 i 3 5b I volts, as in Fig. 5. If a voltmeter across " * the two coils together reads zero, reverse Fig. s. Polarity test the connections of one of the coils. The y a erna ing cur voltmeter should then read 100 volts across method. the two coils together, and 50 volts across each one separately. Terminals A and B are now of one polarity; terminals A' and B' are of the opposite polarity, to be marked with a prime (') or X- Each secondary is then connected in series with one primary, the primary being connected to the 50-volt supply circuit; the secondary in series with it is so connected that the voltmeter reading around the two coils in series is greater (52.5 volts) than the potential from the mains (50 volts). If the reading is less (47.5 volts), reverse the secondary. Secondary terminals are marked with a prime (') or X to correspond with the primary. Small transformers are commonly so wound that, when the primary and secondary leads on one side of the transformer are connected * If the two coils in series were connected to the supply circuit, a burn- out might result if the coils were opposed to each other. 5-A] STUDY AND OPERATION. H3 together, the voltage measured across the two primary and secondary leads on the opposite'side will be the sum of the voltages of the two windings. §29. Polarity; Direct Current Method. — The alternating current method is usually preferred, but sometimes the following method will be. found convenient. The primary is supplied with a direct current sufficient to give a reading on a direct current voltmeter connected to the primary terminals. The voltmeter terminals are then con- nected to what are supposed to be corresponding terminals of the secondary. If, when the primary circuit is closed,* the voltmeter needle is thrown in the same direction as the preceding reading, the voltmeter has been connected to the secondary terminals correspond-, ing to primary terminals; i. e., the voltmeter lead from the primary terminal (') or X is connected to the secondary terminal to be marked (') or X- If the voltmeter needle is thrown in the opposite direction, the reverse is true. § 30. Potential Ratio. — Where one transformer alone is to be tested, the transformer should be supplied with any convenient voltage and the voltage of each circuit measured either by two voltmeters, one of which has been calibrated in terms of the other, or by one voltmeter reading direct on the low potential side and with a multiplier on the high potential side.f When one transformer has been tested in this manner, or a small potential transformer of accurate ratio is available, two transformers can be run in parallel from the same circuit and their secondary voltages on open circuit compared by readings taken with one volt- meter or with two voltmeters whose relative calibration is known. If the secondaries of two similar transformers are connected in series and in opposition, any difference will be shown by a voltmeter connected across the two. § 31. Current Batio. — For commercial testing of the ratio of a transformer, test may be made by comparison of primary and secondary currents instead of voltages. The secondary circuit is short-circuited through an ammeter of low resistance and the *The current should be small so as not to injure the voltmeter by- slamming the needle when the circuit is made and broken. t It is not necessary to run the transformer at full rated potential. When high potentials are used, due caution should be observed. 144 TRANSFORMERS. [Exp. primary and secondary currents measured when a proper voltage (a few per cent, of normal primary voltage) is applied to the primary, so that about the normal current flows. § 32. Circulating Current Test. — As a shop test, after one standard transformer has been tested, other transformers designed for the same ratio may be operated from the same primary mains and tested one at a time by connecting each secondary to be tested in parallel with the secondary of the standard, terminals of the same polarity being connected together. If an ammeter shows a circulation of current through the secondaries, the two transformers are not of the same ratio. Commercially a small difference in ratio is allowable as shown by the circulating current, which, however, should never exceed one per cent, of the rated full-load current. Instead of an ammeter a suit- able fuse may be conveniently used, and more safely where much difference in ratio may exist. APPENDIX II. RELATION BETWEEN FLUX AND ELECTROMOTIVE FORCE. § 33. The fundamental relation between flux and electromotive force is expressed by Faraday's law; that is, in a closed circuit* of S turns embracing a varying flux <^, the induced electromotive force is — S-d(j)/dt. In a transformer, this applies alike to primary or secondary. In the case of a primary coil this induced electromotive force is a counter electromotive force and requires to overcome it an equal and opposite impressedt electromotive force e = Sd<^/dt. § 34. Sine Assumption. — Assuming the wave of electromotive force to be a sine wave, we have * Not limited to a transformer. t (§333)- The actual terminal voltage includes also resistance drop, thus e = Ri-\-S-d'i>/dt. The resistance drop, however, is practically negligible in the primary of a transformer on open circuit. 5-A] STUDY AND OPERATION. HS' Sd^ = £niax. sin tat dt ; The maximum value of the flux is max. *Pinax. — c * and the flux per unit area is E R ■ ^. -^ -^max. ■^max. V>max. • -^ o^ ' To express in terms of effective voltage, substitute V^E for Emax.- Multiplying by lo' to change from C.G.S. units to volts, and remem- bering that 0) is 2-n- times the frequency (n), we have as a working formula for flux density _ V 2£ X lo' _ £Xio° "^''- 2,rnS^ ~ 4.45m5-^ " It follows from this formula that a constant potential transformer is a constant flux transformer. It also follows that, if a certain flux is maintained in the transformer, the voltage in any coil is propor- tional to the number of turns in that coil. For further interpreta- tion, see §§ 24-26. § 35. Without Sine Assumption. — We have the fundamental rela- tion d(j) = edt/S. Integrating for half of a period T, during which time the flux changes from a minus to a plus maximum. 2<^max.=£av.AX7'/2. Writing i/n for T, and multiplying by 10' to reduce to volts, we ^^^^ E Vio' ■Dmax. — and E^^^ = -L^max.- See 1/2 page 37, Bedell and Crehore's Alternating Currents. Form factor is f =:E^j^ -T-E^^ =:!.! for a sine wave. (Form factor was first used by Roessler as E^^ -^^eti.> which for a sine wave is .9.) S-A] STUDY AND OPERATION. 147 A wattm'eter* may be connected in two ways, as follows. § 38. In the first and usual method, Fig. 6, the potential coil is connected between the line wires on the supply side. In this method, \ r^ ^ « tn « (A +; ^ if :S iz 1 ^ w g.-g 5-i a. w ^ S.-S in JL J & JL J Fig. 6. First method. Fig. 7. Second method. Methods for connecting a wattmeter. the wattmeter reading is too large, including not only, the true watts of the load but also the RP loss in the current coil of the wattmeter. This error, which may amount to several watts, can frequently be neglected; correction for it is not easily made. In measuring small amounts of power, in order that the error may be neglected, the current should not exceed one half the rating of the current coil of the wattmeter. With current greater than this, the loss in the series coil, which increases with the square of the current, may be too large to neglect. § 39. In the second method, shown in Fig. 7, the potential coil is connected across the terminals of the load. In this method the watt- meter reading is also too large, since it includes the RP loss in the potential coil of the wattmeter. This error is larger than the error in the first method and should be corrected for by subtracting E' -— R^ from the wattmeter reading, E being the line voltage and 2?^ the * (§373)- Lag Error. — A wattmeter measures £7 X power factor. For accuracy, a wattmeter must have the resistance of the potential circuit so large, compared with its reactance, that the circuit is practically non- inductive. The current in the potential circuit is then practically in phase with the electromotive force; in reality it lags by a small angle, 5^ tan-'CLw-r-i?). The error due to this lag angle is different for dif- ferent power factors, cos , of the load. The true watts are equal to the wattmeter reading multiplied by cos cos ^ cos (0 — 6) In commercial wattmeters, at commercial frequencies, this correction can be neglected. It becomes appreciable on higher frequencies, particularly on loads of low power factor and at low voltages — i. e., when the resistance of the potential circuit is small. 148 TRANSFORMERS. fExp. resistance of the potential coil of the wattmeter. This correction might be, for example, 5 watts in a loo-volt wattmeter, 10 watts in a 200-volt wattmeter, etc. The correction, however, is exact and is readily made, the value oi R^ usually being given with the instrument. In precise work, this method should be used and the correction made. If, however, no correction is to be made, it is better to use the first method, in which the error is smaller. § 40. Use of a Voltmeter with a Wattmeter. — When a voltmeter and wattmeter are used together, the voltmeter should (usually) be connected between the same points as the potential coil of the watt- meter. There are, therefore, the same two methods of connection as with a wattmeter alone. § 41. First Method. — The voltmeter is connected between the lines on the supply side of the wattmeter. The reading of the voltmeter includes the RI drop in the current coil of the wattmeter ; the error is small and may often be neglected. § 42. Second Method. — The voltmeter is connected on the load side of the wattmeter, directly to the terminals of the load. The volt- meter reading is now correct. The wattmeter, however, includes the watts consumed in the voltmeter. The reading of the wattmeter should, accordingly, be corrected by subtracting E' -4- R^, where Ry is the resistance of the voltmeter. The whole correction for the watt- meter is now E'(i/Ry,-{- i/Ry), which is to be subtracted from the wattmeter reading. § 43. Use of an Ammeter with a Wattmeter. — If an ammeter is connected in circuit on the load side of a wattmeter, as the current coil in Fig. 6, the ammeter reads the true load current. The watt- meter reading, however, includes the watts loss in the ammeter — a small error which is neglected. If the ammeter is connected on the supply side of the wattmeter, no error is introduced in the wattmeter reading ; the ammeter reading, however, is too large,* since it includes the current in the potential * (§43^). Ammeter Correction. — The ammeter reading can be corrected by subtracting (i/i?^+ i/R^^W/I. The current which flows in the poten- tial circuits of the voltmeter and wattmeter is E/R^-\-E/R^. This is in phase with the electromotive force and not with the current, and must be multiplied by the power factor of the load W/EI to get its component in phase with the current. 5-Al STUDY AND OPERATION. 149 coil of the wattmeter. This error can be neglected, when the load current is large. The ammeter reading would be correct if the potential coil of the wattmeter (and voltmeter, if one is used) were opened when the ammeter is read; sometimes this is allowable under steady conditions, but simultaneous readings of all instruments are usually more accurate. § 44. Combinations of Instruments. — In the combined use of ammeter, wattmeter and voltmeter, the best method to use depends somewhat upon the conditions of the test. The arrangement of Fig. i is, for most purposes, as good as any; no corrections are made. In the short-circuit test of a transformer, the reading, of the current is most important; hence. Fig. 6, Exp. 5-B, the ammeter for this test can best be connected on the load side of the other instruments. For the open-circuit test, voltage is important and not current; the ammeter is, therefore, in the supply line and the instruments arranged as in Fig. i, Exp. 5-B (requiring a correction) or as Fig. i of this experiment (requiring no correction and hence simpler to use). See § 3a, Exp. 5-B. § 45. Multipliers. — To extend the range of a voltmeter, either a series resistance (called a multiplier) or a potential transformer can be used. The potential range of a wattmeter is extended in the same way. To extend the range of an ammeter, a current transformer is used; the primary of the transformer is connected in series with the line, the secondary being short-circuited through the ammeter. The cur- rent range of a wattmeter is extended in the same way. The ratio of transformation of any potential or current trans- former must be accurately known, and, for a current transformer, this ratio must be known in connection with the particular instru- ment and secondary leads with which it is to be used. Any small phase shifting, due to the fact that primary and secondary quantities are not exactly in phase opposition, introduces no error in the use of instrument transformers with ammeters or voltmeters, but with wattmeters such phase shifting may introduce considerable error and needs to be taken into consideration for accurate work. For a com- plete discussion, see Electric Measurements on Circuits Requiring Current and Potential Transformers, a paper by L. T. Robinson, read at the June, 1909, meeting of the A. I. E. E. 15° TRANSFORMERS. . [Exp. Experiment 5-B. Transformer Test by the Method of Losses. § I. Introductory. — The losses in a transformer are the core loss, which is dependent upon and varies with voUage, and the copper (and load) losses, which are dependent upon and vary with current. The most accurate* and the most convenient method for testing a transformer is to measure these losses sepa- rately, without loading the transformer, and compute the effi- ciency and regulation. This requires two simple tests, each employingf a voltmeter, ammeter and wattmeter: an open-circuit or no-load test for determining the no-load or core loss and the exciting current at various voltages, particularly at. normal voltage; and a short- circuit test at a low voltage (a few per cent, of normal) for determining the copper and load losses and impedance drop for various currents, particularly for normal full-load current. The latter test gives, also, the equivalent resistance and leakage react- ance of the transformer. Measurements are also made of primary and secondary re- 'sistance. §2. This method may be employed in testing any transformer, whether it is intended for constant potential, constant current or other service; the method will be described in detail with refer- ence to its application to a constant potential transformer. * (§ la). This is most accurate for the reasons explained in § ib, Exp. 2-B. It is not practicable to determine' efficiencies accurately by loading a transfofmer (§ i6, Exp. 5-A) and measuring the input and output directly — unless exceeding care be taken — the two quantities measured being so nearly equal. The indirect method of losses is, furthermore, most convenient because no load is required and no high-potential meas- uremerits are necessary. t In many cases the same instruments can be used in the two tests ; com- pare §44. Two similar tests are made in testing alternators; see §9, Exp. 3-B. S-B] TEST BY LOSSES. 151 In a constant potential transformer the magnetization and hence the core loss and exciting current are (siibstantially) the same at all loads, being dependent upon voltage and not upon current. The copper and load losses, on the other hand, depend upon current and vary with the load. (In a constant current transformer, the conditions are reversed; copper losses are con- stant and core loss varies with the load.) PART I. OPEN-CIRCUIT TEST. § 3. With the secondary open, measurements are made on the primary with ammeter, voltmeter and wattmeter, one method for making the connec- => .2 s S o i O I O O -C ^ » Mg Fig. 1. One method of connection for open- circuit test for core loss and exciting current. See § 3a for method of connecting instru- ments requiring no corrections. tions* being shown in Fig. I. Although any coil or combination of coils could be used as a primary in this test, it is most convenient to use a low potential coil (50, 100 or 200 volts) as primary to suit the instruments and supply voltage available; furthermore, there is less danger in working on the low potential side. . For the same degree of magnetization, the exciting current in a 1 00- volt coil is ten times as large as in a 1,000- volt coil, the ampere turns being the same. It is, accordingly, a simple matter *(§3a)- For selection of instruments, see §44. The arrangement of instruments shown in Fig. i should be followed when the highest accuracy- is desired; the wattmeter reading is to be corrected by subtracting £'(i/i?^-|- i/i?y), which is the power consumed in the potential coils of the wattmeter and voltmeter. It is, however, much simpler — and in many cases suificiently accurate — ^to arrange the instruments as in Fig. i of Exp. s-A, and to make no correc- tion. See Appendix III., Exp. S-A. 152 TRANSFORMERS. fExp. to reduce the exciting current measured on a coil of one voltage to its value for a coil of another voltage. The watts core loss is the same measured on one coil as on another, for the same magnetization. CAUTION. I'f two coils are to be connected in parallel or series, to avoid a burn-out it is necessary to first make sure of their polarity, as described in §§ 6 and 28, Exp. 5-A. Be careful of the high-potential terminals in this test. It should be made impossible for loose wires, or for persons making measure- ments, to come in contact with these terminals. Although the testing current and instrument are all of low voltage and although the high- potential coil is open and has no current in it, the potential is there and must be respected. § 4. Data. — At normal frequency, say 60 cycles, vary the volt- age (§4S) from say ^ to i^ normal and determine the core loss Wf„ and exciting current /„ for various voltages. Note the frequency at which the test is made. It is desirable that the frequency be maintained constant, and that the voltage be of sine wave-form. Take very accurate readings at two points (within, say, 5 or 10 per cent, of half and full voltage) by taking at each of these points a series of five readings and averaging. This two-voltage method is very convenient, since normal and half voltage (as 5S/iio or 110/220) are often available — or their equivalent can be obtained by series and parallel connections, as described in § 46. As will be seen later, Figs. 2 and 5, it is very accurate for transformers built of ordinary iron, at normal and higher fre- quencies, but not at frequencies far below normal. For trans- formers with improved iron, the two-voltage method is not cor- rect unless one observed point is taken very near full voltage, little error being then introduced by obtaining values for full voltage from the curves. §5. If possible, repeat the data at a second frequency. If S-B] TEST BY LOSSES. '53 the frequency is higher than normal, complete data can be taken as before. If the frequency is lower than that for which the transformer is intended, the core loss and exciting current will be greater and the voltage should not be raised so that they become excessive for the transformer or the instruments;* W^ should not exceed say twice and I^ four or five times their respective values at normal frequency. It will be understood, however, that these limits are only arbitrary. §6. Curves for Exciting Current. — For each frequency, plot a curve showing the exciting current for different voltages, as in Fig. 2. Locate by heavy black dots the two points accurately Fig. i!. Observed exciting current for varying voltage at different frequen- cies; 2 K.W. transformer, loo-volt coiL To obtain the exciting current for the 2,000-volt coil, divide these values by 20. * (§5a). The current coil of the wattmeter has a certain rated current carrying capacity which should not be exceeded for any length of time. It may, however, be exceeded for a few moments only by 30 or even 40 per cent.; readings are taken quickly and the wattmeter is then cut out. 154 TRANSFORMERS. [Exp. determined at about half and full voltage and note how closely a straight line drawn through them coincides with the curve through the working range. (It will be found that this straight line construction, based on two readings, can not be used when the transformer is worked at a very high flux density, as in the 29-cycle curve of Fig. 2.) § 7. Take from each curve the value of I^ for normal voltage (§48). Resolve the exciting current, /„, into two components: the in-phase power component In (which supplies the core losses due to hysteresis and eddy currents) ; and the quadrature mag- netizing* component, Im- These are determined by the following relations : In = W,-^E. h^-s/In^ + hi\ The value of W^, is taken from curves. Fig. 3 and Fig. 5, described in the next paragraph. At no load, power factor If the transformer under test had a core made of improved steel, with less core loss, the component 7h would be somewhat less than indicated in Fig. 2. On account, however, of the very much greater value of the component Jm (due to the higher flUx density common with such iron), the total exciting current /„ would be greater than shown in Fig. 2. Furthermore, tl;ie point for normal voltage would be near the knee of the curve, so that the straight line construction would not be accurate, as has already been pointed out. Compute, as in Fig. 2, the values of /(,, Is. and /m as per cent, of the normal full-load current of the coil on which the test is made; thus, the full-load current for a loo-volt coil of a 2 K.W. transformer. Fig. 2, is 20 amperes. Expressed as per cent., the results will apply to any coil. * Usage is not fixed in regard to the terms " exciting " and " magnet- izing" currents, these terms being not infrequently interchanged. S-B] TEST BY LOSSES. '55 Fig. 3. Watts core loss for varying voltage at different frequencies ; 2 K.W. transformer, loo-voIt coil. §8. Curves for Core Loss. — The readings of the wattmeter in the open-circuit test (after corrections are made, if there are any, § 3a) gives the core loss plus a small RI^ loss due to the heating effect of the ex- citing current. The latter loss can be computed and deducted from the wattmeter reading; it will generally be found negligible.* The curves showing the change of core loss with volt- age can be plotted on ordinary cross-section paper as in Fig. 3; a derived curve, showing the variation of core loss with frequency, being plotted as in Fig. 4. It is much better, however, to use a logarithmic scale for ordinates and abscissae, in which case the curves become (within limits) practically straight lines. For this purpose, it is convenient to use logarithmic cross-section paper. f Above normal voltage, as higher densities are reached, the curve tends to bend upwards, due to the fact that the hysteretic exponent (which has a value of about 1.6 up to 10,000 gausses) becomes greater. Transformers with improved iron are run at higher densities, so that at normal frequency this bend may be reached at normal, or even below normal, voltage. §9. For each frequency plot a curve on logarithmic paper * (§Sa.) Although any Rio loss should be deducted for obtaining true iron loss, for the calculation of efficiency it is better not to make such a deduction but to include the Rlt^ loss with the iron loss Wa. t This paper can be obtained from the Cornell Cooperative Society, or Andrus and Church, Ithaca, N. Y. The same results can be obtained on plain. paper by plotting the loga- rithms of the observed quantities — a laborious process — or by using a slide rule as a scale. 156 TRANSFORMERS. [Exp. showing the core loss, W,,, for different voltages, as in Fig. 5. Locate by heavy black dots the two points accurately deter- mined at about half and full voltage. Draw a straight line through these points and note that at normal and higher fre- quencies this straight line gives the curve accurately through the working range, so that JVo for normal voltage can be readily obtained from it. At frequencies much below normal, and at high flux- densities, this straight line relation may not hold. <^o- li it was impossible to get data for any curve up to so or*" CO O ^^ a o "20 CO I- I- 20 40 60 80 100 120 140 , , , , FREGUENCT: CYCLES PER SECOND normal valtage, extend the Fig. 4. Watts core loss for diflEerent curve that far as a dotted frequencies at normal voltage; 2 K. W. ,. t-,, . ^ , line. 1 his extension is transformer. quite accurate at frequencies niear normal or higher, but can not be depended upon at fre- quencies way below normal. The curves, however, can be more readily and more accurately extended on logarithmic paper than on ordinary coordinate paper. § lo. The slope of these curves (the actual tangent with the horizontal) is the exponent (a) of £ or 5 in the formula show- ing the law of core-loss variation for different voltages and flux densities at a constant frequency; J^ oc E" oc B". This exponent (o) should be determined and interpreted; see §§49-51- § II. Variation of Core Loss with Frequency. — On the same logarithmic sheet, see Fig. 5, plot a derived curve showing the core loss, W^, at normal voltage for different frequencies. If S-B] TEST BY LOSSES. 157 100' ,20 Frequency; Cycles per Second 40 60 80 100 120 140 160 180 200 90 80 70 60 50 w40r u E : I fe° : iiiZO IQ, 60 80 100 120 140 160 180 200 VOLTS {Logarithmic Scale) '20 30 40 Fig. S- Curves for core loss plotted with logarithmic scale. there are data for only two frequencies and hence only two points on this curve, plot it as a straight line — as it will be practically a straight line through a considerable range. (When the test is made at only one frequency, see § 52.) The slope of this line gives the exponent (b), showing the law of core-loss variation for different frequencies at constant The exponent (b) should be determined; see §51. Plotted on ordinary coordinate paper, this curve appears as in Fig. 4. IS8 TRANSFORMERS. [Exp. § 12. Fig. s also shows a core-loss curve corrected for 60 cycles, although no measurements were made at that frequency. Such a derived curve can be drawn for any desired frequency. § 13. It is seen that core loss becomes less as frequency is increased. A transformer designed for one frequency can, therefore, be operated more efficiently at a higher frequency, the voltage remaining unchanged. Operated at a lower frequency, however, the transformer will have larger core loss and will therefore, heat up more — unless operated at a lower voltage and reduced output. The transformer to which Fig. 5 refers is seen to have the same core loss (41.6 watts), and so would have the same tem- perature rise, when run at 81 volts, 29 cycles; 98 volts, 60 cycles; 100 volts, 66.2 cycles; 118 volts, 140 cycles. The volt-ampere capacity of any transformer is, accordingly, less at lower fre- quencies. For the same capacity, a larger and more expensive transformer is required. If transformers were the only consideration, the frequencies of 125 and 133 cycles in early use would not have been abandoned for lower ones. (See §3, Exp. 3-A.) § 14. Flux Densities. — Compare the flux densities at different frequencies, for normal voltage. The values of flux density can be computed, as in §§ 33-35, Exp. 5-A, if the number of turns and iron cross-section are known. Without calculating the actual values and without knowing the construction data of the transformer, the relative values can be found by the rela- tion B oc E-^n. Thus, if at 60 cycles B is taken as i.o, the flux density at 30 cycles is 2.0, at 120 cycles 0.5, etc. S-B] TEST BY LOSSES. i59 PART II. RESISTANCE MEASUREMENTS. §15. Data. — The primary and secondary resistances are meas- ured by means of a bridge or by direct current, fall-of -potential method (§17, Exp. i-A). Disconnect the voltmeter before the current is thrown off, to avoid damage by inductive kick. Avoid heating the coils and so causing their resistance to increase; the testing current should not exceed 25 per cent, of the full-load current for the coil, or should not be long continued. The range of the ammeter to be used is thus determined from the known value. of full-load current. The range of voltmeter is found by assuming an approxi- mate value for resistance drop; thus, if the resistance drop were one per cent, in primary or secondary for full-load current, this would be 10 volts in a 1,000-volt coil and i volt in a 100- volt coil. If only one fourth of full-load current were used foi testing, the voltage readings would be 2.5 and 0.25 volts, re- spectively. The resistance measurements . by direct current are to be used as a check and for comparison with the results obtained in the short-circuit test. Temperature conditions should be taken account of (§22). § 16. Equivalent Primary Resistance. — From the measured values of R.^ and R^, compute the equivalent resistance* R, * (§ i6a). The equivalent resistance R must have such a value that Rh' = RJ^ + RJ,\ Dividing by I^ and writing the ratio of turns (^i -=- Si) in place of the ratio of currents (/2-7-/1), we have R = R,-\-(S^-^S,yR2. It is obvious, also, that R= (copper loss) -i-Ii'. Any resistance in the secondary, either within the transformer or in the external circuit, has the same effect as though it were multiplied by the square of the ratio of turns and placed in the primary circuit. It may be noted here that the same is true of reactance. i6o TRANSFORMERS. [Exp. which is the joint resistance of the pritnary and secondary in terms of the primary: The value thus determined will be used for comparison with the value (R^Wc^^I") determined from the copper loss in the short-circuit test. PART III. SHORT-CIRCUIT TEST. § 17. Method of Test. — For the short-circuit test, the sec- ondary (low-potential side) of the transformer is short circuited and the primary (high- er 3.= potential side) is supplied with a small difference of potential, just sufficient to cause the full-load or de- FiG. 6. Connections for short-circuit test sired current tO floW. This for copper loss and impedance voltage. . is rarely more than 5 or per cent, of normal voltage. Instruments* are connected in the primary, as shown in Fig. 6. The current might be measured by an ammeter in either circuit, but it is betterf to have the ammeter on the primary side with the voltmeter and wattmeter. * (§ ^7^)- The most important reading to have correct is that of the ammeter, since the wattmeter reading varies as P and the voltmeter read- ing varies as /, and all results are calculated for values of current. For this reason it is well to place the ammeter directly in the primary circuit, as in Fig. 6, in which case no correction is necessary. If the ammeter is connected in the supply line, as in Fig. i, and the instruments are read simultaneously, a small error is introduced (tending, in tliis case, to favor the transformer) unless a correction is applied. See Appendix III., Exp. S-A. Connected as in Fig. 6, the wattmeter usually needs no correction; but for the accurate measurement of small power the method of con- nection shown in Fig. 7, Exp. s-A, should be used and a correction applied. For selection of instruments, see §44. t (§i7b). It is important for accuracy to have the short circuit of the secondary as " short " as possible, i. e., with practically zero impedance S-B] TEST BY LOSSES. i6i In this test, when full-load current flows in the primary, full- load current flows in the secondary; when half-load current flows in the primary, half-load current flows in the secondary, etc. The flux density is very low, so that there is practically no core loss. The wattmeter reading gives, therefore, the total copper losses — both primary and secondary — for any particular current. Included with the copper losses are the load losses. § i8 Load Losses. — Load losses are due chiefly to eddy cur- rents in the copper and are greatest, therefore, in large solid conductors. They have the efifect of causing a greater loss in a conductor when traversed by alternating current than when traversed by direct current, the resistance being apparently in- creased. The term load losses includes all losses* which in- crease with load and depend upon current, over and above the copper losses as determined by direct current. Evidently such losses should be taken into consideration in calculating efficiency,f and the Standardization Rules of the A. I. E. E. so specify. outside of the transformer. An ammeter and its leads in the secondary, sometimes used, tends to give the transformer a poorer regulation and efficiency. It is instructive, however, before taking readings, to insert an ammeter in the secondary, as well as the primary, and to note that the ratio of currents is practically equal to the ratio of turns. If there are two secondaries, it makes no difference whether they be put in parallel or in series; two primaries should be put in parallel or series to suit the range of instruments. No coil should be left idle. * For example, eddy-current loss in the core due to local flux set up by the current in a loaded transformer in addition to the normal core loss. t(§i8a).Very commonly, however, this is not done, copper losses (neglecting load losses) being determined by direct current measurement of resistance. This tends to favor the transformer. In justification of this, it may be said that it has not been fully established that the load losses under actual load conditions are the same as those obtained on short circuit— it being held that they may be less. The two methods serve as a check. If the losses by the wattmeter are only slightly greater than by direct current, the result is satisfactory for the transformer. Any con- siderable difference, however, shows the existence of load losses. For an accurate comparison, great care is necessary in regard to temperature conditions and the calibration of instruments. 1 62 TRANSFORMERS. [Exp. Although these losses may be considerable in large transformers, in small well built transformers they are usually insignificant. § 19. Impedance Voltage. — In an ideal transformer on short circuit, with zero secondary resistance and no magnetic leakage, the only voltage necessary to cause a given current to flow would be i?i/i, to overcome the resistance of the primary. The effect of secondary resistance is to apparently increase the resistance of the primary to and the resistance drop is, accordingly, RI.^. On account of the magnetic leakage, the transformer appar- ently has a reactance X, called leakage reactance;* this causes (in terms of the primary) a reactance drop XI^, in addition to the resistance drop, RI-^. For a given frequency, this reactance due to leakage is a constant of the transformer, the same as resistance. It is the same on open circuit or short circuit, and is the same at no load or full load. The total impedance, which limits the flow of current in the short circuit test, is a combination of the equivalent resistance and leakage reactance, being Z^y/R'^-^X^. The voltmeter reading gives the total impedance voltage, Ez=yRn\+XH\, necessary to overcome both resistance and leakage reactance. § 20. Data. — At rated frequency, take, say, five readings of the impedance voltage (.Ez) and the copper loss {Wc) for various currents from about ^ to i| full-load current. §21. Readings at various currents are chiefly for illustration and are not essential. When facilities for varying the current are lacking, one accurate reading (or better the mean of five * Discussed more fully in Exp. S-C. S-B] TEST BY LOSSES. 163 readings) at any convenient value of current is sufficient for all results. Slight changes in wave form are immaterial and a series resistance or any other means may be used for adjusting the current (§45). §22. These readings vary with temperature, being dependent upon resistance, and should be taken at some definite tempera- ture or under some definite temperature conditions to be specified in the report, as hot after a heat run of a certain duration, or cold before the transformer is heated up. In this latter case readings must be taken quickly to avoid rise in temperature due to the testing current. Commercial tests should be under speci- fied service conditions, commonly after a three hour heat run at full load or the equivalent, the room temperature being 25° C. ; in this the A. I. E. E. Standardization Rules should be consulted and followed. §23. At a second frequency repeat the readings. (In a com- mercial test, readings would be taken at rated frequency only.) It will be found that the copper loss and apparent resistance vary but slightly with frequency and that the leakage reactance is proportional to frequency. Known for one frequency, it can be computed for any other. § 24. Results. — The impedance voltage, Ez, and the copper losses, Wc, can be plotted directly from the voltmeter and watt- meter readings, with primary current as abscissae, as in Fig. 7 and Fig. 8. It is better, however, to proceed as follows: § 25. From the readings of the ammeter, voltmeter and watt- meter in the short-circuit test, compute, for each observation, the values of Z, R and X, as given below, and determine an average value for all the observations. Impedance : Z = £z -^ A. Resistance : R = Wc -J- Ii"- Reactance: X=^/Z' — R'' i64 TRANSFORMERS. [Exp. These are equivalent or apparent values, in terms of the primary, and include the effect of both primary and secondary. The effect of load losses is included in the values of Wc and R. 0.5 0.75 AMPERES, PRIMARY 1.05 Fig. 7. Short-circuit test, 2 K.W. transformer, 2,000-volt coil. Voltage drop due to impedance resistance and leakage reactance. §26. Compare this value of R with the value found from resistance measurements of i?i and R^ in § 16. § 27. Where tests are made at different frequencies, compute the mean value oi X-i-n for each frequency; the value should be about the same for all frequencies. §28. The values of X affect regulation but not efficiency; while the values of R affect not only regulation (on account of RI drop) but also efficiency (on account of RP loss). § 29. Curves for Voltage Drop. — Using the values of Z, R and X thus determined (§25), plot curves for ZI^, RIj_ and XI.^ drops for different values of primary current, as in Fig. 7, these curves being straight lines. Compare th ese curves with the cor- responding curves for an alternator. Fig. 2, Exp. 3-B. l^ For normal fnll-lpad girrpnt, mark the value of each drop in volts and as per cent, of normal full-load voltage. The per 5-B] TEST BY LOSSES. i6s cent, impedance drop is also called impedance ratio (§13, Exp. 3-B). § 30. Curves for Copper Losses. — Calculate RI^^ for ■^, \, I, %, I and li load. Plotted as in Fig. 8, these give the values of 95.38 121.9 80.3 41.6 -0.25 0.5 0.75 1.0 125' AMPERES, PRIMARY. Fig. 8. Losses and efficiency of a 2 K.W. transformer. the copper loss, including load losses, for different currents; the curve is a parabola. It is seen that the copper loss, in watts, for a given load is pro- portional to the copper drop, in volts. The copper loss, expressed as a per cent, of rated volt-amperes, is equal to the copper drop, expressed as a per cent, of rated volts. Per cent, copper loss =: RPIEl. Per cent, copper drop ^ RI/E. PART IV. RESULTS. EFFICIENCY AND REGULATION. §31. Efficiency. — Efficiency is equal to output divided by in- put and is readily determined when the losses are known. For a particular frequency and normal voltage, take the value of core loss from the curves already determined. Figs. 3, 4 and 5. 1 66 TRANSFORMERS. [Exp. Thus, let W(, = 4i.6 watts. The total losses are found by adding this constant core loss to the copper losses for each load, as in Fig. 8. The efficiency* should be computed for -^j ^, ^, f, i and I J load. The computations for full load and half load are as follows: At full load. Core loss = 41.6 Copper loss^ 51.4 Total loss = 93.0 Output = 2,000.0 Input == 2,093.0 T-. ,1 V total loss , , 9'? Per cent, loss = 100 X — = 1 — = 100 X — ^^^— = 4-44' mput 2,093 „^ . , ^^ total loss Emciencyt = 100 — 100 X At half load. mput = 100 — 4.44=^95.56. Core loss =41.6 Copper los's^ 12.85 Total loss = 54-45 Output = 1,000. Input = 1,054.45 Per cent, loss = 100 X —^^— = S.i6. 1,054.45 Efficiency = 100 — 5.16 ^94.84. * (§3ia). The efficiency will be different for different frequencies and for different rating of voltage and current ; see § 48. See § 57 for a more exact method of determining Wo for full-load voltage. Referring to Fig. S, the core loss at 100 volts is 41.6 for the frequency (66.2 cycles) used in the test. Corrected for 60 cycles, Wo := 43.3, it being possible to thus determine the efficiency for a frequency not used in the test. This is useful in comparing guarantees. t (§3ib). This formula will be found much better for making computa- tions than the equivalent and more usual form. Efficiency =: Output -=- Input. S-B] TEST BY LOSSES. 167 § 32. Maximum efficiency* occurs at such a load that the cop- per loss is equal to the core loss; in Fig. 8, this is at 0.9 full load. Note the similarity between the curves for a trans former, shown in Fig. 8, and the corresponding curves for a shunt motor, Fig. 3, Exp. 2-B. § 33. All-day efficiency is computed on some assumption, as 5 hrs. full load and 19 hrs. no load. Other assumptions can be made to suit specific service conditions. Except under special con- ditions, the term "all-day efficiency" has no useful significance. § 34. Regulation. — The regulation of a constant potential transformer is the per cent, increase in secondary voltage in going from full load to no load. See Appendix I., Exp. S-C. There are various graphical methods for determining regulation, which are necessarily unsatisfactory on account of the small values of some of the quantities and the consequent difficulty in making an accurate drawing to scale. There are also various analytical methods, many of which are equally unsatisfactory on account of their involved character and the unnecessary labor required in using them. The regulation of a transformer can be determined for all power factors— current lagging or leading— by the same method as is used in determining the regulation of an alternator by the electromotive force method (§§ 16-22, Exp. 3-B), either graph- ically or analytically. A modified method, however, is easier to apply to a trans- former on account of the fact that the resistance and reactknce drops in a transformer are comparatively small. §35. What the writer believes to be the simplest and most practicable methodf for determining the regulation of a trans- * See § 28, Exp. 2-B. t (§3Sa). From a paper "Transformer Regulation," by F. Bedell, Elec. World, Oct. 8, 1898; the term with in is now dropped on account of difference of definition (see Appendix I., Exp. S-C). 1 68 TRANSFORMERS. [Exp. former is given below, any errors being less than the usual errors of observation. Regulation is to be computed for non-inductive load and for loads of various power factors, with current lagging and lead- ing. (It is suggested that the reader compares the results ob- tained by this method and by other methods with which he may be familiar, and that he also compares the labor required in applying the different methods.) Let r be the per cent, resistance drop arid x the per cent, reactance drop, as determined by the short-circuit test. Thus, in Fig. 7, r^2.$y and ^.-1=1.76 (not .0257 and .0176). § 36. Non-inductive Load. — The regulation on non-inductive load is computed as follows: Per cent, regulation = r -1 — -. ; — r. ° 2(100 + r) For all practical purposes, as a glance at the numerical ex- ample will show, this may be written x' Per cent, regulation = r -1 . For example, when r^^2.57 and .^=1.76; In-phase drop= r =2.57 per cent. x^ Effective quadrature drop= =0.015 per cent. Regulation = 2.585 per cent. It is seen that the regulation is practically determined by the resistance drop; the effect of reactance drop on non-inductive load is nearly negligible. This is seen in Fig. 9 which is dis- cussed lat^r. In computing the regulation, therefore, the accu- racy of the results depends directly upon the accuracy with which the resistance is determined. Regulation varies with tem- perature and to be definite must be for a specified temperature. § 37. For Lagging Current. — When the load has a power S-B] TEST BY LOSSES. 169 factor (cos 6) less than unity and the current is lagging, the regulation is practically* as follows: Per cent, regulation = r cos 6-\-x sin d. For example, let cos $ = 0,866; sin 6 = 0.5; In-phase resistance drop^r cos 6 = 2.57X0.866=2.23 per cent. In-phase reactance drop ^j; sin g= 1.76 X 0.500=0.88 per cent. Regulation = 3. r i per cent. § 38. For Leading Current. — When the load has a power factor (cos d) less than unity and the current is leading, the regulation is practicallyf Per cent, regulation = r cos — x sin 6. For example, let cos 6 = 0.866; sin 6 = 0.5; In-phase resistance drop = r cos 6 = 2.57X0.866^2.23 per cent. In-phase reactance drop = j;sin 6 ^1.76X0.500=0.88 per cent. Regulation := 1.35 per cent. § 39. Proof. — Fig. 9 shows a simple graphical method for obtain- ing regulation at non-inductive load. (Compare also Fig. 11, Exp. 5-C, in which the same lettering is used, and Fig. 3, Exp. 3-B.) Referring to Fig. 9, lay off AL equal to the secondary full-load voltage £j=ioo per cent. (A scale of volts could be used, if Fig. 9. Method for determining regulation ; r = per cent, resistance drop ; a: = per cent, reactance drop ; regulation ^ £„ — £. ^ 2.585. (This Fig. is not drawn to scale.) *(§37a)- For greater accuracy, a term for effective quadrature drop (g"-^20o), should be added, §42. In the present example this term is only .0003, making the regulation 3.1103. In any ordinary case, on lagging current, this term can be neglected. t (§382). For greater accuracy, a term g''-i-200 should be added, §43; in the present example, this term equals .039. 17° TRANSFORMERS. fExp. desired, instead of per cent.) Lay off LJ = r and, JK = x. Then AK = E„, the secondary terminal voltage at no load. Per cent, regu- lation = £„ — £2 = 2.585. As already stated, the graphical method can not be accurately applied on account of the small values of r and x. From the graphical method, an analytical method is derived as follows. § 40. Analytically, we have from Fig. 9, or, more simply (see §41), £„=ioo + »-4 2(100 -!-»') Transposing, we have Regulation =E. — 100 ^r A — -, ; — rr- . " ° ' 2(ioo + r) §41. Expressed more generally, let />^per cent, in-phase voltage drop, g^per cent, quadrature voltage drop. £. = V(ioo + /,r + g'=ioo + ^ + ^-^-^. This practical identity* can be seen by squaring, or by solving a numerical example. Transposing, we have a Regulation = E. — 100 = * -I -. — . ,. , " " "^ ' 2(100 + />) or, for practical purposes, = p+ (--]-[?~(r:)"]- The above equations can be derived as follows: Eddy current loss, irrespective of frequency and wave form (§49), varies as E' and equals oB^ where o is a constant. Similarly, for any wave form, hysteresis loss equals bnB^, where b and x are constant; no assumption is made that X = 1.6. At the two frequencies the total losses are (i) W" =a(^E'y + bn' (B'r; (2) W" = a(iE"y-\-bn"(B")\ For B", write B', this being the condition of the test; for E", write E'(,E"-^E'). Multiply (2) by n'-^n", subtract from (i) and solve for eddy current loss a (£')'■ When the wave form of electromotive force is the same at the two frequencies, (£"-=-£') = (n"-Hn'). The separation of losses by measurements at two frequencies was first made by Steinmetz ; the influence of form factor was introduced by Roessler. There are various methods for making the calculations, differ- ing somewhat in detail. The formulae here given are from a paper by the author before the Cornell Electrical Society, May 4, 1898. Note § 35, Exp. S-A, and Appendix I., Exp. 2-B ; also Bedell's Transformer, p. 312 et seq. (Some of these references, following Roessler, use form factor as the reciprocal of /, as defined above.) M. G. Lloyd has recently pub- lished a very complete investigation of the subject; see Bull. Bureau 0/ Standards, February, 1909. 5-B] TEST BY LOSSES. i77 § 54- Insulation and Temperature Tests. — These tests are of com- mercial importance but need no full discussion here. The Standardi- zation Rules specify fully the conditions under which they are to be made; details of the tests are described in the usual handbooks. § 55. Insulation. — The insulation is tested between each winding and all other parts. The applied voltage is increased gradually, so as to avoid any excessive momentary strain. This is usually done by some means of primary control in a special testing transformer. Various companies make testing transformers for obtaining high potential for this test and furnish detailed instructions for their use. The voltage is preferably measured by means of a spark gap with a high protective resistance in series with it. The test consists in seeing that the apparatus withstands a specified over-voltage for a specified time without breaking down. § 56. Heat Runs. — These are made under full-load voltage and full-load current for a specified time, temperatures being found by thermometers and resistance measurements. The heat run could be made by actually loading the transformer, but is usually made by some kind of opposition or pumping back method, of which there are several. No load is then required and no power, except enough to supply the losses. A common form of opposition run employs two similar trans- formers: the two secondaries (low potential side) are connected in parallel to source A, of normal frequency and normal voltage, which supplies the core loss; the two primaries are connected _ in series, opposed to each other, and are then connected to source B, which supplies the normal full-load current. (Source B requires a voltage equal to twice the impedance voltage of one transformer and can be of any frequency, i. e., it may or may not be the same frequency as /J.) All windings now have full-load current and normal voltage. Instruments in A will give, if desired, the core loss and exciting current; instruments in B will give copper loss and impedance voltage. (The two transformers need not be identical.) Instead of connecting source B in the high potential side, a com- mon modification is to connect the high potential windings of the two transformers directly in opposition and to insert source B in series with the low potential winding of one of the transformers. This has 13 178 TRANSFORMERS. [Exp. the advantage that all connections with supply lines and instruments are at low potential; see Electric Journal, p. 64, Vol. VI., and Fig. 322, Karapetoff's Exp. Elect. Engineering. A modified form of opposition test can be applied to a single transformer; see Foster's Handbook. § 57. Note on Efficiency. — If the rated secondary voltage is £j,= ioo, the customary and most simple procedure is to take the core loss for this voltage from Fig. 5 (thus, W„ = 4i.6) and to compute the full load efficiency as in § 31. To be accurate, however, the secondary core voltage or flux voltage, E^, should be taken as Ej plus the secondary RI drop. Taking this drop as 1.28 (,= ir in §35)' we have £5:= 101.28 and the corresponding core loss, W„ = 42.5; this gives the correct efficiency of 95.51 .instead of 95.56. The difference between these values is so little that the method of § 31 is usually sufficiently correct. Approached in another way, we might consider Eg = 100 and W„^4i.6. Then £2= 100 — 1.28 = 98.72. To get the rated out- put of 2 K. W., since E^ is decreased, the current must be increased by the factor i -4- 98.72. The copper loss must then be increased by the factor (i-;-98.72)^ giving an efficiency of about 95.5. S-C] CIRCLE DIAGRAM. 179 Experiment 5-C. Circle Diagram for a Constant Poten- tial Transformer. § I. Introductory. — It has been seen, Exp. 4-B, that when the resistance is varied in a series circuit with constant reactance the vector representing the current follows the arc of a circle as a locus. In a similar manner, the primary current of a con- stant potential transformer follows the arc of a circle as a locus when the secondary resistance is varied. The same is true for an induction motor when its load is varied, and use is made of this fact in practical motor testing. The following experiment will, accordingly, serve to make clear certain principles of the induction motor as well as of the transformer; upon these prin- ciples is based the method of transformer testing developed in detail in Exp. S-B. In Part I. the general principles governing the action of a transformer will be discussed; in Part II. these principles will be applied in constructing a circle diagram. The practical re- sults, so far as commercial testing is concerned, are all given in Exp. 5-B. The actual construction of a diagram to scale gives one a definite and concrete idea of what might otherwise be vague and abstract. Furthermore, the abstract diagrams given here (Figs, i-ii) and elsewhere are so grossly exaggerated that they give very wrong ideas of real values. Even Fig. 12, which is more nearly to scale, is much exaggerated. § 2. Data. — The same data are required as in Exp. 5-B. See § 25 of this experiment. PART I. GENERAL DISCUSSION OF THE ACTION OF A TRANSFORMER. § 3. The action of a transformer will be most readily under- stood by considering its action first without a load — i. e., on open circuit — and then with a load. TRANSFORMERS. [Exp. § 4. Transformer on Open Circuit. — When a transformer is on open circuit, the secondary winding has no current flowing in it and it accordingly has no magnetizing effect on the core. A small current flows in the primary which magnetizes the core. Let us see what determines the magnitude and phase of this open-circuit primary current. § 5. Assuming No Core Loss. — The open-circuit diagram for a perfect transformer, in which there are no losses, is shown in Fig. I. The primary electromotive force £p causes a current /fl to flow and this current sets up a flux (j>. This flux, being alternating, causes a counter-electromotive force opposed to the primary impressed electromotive force. When the primary cir- cuit is closed, the current /„, and the flux <^ which it sets up, assume such values that the counter-electromotive force is just equal* to the impressed electromotive force. This primary counter-electro- motive force has, at any instant, the value e' = — S^{d^-^dt), the equal and opposite im- pressed electromotive force be- ing ev=^Si{d4>-^dt). It will be seen that the electromotive force is zero when the flux is a maximum and that the flux ^ lags 90° behind the impressed electromotive force £p, as in Fig. I. § 6. In the absence of core loss, the current /„ is in phase with the flux <^, which it produces. When permeability is constant, magnetizing force H is proportional to I^ and is in phase with *The primary resistance on open circuit is very small and can be neglected. , ■ 'Bp Flux^ _ Phase of ~B and R Fig. I. Open-circuit diagram for a transformer with no core loss. S-C] CIRCLE DIAGRAM. i8i ^H and proportional to the flux density B. The B-H curve is a straight line, instead of the familiar hysteresis loop, and there is no hysteresis loss. The current Iq, as shown in Fig. i, is in quadrature with the electromotive force and is wattless. § 7. The flux throughout this discussion refers to the flux which links with both primary and secondary, and £p and Es are the induced or flux voltages,* proportional to <^. In an ideal transformer there is no other flux, but in an actual trans- former there is, in addition to this main flux, a relatively small local or leakage flux, which links with the turns or part of the turns of one winding only and causes a reactance called leakage reactance. On account of the drop due to leakage reactance and the drop due to the resistance of the transformer windings, as discussed later, the terminal voltages, E^ and E^, are slightly dif- ferent from the flux voltages Ep and £s. * (§8a). Strictly speaking E-p is not the flux voltage but is equal and opposite thereto. B.r Core-loss /^ Component O ^% Magnetising Con:^ponent n. O -B Fig. 2. Open-circuit diagram for a transformer with core loss, show- ing the two components of exciting current and the angle a of hysteretic advance. iS2 TRANSFORMERS. [Exp. § 9. With Core Loss. — A transformer with an iron core differs from the ideal transformer just discussed because there is a loss in the iron due to hysteresis and eddy currents. The open- circuit diagram now becomes as shown in Fig. 2. The flux ij> is still in quadrature with Ep and £s, in accordance with Faraday's fundamental law of induced electromotive force, ^ = — S (dfj) ~- dt) . The exciting current /„> however, can no longer be a wattless quadrature current, for it must have an in-phase power component to supply the core losses due to hysteresis and eddy currents. This core loss component is Jh^ watts core loss-f-£p. The exciting current /,, is, accordingly, advanced in phase by an angle a, called the hysteretic* angle of advance. It is seen, therefore, that /„ consists of two components — the core loss component /h and the true magnetizing component /m which is wattless and in phase with the flux. The total ex- citing currentf is the vector sum of these two components : § 10. A constant potential transformer (one in which £p is constant) is a constant flux transformer. It therefore follows * (§9a). As here defined, this angle includes the effect of eddy currents. t(§9b). The exciting current of a transformer is distorted, i. e., has a wave form different from that of the electromotive force, on account of harmonics introduced by hysteresis. (See Appendix II., Exp. 6-A.) These harmonics — currents of 3, S, 7. etc., times the fundamental fre- quency — are necessarily wattless. They do not appear, therefore, in the power component In, but are included in the wattless componeflt /m. Strictly speaking, alternating currents in which harmonics are present can not be represented by vectors in one plane; for practical purposes, how- ever, the plane vector diagram, as here given, is sufficiently accurate. (See §47, Exp. 6-A; also "The Effect of Iron in Distorting Alternating Current Wave Form," by Bedell and Tuttle, A. I. E. E., Sept., 1906; and " Vector Representation of Non-Harmonic Alternating Currents," by B. Arakawa, Physical Review, 1909.) These harmonics have the same value at all loads; at full load they form such a small part of the total current that the distortion which they produce is very small. 5-C] CIRCLE DIAGRAM. 183 that la. In and /m are constant, and loads. This would not be quite true the primary line voltage E^ (and not £p) is constant; the difference in the two cases is very small. § II. Transformer Under Load. — The complete diagram for a constant potential transformer un- der non-inductive load is shown in Fig. 3. This will be seen to be exactly the same as Fig. 2, the open-circuit diagram, with certain additions. As in Fig. 2, we have the electromotive forces £p and £s opposite to each other in phase and in quadrature with the con- stant flux the former in quadrature and the latter in phase with I^. For a non-inductive load, the secondary cur- rent, 1 2, is in phase with the terminal voltage, E^. (For an inductive load, I^ would lag behind E^ by an angle d, where cos 6 is the power factor of the load.) §13. In the secondary, it is seen that Es is constant (flux being constant) and the secondary may, accordingly, be treated as a simple constant potential circuit. The locus of the secondary current, as the load resistance varies, is, accordingly, the arc of a circle, as in any constant-potential circuit with constant react- ance. (See Exp. 4-B.) § 14. Primary Quantities. — It has been seen that on open cir- cuit the primary current assumes a certain value /„, so as to produce a flux that generates a counter-electromotive force just equal and opposite to the impressed electromotive force. When a secondary current I^ flows, it disturbs this equilibrium by tending to demagnetize the core. This allows more current to Bow in the primary. The primary current increases until (in addition to /„) a current /(j) flows in the primary, the magnet- izing effect of which (ampere turns) just balances the magnet- izing effect of the current I^ in the secondary. The magnet- izing effect of the secondary being thus neutralized, the flux has the same constant value as before (as though produced by I^ alone), so that the counter-electromotive force produced by the flux continues to be just equal and opposite to the impressed elec- tromotive force. In Fig. 3, the total primary current I^, is seen to be composed of the constant /„ (which is small) and the load current I^^-,, which is opposite to the current I^ in the secondary and equal to 1 2 multiplied hy (S^-^S-^). In a i : i transformer, the pri- mary load current I^2^, is equal to the secondary current 1 2- -C] CIRCLE DIAGRAM. 185 § 15. Fig. 4 shows that, in a loaded transformer, the resultant ampere turns are constant; hence the flux is constant, so ihat the counter-electromotive force — irrespective of load — equals, the impressed electromotive force. As the load changes, the primary cur- rent assumes such a value that the resultant ampere turns remains constant and this condition of .'7 equilibrium is maintained. § 16. The primary electromotive force, thus balanced by the counter-electromotive force, is £p. Referring to Fig. 3, it will be seen that the terminal impressed electromotive force, E^, is a little greater (say one per cent, greater) than £?, on account of the R^Ix and XJ.^ drops, due to primary resistance and to leakage react- *""^°^' ^^^ resultant ampere turns are constant. ance. § 17. The locus of the secondary current I^ is the arc of a circle (§13). Hence the locus of the primary load current, 7(2) in Fig. 3, is the arc of a circle. The total primary current, /i, measured from O to P, follows this same locus. Some simplified diagrams will now be discussed. § 18. Representation of Transformer Circuits. — From the foregoing discussion, it will be seen that the circuits of a trans- former may be represented as in Fig. 5, in which the resistance and leakage reactance of the two windings are considered as external to the transformer. Furthermore, the exciting cur- rent, /fl, is considered as flowing in a shunt circuit, also external to the transformer. This shunt circuit consists of two branches : Fig. 4. Diagram of ampere 1 86 TRANSFORMERS. [Exp. a non-inductive branch for the in-phase component, Is, and an inductive branch (without resistance) for the wattless quadra- ture component /m- The currents which would flow in such equivalent shunt circuits correspond exactly to the currents /(,, /h and /m which actually flow in a transformer. —*-'\/\/\fr-^Wsir^ j-T'T'5WM/WVTr-| Load Fig. s. Complete equivalent of a transformer. The exciting current /» is considered as flowing in a shunt circuit. The resistance and leakage react- ance of primary and secondary are considered as external. Corresponds to Fig. 3. §19. The transformer proper, in Fig. 5, is considered as ideal, all the losses being treated as external; /(2)=/2(5"2-f-5'i) ; and Ef = Es{Si-^S2). The voltage at the primary terminals, £1, is more than £p on account of the drop in X^ and in i?i. Likewise, the voltage at the secondary terminals, E2, is- less than Es on account of the drop in X^ and in R^. — jHW/VTMS^ 'TTOBir'-^/vvv*— I s 1 Fig. 6. Equivalent circuits as level (1:1) transformer. Corresponds to Fig. 7. The total primary current I^ is seen to be equal to the load current I^^-,, plus (vectorially) the small no-load current /„. § 20. Equivalent Circuits. — The circuits of a transformer may be represented more simply by the equivalent circuits of Fig. 6, s-c] CIRCLE DIAGRAM. 187 in which all quantities are expressed in terms* of the primary. This will be most readily understood by treating the transformer as a "level" (1:1) transformer; we have then, £p^£s; and •'(2)=-'2- The diagram corresponding to Fig. 6 is shown in Fig. 7 and is seen to be the same as Fig. 3 with all secondary quantities expressed in terms of the pri- mary and drawn in the first quadrant. § 21. Simplified Circuits. — The equivalent circuits so far considered (Figs. 5 and 6) and the corresponding diagrams (Figs. 3 and 7) are prac- tically exact and may be used for the accurate solution of any transformer problem. It will be noted that the resistance and reactance for the two wind- ings are treated separately, 7?iXi in the primary and i?2^2 in the secondary. By com- bining these into a single equiv- alent R and X, the trans- former circuits can, with little error, be simplified in either of two ways : *(§2oa). To express secondary quantities in terras of the primary: multiply current by (.S^-i-Si) ; multiply voltage by (Si-^Ss) ; multiply X and R by (Si^S^y. See § i6a, Exp. S^B. It will be understood that secondary quantities thus represented in the primary are not the real secondary quantities but the equivalent primary quantities which could . produce the same results ; thus, in a 10 : i transformer, i ohm in the primary is equivalent to o.oi ohm in the secondary. To express primary quantities in terms of the secondary, divide instead of multiply by these factors. -J ^ Fig. 7. Exact diagram as level transformer, corresponding to Fig. 6, The same as Fig. 3 with secondary quantities expressed in terms of the primary. i88 TRANSFORMERS. [Exp. I. All the resistance and leakage reactance are considered to be in the primary, as in Figs. 8 and lo. -is,x- B. ^r-^/\/w^^R^R^'— -TJOSiNVVVV >^x Fig. 8. Simplified circuits ; R and X all in primary. Corresponds to Fig. lo. 2. All the resistance and leakage reactance are considered to be in the secondary, as in Figs. 9 and 11. Each of these simplifications differ very little from the more exact representations already discussed. In the actual transformer, as represented in Fig. 6, it is seen that the current which flows through XJi^ is /(j), while a dif- ferent current 7^ (slightly larger, due to /(,) flows through XJi^. In the simplifications, the same current is considered to flow through i?i^i and R^X^ which are now combined into a single -iJ,X- ? p? ■y-^tN\r'^ssss^-''ms^-^m^ s. Fig. 9. Simplified circuits; R and X all in secondary. Corresponds to Fig. 11. R and X, this current being either I^ (as in Fig. 8) or 7(2, (as in Fig. 9). If 7(, were zero, Figs. 8 and 9 would not differ from Fig. 6, and all the representations would be identical. In fact, !„ is so small that either simplification and its resultant diagram. Fig. s-c] CIRCLE DIAGRAM. 189 10 or II, may, for most practical purposes, be considered as correct. This makes it possible to use the single equivalent values for R and X obtained by the short-circuit test of Exp. 5-B, and does not require separate values of R and X for the primary and sec- ondary circuits. § 22. Again, the voltage which causes ^^ /(, to flow is £p, as is seen in Fig. 6. In the simplifications, this voltage is taken as £2 (Fig- 8) which is, say, i per cent. less, or as fi^ (Fig- 9) which is, say, I per cent, more than the value of £p in the actual cases of Fig. 6. This would make an insignificant change in the value of /„ which is itself small. In the latter case /„ depends only upon line voltage and is independ- ent of load. §23. Diagrams Compared.— Let us pic. 10. Simplified dia- compare the exact diagram, Fig. 7, with gram; R and X all in pri- ., .,.-,. „. , mary. Corresponds to Fig. 8. the Simplifications, Figs. 10 and 11. In Fig. 7, the primary and secondary RI drops are in phase with /i and /(j,, respectively, the XI drops being in quadrature. The phase difference between I^ and 1(2) is small — much smaller in fact than shown in the figure. The primary and secondary drops may, accordingly, be combined with little error. This may be done by taking the combined resistance drop in phase with /i (Fig. 10), or in phase with /(j, (Fig. 11). The com- bined reactance drop is, in each case, at right angles to the combined resistance drop. In an actual case little error is introduced by these simplifications and either may be used, as is most convenient. 190 TRANSFORMERS. [Exp. PART n. THE CIRCLE DIAGRAM AND ITS CONSTRUCTION. § 24. The circle diagram for a transformer shows the varia- tion in the primary current for different values of load resist- ance with constant impressed voltage. In Fig. 11, the primary Fig. 1 1. Simplified diagram ; R and X all in secondary. Corresponds to Fig. 9. current is OP, being composed of the no-load current OA and the load current AP. As the load resistance is decreased from in- finity to zero, the point P will trace the arc of a circle, and will take the position P" on short circuit.* If it were possible to elimi- nate the resistance of the transformer windings, the point P * (§24a). If a transformer is constructed so as to have a large leakage reactance (or if a reactance is included in the circuit external to the transformer), the short-circuit current and the diameter, Ei-{rX, are reduced. The transformer may then be operated at or near short circuit, in which case ,the current will be nearly constant. This method is used for obtaining constant current from a constant potential line. (See § 4a, Exp. 5-A.) Large reactance or magnetic leakage in any apparatus tends towards constant current operation. See § 8, Exp. 3-A, §§ 27, 27a, Exp. 3-B, and § 14, Exp. 4-B. s-ci CIRCLE DIAGRAM. would complete the semi-circle and assume the position P'", the current in this case {E^-^X) being limited only by the leakage reactance, X. The short-circuit current of a transformer oper- ated at full voltage would be, however, greatly in excess of the carrying capacity of the transformer windings, and, in actual operation, the point P does not go far beyond the full-load point P'. See also Fig. 12, which is more nearly to scale. §25. Data Necessary. — The data necessary are the values of /(,, /h and /m, to locate the point A, and the leakage reactance X, to determine the diameter of the semicircle. , These data are obtained from the open-circuit 1 and short-circuit tests of Exp. 5-B. All quantities are to be in terms of the pri- mary (high-potential) side; thus, in Fig. 2. Exp. 5-B, the values of /„, In and /m, meas- ured on the lOO-volt coil, are divided by 20 to obtain the corresponding values for the 2,000- volt primary. This gives us : /„ = .03025 ; Is. = .0208 ; 7m = .0220. The reactance X for the same transformer, is 35.2 ohms; see Fig. 7, Exp. 5-B. § 26. Construction of Diagram from Experi- mental Data. — From the data given above, lay off (Fig. 12) : OB^Ih; BA=^Im; 0A=I„. Fig. 12. Con- struction of cir- cle diagram. The diameter of the circle is £1^-^:^2,000-^35.2 = 56.8 amperes. The radius p ^ £1 -=- 2 Z = 28.4. These values are large compared with /„ == -03 ^"d full-load current I^2■, = i ampere. It is, accordingly, not practicable to construct the whole semicircle, as in Fig. 11, which is not at all to scale. 192 TRANSFORMERS. [Exp, For a working range it can be readily constructed, as in Fig. 12, which is more nearly to scale, as follows : Lay ofif AD=Ii^., ior-jl^, ^, i, I, f, i and i^ load. (It is to be noted that, in Fig. 12, the angle DAP is small ; hence AD is taken as practically equal to AP or /(a).) Thus, for a 2,000- volt, 2 K.W., transformer, AD is laid off, successively, equal to .01, 0.1, .25, .50, .75, i.o and 1.25 amperes. For each value of AD, the point P is located by laying off DP=p~Vp^ — Aff which can be derived from the figure and is the equation of a circle referred to A as an origin. The line DP represents the quadrature component of primary current due to leakage re- actance. This is always small and would be zero when X = o, for the diameter of the semicircle (see Fig. 11) is then infinite. The power component AD is, therefore, practically equal to AP. It is to be noted that CP = DP + BA; 2.nA0C=0B-\-AD. From these values, compute* for different loads Primary current =0P = T^OC'-f- CP\ Power factor= OC— OP. The curves in Fig. 4, Exp. 5-A were thus computed. Note §4ia, Exp. 5-B. * (§26a). It will be seen, also, that Watts input, Wt=OCXE^; Watts output, W-i = Wi — losses ; This gives a possible method for determining the total voltage drop. 5-C] CIRCLE DIAGRAM. '93 APPENDIX I. NOTE ON REGULATION. §27. Definition of Terms. — Regulation is defined by the Institute as follows : In constant-potential transformers, the regulation is the ratio of the rise of secondary terminal voltage from rated non-inductive load to no load (at constant primary impressed terminal voltage) to the secondary ter- minal voltage at rated load. (Compare §34, Exp. S-B.) If the secondary terminal voltage is £„ at no load and E^ at full load, the regulation is, Regulation = (£„ — E^) -^ E^ The drop on which regulation depends is E„ — E^, which we may term the regulation drop. This drop, expressed as per cent, of E„ gives the regulation. § 28. The total voltage drop, in terms of a 1:1 transformer, is £1 — E^ and is a Httle more than the regulation drop, because E^ is a little more than £„ on account of the drop due to exciting current in the primary winding. The per cent, voltage drop is {E^ — E^) -v-E„ taking E^ as 100 per cent.; or, (£, — E^) -^E^, taking E^ as 100 per cent. § 29. Numerically, the difference between " regulation " and " per cent, voltage drop " is small. In earlier usage,* the term regulation was commonly employed to designate " per cent, voltage drop." This confusion is one cause for the differences between various methods which have been used (and still are used) for determining regulation. A difference arises, also, according to whether E^ or E^ is taken as IQO per cent. * (§29a). See the following articles, in the Elec. World, on the pre- determination of transformer regulation : Bedell, Chandler and Sherwood, August 14, 1897 ; A. R. Everest, June 4, 1898 ; F. Bedell, October 8, 1898. See also Foster's Electrical Eng. Pocket Book, p.- 492, fifth edition, 1908. 14 194 TRANSFORMERS. [Exp. § 30. An illustration will make this clear. Let £,= 100; £„ = 99.9; £, = 97. Regulation drop = £„ — £2 ^ 2.9 volts. Per cent, regulation = 2.9 h- 97 = 2.99 per cent. Total voltage drop =E^ — -Ej ^ 3 volts. Per cent, voltage drop ^ 3 -=- 100 = 3 per cent. ; or = 3 ^- 97 = 3.1 per cent. § 31. Regulation drop depends upon the difference between E, and E^. This drop is due to load current, and does not include any drop due to exciting current, which affects £„ and E^ alike and, practically, does not affect their difference. The total voltage drop depends upon the difference between £, and E^. This drop is chiefly due to load current, but includes, in addition, a small drop due to exciting current which affects E^ but not E^ and so directly affects their difference. § 32. Computations. — To compute regulation drop, we have the problem: Given E„ to compute £„. To compute total voltage drop, we have the problem: Given £„ to compute £j. § 33. Regulation. — For determing regulation, we compute E, = V(£. + ^+?^ ; where in-phase drop =/> = i?7(,j, quadrature drop^=q = XI^^y It is seen that exciting current does not enter. The various drops may be expressed either in volts or in per cent. The working details of the method are discussed in §§ 34-43, Exp. 5-B. § 34. Total Voltage Drop. — For determining total voltage drop, we compute E,=vT£H^fr+?. The in-phase drop p, consists principally of i?/(,), but includes the small additional terms RJ^ and XJn, which are drops caused by the two components of the exciting current flowing through X^, R^. Without much error, X^, R^ may be taken as half of X, R. Hence S-C] CIRCLE DIAGRAM. '95 In a like manner q = -X^^cz) + XJn — i?j7M, = XI^,, + iXlH — iRIu. The last two terms are small and nearly cancel each other. § 35. Other methods of analysis may be employed for determining the total voltage drop, and the form in which the results are ex- pressed will vary according to the manner in which the various terms are combined and the approximations which are introduced. In any case some small and troublesome terms are introduced, which affect the result very little and which do not enter in the determination of " regulation," as defined by the Institute. The results are affected less by these small terms than by variations in the value of R, depend- ing upon whether, in its determination, load losses were included or not, and whether steady temperature conditions were maintained during the test. § 36. It might well be held that regulation should be so defined that magnetising current should enter into its determination, particularly since magnetising current has been much increased by the use of improved iron worked at higher densities ; on the other hand, it is much simpler to define regulation independently of magnetizing cur- rent and to specify the value of the magnetizing or exciting current as a separate item. CHAPTER VI. POLYPHASE CURRENTS. Experiment 6-A. A General Study of Polyphase Cur- rents.* PART I. § I. Introductory. — In a polyphase system, several single- phase currents differing in phase from each other are combined into one system. The circuits for each phase may be independ- ent, without electrical connection, or interconnected. The phase difference between the currents of the several phases is usually 90° or 120°, the corresponding systems being called two-phase or three-phase. j A' A •''inioooooiiiio^* B'^ (a) Fig. I. Two-phase connections for generator or receiver circuits, a, 4-wire system with independent phases, h, 3-wire system, c. Quarter-phase, star- connected ; or, 4-wire system with interconnected neutral, d. Quarter-phase, mesh-connected ; or, ring-connected. To form a polyphase system we must have several sources of single-phase electromotive force which differ in phase by proper amounts. For a symmetrical polyphase system these electro- motive forces must be equal and differ from each other by equal phase angles, as in the 3-phase and quarter-phase systems soon * (§ la). In making polyphase measurements, some form of voltmeter and ammeter switches will be found convenient, so that all readings can be made with one voltmeter and one ammeter. The same switches will serve to transfer one wattmeter from one circuit to another. 196 6-A] GENERAL STUDY. i97 to be discussed. The sources of these electromotive forces are in principle several rigidly connected single-phase generators, but in practice they are generator coils on a single armature. The secondary coil of a transformer may be considered as a generator coil. The currents from these sources may be utilized separately as single-phase currents (as in lighting), or jointly as polyphase currents (as in an induction motor). Fig. ii. Three-phase connections for generator or receiver circuits, a, Inde- pendent circuits ; see § 3a. b. Star- or y-connected. c, Mesh- or delta- (A) connected, d, T-connected. e, K-connected ; or, open delta. § 2. The load on a polyphase system is balanced when each phase has an equal load with equal power factor. In a balanced polyphase system the flow of energy is uniform, which is a bet- ter and more general definition of such a system; (see Steinmetz, Alternating Current Phenomena). In a single-phase system or unbalanced polyphase system, the flow of energy is pulsating, — discussed further in § i, Exp. 7-A. The torque is, accordingly, pulsating in all single-phase machinery; whereas it is uniform in polyphase motors and in polyphase generators on balanced load. Furthermore, a polyphase induction motor on account of its rotating field can be given a good starting torque, whereas a single-phase induction motor has none in itself and has only a small starting torque when auxiliary starting devices are used. Polyphase machinery has a greater output than single phase for a given size, or has a smaller size for a given output. These features, together with the copper economy of 3-phase as com- pared with single-phase transmission, all favor the use of poly- phase systems ; see Appendix III. 198 POLYPHASE CURRENTS. [Exp. §•3. Methods of Connecting Phases. — Generating or receiving coils or circuits may be combined in various ways, the common ones being shown* diagrammatically in Figs, i and 2. In Figs. I and 2, the relative positions of the various coils represent the relative phase positions of their several electromotive forces.f The black dotsf may be taken as line wires in cross-section. On paper the distance between any two dots is the difference of potential between them ; phase, as well as magnitude, is shown in this way. To the same polyphase system, a number of differently con- nected polyphase generators and receivers may be connected at the same time ; thus, on a 3-phase system, some apparatus may be delta- and some star-connected. From a 4-wire 2-phase system, induction motors may be run simultaneously when connected as (o), (c) or {d), Fig. i. Connection (b) can be combined on the same system with (a), but not with (c) or {d). This is an objection to 3-wire 2-phase distribution, inasmuch as synchron- ous motors and converters as well as generators are frequently wound quarter-phase and so cannot be run from a 3-wire system. A further objection, that the line drop in the common wire makes the voltages unsymmetrical, is discussed later, § 14. § 4. Object. — In performing this experiment, the object is to gain a knowledge of the connections of polyphase circuits and polyphase apparatus, and to understand their electrical relations and various diagrammatic methods for representing them. Make a study of whatever polyphase supply circuits are available and by means of transformers obtain, so far as possible, all the systems indicated in Figs, i and 2. * (§3a). The arrangement of Fig. 2 (o) is never used for independent 3-phase circuits; it is used only for connecting transformer secondaries to so-called 6-phase synchronous converters, § 27. t(§3b). Although the diagram of connections can not in general be taken as the vector diagram of electromotive forces, this can be done in the simpler cases and makes the introduction to the subject more clear. t (§3c). This representation by dots is called by Steinmetz the topo- graphic method. 6-A] GENERAL STUDY. 199 Note also the connections on various pieces of polyphase apparatus (as generators, motors, etc.) which may be available, and note for what kind of polyphase system the apparatus is intended. PART II. § 5. Two-phase Measurement. — Take two transformers* with the same ratio of transformation (say 1:1). Connect the primary of one transformer to phase ^ of a 2-phase circuit,t and the primary of the other transformer to phase B. Measure the secondary voltages when the secondary circuits are inde- pendent, thus forming a 4-wire system with independent phases. Fig. I (a). § 6. Addition of Electromotive Forces. — Connect the two sec- ondaries as a 3-wire system. Fig. i (&), and measure the voltage of each phase (Ea. and £b) and the voltage E between outside wires. Lay off these voltages as a triangle and note how nearly Ea and Eb are at right angles, so making a true 2-phase system. This triangle may be drawn as in Fig. 3, 4 or 5. n, Fig. 3. Topographic method. \ \. Fig. 4. Addition method. Ua \ v \ O Ha B Fig. 5. Subtraction method. § 7. If we use the topographic method of Steinmetz and omit arrows, we can represent the electromotive forces of the 2-phase 3-wire system by Fig. 3 (see Appendix I.). This electromotive * It is preferable that each secondary consists of two equal coils : thus, we might have primary no volts; secondaries SS volts each, giving in series no volts with a middle or neutral point. Note the various possible voltage transformations for each transformer. t It matters not whether the supply circuit is 3-wire or 4-wire, or how connected. If several kinds of supply circuits are available, use each one in turn. Compare Fig. 6. 200 POLYPHASE CURRENTS. [Exp, force diagram is seen to be similar to the circuit diagram (&) of Fig. 1. § 8. If we take one outside line, say B, as our starting point (imagining if we wish that it is grounded, but this is unneces- sary), we have the electromotive forces £b and £a represented by the vectors, BO and OA, in the direction sh5wn by arrows in Fig. 4. The sum of these two vectors is BA: § 9. If, as is common, we take the neutral O as the starting point (say ground), the differences of potential between the side wires and ground are OA and OB, the direction of the vectors being from the starting point as in Fig. 5. The difference in potential between A and B is now the difference between OB and OA, Fig. 5 ; which is the same as the svim of BO (equal — OB) and OA, Fig. 4. § 10. In general, if we take electromotive forces in sequence — as BO, OA — they must be added (Fig. 4) ; if, however, we con- sider each electromotive force in a direction away from a com- mon joining point — as OB, OA — ^they must be subtracted (Fig. 5). For the simple case at hand, involving only two elec- tromotive forces connected to a common point, the difference method may be readily applied. For more complicated networks the addition method is used, as it is capable of more general application; it is based on the statement of Kirchhoff's Law (§ 32, Appendix I.) that the differences of potential around any mesh add up to zero. For this addition method, all arrows are taken consecutively, from feather to tip. § II. If each transformer has a secondary winding, consisting of two equal coils, connect the secondary coils of the two trans- formers so as to form a star-connected and a mesh-connected quarter-phase system, as in (c) and {d) of Fig. i. Measure all voltages and draw diagrams of voltages for the star and for the mesh connection. In the mesh connection, the two secondaries of one trans- former are connected as the opposite sides of a square, due "2 Aa hi — th • 6-A] GENERAL STUDY. 201 attention being given to polarity; the two secondaries of the other transformer form the remaining two sides. Before clos- ing the square, connect a voltmeter between the two points about to be connected and proceed to connect them only in case the voltmeter reads zero. This precaution should be taken in mak- ing any mesh connection. § 12. A convenient laboratory supply board is obtained from 2-phase secondaries, the secondary circuit on each phase consist- ing of four equal coils in series so as to form a 5-wire system on each phase. With the neutrals of the two phases intercon- nected,, this gives supply voltages, as Fig. 6. If the total volt- age of each phase is 220 volts, this gives 2-phase voltages as follows: 4-wire no and 220 volts; 3-wire 55, no, 77.8 and 155.6 volts; also additional single-phase ^1 voltages, 123 and 165 volts. The voltage between any two points can be scaled off from the drawing in Fig. 6, as shown in *Bj the discussion of Figs. 3, 4 and 5. When ^^<^- 6- Two-phase the transformer secondaries cannot be so ^ °" °'^^ ^"^^ ^ '° ages. subdivided, the result can be obtained by connecting across each phase of a 4-wire system an autotrans- former made of four equal coils. Verify these voltages by calcu- lation or by measurement. The preceding study has brought out the fact that in poly- phase circuits, the single-phase voltages of interconnected gen- erator or receiver coils are combined geometrically to give re- sultant voltages. Although this was shown particularly for 2-phase circuits, it will be understood to be general and to apply as well to a 3-phase circuit or to any circuit whatsoever. § 13. Addition of Currents. — Currents, also, when of differ- ent phases, are added* vectorially to obtain the resultant current. To show this proceed as follows : * Branch currents, flowing to or from a common point, always combine by addition — not by subtraction — to give the total current. See Appendix I. 202 POLYPHASE CURRENTS. [Exp. /=V/a= + /b=' A From a 3-wire 2-phase supply, connect two resistances as load, one on each phase. Measure the currents, /a and /b, in each resistance and the total current / in the common conductor. If the two currents /a and I3 differ in phase by 90°, we will have This will be true for an inductive as well as for a non-inductive load, provided the load on each phase has the same • power factor, — i. e., 6a=^0b- If £a and £b are not at right angles, or 6a and Ob are not equal, the currents /a and /b will no longer be at right angles ; the branch currents will still, however, add as vectors to give the total current, as in Fig. 7. § 14. Line drop. — To illustrate line drop, with the same circuits and re- sistances just used, insert a small additional non-inductive resist- ance in the supply wires to represent resistance in a long supply line. Construct a triangle OAB for the supply voltage and O'A'B' for the delivered voltage for the following three cases: Fig. 7. Addition of currents. B' B Fig. 8. Resistance in lines A and B, Fig. 9. Resistance in common conductor. B' B Fig. 10. Resistance in all three lines. With resistances in lines A and B only. Fig. 8; With a resistance in the common conductor only. Fig. 9; With resistances in all three lines. Fig. 10; in this third case measurements of voltages O'A and O'B are also to be taken. For the first case (Fig. 8), the supply voltages, OA and OB, '6-A] GENERAL STUDY. 203 shrink to the delivered voltages, OA' and OB'; the drop due to resistance in lines A and B is in phase with the currents /a and /b. There is the same phase difference (90° ) between the deliv- ered voltages as between the supply voltages. For the second case (Fig. 9), the line drop in the common conductor is in phase with /, and it is seen that, on account of this drop, the phase angle between the delivered voltages is greater than between the supply voltages. This, also, is true in Fig. 10. This lack of symmetry in delivered voltages is one dis- advantage of the 3-wire system; see § 3. § 15. These diagrams illustrate the topographic or mesh method for representing electromotive forces. The direction assigned to any line depends upon the sense in which it is taken. Resistance drop consumed by resistance is in phase with cur- rent; resistance drop produced by a resistance is opposite to. the current, as discussed in Exp. 4-A. It is taken in this latter sense in applying the mesh principle, — Law (i) of Appendix I. — that the electromotive forces around any mesh have a vector sum of zero and can be represented as a closed polygon. Thus, in Fig. 10, proceeding around the mesh OAA'O', we have the fol- lowing electromotive forces : OA produced by the generator ; AA' produced by resistance in line A and opposite to 7a; A'O' the counter electromotive force produced by the load (the elec- tromotive force delivered to the load being O'A') ; O'O pro- duced by resistance in the common line and opposite to I. The line drop for a single-phase circuit can be similarly repre- sented. With inductance in the lines, besides the resistance drop just discussed, there is a reactance drop at right angles to the cur- rent; this reactance drop is 90° ahead of the current when con- sidered as consumed by reactance, and 90° behind the current when considered as produced by reactance. (See § 180 and Fig. 2, Exp. 4-A, and Figs. 3, 4 and 5, Exp. 3-B.) § 16. The line drop diagram. Fig. 10, is true for any 3-wire 204 POLYPHASE CURRENTS. [Exp, system, and may be applied to a 3-phase system by making the triangles more or less equilateral. A 4-wire system or any other system can be treated in a similar manner. Furthermore, the method just discussed for treating the effect of resistance drop and reactance drop in line conductors is not limited to non-inductive loads, but is applicable as well to other loads, either with leading or lagging currents. (See § 56, Exp. 3-B.) § 17. Conclusion. — In the main it has been seen that 2-phase circuits are essentially the same as two single-phase circuits and can be so treated. Three-phase circuits are likewise essentially three single-phase circuits and the conception of polyphase cir- cuits is thus made simple. In any polyphase circuit the funda- mental principles for the vector addition of currents and electro- motive forces apply as in single-phase circuits. For 3-phase circuits, however, there are modified forms of treatment that are found practically convenient; these will now be considered. Line Vohage PART III. § 18. Three-phase Measurement. — The most important 3-phase connections (Fig. 2) are the star and delta connections, the elec- trical relations of which will first be studied. Other 3-phase connections will then be stud- ied with reference to various arrangements of transformers on 3-phase circuits. § 19. Star-connection. — On a 3-phase line, connect three approximately equal resist- ances* in star-connection; see Fig. II. Measure the line voltages XY, YZ and ZX; these are also called delta voltages * Some measurements should also be made with unequal resistances. Fig. Star- or F-connection of load resistances. "6-A] GENERAL STUDY. 205 and for clearness may be designated by the subscript d — thus, £d. When nothing further is specified than the voltage E of a 3-phase line or machine, it is this delta or line voltage that is meant. Measure* the star voltage Es (called also voltage per phase or phase voltage, § 30) from each line to the junction O, Fig. 11. Also measure the star current Is for each phase. The line cur- rent is always the star current, as is evident for this case. Compare the measured values of Ed and £s with the expres- sion (which should be proved) ■Ed=V3 -Es. § 20. Compute the power for each resistance. This is obvi- ously, as in a single-phase circuit, equal to the product of volts X amperes (for a non-inductive load), i. e., the product of star voltage and star current (£s/s) for each phase. For an inductive load in which the current lags by an angle 6, as in Fig. 12, the power for each star circuit is £s Is cos 6. When Es, Is and 6 are the same for each phase, we can multiply the power for each phase by 3 to obtain the total power; thus, But Total power = 3Es/s cos 6. £s = £D-f- V3; Fig. 12. Currents and voltages in a star-con- nected 3-phase circuit, — radial method of represen- tation. hence Total power =: Vs -Ed/s cos 6. *(§i9a). If the neutral point of the supply is available, measure the voltage between it and O, and test with a telephone as described in Appen- dix II., §44. This can be done either in connection with the present test or later in connection with Appendix 11. 206 POLYPHASE CURRENTS. [Exp. Since line voltage is Ed and line current is 7s, we may drop the subscripts and write Total power:^ 1/3 EI cos 6== Vs EI X power factor, where E is line voltage and I is line current. This is the custo- mary formula for power in any balanced 3-phase system, no . , ^, , matter how connected. In the Line Voltage -*.^ next paragraph it will be de- rived for a delta-connection. §21 Delta-connection. — Con- nect the same three equal* re- sistances in delta to a 3-phase supply, as in Fig. 13. Measure the current and voltage for each resistance, — namely the delta current /d and the delta (line) voltage Ej). Also measure the line current / and the star voltage £s, if the neutral of the supply system is accessible. It is seen, as above, Es = Eo-~ V3- Compare the measured values of / and lo with the expression (which should be proved) Compute the power for each resistance £d/d, and compare with the power found for the same resistances in star-connection. For an inductive load, we should multiply by cos 6 to obtain the true power in each resistance. If £d, /d and 6 are the same for each phase, we find total power by multiplying by 3; hence Fig. 13, Delta- or mesh-connection of load resistances. Total power = 3£d/d cos 0. But 7d = 7--V3; hence * Some measurements should also be made with unequal resistances. 6-A] GENERAL STUDY. 207 Total power = V3 EbI cos 6, = V3 EI cos 0, = V3 EI X power factor, where E and / are line voltage and line current. This is the customary power formula for any balanced 3-phase system, as has already been found for the star-connection. § 22. The currents and voltages for the delta-connection can be laid off by the radial method (see Appendix I.) from a com- mon center, giving a diagram similar to Fig. 12. Another method is shown in Fig. 14, in which the voltages are laid off as a triangle (polygon method) and the currents radially from the corners. The cur- rents in Fig. 14 are drawn as lagging. These currents are Ixy (from X to Y), lyz (from y to Z), and 7zx (from Z to X). With sign reversed, the latter becomes /xz, meas- ured from X to Z. The sum* of /xY and /xz gives /. If we wish to select signs so that the sum of these three vectors is zero, we must reverse the sign of / so as to give the line current /' ; we now have /', /xy and /xz all measured from X, so that Law (3) of Appendix I. is satisfied. § 23. Transformer-connections on 3-Phase Circuits. — Trans- former secondaries and primaries — like any generating or re- ceiving circuits — can be connected to a 3-phase circuit by A-, Y-, T- or F-connections,. shown in Fig. 2. *(§22a). The current / is the sum of /xz and /xv (both measured from X), or the difference between /zx and /xy (measured one towards and the other away from X). See Laws (,^) and (4), Appendix L Fig. 14. Currents and voltages in a delta-connected 3-phase circuit, — poly- gon or mesh method of representation. 2o8 POLYPHASE CURRENTS. [Exp. The most convenient and instructive method for studying the electrical relations of these connections is to use three trans- formers with the same ratio of transformation (say i: i), the primaries and secondaries of which can be connected in any desired manner. With three such transformers and with a 3-phase supply given, make connections in the following six ways: With three transformers: — (i) Primaries star-connected. Secondaries star-connected. (2) " " " " delta (3) " delta " " star (4) " " " " delta With two transformers : — (5) Primaries T-connected. Secondaries T-connected. (6) " V " " V In each case measure all electromotive forces and construct electromotive force diagrams, ^. I1 comparing computed and meas- ' ured results. The star- and delta-connec- tions have already been dis- cussed; the special relations of the T- and F-connections, will now be considered. *■'■ § 24. T-connection. — For the Fig. is. Relation between currents rr, ,. ., , , ,, . „ ,. J -connection, measure the volt- and voltages m a T-connection. age OZ, Fig. 15, and note that it is 86.6 when XY is 100. For a balanced load, the'three cur- rents, 7x, Iy, Iz, are equal. For a non-inductive* load, Fig. 15, the current in transformer XY is out of phase with the elctro- motive force by 30° and the power factor (cos 30°) is 0.866; * (§243). For an inductive load, the currents take the positions shown by dotted lines in Fig. is; /x is now out of phase more than 30°, and /v less than 30°. 6-A] GENERAL STUDY. 209 in transformer OZ the current is in phase with the voltage, giving unity power factor. For a current of 100 amperes, on non-inductive load, we have Volt- E / Power Factor. Watts. amperes. Transformer XY 100 100 0.866 8,666 10,000 Transformer OZ 86.6 100 I.OO 8,666 8,666 17.333 18,666 This shows that the power output of each transformer is the same ; for non-inductive load the two transformers require about 8 per cent, more transformer capacity (volt-amperes) than watts power transmitted. For delta- and star-connection, on non- inductive load, no excess of transformer capacity is needed. The T-connection is discussed further under Polyphase Trans- formation, Exp. 7-A, where it is shown (§8) that, for good regulation, the windings OX and OF on one transformer must be interspaced so as to reduce the magnetic leakage between them. The neutral point in a T-connection can be obtained by a tap at N in the coil OZ (see Fig. 9, Exp. /-A), dividing the coil into i and |. § 25. V-connection or Open Delta. — Draw a diagram similar to Fig. 15, for the F-connection, and from the power factor of each transformer show that for non-inductive* load this connec- tion requires 15J per cent, more transformer capacity than power transmitted. Obviously a F-connection can be replaced to ad- vantage by a T-connection ; even using the same two transform- ers, there will be an advantage in the change, for there will be less voltage on one of the transformers and hence less core loss. It is seen that, as a general principle, apparatus in which cur- rents and voltages are out of phase require greater volt-ampere * (§ 2Sa) . Note that an inductive load will cause the power factor for one transformer to become more and for the other less than cos 30°, which will make the regulation better on one and worse on the other. 2IO POLYPHASE CURRENTS. [Exp. capacity for the same power than apparatus in which currents and voltages are in phase. § 26. A Comparison. — In comparing the relative advantages of transformer-connections, it is to be borne in mind that three transformers (even though of somewhat smaller aggregate capacity) will usually cost more than two. The F-connection gives the least voltage per transformer and the least insulation strain, particularly if the neutral is grounded; for this reason it is to be preferred on high potential lines, say, 20,000 volts or over. On the other hand, the delta-connection has the advan- tage that, if one transformer breaks down, the remaining two will operate F-connected; at moderate voltages (say, under 20,000 volts) the delta-connection is accordingly to be preferred. In the delta-connection, if one transformer breaks down, each remaining transformer will have J instead of ^ of the whole power and will have to carry the line current instead of the delta current. By what per centages are current and power in each transformer thus increased? This increase would cause abnormal heating. For the same heating (same current) show that the two transformers F-connected will carry 57! per cent, as much load as the three delta-connected transformers. With transformers delta-connected, the voltage of the system can be increased by using the same transformers F-connected. In a new system, the delta-connection is sometimes installed with a view to changing later to a F-connection and a higher voltage. A single 3-phase transformer requires less material than three single-phase transformers of the same aggregate capacity, and is more efficient. (See Handbooks.) The three single-phase transformers may be cheaper or more readily obtained because more nearly standard, and in case of breakdown one third and not all the equipment needs be replaced ; in other respects the single 3-phase transformer is preferable and is coming more and more into use. § 27. Six-Phase Circuits. — ^A 6-phase circuit is a 6-wire cir- 6-A] GENERAL STUDY. 211 cuit, the potential diagram of which forms a hexagon. Its only use is in connecting transformer secondaries to 6-phase syn- chronous converters.* The usual and best method for obtaining a 6-phase circuit is by means of the diametrical-connection, as follows. Three transformers have primaries connected to a 3-phase circuit. The six wires of the 6-phase circuit may be represented by the apices of a hexagon; the three transformer secondaries, Fig. 2 (a), are connected so as to form diagonals or diameters of the hexagon. The three neutral or middle points of the secondaries may, or may not, be interconnected. Connect transformers in this manner, with the neutrals interconnected, and test with a voltmeter; for present purposes this one test will be sufficient. If each transformer has two separate secondaries of equal voltage, these six coils can be used as a 6-phase supply by a ring- or mesh-connection (each coil forming diagrammatically one side of a hexagon) ; or, a 6-phase supply can be obtained, by a double T or double delta, one T or delta being reversed with respect to the other. A double F-connection is the same as the diametral-connection. One advantage of the diametral-con- nection is that it gives a neutral which may be used as a " derived neutral " for a 3-wire system on the direct current service from the converter; this is particularly useful in lighting systems. PART IV. § 28. Equivalent Single-phase Quantities. — Polyphase quan- tities are sometimes reduced to equivalent single-phase values for * (§273). A 3-phase converter may be increased in rating 40 or so per cent, with no increased losses and with a corresponding higher efficiency when changed to 6-phase by the addition of three more collector rings and (if necessary) an extension of the commutator. A most valuable paper on this subject is one by Woodbridge (A. I. E. E., February 14, 1908), who states that of 1,000,000 K. W. of railway converters, one third are 6-phase ; above 500 K. W. one company makes all converters 6-phase. See also .Chap. XL, Alternating Current Motors, by A. S. McAllister, where 6-phase transformer connections are given in detail. 212 POLYPHASE CURRENTS. [Exp. simplicity in working up and comparing data relating to poly- phase machinery. The equivalent single-phase current I' (sometimes called total current) in any balanced polyphase system is the current which, multiplied by the line voltage and power factor, gives the true (total) power; hence Total power =■ EI' X power factor. For a 2-phase circuit, the equivalent single-phase current /' is evidently twice the line current. For a 3-phase circuit, the equivalent single-phase current is V3 times the line current. (In a delta-connection, it is seen that this is three times the delta current, — hence the significance of total current.) § 29. Equivalent single-phase resistance R' is the resistance which, multiplied by the square of the equivalent single-phase current, gives the total copper loss {=R'I'^). It will be found* that for star- or mesh-connection, or any symmetrical combination of star and mesh, — ^2-phase as well as 3-phase, — R' is one half the resistance measured between lines of one phase. For a 2-phase circuit, this becomes apparent upon inspection. For a 3-phase circuit, with the three equal resistances r under test connected star and connected delta, determine R' and /'; in each case compare R' with r and with the resistance measured between any two line-wires. Equivalent single-phase reactance and impedance are likewise one half the measured values between lines of one phase. § 30. Current and Voltage per Phase. — Current per phase and voltage per phase (or phase voltage) are more commonly used than equivalent single-phase quantities; the meaning is not so definite, but can generally be told from the context. The terms * See Standard Electrical Handbook; or Alternating Current Motors, by A. S. McAllister, in which equivalent single-phase quantities are exten- sively used. • 6-A] GENERAL STUDY. 213 are usually so used that the total power in a 2-phase circuit is twice the product of current per phase, voltage per phase and power factor; the total power in a 3-phase circuit is three times the product of current and voltage per phase, and power factor. In a 2-phase system, there is little chance for ambiguity. In a 3-phase system, the current and voltage per phase (as defined above) may be either the star (line) current and star voltage, or the delta current and delta (line) voltage. In either case, the total power is three times the power per phase. Using line current, we must Hse star voltage; using line voltage, we must use delta current. It will be remembered that, if line cur- rent and line voltage are used, the total power is V3 times their product multiplied by power factor. APPENDIX I. VECTOR ADDITION OF ALTERNATING CURRENTS AND ELECTRO- MOTIVE FORCES IN A NETWORK OF CONDUCTORS. § 31. Laws of Vector Addition and Subtraction. — Any hill may be considered to be up or down according to the direction in which one is walking; the difference in level may be considered positive or negative. In the same way difference of potential may be considered as positive or negative according to the sense in which it is taken — that is, according to the direction one takes in proceeding around a circuit or from point to point in a circuit. Consider a network of highways in a hilly country. If from any starting point one proceeds by any route or circuit back to the starting point, he will find himself at the original level — the plus hills and the minus hills adding up to zero. On different trips he may traverse the same hill in opposite directions, giving it one time a plus and the other time a minus sign. This would be true at any instant, even if the surface were rising and falling, as in an imaginary earthquake or on the surface of the ocean. Consider now a network of conductors. If from any starting point one proceeds by any route or circuit back to the starting point, he will 214 POLYPHASE CURRENTS. [Exp. reach the original potential ; the algebraic sum of the potential differ- ences at any instant, taken in the proper sense, adding up to zero. For an alternating current circuit in which currents and potential differences vary harmonically and can be represented by vectors, algebraic addition is used for instantaneous values and vector addi- tion for maximum or for effective values; hence, for maximum or effective values we have the modified statement of Kirchhoff's Law: §32. Law (/). Vector Addition of Electromotive Forces: Gen- eral Law. — In proceeding completely around any mesh or number of meshes in an alternating current system of conductors, the vector sum of all the differences in potential is zero; such vectors form a closed polygon. For this vector addition, electromotive forces are represented by arrows, the tip of one to the feather of the next, which must be in sequence according to the direction in which we proceed around the circuit. A coil xy may have an electromotive force represented by a vector XY, as measured from x to y. Taken in the opposite sense (by traversing the circuit in the opposite direction) the electromotive force would be YX, the same vector with arrow reversed. To illustrate* further this addition, from a point O on the side of a hill, let two paths ascend: one to the point A (elevation 100) ; the other to B (elevation 90). If a man starts at A, descends to O, ascends to B and back to A, the ascents and descents add to zero (—100; +9o;-j-io). To illustrate the special case of subtraction, if the sense or sign of one quantity be reversed: let two men start from O, one ascending to A (+ 100) and the other to B (+90). The difference in their level is now the difference between -\- 100 and + 90, which illustrates the following law : §33. Law (2). Vector Subtraction of Electromotive Forces: Special Law. — In an alternating current system, if two electromotive forces are separately measured away from a common point (as OA and OB) the difference in potential between their outer ends {A and B) will be the vector difference of the two electromotive forces {OA and OB). * For unvarying potentials or instantaneous values of varying poten- tials this is a correct analogy ; for the vector addition of varying quantities it is merely an illustration. 6-A] GENERAL STUDY. 215 The discussion of Figs. 3, 4 and 5 illustrates the application of Laws (i) and (2). The modified form of Kirchhoff's Law for current becomes : §34. Law (j). Vector Addition of Ciirrents: General Law. — At any point in an alternating current system the vector sum of the currents measured all towards or all away from that point is zero; such vectors form a closed polygon. § 35. Law (4). Vector Subtraction of Currents: Special Law. — At any point in an alternating current system where three currents come together, if one current is measured towards and the second away from that point, the third current will be the vector difference of the two. The discussion of Fig. 14 illustrates the application of Laws (3) and (4). § 36. Notation.' — There is no universally adopted notation for poly- phase circuits. The most complete and least ambiguous method is to letter every junction or point on the diagram of connections and to use two letters (as subscript if desired) in the proper sequence to designate the vector current or electromotive force between, two points. Thus, from X to, Y we may have electromotive forces XY or £xY ; in the reverse sense, YX or £yx ; similarly, we may speak of the currents XY or /xy and YX or /yx. This makes definite the direction or sign of the vector quantity in every case. In some cases, particularly the simpler ones, the complete definiteness is not needed (being unessential or obvious) and a single subscript is then simpler, as Ed, Es, I a, /b. In general the double-subscript notation is to be recommended on account of its exactness, as illustrated in the dis- cussion of Fig. 14. §37. In applying Law (i) it is necessary, in order to obtain a vector sum of zero in proceeding from a generator around a circuit and back to the generator, to take the generated electromotive forces or counter electromotive forces in each part of the circuit: thus, the electromotive force produced by self-induction 90° behind the cur- rent (not that to overcome self-induction 90° ahead of the current) ; and the electromotive force produced by resistance, in direction exactly opposite to the current (not the electromotive force to over- come resistance which is in phase with current). This becomes obvious upon inspection of the triangle for the electromotive forces 2i6 POLYPHASE CURRENTS. [Exp. in a simple circuit, the hypotenuse of which is E, one side Rl and the other side XI; the principle is applied in the discussion of Fig. lo. . § 38. Polygon or Mesh Method of Representation. — As applied to electromotive forces, there is in this method of representation a cer- tain sirnilarity between the diagram of connections and the diagram for electromotive forces. It seems a natural method to apply in many cases, as in Figs. 8, 9, 10. There is no essential difference between it and the topographic method. Law (i), above, applies directly and the electromotive forces around any mesh have a vector sum of zero, introducing arrows with feather to tip in sequence. (Compare analogy of network of highways, § 31.) As applied to currents, the three currents drawn radially in Fig. 12 may be drawn as a closed polygon. So also in Fig. 7. Compare likewise Fig. 14. § 39. Radial Method of Representation. — In this method all vectors for currents and electromotive forces are drawn radially from a common center. This method is advocated by some for all cases (Porter, Electric Journal, September, 1907), together with the double subscript notation, in order that in involved problems ambiguity can be minimized. For a star-connection the application is obvious. For a delta-connection, we have the same radial diagram as for the star-connection. See Fig. 12. A modified radial method, with vectors from several centers, is illustrated in Figs. 14 and 15, and for particular cases, as in those illustrated, possesses some advantages. § 40. Preferred Method. — It is not proposed to advocate here a par- ticular convention but rather to assist in making underlying principles clear. One may choose or develop one method and apply it in all cases;' or he may select the method which is simplest or clearest for each particular case. The important point is to see clearly the sig- nificance of whatever method is used. 6-A] GENERAL STUDY. 21? APPENDIX II. TRIPLE HARMONIC IN DELTA AND STAR CONNECTION. § 41. In a circuit supplying current to a transformer, induction motor or similar apparatus with iron, hysteresis in the iron intro- duces* in the exciting current odd harmonics of 3, 5, 7, 9, etc., times the fundamental frequency. In ■ a 3-phase system, if three transformers have their primaries either star- or delta-connected, the currents in the three transformers will have a phase difference of one third of the fundamental period. The third harmonic due to hysteresis will accordingly have the same phase in each of the three transformers. This will be seen by sketching curves for the fundamental and third harmonic, and shift- ing the curves to left or right one third of the fundamental period, which is one full period for the third harmonic. In a 3-phase system all harmonics divisible by 3, as the 9th, isth, etc., will likewise have the same phase in each transformer. For a 5- or 7-phase system, the harmonics thus appearing would be 5, 15, 25 and 7, 21, 35, etc., respectively. In an even phase (single- or 2-phase) system, even harmonics only could appear; but no even harmonics are produced by hysteresis. These facts can be shown by curves taken by the method of instan- taneous contact or the oscillograph. A set of such curves has been published and discussed by E. J. Berg (Electrical Energy, p. 154). § 42. Thirdt Harmonic in Delta-connection. — If the transformer primaries are delta-connected, the harmonics due to hysteresis for the three transformers are in phase and form a current which circu- lates around the delta but does not appear in the line. The delta current Id may accordingly be 5 or 10 per cent, more -than the line current /, divided by \/3- If H is the current (third and higher har- monics) caused by hysteresis, we have:): /d ^ V (j^ -^- Vs)" + H'- * Compare " The Effect of Iron in Distorting Alternating Current Wave Form," A. I. E. E., September, 1906, and its discussion by Steinraetz. tThe third harmonic is mentioned, being most important; it will be understood that the ninth, fifteenth, etc., are included when only the third is mentioned. t (§ 42a). If A and B are currents or voltages of any two frequencies, the total effective value is ^A' -f- S^ This is easily shown experimentally 21 8 POLYPHASE CURRENTS. [Exp. Suppose, for example, / is 173 and Id is 105 instead of 100 (an increase of 5 per cent.); then H = ^io5' — 100'' = 32. In the laboratory measure / and Id and calculate H. Compute H as per cent, of /-=- Vs; also compute the per cent, increase in Id over / H- V3- Although noticeable at no load, the percentage difference practically disappears under load, for H remains constant and hence is relatively smaller when / and Id become large. § 43. Third Harmonic in Star-connection. — If the transformer pri- maries are y-connected, the third harmonic caused by hysteresis will be in the same phase in the three transformers and will tend to flow to or from the neutral simultaneously in the three. The star voltage Es will thus be more than the line voltage E divided by V3, thus Es = '\/{E-i-\/^y-\-EH, where En is voltage due to hystere- sis. If the neutral is insulated no current due to these harmonics can flow. If there is a return circuit frdm the neutral, through ground or a fourth wire, a current of triple frequency will flow; but no current of fundamental frequency will flow in the neutral if the line voltages are symmetrical. § 44. The third harmonic in the neutral can be prettily shown in the laboratory by means of a telephone, which should be protected by a resistance in series, or in shunt, or both, or by connecting through a transformer. Connect to a 3-phase supply three transformers or other coils* with iron; the more nearly similar these are the better. Let O' be the neutral of the three coils and let O be the neutral of the supply system. (If the supply system has no neutral, one may be obtained by three y-connected resistances.) Connect the telephone between and 0'. If the coils are well balanced, the fundamental will be perhaps scarcely discernible; the third harmonic will sound very clearly an octave and a fifth {do to sol) above the fundamental; the ninth, if discernible, is the same interval above the third.. On a 64-cycle circuit, the fundamental is C with harmonics g, d", b", etc. If there is any question as to what is the fundamental, it can usually be told by listening to various apparatus in the laboratory; or by connecting the telephone, with a series resistance, to the supply circuit. by measuring the total and separate voltages when two sources are put in series. Do not short circuit one source on the other. * Shunt choking-coils used for series lighting are suitable for this. 6-A] GENERAL STUDY. 219 If, instead of coils with iron, three resistances are used, the har- monics cannot be heard; the fundamental will no doubt be heard due to lack of perfect symmetry, and will become louder if the circuits are thrown more out of balance. § 45. If an electrostatic voltmeter is available, connect ' it between O and O' (in place of the telephone) and measure the hysteresis voltage, En. Measure also from one line wire X, the voltages OX and O'X, the latter being the larger on account of hysteresis harmonics. Compute O'X from the formula O'X = VjpX) °- ■+ En" and com- pare with the measured value. It is to be noted that OX = £ -h V3 and O'X =£s. By what per cent, is £s greater than £^-V3? What per cent, is En of £ h- V3 ? § 46. With a voltmeter* measure the line voltage XYZ and con- struct a triangle as in Fig. 16. Measure also the three star voltages O'X, O'Y, O'Z and lay them off as shown, each one twice. A supply neutral is not necessary for this test. Cut out the diagram on the heavy lines and fold on the light lines, bringing the three points 0' together so as to form a pyramid. The height of the pyramid represents the voltage En due to hysteresis harmonics. § 47. The foregoing illustrates the fact that vectors in a plane can exactly represent only currents and electromotive forces which are simple sine functions; the error due to harmonics is commonly neglected. § 48. Generator Coils. — If there is a third harmonic in the generated electromotive force, with the generator coils delta-connected it cannot appear in the line but will appear as a circulating current in the delta. This may cause appreciable heating if the harmonic is large. § 49. The third harmonic can appear on the line only in case the generator coils are F-connected and have the neutral connected to ground or a 4th wire. If the line is not grounded also at the receiv- * Use an electrostatic voltmeter ; although this is not important with large transformers, it becomes necessary in case the coils or transformers are small, as the current taken by an ordinary voltmeter may cause con- siderable error. 220 POLYPHASE CURRENTS. ■ [Exp. ing end or a 4th wire return used, the potential of the line as a whole will be raised by this electromotive force of triple frequency. APPENDIX III. COPPER ECONOMY OF VARIOUS SYSTEMS. § 50. In figuring copper economy, it is to be assumed that all sys- tems compared are to have the same line loss and per cent, resistance drop. As a general principle, in any given system, the amount of copper necessary varies inversely as the square of the voltage; thus, if the voltage is doubled, the current will be halved and the copper reduced to one fourth, increasing R four-fold. This gives the same RP loss in the line and the same per cent. RI drop. Any comparison of systems should, therefore, be made on the basis of equal voltage; this may mean either the greatest voltage between any two line wires or the voltage between any wire and the neutral. This latter becomes more significant when the neutral is grounded. § 51. On the Basis of the Same Voltage £s from the Line Wire to Neutral. — On this basis all symmetrical alternating systems give the same copper economy, as will be seen from the following. Let us consider all wires to be of a given size and to carry a given current /, thus giving the same drop and loss per wire. We then have Single-phase, 2 wires : amount of copper 2 ; power = 2 Esl. Three-phase, 3 wires : amount of copper 3 ; power = 3 £s/. Quarter-phase, 4 wires: amount of copper 4; power = 4 £s/. w-phase, n wires : amount of copper n ; power = n Esl. The amount of power is seen to be proportional to the amount of copper, giving therefore equal copper economy for all systems on the basis of equal voltage between the line and the neutral or ground. § 52. On the Basis of the Same Voltage Between Line Wires. — Between line wires the voltage is 2£s for the single-'phase (or quarter-phase) system and ^/^Es for the 3-phase system. To make the voltage between line wires equal in these systems, the voltage in the 3-phase system can be increased in the ratio V3 : 2. The amount of copper can accordingly be reduced (see § 50) inversely as the square of this ratio, namely, 4 : 3. Hence, for the same line voltage, a 3-phase system requires 75 per cent, as much copper as a single- phase or quarter-phase system. 6-A] GENERAL STUDY. 221 § 53. Direct Currrent System.— A direct current system has the same copper economy as a single-phase system, when the direct cur- rent voltage is made equal to the effective (sq. rt. of mean sq.) value of the alternating voltage. If, however, the direct current voltage is increased so as to equal the maximum value of the alternating current voltage, the direct current voltage is increased in the ratio of i wJt. and the copper is decreased as the inverse square of this ratio. The direct current system then requires only one half the copper of a single-phase or two thirds the copper of a 3-phase system, on the basis of equal volt- age between wires. A direct current system would, therefore, be more economical of copper than any other system, at the same voltage. § 54. Choice of Systems. — On account of copper economy and the simplicity due to the use of only two wires, direct current would be superior to any alternating current system, if it were not for lack of simple and suitable means for transforming direct current so as to obtain the advantage of high potential transmission with low potential generation and utilization. In the case of alternating cur- rents, these means are provided for by the transformer which makes alternating current systems so flexible that they are practically always* used for long distance transmission, instead of direct current. In comparing alternating current transmission systems, the choice is to be made between single-phase — with its simpler line construction, fewer insulators, etc. — and 3-phase, requiring only 75 per cent, as much copper. If these were all the factors, single-phase transmission systems would be more common than they are, the simplicity offsetting the poorer copper economy. An important and perhaps a determin- ing factor, however, is the superiority of polyphase as compared with single-phase machinery (§ 2) ; for this reason a polyphase system is commonly preferred, quite aside from considerations of copper econ- omy. Of polyphase systems, the 3-phase system is most economical and is therefore the system in general use. * (§543)- In a few cases high potential direct current has been used for power transmission, notably in the Thury system. This is essentially a constant current system. The high potential is obtained by generators in series ; the motors are likewise in series. See Land. Electrician, March 19, 1897 ; New York Elect. Rev., January, 1901. 222 POLYPHASE CURRENTS. [Exp. Experiment 6-B. Measurement of Power and Power Fac- tor in Polyphase Circuits. PART I. GENERAL DISCUSSION. § I. Preliminary. — For measuring power in any 3-wire sys- tem, the best method is the two-wattmeter method § 23 ; for the particular case of a balanced 3-phase load, some one-wattmeter method, §§ 32-9, may be used. For measuring power in systems with more than three wires, the n — I wattmeter method of § 16 is correct for all cases; for the particular case of a balanced 2-phase load, on a 4-wire sys- tem, the method of § 10, employing two wattmeters, may be used. An unknown load should not be assumed to be balanced. It will be understood that, in cases where several wattmeters are described as being required, a single instrument may be used and shifted by suitable switches from circuit to circuit, readings being taken successively in the different positions. § 2. Separate Phase Loads. — In any single-phase system power is measured by means of a wattmeter, the current coil being connected in series and the potential coil in parallel with the circuit, as discussed in Appendix III., Exp. 'S-A. An exten- sion of this method can be applied to a polyphase system, if the phases are separately accessible so that the load of each phase can be separately measured. A wattmeter is then used for each phase load, with current coil in series and potential coil in parallel with the particular load being measured, the total power being the arithmetical sum of the several wattmeter readings. For example, to measure the power in three star-connected re- sistances on a 3-phase circuit by this method, three wattmeters would be required, each current coil carrying the star (or line) current and each potential coil being subjected to the star voltage. With three resistances delta connected, three wattmeters would &-B] MEASUREMENT OF POWER. 223 also be required, each current coil carrying the delta current and each potential coil" being subjected to the delta (or line) voltage. § 3. This method of measuring the separate phase loads is simple in principle and is commonly used on a 2-phase circuit (§6), but it is not capable of general application inasmuch as phase loads are not always separable. On a 3-phase circuit — in testing, for example, a 3-phase induction motor — it may be impossible to measure delta current or star voltage, so that some method not requiring either of these measurements becomes necessary; furthermore, the method is open to objection on account of the number of measurements required, — unless the assumption is made that all phases are alike, so that measure- ments are necessary on one phase only. § 4. Polyphase Power Factor. — ^A polyphase system is a com- bination of single-phase elements. If E, I and W are, respectively, the voltage, current and power for any separate element, the power factor for that element is W-^EI, by definition. When the separate elements or phases of a polyphase system have the same power factor, this is the power factor for the whole system. §5. When, however, the separate elements have different power factors, there is no one power factor that has a definite value or physical significance for the whole system. It is convenient, however, to obtain a kind of average power factor for the system, the value of which will depend upon the method used in its, determination.* An average power factor may be satisfactorily determined when the separate phases are nearly alike, but has little meaning when they are widely different. § 6. Two-phase Load. — Two-phase power is usually measured by two wattmeters, one on each phase, as just described. §7. When the phases are independent, as in a, Fig. i, Exp. 6-A, the measurements differ in no respect from measurements made on single-phase circuits. *(§Sa). See A. S. McAlIiser, Alternating Current Motors, p. 12; A. Burt, Three-phase Power Factor, A. I. E. E., p. 613, Vol. XXVIL, 1908. 224 POLYPHASE CURRENTS. [Exp. § 8. On a 3-wire, 2-phase circuit, as in b. Fig. i, Exp. 6-A, the same method may also be used, the two wattftieter current coils being located in the two " outer " conductors, A and B, respectively. With the wattmeters thus located, the sum of their two readings will give the true power (§23) for any load whatsoever, even when part of the load is between A and B. (These connections are seen in Fig. i, in which X and Y are the outer conductors and Z is the common conductor or return.) § 9. When the load in a 3-wire 2-phase system is balanced and there is no load between the two outer conductors A and B, one wattmeter may be conveniently used by connecting the current coil in the common conductor; one end of the potential coil is connected to the common conductor and the other end connected first to one and then to the other outer conductor. A reading* is taken in each position and the algebraicf sum gives the total power. (The connections are seen in Fig. 7, in which Z is the common conductor.) A 3-wire 2-phase circuit is Hkely not to be balanced (§ 14, Exp. 6-A) and the method should be used with caution. § 10. On a 4-wire, quarter-phase, 2-phase system, as in c and d, Fig. I, Exp. 6-A, two wattmeters, one on each phase, will give the correct power only when the load is balanced. The method may be used for testing a smgle machine, but not for measuring the power of a circuit when the character of its load is unknown. * (§9a). For a balanced load, power can be determined from a single reading of the wattmeter by connecting the current coil in the common conductor and connecting the potential circuit from the common conductor to the middle point of two approximately equal non-inductive resistances, i?i i?2, connected across the two outer conductors as in Fig. 5. A sino-le reading of the wattmeter gives one half the total power, if the wattmeter, is calibrated as a single-phase instrument with i?, and R2 connected in parallel with each other and in series with the potential circuit (§36a). See also § 33a. t (§9b). For low power factors, when e exceeds 45°, the reading of the wattmeter in one position is negative. The similar case for a 3-phase circuit is fully discussed later. 6-B] MEASUREMENT OF POWER. 225 That the method is not generally correct will be seen by assum- ing the current coils of the two wattmeters to be connected in two of the lines, as A and B; neither wattmeter would then record a single-phase load drawing current from the other two lines, A'B'. On a 4-wire system, with unbalanced load, at least three watt- meters must be used, § 16. § II. Power Factor in a Two-phase Circuit. — If E, I and W are measured on one phase of a 2-phase circuit, W -^ EI is the power factor for that phase, § 4. This may be called the cosine method for determining power factor, since W -i- EI ^ cos 6 when currents and electromotive forces are represented by sine waves. § 12. The following tangent method for determining power factor from two readings of the wattmeter will be found simple and often convenient. The current coil of the wattmeter is connected in one line of phase A ; the potential coil is connected across phase A, whose voltage is Ek- The wattmeter now reads the power volt-amperes or true watts (i) W,=E Jocose. Transfer the potential coil to phase B, whose voltage is £b. The wattmeter now reads the wattless or quadrature volt- amperes (sometimes called wattless, or quadrature, watts), (2) W^^E^I A sine. Dividing the second reading by the first, (3) wr'E.'^''^- Tan^, and hence power factor (cos 6), is determined by the ratio of the two readings. Usually Eb^Ea, so that tan 6 = IV2 -^ f^i- The power factor thus determined is the power 16 226 POLYPHASE CURRENTS. [Exp. factor of phase A; 6 is the phase difference between 7a and Ea- The method assumes that £a and £b differ 90° in phase and that electromotive forces and currents follow a sine law. The advantage of the tangent method is its simplicity and inde- pendence of the calibration of instruments. The method can be used for determining the power factor of a single-phase load, drawn from a 2-phase supply, and a somewhat similar method can be used for determining the power factor of a 3-phase load, §§28,38,41. § 13. The value of 9 and power factor can be found by the sine method directly from (2); thus, sin 6 = H^2 ~^ -Sb/a- For a single-phase or 2-phase load there is little advantage in this method, which is useful, however, on 3-phase circuits, § 43. § 14. The " cosine " method gives correct power factor by definition and is general, being independent of wave form. The "tangent" and "sine" methods are based on the assumption that voltages and currents follow a sine* law. The " cosine " and " sine " methods require carefully cahbrated instruments. § 15. The three methods are seen to be based on the relation, . power volt-amperes total volt-amperes . . wattless volt-amperes sm 6 = -. ; . total volt-amperes wattless volt-amperes tan ^= ; . power volt-amperes §16. General Method for Measuring Power; n — i Watt- meters. — This method consists in selecting any one conductor of a system and considering it as a common return for all the others. One wattmeter is then used for each conductor, except *(§l4a). With non-sine waves, the value of power factor by the tangent method would, theoretically, be a little larger than the true value by the cosine method; the value by the sine method would be a little larger than the value by the tangent method. 6-B] MEASUREMENT OF POWER. 227 this common return. No wattmeter is required for a return circuit; thus, for a 2-wire system, one wattmeter only is needed, no wattmeter being needed in the return conductor; in a 3-wire system, two wattmeters are used, none being needed in the re- turn conductor, etc. If m is the number of line conductors, n — I wattmeters are, accordingly, required. For a 3-wire sys- tem, the connections are shown in Fig. i. To measure power in any system, connect a wattmeter in every line circuit except one {considered as the return conductor^, each wattmeter having its current coil in series with one of the lines and its potential coil connected from this line to the return conductor. One less wattmeter is required than the number of line wires; the total power is the algebraic sum of the individual wattmeter readings. § 17. To read positive power each wattmeter is to be connected in the positive sense, — that is, connected in the same way as for measuring power in a 2-wire system, direct or alternating. If, when connected in this mariner, the needle of any wattmeter deflects the wrong way, the connections of its potential or current coil are to be reversed and its reading is to be considered negative. Compare § 25. § 18. This method of measuring power is absolutely general ; the current may be direct or alternating and may vary by any law whatsoever; the system may be single-phase or polyphase, balanced or unbalanced, symmetrical or unsymmetrical. As a. particular case, the two-wattmeter method for a 3-wire system is of special importance with reference to 3-phase circuits and will be considered later (§23) in detail. § 19. The foregoing method has been explained by considering one conductor as a common return for all the others, and for most purposes this explanation is sufficient. The method with n — I wattmeters can be rigorously established (§22) by first developing the method with n wattmeters, § 20. 228 POLYPHASE CURRENTS. [Exp. §20. General Method, n Wattmeters. — In any star-connected system, if a wattmeter is connected in each line — the current coil connected in series with the line and the two ends of the potential coil connected, respectively, to the line wire and to the junction or neutral point of the system — the total power of the system will be the sum of the separate wattmeter readings, as discussed in § 2. §21. This arrangement of wattmeters, however, is not limited to star-connected circuits ; nor is it necessary to have the neutral point accessible. The true power of any system whatsoever may he measured by connecting one wattmeter in each line, with cur- rent coil in series with the line and potential coil with one end connected to the line and the other end to any point P of the system, which may or may not be the neutral. To this potential point P is connected the potential coil of every wattmeter. The algebraic sum of the wattmeter readings gives the true power. A general proof of this is given in § 53 ; it can be verified by ex- periment, §§ 45-49- § 22. The fact that any point of the system may be taken as the potential point P leads to the practical simplification by which one wattmeter is omitted. In a system of line wires, a,b,c ■•• n, let the line wire n be taken as the potential point. Wattmeters A, B, C, etc., will have current coils connected in series with a, b, c, etc., and potential coils connected from a to n, from h to n, etc. Wattmeter N would, accordingly, have its potential coil connected from w to m; as both ends of the pressure co;! would thus be connected to the same point, this wattmeter would always read zero and, accordingly, can be omitted. The method of n — I wattmeters, § 16, is thus established. §23. Two Wattmeter Method for any 3-wire system. — This is the method generally used for measuring 3-phase power. Be- ing a particular application of the n — i wattmeter method, § 16, the two-wattmeter method can be applied to any 3-wire 6-B] MEASUREMENT OF POWER. 229 system* and is independent of any assumptions as to wave form or the nature of the load. § 24. The arrangement of instruments is shown in Fig. i. The wattmetersf are inserted in any two lines, as X and Y, the third wire Z being considered as a common return. ULL w. Z-2 W. Cn Fig. I. Two-wattmeter method for measuring power in any 3-phase or other 3-wire circuit. The total power is the algebraic sum of the readings of the two wattmeters. For high power factors (more than 0.5) this will be the arithmetical sum, both wattmeter readings being posi- tive. For low power factors (less than 0.5), the reading of one wattmeter is to be considered negative, the total power in this case being the arithmetical difference of the two readings, as shown later in §31. § 25. There are several ways for telling whether one reading is negative or not, the principal ones being as follows : (a) From the sense of the connections, § 17. * (§233). If each end of a 3-phase line has its neutral well grounded, it becomes virtually a 4-wire system; the ground circuit can not be neglected unless the load is practically balanced. ^ (§ 24a). Polyphase Wattmeter. — Instead of two single-phase watt- meters, a single instrument combining the two is commonly used. This consists of two wattmeters, one above the other, with the moving elements mounted upon a common shaft. The reading of such an instrument gives the total power. The electrical connections are the same as for two separate instruments. 230 POLYPHASE CURRENTS. [Exp. (b) For the given load substitute a load that is non-inductive or is known to have high power factor; if, with certain connec- tions, both wattmeters deflect properly, their readings for these connections are positive. When one connection needs to be re- versed to obtain proper deflection, one reading is negative. (c) Disconnect one* potential circuit from the middle wire Z and connect it to the outside wire, X or Y ; ii the wattmeter re- verses, the readings of one of the wattmeters must be considered negative. Method (c) can be readily applied during test, when using the two-wattmeter method on a 3-wire system, but does not apply to a system with more than three wires. Method (a) is general and can be applied to a system with any number of wires. The polarity of the wattmeter circuits may be marked once for all, instruments of one make being similar. The instruments can be properly connected in the posi- tive sensef in advance and confusion during the test avoided. §26. Two-wattmeter Method with Balanced Three-Phase Load. — As has been already stated, the two-wattmeter method is general for any kind of 3-wire circuit. Detailed proof for each particular case is, accordingly, unnecessary. A discussion of its application to measuring a balanced 3-phase load will, however, prove 'instructive as an illustration and will serve to make clear the negative reading of one wattmeter at low power factors. Furthermore, it will show a method for obtaining 3-phase power factor. § 27. Fig. 2 is the diagram for a balanced 3-phase load, it being assumed that currents and voltages follow a sine law. For unity *(§2Sa). On a 3-phase circuit it is sufficient to do this with one potential circuit only; but in general it should be done, successively, with each potential circuit, a reversal of either instrument indicating that one reading is negative. t(§2Sb). This also indicates the direction of the flow of power; see " Polyphase Power Measurements," by C. A. Adams, Elect. World, p. 143, January 19, 1907. 6-B] MEASUREMENT OF POWER. 231 power factor {6^0), the three Hne currents are shown by the heavy arrows Ix, Iy, Iz. The dotted arrows show these currents for lower power factors, ^ = 30°, ^^60" and 5 = 90°. Fig. 2. Currents and voltages in a balanced 3-phase system. If two wattmeters are connected as in Fig. i, wattmeter (i) has a current Ix in its current coil and a voltage £xz across its potential coil, the phase difference between this current and voltage being 6 — 30". The component of Ix in phase with Exz is, accordingly, Ix cos {9 — 30°) ; hence — ^writing E for Exz and / for /x^wattmeter (i) reads W^ = EI cos (e — 3o°)=-E/ (cos 30° cos + sin 30° sin ^). In a like manner, wattmeter (2) has a voltage Eyz and a cur- rent Iy, having a component Iy cos {6 -\- 30°) in phase with Eyz- Hence — writing E for Eyz and / for /y — wattmeter (2) reads W^ = EI cos (e + 30°)=£/ (cos 30° cos (9 — sin 30° sin e). Adding W2 to W.^, we have W -,_-{- W 2=^ 2EI cos 30° cos = \/2EI cos 0, 232 POLYPHASE CURRENTS. [Exp. which is seen to be the expression for the total power in a 3-phase circuit (§§20, 21, Exp. 6-A). The two-wattmeter method for a balanced 3-phase load is thus established. § 28. Power Factor. — Subtracting W^ from W^,, we have W^—W^ = 2EI sin 30° sin e = EI sin 6. Hence, by dividing, we have W^ — W, tan^ ^x + ^. V3" The value of 6 and of power factor (cos 6) for a balanced 3-phase circuit is, accordingly, determined by the tangent formula tan^=V3 W, — W, w^ + w,- The larger reading is W^, and is always positive; the smaller reading, W^, may be positive or negative. § 29. To save labor in com- putation, it is convenient to plot a curve. Fig. 3, with power factor (cos 6) as or- dinates and the ratio of watt- meter readings, W2 -j- W^, as abscissae. Points on this curve are determined by the relation W^ __ cos {6 4- 3 0°2 f^i^cos \e—zo°y 1.0 /" /' / / % ^ / D- / / y / y 11 sNe iative TTs 'osit i/e -1.00-.60 -.60 -.40 -.20 Ratio of Wattmeter Read :adrn.:;i:^'Vr°"-'"By means of this curve, the Power factor of balanced P°wer factor for a balanced load is readily determined from the ratio of the two wattmeter readings. For plot- ting the curve in Fig. 3, the following points were determined: Fig. 3, 3-pliase circuit for different ratios of wattmeter readings in two-wattmeter method. 6-B] MEASUREMENT OF POWER. 233 Wi-i-Wi — I. — .80 — .60 — .40 — .20 O +.20 +.40 +.60 +.80 +1. COS o .064 .143 .240 .359 .5 .6ss -803 .918 .982 i. Intermediate values can be found by interpolation. It is seen that the curve is not symmetrical. § 30. Errors of calibration are avoided if one wattmeter is used, successively, in the two positions to determine ^^ and W^. Since the assumption is made that the current in the wattmeter is the same for the two readings (/x = /y = /), greater accuracy is obtained if the current in the two cases is actually the same current. This is accomplished by using the one wattmeter method of Fig. 7, which is more accurate for determining power factor than is the two-wattmeter method. In either case, corrections may be made (§42) for slight variations in voltage. § 31. Negative Reading of Wattmeter. — Referring to Fig. 2, it is seen that for all values of 6 from o to 90°, the projections of /y upon £zx have the same sign; the wattmeter reading W^ is, therefore, in all cases positive. The projection of /y upon Eyz decreases as 6 increases, be- comes zero when 6^60°, and then changes sign. The watt- meter reading W^, accordingly, changes sign, being positive when 6 is less than 60° (power factor more than 0.5) and negative when 6 is more than 60° (power factor less than 0.5). In all cases W-i^ is the larger, and W^ is the smaller, reading. On non-inductive load, 9 = o and Wi^W^; each wattmeter reads half the total power. When ^ = 90°, Wj^=^ — W2 and the total power is zero. § 32. Three-Phase Power with One Wattmeter. — ^With only one wattmeter, 3-phase power can be measured by the two-watt- meter method (§23) by using suitable switches for throwing the wattmeter from one position to the other. This procedure gives the true power for unbalanced as well as balanced loads and is generally the best one to follow. The transfer of the current coil of the wattmeter from one line to another is not always convenient or possible and, when 234 POLYPHASE CURRENTS. TExp. the load is balanced, the power in a 3-phase system can be measured with only one wattmeter without such transfer by one of the following methods. § 33- With Neutral Available. — When the neutral is available, the current coil of the wattmeter can be connected in any one line circuit and the potential coil connected from that line to the neutral. For a balanced load, the total power -will be three times the reading* of the wattmeter. The power factor is equal to W-i-EI, where / is the line current and E is the star voltage. When the load is not balanced the total power will be the sum of three readings, one on each phase. § 34. With Artificial Neutral. — ^When the neutral is not avail- able, an artificial neutral can be created, as by means of three equal star-connected non-inductive resistances, R-^, R^, R^ in Fig. 4. The method of § 33 can then be applied. It is necessary that these resistances be relatively low, as com- pared with the resistance of the potential circuit Rw of the wattmeter. The current in them will then be relatively large, so that the potential of the neutral will not be dis- turbed by the connection of the potential circuit of the wattmeter. The power taken in the resistances may be in- cluded or not in the measured power as desired; correction for this power can be made when necessary. § 35. Strictly speaking R^ and R^ should each be equal to the joint resistance of R^ and Rw in parallel. In this case there is no need of making the resistances low ; this leads to the method of § 36 in which R^ is omitted entirely, that is, R^^=iaa. *(§33a). Calibration for Total Power. — In this method, or in any method depending upon a single reading, the wattmeter can be cahbrated to read total power. Fig. 4. Measuring power with one wattmeter connected to the neutral in a balanced 3-phase circuit. 6-B] MEASUREMENT OF POWER. 235 X -mMW- ^ Fig. 5. Measuring power with one wattmeter and a Y-multiplier in a bal- anced 3-phase system. § 36. With Y-Multiplier. — ^With a balanced load and with one wattmeter, the current coil of the wattmeter is connected in one line. One end of the potential circuit is connected to the same line, the other end being con- nected to the junction of two resistances i?i and R^, which are connected to the other two lines, as shown in Fig. 5. The resistances R^ and R^ are non- inductive and are each equal* to Rw, the resistance of the potential circuit of the watt- meter. True power is three times the reading of the wattmeter, cali- brated as a single-phase instrument. The resistances are some- times put up in a special volt-box or Y-multiplier for 3-phase cir- cuits; the instrument may then be calibrated so as to read total power, §33a. § 27- ^y Means of T-connection. — In a 3-phase system, with three lines X, Y, Z, connect the current coil of the wattmeter in any one line, as Z, Fig. 6. Connect the. potential coil from Z to a point 0, the middle point of a transformer coil across XY. See Fig. 15, Exp. 6-A. In any balanced 3-phase system, how- ever the load is connected, the wattmeter will now give one half the total power. (This may be seen as follows : If the wattmeter *(§36a). Provided Ri and R2 are approximately equal to each other, this same method may be used without having i?i and R^ equal to R^^, The instrument is calibrated as a single-phase virattmeter with R^ and R^ in parallel with each other and in series with i?w; a single reading then gives one half the total power. Compare §§ pa, 33a. X X Iz z-s 3 ■ of ' LA w ^ Y > Y Fig. 6. Measuring power with one wattmeter, T-con- nected, in a balanced 3- pliase circuit. 236 POLYPHASE CURRENTS. [Exp. were connected with its potential coil on the star voltage, the watt- meter would read one third the total power; with its potential increased 50 per cent. — see Fig. 2, — it will read one half the total power.) § 38. The power factor is fF -^ £oz/z. The power factor can be found from the tangent formula, by taking one reading, Wi, of the wattmeter with the connections as described and a second reading, W^, with the potential circuit of the wattmeter transferred to XY. W-i_ Eoziz cos 6 W^. § 39. Two-reading Method. — This is one of the simplest and most satisfactory methods for measuring power and power fac- tor with one wattmeter in a balanced 3-phase circuit. The current coil is connected in one line, as Z, Fig. 7, one end of the potential circuit being connected to the same line. The n r other end of the potential coil is con- w nected, successively, to X and Y, and a Y I Y rcading taken in each position. The Fig. 7. Measuring power algebraic sum of the two readings gives by two readings of one the total power. (The smaller readings, wattmeter in a balanced j^ jg considered negative whenever it 3-pnase circuit, is necessary to reverse the potential or current coil of the wattmeter to obtain a proper deflection.) § 40. The proof of the method will be seen by referring to Fig. 2, which assumes that voltages and currents follow a sine law. The two readings of the wattmeter are Wi = EI cos (e — 30°); W^ = EI cos (i9-f 30°). Hence, the sum of the two readings gives the total power, § 27. § 41. The power factor (cos 6) is determined from the tangent formula, § 28, 6-B] MEASUREMENT OF POWER. 237 By referring to Fig. 3, power factor can be found directly from the ratio W^ s- Wi. § 42. When there is an appreciable difference between the phase voltages (which we may term E^ and E^) across which the potential circuit is connected when W^ and W^ ^^e read, a more accurate value of power factor will be obtained by correcting W.^ or IV2 by direct proportion to obtain values correspond- ing to equal voltages. The ratio M-^^ -^ ^1 then becomes EJV2 -^- EJV.^. The power factor thus determined is quite accurate, being independent of the calibration of any instrument and of any slight inequality in the phases. Even for an un- balanced load, it gives accurately the value of cos 6 for Iz, where is the phase difference between 7z and the voltage OZ (Fig. 2) midway in phase between XZ and YZ. The method is more accurate with one than with two wattmeters, § 28. § 43. Power Factor by Sine Method. — The power factor of a balanced 3-phase circuit can be determined by the sine method (§ 13) with only a single reading of voltmeter, ammeter and wattmeter. The method does not require the neutral to be available, nor does it require any auxiliary resistances or other devices. Representing the three line wires as X, Y and Z, the ammeter and the current coil of the wattmeter are connected in one line, as Z. The voltmeter and the potential coil of the wattmeter are connected across the other two lines, X and Y. The watt- meter reading gives the wattless or quadrature volt-amperes, W^ExyIz sin 6, from which d and cos 9 are determined. 238 POLYPHASE CURRENTS. [Exp. PART II. MEASUHEMENTS. § 44. Many of the methods just described for measuring poly- phase power and power factor can best be taken up as occasion arises for their use. Without undertaking in the present experi- ment to subject all of these methods to test, it will be well to select a few of them for trial in the laboratory in order to illus- trate and make clear the methods as a whole. For this the following tests are suggested. § 45. Verification of Methods for Measuring Polyphase Power. — With a single-phase non-inductive load, forming a 2-wire sys- tem, measure the total power with two wattmeters. Each line is to contain the current coil of one wattmeter, the potential coil of which is connected from the line to a common point P, as in § 21. The experiment consists in connecting P to different parts of the circuit, of various potentials, and noting that the algebraic sum of the two wattmeter readings is constant. When the power indicated by one wattmeter becomes greater, as P is changed, the power indicated by the other wattmeter be- comes less.* § 46. For example, let the supply lines be a-^a^, as in Fig. 6, Exp. 6-A. Connect P, successively, to points of different po- tential, as fli, a^, the neutral O, A^ and A^, these points being all on phase A. When phase 5 of a two phase supply is avail- able, proceed, also, to connect P successively to points 5^, b^, &2, B^ on phase B. § 47. Repeat with an inductive load. §48. Repeat in some modified manner, as by using Oj&i as supply lines and connecting P, successively, to various points as described above. § 49. When points, as in Fig. 6, Exp. 6-A, are not available, a resistance can be bridged across the circuit and the point P * A positive reading decreases ; a negative reading increases. b-B] MEASUREMENT OF POWER. 239 connected to different points on this resistance. The load re- sistance itself can be thus utilized. The experiment might be extended to using 3 wattmeters on a 3-wire system, 4 wattmeters on a 4-wire system, etc., but this seems hardly necessary. The method of n wattmeters, n — i wattmeters and two wattmeters may, in this way, be experi- mentally verified. § 50. Two-phase Power Factor. — From one phase, A, of a 2-phase supply draw a single-phase load. Take measurements with a voltmeter, ammeter and wattmeter and determine the power factor by the " cosine method," § 14. Transfer the voltmeter and potential coil of the wattmeter to the other phase, B, and determine the power factor by the "sine method," §13, and by the "tangent method," § 12. §51. Three-phase Power and Power Factor. — With a 3-phase balanced load supplied from a 3-phase circuit, take two readings of a wattmeter connected as in Fig. 7. Determine the total power ; calculate the power factor by the tangent formula, § 28, and by the ratio of wattmeter reading. Fig. 3. § 52. Transfer the potential coil of the wattmeter to the third phase, so as to read the "quadrature" volt-amperes; take the necessary readings of the wattmeter, voltmeter and ammeter, and determine power factor by the " sine method," § 43. APPENDIX I. MISCELLANEOUS NOTES. § 53. General Proof. — In any system, with any number of con- ductors a, b, c, etc., let the instantaneous values of the currents in these conductors be ia, H, ic, etc. Designate by e„, ej,, e^, etc., the instantaneous values of the potentials of the several conductors. The currents and electromotive forces may vary in any manner what- soever. There, is no limitation as to the arrangement or method of connection of the generator and receiver circuits. 24° POLYPHASE CURRENTS. [Exp. The total power at any instant is (i) w = eaia + eiij, + ecic ■ . ■ =%ei. Let ep be the instantaneous potential of any point P of the system. Since it is known that S» = o, it follows that (2) V"o+V'i> + Vo • • ■ =ep%i = o. Since (2) is equal to zero, it may be subtracted from (i) without affecting its value ; hence (3) ii'={ea — ep)ia+ (ei — ej,)it,+ (e„ — ej,)ia . . =-S,{e — ej,)i. The total power at any instant is seen to be the sum of the products of the instantaneous currents in each conductor and the instantaneous differences of potential between the respective conductors and the point P. The mean power W is found by integrating the instantaneous power over a time equal to one period, T, and dividing by T. ^= |'X%a-^|.>y'+ tSo^^'"-'^^'^'^^^ YS^^'-'i>¥*- But each one of these terms represents the power, as read by a wattmeter with current coil in series with one conductor and with potential coil connected from that conductor to the common point P, and the total power is the sum of the several wattmeters so con- nected. For an w-wire system, n wattmeters are required, the total power being When the point P coincides with one conductor, the wattmeter for that conductor reads zero and can be omitted; « — i wattmeters are then required. The method for n wattmeters, for n — i wattmeters, and for two wattmeters, is, accordingly, proved without reference to wave form or the character of the load. This general proof was first given by A. Blondel, p. 112, Proceedings International Electrical Congress, Chicago, 1893. CHAPTER VII. PHASE CHANGERS, POTENTIAL REGULATORS, ETC. Experiment 7-A. Polyphase Transformation. § I. Possible Kinds of Transformation. — The transmission of power in a single-phase system is pulsating, no matter what the character of the load. This can be readily seen by sketching assumed curves for the instantaneous values of an alternating electromotive force and current, and plotting the products of the ordinates from instant to instant as a power curve.* In a bal- anced polyphase system, however, the pulsations of power in the different phases are seen to so combine that the total transmis- sion of power is uniform,t without pulsation. (See §2, Exp. 6-A.) § 2. Polyphase to Single-phase Transformation not Possible. — In a transformer, neglecting the slight modification due to losses, the power given into the primary at any instant is equal to the power taken out of the secondary at that instant. It is not pos- sible, therefore, simply by means of transformers to change a * (§ la). The area included between the power curve and time axis rep- resents energy, this energy being positive (supplied to the line) or nega- tive (returned from the line) according to whether the current and electro- motive force have, at the time, like or unlike signs. It is instftictive to sketch curves for currents differing in phase from the electromotive force by 0. 45 and go degrees. t (§ lb). This can be shown for a 2-phase system by drawing, for each phase, sine curves for electromotive force and current and plotting the product as a power curve. Two power curves are thus obtained, one for each phase, and it will be seen that the crests of one correspond to the hollows in the other, the algebraic sum of the two power curves being constant. The sum of the three power curves for a 3-phase circuit can be shown to be constant in the same way. 17 241 ^42 PHASE CHANGERS, ETC. [Exp. pulsating into a non-pulsating transmission of power, or vice versa. It is accordingly not possible, by means of transformers, to draw single-phase current from a polyphase system and draw from the several phases equally, so that the flow of energy is non-pulsating. To accomplish such a transformation, use is made of a motor-generator* consisting of a polyphase motor driving a single-phase generator. The moving parts act as a flywheel, storing and restoring kinetic energy, thus accounting for the momentary difference between the pulsating output and non-pulsating input of electric energy. This method is advo- cated for running single-phase railway feeders from a polyphase transmission line. § 3. Single-phase to Polyphase Transformation not Possible. — It is likewise not possible, by means of transformers, to change a single-phase into a balanced polyphase system. This too can be done by means of a motor-generator, or by running a poly- phase induction motor on a single-phase circuit, — a 2-phase or a 3-phase motor giving 2-phase or 3-phase currents. Various sta- tionary phase-splitting devices will give difference in phase suffi- cient for starting induction motors on single-phase circuits, but such devices cannot give balanced polyphase currents. § 4. Polyphase Transformation Possible. — It is possible, how- ever, by various arrangementsf of transformers to change from one balanced polyphase system to any other balanced polyphase system, the flow of energy in each system being uniform. This is termed polyphase transformation and its study is the object of this experiment. The various methods of polyphase trans- formation are similar in principle, use being made of the factj * Such a motor-generator has been installed in the chemical laboratory of Cornell University to supply 2,000 or 3,000 amperes of single-phase current for the electric furnace. t(§4a). Two transformers, only, are necessary; but more than two are used in some arrangements, as Fig. 4. X Fully discussed in Exp. 6-A. 7-A] POLYPHASE TRANSFORMATION. 243 TWO PHASE PHASE A A A' that, if two coils with electromotive forces differing in phase are connected in series, the electromotive force across the two coils is the vector sum (or difference) of the two separate electro- motive forces. A resultant electromotive force of any desired phase can thus be obtained from a polyphase supply by means of two transformers. The transformation from 2-phase to 3-phase, or vice versa, is most important on account of the copper economy* in 3-phase transmission and the sometime advantage of 2-phase generation or utilization. § 5. Two-phase to Three-phase Transformation (and vice versa) by T-connection. — This method, first published by Mr. C. F. Scott, is shown diagrammatically in Fig. I, in which A and B are the two phases of a 2-phase system; X, Y and Z represent the three line wires of a 3-pha&e system. Let the transformation be from 2- phase to 3-phase. Two transformers are used. One has a primary AA' on phase A of the 2-phase system and has a sec- ondary (XY) wound, let us say, for 100 volts with a middle tap at O, dividing the coil into two parts of 50 volts each. The second transformer has a primary BB' on phase B of the 2-phase system and has a secondary (OZ) wound for 86.6 volts (86.6=100 X 4V3). which has one end connected, as shown, to the middle tap of the first trans- former. It will be found that the three vohages, XY, YZ and ZX, are equal and differ in phase from each other by 120°, thus making a 3-phase system. These voltages are represented in Fig. 2. They should be interpreted as in Exp. 6-A; see also Appendix I. to this experiment. * See Appendix III., Exp. 6-A. c •-ry55ffVT5555">-« ? in ca < u 0- < PB' Fig. I. Transformation from 2-phase (.AB) to 3- phase (XYZ), or vice versa. 244 PHASE CHANGERS, ETC. [Exp. Y^X/ Fig, ♦2. 2. Voltage and cur- rent relations. § 6. This method of transformation is reversible ; i. e., if a 3-phase system be connected (see Fig. i) to XYZ as primary, 2-phase circuits may be taken from A A' and BB' as secondary. § 7. Double Transformation. — In Fig. 3 is shown a double transformation, from the 2-phase gener- ating circuits A, B to the 3-phase trans- mission circuits X, Y, Z, and from these to the 2-phase receiving circuits A, B. The receiving circuits, A and B, may be used together, as for operating polyphase motors, or separately as for lighting. § 8. As a further explanation of the T-connection, referring to Fig. 3, suppose the connections OZ', OZ' were left out and that, instead, a fourth wire 2' (not shown) were used to connect Z' and Z' ; each phase would then have its independent 2-wire transmission circuit, — ^wires xy for phase A and zz' for phase B. In making the T-connection of Fig. 3, the fourth wire z' is omitted* and in its place use is made of the two wires x and y, acting in parallel as a single conductor. The current from the coil ZZ' flows to O and divides, passing through OX and OY differentially, so as to have no magnetizing effect on the core of XY. With respect to the current from Z', the two parts of the coil XY are wound non-inductively. They should be inter- spaced so as to have the least possible magnetic leakage and con- sequent leakage reactance, which would give poor regulation on phase B. This precaution is necessary in winding any T-con- nected transformer. The regulation of phase A and of phase B are as independent of each other with three wires (Fig. 3) as they would be with four wires making separate circuits ; phase B may have a heavy *(§8a). There is obvious copper economy in this case in changing from a 4-wire 2-phase to a 3-wire 3-phase transmission; see Appendix IH., Exp. 6-A. 7-A] POLYPHASE TRANSFORMATION. HS motor load with 50 per cent, drop, while A has a lighting load with, say, 2 per cent, drop, unaffected by the starting and stop- ping of the motors on B. They are absolutely independent of each other. § 9. Composite Transmission. — If the phases A and B, Fig. 3, were generated and utilized separately, it would not be necessary for B to differ from A A'gg JljO °MW ll ^y ninety degrees; B r could have any phase, even the same phase as "f^^^"^ "^ A. Again A and B Fig 3. Two-phase generator and receiving might be of different circuits with 3-phase transmission. . ■ r , .1 frequencies ; m fact they can be treated as two independent transmission systems* whether of the same or of different frequencies. In the same way a direct and alternating current can be combined with economy of copper and independence of regulation. § 10. Test.- — -First note the single-phase transformations which can be made with the transformers to be used, and determine whether or not the transformers are suitable for the purpose. Connect the transformers so as to transform from a 2-phase system to a 3-phase system and make measurements of the primary and secondary line voltages, and the voltage of the T-connected coil, checking all by computation. Make corresponding transformation from a 3-phase to a 2-phase system. If the transformers are provided with two sets of coils, for parallel and series connection, make the polyphase transforma- tions with all possible voltage ratios. Compute the volt-ampere *(§9a). Various methods of composite transmission will be found in the followiiig: Elect. World, February 28, 1903,' pp. 347 and 3Si, Vol. XLI., No. 9; Am. Electrician, April, 1903, pp. 189 and 177, Vol. XV., No. 4; Elect. Review (New York), March, 1903, p. 362, Vol. 42, No. 11; Elect. Age, March, 1903, p. 179, Vol. XXX., No. 3; Mill Owners, April, 1903, p. 14- 246 PHASE CHANGERS, ETC. [Exp. capacity for the 3-phase side of each transformer when the total power output, on non-inductive load, is 100 watts ; see § 24, Exp. 6-A. § II. Instructions for Special Transformers. — These instructions relate to two transformers, DEF and a^SyS. Each transformer has two primaries, which may be connected in series or in parallel. The windings are as follows: Primary a, no (or 165) vojts. Primary D, no (or 165) volts. Primary |8, no (or 165) volts. Primary £, no (ori6s) volts. Secondary y, 368 (or 55) volts. Secondary P, 63.5 (or 95.25) volts. Secondary 8, 36! (or 55) volts. The first number gives normal voltage for highest efficiency at 60 cycles ; the number in parenthesis is 50 per cent, above normal voltage. These transformers were specially made for use at either voltage. With D and E in parallel on one phase, and a and p in parallel on the other phase of a 2-phase system, connect F to the middle point of y and 8 connected in series, thus making a 7"-connection. From ^-phase circuits of no volts (and also 165 volts) obtain 3-phase secondary voltages by computation and measurement. Make the primary connections from no-volt 4-wire 2-phase system, and also from I lo-volt 3-wire 2-phase system. Repeat with primaries in series instead of in parallel ; compute and measure secondary 3-phase voltages. Perform corresponding transformation from no-volt 3-phase to 2-phase. What two 2-phase voltages can be thus obtained? § 12. Monocyclic Transformation. — In the monocyclic system (no longer being installed) a single-phase voltage is combined with a quadrature voltage of one fourth its value ; thus, in Fig. 6, Exp. 6-A, a monocyclic voltage is obtained from A-^A^b^. It is an unsymmetrical 2-phase system. If two i : i transformers are used, the primary of one being connected to A^b^, of the other to A^b^, the secondaries (with two ends together for a 3-wire system) will give a monocyclic voltage the same as the primary. This secondary voltage is an open delta with one side reversed. Test this with a voltmeter and draw a diagram of voltages. 7-A] POLYPHASE TRANSFORMATION. 247 Reverse the primary or the secondary coil of one transformer; the secondary voltages now form an open delta, making very nearly an equilateral triangle, and hence forming a nearly sym- metrical 3-phase system. Compute these voltages and verify by measurement. This method was used for obtaining polyphase current for operating 3-phase motors upon what was initially a single-phase circuit with a so-called "tea- zer" circuit {b^) added. It was introduced by Steinmetz; the name indicated a pul- pj(,_ ^_ Transforma- sating (monocyclic) flow of energy instead tion from 3-phase of a non-pulsating (polycyclic) flow. The JJ°JJ *° ^"P''^" transformation is instructive, even though its introduction has been discontinued. § 13. Miscellaneous Transformations. — Several other trans- formations are here indicated. Try these by experiment, so far as time and facilities permit. Compute for each case the excess of volt-amperes over watts ; see § 24, Exp. 6-A. § 14. Fig. 4 shows a method for transforming from 3- to 2- phase, with three transformers. The primaries are connected to a 3-phase supply. The secondaries of the first two transformers are OX and OY. The third transformer has two secondaries, whose vol- tages are AX and YB. What must be the values of these, in order that AOB will give true 2-phase voltages? Fig. s. Trans- § IS- Fig- 5 shows a method, occasionally formation from 3- of laboratory use, of using three auto-trans- p ase t ^ ^ ° ^' formers for ^-phase transformation from phase (xys). ^ '^ XYZ to xys. What ratio of transformation will be obtained by using three auto-transformers with a middle tap? § 16. Fig. 6 shows two F-connected auto-transformers for 3-phase transformation from XYZ to Xyz. This is a method z 248 PHASE CHANGERS, ETC. [Exp. commonly used for obtaining a low starting voltage for 3-phase motors and converters. The taps y and z can be located where desired. It is to be noted that the voltage is changed, but not the phase. A third auto-transformer, YZ, A might be used, with a tap at O. Although y/-..\z better for continuous operation, this would / \ have the disadvantage of requiring an addi- Y Q ^2, tional auto-transformer; furthermore, this ar- FiG. 6. Auto- rangement could not give less than half vol- transformers on 3- ^^gg^ ^jj^ would make a reversal of phase in phase circuit. , . , changmg from low to high (startmg to run- ning) voltage, which is not desirable in starting a synchronous machine from the alternating current end. § 17. Two-phase to Six-phase Transformation. — The most practical method for this consists in transforming from a 2-phase primary circuit to two sets of T-connected secondaries, one set being inverted; (thus T and I,). Two or four transformers can be used. It is not necessary to make this transformation in the laboratory. (Detailed connections are given in McAllister's Alternating Current Motors; see also § 27, Exp. 6^A.) APPENDIX I. MISCELLANEOUS NOTES. § 18. Further Interpretation of T-connection. — A general discussion of the vector combination of electromotive forces is given in Exp. 6-A (particularly Appendix I.), and the general principles there given can be applied to the T-connection. The following is a more detailed discussion of this particular case. The electromotive force of any alternating current coil may be repre- sented by a vector in a certain direction. If this coil is the secondary of a transformer connected to a secondary line, as in the present case, the electromotive force impressed upon this line will be represented by the same vector. If in connecting the coil to the line the termi- nals are reversed, the vector representing the electromotive force 7-A] POLYPHASE TRANSFORMATION. 249 100 vv '00 xn 50 ^ YO OX ^ Fig. 7. Two senses in which vectors can be considered. the impressed upon the line is likewise reversed. Thus, in Fig. I, the electromotive force of the coil XY may be a vector XY ; when it is connected in the opposite sense, the electro- motive force of the coil YX is the vector YX. Fig. 7 shows these vectors for the secondary- coils of Fig. I. From these elementary prin- ciples can be shown the delta and the star equivalents of a T-connection. § 19. Delta Equivalent of T-connection. — ■ The electromotive forces between the terminals XYZ of Fig. I or Fig. 2, should be considered in a certain order, XYZ or ZYX. Let us consider them in XYZ order, as shown in Fig. 8. Going from X to Y, we have the vector XY as shown. From Y to Z, we have the vectors YO and OZ (compare Fig. 7) which combine to give YZ. From Z to X, we have the vectors ZO and OX which com- bine to give ZX. The three resultant vectors are thus shown to be equal and to differ in phase by 120". The line voltage, thus obtained by the T-connection, is accordingly the same as would be obtained by three 3-phase genera- tor coils, XY, YZ, ZX, connected in delta. § 20. Star Equivalent of T-connection. — Suppose the neutral point A^ (either actual or imaginary) in the coil OZ divides its voltage into i and I; thus, in Fig. 9, we have 0N = 28.9 and NZ = 57.7, with arrows down ; NO = 28.9, with arrow up as drawn. From the neutral A'' we have the vector NY, the resultant of NO and OY; and NX the resultant of NO and OX. It follows that NX, NY and NZ are each equal to S7-7 volts (57-7= 100 -f- Vs) and differ in phase from each other by 120°. The line voltage, thus obtained by the T-connection, is the same as would be obtained by three 3-phase generator coils NX, NY, NZ, when star- connected. Fig. 8. Delta equivalent of T-connection. Fig. g. Star equivalent of T-connection. 25° PHASE CHANGERS, ETC. [Exp. Experiment 7-B, Induction Regulators.* § I. Types of Potential Regulators. — When a generator sup- plies current — either alternating or direct — to a single line, the desired voltage at a distant receiver can be maintained by vary- ing the excitation of the generator, either by a hand-operated rheostat or by some automatic device as the Tirrell regulator (§3a, Exp. i-B). This is also accomplished, to a certain ex- tent, by a compound winding (§4, Exp. i-B) or composite winding (§ iia, Exp. 3-A) on the generator. When, however, a generator (or several generators in paral- lel) supplies several lines or feeders, with independently varying loads, this simple method of regulation is no longer possible; for at any particular time the voltage on one feeder may be too high, while the voltage on another feeder is too low, and there is no change of station voltage which can be made which will bring the delivered voltages on all feeders to their proper values. In direct current distribution systems, the proper voltage can be approximately maintained by inter-connecting the various feeders and proportioning the amounts of copper according to average load conditions. In large stations, a step further is taken by maintaining in the station several sets of bus bars at different voltages, so that feeders may be supplied with the proper voltage according to conditions, long feeders being sup- plied with a higher voltage than short ones. Use is also made of auxiliary batteries, motor-driven boosters, etc. In early sta- tions, wasteful series resistances in each feeder were sometimes used. § 2. In an alternating current system, the most satisfactory resultsf are obtained by the use of a potential regulator in each * Note that an induction motor with wound secondary can be used as an induction regulator ; see § 4. t (§2a). Series resistances and reactances have been used for this pur- pose. To use the former is not good practice on account of the energy ■V-B] INDUCTION REGULATORS. 251 feeder. The potential regulator is a variable-ratio transformet or auto-transformer used to raise the voltage as a booster, 01 to lower the voltage as a negative booster. The regulator may be operated either manually, or automatically by means of a small motor which is controlled* by potential wires from any desired point in the system. Alternating current potential regulators are of two types, the step-by-step regulator and the induction regulator. § 3. The Step-by-step Regulator. — ^The step-by-step regulator is an auto-transformer (or transformer) with switching arrange- ments for changing the number of turns. In principle it is the same as any auto-transformer (Exp. 5-A). In practice, the switching arrangement may consist of a number of individual knife switches, but more usually is either of a drum or a dial pattern, operated either manually or automatically. In the drum or dial pattern, resistance or reactance in the contact leads is sometimes used so that the contact arm can temporarily bridge two contact points without disastrous short circuit. The step-by-step regulator is not easily made automatic. The contacts deteriorate, even when arcing tips are used, and hence this type of regulator is better for occasional than for constant adjustment. For continuous automatic adjustment, the induc- tion regulator is generallyf used. § 4. The Induction Regulator. — ^An induction regulator is a stationary transformer with a movable primary or secondary which may be set in different positions for obtaining potentials wasted. Reactances are satisfactory for some cases ; to be effective, how- ever, they must be large and expensive. * (§2b). The motor may be either direct or alternating and is usually controlled through a relay, one form oi Tirrell regulator being made for this purpose. t(§3a)- In cases where very rapid continuous automatic adjustment is required, the induction type can not be used on account of the heavy mass to be moved. An automatic regulator of the step-by-step type is better for this rapid adjustment, because the moving part is only a light contact arm. 252 PHASE CHANGERS, ETC. [Exp. of different values or of different phase. A form* of apparatus in common use is essentially an induction motor with wound secondary brought out to terminals, and any induction motor so constructed can be used as an induction regulator. Such an apparatus may be used : 1. As a single-phase potential regulator; used on lighting feeders. 2. As a phase shifter; used in laboratory testing. 3. As a polyphase potential regulator ; used on polyphase lines, particularly in supplying current to synchronous converters. §5. (i) Single-phase Potential Regulator. — Supply the primary (or one phase of the primary if a polyphase induction motor is used) with a constant single-phase voltage not exceeding the normal voltage of the apparatus. PosuUMofBata, The Secondary voltagc may be varied by turn- Fig. I. Use as ing the rotor by hand to any desired position, smg e-p ase rans- ^^^ current may be drawn up to the full-load former ; secondary "^ -^ voltage for differ- rating of the Secondary. The apparatus is ent positions of ro- used in two ways : (a) as a transformer with primary and secondary not connected together ; {b) as an auto-transformer with the two coils connected, as in Fig. 2. § 6. (a) Use as Transformer. — Place a voltmeter across the open secondary and revolve the rotor step by step, so that the secondary potential changes between zero and a maximum. (On open circuit the data for methods (a) and (&) can be taken * (§4a). An earlier form of regulator had stationary primary and sec- ondary coils located at right angles, and a movable iron core which formed part of the magnetic circuit and permitted more or less of the primary flux to pass through the secondary. This device is sometimes referred to as a " magnetic shunt." For a description of different forms of regulators, see: "Alternating Current Feeder Regulators," by W. S. Moody (a paper before the Toronto Section, A. I. E. E., February, 1908) ; " Alternating Current Potential Regulators," by G. R. Metcalfe, Electric Journal, August, 1908. 7-B] INDUCTION REGULATORS. 253 simultaneously.) Plot a curve, as Fig. i, showing the secondary- potential for various angular positions of the rotor. Note the ratio of maximum secondary to primary potential. § 7. (b) Use as Auto-transformer. — The second and commer- cially preferred method for use as a single-phase regulator is to supply the primary with single-phase constant voltage as before, and to connect the secondary in series with the load, as in Fig. 2, POSITION OF ROTOR Fig. 2. Connections. Fig. 3. Delivered voltage. Single-phase potential regulator used as auto-transformer. SO that the delivered potential E taken from the machine, now acting as an auto-transformer, is equal to the primary potential El, either increased or decreased by the potential E^ of the sec- ondary. This is either additive or subtractive, the apparatus being a booster or a negative booster, according to the position of the rotor. In this manner, the potential may be varied be- tween the limits of £1 -j- E^ and E-^ — E^. Measure E^, E^ and E. Plot a curve, as in Fig. 3, showing the delivered potential for various positions of the rotor. Compare E with the algebraic sum of £1 and E^- What relation is there between the curves in Figs. I and 3? § 8. /4 Comparison. — In method (a), the output of the regu- lator is equal to the volt-ampere capacity of the secondary; in method (b), the output of the same regulator is much greater. Take, for example, a regulator with primary for 100 volts X 100 254 PHASE CHANGERS, ETC. [Exp. amperes, and secondary for lo volts X looo amperes. In method (o), the secondary output is limited to lo volts X lOOO amperes, or lo kilowatts. In method (b), the potential may be varied from 90 to no volts, which with 1000 amperes gives an output of about 100 kilowatts. In practice, method (b) is therefore used; in the laboratory, method (a) is often convenient when the range of delivered voltage desired does not exceed E^. §9. Again, to take care of a certain load as a transformer, the regulator must have a capacity {EJ^) equal to the load, 1 2 being load current and E^ being load voltage. As an auto-trans- former, the regulator will have a capacity {EJ^) which is much smaller, I^ being load current and E^ the increase or decrease of voltage (see Fig. 2) ; thus, if the voltage is to be raised or lowered 10 per cent., the capacity of the regulator needs to be only 10 per cent, of the full load of the feeder. § 10. Further Experiments. — The regulator may be tested under load, either inductive or non-inductive, as in Exp. S-A; or its performance can be predetermined, as in Exps. S-B and S-C. The air gap necessitates a larger magnetizing current than in a trans- former with a closed magnetic circuit; and, on account of larger leakage reactance, gives poorer regulation and a smaller diameter to the circle diagram. § II. Tertiary Coil. — ^As the secondary coil moves away from the influence of the primary and comes more nearly to the neutral position, it includes less and less of the primary flux; the secondary leakage flux now causes the secondary to act more and more as a choke-coil in series with the load, thus giving a low power factor. This has been overcome by a short- circuited tertiary coil, wound midway between the primary windings, so located that it is not cut by the primary flux. As the secondary moves away from the influence of the primary, the short-circuited winding comes into play, acting similarly to a short-circuited secondary on a transformer, so that the choking effect of the secondary or series coils becomes less and less, and is practically zero in the neutral position. (See citations, §4a; also Standard Handbook, 6-158.) In a polyphase induction regulator no tertiary coil is needed {Standard Handbook, 6-161). ' § 12. (2) Induction Regulator as Phase-shifter. — Supply the primary with polyphase current at normal constant voltage. The secondary voltage will be found to be constant in value for all 7-B] INDUCTION REGULATORS. 255 positions of the rotor (instead of varying as in the preceding tests), but to be of varying phase, having a definite phase posi- tion for each position of the rotor. This should be explained and demonstrated experimentally. To do this, connect one primary circuit and one secondary cir- cuit in series as before, measure E^ and E^ separately, and E the sum of the two for each position of the rotor, and construct triangles on a common base £i, as in Fig. 4, which illustrates the varying phase* of E^. Observe the relation between mechanical and electrical degrees, noting, for example, the mechanical angle through which the rotor is „ ° Fig. 4. Voltage relations as phase turned to shift the phase of E^ shifter. by 45 electrical degrees. Although of little commercial use, this method is extremely useful in the laboratory. If the primary supply is symmetrical and of constant voltage, the secondary voltage on open circuit will be constant and its phase angle will vary exactly with the position-angle of the rotor, which can be read with a suitable scale. A secondary load will, however, distort these conditions, so that the scale reading will not give the phase exactly. The varying resultant potential E, in Fig. 4, shows that with polyphase supply the apparatus can also be used as a potential regulator, to be discussed in the next paragraph. § 13. (3) Polyphase Potential Regulator. — The primary is supplied, as in (2), with polyphase current at normal constant voltage. The secondary coils for each phase must be separate from each other, one secondary coil being connected in series with each delivery circuit. For a 3-phase regulator (or 3-phase motor used as a regulator) the connections are shown in Fig. 5. The supply circuit is connected to the terminals i, 2, 3 of the primary — ^which may be star-connected, or delta-connected as * This can also be shown by a phase-meter. 256 PHASE CHANGERS, ETC. [Exp. shown. The deUvered currents are taken from a, b, c. The voltage relations are shown in Fig. 6, where a', b', c' gives the maximum delivered voltage. As the rotor is turned, this becomes a", b", c", etc., until the minimum a'", b"', c'" is obtained. <>>— -^ Fig, 5. Connections. Fig. 6. Voltage relations. Polyphase potential regulator : supply voltage, i, .i, 3 ; secondary coils, *■, y, z, in series with load ; delivered voltage, ahc. With a voltmeter, show that the secondary voltages x, y and s do not change in value with a change in position of the rotor; also show that the three delivered voltages, ab, be, ca, are sub- stantially equal for any one position of the rotor. (That x, y, z do not change and ab, be, ea change simultaneously, as the rotor changes, is well shown by incandescent lamps.) Measure one delivered voltage, as ab, for different rotor posi- tions, noting particularly the positions for maximum and mini- mum values, and plot a curve, as in Fig. 3. Construct a diagram to scale, as in Fig. 6, making the triangle I, 2, 3 equal to the primary voltage; the circles have radii equal to the secondary voltages, x, y and z. From this diagram pick off values of delivered voltage, a'b', a"b", etc., for different rotor positions and plot these values as a second curve, to be compared with the first curve already plotted from measurement. The limiting values of delivered voltage are shown to be E^ ± 2E^ cos 30°. CHAPTER VIII. INDUCTION MOTORS. Experiment 8-A. Preliminary Study of an Induction Motor and the Determination of its Performance by Loading. / PART I. INTRODUCTORY. ^ § I. Use. — The induction motor is the form of alternating cur- rent motor in most general use. It is practically a constant speed motor, the speed at full load being a few per cent, less than the speed at no load, and in this respect it resembles the shunt motor. On account of its simplicity and nearly constant speed, it is well adapted for operating machinery, but is not so well adapted for use in traction,* or similar service, where there is frequent accel- eration. (For variable speed motors, see §59.) Induction motors may be single-phase or polyphase, large motors being generally 3-phase or 2-phase. ,_, D- --^ 1 A N • >B \ ^ Each of these members is built of laminated iron and has on it a winding disposed in slots. These windings are termed the primary and the secondary, for the induction motor is a form of transformer. The primary receives current directly from the line, while the secondary is short-circuited upon itself and has no electrical connection with the line or with the primary, the current in it being set up by induction, as in a transformer. Usually, the primary winding is on the stator and the second- ary is on the rotor. This gives the simplest form of motor; the primary can be connected directly to the supply line and there is no need of brushes or slip rings. Sometimes, however, the secondary is on the stator and the primary is wound on the rotor, the current from the line being in this case intro- duced through slip rings. The principle of operation is the same in the two cases. §3. Rotating Field.— The operation of an induction motor will be most readily understood by a consideration of its rotating magnetic field. Fig. I. Stator of a 3-phase motor Let US consider a polyphase* with 4 poles per phase, — often termed a . . r j r 4-pole motor. (This diagram is iiiustra- motor in which the primary tive and does not represent construction.) .... . is stationary.y Fig. I illustrates the primary or stator of a 3-phase motor in which there are four poles per phase. When the current in Phase I. is a positive maximum, AA form north poles and A'A' * (§3a). For a single-phase motor, see §56. t (§3b). In this case, with the primary stationary, the idea of the rota- ting field is most simply seen. The relation between primary and secon- dary, however, and the torque produced, will be the same, irrespective of which winding is stationary and which is moving. 8-A] OPERATION AND LOAD TEST. 259 form south poles. A little later (after % cycle) the current in Phase II. becomes a maximum, so that BB form north poles and B'B' south poles. {A and C are now also north but weaker, the maximum field being under B.) Later (after % cycle) the cur- rent in Phase III. becomes a maximum and north poles are formed under CC. Each pole is accordingly seen to progress and to be successively under A, B, C, A', B', C, etc. The. primary or stator winding thus produces a revolving or rotating field which tends to drag the rotor around with it, § 50. With the usual distributed winding, the field is uniform and revolves with uniform speed. §4. In a 2-pole model (having two poles per phase) the field makes one revolution in one cycle; in a 4-pole model, as Fig. i, one revolution in two cycles; in a 6-pole model, one revolution in three cycles, etc. It is seen that, if n is the frequency in cycles per second and p is the number of pairs of poles (per phase), the rotating field makes n^-p revolutions per second. This is known as the syn- chronous speed of the motor; compare § i, Exp. 3-A. § 5. In revolutions per minute. Synchronous speed = 60 w -f- ^ ; Pairs of poles per phase := 60 m -h synchronous speed. § 6. Speed and Slip. — The rotor of an induction motor revolves at a little less than synchronous speed; for, at synchronous speed it would revolve at the same speed as the magnetic field, in which case there would be no cutting of lines of force, no secondary current and hence no torque. The actual speed of an induction motor is, therefore, less than synchronous speed by a few per cent, called the per cent. slip. The slip increases with the load, thus increasing the cutting of lines of force, the current in both secondary and primary and the torque. § 7. Primary Winding. — An induction motor is not commonly 26o INDUCTION MOTORS. [Exp. constructed with separate poles, as in Fig. i. The primary wind- ings a're usually distributed in slots and are so arranged in groups that a series of poles are produced which correspond to those shown in Fig. i. There are a number of slots for each pole. A lap or a wave winding (see § 3b, Exp. i-A) can be used, accord- ing to various winding schemes, as described in text-books. ■/ § 8. Squirrel Cage Secondary. — The secondary winding usually consists of a squirrel cage, made up of parallel copper bars set in slots with ends connected by two short-circuiting rings. Such a construction is strong and simple; it makes possible a low secondary resistance which gives high efHciency and good speed regulation but low starting torque. The squirrel cage secondary is self-contained and has no outside connection. ■^ §9. Phase-wound Secondary. — To obtain a higher secondary resistance and hence a greater starting torque,* the secondary is sometimes wound, or phase-wound as it is called. Extra sec- ondary resistance, either internal or external to the motor, may be included in the circuit on starting to increase the starting torque, this resistance being cut out as the motor speeds up so as to give higher efficiency and better speed regulation while run- *(§9a). Maximum Torque. — Torque is proportional to secondary electrical input, £2/2 cos ^2, see § 54. Starting torque is proportional to secondary input at standstill and is a maximum when R^^X2, for h then lags 45° behind £2 and has a maximum power component in phase with £2. (In any constant potential circuit with constant reactance, cur- rent has a maximum power component when the lag angle is 45°, as can be seen in Fig. 2, Exp. 4-B.) Increasing R2 beyond a certain amount will decrease the starting torque. When Tunning with a slip s, the secondary reactance becomes SX2 and the torque is a maximum when R2 — SX2. The secondary resistance can be given such a value as will give the motor its maximum torque at any desired slip, as for example at standstill when s=: i.oo. The maximum value which the torque can have, irrespective of speed, is independent of R2, being dependent solely upon the input E-i-2X, except for the small effect of primary losses ; see §§ 12a, 22, Exp. 8-B. Changing i?2 can not alter this maximum, but can cause it to occur at any desired speed. / 8-A] OPERATION AND LOAD TEST. 261 ning. (High resistance in a squirrel cage armature is some- times obtained by making the short-circuiting rings with high resistance.) § 10. The secondary resistance, besides increasing the torque, acts as a starting resistance (§ 14) and serves to reduce the starting current taken from the Hne, in the same manner as the starting resistance of a direct current motor. The secondary resistance may also be used to vary the speed of an induction motor when running, but this is inefficient, as is the case when the speed of a shunt motor is varied by means f a resistance in series with the armature. See § 60. §11. Starting Polyphase Motors. — Polyphase induction motors are self-starting and require no special starting devices in order to obtain a starting torque. There is, however, objection to starting a motor by connecting it directly to the line on account of the excessive current which would thus be taken. Small motors — say under 5 horse-power — may be started without load in this manner, inasmuch as such a motor forms but a small part of the total load of a system and cannot seriously affect its regu- lation. Furthermore, a small motor comes up to speed so quickly that the momentary excessive current does not overheat the motor. All large motors, however, and small motors when loaded, require some starting device to limit the starting current. §12. Starting Corn^nsatar^ or__Auto-transformer. — The most common method of starting is to connect the motor .to a low vol- tage for starting and to throw it on to full line-voltage after it has reached, say, half speed. The low starting-voltage is obtained by means of auto-transformers connected across the line and provided with suitable intermediate taps, as in Fig. 2, Exp. 5-A. On a 2-phase circuit, two auto-transformers are used, — one on each phase. On a 3-phase circuit, two auto-transformers are also used, these being F-connected, as in Fig. 6, Exp. 7-A. The same starting box is, accordingly, suitable for either a 2-phase or 262 INDUCTION MOTORS. [Exp. a 3-phase motor. To save losses, the auto-transformers are cut out when the motor is running. The motor starters are often arranged so as to throw the motor step-by-step on suc- cessively higher voltages, thus avoiding too sudden acceleration and rush of current at any one step. § 13. Any auto-transformer serves to step up the current at the same time that it steps down the voltage and so requires less current from the line; thus, if at % voltage* a motor takes a starting current of 90 amperes, the current drawn from the line is only 30 amperes.f The auto-transformer, therefore, not only reduces the starting current taken by the motor itself on account of the reduced voltage but makes a further proportional reduc- tion in the line current. It serves admirably as a motor starter for all cases except those in which a large starting torque is necessary. § 14. Se condary Star ting Resist ame^ — When a large starting torque is necessary, it is best obtained by using a phase-wound secondary with additional resistance for starting; see §9a. As the motor speeds up, this resistance is cut out, either all at once or gradually so as to control the acceleration and current of the motor. In motors with revolving secondaries, this extra resist- ance may be contained within the rotor and controlled by a lever bearing against a sliding collar, or it may be external to the motor with leading-in wires connected through slip rings. § 15. Some motors are provided with a centrifugal device for cutting out this resistance automatically when a certain speed is reached. § 16. Starting Single Phase Motors.— jWhen supplied with single-phase current an induction motor has no starting torque, although it will run satisfactorily when once started (§56). In some cases small motors may be started by hand, but in gen- * (§i3a). At full voltage the current would be 270 amperes, t (§ 13b). If a series resistance were used to reduce the voltage to 1/3, the current drawn from the line would be 90 amperes. 8-A] OPERATION AND LOAD TEST. 263 eral some special provision for starting a single-phase motor must be made. Whatever means are used for starting, the starting torque is small and it is preferable to start the motor without load. The load may be assumed after starting, by means of a loose pulley or clutch. § 17. Shading Coils. — One method for making small single- phase motors self-starting depends upon the use of shading coils. Each magnetic pole of the field of such a motor is divided into two parts. Around the leading portion of each pole is wound a short-circuit coil of low resistance, called a shading coil. When the flux is changing in any pole, its increase or decrease in the leading part is retarded by the short-circuited coil. In consequence of this action, the leading part of each pole attains its maximum magnetization after the other part, so that a revolv- ing field is produced and the rotor is drawn around as in a poly- phase motor. § 18. Repulsion Motor. — A single-phase motor is often made to start as a repulsion motor and is converted into an induction motor when nearly full speed is attained. Such a motor has a wound rotor provided with a commutator and brushes, as the armature of a direct-current generator or motor. The brushes are connected together by a heavy conductor. In a 2-pole model, the brushes are opposite each other. In a 2-pole model, suppose the plane of the brushes (or rather the plane of the coils to which they are connected through the commutator) to be in the center line of the poles. There would be a large flow of current through the rotor windings and the brushes, but from symmetry there would be Jio torque ; half the conductors would tend to turn in one direction and half in the other. If the plane of the brushes were turned at right angles, there would be no current flowing and therefore no torque. If, however, the brushes are set obliquely, a current will flow and there will be a resultant torque, for the current in the conductors 264 INDUCTION MOTORS. [Exp. under opposite poles will flow in opposite directions so that the conductors under both poles will tend to give rotation in the same direction. After a certain speed is reached, a centrifugal device is com- monly used to lift the brushes, thus saving friction and wear, and to short-circuit the rotor windings so that the motor runs as an induction motor. It then has the characteristics of an induction motor and not the characteristics of a repulsion motor, which are similar to those of a series motor. § 19. Phase Splitters. — A single-phase motor may be made to start as a 2-phase or 3-phase motor by means of polyphase cur- rents temporarily derived from a single-phase circuit. Polyphase currents sufficient for this purpose can be obtained from a single- phase line by various arrangements of resistances and reactances ; such devices, termed phase-splitters, are in common use and will be discussed later (§§26-32). Polyphase currents so derived can never be balanced (§§ 1-3, Exp. 7-A) and are only used for starting. PART II. PRELIMINARY STUDY. ^ 20. Structure and Rating. — Study the general structure and windings* of the motor ; note the rated full-load speed and output and the frequency, voltage and kind of circuit (single-phase, 2-phase 3-wire, 2-phase 4-wire, 3-phase, etc.) for which. the motor is designed. §21. Compute the watts input and the current per line at full load, assuming a certain efficiency (say 80 per cent.) and a certain power factor (say 80 per cent.). For a polyphase motor, com- *(§2oa). A detailed study of the windings may be made when the necessary data are obtainable. The study may include a diagram of winding, and the following data : — number of poles or groups of coils per phase; number of coils per group and turns per coil; size wire; current density for full-load current; total resistance; letigth of wire (per coil and total) computed from dimensions and from resistance measurement; type of secondary ; number and size of conductors ; number of slots, etc. &-A] OPERATION AND LOAD TEST. 265 pare §§20, 21, Exp. 6-A. From the speed and frequency, com- pute the number of poles per phase, § 5. Note the manner in which the motor is to be connected to the supply circuit and any special provision there may be for starting. § 22. Polyphase Motor. — Connect in one line circuit an amme- ter with a range, say, 50 or 100 per cent, in excess of the full-load current. Start the motor in the regular manner, without load, and note the starting current and the current when the motor is running at full speed. Note the change in current as the motor gains speed and as the motor is changed over from the starting to the normal running conditions. Repeat the test with the motor belted to some load (belted, for example, to a generator) and note the current taken to start the motor and note that more time is required to attain full speed. A 3-phase motor can be run on any 3-phase circuit, of proper frequency and voltage, irrespective of whether the circuit is star, delta, T or V connected. A 2-phase motor with two independent circuits can likewise be operated on any 2-phase circuit. If, however, a 2-phase motor has its circuits brought out to only three terminals, as (b) in Fig. I, Exp. 6-A, it can only be operated on a 3-wire circuit and cannot be operated on a quarter-phase system (as c or rf). Con- versely, if the motor is connected as c or d, it cannot be operated on a 3-wire circuit, as b. § 23. Polyphase Motor Started with Secondary Resistance. — If a secondary starting resistance is provided, start the motor with this resistance in circuit and cut it out, either gradually or in one step, as the motor attains full speed. Note that the primary cur- rent is increased by cutting out the starting resistance. Unless specially designed for continuous operation, this resistance will overheat if kept long in circuit. If a half-voltage supply is available, start the motor, with no load, without the starting resistance, noting that the starting cur- 266 INDUCTION MOTORS. [Exp. rent is much greater than with the resistance in circuit. (This may be done at full voltage, if the motor is not too large.) With- out the starting resistance, the starting torque is and that NK is proportional to AN, we have NK « /'<=>. Q. E. D. X It will be remembered that R2 and /«> are the values of secondary resistance and current, respectively, in terms of the primary. II (§ 153)- The primary copper loss at no load is R\h^. For a primary current U, at any load, the primary copper loss is RJ'^RiI^-\- R1I1.21'' — 2RJJ2COSPAO. Strictly speaking, therefore, the last two terms are the "added" primary copper loss due to /<•). We may consider, either (i) 284 INDUCTION MOTORS. [Exp. equal to RJli-^ and can be calculated. (For a 3-phase motor, this loss is zRJl2y) § 16. Separation of Primary and Secondary Losses. — If the primary copper loss is calculated and laid off as NS, we have SK equal to the secondary loss. This needs to be done for one point only, the line ASG being then drawn as a straight line. The point selected is usually the point G, corresponding to the locked posi- tion, and this can be located in several ways. The various meth- ods for doing this are only approximate* and give slightly vary- ing results, — sufficiently accurate, however, for practical results. § 17. First Method. — One procedure is to calculate the added primary loss for the locked position and lay this off as HG, thus locating the point G. The line AG is then drawn. For example, in the present test, I^2) on short circuit is /4P"^5o.5 amperes; i?i = 0.255 ohms. The point G is located so that HG = ^ X 0.255 X 503''= 1,950 watts. This procedure was used in con- structing Fig. I. § 18. Second Method. — With the motor at standstill, JH has no particular significance; neither has /„ (see §32). Without involving either of these, G is readily, and perhaps more accu- rately, located from the short circuit current Is, by laying off JG = RJs^, multiplying by 3 for a 3-phase motor. In the present example, 7G = 3 X 0.255 X 56.6' = 2,451 watts. § 19. The same location for G as found in the preceding para- graph can be obtained by dividing JP" in the ratio R^ : R^, where i?2 is calculated as in § 33. Then JG : GP"=Rj^ : i?2=o.255 : 0.453. This does not involve any special scale for JG and GP", and it is that the last term is neglected as small in the working range of the motor (heing zero when h and 7(2) are at right angles to one another), or (2) that the loss represented by the last term is included in MN and com- pensates for the decrease in friction and windage as the motor slows down with load. * There is no method for determining secondary loss which is exact for all loads' and all types of motors ; compare § 32. The use of three straight lines AH, AG and AP" radiating from A for defining the various losses is convenient but not exact. 8-B] CIRCLE DIAGRAM. 285 not necessary to calculate their values in watts, which may prove a convenience. § 20. Torque. — Torque is equal to the secondary input divided by the synchronous speed ; see § 54, Exp. 8-A. The secondary input, in Fig. i, is seen to be SP, being the primary input MP, less the no-load losses MN and primary copper loss NS. § 21. In synchronous watts, torque is equal to SP, in watts, divided by the synchronous speed of the motor in revolutions per minute. If this is multiplied by 7.04, we have the torque in pounds at one foot radius. (Compare §3b, Exp. 2-A.) To get torque in synchrofaous horse-power, SP is measured in horse-power and divided by the synchronous speed in revolutions per minute. § 22. The maximum or " pull-out " torque is seen to occur at Fg where a tangent to the circle is parallel to AG. The maxi- mum occurs at such a slip, s, that R^ = sX^ ; § 9a, Exp. 8-A'. § 23. Ratios. — The following results, being ratios usually ex- pressed as percentages, are independent of scale. They may be found by division, or graphically as in Appendix I. §24. Power Factor. — The power factor at an^^ load is equal to CO divided by OP ; it is a maximum at F^, where a line from O is tangent to the circle. § 25. Efficiency. — The efficiency is equal to the output divided by the input, namely, KP divided by MP. §26. Slip. — The slip is equal to the secondary copper loss, divided by the secondary input (see §§ 54, 54a, Exp. 8-A) ; namely, SK divided by SP. 286 INDUCTION MOTORS. [Exp. APPENDIX I. GRAPHICAL CONSTRUCTION FOR OBTAINING POWER FACTOR, SLIP AND EFFICIENCY. § 27. Certain results— power factor, slip and efficiency — depend upon ratios; these are usually expressed in per cent, and are determined by dividing one quantity by another, as in §§ 23-26. Some prefer to obtain these ratios by using a slide rule ; others, by using the graphical construction of Fig. 3 which is in common use, and is particularly convenient when one has many motors to test. The reader, however, should bear in mind that this construction is only a convenience for computing and not an essential for the proper understanding of the circle diagram. § 28. Power Factor. — In the direction of OE, lay off an arbitrary scale of 100 parts of any convenient length. From the lootli division, draw the quadrant of a circle with the center at O. To obtain the power factor for any primary current OP, extend OP to R; from R draw a horizontal line to the point p, which gives the power factor. The construction is obvious ; it may be used in any alternating current problem. § 29. Slip. — From the point A, draw AA' parallel to OE. The scale aa', of 100 equal parts, is drawn parallel to AG at any convenient distance. The point a on AA' is marked zero ; the point a' on AP" is marked 100. To determine the slip corresponding to any primary current OP, locate the point 5 where the line AP cuts the slip scale. The per cent, slip* is sa. (If the measured value of the slip at no load is appre- ciable, the scale should be given this value, and not zero, at the point a. Ordinarily this refinement is unnecessary.) *(§29a). Proof of Slip.— For any point P, the slip is KS -^ PS. The triangles KAS and Aa'a are similar; hence KS:AS — Aa:a'a. (i) The triangles APS and sAa are similar; hence PS ■.AS = Aa -.sa. (2) Dividing (i) by (2), we have the slip KS/PS = sa/a'a. 8-B] CIRCLE DIAGRAM. 287 §30. Efficiency.— Extend the line AP" back to L and draw LU parallel to OE. The scale W, of 100 equal parts, is drawn parallel to AP'" at any convenient distance. The point / on LU is marked 100; the point I' on AP" is marked zero. To determine the efficiency corresponding to any primary current J 100 -90 l' ^^~^^ 80 1 \ \ p \Tl^ ~~~" .0 i 1' P '//la 60 _ 5tf\_ 40 30 _ 20 _ 10 0^^ \ s a. c 0. 1 a' — 1/// 30 Per cejit efficfency go ^£_ 40 60__J^y-i£-— '--^ 100^ ^^ G \ H f l/"" A ( 3 L "^ I H J Fig. 3. Graphical method of obtaining power factor, slip and efficiency. OP, locate the point e where the line LP cuts the efficiency scale. The per cent, efficiency* is I'e. *(§ 30a). Proof of Efficiency.— For any point P, the efficiency is PK -^ PM. The triangles LKM and I'Ll are similar ; hence KM:LM = Ll:n. The triangles LPM and eLl are similar; hence PM ■.LM = Ll -.el. Dividing (i) by (2), we have KM/PM=cl/n. But PM/PM=n/l'l. Subtracting (3) from (4), we have (PM — KM) H- PM= (I'l — el) -- VI; whence PK/PM=l'e/l'l, which is the efficiency. (i) (2) (3) (4) . 288 INDUCTION MOTORS. [Exp. APPENDIX II, MISCELLANEOUS NOTES. § 31. Basis of the Circle Diagram. — In any circuit or apparatus with constant reactance and variable power consumption the current will have a circle locus if the supply voltage is constant. Exp. 4-B shows this experimentally for a particular case. This was first shown by Bedell and Crehore in 1892. That the induction motor nearly fulfills these conditions and that its current locus is practically the arc of a circle, was first shown by Heyland* in 1894. § 32. Accuracy. — No circle diagram for an induction motor is exactly correct, either in theory or in practice, for there are various factors that cannot be accurately taken into account; for example,-)- the effect of changes in the reluctance of parts of the iron under changing conditions, the effect of wave distortion, the uncertainty as to load losses, etc. The circle diagram, therefore, is theoretically correct only in case certain assumptions are made. In practice, however, the circle diagram is found to give results that are approximately correct and within the usual range of engineer- ing accuracy. This is partly due to the fact that some of the errors tend to cancel each other. The variations in the methods given by different writers for constructing and using the circle diagram have arisen from a difference in the selection of the errors to be eliminated or minimized, the remaining errors being neglected. If the no-load current /„ were zero and there were no primary resistance loss, most of the errors would disappear and some of these different construc- tions would become identical. Generally speaking the effect of errors in using the circle diagram becomes less as the size of motor increases, so that the method is reasonably accurate on motors larger than, say, 5 or 10 H.P. For motors under 5 H.P. the results, although not so accurate, are fairly * Elektrotechnische Zeitschrift, Oct. 11, 1894; published later in book form and translated into English by Rowe and Hellmund. t (§323). Behrend shows, for example, that there is a departure from a true circle locus when the slots are bridged or closed; the short-circuit characteristic between volts and amperes, under these conditions, is a curve and becomes a straight line only if the leakage path contains no iron. See pp. 20, 21, The Induction Motor, by B. A. Behrend. 8-B] CIRCLE DIAGRAM. 289 satisfactory, while for motors under i H.P. the results are of little value, unless refinements* are introduced in the construction of the diagram. It is, however, in testing large motors that a method of testing without load, as by the circle diagram, is particularly desirable; small motors can be readily tested by brake or other load methods. § 33. Calculation of Secondary Resistance. — The short-circuit watts Wsj are chiefly due to copper losses in the primary and secondary. In reality various load-losses are included,! which cannot be sepa- rately determined. These copper losses are (i^,-fi^,)7s^ per phase, where RJs' is the primary copper loss and RJs' is the secondary copper loss. Here R^ is the secondary resistance in terms of the primary (§§ 16, i6a, Exp. s-B ; § 20a, Exp. S-C) and is the quantity to be determined. Per phase, we have Copper loss= (i?,-f i?,)7s'; hence, i?i + i?j = copper loss -f- Is'. Since R^ is known, R. is thus determined. For a 3-phase motor, Ws = 3(iR, + R,)Is''; and, R^ + R^=i/2(Ws-^h'). In the present test ■i?i-fi?j^ 1/3 (6,810 -7- 56^') = 0.708 ohms; hence, R. = 0.708 — 0.255 = 0453 ohms. § 34. Leakage Bieactance. — The leakage reactance X, of an induc- tion motor, both primary and secondary in terms of the primary, can * (§32b). See a comprehensive article by H. C. Specht, Elec. World, p. 388, Feb. 25, igos, in which it is said the modifications introduced give a diagram applicable to induction motors of all sizes, single-phase or poly- phase. To correct for error due to primary resistance, Specht tips his diagram slightly, by dropping A and raising O' a small amount. Such a correction was pointed out by Heyland, p. 23 of the English translation. t (§ 33a). This gives to R2 a value somewhat greater than the value that would be determined by direct current resistance measurement. 20 290 INDUCTION MOTORS. [Exp. be calculated* from the diameter of the circle locus AP'", which is equal to Eh-.X' amperes. Thus in Fig. i, AP'" = ^^.^ amperes; the leakage reactance per phase is £-7-/4P"'^ 107-^-55.4^ 1.93 ohms. § 35. Leakage Coefficient. — The leakage coefficient, or leakage factor, is defined by the ratio BA -=- AP'". To have this quantity small neces- sitates a small air gap. In Fig. i, 5^ -=-^F"'== 5.8 -^55.4 = 0.105. •* (34a). The leakage reactance can also be calculated as in §25, Exp. S-B; thus, 2 = £-^/s=io7-T-56.6= 1.89 ohms; X = yZ' — K^ = V (1.89)''— (.708)' = 1.75 ohms. The two methods of calculation agree only when h is zero. CHAPTER IX. INDUCTION MACHINES: FREQUENCY CHANGERS AND INDUCTION GENERATORS. Experiment 9-A. Operation and Test of a Frequency Changer (Secondary Generator). § I. Principles of Operation. — The frequency of the current in the secondary of an induction motor depends upon the speed of rotation of the rotor; within limits any desired secondary fre- quency can be obtained by giving the rotor the proper speed. The usual form of frequency changer or frequency converter consists merely of an induction motor and a separate driving motor for driving the rotor at the proper speed. The primary is furnished with polyphase current and produces a rotating magnetic field as in any induction motor. The second- ary is phase-wound and delivers, usually, polyphase current to the receiving circuit. The driving motor may be of any type, but in commercial practice a direct-connected synchronous motor is commonly used, so that the delivered current has a definite fixed frequency. (Since the induction machine takes a lagging current of low power factor, the synchronous motor by taking leading current will raise the power factor of the set.) § 2. The secondary frequency is increased or decreased accord- ing as the rotor is turned in the opposite direction or in the same direction as the rotating field ; the secondary voltage is increased or decreased in proportion to the secondary frequency. Aside from losses, the secondary current (with a i : i ratio) is equal to the primary current, either for an increased or decreased fre- quency. Frequency changers are generally used to change from a lower to a higher frequency, — for example, from a low-frequency power circuit to a 60-cycle lighting circuit. 291 292 INDUCTION MACHINES. [Exp. § 3. The secondary frequency n^ varies with the sHp s ; that is, M2 = JMi. At synchronous speed, ,s=o and M2=o; at stand- still, s=i and n^^n^. When, the rotor is driven against the rotating field, with a speed equal to synchronous speed, s = 2 and M2 = 2Mi; etc. In this case the frequency is increased and the driving motor is supplying power proportional to the increase in frequency and voltage. Thus, when the frequency is increased 50 per cent., two thirds of the power is supplied by the primary of the induction machine and one third is supplied by the driving motor; losses are here neglected. When the frequency is decreased, the rotor revolves in the same direction as the field at less* than synchronous speed; the secondary voltage and power are decreased so that the electrical power supplied to the primary is more than the power given out by the secondary. If the surplus power is more than enough to supply the losses, the induction machine runs as a motor and furnishes mechanical power. Frequency changers are not com- monly used to decrease the frequency. §4. Referring to the circle diagram, Fig. i, Exp. 8-B, the short-circuit point has the position P" when the rotor is at stand- still (j==i) ; the machine then acts as a stationary transformer, the range of working as the secondary external resistance changes being from A on open circuit to P' at full load and P" on short circuit. The excessive current, however, prohibits going much beyond full load, as in any transformer. When the rotor is turning with the field (j < i), the short- circuit point P" shifts to some point as 8 and when turning against the field (j> i) to some point as 9, the full range of working being from A on open circuit to the short-circuit point * (§3a). The rotor could be driven above instead of below synchronous speed, with a negative instead of a positive slip. Electrical power would then be given out by the primary as well as the secondary, the machine being simultaneously a primary and secondary generator (see Exp. 9-B), but it is doubtful whether there is any useful application for such operation. 9-A] FREQUENCY CHANGER. 293 P" wherever located. The short-circuit point P" may be located anywhere on the semi-circle, from A (when s = o) to P'" (when s=x), according to the slip. P'" would be reached only if the speed were infinite, or, if the secondary resistance, internal as well as external, were zero. See also Fig. i, Exp. 9-B. § 5. Apparatus. — When a commercial frequency changer is available, it should be run under its rated conditions and appro- priate measurements made of input and output. In the laboratory, any phase-wound induction motor can be conveniently used as a frequency changer. It may be belt-con- nected to a direct-current shunt machine which may be driven at different speeds and will serve as a driving motor or as a generator. § 6. Preliminary Test. — ^With primary voltage constant, make a run at different speeds from synchronous speed in the same direction as the rotating field to the same speed in the opposite direction. (In changing the relative direction of rotation, it may be simpler to reverse the direction of the rotating field by changing primary connections than to reverse the direction of the driving motor.) For different speeds, note the frequency (which may be computed) and the voltage of the delivered cur- rents. Plot voltage for different speeds, frequencies or slip. §7. Load Run at Constant Speed. — (These tests may be cur- tailed or expanded as desired.) With the rotor driven against the field, so as to convert from a low to a high frequency, vary the load of the receiver circuit consisting of non-inductive resist- ances. Measure the primary input and secondary output of the induction machine, and the input of the driving motor. The mechanical power supplied by the driving motor is the motor input less losses determined as in Exp. 2-B. This mechanical power added to the measured primary power gives the total power supplied to the induction machine. Compute the efficiency of the induction machine alone and of 294 INDUCTION MACHINES. [Exp. the complete set. Note the voltage regulation with load. Note how the relative amounts of power supplied by the driving motor and by the primary are related to the change in frequency. § 8. Repeat the test, converting from a high to a low frequency. The driving motor, when running as a generator, may pump power into the line. 9-B] INDUCTION GENERATOR. 295 EXPERIUENT 9-B. Operation and Test of an Induction Gen- erator (Primary Generator). PART I. INTRODUCTORY. § I. Principle of Operation. — ^As the speed of an induction motor approaches synchronism, the shp decreases so that there is less cutting of magnetic flux by the secondary conductors. There is accordingly less electromotive force induced in the secondary, less secondary current and less torque. At synchronous speed the slip becomes zero, and there is no secondary current and no torque. When the rotor is driven above synchronous speed, the slip becomes negative; the secondary current and the torque are now reversed* and mechanical power is required to drive the rotor. The machine has become a generator and supplies elec- trical power to the line. (This condition occurs when an electric train, equipped with induction motors, runs down hill.) As the induction generator does not operate at synchronism, it is fre- quently described as non-synchronous or asynchronous. §2. When the secondary current is reversed, the primary " added " current 7(2) due to the secondary is also reversed (see § la), but it still follows the circle locusf as shown in Fig. I. The point P follows the upper semi-circle (as Pm) when the machine is operating as an induction motor, the lower semi-circle (as Pg ) when it is operating as an induction generator. In Fig. I, the impressed or line electromotive force is repre- sented as E; the generated or counter electromotive force, as E'. § 3. As a motor, the primary current is /j ^= OPu, consisting of the exciting current, Io = OA, and the added current Ii2-,=AFu- (The lines for a motor are not all shown in Fig. i ; see also Fig. i, Exp. 8-B.) The primary current is always lagging with respect * (§ia). Strictly speaking only the power component of current is re- versed, as will be seen later. fit will be understood that for an induction generator, as for an induc- tion motor, the circle locus is approximate and not exact. 296 INDUCTION MACHINES. [Exp. to E; for a definite load and slip it has a definite power com- ponent OC in phase with E and a definite wattless component CFm lagging 90° behind E. The power factor, for a given load and slip, is likewise definite. As a generator, the primary current is I^ = OPq, consisting of Fig. I. Circle diagram showing the operation of an induction machine as a motor and as a generator, (In the range Aa the machine gives out no power.) the same exciting current as before, /„ = OA, and the added cur- rent I(^-)=APq. The primary current is always leading with respect to E' ; for a definite load and slip it has a definite power component OC in phase with £' and a definite wattless component C'Pq, which is 90° ahead of £'. The power factor for a given load and slip is definite. § 4. It is seen that in an induction generator, as well as in an induction motor, all the current can not be power current. There 9-B] INDUCTION GENERATOR. 297 must be a flow of wattless magnetizing current* to set up the flux ; otherwise the generator can produce no voltage. This means that an induction generator will give voltage only when it is connected to a circuit that allows the proper wattless current to flow ; it can not operate when connected to a resistance load (or other load taking power current only) unless there is connected in parallel some device, as a condenser or synchronous machine, that takes leading current. § 5. The size and cost of condensersf being prohibitive, the indiiction generator in practice is used (a) in parallel with a synchronous generator, or (b) in parallel with an over-excited synchronous motor or converter. § 6. When an induction generator is used in parallel with a condenser, synchronous motor or converter, the wattless current is a leading current supplied by the generator to the condenser or synchronous machine. Commonly but less logically, however, this wattless current is described as a lagging current supplied to the generator by the condenser or synchronous machine. The synchronous machine is said to "supply the excitation" for the generator. When the synchronous machine is a generator, there are two generators in parallel and the current which circulates between them is due to the combination of their two electromotive forces, the current being lagging with respect to one and leading with respect to the other. §7. Uses. — The induction generator has been but little used, due no doubt to its inability to supply lagging current and the * (§ 4a) . Aside from saturation, the voltage of an induction generator at no load is proportional to the wattless magnetizing current. As the load increases, the wattless current is increased from BA to C'P^ on account of leakage reactance, as in a transformer or induction motor. The diam- eter of the circle, E -¥X, becomes greater as X diminishes and would be infinite when X = o. t For operation with condensers, see McAllister's " Alternating Current Motors." 298 INDUCTION MACHINES. [Exp. necessity of using a synchronous machine in conjunction with it. It has the advantage* of rugged construction, with no commu- tator, brushes or slip rings. The squirrel-cage rotor requires no moving coils of wire and practically no insulation. The machine gives a smooth wave of electromotive force and tends to damp out, rather than to produce, harmonics and surges. On short circuit the machine gives no voltage, which is an advantage in operation. §8. When an induction generator is operated in parallel with a synchronous generator, the frequency and voltage are determined by the latter. The load taken by the induction generator depends upon its slip, — that is, its speed with reference to the speed of the synchronous machine. This characteristic may prove desirable or not according to cir- cumstances. It would, for example, be obviously undesirable if the induction and synchronous generators were driven at constant speed; for, as the load increased, the induction generator would not take its share. On the other hand it would prove desirable if a station with induction generators driven by water power were connected in parallel with a station composed of syn- chronous machines driven by steam power. It could be so ar- ranged that the induction generators would tend to speed up and take all the load up to the limit of the water power, the steam-driven synchronous machines carrying only the excess of load. §9. When an induction generator is operated in parallel with a synchronous motor or converter, the frequency will depend upon the speed of the generator (see Fig. 5) but will vary also with the slip, that is, with the load. At constant speed the frequency would diminish with the load; or, for constant frequency, it would be necessary for the speed to increase with the load. The voltage * For a discussion of the induction generator and its use, see a paper by W. L. Waters, A. I. E. E., Vol. XXVIL, pp. 157-180 and the discussion pp. 217-254. 9-B] INDUCTION GENERATOR. 299 depends upon the speed of the generator, Fig. 5, and the field excitation of the synchronous machine, Fig. 4. One synchronous machine, either in the station or in a substa- tion, is sufficient for the operation of several induction generators. In the case of a long transmission line, it has been proposed to locate the synchronous machine at the receiving end so that the leading current supplied to it v/ill improve the regulation of the line. PART II. TESTS. § 10. (a) Operation in Parallel with a Synchronous Generator or Supply Line. — The induction generator is driven by mechanical power. As a driving motor, a shunt motor will be found con- ;-t.-T^ + Synchronous Generator, Motor or Converter FiQ. i!. Connections for operating an induction generator. The switch ^2 may connect to a supply line. venient since its speed can be easily varied. The connections* are shown in Fig. 2. § II. Loading Back Test. — No load is used, the switch S^ being *(§ioa). The connections shown are for single phase; three ammeters and three wattmeters may be used, or one ammeter and one wattmeter can be switched from circuit to circuit. When polyphase apparatus is used, as a laboratory test it may be operated single phase for simplicity. When operated polyphase, the usual disposition of instruments should be made; care should be take that the polyphase connections are so made that the induction machine would run as a motor in the same direction it is driven as a generator. 300 INDUCTION MACHINES. [Exp. open. The induction machine is driven at about normal speed and is then connected (by the switch S^) in parallel with a syn- chronous generator or supply line. Note that the switch ^"2 may be closed when the induction generator is running either above or below synchronous speed. After the switch is closed, the induc- tion machine continues to run as a motor below synchronous speed or as a generator somewhat above synchronous speed. § 12. Vary the speed of the induction machine, by varying the speed of the driving motor, and note the wattmeter W^. Below synchronous speed, the in- duction machine takes power from the line as a motor. At synchronous speed, corresponding to the point A in Fig. i or 3, the wattmeter shows only the no- load losses which are all supplied elec- trically. A little above synchronism (between A and a) the wattmeter reading de- creases, some of the losses being sup- plied mechanically by the pulley. At the point a all the losses are supplied mechanically and the wattmeter read- Fuii load '"S becomes zero. Above this speed, generator the wattmeter reverscs and shows elec- tric power given to the line by the in- duction machine &s a generator. Note that the frequency and voltage in all cases are determined by the syn- chronous alternator or supply circuit. § 13. Take readings of volts, watts and amperes through the full range between no load and full load with the machine oper- ating as a motor and as a generator. (It is instructive also to measure speed or slip and to plot slip — ^positive and negative — for different amounts of power.) Fig. 3. Current taken in by an induction machine as a motor and given out as a generator. 9-B] INDUCTION GENERATOR. 301 § 14. For each reading, calculate Power current, Ip=I cos 0=^W-^E; Wattless current, 7q = 7 sin ^= V^^— -7p^ Plot results as in Fig. 3 by laying off wattless current as abscissae and power current as ordinates. Compare Fig. 5, Exp. 8-A. § 15. Load Test. — Connect to a non-inductive* load by closing the switch S^, Fig. 2. With load constant, vary the speed of the induction generator and note instruments. The load receives power from both generators or from only one ; W^ = W^ + W^. When Wj^ and W^ are positive, both machines are supplying power as generators. When IV-^ or W^ is zero, the corresponding ma- chine is neither supplying nor taking power. When W^ or W2 is negative, one machine is taking power as a motor, all the power being supplied by the other machine. For several sets of readings, compare the values of I cos 6 and 7 sin 5 as calculated for each of the three circuits. In commercial use, the machines would be so operated that both machines are supplying power, the division of the load depending upon their relative speeds. § 16. Tests can be made with variable load under any desired arrangement of conditions. § 17. (b) Operation in Parallel with a Synchronous Motor or Converter. — In this test the synchronous machine, hereafter re- ferred to as the converter, may be either a motor or converter. The connections are as shown in Fig. 2. The converter can be readily brought up to speed with direct current, as a direct cur- rent motor. After the induction generator and the converterf have been * Loads that are not non-inductive can be made the' subject of special investigation. t (§i7a). When a synchronous motor is used, a good procedure is to close ^2 and bring the machines to speed before exciting the motor field. The motor field current is then gradually increased until the induction generator gives the desired voltage. 302 INDUCTION MACHINES. [Exp. brought up approximately to speed (the exact speed is not neces- sary) the two machines are connected together by closing the switch 5*2, the power supply used in bringing the converter to speed being cut off. § 1 8. No-load Excitation Curve. — The load switch S^ is open. Vary the field current of the converter and measure watts, amperes and volts. The induction generator is driven so as to give rated frequency at rated voltage; hold speed, or frequency, con- stant during the test. The converter may be sepa- rately or self excited. When the field current is reduced be- low a certain value, the con- verter goes out of step and stops; the induction generator then gives no voltage. § 19. For various field cur- rents, plot voltage as in Fig. 4 ; 6 810 1.4 1.8 Z.2 2-6 slso 7 sln 9, thc wEttlcss compo- Field current of Synchronous Converter nCUt of line CUrrCUt. Itisthc Fig. 4. ^ange i„ voltage and w^^^^^^ increase in 7 sin ^ that increases less current of an induction generator with the excitation of synchronous the excitation of the induction machine. generator and so increases the generated voltage. This relation will be seen by plotting voltage for different values of 7 sin $. § 20. No-load Speed Characteristics. — Separately excite the field of the converter and keep the field current constant. Vary the speed of the induction generator and measure line voltage and fre- quency. Begin with a speed of say 10 per cent, above normal and decrease speed until the converter stops. Converter speed may be measured instead of frequency. §21. Results are plotted as in Fig. 5. The lower curve shows 9-B] INDUCTION GENERATOR. 303 that, in order to give a frequency of 60 cycles, the machine to which the curve refers must be driven at 1823 R.P.M., correspond- ing to a negative slip of 1.3 per cent, the synchronous speed being 1800. §22. Repeat with the con- verter self excited. § 23. Load Test. — Connect a non-inductive load by closing ^■g. Compare the values of / cos 9 and I sin 6 for the three circuits. All the power, and hence all the power current, is derived from the ~ 70 ^ r«„ E) and for under-excitation (£' < E) . § 8. Current Loci by Test. — With constant excitation and vary- ing load determine the current loci for several excitations and compare with the predetermined loci. §9. Further Investigations. — The investigation may be ex- tended to include a predetermination of 0-curves and F-curves (as by McAllister), measurement of angular armature position (as by Bedell and Ryan), study of the frequency of hunting, limits of stability, etc. See § la ; also § 24, Exp. lO-A. lo-C] SYNCHRONOUS CONVERTER. 321 Experiment io-C. Study of a Synchronous Converter. § I. Methods for Obtaining Direct from Alternating Current. — Direct current is usually distributed from substations and is ob- tained by means of synchronous converters which are operated by alternating current transmitted from a distance. For economy the transmission is usually 3-phase (§54, Exp. 6-A). The con- verters may be single-phase or polyphase, in practice being usually either 3-phase or 6-phase (§§27, 27a, Exp. 6-A). The growth of electric traction has been coincident with, if not indeed depen- dent upon, the general use of the synchronous converter. While more generally used on circuits of low frequency (25 cycles), its use at 60 cycles (§3, Exp. 3-A) is common. §2. The synchronous converter is essentially a synchronous motor and direct-current generator combined in one machine;* it has one field, which is self- excited by direct current, and one armature winding which is provided with collector rings for receiving alternating cur- rent and with a commutator for delivering direct current. The armature connections for a 3- FiG. I. Armature connections for a phase converter are sHown in 3-phase converter, 2-pole model. _. , . , . , ,. Fig. I, which IS the diagram for a 2-pole model. Each collector ring is tapped into the arma- ture winding at one point in a 2-pole model, two equidistant points in a 4-pole model, three equidistant points in a 6-pole model, etc. *(% 2a). Dynamotors and Motor-Generators. — Provided with indepen- dent armature windings, but with a common field, the machine would be a dynamotor; with independent fields as well as armatures the machine would be a -motor-generator with motor and generator separate, — a more flexible arrangement in regard to control and regulation but more costly in con- struction Hud less efficient in operation. 322 SYNCHRONOUS MACHINES. [Exp. In a synchronous converter, the armature revolves while the field and brushes are stationary; for mechanical reasons the re- verse arrangement — with revolving* field and brushes and with stationary armature — is undesirable. Connections for operating a synchronous converter are shown in Fig. 2. §3. Other devices for deriving direct from alternating cur- rents are: synchronous commutators^ (which have proved short- lived both as individuals and as a class) ; and rectifiers that depend upon a valve effect, as the aluminum rectifier, mercury-arc recti- fier, etc., the latter being the only one of these with high enough efficiency to warrant extensive use. § 4. A synchronous converter is normally used to receive alter- nating and to deliver direct current, but may be used as an in- verted converter — to receive direct and to deliver alternating cur- rent — or, as a double current generator driven by power and delivering both direct and alternating currents. § 5. Voltage Ratios. — Terminal voltages in any machine differ somewhat from the induced or generated voltages on account of drop in the windings which varies with the load. The ratio of generated voltages in a converter may be computed as follows: Consider a converter driven as a generator, delivering direct current and single-phase alternating current. When the brushes are properly set, the D.C. voltage will be equal to the maximum value of the A.C. voltage. The effective A.C. voltage will depend upon wave form, being for a sine wave 1/V2 times the maximum, or direct current, value. * (§2b). Permutators. — The rotating field may be produced electrically, in which case both field and armature windings are stationary, the brushes being driven at synchronous speed by a light driving mechanism. A modi- fication of this arrangement is the permutator, the introduction of which has no doubt been prevented by the difficulties introduced by the revolving brushes. See: Elektrotech. Zeit. (Vienna), Aug. 28, 1898; L'Industrie Electrique, Feb. 10, 1902, Nov. 25, 1905 ; Land. Electrician, Dec. 9, 1905, Dec. 10, 1906 ; Elect. Age, Nov., 1908 ; Sibley Journal of Eng., June, 1909. t For a test of such a commutator, see paper by J. B. Whitehead and L. O. Grondahl, Elec. World, pp. 896 and 914, April 15, 1909. ^0-C] SYNCHRONOUS CONVERTER. 323 Hence, for a single-phase (or 2-phase) machine, the A.C. voltage is £a.c.= (I/V2)£D.c.■=.707•ED.c.• The star voltage £s, measured between one alternating current line and the neutral, is Es = y2{ 1/ V2) £d.c. = o.3S4£d.c., which is true for a polyphase, as well as for a single-phase, ma- chine. From the star voltage, the line voltage in any case is readily computed; thus, — In a single-phase or 2-phase machine, the line voltage is twice the star voltage. In a 3-phase machine (§ 19, Exp. 6-A), the line voltage £3 is £3 = Vj-Es =-^ Ejy.c. = .612ED.C.. 2V2 In a 6-phase machine, the line voltage E^ is £a = -Es = o.3S4£D.c.. § 6. Current Ratios. — ^Assuming a certain eificiency and power factor, the alternating current in each supply line corresponding to any particular value of direct current output can be computed. Thus, if power factor and efficiency are i. 00, each ampere of direct current requires an alternating current of \/2 amp. (^1.414) single-phase, % V2 amp. (=0.707) 2-phase, %V2 amp. (=0.943) 3-phase, %V2 amp. (^=.472) 6-phase. §7. Rating. — In a converter, each armature conductor carries an alternating current and a rectified direct current, giving an irregular wave form, with a chopped up appearance, diiifering in the various conductors according to the time that has elapsed since each conductor has passed under a brush. The rating* of a con- *For a good discussion, see paper by O. J. Ferguson, Elect. World, p. 214, Jan. 21, 1909, where ratings are derived for various power factors, based upon hottest and coolest coils as well as upon average heating. See also paper by W. L. Durand, Elect. World, p. 23s, Jan. 26, 1911. 324 SYNCHRONOUS MACHINES. [Exp. verter depends upon armature heating and may be based upon several assumptions. Based upon average armature heating and the assumption of unity power factor, the relative capacities of a converter are single-phase, 0.85 ; 3-phase, 1.33; 6-phase, 1.93 ; the capacity as a direct current generator being unity. The advantage of the 6-phase converter is obvious (§27a, Exp. 6-A), but there is little advantage in more than six phases, the capacity for infinite phases being only 2.3 as compared with 1.93 for six phases. §8. Voltage Control.* — The D.C. voltage of a converter is usually controlled by altering the A.C. voltage supplied to the collector rings, and this is commonly done : ( i ) by means of an induction regulator (or some other form of potential regulator, Exp. 7-B) ; (2) by means of reactance placed in series with the converter on the A.C. side, as discussed later; or, (3) less com- monly by a synchronous, booster.'\ Another method, used in the split-polej converter, controls the D.C. voltage without altering the A.C. voltage. * (§8a). For a discussion of converter construction and operation, with particular reference to voltage control, see the following papers and their discussions: A. I. E. E., Vol. XXVIL, C. W. Stone, p. 181; J. E. Wood- bridge, p. rgi ; C. A. Adams, p. 959 ; also Elect. Journal, Vol. V., F. D. Newbury, pp. 615, 616. t(§8b). Synchronous Booster. — A small auxiliary alternator, mounted on the same shaft as the converter is connected in series with it as a booster on the A.C. side. The A.C. voltage supplied to the converter depends, therefore, upon the excitation of the booster, which may be controlled by a suitable regulator. Some of the field windings of the booster may be put in series with the D.C. load, thus giving an increasing excitation with load. X (§8c). Split-pole Converter. — In this converter each pole is divided into sections which can be given different excitations so as to vary the flux distribution. This shifts the flux, which is practically the same as shifting the brushes, and so changes the D.C. voltage; or, it alters the wave form of electromotive force and so changes the ratio of the D.C. to the A.C. voltage; or, it both shifts the flux and alters the wave form. lo-C] SYNCHRONOUS CONVERTER. 325 § 9. Reactance Control. — ^A lagging current through a reactance always causes a drop in voltage; but a leading current, when sufficiently in advance of the electromotive force, will cause a rise in voltage. With reactance in a circuit, therefore, the voltage delivered will depend upon how much the current is lagging or leading, and this — in a synchronous motor or converter — is con- trolled by the excitation. § 10. Fig. 2 shows a reactance X located in one line; in practice, an equal reactance is located in each line (§20) either as separate coils or as leakage reactance* in transformers. Let £0 be the line voltage and £t be the terminal voltage at the collector rings of the converter. When the current / is in phase with Et, the reactance causes a voltage drop in quadrature with I so that £t is somewhat less than Eo, as shown in Fig. 3, Exp. 3-B. This corresponds to a particu- lar excitation of the converter. When the current / is lagging, the drop through the reactance is more effective and Et is much less than £0, as in Fig. 4, Exp. 3-B. When the current / is leading — corresponding to an increased converter excitation — Et is increased, as in Fig. 5, Exp. 3-B. It is seen, therefore, that the A.C. voltage supplied to the con- verter (and so the D.C. voltage delivered) can, when series react- ance is used, be increased or decreased by increasing or decreas- ing the motor excitation. § II. In the laboratory, and sometimes in practice, this change in excitation can be made by a hand adjustment of the field rheo- stat. In practice, however, it is usually made automatic by com- pounding the converter, by means of series turns carrying the D.C. load current. As the load increases or decreases, this in- creases or decreases the excitation and the voltage. By properly proportioning the reactance and the series winding (or a shunt *.(§ loa). The unavoidable reactance of the generator, line, transformers and converter may be sufficient. 326 SYNCHRONOUS MACHINES. [Exp. around it) the converter may be over-, under-, or flat-compounded in the same manner as a D.C. generator (§§ i, 23, Exp. i-B). § 12. Derived Neutral. — ^An interesting feature of a converter or double-current generator is that the potentials of the neutral of the A.C. system and of the D.C. system are alike. The neutral may be considered as a (usually fictitious) middle point of the armature. Taking this neutral potential as zero, the positive D.C. brush is at a constant positive potential and the negative D.C. brush is at a constant negative potential, the mean of these being the potential of the neutral. Referring to a single- phase machine, the two A.C. brushes have alternating potentials that are equal but of opposite sign, the mean of which is the potential of the neutral. The A.C. neutral is readily obtained from a tap in the middle of a transformer or choking coil across the A.C. lines of a single- phase system, or opposite lines of a 2-phase or 6-phase system. The D.C. neutral, otherwise not easily obtained, can be readily "derived" from the A.C. neutral, i. e., the neutral of the A.C. side is used as the neutral for the D.C. system. This is one advantage of a 2-phase or 6-phase converter with diametral con- nections (§27, Exp. 6-A). § 13. Direct-current generators are often constructed with A.C. collector rings across which are placed choking coils for the pur- pose of deriving a neutral for a 3-wire D.C. system. Any direct current returned to the generator by the neutral or third wire, due to an unbalanced load, passes into the middle tap of such a chok- ing coil and thence differentially through the two halves of the coil with no magnetizing effect upon the core. Special 3-wire generators are constructed with the choking coils contained within the armature, the outside connections being made through the commutator and one slip ring. lo-C] SYNCHRONOUS CONVERTER. 327 PART II. OPERATION AND TESTS. § 14. Synchronizing, — A synchronous converter can be brought up to speed and synchronized by any of the means used for start- ing and synchronizing a synchronous motor (Exp. 10-A). It can also be brought to speed as a direct-current shunt motor (Exp. 2-A) by means of direct current supphed to the commu- tator end of the converter, a starting resistance being used in series with the armature to limit the starting current; the field rheostat regulates the field current and so controls the speed in synchronizing. In the laboratory, direct current starting is usually the most convenient. In practice, alternating current starting with low starting volt- age is most common ; a " break-up " switch is used for separating the field spools (see § 11, Exp. lo-A). A D.C. voltmeter on the D.C. side shows when synchronism is reached by ceasing to beat and by assuming a steady reading, either positive or negative. If the polarity is not the one desired, the machine must be synchron- ized again or allowed to slip a pole by opening and closing the main switch. Another way to slip a pole is to reverse the field connections and, after the D.C. voltmeter has come to rest near zero, to again reverse the field; the converter then locks in step with the proper polarity. § 15. Voltage Ratio. — On open circuit, measure the A.C. and D.C. voltage and compare their ratio with the calculated ratios, § 5. § 16. Tests (Without Series Reactance). — On the A.C. side, connect* a circuit-breaker, voltmeter, ammeter and wattmeter; on the D.C. side, connect an ammeter and voltmeter and a variable resistance for a load. No reactance X is in the line ; otherwise the connections are as shown in Fig. 2. Line voltage should, if possible, be kept constant. The following scheme of tests may be followed or modified as seems desirable. § 17. No-load Excitation Test. — Take the same no-load exci- * For a polyphase machine, see § 21a, Exp. lo-A. 328 SYNCHRONOUS MACHINES. [Exp. tation characteristics as for a synchronous motor (§§21, 23, 24, Exp. lo-A), § 18. Full-load Excitation Test. — Repeat the test, keeping the direct current constant at rated full-load value, or other selected value. § 19. Load Run; Excitation Constant. — ^With constant excita- tion* (for which separate excitation may be convenient), vary the D.C. load and take simultaneous readings of all instruments. Make runs with over-excitation, normal-excitation and under- excitation. § 20. Tests with Series Reactance. — Place a reactance in series with each line (for a single-phase converter, reactance in one line is suflScientf) and repeat the preceding tests. Read all instruments, as shown in Fig. 2; also read voltage VlJW ^ J \ (k ^' Q r ] Converter 1 /Ti \ / Fig. 2. Connections for operating a synchronous converter. drop around the coil X, which should be so designed that the drop does not exceed, say, 25 per cent, of the terminal voltage. Measure the resistance and reactance of the coil, plotting results as in Fig. 7, Exp. 5-B. §21. Results. — Show all results by curves and compare curves with and without series reactance, — particularly the curves show- ing voltage variations. Efficiencies are computed from input and output. Construct diagrams, as Figs. 3, 4, S, Exp. 3-B. * As a modification, with the converter self -excited, leave the field rheostat in one position. t When supplied from independent transformers, 2 reactances are sufB- cient for a 2-phase converter; 3 reactances, for a 6-phase converter. lo-C] SYNCHRONOUS CONVERTER. 329 §22. Inverted Converter. — When a converter js driven by direct current, its speed depends upon the field excitation, as in the case of a D.C. motor. If the field is weakened through any cause (by a decrease in field current or by the demagnetizing effect of armature reaction), the speed increases; likewise, if the field is strengthened (by an increase in field current or by the magnetizing effect of armature reaction), the speed decreases. When alternating current is being delivered by the converter at unity power factor, the magnetizing or demagnetizing effect of the armature current is insignificant. At other power factors, however, a lagging current weakens and a leading current strengthens the field, as in a generator, and makes a corresponding change in speed and in frequency. A machine designed for operation as an inverted converter should, therefore, be designed with a magnetically weak arma- ture ; or, some device should be provided for controlling the exci- tation and maintaining the speed constant. This is sometimes done by using an exciter mounted on the same shaft so that any increase of speed of the converter is checked by the increase of exciting current which it produces. §23. Test. — Operate an inverted converter with an inductive load* and with a non-inductive load adjusted for the same value. (Caution: Be careful to avoid excessive speed.) With constant field current, compare the speeds in the two cases. (Complete curves from no load to full load may be taken when desired.) § 24. Note the change in field current necessary to produce the same speed in the two cases. §25. Compounding with Series Reactance. — With a given series reactance, determine the number of series turns needed to give the same D.C. voltage at full load as at no load (§11); proceed as with a D.C. generator (§28, Exp. i-B). * An induction motor, locked, may be found convenient for this. 330 SYNCHRONOUS MACHINES. [Exp. §26. With a given series winding, flat-compounding can be obtained by trial by adjusting the series reactance or adjusting a shunt around the series turns (§23, Exp. i-B). §27. Derived Neutral. — Obtain a derived neutral (§12) and test as a converter or as a generator with unbalanced load. CHAPTER XL WAVE ANALYSIS. Experiment ii-A. Analysis of a Complex Wave by the Method of 1 8 Ordinates. § I. Introductory. — ^An alternating current or electromotive force is rarely an exact sine wave ; in addition to the fundamental wave or first harmonic it usually comprises odd harmonics of 3, 5, 7, etc., times the fundamental frequency. (Even harmonics* are never present when the negative half-wave is a repetition of the positive half-wave.) If we are given the ordinates — or certain ordinates — of a complex wave, we can "analyze" it into its components, that is, we can find the fundamental and the harmonics of higher frequency of which it is composed, each component wave being defined by its amplitude and its phase position with respect to the fundamental. § 2. Any complex wave in which there are no even harmonics (the negative half -wave being a repetition of the positive) can be represented by a Fourier's series consisting of the following sine and cosine terms : 3; = ^1 sin JT -|- /ig sin 3;ir -|- ^5 sin 5;ir • • • -)-5iCOs;t: + 53cos3^ + -B,coss.r.-.. (i) X is an angle varying with time ; thus x = sin Ar-f-sin cos x) = A sin x-{-B cos x, where A = VA' + B' cos . This will be seen more clearly by constructing a right triangle with C as the hypothenuse and A and B as the two sides. 331 332 WAVE ANALYSIS. [Exp. y = C^sm(x + ,f,^)+CsSm(:ix + ^)+C^sini5x-{-,t>i)---' (2) where* C, = + VA,' + B,'; C,= +VA,'' + B,'';etc. (2a) <3!>i=tan-i-^; <^3 = tan-i-^; et^. (2b) The first term in (2) represents the fundamental; the remain- ing terms represent the harmonics of 3, 5, 7, etc., times the fundamental frequency. Their ampHtudes are given by C^, C^, C^, etc., and their relative phase positions by ^j, ^3, <^^, etc. The abso- lute values of <^i,<^3, <^g, etc. (and the corresponding values of A^,A^,As, etc., and B^,B^,B^, etc., but not of C^,C3,C^, etc.), depend upon the origin or point of reference from which angles are measured. In the following analysis the origin from which the angles <^i, ^3, c^g, etc., are measured is determined by the selec- tion of the initial ordinate (see Fig. 2) where y=3'o when x = o°. It is convenient, when plotting, to measure time or angle from the zero of the fundamental wave. We therefore rewrite (2) by substituting x — ^^ for x; thus, y=C^smx + C^smi\x+-j-^A+C^smS\x+ y-<^ij (s) or, y=C^smx+ 6*3 sin i{x + a^ + C^ sin 5(jr + aj (4) where «3=y-<^i; «5 = -y--<^i; etc. (4a) * (§ 2b). In computing , note the signs of B and A ; thus ■o with + B and + A, we have =: -|- tan"' — • A n with — B and — A, we have ^ -|- tan""^ _-t-i8o°; with — B and -{-A, we have 0= — tan"' — ; n with + B and — ^, we have = — tan"' — ± 180° ; where +tan"' -3 is a positive angle (from 0° to +90°) and — tan"' ;^ is a negative angle (from 0° to — 90°). ii-A] WAVE ANALYSIS. 333 § 3. Equation (4) is the most convenient for plotting and for general use. The phase angles a^, a^, etc., are measured from the zero of the fundamental in the same angular scale as the funda- mental wave, as in Fig. i ; that is, 180° always represent half a wave of the fundamental and not half a wave of each particu- lar harmonic. (On the other hand, 1^3, 5, etc., are measured each to the scale of the particular harmonic; the scale for 3 measures 180" for a half-wave of the third harmonic, the scale for ^5 measures 180° for a half-wave of the fifth harmonic, etc.) A positive phase angle indicates a leading wave, as the third y=Ci Binas+Casrn 3 (a;H-a:s)+C5 sm SCai+as) = 100 sin CJ! + 30sin3(a;+30°) + I5sinS(a;— 15°) Fig, I. Complex wave composed of a fundamental, a leading third and a lagging fifth harmonic. Phase angles a^ and a^ are measured from zero of the fundamental. harmonic in Fig. i ; a negative phase angle indicates a lagging wave, as the fifth harmonic in Fig. i, this being the usual notation in alternating currents. Note that in measuring to or from a zero of a wave the zero selected is always one where the wave changes from negative to positive. 334 WAVE ANALYSIS. [Exp. §4. We are given the ordinates of a complex wave. The process of analysis consists first in determining the values of Ai,A^,A^, etc., and B^,B^,B^, etc. These values may then be substituted in (i) ; or, as is more useful, the values of Cj, C3, C^, etc., i, <^3, s, etc., and a^, a^, etc., are computed and substituted in (2), (3) or (4). § 5. When the wave to be analyzed is given in the form of a curve — as an oscillograph record, for example — the values of the ordinates necessary for computation are measured from the curve. In some cases, however, the values of the ordinates are determined directly by experiment — as in the determination of an alternating current wave by the point-by-point method of instan- taneous contact — and in this case plotting the curve is unnecessary. The ordinates used must be equi-distant and must be known Fig. i. Showing 18 ordinates taken in a half wave. for one half of a wave. The following method is based upon 18 ordinates for one half wave and is sufficient for determining the amplitude and the phase of the odd harmonics up to and includ- ing the seventeenth ; even harmonics are assumed to be absent. Ji-A] WAVE ANALYSIS. 335 For the origin of the method, see Appendix I. ; for the determina- tion of even as well as odd harmonics, see Appendix II. §6. Procedure. — Ascertain the values of i8 equidistant ordi- nates distributed over any interval of i8o°, as yo>yi>y2'" yn ^^ Fig. 2, the ordinates being determined for every lo degrees. The initial ordinate y^ may have any position whatsoever, without reference to the zero or maximum of the curve; 3'i8,3'i9, etc., are repetitions of J'o, 3'i, etc., and are not used, unless 3;,, and y^a, y^ and 3)19, etc., are averaged for greater accuracy. In all cases care should be taken to note the algebraic sign ; all additions, subtrac- tions and multiplications are algebraic. § 7. Scheme. — ^Arrange the 18 ordinates in a scheme as shown. Scheme (18 ordinates). Sums Write the algebraic sums and differences as indicated, where Ji=yi + 3'iT;----y9=3'o; do=yo; ds=ys—yio- Certain of the values thus obtained are further combined, alge- braically, in the following manner : yo yi ^2 ^3 ^4 y^ 3*6 y7 ys 3'9 y.7 3'l6 yi5 3'l4 yis 3'l2 yii yio Si Jj •^3 •^4 •^5 •^6 ^7 •fs \ do d. d. ^3 d. d. d. d, d. •?l+-f5— -^T^-fl' d,-d, = d,' •f 2 ~\~ ■^i -^8 ^^= ^^2 d,- -d,-d,=d,' J3 J9 = Sg d,- -d,-d,^d,' •f 1 — •>3 — -^ d,'-d,'^d". 336 WAVE. ANALYSIS. [Exp. §8. Tabulating. — These values should be placed in the Table, after being multiplied by the sine of the angle shown in the first column; thus, Si denotes the algebraic product of s^ and the sine of io°, or, 81 = ^1X0.1736, etc. Write the algebraic sums of the first columns (i, 3, 5, 7, 9) on line I., and of the second columns (17, 15, 13, 11) on line II. The line (I. -f-H-) is found by adding line II. to line I.; the line (I. — II.) is found by subtracting. Dividing these results by 9 gives the values of A.^, Ag, ■ ■ ■ A^^ and B^, B^, ■ ■ • B-^^, as shown in the last two lines, which may be substituted in equation ( i ) . § 9. Check. — ^As a check on the computation, the following rela- tions should hold, each constant being given its proper sign : A^ — A^ + A, — A,+A, — A,^ + A^, — A,,+A^, = y^; B, + B, + B, + B, + B, + 5ii + 5i3 + B,, + B^,=y,. Table (18 ordinates). Sine Components. Cosine Components. Harmonic. I 17 3 15 5 13 7 " 9 I 17 3 15 5 13 7 " 9 sin 10° sin 20° sin 30° sin 40° sin 50° sin 60° sin 70° sin 80° sin 90° Si s., S3 Si S5 Se S7 Sg Ss s/ s/ S3' -S7 -S4 S3 Ss Si -Se -Ss Sz S9 -Ss -Ss -Ss s, Se Si -s, -s. s" ^' d d ' d * ' d do '' d/ d/ do' d ~^' d *' '^* d d ' do ^ A d ' ^ d A ^' do d" I. Sum 1st col. II. Sum 2d col. I. +11. I.— II. ,(I. + II.) J{I.-II.) a' a' As Al3 A, All A, B3 Bl5 Bl3 b' Bii B, Calculate Ci, C3, Ci, etc., as in equation 20. Calculate ^i, ^3, c, etc., as in equation 26. Calculate as, a,, etc., as in equation 4a. ii-A] WAVE ANALYSIS. 337 § lo. Example. — Required to analyze a wave of which the fol- lowing i8 ordinates at io° intervals are known (see Fig. 2) : yo =— 18.0 311= — g.o yj =+ i.o 3/3=+ 14.0 jii =+ 46.0 3^=+ 72.0 yi>—+ 99.0 JiT = + 119.0 ys =+127.5 > =+128.0 3'" = + 123.0 3)ii = + ii6.o 3ii2= + lo8.o 3;is = + 97.5 jii4 = + 85.0 yu — + 73.0 3r„ = + 57.S 3;i, = + 41.0. These are arranged according to the scheme of §7. Sums Differences Sums Differences Scheme. — 18.0 — 9.0 + 1.0 + 14.0 + 46.0 + 41.0 + S7.S + 73.0 + 85.0 — 18.0 + 32.0 — 50.0 + 58.5 - 56.S + 87.0 — 59.0 + 131.0 — 39-0 + 72.0 + 97.S + 99.0 + 108.0 + 1 19.0 + 1 16.0 + 127.5 + 123.0 + 128.0 + 169.5 — 25. s + 207.0 — 9.0 + 235.0 + 3.0 + 250.5 + 4.5 + 128.0 si' = + 32.0 + 169.S — 235.0 = — 33.5 i=' = + 58.5 + I3I.0 — 250.5 = — 61.0 j3' = + 87.o — 128 = — 41.0 •f" = — 33.S+ 41 =+ 7-5 da' = — 18.0 +9.0 = — 9.0 di = — 50.0 + 25.5 — 3.0 = — 27.5 (k' = — 56.5 + 39.0 — /f.S = — 22.0 d" = — 90 + 22.0 = + 13.0 Table. Sine Components. t 17 3 15 5 13 7 " 9 sin 10° + S.S6 — 40.81 — 29.43 sm 20° + 20.01 — 44.S0 - 85.67 sin 30° + 43.50 - 16.7s + 43.50 — 43.50 sin 40° + 84.21 +I6I.0Z + 37.60 sin so" + 129.84 + 24.51 + 180.02 sin 60° + 179.27 — 52.83 —179.27 + 179.27 sin 70" +220.83 -159.28 + 30.07 sin 80° + 246.69 + S7.6i — 129.01 sin 90° +128.00 — 41.00 +128.00 —128.00 +7.50 I. Sum 1st col. + 527.73 — 57.75 —4.08 +9.16 +7.50 II. Sum 2d col. + 530.18 -52.83 —5-44 + 2.19 +7.50 I. +11. + 1057.91 — 110.58 -9.52 + 11.35 + 7.50 I. — II. - 2.45 — 4.92 +1.36 + 6.97 i{i.+n.) ^i=-+ii7.55 ^3 =—12.29 -45=— 1.06 ^,=+1.26 ^,=+0.83 h (1.-11.) ^„=— 0.27 -^16=— °-S5 ^,8=+o.i5 ^i,=+o.77 Check : Ai — A3 + As — A,-\ \-Aij= 128.01 ; jid = 128.00. 23 33^ WAVE ANALYSIS. [Exp. Cosine Components. I 17 3 IS s 13 7 " 9 sin 10° + 0.78 -r 9.81 -6.77 sin 20° + 1.03 + 8.72 — 17.10 sin 30° — 4.50 — 11.00 — 4.50 — 4.50 sin 40° —16.39 —32.14 — 1.93 sin 50° -29.87 + 3.4s +43.28 sin 60° -SI. 09 —23.81 +51.09 +51.09 sin 70° -53.09 +36.65 — 4.23 sin 80° —49.24 + 2.95 —25.11 sin 90° —18.00 — 9.00 —18.00 — 18.00 + 13.00 I. Sum 1st col. —104.68 — 20.00 +27.41 +9.78 + 13.00 II. Sum 2d col. -115.69 -23.81 +30.62 +6.95 I.-F II. -220.37 -43.81 +58.03 +16.73 + 13.00. I.— II. + II.OI + 3.81 — 3.21 + 2.83 i{i.+ u.) ^1 =—24.49 ^,=-4.87 ^6 =+6.45 .5- =+1.86 ^,=+1.44 HI— n.) Al=+ 1-22 ^,6=+0.42 -»i3=-o-36 ^,,=+0.31 Check: Bi + Bi + B^ + B, h Bit = — 18.02 ; yo^- 18.00. Ci == 120.07 =— 11.8° 0° Cy =: 2.25 ■^7= + 55-9° Ci3= 0.39 and a. Added Note. — For a convenient method of constructing schedules for even as well as odd harmonics, see H. O. Taylor, Phys. Rev., 1915. 340 WAVE ANALYSIS. [Exp. (m — i) harmonic, assuming that higher harmonics are negligible; thus, i8 ordinates determine harmonics to the seventeenth, 6 ordinates determine harmonics to the fifth, etc., assuming that no higher har- monics are present to an appreciable extent. Reducing the number of ordinates used, however, not only reduces the number of harmonics that can be determined but reduces the accuracy with which these are determined. For example, if the seventh or ninth harmonic has considerable amplitude, the method with 6 ordinates in general would not accurately determine even the third and fifth harmonics. The more ordinates used, therefore, the more accurate is the method, but the labor is likewise increased. Except in work of the greatest precision, the use of more than i8 ordinates in a half-wave is hardly worth while, for the accuracy of the data will rarely warrant it. On the other hand, it does not pay to make an analysis with too few ordinates and to risk errors in the result, unless only an approximate analysis is desired. For simplicity, the following discussion is limited to the method using 1 8 ordinates, but it can be readily modified so as to apply when more ordinates, or less, than i8 are used. § 13. Development of Method. — For practical use the infinite series in equation ( i ) must be limited to a finite number of terms ; thus, ex- cluding all harmonics above the seventeenth, we have y = /4i sin ;tr + A, sin 3^ + A^ sin Sx ■■■ A„ sin ijx + B^ cos x-\-B, cos 2,x + B, cos c,x ■■■ B„ cos lyx. '(S) Substituting for x 18 known consecutive values (0°, 10°, 20°, etc.) and for y the corresponding 18 known values {y„ y^, y„ etc.), we have 18 simultaneous equations of the first degree which may be solved for the 18 unknown coefficients A^ to ^„ and B, to B„. The coefficients of the Mth harmonic may be written as summations, in which the values of k vary from to 17; thus 2 *=" A„ = —^ jC,sin«/6io°, (6) 2 *='' B„= —^ % y^cosnk 10°. 10 i=n ii-A] WAVE ANALYSIS. 34 1 § 14. Proof. — The foregoing expressions may be derived* as follows. To determine a coefficient A^, multiply the first of the 18 equations by sin 0°, the second by sin n 10°, the third by sin 2« 10°, etc., and add. The sumf of all terms that contain An on the right hand of the equations (after multiplying) is gA^; the sum of the other terms is zero. The sum of the left hand terms may be written as a summa- tion. Thus, 2 y^ sin nkio°= gA^. (7) * = o Transposing, we have the value of A„, as in (6). The value of 5„ is similarly found by multiplying by cosines instead of sines. § 15. Determining Values for Individual Coefficients. — Given the general expression (6) for An and B„, the next step is to find parti- cular expressions for A^, A^, A„ etc., which may be conveniently used in a numerical solution. It will suffice to determine A^ as an. illustra- tion. In (6) or (7), let w = 3 and assign values for k from o to 17. We then have gA, = + y, sin 0° + y^ sin 30° -|- y, sin 60° -|- y, sin 90° + y„ sin 180° + 3^5 sin 150° + y^ sin 120° -\- y, sin 210° + 3)5 sin 240° + % sin 270° 4- y,, sin 360° + 3i„ sin 330° + y,, sin 300° 1 (^) + 3;^ sin 30°-|-3f„sin 6o°-|-3i„sin 90° + 3;„ sin 150° + y,^ sin 120° Since sin 150° = sin 30°, sin 210°= — sin 30°, sin 330°=. — sin 30°, sin 120° = sin 60°, sin 240°^ — sin 60°, sin 300=: — sin 60, sin 270°^ — sin 90", we may write (8) as follows: 9^8 = (^1 + y,r + y. + yn — y, — yn) sin 30° + (y^ + 3'ie + ^4 + 3-1. — ^8 — ^lo) sin 60° 4- (ys + yis — 3'.) sin 90° * (§ 14a). General Expression for Coefficients. — Determined more gen- erally for an infinite number of terms, the coefficients of the Mth order are 2 /*T 2 f^ A^= — I y sin nx ■ dx; B^ = — I y cos nx IT Jo irJn dx. (See Byerly's Fourier's Series and Spherical Harmonics, Qiap. II. ; Tod- hunter's Int. Calculus, Chap. XIII. ; Greenhill's Int. Calculus, § 183 ; etc.) t If there are m terms and m equations instead of 18, this sum is imAn, the average value of the sine of an angle, squared, being ^. 342 WAVE ANALYSIS. ' [Exp. A^ = 1/9 ( j/ sin 30° + j/ sin 60° + s,' sin 90° ) , which is the form used in the Table, § 8. § 16. A^ is developed in terms of sin k 30° while ^„ is developed in terms of sin fe 150°. The latter can be written sin fe (180° — 30°), which expanded gives sin k 180° cos k 30° — cos k 180° sin k 30°. It is evident that, when k is odd, sin k 30° = sin k 150°, while when k is even sin fe 30° = — sin ^ 150°; hence, A^^ can be determined from A^ by changing the signs of all terms in the third column of equation (8). Hence A^ = 1/9 [j/ sin 30° — j/ sin 60" + j/ sin 90°]. The other coefficients are determined in a similar manner. §17. Check. — In equation (5), 3» = 3'8 when x^=<)0°; and y^y, when x=^o°. Substituting these values, we have A^ — A, + A, — A,----^A^, = y^; After analyzing a wave, these equations may be used to check the values of A^, A^, A„ etc., and B^, B„ B^, etc. APPENDIX II. ANALYSIS OP WAVE WHICH MAY HAVE EVEN AND ODD HAR- MONICS AND A CONSTANT TERM. § 18. Method of 12 Ordinates. — In the most general case,* when the negative part of the wave is not equal to the positive and is not a repetition of it, both even and odd harmonics may be present and also a constant term, B„. To analyze such a wave, see Fig. 3, equidistant ordinates must be taken over an entire period, or 360°, and not merely for a half period, as in the preceding pages when odd harmonics only were considered. Let ordinates be takenf at intervals of 30°, i. e., there are 12 known ordinates in a complete wave. * This general method, which applies whether there is a constant term or not and whether odd or even harmonics are present or not, is taken directly from Runge's article, where the method is given in detail for 12 and for 36 ordinates. fFor greater accuracy (§ 12) more ordinates must be used. ii-A] WAVE ANALYSIS. 343 Arrange these in the scheme, as shown ; then compile the table, after multiplying by the sine of the angle indicated : thus, a^ = o, sin 30°. Fig. 3. Wave with even and odd harmonics and a constant term ; 12 ordinates used. Scheme (12 ordinates). Ordinates : y« y. y. y. y. y. yii Vro y. y> y. y. Dif?.: d. d. d. d. d. Slim: \ ■Ti ^2 ■fs s. ■f. s. < di ^. •fo \ J-. •^3 «i K K h 4 d. Sum: ^6 h h Diff.: «3 K Sum : h h Sum : a^ a- «» ^ K K \ a" b" ^0 ^1 Diff.: a' a,' Diff.: K i: i.: Table (12 ordinates). Sine Components, Cosine Components. ' S 2 4 3 J 5 2 4 3 6 sin 30° = 0.500 sin 60° =0.866 singo° = i.ooo ai aa as a/ a/ a" b/ V -b, b. b„ -bs b" Co C, I. + II. I. — II. i(I. + II.) ^5 % ^8 1^ 1: ^. 2^0 25. Check: A^ — A,A-A, + B, — B^ + B, — B, = y,; —A, + A, — A, + B„-B, + B,-B, = y,; B, + B, + B, + B, + B, + B, + B, = y,; B,-B, + B,-B, + B-B, + B, = y^ 344 WAVE ANALYSIS. [Exp. § 19. Example. — Required to analyze the wave shown in Fig. 3, the following 12 ordinates at 30° intervals being known : 3'o = +-i4-o 3/1 = + 15.8 y,=+i2.o ys= + 7-7 y. = + 4-3 3'= = + 1-4 y,=— 4-6 y,= — 6.8 y, = — 8.2 y, = — 9-3 3'io = — 9-9 3'u = — 6.8 Scheme. Ordinates : + 14.0 + 15.8 + 12.0 + 7.7 + 4.3 + 1.4 — 6.8 — 9.9 — 9.3 — 8.2 — 6.8 — 4.6 Diff. : + 22.6 + 21.9 + 17.0 + 12.5 + 8.2 Sum: +14.0 + g.o + 2.1 — 1.6 — 3.9 — 5.4 — 4.6 + 22.6 + 8.2 Sum : + 30.8 Diff. : + 14.4 + 30.8 + 17.0 + 21.9 + 12.S + 34-4 + 9-4 + 18.6 + 6.0 Diff. : + 13.8 + 12.6 + 17.0 + 14.0 + 9.0 + 2.1 — 1.6 — 4-6 — .5-4 —3-9 + 17.0 Sum : + 9-4 + 3-6 Diff. : + 18.6 + 14.4 + 9.4 +■ 3-6 — 1.8 — 1.6 Sum: + 7.6 + 2.0 — 1.8 + 6.0 — 1.6 Table. Sine Components, Cosine Components. I s 2 4 3 ' S -' 4 3 6 sin 30° =0.500 sin6o°=o.866 sin 90° =1.000 +15.4 +29.8 +17.0 + I2.S+8.I + 13-8 + 3.0 + I2.S + 18.6 +0.9 +1.8 +9.4 +1.6 +12.6 +7.6+2.0 I. Sum 1st col II. Sum 2d col. +32.4 +29.8 + I2-S + 8.1 + 13.8 + 21.6 + 12.3 + 10.3 + 3-4 + 12.6 +7.6 + 2.0 I.+ II. I.— II. +62.2 + 2.6 +20.6 + 4.4 + 13-8 + 34.1 + 9.1 + 13.7 + 6.9 +12.6 +9.6 +5.6 HI-+"-) ^(i.-ii.) ^i=+io.4 ^6=+ 0.4 ^.=+3-4 ^4=+0-7 ^s=+^-3 ^i=+S.7 ^6=+ '-5 ^»=+2.3 ^,=+1.1 5,=+2.I ^„=+o.8 ^6=+0-S Check: A,-A, + A, + B„-B, + B,-B,= -A, + A, — A, + B,-B, + B, — B, = - Check: B, + B, + B, + B, + B, + B, + B„ = B,-B, + B,_-B, + B,-B, + B, = - 7.6; y,= 7-7 ; 94; y,=— 9-3; 14.0; 3'o= 14-0; 4.6; % = — 4-6. CHAPTER XII. PROBLEMS. Many who are efficient in carrying out standardized experi- ments are not so efficient in carrying out experiments for which no instructions are given. It is very important to possess such ability and it can be acquired only by attacking problems which demand initiative and responsibility. It is futile to prepare a schedule of problems of this sort with any expectation of its being adequate or complete; some of the problems here given, however, may prove useful or suggestive. Various reference books and periodicals should be consulted in most cases before proceeding with experimental work. A familiarity with original sources and the ability to give proper weight to different authorities is highly desirable. § I. Given an over-compounded D.C. generator. Determine a shunt to go in parallel with the series coils to produce a definite regulation (as flat compounded or, say, 5 per cent, over compounded) for a cer- tain speed and voltage. Determine how the result would be affected, and the cause for it, if the generator is operated at the same speed, but at a different voltage (say 100 instead of 125 volts) ; or at the same voltage but different speed. If the source of power is an induc- tion motor, how will its slip enter into the problem ? § 2. Determine the relation between electromotive force and speed in a separately excited generator and in a self-excited shunt generator. §3. Determine the relation between line voltage and speed in a shunt motor. § 4. Operate two D.C. generators in parallel, first as shunt and then as compound machines, and ascertain how any desired division of the load is obtained. In the case of compound generators, an equalizing bus-bar is necessary connecting the two brushes (one on each machine) to which the series coils are connected. § 5. Explore the field of a dynamo-electric machine by determining the distribution of the flux in the air gap. 34S 346 PROBLEMS. [Chap. § 6. Determine the relation between the total flux set up by the field windings of a dynamo-electric machine and the useful flux that passes through the armature. The ratio of the former to the latter is the dispersion, or leakage, coefficient. § 7. Given a separately excited D.C. motor the armature of which is supplied with current from a series generator. Investigate and explain the conditions affecting the direction of rotation of the generator and the conditions under which the direction will periodically reverse. The motor should have brushes which will not damage the commutator when the direction of rotation is reversed. § 8. In a separately excited motor in which the armature resistance drop is so large that the counter-electromotive force is practically negligible (as in certain watthour meters), determine the relation between speed and field excitation when line voltage is constant. § 9. Design, have made, and test commutating interpoles for some machine which commutates badly. § 10. Find the relation of potential drop to current density between brushes of various materials and slip rings at usual speeds; also at very high speeds. § II. Analyze all the losses in a given machine or apparatus. § 12. Make a study of the temperature rise in a machine or appa- ratus by thermometer and by resistance measurements. § 13. Given a differential D.C. motor which runs too fast at full load. Determine a shunt to go in parallel with the series coil that will give a certain speed at no load and full load. Determine whether the same shunt will do for a different speed, and report as to why it will or will not. § 14. Take some point concerning which you find your knowledge inadequate, on some subject you have already studied, and if possible plan an experiment to settle the matter to your own satisfaction. § 15. Take a technical article which proves of interest (as the paper or papers of some A. I. E. E. meeting) and investigate such points as you can in the laboratory. The Digest of the Electrical World and the Question Box of the Electric Journal, can be used to advantage as a source of timely practical problems. § 16. Given a patent specification and claims. Investigate the inven- tion by experiment and study, and report on one or more of the fol- lowing: (i) Its usefulness (from the standpoint of a possible user or XII.] PROBLEMS. 347 purchaser) ; (2) its apparent novelty, including points which differ- entiate this invention from other methods or apparatus for securing similar ends; (3) its operativeness without further invention. (To be valid, a patent must be new, useful and operative.) § 17. Determine the insulation resistance of a machine or line by means of a voltmeter. § 18. Make a study of a Tirrell or other voltage regulator. § 19. Study the electrolytic or " pail forge " method of heating rods for welding. The following solution may be used: 10 gal. water; | lb. borax; 3^ lb, sal soda; i lb. salt. A D.C. dynamo of 200 to 300 volts has one terminal connected to a submerged lead plate in the solution. The other terminal is connected to the rod or to a horizontal piece of copper upon which the rod rests when contact is desired. The rod becomes heated when submerged, if the current flows in the proper direction. § 20. Determine the torque of a machine by the electrical method of McAllister, using a shunt motor as load ; see Standard Handbook and McAllister's Alternating Current Motors. § 21. Investigate various methods for obtaining a neutral on a 3-wire D.C. system. § 22. Operate two alternators in parallel and study the conditions that determine the division of the load between the two machines. § 23. Determine the characteristics of a high frequency alternator and note the effect of lagging and leading currents upon the terminal voltage. With condensers in parallel with the load, no field excitation may be necessary. Care is necessary in this test as there is danger of excessive voltage. § 24. Select and use one or more methods for determining induc- tance and capacity with a fair degree of accuracy. § 25. Connect in series two electromotive forces, one alternating and the other direct (or alternating of a different frequency). Meas- ure the combined voltage and determine the relation between it and the separate voltages. § 26. Superpose in a conductor with resistance R an alternating cur- rent /, and a direct current I^ (or an alternating current of different frequency). Determine the relative values of the copper loss {RI^ and RI') for each current alone and {RP) for the total current, I. Determine the relation between the effective values of /, I^ and L. 348 PROBLEMS. [Chap. § 27. Given a transformer with an open magnetic circuit, as the now obsolete " Hedgehog " transformer. Investigate the transformer with a view to bringing out the characteristic differences between it and a transformer with a closed magnetic circuit. Report on the relative advantages and disadvantages of the two types. At one time this subject was much debated. §28. Required to find at what frequency, current and voltage a given transformer will give the highest efficiency. The voltage should not exceed a certain specified value; the temperature rise should not exceed the limit set by A. I. E. E. Standardization Rules ; assume that any frequency, from say 20 to 125 cycles, is available. Outline com- pletely the method of procedure before making tests. § 29. Make a comparison of loading-back methods for testing trans- formers. § 30. Make a general study of a series current transformer ; make a particular study of the accuracy of its ratio at different frequencies and of phase errors when used with a wattmeter. § 31. Determine the effect of different wave forms upon transformer losses, regulation and magnetizing current. § 32. Study the effect of wave form upon circle diagrams and other vector diagrams. § 33. Given a 2-phase 4-wire supply from two independent trans- formers. Insert between the two neutrals an additional source of electromotive force of different frequency (or, direct current). Meas- ure all voltages and construct a vector model. What must be the value of the inserted electromotive force to cause all voltages between line wires to be equal (see §93, Exp. 7-A). § 34. Determine the variation in the starting torque of an induction motor for different values of secondary resistance. § 35. An induction motor was purchased for a certain frequency. The central station equipment has been changed to a different fre- quency. Can this motor be used? If so what voltage will be best? Will any benefit come by changing its 3-phase primary from star to delta or vice versa? § 36. Test an induction generator with condenser excitation. § 37. Test the method of obtaining polyphase from single-phase current by means of an induction motor (§ 3, Exp. 7-A) and a similar method using a synchronous motor. XII.] PROBLEMS. 349 § 38. Make a study of devices for indicating synchronism and of methods for synchronizing. § 39. Construct a shunt for an A.C. Wattmeter so as to extend its range ; it should be designed so as to be correct at different fre- quencies. § 40. Determine the losses of a machine by the retardation method ; see Standard Handbook. § 41. Determine the hysteresis loss in different dielectrics. § 42. Set up and test an electrolytic or a mercury arc rectifier. § 43. Make a study of potentiometer methods for measuring alter- nating currents and voltages. § 44. Make a comparison of the behavior of various types of instruments at commercial frequencies and at high frequencies. § 45. Determine the current efficiency and energy efficiency of a storage cell. § 46. Adapt the Ryan-Braun Tube to the measurement of power — particularly of small power. § 47. With an oscillograph, study the behavior of fuses and circuit breakers with D.C. and A.C. loads, inductive and non-inductive. § 48. Determine the instantaneous values of resistance, for example, of a lamp with alternating current, or with direct current at brief intervals of time after the circuit is closed. § 49. Determine the instantaneous values of flux in a transformer, either indirectly (from instantaneous values of current and voltage) or directly by some special device; see Bulletin of the Bureau of Standards, Vol. IV., p. 467. § 50. Make a study of different methods and apparatus for measur- ing one of the following quantities, with a view to determining their relative advantages and developing modifications of new methods: (a) slip; (&) frequency; (c) speed; (d) phase; (e) form factor; (/) wave form; (g) power factor; (h) reactive factor; (i) very small (or large) alternating current, voltage or power. § 51. Make a study of an interpole motor, repulsion motor, series A.C. motor, or other particular machine. § 52. Make a study of " concatenation," or other method of speed control for induction motors. § 53. Determine the wave form of alternating electromotive forces and currents by means of the Pierce Analyzer (§ iia, Exp. ii-A) and by other methods. INDEX. Acyclic dynamo, 2 Adams, C. A., on polyphase power measurements, 230 on synchronous motors, 316 Admittance, 104, 115 Aging of transformer iron, 174 Alternators, armature reaction of, 94 auxiliary field winding, 69 characteristics of, 62—72 components of magnetic flux in, 74 composite winding, 69 constant current, 67, 88 constant potential, 67, 88 design as affected by the steam turbine, 62 determining efficiency of, 71 electromotive force method for predetermining characteristics of, 7S, 80-90 impedance ratio of, 80 Institute rule for regulation of, 93 magnetomotive force method for predetermining characteristics of, 75, 91 predetermining characteristics of, 73-101 regulation of, 66, 69 regulation at lower factors, 99 split-field method of testing, 98 synchronous impedance of, 79 synchronous reactance of, 80 tests at low power factors, 98 types defined, 63 variation of characteristics with power factor, 70 Aluminum rectifier, 322 Ammeter, correction when used with a wattmeter, 148 current transformers for, 149 methods of connecting when used with a wattmeter, 148 Ampere-turn method of testing alter- nators, 75 method for synchronous motors, 307 Analysis of complex waves, 331 Apparent power, 113 35' Arakawa, B., on vector representa- tion of non-harmonic alternating currents, 182 Armature characteristic, of A. C. generators, 69 Armature characteristic, of D. C. generators, 23 Armature insulation, drying by short- circuit current, 78 Armature reactions, demagnetizing and cross-magnetizing effect of, 6 effect on alternator regulation, 73 effect on brush position, 6 effect on series characteristic, 9 in alternators, 74, 94 in D. C. generators, 19 in motors, 31 in synchronous motors, 306 local self-induction of armature conductors, 19 Armatures, closed coil, open coil, lap or parallel winding, wave or series winding, 3 function of, in generators, i peripheral speed of, 6 resistance of, 12, 42, 76 Asynchronous machine, 62, 29s Automatic synchronizers, 313 Auto-transformers, 134-136 advantages of, 133 as boosters, 135 for starting induction motors, 261 induction regulators as, 253 phase relation of primary and secondary currents, 13s step-up and step-down, 135, 136 Auxiliary field winding, 69 Average value of a sine wave, 146 Balance coil, 13s Balanced load, 197 Bedell, F., on separation of iron losses, 176 on transformer regulation, 167, 193 Bedell and Crehore, on equivalent re- sistance and inductance, 119 INDEX. 351 Bedell and Crehore, on effective and average values of a sine wave, 146 on current locus when resistance is varied in an inductive cir- cuit, 123, 288 on three-voltmeter method of measuring power, 118 Bedell and Ryan, on synchronous motors, 316 Bedell and Tuttle, on the effect of iron in distorting alternating cur- rent, 44, 182 Bedell and Steinmetz, on reactance, 115 Behrend, B. A., on circle diagram for induction motors, Behrend, B. A., alternator regulation, 75, 97, 100 split field method of testing, 98 Belt losses, 55 Berg, E. J., harmonics in alternating currents, 217 Berson, S., on harmonic analysis, 339 Blondel, A., alternator regulation, 100 loading back method, 56 on polyphase power, 240 on synchronous motors, 316 Brush positions, in generators, 6, 9 in motors, 31 Burt, A., polyphase power factor, 223 Capacity, circuits with, 120, 121 Cascade operation of induction mo- tors, 277 Characteristics of alternators, 62-101 armature, 65, 69 electromotive force method for determining, 73-96 external, 65, 67, 88 full-load saturation curve, 65-67, 77, 89 magnetomotive force method for determining, 73-96 no-load saturation curve, 65, 66, 77, 89 predetermination of, 73-101 Characteristics of compound genera- tors, 13-26 armature, 17, 23 compound, 20, 21 differential, 17, 21, 22 full-load saturation curve, 26 no-load saturation curve, 14 series, 17, 21, 23 shunt, 17, 18, 21 Characteristics of D.C. motors, 27-40 Characteristics of D.C. Motors, com- pound, 37, 38 differential, 37, 38 series, 39, 40 shunt, 37, 38 Characteristics of frequency changers, 291-294 Characteristics of induction genera- tors, 298-303 Characteristics of induction motors, 268, 281 Characteristics of series generators, S-12 external series, 7, 8, 9 magnetization curve, s, 9 total series, 8, 9 Characteristics of shunt generators, 14-20 armature, 17, 23 external shunt, 17, 18 full-load saturation curve, 26 no-load saturation curve, 14 total shunt, i!s, 19 Characteristics of synchronous con- verters, 327-330 ' Characteristics of synchronous motors, 314-31S Charters and Hillebrand, on capacity of induction motors, 279 Circle diagram, for circuits with re- sistance and reactance, 123-7 for constant potential transfer- ers, 179-igs for frequency changers, 292 for induction generators, 296, 300 for induction motors, 278-290 for synchronous motors, 320 Circular mils, s Closed coil armature, 3 Close regulation, 66 Coefficient of self induction, 103 Coil voltages, 3 Coils, polarity tests of transformer, III, 132-134, 142-143 Commutator, 3 Commutation, line or diameter of, 6 Commutating poles, 33 Compensated winding, 33 Compensator, 135 for starting induction motors, 261, 266 Composite field, 69 Composite transmission, 245 Composite currents, 347, 348 Compound generators, 13-26, 345 characteristics of, see Character- istics 352 INDEX. Compound generators, efficiency of, s8 uses of, 4 Compound motors, 27-61 characteristics of, see Character- istics Condenser excitation of induction gen- erator, 297 Condensers for starting single-phase motors, 267 Conductance, 115 of parallel circuits, 120 Constant losses, 47 Converter, synchronous, see Synchro- nous converter Copper economy of various systems, 220 Copper losses, in alternators, 72 in generators and motors, 45-49, 57 in induction motors, 272, 275, 284 in transformers, 129, 151, 161, 16s, 220-221 Core losses, in transformers, 129, 151, 155-158, I73> 174 Cosine method, of measuring power factor, 225 Cotterill, belt losses, SS Counter-electromotive force, in a motor, 28-31, 34, 50 in a synchronous motor, 305 Crehore, A. C, see Bedell and Cross-magnetizing flux, in alterna- tors, 74, 95 in generators, 6 Current, apparatus for obtaining con- stant, 126 in a D. C. circuit, 103 .in an A. C. circuit, 103, 115, 123 method of adjustment, 10 per phase, 212 Currents, addition of alternating, 201- 202 Current locus, when resistance is varied in an inductive circuit, 123, 125-126, 187, 190-192 for induction motor, 278 for synchronous motor, 320 Current ratio, for converters, 323 Cycle, 62 Damping of hunting in synchronous motors, 309 Delta connection, 197, 206-208 Delta voltage, 204 Derived neutral, 326 Diameter of commutation, 6 Diametrical 6-phase connection, 211 Differential generator, characteristics of, 22 Differential motor, equivalent shunt excitation, 59 speed and torque of, 36 test of, 59 uses of, 4 Direction of rotation, effect on pick- ing up of shunt machine, 1 5 Dispersion coefficient, 346 Double current generators, 322 Double delta connection of 6-phase circuits, 211 Double T Connection of 6-phase circuits, 211 Double transformation, 244 Drying armature insulation by short- circuit current, 78 Durand, W. L., on rating of synchro- nous converters, 323 Dynamo, see Generator Dynamotors, 321 Eddy current loss, in generators and motors, 46, 50-53 in transformers, 173-176 Effective value of a sine wave 105, 146 Efficiency of alternators, 71 of generators and motors, 41-61 of transformers, i66, 178 Electrolytic welding, 347 Electromotive force, how generated, I, 2, 28 method of alternator testing, 75, 80-90 methed for synchronous motors, 306, 316, 320 non-sinusoidal, 121, 122 of self induction, 107 Electromotive forces, addition of, in a series circuit, 109, 114 addition of, in polyphase circuits, 199-201, Z13-214 Electrical degree, 105 Equivalent inductance, 119 Equivalent reactance, 150 Equivalent resistance, 119, 150, 159- 160, 271, 272 Equivalent sine wave, 105 Equivalent single-phase quantities, 211-213 Even harmonics, 342 Everest, A. R., on transformer regu- lation, 193 INDEX. 353 Excitation characteristic, see Char- acteristics External characteristics, see Char- acteristics External revolving field, 63 External series characteristics, see Characteristics External shunt characteristics, see Characteristics Faraday's disk dynamo, z Faraday's principle, i, 107, 133, 144, 181 Ferguson, O. J., on rating of con- verters 323 Field compounding curve, see Char- acteristics (armature) definition of, 23 Field magnets, i, 4 Fischer-Hinnen, on wave analysis, 339 Flather, dynamometers and the meas- urement of power, 41 Fleming, J. A., on wave analysis, 339 Flux, magnetic, 2 relation to electromotive force, 144 unit of, 16 Flux density, see Saturation in transformers, 139, 158 Form factor, definition of, 146 effect on iron losses, 176 Fourier's series, 331 Franklin and Esty, on acyclic homo- polar dynamos, 3 on maximum efficiency, 57 on predetermining alternator characteristic!!, 91 on synchronous motors, 319 on regulation of alternators, 100 Frequency, best frequency for differ- ent machines, 64 changers, 291-294 effect on exciting current, 153 effect on transformer losses, 156 relation to speed, 62, 6s Friction and windage, law of, 47 in series motors, 61 Full-load saturation curve, see Char- acteristics definition of, 26 Gauss, 16 Generators, alternating current, see Alternators armature reactions in, 6, 19 24 Generators, asynchronous and syn- chronous, 62 brush position, 6 characteristics of, see Character- istics compounding by added turns, 2 compounding by armature char acteristic, see Characteristics constant current, 127 direct current, 1-26 efficiency of, see Efficiency fields of, I, 4 loading back, 55 number of poles in, 27 study of, i-s tests on polyphase, 70 torque in, 27, 28 Gilbert, unit defined, 16 Guilbert, on alternator regulation, loa Harmonic analysis, 339 Harmonics in delta and star connec- tions, 217-219 Harrison, J., on wave analysis, 339 Hedgehog transformer, 130 Henderson and Nicholson, on regu- lation of alternators, 100 Henry, unit of self induction, 108 Herdt, L. A., on regulation of alter- nators, 100 Hobart and Punga, on regulation of alternators, 100 Homopolar dynamo, 2 Hopkinson, on motor testing, 56 Housman, R. H., on separation of losses in a motor, 50 Houston, E. J., on wave analysis, 339 Hutchinson, on motor testing, 56 Hutchinsin, C. T., on the induction motor for traction, 257 Hysteresis, coefficient of, 174 effect of aging on, 174 separation from eddy current loss in motors, 50-53 separation from eddy current loss in transformers, 175 variation with speed and fliix, 46 Hysteretic angle of advance, 182 Impedance, 103, us Impedance ratio, 80, 165 Impedance triangle, 109 Impedance voltage, 162 Inductance, 103-121 calculation by impedance method, 116 354 INDEX. Inductance, calculation by three- voltmeter method, 117 calculation by wattmeter method 116 Induction generator, 62 test of, 295-303 circle diagram for, 296, 300 excitation of, 297, 301 uses of, 297 Induction regulators, 230-236, 324 Induction motors, 257-277 " added " current in, 283 ■' added " losses in, 283 as frequency changer, 293 as potential regulator and phase shifter, 252 auto-transformers for starting, 261 best frequency for, 64 cascade operation of, 277 circle diagram for, 278-290 compensator for starting, 266 condensers for starting, 267 copper losses in, 272 equivalent single-phase resistance of, 271 graphical construction for effi- ciency of, 287 graphical construction for power factor of, 286 graphical construction for slip of, 285, 286 leakage coefficient of, 290 leakage factor of, 290 leakage reactance of, 289 multispeed, 276 output, measurement of, 269, 270 phase splitters for starting sirigle- phase, 264, 266 predetermination of characteris- tics of, 278-290 primary winding of, 259 resistance measurements of, 271 rotating field, 258 rotor of, 257 secondary losses in, 274 secondary resistance of, 289 secondary resistance for starting, 262, 265 separation of losses in, 284 shading coils for starting single- phase, 263 single-phase, 266, 276 slip of, 62, 259, 274 speed of, 259 speed control by secondary resis- tance, 277 Induction motors, speed control by varying frequency, 277 speed control by varying poles, 277 squirrel cage secondary, 260 starting as repulsion motor, 263 starting compensator for, 261 starting polyphase, 261 starting single-phase, 262 starting torque of, 260 stator of, 257 structure of, 257 phase-wound secondary, 260 tandem operation of, 277 torque of, 274, 285 torque maximum of, 260 variable speed, 276 Inductor generator, i, 62, 63 Insulation tests, of transformers, 177 Internal revolving field alternators, 63 Internal shunt characteristic, see Characteristics Interpole motor, 33 Inverted converter" 322, 329 Iron, aging of, in transformers, 130 Kapp, G., on motor testing, 50, 56 Karapetoff, V., on heat runs of trans- formers, 178 on motor testing, 56 on alternator regulation, 100 Kelley and Spoehrer, on variation of transformer core loss, 175 Kennelly, A. E., on temperature co- efficient of copper, II on wave analysis, 339 Kintner, S. M., on wave analysis, 339 Kirchhoff's law, 200 Lamps for synchronizing, 310, 313 Langsdorf, A. S., on wave analysis, 339 Lap winding, 3 Leakage coefficient, 290, 346 Leakage factor, 290 Leakage reactance, 129, 163, 289 Lincoln, P. M, on wave analysis, 339 Line current of 3-phase system, 205 Line drop in polyphase system, 202- 204 Line of commutation, 6 Line voltage of 3-phase system, 204 Lloyd, M. G., on iron losses, 176 Load losses, defined, 46, 72 in transformers, 161 INDEX. 355 Load losses, Institute rule for esti- mating, 72 Loading back, a generator, 53 a transformer, 177 Local armature reaction, 74, 306 Long shunt, compound generator, 21 Magnetic flux, 2, 16 Magnetic leakage in transformers, 137 Magnetic shunt, potential regulator, 252 Magnetic units, 16 Magnetization curve, see Character- istics Magneto-generators, 4 Magnetomotive force method of al- ternator testing, 75, gi method for synchronous motors, 307 Maximum efficiency, in motors, 49, 56 in induction motors, 282 in transformers, 16,7 Maximum torque, in induction mo- tors, 260, 282 Maxwell, unit defined, 16 McAllister, A. S., on changing from 3-phase to 6-phase converters, 211 on equivalent single-phase quan- tities, 212 on measuring torque, 40, 347 on power factor of unbalanced systems, 223 on synchronous motors, 316, 320 on 2- to 6-phase transformation, 248 Mercury arc rectifier, 322 Mesh-connected systems, 196, 197,211 Mesh method of representing alter- nating currents and electromotive forces, 216 Metcalfe, G. R., on alternating cur- rent potential regulators, 252 Michelson, A. A., on wave analysis, 339 Monocyclic transformation, 246 Moody, W. S., on alternating current feeder regulators, 252 Mordey, W. M., on divided armature method of alternator testing, 98 on separation of losses in motors, SO Motor generators, 54, 321 Motors, armature reactions in, 31 asynchronous, 62 best frequency for induction, 64 best frequency for series, 64 brush position in, 31 Motors, compound wound, 4 copper losses in, 46 damage of, by field discharge, 35 differential wound, 4 effect of breaking field of, when running, 34 efficiency of, by the measurement of losses, 41-61 interpole, 33 iron losses in, 46 operation and speed characteris- tics of, 27-40 rotation losses in, analyzed, 50 separation of losses in, 51-54 series wound, 4 shunt wound, 4 speed control of, 32 speed equation of, 30 speed of shunt, 31 speed regulation of, 37 stopping of, 34 synchronous, see Synchronous torque in, 28, 30 Motor generator, efficiency of, 54 Motor starters, automatic release, 34 multiple switch, 34 Multispeed induction motors, 276 Multipliers, for ammeters, 149 for voltmeters, 149 for wattmeters, 149 Neutral position of brushes, 6 Noeggerath, acyclic homopolar dy- namos, 3 No-load saturation curve, see Char- acteristics Non-inductive circuit, 103 Notation for polyphase circuits, 215 0-curves for synchronous motor, 315 Oersted, unit defined, 16 Odd harmonics, 331 Open coil armature, 3 Open delta in 3-phase system, 197, 209 Pail forge, 347 Parallel operation of generators, 345 Parallel winding, 3 Peripheral speed, of armatures, 6 Permutators, 322 Perry, J., on wave analysis, 339 Phase splitters, 264, 266 Phase wound induction motors, 260 Pierce, C. A., on harmonic analyzer, 339, 349 356 INDEX. Polygon method of representing al- ternating currents, 207, 216 Percentage of saturation,, 16 Phase shifters, induction regulators as, 254 Polyphase currents, 196-221, 222-240 Polyphase generators,, 70 Polyphase systems, 196-241 . addition of currents in, 201 addition of electromotive forces in, 199 copper economy of, 220 current and voltage per phase fn, 212 equivalent single-phase quanti- ties in, 211 harmonics in, 217 line drop in, 202 mesh method of representing A. C. quantities in, 216 methods of connecting, 196 polygon method of representing A. C. quantities in, 207 power factor of, 223 . radial method of representing A. C'.-quantities. in, 216 i 6-phase systems, 210 uniform flow- of power.ip, S41 Polyphase transformatiop, '241.-249 Polyphase wattmeter, 229 Porter, on radial method of repre- senting alternating current quanti- ties, 216 .it Potential regulators, 250-256 use of induction motors as, 252 induction regulators, 251 magnetic shunt, 252 polyphase, 255 single-phase, 252 step-by-step type, 251 use as transformers, 252 Potier, on motor testing, 56 Power, definition of, 116 in a non-inductive resistance, 113 in an inductive circuit, 113- measurement by three-ammeter method, 118 measurement by three-voltmeter method, 118 measurement of, in polyphase systems, 222-240 n wattmeter method of measur- ing, 228 n-i wattmeter method of meas- uring, 226 one wattmeter method of meas- uring 3-phase power, 233 Power, proof- of wattmeter methods of measuring power, 240 three-phase, 205, 206 two wattmeter method . for any 3-wire system, 228 Power current, iij Power factor, 103, 104, 114 by sine method, 237- by, tangent method,- 237 for equivalent sine waves, 105 for non-sine waves, 226 polyphase, 223, 225, 232, 237 Predetermination, of alternator char- acteristics, 73-101 Problems, 345 Prony brake, 41, 266, 268 Puffer, testing of > motors by opposi- tion method, 56 Quadrature relation of current and electromotive force, 106 Quarter-phase 2-phase sysieoj, 196' Radial method of represeivting alter- nating currents and electromotive -forces, 216 ,- - . Reactance, 103, 115 -v.. effect of frequency .on," i^l-jr. 122 by impedance method, ijfi , .. . - of a circuit coii^taining inductance and capacity, 121 , of a series -circuit, 109 .^ by three-voltmeter method, -i-*?. by wattmeter method, ij6 Reactance drop, in polyphase sys- tems, 203 ,, ^ in transformers, 129 in alternators, 6t, 74, 8o-g6 Reactance method of testing alterna- tors, see Electromotive force method Reactive factor, 114 Rectifiers, 322 Regulation, Institute rule on compu- tation of, 93 of alternators, (>(i, 67, 69, 71, 73, 82-84, 87, 99-100 of generators, 19, 22 of motors, 32 of transformers, 167, 193 . _ of transmission lines, loi .. . Reist and Maxwell, on multispeed in- duction motors,. 277 ,, Reluctance, 2 Repulsion motor, 263 Residual magnetism, effect on picking up of series machine, 8 INDEX. 357 Residual magnetism, effect on picking up of shunt machine, is Resistance, 109, iis Resistance' drop, in alternators, 66 73, 76, 79, 80, 81-83, 86 in generators, 8, 19, 24" Resistance drop, in motors, 29, 31 in polyphase systems, 202 in transformers, 129, 162, 165, 168-169, 185 Resistance measurements, by fall-of- potential method, 11 by substitution, 12 Resonance, current, 121 voltage, 121 Reversing motor, 346 Revolving armature, 2, 63 Revolving field, 2, 63, 258, 276 Ring connected 2-phase system, 196 Robinson, L. T., on electric meas- urements with current and poten- tial traflsformei-s, 14-9 Roessler, on form factor, 146 on influence of form factor- on iron losses, 176" Rotary converter, see Synchrolious converter. Rotation losses, 47, Jo, 60 Rotor, 257 ■ . • • . Rovlre and 'Hellmend, on induction motors, 288 - Runge, C, on v/zve analysis, '339 Rushmore, D. B., on regulation of alternators, 91, 100 on devolving field, alternators, 63 Ryan, H. J., on compensated wind- ings, 33 on synchronous motors, 316 Saturation, effect of, on compound- ing, 13, 14 effect of, on regulation of com- pound generator, 22 effect of, on regulation of shunt generator, 20 effect of, on series characteristic, 7 effect of, on speed of series motor, 46 Saturation curve, 14 Saturatiofi factor, 16 Scott, C. F., on two- to three-phase transformation, 243 Secondary losses, in induction motors, 274 Secondary resistance, for starting in- duction motors, 262, 265 Separation of losses, in generators and motors, 50-54 in transformers, 175-176 Series A. C. circuits, 102-1Z2 Series A. C. motors, best frequency for, 64 Series characteristic of compound generators, see Characteristics Series generators, 1-12. armature reactions, 6 brush position, 6 characteristics of, see Character- istics uses of, 4 Series motors, 4, 39-40, 60-61 Series turns, determination of, 24-26 Shading coils, for starting single- phase motors, 263 Sheldon, Mason and Hausman, on synchronous motor, 319 Short shunt connected generator, 21 Shunt generators, 13-20 armature reactions in, 19 characteristics of, see Character- ■ ■•. istics compounding of, 23-24, 26 direction of .rotation of, 15 efficiency of, 41-59 ^regulation of, ig; 20. uses of, 4, 13 Shunt motors, 27-61 armature reactions in, 31-32 brush position of, 31-32 efficiency of, 41-46 operation of, 27-40 speed characteristics of, 37-40 speed control of, 32-84 stopping of, 34 Shunt turns, determination of, 25 Sine method, of measuring power fac- tor, 226, 237 Single-phase currents, 102-122 Single-phase induction motors, 262, 276 starting as repulsion motor, 263 starting with phase splitter, 264 266 starting with shading coils, 263 Six-phase circuits, 210-211 Slip, in induction motors, 259,- 274, 29 S Smith, S. P., on alternator regula- tion, 99 Space degrees, 106 Sparking at brushes, 6, 22 Specht, H. C, on circle diagram for induction motors, 289 358 INDEX. Specht, H C, on speed control of in- duction motors, 277 Speed control, of D. C. motors, 32 of induction motors, 277 Speed equation of motors, 30 Speed regulation, 37 Speed, relation to frequency, 65 Split-field method of alternator test- ing, 98 Split-pole converter, 324 Squirrel cage winding, 260 Star-connected 2-pliase system, 196 3-phase system, 197, 204 Star current, 205 Star voltage, 20s Starting boxes, 34 Starting torque, of induction motors, 260 Static torque, 29 Stator, of induction motor, 257 Steam turbine, influence on design of alternators, 62 Steinmetz, C. P., on choice of fre- quency, 64 on definition of a balanced poly- phase system, 197 on form of external alternator characteristics, 89 on hysteresis exponent, 174 on monocyclic transformation, 247 on separation of iron losses, 176 on topographic method, 198, 199 on wave form, 217 Steinmetz and Bedell, on reactance, IIS Stratton, S. W., on wave analysis, 339 Stray power, 47 method of motor testing, 41-61 Susceptance, 115 of parallel circuits, 120 Swenson and Frankenfield, on motoi testing, 56 Symmetrical polyphase system, 196 Synchronous converters, 321-330 best frequency for, 64 current ratios of, 323 induction generators with, 301 inverted converter, 322, 329 rating of, 323 synchronizing of, 310, 311, 327 voltage control of, 324, 325 voltage ratios of, 322 Synchronizers, automatic, 313 Synchronizing, methods for, 310, 327 Synchronous booster, 324 Synchronous commutator, 322 Synchronous generators, see Alterna- tors Synchronous impedance, 79, 80, 306, 316 Synchronous machine, definition, 62 Synchronous motors, 304-320 armature reaction of, 306 auxiliary winding for starting, 304 circle loci of, 320 counter-electromotive force in, 305 current locus of, 319 electromotive force method of treating, 306, 316, 320 hunting of, 308 induction generators with, 301 local reactance in, 306 magnetomotive force, method of treating, 307 maximum power of, 319 O-curves of, 315 power of, 307 pulsating power and torque, 308 starting with alternating current, 309 synchronizing of, 310, 311, synchronous impedance of, 306, 316 synchronous reactance of, 316 synchroscopes for, 313 torque of, 307 v-curves of, 314 Synchronous reactance, 80, 316 Synchronous speed, 259 Synchronous watt, 29 Synchroscopes, 313 Tandem operation of induction mo- tors, 277 T-connected transformers, delta equiv- alent of, 249 for 2- to 3-phase transforma- tion, 243 star equivalent of, 249 voltage and current relations, 248 T-connection of 3-phase circuits, 197. 208 Tangent method of measuring power factor, 22S, 232, 237 Teaser circuit of monocyclic sys- tem, 247 Temperature coefficients of copper, 11 Temperature corrections, formula for, 10 INDEX. 359 Temperature of transformers, see Transformers Temperature rise, computed from change in resistance, ii Third harmonic, in delta-connections, 217 in generator coils, 219 in star-connections, 218 Thompson, S. P., on regulation of alternators, 100 on wave analysis, 339 Thomson, Sir William, on wave an- alysis, 339 Three-phase systems, 197 delta and star currents and vol- tages, 204 measurement of power in, 228 230, 233 power in, 205 transformation to 2-phase, 243 Thury system of direct current power transmission, 221 Time degrees, 106 Tirrell regulator, 13, 70, 251 Topographic method, 199, 203 Torda-Heymann, on regulation of alternators, 100 Torque, expressions for, 28 electrical measurement of, 40 how created, 28 in a generator, 27 in a motor, 30 in compound motor, 35 in differential motor, 36 in induction motor, 274, 280, 28s in series motor, 39, 40 in single-phase and polyphase machinery, 197 in synchronous motor, 307 static, 29 Total characteristics (see Character- istics), 7, 8, 19 Tuttle, E, B., on the effect of iron in distorting wave form, 182 on wave analysis, 339 Transformer, adjustment of voltage in testing, 172 aging of iron in, 130 all-day efficiency, 167 auto-transformers, 134-136 best frequency for, 64 circle diagram, 179-195 circulating current test, 144 computation of efficiency, i66 constant current, 127 constant current from constant potential, 190 Transformer, copper loss, 163 core loss, 155 current density in, 140 current ratio of, 143 design data, 139-142 electromotive force and flux, 144, 146 efficiency, 139, 178 equivalent circuits of, 186-192 equivalent leakage reactance of, 150 equivalent primary quantities, 187 equivalent resistance of, 150, 159, 160 exciting current, 137, 151, 153, -154 flux density in, 140 form-factor, effect of, 176 general discussion of, 179-189 harmonics due to hysteresis, 182 heat runs, 177 heating of, 130 hysteresis in, see Hysteresis insulation tests, 177 impedance, 163 impedance ratio, i6s impedance voltage, 162 instruments for testing, 171 load losses in, 161 loading back method, 177 losses in, 129 magnetic densities in, 141 magnetic leakage in, 137 magnetizing current in, 134, 180 maximum efficiency of, 167 net and gross cross-section, 140 normal current and voltage in, 173 open circuit test, 151 operation and study of, 128-149 phase of primary and secondary electromotive forces and cur- rents, 134 polarity of coils, 132, 133 polarity test by alternating cur- rent, 142 polarity test by direct .current, 143 polyphase, 131, 210 potential ratio of, 143 ratio of transformation of, 133 reactance drop in, 129 reactance of, 163 regulation of, 139, 167, 193, 194 resistance drop in, 129 resistance of, 163 secondary quantities in terms of primary, 187 36o INDEX. Transformer, separation of hysteresis and eddy current losses in, 173 series, 131 short-circuit test, 160 step-up and step-down, 128 systems of polyphase connections, 210 T-connection of, 243 test by the method of losses, 130^178 total voltage drop in, 194 tub type, 131 types of, 130 variation of core losses in, 156-158, 173-176 voltage and current transforma- tion, 128 volts per turn, 141 weight of copper and iron in, 141 Transmission lines, regulation of, loi Two-phase system, 196 laboratory supply, 201 power factor, 225 transformation to 3-phase, 243 transformation to 6-phase, 248 Unipolar dynamo, 2 Variable speed induction motors, 276 V-connection, of 3-phase circuits, 197, 209 of auto-transformers for starting motors, 248 V-curves, of synchronous motor, 314 Vectors, addition and subtraction of, 213-216 direction of rotation of, 105 for representing admittances, 118 for representing currents and electromotive forces, 105 for representing impedance, re- sistance and reactance, 109 for representing non-harmonic quantities, 122 Vectors, relative accuracy when ap- plied to inductive and capacity circuits, 122 significance of 105 Voltage adjustment, 10 Voltage ratio of converters, 322 Voltage per phase, 212 Voltmeters, damage due to induced electromotive force, 12 methods of connecting when used with a wattmeter, 148 multipliers for, 149 power consumed in, 148 Waters, W. L., on induction genera- tors, 298 Wattless current, 115 Wattless power, 107 Wattmeters, correction for power consumed in instrument, 151 errors in, 146-149 multipliers for, 149 n wattmeter method of measuring power, 228, 240 n-i wattmeter method of meas- uring power, 226, 240 negative reading of, in 3-phase power measurements, 233 one wattmeter methods of meas- uring 3-phase power, ^33 polyphase, 229 two wattmeter method of meas- uring power, 228 Wave analysis, 331 Wave winding, 3 Welding, electrolytic, 347 Whitehead and Grondall, on synchro- nous commutators, 3224 Woodbridge, on converters, 211 Workman, on motor testing, 56, 61 Y-connected, see Star-connected