Cornell University Library 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/cu31924004437079 CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1891 BY Cornell University Library TK 2712.L41 Principles of a"ernatirig jju^"* Sjj^ 3 1924 004 437 079 PRINCIPLES OF ALTERNATING CURRENT MACHINE ELECTRICAL ENGINEERING TEXTS A Series of Textbooks Outlined by the Following Committee. Harbt E. Cl/iFroRD, Chairman and Consulting Editor, Gordon McKay Professor of Electrical En- gineering, Harvard University, and Massa- chusetts Institute of Technology. Murray C. Beebe, Professor of Electrical Engineering, University of Wisconsin. Ernst J. Berg, Professor of Electrical Engineering, Union College. Paul M. Lincoln, Engineer, Westinghouse Electric and Man- ufacturing Company, Prtrfessor of Electrical Engineering, University of Pittsburgh. Henry H. Norris, Associate Editor, Electric Railway Journal, Formerly Professor of Electrical Engineering, Cornell University. George W. Patterson, Professor of Electrical Engineering, University of Michigan. Harris J. Ryan, Professor of Electrical Engineering, Leland Stanford Junior University. Elihu Thomson, Consulting Engineer, General Electric Co. William D. Weaver, Formerly Editor, Electrical Wffrld. ELECTRICAL ENGINEERING TEXTS PEINCIPLES OP ALTERNATING CURRENT MACHINERY BY RALPH R. LAWRENCE ASBOCIATB PBOPESSOR OF ELECIRICAIi ENGINEERING OP THE MASSACHUSETTS INSTITUTE OP TECHNOLOGY AND HARVARD UNIVERSITY; MEMBER OP THE AMERICAN INSTITUTE OP ELECTRICAL ENGINEERS, ETC. First Edition McGRAW-HILL BOOK COMPANY, Inc. 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., Ltd. 6 & 8 BOUVEEIE ST., E. C. 1916 Copyright, 1916, by the McGraw-Hill Book Company, Inc. THBJ AI^PI,S} PSBSs T O H HI PA PREFACE This book deals with the principles underlying the construc- tion and operation of alternating-current machinery. It is in no sense a book on design. It is the result of a number of years' experience in teaching the subject of Alternating-current Machinery to senior students in Electrical Engineering and has been developed from a set of printed and neostyled notes used for several years by the author at the Massachusetts Institute of Technology. The transformer is the simplest piece of alternating-current apparatus and logically perhaps should be considered first in discussing the principles of alternating-current machinery. Experience has shown, however, that students just beginning the subject grasp the principles of the alternator more readily than those of the transformer. For this reason the alternator is taken up first. No attempt has been made to treat all types of alternating- current machines, only the most important being considered. Certain types have been developed in considerable detail where such development seemed to bring out important principles, while other types have been considered only briefly or omitted altogether. No new methods have been used, but it is believed that bringing together material which has been much scattered and making it available for students is sufficient reason for the publication of the book. Mathematical and analytical treatment of the subject has been freely employed where such treatment offered any advantage. The symbolic notation has been used throughout the book. The author wishes to express his sincere thanks to Professor W. V. Lyon of the Massachusetts Institute of Technology for many suggestions and especially to Professor H. E. Clifford, Gordon McKay, Professor of Electrical Engineering at Harvard University and the Massachusetts Institute of Technology, who critically read the original manuscript and offered many sugges- tions. The author also wishes to express his thanks to Mr. N. S. vi PREFACE Marston for his care in reading the proof, and to the Crocker- Wheeler Company, the General Electric Company and the Westinghouse Electric and Manufacturing Company who fur- nished photographs from which the drawings of machines were prepared. Ralph R. Lawrence. Massachusetts Institute of Technology, Boston, September, 1916. NOTATION In general fhe notation recommended by the American Institute of Electrical Engineers has been followed. Throughout the book E has been used to denote a voltage generated or induced. V has been reserved for a terminal voltage which could be measured by a voltmeter. V differs from E by the impedance drop in the machine or part of the machine con- sidered. The line which is often used over quantities in equations to indicate that they are to be considered in a vector sense has been omitted in all but one or two cases. Most of the equations in the book are to be considered in a vector sense. Those which are purely algebraic are readily distinguished. In general the letters used have the following significance: A = Armature Reaction, generally expressed in ampere turns per pole. A' — Fictitious Armature Reaction, including real armature reaction and the effect of the leakage reactance, generally expressed in ampere turns per pole. a = Ratio of Transformation. (B = Flux Density. b = Susceptance. E = Induced or Generated Voltage. El = Primary Induced Voltage of a transformer or of an induction motor. El = Secondary Induced Voltage of a transformer or of an induction motor. F = Impressed Field of a synchronous generator or motor, generally expressed in ampere turns per pole. [F = Magnetomotive Force. / = Frequency. / = Function. g = Conductance. I = Current, /p = Magnetizing Current of a transformer or of an induction motor. h+e = Core-loss Current of a transfornler or of an induction motor. /„ = Exciting Current of a transformer or of an induction motor, /'i = Load Component of Primary Current of a transformer or of an induction motor, /i = Primary Current of a transformer or of an induction motor. I2 = Secondary Current of a transformer or of an induction motor. $ = Moment of Inertia. j = Operating Factor which rotates a vector anti-clockwise through ninety degrees. kb = Breadth Factor, fcj, = Pitch Factor. N = Turns. « = Speed, or Number of Phases. viii NOTATION P = Power, p = Number of poles. p.f. = Pdwer Factor. R = Resultant Field of a Synchronous generator or motor, generally expressed in ampere turns per pole. (R = Reluctance. r = Resistance. ri = Primary Resistance of a transformer or of an induction motor, rs = Secondary Resistance of a transformer or of an induction motor. r« = Effective Resistance or Equivalent Resistance of a transformer, s = Slip or Number of Slots per Phase. T = Torque. V = Terminal Voltage. Vi = Primary Terminal Voltage of a transformer or of an induction motor. V2 = Secondary Terminal Voltage of a transformer. X = Reactance. Xo = Leakage Reactance of a generator or motor. Xi = Primary Reactance of a transformer or of an induction motor. X2 = Secondary Reactance of a transformer or of an induction motor. Xe = Equivalent Reactance of a transformer. Xj = Synchronous Reactance. y = Admittance. Z = Number of Inductors. z = Impedance. Zs = Synchronous Impedance. a = Angular Acceleration or a Phase Angle. 7j = Efficiency or Hysteresis Constant. 6 = Angle of Lag. p = Coil Pitch. S = Summation. (P = Fhix. 10 = Angular Velocity or 2^/. Where the letters given in the preceding table are used with other sig- nificance than as just indicated, it is so stated in the text. Where other letters are used, their meaning is stated in the text. CONTENTS Paqe Preface v Table of Symbols vi SYNCHRONOUS GENERATORS CHAPTER I Types of Alternators — Frequency — Armature Cores — Field Cores — 1 Armature Insulation — Field Insulation — Cooling — Filtering or Washing Cooling Air — Permissible Temperatures for Different Types of Insulation. CHAPTER II Induced Electromotive Force — Phase Relation Between a Flux and the 20 Electromotive Force It Induces — Shape of Flux and Electro- motive-force Waves when Coil Sides are 180 Electrical Degrees Apart — Calculation of the Electromotive Force Induced in a Coil when the Coil Sides are not 180 Electrical Degrees Apart. CHAPTER III Open- and Closed-circuit Windings — Bar and Coil Windings — Con- 27 centrated and Distributed Windings — Whole- and Half-coiled Windings — Spiral, Lap and Wave Windings — Single- and Poly- phase Windings — Pole Pitch — Coil Pitch — Phase Spread — Breadth Factor — Harmonics — Pitch Factor — Effect of Pitch on Harmonics — Effect on Wave- Form of Distributing a Winding — Harmonics in Three-phase Generators. CHAPTER IV Rating — Regulation — Magnetomotive Forces and Fluxes Concerned 51 in the Operation of an Alternator — Armature Reaction — Armature Reaction of an Alternator with Non-salient Poles — Armature Reac- tion of an Alternator with Salient Poles — Armature Leakage React- ance — Equivalent Leakage Reactance — Effective Resistance — Factors which Influence the Effect and Magnitude of Armature Reaction, Armature Reactance and Effective Resistance — Con- ditions for Best Regulation. X CONTENTS CHAPTER V Paob Vector Diagram of an Alternator with Non-salient Poles — Vector Dia- 84 gram Applied as an Approximation to an Alternator with Salient Poles — Calculation of the Regulation of an Alternator from Vec- tor Diagram — Synchronous-impedance and Magnetomotive-force Methods for Determining Regulation — Data Necessary for the Application of the Synchronous-impedance and the Magneto- motive-force Methods — Examples of the Calculation of Regulation by the Synchronous-impedance and Magnetomotive-force Methods — Potier Method — -American Institute Method — Example of the Calculation of Regulation by the American Institute Method — Value of A' of the Magnetomotive-force Method for Normal Saturation — Example of the Calculation of Regulation by the Magnetomotive-force Method Using the Value of A' Obtained from a Zero-power-factor Test — Blondel Two-reaction Method for Determining Regulation of an Alternator — Example of Cal- culation of Regulation by the Two-reaction Method. CHAPTER VI Short-circuit Method for Determining Leakage Reactance — Zero- 118 power-factor Method for Determining Leakage Reactance — Potier Triangle Method for Determining Reactance — Determination of Leakage Reactance from Measurements made with Field Structure Removed — Determination of Effective Resistance with Field Structure Removed. CHAPTER VII Losses — Measurement of the Losses by the Use of a Motor— Measure- 123 ment of Effective Resistance — Retardation Method of Determining the Losses — Efficiency. CHAPTER VIII Short-circuit Current jqq CHAPTER IX Conditions and Methods for Making Heating Tests of Alternators 136 without Applying Load. CHAPTER X Calculation of Ohmic Resistance, Armature Leakage Reactance, 142 Armature Reaction, Air-gap Flux per Pole, Average Flux Density in the Air Gap and Average Apparent Flux Density in the Arma- ture Teeth from the Dimensions of an Alternator— Calculation of Leakage Reactance and Armature Reaction from an Open-circuit Saturation Curve and a Saturation Curve for Full-load Current at Zero Power Factor- Calculation of Equivalent-leakage Flux per CONTENTS XI Page Unit Length of Embedded Inductor and Effective Resistance from Test Data — Calculation of Regulation, Field Excitation and Effi- ciency for Full-load Kv-a. at 0.8 Power Factor by the " A. I. E. E. Method. STATIC TRANSFORMERS CHAPTER XI Transformer — Types of Transformers — Cores — Windings — Insulation 151 — Terminals — Cooling — Oil — Breathers. CHAPTER XII Induced Voltage — Transformer on Open Circuit — Reactance Coil 164 CHAPTER XIII Determination of the Shape of the Flux Curve which Corresponds to a 169 Given Electromotive-force Curve — Determination of the Electro- motive-force Curve from the Flux Curve — Determination of the Magnetizing Current and the Current Supplying the Hysteresis Loss from the Hysteresis Curve and the Curve of Induced Voltage — Current Rushes. CHAPTER XIV Fluxes Concerned in the Operation of a Transformer and No-load 179 Vector Diagram — Ratio of Transformation — Reaction of Second- ary Current — Reduction Factors — Relative Values of Resistances — Relative Values of Reactances — Calculation of Leakage Reactance — Load Vector Diagram — Analysis of Vector Diagram — Solution of Vector Diagram and Calculation of Regulation. CHAPTER XV True Equivalent Circuit of a Transformer — Graphical Representation of 195 the Approximate Equivalent Circuit — Calculation of Regulation from the Approximate Equivalent Circuit. CHAPTER XVI Losses in a. Transformer — Eddy-current Loss — Hysteresis Loss — 200 Screening Effect of Eddy Currents — Efficiency — All-day Efficiency. CHAPTER XVII Measurement of Core Loss— Separation of Eddy-current and Hystere- 213 sis Losses — Measurement of Equivalent Resistance — Measurement of Equivalent Reactance, Short-circuit Method — Measurement of Equivalent Reactance, Highly-inductive-load Method — Opposition Method of Testing Transformers. xii CONTENTS CHAPTER XVIII Page Current Transformer— Potential Transformer— Constant-current Trans- 222 former — Auto-transformer — Induction Regulation. CHAPTER XIX Transformers with Independently Loaded Secondaries; Parallel Opera- 241 tion of Single-phase Transformers. CHAPTER XX Transformer Connections for Three-phase Circuits Using Three Trans- 252 formers — Three-phase Transformation with Two Transformers — Three- to Four-phase Transformation and Vice Versa — Three- to Six-phase Transformation — Two- or Four-phase to Six-phase Trans- formation — Three- to Twelve-phase Transformation. CHAPTER XXI Three-phase Transformers — Third Harmonics in the Exciting Cur- 272 rents and in the Induced Voltages of Y- and A-connected Trans- formers — Advantages and Disadvantages of Three-phase Trans- formers — Parallel Operation of Three-phase Transformers or Three-phase Groups of Single-phase Transformers — V- and A-con- nected Transformers in Parallel. CHAPTER XXII Ratio of Transformation, Flux and Flux Density — Primary and 288 Secondary Leakage Reactances, Equivalent Reactance, Primary and Secondary Resistances Calculated from the Dimensions of a Trans- former — Core Loss — Component of No-load Current Supplying Core Loss, Magnetizing Current and No-load Current Calculated from Dimensions of Transformer and Core Loss and Magnetization Curves — Equivalent Resistance and Equivalent Reactance from Test Data — Calculated Regulation and Efficiency. SYNCHRONOUS MOTORS CHAPTER XXIII Construction — General Characteristics — Power Factor — V-curves 297 Methods of Starting — Explanation of the Operation of a Syn- chronous Motor. CHAPTER XXI Vector Diagram — Magnetomotive-force and Synchronous-impedance 304 Diagrams — Change in Normal Excitation with Change of Load — Effect of Change in Load and Field Excitation. CONTENTS xiii CHAPTER XXV Paqh Maximum and Minimum Motor Excitation for Fixed Motor Power and 309 Fixed Impressed Voltage — Maximum Motor Power with Fixed Ea,', V, r, and a;,; Maximum possible Motor Excitation with Fixed Impressed Voltage and Fixed Resistance and Reactance — Maxi- mum Motor Activity with Fixed Impressed Voltage and Fixed Reactance anji Resistance. CHAPTER XXVI Hunting — Damping — Stability — Methods of Starting Synchronous 314 Motors. CHAPTER XXVII Circle Diagram of the Synchronous Motor — Proof of Diagram — Con- 330 struction of Diagram — Limiting Operating Conditions — Some Uses of the Circle Diagram. CHAPTER XXVIII Losses and Efficiency — Advantages and Disadvantages — Uses 338 PARALLEL OPERATION OF ALTERNATORS CHAPTER XXIX General Statements — Batteries and Direct-current Generators in 341 Parallel — Alternators in Parallel — Synchronizing Action, Two Equal Alternators — Synchronizing Current — Reactance is Neces- sary for Parallel Operation — Constants of Generators for Parallel Operation need not be Inversely Proportional to Their Ratings. CHAPTER XXX Synchronizing Action of Two Identical Alternators — Effect of Paral- 353 lehng Two Alternators through Transmission Lines of High Imped- ance — ^the Relation between r and x for Maximum Synchronizing Action. CHAPTER XXXI Period of Pha,se Swinging or Hunting — Damping — Irregularity of 361 Engine Torque during Each Revolution and Its Effect on Parallel Operation of Alternators — Governors. CHAPTER XXXII Power Output of Alternators Operating in Parallel and the Method of Ad- 370 justing the Load between Them — Effect of Difference in the Slopes of the Engine Speed-load Characteristics on the Division of the Load between Alternators which are Operating in Parallel — Effect of Changing the Tension of the Governor Spring on the Load Car- ried by an Alternator which is in Parallel with Others. xiv CONTENTS CHAPTER XXXIII Page Effect of Wave Form on Parallel Operation of Alternators 377 CHAPTER XXXIV A Resume of the Conditions for Parallel Operation of Alternators— 382 Difference between Paralleling Alternators and Direct-current Generators — Synchronizing Devices — Connections for Synchron- izing Single-phase Generators— A Special Form of Synchronizmg Transformer — Connections for Synchronizing Three-phase Gen- erators Using Synchronizing Transformers — ^Lincoln Synchronizer. SYNCHRONOUS CONVERTERS CHAPTER XXXV Means of Converting Alternating Current into Direct Current 393 CHAPTER XXXVI Voltage Ratio of an n-phase Converter — Current Relations 396 CHAPTER XXXVII Copper Losses of a Rotary Converter — Inductor Heating — Inductor 403 Heating of an n-phase Converter with a Uniformly Distributed Armature Winding — Relative Outputs of a Converter Operated as a Converter and as a Generator — Efficiency. CHAPTER XXXVIII Armature Reaction — Commutating Poles — Hunting — Methods of 414 Starting Converters. CHAPTER XXXIX Transformer Connections — Methods of Controlling Voltage — Split-pole 422 Converter. CHAPTER XL Inverted Converter — Double-current Generator — 60-cycle Versus 429 25-Cycle Converters— Motor Generators Versus Rotary Convert- ers. CHAPTER XLI Parallel Operation , 43g CHAPTER XLII Field Excitation and Efficiency Calculated from Armature Resistance, 437 Winding Data, Open-circuit Core Loss and Open-circuit Satura- tion Curves. CONTENTS XV POLYPHASE INDUCTION MOTORS CHAPTER XLIII Page Asynchronous Machines — Polyphase Induction Motor — Operation of 443 the Polyphase Induction Motor — Slip — Revolving Magnetic Field — Rotor Blocked — Rotor Free — ^Load is Equivalent to a Non- inductive Resistance on a Transformer — Transformer Diagram of a Polyphase Induction Motor — Equivalent Circuit of a Polyphase Induction Motor. CHAPTER XLIV Effect of Harmonics in the Space Distribution of the Air-gap Flux 455 CHAPTER XLV Analysis of the Vector Diagram — Internal Torque — Maximum Internal 460 Torque and the Slip Corresponding Thereto — Effect of Reactance, Resistance, Impressed Voltage and Frequency on the Breakdown Torque and Breakdown Slip — Speed-torque Curve — Stability — Starting Torque — Fractional-pitch Windings — Effect of Shape of Rotor Slots on Starting Torque and Slip. CHAPTER XLVI Rotors, Number of Rotor and Stator Slots, Air Gap — Coil-wound Rotors 468 — Squirrel-cage Rotors — Advantages and Disadvantages of the Two Types of Rotor. CHAPTER XLVII Methods of Starting Polyphase Induction Motors — Methods of Vary- 471 ing the Speed of Polyphase Induction Motors — Division of Power Developed by Motors in Concatenation — ^Losses in Motors in \ Concatenation. CHAPTER XLVIII Calculation of the Performance of an Induction Motor from Its Equiva- 484 lent Circuit — Determination of the Constants for the Equivalent Circuit. CHAPTER XLIX Circle Diagram of the Polyphase Induction Motor — Scales — Maximum 490 Power, Power Factor and Torque — Determination of the Circle Diagram. CHAPTER L General Characteristics of the Induction Generator — Circle Diagram 497 of the Induction Generator — Changes in Power Produced by a Change iri Slip— Power Factor of the Jnduction Generator — Phase xvi CONTENTS Page Relation between Rotor Current Referred to the Stator and Rotor Induced Voltage, £2— Vector Diagram of the Induction Generator— Voltage, Magnetizing Current and Function of Syn- chronous Apparatus in Parallel with an Induction Generator- Use of a Condenser instead of a Synchronous Generator m Parallel with an Induction Generator— Voltage, Frequency and Load of the Induction Generator— Short-circuit Current of the Induction Generator— Hunting of the Induction Generator- Advantages and Disadvantages of Induction Generators— Use of Induction Generators. CHAPTER LI Calculation of the Constants of a Three-phase Induction Motor for the 505 Equivalent Circuit— Calculation of Output, Torque, Input, Effi- ciency, Stator Current and Power Factor from Equivalent Circuit for a Given Slip. SINGLE-PHASE INDUCTION MOTORS CHAPTER LII Single-phase Induction Motor — Windings — Method of Ferraris for 511 Explaining the Operation of the Single-phase Induction Motor. CHAPTER LIII Quadrature Field of the Single-phase Induction Motor — Revolving 516 Field of the Single-phase Induction Motor — Explanation of the Operation of the Single-phase Induction Motor — Comparison of the Losses in Single-phase and Polyphase Induction Motors. CHAPTER LIV Vector Diagram of the Single-phase Induction Motor — Generator 527 Action of the Single-phase Induction Motor. CHAPTER LV Commutator-type, Single-phase, Induction Motor — Power-factor Com- 53S pensation — Vector Diagrams of the Compensated Motor — Speed Control of the Commutator-type, Single-phase, Induction Motor — Commutation of the Commutator-type, Single-phase, Induc- tion Motor. CHAPTER LVI Methods of Startmg Single-phase Induction Motors 545 CHAPTER LVII The Induction Motor as a Phase Converter , , , 551 CONTENTS xvii SERIES AND REPULSION MOTORS CHAPTER LVIII Page Types of Single-phase Commutator Motors with Series Characteristics 555 —Starting — Doubly Fed Motors — Diagrams of Connections for Singly and Doubly Fed Series and Repulsion Motors — Power- factor Compensation. CHAPTER LIX Singly Fed Series Motor — Vector Diagram — Approximate Vector 559 Diagram — Over- and Under-compensation — Starting and Speed Control — Commutation — Inter-poles — Construction, Efficiency and Losses of Series Motors. CHAPTER LX Singly Fed Repulsion Motor — Motor at Rest — Motor Running — 570 Vector Diagram — Commutation — Comparison of the Series and Repulsion Motors. CHAPTER LXI Compensated Repulsion Motor — Diagram of Connections — Phase 583 Relations between Fluxes, Currents and Voltages — Power-factor Compensation — Commutation — Vector Diagram — Speed Control and Direction of Rotation — Advantages and Disadvantages of the Compensated Motor. CHAPTER LXII Doubly Fed Series and Repulsion Motors — Doubly Fed Series Motor — 595 Approximate Vector Diagram of the Doubly Fed Series Motor — Commutation of the Doubly Fed Series Motor — Starting and Operating tiie Doubly Fed Series Motor — Doubly Fed Repulsion Motor — Doubly Fed Compensated Repulsion Motor — Regenera- tion by the Doubly Fed Compensated Repulsion Motor — Advan- tages of the Two Types of Doubly Fed Motors — Compensation and Commutation of the Doubly Fed Compensated Repulsion Motor — Starting and Speed Control of the Doubly Fed Compen- sated Repulsion Motor. Index 605 PRINCIPLES OF ALTERNATING- CURRENT MACHINERY SYNCHRONOUS GENERATORS CHAPTER I Types of Alternators; Frequency; Armature Cores; Field Cores; Armature Insulation; Field Insulation; Cool- ing; Filtering or Washing Cooling Air; Permissible Temperatures for Different Types of Insulation Types of Alternators. — Alternating-current generators do not differ in principle from generators for direct current. Any direct- current generator, with the exception of the unipolar generator, is, in fact, an alternator in which the alternating electromotive force set up in the armature inductors is rectified by means of a com- mutator. Although any direct-current generator, with the ex- ception of the unipolar generator, may be used as an alternator by the addition of collector rings electrically connected to suit- able points of its armature winding, it is found more satisfactory, both mechanically and electrically, to interchange the moving and fixed parts when only alternating currents are to be gener- ated. It is not only a distinct advantage mechanically to have the more complex part of the machine stationary, but it is, more- over, easier with this arrangement to protect and insulate the armature leads which usually carry current at high potential. The only moving contacts required are those necessary for the field excitation and these carry current at low potential. Alternating-current generators may be divided into three classes which differ mainly in the disposition and arrangement of their parts. The three classes are : (a) Alternators with revolving fields. " , . (&) Alternators with revolving armatures, (c) Inductor alternators. —. . 1 2 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY All modern alternators with very few exceptions belong to the first class for reasons which have already been stated. Inductor alternators differ from the other two types by having the varia- tion in the flux through their armature windings produced by the Fig. 1. Fig. 2. rotation of iron inductors. The windings of both the armature and the field of this type of alternator may be stationary. A distinguishing feature of an inductor alternator is that any one set of armature coils is subjected to fiux of only one polarity. This fluctuates between the limits of zero and maximum, but does SYNCHRONOUS GENERATORS 3 not reverse. Figs. 1, 2, and 3 illustrate the three classes of alternators in their simplest forms. Fig. 3 shows two views of one type of inductor alternator. The left-hand view is a portion of a section taken parallel to the shaft about which the inductor revolves. The other half of the figure is a side view. The letters on this figure have the following significance : /^— Field coil A— Shaft CC — Armature coils NIS — Inductor. By referring to Figs. 1 and 2 it will be seen that both sides of coils on the armatures of the revolving-field and the revolving- armature types of generators are in active parts of the field at the Fig. 3. same time, and, since the opposite sides of the coils are under poles of opposite polarity at each instant, the electromotive forces induced in them will be in phase with respect to the coil. The conditions are different in the case of the inductor type of alter- n,ator. In this alternator only one side of an armature coil is in an active part of the field at any time, the other side being be- tween two poles. Therefore, either the turns or the flux must be doubled in order to get the same voltage as would be obtained if the flux through the armature winding reversed as it does in the other two types of alternators. Inductor alternators are usually characterized by large armature reaction, relatively high magnetic 4 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY density, small air gap and greater weight than alternators of the other types. The difficulties in the design of a satisfactory induc- tor alternator have caused this type of alternator to go out of use. Frequency.— The commercial frequencies which are most com- mon in America are 60 and 25 cycles per second. In Europe both 50 and 40 cycles are used. Twenty-five cycles is used for long-distance power transmission, but so low a frequency is not suitable for lighting on account of the very noticeable flicker produced by it on arc lights and all incandescent lamps except those with filaments of large cross-section. A frequency of 25 cycles or less is best adapted for single-phase motors of the series or repulsion type such as are used for traction purposes. The frequency given by any alternator depends upon its speed and number of poles and is equal to f = -^ (1) •' 2(60) ^ ^ where/, p and n are, respectively, the frequency in cycles per sec- ond, the number of poles and the speed in revolutions per minute. The speed, and therefore the number of poles for which an alter- nator for a given frequency is designed, depends upon the method of driving it. Engine-driven alternators as well as alternators driven by water wheels operated from low heads must run at relatively low speeds and, consequently, they must have many poles. On the other hand, alternators driven by steam turbines operate at very high speeds and must have very few poles, usually from two to six according to their frequency and size. Low- frequency alternators are always heavier and therefore more expensive than high-frequency alternators of the same kilovolt- ampere rating and speed, but the advantages of low frequency for certain classes of work, notably power transmission and traction, usually more than balance the higher cost of the low- frequency alternators. Armature Cores. — The armature cores of all alternators are built up of thin sheet-steel stampings with slots for the armature coils on one edge. The opposite edge usually has either two or more notches for keys which are inserted in the frame in which the laminations are built up, or projections which fit in slots cut in the frame. Notches cut in the sides of the teeth serve to hold SYNCHRONOUS GENERATORS ■ Fig. 4. Fig. 5. Fig. 6. 6 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the wedges driven between adjacent teeth to keep the coils in place. Typical armature stampings are shown in Figs. 4 and 5, which illustrate, respectively, stampings for a slow- or moderate- speed alternator and a turbo alternator. The holes through the laminations for the turbo alternator form passages, when the laminations are built up, through which air is forced for cooling the armature. Fig. 7. The armature stampings are built up with lap joints in a frame or yoke ring, usually of cast steel, and are held from slipping either by keys inserted in the frame or by projections on the laminations. They are securely bolted together and to the frame between end plates. These plates usually have projecting fingers to support the teeth. Fig. 6 shows a typical frame for an engine-driven SYNCHRONOUS GENERATORS 8 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY alternator with one lamination in place. Fig. 7 gives a view of a portion of an armature core and frame and illustrates one form of end plate and a method of bolting the laminations together. The frame or yoke which supports the laminations is hollow arid is provided with openings for ventilation. The armature lamina- tions are separated in two or more places by the insertion of Pig. 9. spacing pieces in order to provide radial air ducts for cooling the armature. Except for very small generators, frames or yoke rings are made in two or more sections bolted together which may be separated for transportation. A complete engine-driven gen- erator is shown in Fig. 8. A typical frame for a turbo alternator with the laminations in SYNCHRONOUS GENERATORS 9 place is shown in Fig. 9. As turbo alternators require forced ventilation, they must be completely enclosed. Field Cores. — All slow-speed alternators of standard design have laminated salient or projecting poles built up of steel stamp- ings. These are bolted together and either keyed or bolted to a c o o o O u^^ ~::^ ^2^ Fig. 10. steel spider which is itself keyed to the shaft. Fig. 10 shows typical pole stampings. Fig. 11 shows the core of a complete pole of the bolted-on type both with and without the winding. This type is\lso illustrated in Fig. 8. A complete field with the poles and winding in place is shown in Fig. 12 Fig. 11. The field structure of a high-speed turbo alternator does not have projecting poles. It is cyUndrical in form and has slots cut in its surface for the field winding. Such alternators have from two to six poles according to their size and the frequency. Pro- jecting or salient poles would cause excessive windage losses and in addition would make a high-speed alternator very noisy. 10 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Moreover, it would be difficult if not impossible to make a field structure with salient poles sufficiently strong to safely stand the high speeds used for turbo generators. The field structure for a small or moderate-size turbo alter- nator is often a soUd steel forging. For a large machine it is btdlt up of thick discs cut from forged steel plates. The shaft, Fig. 12. except in small machines, does not as a rule pass through the core, as the hole required in the core for this would remove too much metal back of the slots which receive the field winding and thus weaken the structure. The shaft is in two pieces fastened to end plates securely bolted to the core. The distribution of flux over the pole faces is determined by the distribution of the field coils which are placed in slots cut in the core. SYNCHRONOUS GENERATORS 11 There are two types of cylindrical field cores which differ in the way in which the slots for the field winding are cut. These are the radial-slot and the parallel-slot types. They are illus- trated in Figs. 13 and 14, respectively, both of which show two- pole fields. When parallel slots are used for fields with more Fig. 13. than two poles, they produce the effect of salient poles. Four- pole parallel-slot fields are seldom used. The radial-slot type is the better in most cases, even when there are only two poles, as its teeth are subjected only to radial stresses. The teeth of the parallel-slot type of field, in addition to the radial stress, have to support a lateral stress arising from the centrifugal force on them and on the field coils. Fig. 14. Field cores with parallel slots have the slots cut across their ends as well as on their faces, permitting the exciting coils to be completely embedded. It is obvious, under these conditions, ■that the end plates which carry the shaft cover the end connec- tions of the winding and that no external support is required to 12 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY hold them in place. The end connections on each end of a radial- slot type of field are held in place by a steel ring of high tensile strength which covers them. With the two-pole parallel-slot type of field, the shaft must be made in two parts and bolted on, but the shaft may be integral with the core when more than two poles are used. Armature Insulation. — The conductors which form the coils of alternators, as well as the coils themselves, must be insulated in much the same way as the conductors and coils of direct-current generators. On account of the high voltages at which alterna- tors usually operate, they require much more insulation than direct-current machines. The materials used in the insulation of direct-current generators are often not suitable for alternators on account of the higher voltages of the latter and the higher temperatures often reached by their windings. Vulcanized fiber, horn fiber, fish paper, varnished cambric and paper, mica and other similar materials are used in the insulation of alternators. When high temperatures do not have to be re- sisted by the armature windings, double- or triple-cotton-covered wire is used for the coils. These are thoroughly impregnated with insulating compound after being wound and are then given several layers of varnished cambric or of some other similar material. With such coils it is necessary to insulate the slots with fiber or mica. Vulcanized fiber has a tendency to absorb moisture which causes it to expand and also reduces its insulating properties. For this reason it should not be relied upon to insu- late high-voltage machines. Mica is the only reliable insulation when high voltage or high temperature is to be encountered. The objection to mica is its poor mechanical properties, and for this reason it has to be used with other materials. For insulating slots it is split into thin flakes which are built up with lap joints into sheets with varnish or bakelite to cement the mica together. It is then baked under pressure. When built up in this way, the finished sheet may be moulded hot in U-shaped troughs or into other shapes for insu- lating the slots or other parts of the machine. Mica is now almost exclusively used for the insulation of high- voltage alternators and especially for the insulation of turbo alternators. With the high speeds necessary for turbo alterna- SYNCHRONOUS GENERATORS 13 tors, comparatively few armature turns are required for a given voltage. This increases the voltage per turn and necessitates more insulation between turns. Large machines often have only one or two turns per coil. Although fibrous insulation could still be applied which would withstand the higher voltage between turns, mica is the only substance which will withstand the high temperatures reached by certain parts of the coil which are em- bedded in the armature iron. The portions of the inductors and coils which are embedded in the slots are insulated with mica which is built up with varnish upon thin cloth or paper and applied to the straight portions of the conductors which lie in the slots as well as to similar portions of the finished coils. It is found in practice that the supporting cloth or paper may be destroyed by heat without impairing the insulation of the coils, provided they are firmly held in place in the slots. The percent- age of the space occupied by the cloth or paper is small as com- pared with the space occupied by the mica, and experience has shown that the complete carbonization of the cloth or paper by maintained high temperature does not cause the coils to loosen in the slots. The portions of the coils which lie without the slots, i.e., the end connections, are insulated with varnished cambric, mica tape or some similar material. The mica insulation is now generally applied and rolled hot on the straight portions of the conductors and coils by the Haefly process, which was developed in Europe and is extensively used in America. By this process the mica wrappers are so tightly rolled on the coil that they form a solid mass of insulating mate- rial of minimum thickness free from air spaces and having good heat conductivity. Mica insulation applied in the ordinary way has a heat conductivity of only 60 or 70 per cent, of that of var- nished cambric and similar materials. The static discharge which was often encountered between the armature copper and iron in the earlier high-voltage alternators is avoided when roUed-on mica insulation is used. As would be expected, the effect of the static discharge was most marked where there were sharp edges, as at the edges of the radial venti- lating. ducts. Its effect is to eat holes in and to pit the outside insulation of the coils, weakening or even destroying the insula- tion. One method of avoiding this static discharge, which has 14 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY been used with some success, consists of wrapping with tin foil the portions of the insulated coils which pass through the slots before giving them then- protecting layer of tape, and grounding the metallic sheath thus formed to the core. Field Insulation.— Since the fields of alternators are always wound for low voltage, 125 to 250 volts, the problem is not so much one of insulation as of providing a mechanical separation between the turns which shall be mechanically strong and shall withstand high temperature. Neither the problem of mechan- ical' strength nor of high temperature is serious in the case^ of slow-speed alternators, since the stresses and temperatures which have to be withstood by the field windings of such machines are not great. In case an alternator becomes short-circuited, the field winding may be subjected to high voltage during the initial rush of arma- ture current due to the transformer action which takes place be- tween the armature and field. This action as a rule is not serious. It is least in alternators with non-salient poles and low field react- ance. Sufiicient insulation must be provided on the field winding to guard against breakdown due to this cause. Generators with salient poles usually have their fields wound with double-cotton-covered wire with insulating strips between layers. After being wound, the coils are impregnated with in- sulating compound and taped. They are then placed on insu- lating spools of fiber or similar material and slipped over the pole pieces. Fields are often wound with flat strip copper wound on edge. In this case the successive turns are insulated from one another by insulating strips of thin asbestos paper or other ma- terial. The copper at the outside surface of edgewise-wound field coils is left bare to facilitate cooling. The windings of cylindrical fields, such as are used for turbo alternators, are subjected to much greater stresses, on account of the high speed at which they operate, than the windings of fields having salient poles. At times of short-circuit the stresses in the field windings of large turbo generators become very great. Ordinary cotton insulation would not have sufficient strength to withstand the severe crushing stresses at such times, especially if the insulation had become slightly carbonized by the high temperatures at which the fields of such machines generally op- SYNCHRONOUS GENERATORS 15 erate. The only material which will withstand the high tem- perature, and which is at the same time sufficiently strong, is mica. The slots of the cylindrical fields of turbo alternators are insulated with mica troughs and the separate turns of the field windings, which consist of flat strips of copper laid in the slots by hand, are separated from each other by thin strips of asbestos or mica paper. Cooling. — All generators are air cooled either by natural or by forced ventilation. There are four things which must be considered in the cooling or ventilation of any generator, namely: the total losses to be dissipated, the surface exposed for dissipat- ing these losses, the quantity of air required and the temperature of the cooling air. The rate at which heat is lost from any heated surface depends upon the difference in temperature between the heated surface and the cooling medium, which in the case of gen- erators is always air. If the quantity of air supplied is too small, the cooling air will reach a temperature which is nearly the same as the temperature of the surface to be cooled and little heat will be carried off. If, on the other hand, the quantity of air is large, its temperature will be only slightly increased. Any increase in the volume of air beyond this point will produce very little further gain in cooling and is wasteful. There is little difficulty in cooling slow-speed engine-driven generators. By providing proper ventilating ducts in the arma- ture laminations and openings in the frame, with, in some cases, fans added to the rotors, the cooling of such generators can be handled without difficulty. The conditions are, however, very different in the case of high-speed turbo alternators. The output of turbo alternators is very great per unit volume and the quantity of heat which must be dissipated per unit area of available cooling surface is very large. Forced ventilation must be used for such generators, and even with this it is exceed- ingly difficult to get sufficient air through the air gap and such other passages as can be provided. For this reason very large turbo generators must operate at a higher temperature than low- speed generators of smaller output and the insulation used in their construction must be such as to withstand the higher temperature. Mica insulation is universally used for such machines. 16 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY One kilowatt acting for one minute will raise the temperature of 100 cu. ft. of air approximately 18°C. Assuming a 20,000- kv,-a. generator with an efficiency of 97 per cent., 600 kw. will have to be taken up by the cooling air. If the increase in the temperature of the air in passing through the machine is not to exceed 20°, 600 X 100 X ^Mo = 54,000 cu. ft. of air will be re- quired per minute. If this has a velocity of 5000 ft. per minute, ventilating ducts of nearly 11 sq. ft. cross-section will be required. Since, in the case of such a machine, the air would be passed in from both ends, ducts of only half this cross-section will be required. With the cooling air passed in from both ends and with velocities as high as 5000 to 6000 ft. per minute, such as are actually used in practice, it would be exceedingly difficult to provide ventilating ducts of sufficient cross-section. The ven- tilating duct formed by the air gap between the field and arma- ture alone would not begin to be sufficient. To use forced ventilation it is obviously necessary to completely enclose a machine. There are three methods of artificially ventilating turbo alternators which are designated according to the way the cooling air is passed through the machine. These are radial ventilation, circumferential ventilation and axial ventilation. Air-gap ven- tilation is used in conjunction with these. Radial Ventilation. — In the radial method of cooling large alternators, the cooling air is passed in along the air gap from both ends and out through radial ducts made in the armature core by inserting spacing pieces between the armature lamina- tions. As a rule, when radial-slot rotxjrs are used they are provided with radial ducts. Air is passed through the rotor under the slots, out through these radial ducts and thence through the stator ducts. All of the air passes out through the radial ducts in the stator. The air gap alone, with any reason- able air velocity, is not sufficient in most cases to allow the pas- sage of sufficient air for cooling the stator. Radial ventilation has been used with success, but when applied to large gener- ators it is difficult to pass sufficient air to keep the stator cool. There is no difficulty in cooling the rotor, but the losses in it are not over 10 or 15 per cent, of the total losses to be taken care of. SYNCHRONOUS GENERATORS 17 Circumferential Ventilation. — When the circumferential method of ventilation is used, the air for cooling the stator is supplied to one or more openings in the circumference of the stator and passes around through ducts in the stator core in two directions from each opening and out other openings also in the circumference of the stator, without entering the air gap. If air is admitted at only one point on the circumference, it passes out at a point diametrically opposite. In addition to the air for cooling the stator, air must also be supplied to the air gap for cooling the rotor. Axial Ventilation. — A common objection to both the radial and circumferential methods of cooling is that the heat developed in the stator must pass transversely across the laminations to the air ducts in order to be carried off by the cooling air. The rate of heat conduction across a pile of laminations is not over 10 per cent, as great as along them. Since in both the radial and circumferential methods of cooling the heat must pass across the laminations to the air ducts, neither of these methods is as efficient as one where the heat passes along the laminations to the air ducts. This is the way the heat passes in the axial method of cooling. For this method, numerous holes are punched in the armature stampings. When the stampings are built up, these holes form ducts in the armature core which are parallel to the axis of the machine, and which may extend either uninterruptedly from one side to the other or from each side to one or more large central radial channels or ducts which form the outlet. The stator and the armature stamping shown in Figs. 10 and 5, respectively, are for axial ventilation. Air-gap ventilation is used for cooling the rotor. Filtering or Washing the Cooling Air. — The quantity of air which passes through a large turbo generator is very great and may reach 50,000 to 75,000 cu. ft. per minute. Even if the cooling air is reasonably clean, enormous quantities of foreign matter must be carried by it through the ventilating ducts in the course of a year and the deposit of even a small percentage of this is serious. Fortunately,' the high air velocity which it is necessary to use in the ducts tends to make generators self- cleaning. If, however, any moisture or more especially any oil gets into the passages, it will quickly collect foreign matter. 18 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Certain types of generators require cleaning at more or less frequent intervals in order to keep their ducts free, and it is advisable to clean all types occasionally. With the types of alternators used in America, it has not been necessary to clean the cooling air except when the. conditions are particularly bad, as, for example, when turbo generators are operated near coal mines or in a smelting plant where the air contains enormous quantities of dust. The most satisfactory method of cleaning the air is by washing it by passing the air through sprays of water before it enters the generator. This method of cleaning the cooling air has the double advantage of increasing its humidity and at the same time cooling it. A decrease of even 5° or 10° in the air enter- ing a generator will very appreciably increase its permissible maximum output. Permissible Temperatures for Different Types of Insula- tion. — All insiilating materials are injured or destroyed by high temperatm-e. The continued application of a temperature which would not injure an insulating material if applied for a short time will cause it to slowly deteriorate and ultimately to be destroyed. The continued application of even quite moderate temperatures to cotton, silk, shellac, varnishes and other similar materials commonly used for insulating electrical apparatus causes them to carbonize and to lose their insulating qualities and mechanical strength. Mica alone is the one substance used for insulating electrical apparatus which will stand high temper|£- tures, but mica can seldom be used alone without being built up into sheets or strips with shellac or some form of varnish as\a binder. Except where the binder is used only for structural purposes and where its destruction, when the insulation is once in place, does not decrease the insulating properties or the mechanical strength of the built-up material, mica insulation cannot be used for much higher temperatures than cotton or silk. In many cases where built-up mica insulation is employed, . as, for example, for insulating the slots and the straight parts of armature coils which are embedded in the iron, the binder may be destroyed without injury to the insulation, provided the coils are held firmly in place. SYNCHRONOUS GENERATORS 19 The temperature limits recommended- in the revised Standardi- zation Rules (1914) of the American Institute of Engineers are: Ai. For cotton, silk and other fibrous materials not treated to increase their thermal limit, 95°C. Ai. For the substances named under Ai but treated or impregnated, and for enameled wire, 105°C. ' JSi. Mica, asbestos, or other materials capable of resisting high tempera- tures in which any class A material or binder if used is for structural purposes only, and may be destroyed without impairing the insulating or mechanical qualities, 126°C. CHAPTER II Induced Electromotive Force; Phase Relation between A Flux and the Electromotive Force it Induces; Shape op Flux and Electromotive Force Waves when Coil Sides are 180 Electrical Degrees Apart; Calculation of the Electromotive Force Induced in a Coil when THE Coil Sides are not 180 Electrical Degrees Apart Induced Electromotive Force. — The electromotive force in- duced in a direct-current generator depends upon its speed, the number of armature inductors connected in series between brushes and the total flux per pole, and is independent of the manner in which the flux is distributed, provided the brushes are in the neutral plane. In the case of an alternator, however, the electromotive force depends upon the way in which the flux is distributed. The same total flux can be made to give different values of maximum and of root-mean-square electromotive forces by merely changing its distribution. The value of the electro- motive force will also depend upon the arrangement of the arma- ture winding such as its pitch and coil spread. The electromotive force induced in any coil on the armature of an alternator is given by where N,

^ cos cot, where w is the angular velocity of the armature in electrical radians per second and t is the time in seconds required for it to move through the angle a = wt. e = o}N) Fig. 16. voltages in them will not be in phase at every instant when con- sidered around the coil. The voltage in the coil, however, will still be equal to the vector difference of the voltages induced in its active sides. If the distribution of the flux is not sinusoidal but its distribu- tion in the air gap is known in terms of a fundamental and a series of harmonics, the voltage in the coil at each instant may SYNCHRONOUS GENERATORS 25 be found by taking the vector differences of the voltages induced in the inductors of the coil by the fundamental and each of the hairmonics separately. Fig. 16 gives a distribution of flux which contains a fundamental and third and fifth harmonics. Inspec- tion of this curve should make it clear that any change, a, in the angular distance between the two inductors of a coil corresponds to a change in phase between the voltages in the two inductors of a for the fundamental and na for the nth harmonic. Let the space distribution of flux in the air gap of an alternator measured from a point midway between the poles be (B = (Bi sin a + (B3 sin So; + (85 sin 5a Fundamentals, 3rd Hannonlcs, Fig. 17. 6th Harmonics, where the (B's represent the maximum flux densities for the funda- mental and the harmonics, and a is the angular distance measured around the gap from the reference point midway between the poles. If the inductors of the coil are 160 electrical degrees apart, the fundamentals of the voltages in the two inductors will be 20 degrees out of phase opposition, the third harmonics 3 X 20 = 60 degrees and the fifth harmonics 5 X 20 = 100 de- grees. The vectors for the fundamentals and the harmonics are 26 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY shown in Fig. 17. In this figure the E's are the resultant vol- tages. 1 and 2 are the voltages in inductors 1 and 2 respectively. If the coil contains N turns and moves with a velocity of v cm. per second, and the length of the inductors which cut flux is L, the voltage in volts induced in the coil referred to the voltage in inductor 1 is e = 2LvNl(lr^(S,i cos 10° sin (a + 10°) + (Ba cos 30° sin {da + 30°) + (Bb cos 50° sin (5a + 50°) ) The root-mean-square value of this voltage is equal to the square root of one-half the sum of the squares of the maximum values of the fundamental and the harmonics. CHAPTER III Open- and Closiid-ciecuit Windings; Bab and Coil Windings; Concentrated and Distbibuted Windings; Whole- and Half-coiled Windings; Spiral, Lap and Wave Windings; Single- and Polyphase Windings; Pole Pitch; Coil Pitch; Phase Spread; Bbeadth Factoe; Harmonics; Pitch Factor; Effect op Pitch on Habmonics; Effect ON Wave Fobm op Distributing a Winding; Harmonics IN Three-phase Generators Open- and Closed-circuit Windings. — All alternating-current windings may be divided into two general groups : I. Open-circuit windings. II. Closed-circuit windings. An open-circuit winding, as its name signifies, is not closed on itself. In an open-circuit winding there is a continuous path through the conductors of each phase on the armature which terminates in two free ends. A closed-circuit winding has a continuous path through the armature which re-enters on itself, forming a closed circuit. All closed-circuit windings have at least two parallel paths between their terminals. All modern direct-current windings are closed-circuit windings. Either open- or closed-circuit windings may be employed for alternators but, except in a few special cases, open-circuit wind- ings are better adapted for alternators and are universally used. Multipolar alternator armature windings may have two or more parallel paths through their armatures, but such windings are not re-entrant, i.e., closed-circuit, windings. Windings with two parallel paths between terminals are called two-circuit wind- ings or, in general, windings with two or more parallel paths between terminals are called multicircuit windings. Such windings are sometimes used for low-voltage alternators. A continuous-current winding may be used for an alternator. 27 28 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY but an alternating-current winding, since it is not re-entrant, cannot be used for a direct-current generator. Bar and Coil Windings.— Armature windings may be divided into two general classes according to the way in which the coils are placed in the slots, namely: bar windings and coil windings. In the former, insulated rectangular copper bars are laid in the armature slots and are then suitably connected by brazing, welding or bolting. In the latter type of winding, coils of rec- tangular or of round insulated wire are wound on forms in lathes, are insulated and then placed in the slots. When closed or nearly closed slots are used, it is sometimes necessary to wind the coils by hand directly on the armature by threading the wire through the slots. Form-wound coils are more reliable and are generally used, except where nearly closed slots are required. SYNCHRONOUS GENERATORS 29 Whether bar or coil windings are employed, the slots must be properly insulated by press board, mica or other suitable material. A bar wave winding with two bars per coil and four bars per slot is shown in Fig. 18. Fig. 19 shows two types of coils for coil-wound armatures. When the type of coil shown on the left is used, all the armature coils are the same size and shape irre- spective of the phase they are in or their position on the armature. Two different shapes of coils are required for the other type of winding. Moreover, it does not permit as good bracing of the end connections as the first. Fig. 19. Concentrated and Distributed Windings. — Concentrated wind- ings have all of the inductors of any one phase, which lie under a single pole, in a single slot. Better results can usually be obtained by distributing the inductors among several slots. Such windings are called distributed windings. They are com- pletely distributed or partially distributed according as they are spread over the entire armature surface or over only a portion of it. Distributed windings diminish armature reactance and armature reaction, give a better wave form and a better distri- bution of the heating due to the armature copper loss than con- centrated windings. 30 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Whole- and Half -coiled Windings.— The one common require- ment for all windings is that all conductors must be connected together in such a way that their electromotive forces shall assist. Fig. 20 shows a six-pole alternator with two inductors per pole. The short lines over the poles represent diagrammatically the armature inductors, and the arrows on these lines represent the direction of the electromotive forces induced in them for the clockwise rotation of the field. An inductor extending into the Fig. 20. paper is represented by a line drawn radially outward. Each slot on the armature is assumed to contain two inductors. These are shown side by side in Fig. 20. They may be connected in two ways as illustrated by Figs. 21 and 22. Electrically the con- nections shown in Figs. 21 and 22 are identical. The lower halves of these figures represent the connections on the backs of the armatures as they would actually appear. Fig. 21 represents what is known as a whole-coiled winding. Fig. 22 shows a half-coiled winding. Whole-coiled windings have as many coils per phase as there are poles. Half-coiled windings have only one coil per phase per pair of poles. The two turns per pair of poles shown in Fig. 22 would be in a single coil. The only real difference between the two types of winding lies SYNCHRONOUS GENERATORS 31 Fig. 21. 32 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Fig. 22. SYNCHRONOUS GENERATORS 33 in the method of making the end connections between the inductors in the slots. The connections between the coils for a half -coiled winding are simpler than for a whole-coiled winding. When a half-coiled winding is used on a generator, the armature frame or yoke may be split into two or more sections for ship- ment without disturbing many end connections. Spiral, Lap and Wave Windings. — When the windings are dis- tributed they may be connected in three different ways giving what are known as: (a) Spiral windings. (6) Lap windings. (c) Wave or progressive windings. Fig. 23. Lap and wave windings may also be used for concentrated windings. The difference between these three types of windings will be made clear by referring to Figs. 23, 24 and 25, which show respectively a spiral winding, a lap winding and a wave winding. All three figures show distributed single-phase windings with eight slots per pole. 34 PRINCIPLES OF ALTERNATINO-CURBENT MACHINERY Fig. 24. Fig. 25. SYNCHRONOUS GENERATORS 35 The lap winding lends itself better to the use of lathe-wound formed coils than the spiral winding as in the former all of the coils will be the same. If formed coils are uspd for a spiral winding, there will have to be as many different widths of coils, i.e., coil pitches, as there are slots per pole per phase for a half- coil winding, but only one-half as many for a whole-coil winding. Single- and Polyphase Windings. — A single-phase winding has only one group of inductors per pole. These may be in a single slot or in several slots according to whether the winding is con- centrated or distributed. A polyphase winding may be con- sidered to consist of a number of single-phase windings displaced by suitable angles from one another. The electrical space dis-: placement between the single-phase elements must be the same as the phase differences between the voltages to be induced. For example, the corresponding elements ,of the winding of a three-phase alternator must be displaced 120 electrical space degrees from one another. Although the single-phase windings which make up the polyphase winding are independent of each other, the windings are always interconnected in either star or mesh. The number of leads brought out will be equal to the number of phases, except when star connection is used when an additional lead may be brought out from the common junction or neutral point of the phases. In the case of three-phase alter- nators, the star and mesh connections are, respectively, the Y and A connections. Most modern alternators are connected in Y. Y connection permits the neutral point to be grounded and gives a higher voltage between terminals for the same phase voltage than the A connection. It also gives a higher slot factor, i.e., the ratio of copper to insulation for a given size slot is greater for a given thickness of insulation than for the A connection. High-voltage alternators are invariably F-connected as with this connection the strain on the slot insulation is only — — as great as it would be with A connection for the same terminal voltage. When there is no consideration such as high voltage to determine whether F or A connection should be used, the method of con- necting the phases is sometimes fixed by the number of slots in the standard armature stampings which are available, the fre- quency, the voltage and the permissible range of flux density. 36 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY For the same voltage between terminals and line current, Y and A connections require the same amount of copper, but the Y connection requires fewer total turns than the A connection — = 0.58 as many), and since the thickness of insulation required on the wires depends upon the voltage and not upon their size, the ratio of space occupied in a slot by the copper to the space occupied by insulation will be greater for the Y than Fig. 26. for the A connection. In other words, the slot factor of a F-con- nected alternator will be higher than the slot factor of a A-con- nected alternator. Therefore, smaller slots can be used for Y connection than for A connection. Fig. 26 shows a simple six-pole, two-phase half-coiled wind- ing with two inductors per slot. Fig. 27 shows a similar three- phase winding. The phases 1, 2 and 3 of the three-phase winding are indicated by full lines, dashed-and-dotted lines and dotted lines respectively. SYNCHRONOUS GENERATORS 37 The arrangement of the coils of a three-phase alternator having one slot per pole per phase and a half-coiled winding is shown in Fig. 28. Fig. 29 shows an end view of a large turbo alternator and illustrates one of the most satisfactory methods of bracing the end connections to resist the severe stresses to which they are subjected at times of short-circuit (Chapter VIII). Pole Pitch. — The pole pitch is the distance between the cen- ters of adjacent north and south poles. Coil Pitch. — The distance between the two sides of any arma- ture coil is called the coil pitch. Coil pitch is usually expressed as a fraction of the pole pitch, but it is sometimes convenient to express it in electrical degrees or in slots. For example: a coil pitch of % would be a pitch of 120 electrical degrees or, if there were twelve slots per pole, a pitch of eight slots. A winding having a coil pitch of less than 180 electrical degrees or unity is called a fractional-pitch winding. Since the two sides of a coil of a fractional-pitch winding do not lie under the centers of ad- 38 PRINCIPLES OF ALTERNATINO-CURRENT MACHINERY jacent poles at the same instant, the electromotive forces in- duced in them are out of phase. The voltage produced by a fractional-pitch winding is, therefore, less than that produced by a full-pitch winding having the same number of turns. Frac- tional-pitch windings are often used. They decrease the length of the end connections and thus the amount of copper required. Pig. 28. They also somewhat reduce the slot reactance and give a means of eliminating any one harmonic from the electromotive-force wave and reducing the others. They require a few more turns or a greater flux for the same electromotive force than a winding having a full pitch. Phase Spread. — The phase spread of a winding is the percent- age of the periphery of the armature over which the windings of a single phase are spread. For example: a single-phase winding which covers the entire surface of an armature has a spread of SYNCHRONOUS GENERATORS 39 unity. Phase spread may also be expressed in electrical degrees. A phase spread of unity is a phase spread of 180 degrees. Breadth Factor. — The voltages induced in the separate coils of a distributed winding are not in exact phase and their resultant is, therefore, less than would be given by a concentrated winding having the same number of turns. The ratio of the voltages produced by distributed and concentrated windings having the same number of turns is called the breadth factor. The breadth factor for any form of winding may be found by calculating the voltage induced in each turn or group of turns occupying a single pair of slots and then adding vectorially the voltages produced 40 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY in all pairs of slots over which the phase is distributed. The ratio of this voltage to the voltage which would be produced if all the turns were concentrated in a single pair of slots is the breadth factor. Consider a three-phase generator having six slots per pole, that is, two slots per pole per phase. Let N be the number of turns per coil. The voltage per coil is, equation (2), page 21, E = 4.44Ar/« + oib) + . . . (6) and i = 7i sin (w( -|- ai — fli) -1- I3 sin (3w< + 0.3 — 63) + h sin {5cot + as - 06) + (7) where the E's and 7's are the maximum values of the different harmonics and the d's q,re the angles of lag between the currents and voltages of the corresponding harmonics. Pitch Factor. — The voltage generated in any single turn on the armature of an alternator is the vector difference of the voltages generated in the two inductors which form the active sides of the turn. With "a full-pitch winding, these two voltages are in phase when considered around the coil. In the case of a fractional-pitch winding, the active sides of the coil are less than 180 electrical degrees apart and the electro- motive forces generated in them, therefore, will be out of phase when considered around the coiL- If p is the pitch expressed in electricaPdegrees, the difference in phase for the fundamental of the two voltages will be 180 — pr^ In general, since the displace- ment for any harmonic such as the nth must be n times the phase displacement for the fundamental, the difference in phase between the harmonics of any order, such as the nth, generated in the two active sides of any coil of a fractional-pitch winding, will be (180 - p)n. Since the voltage in a coil is the vector difference of the voltages generated in its active sides, the voltage. En, of the nth harmonic generated in a coil is equal to ^ T., (180 - p)n En =-2 E'n cos ^^ ij-^^ 44 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY where E'„ is the value of the nth harmonic voltage in the coil side. The pitch factor is the ratio of the voltage, En, induced in a fractional pitch winding to the voltage, 2 E'n that would be in- duced if the winding had a full pitch. The pitch factor for the fundamental is therefore kp = cos 2 — ■ ^^^ Effect of Pitch on Harmonics.— Any harmonic may be elimi- nated from the voltage generated in a coil by choosing the proper pitch. To eliminate a harmonic, the pitch must be such that (180 — p)n = 180, or the pitch must be p = 180^^^^ (9) n A ^ or 120-degree pitch will eliminate the third harmonic. A ^ or 144-degree pitch will eliminate the fifth harmonic, and a ^ or 154.3-degree pitch will eliminate the seventh. Eliminating any one harmonic from the voltage induced in the armature coils of any alternator not only eliminates that particu- lar harmonic, but diminishes the others, usually by different amounts, and changes their phase with respect to the fundamental. For example, let the voltage generated in the active sides of any coil be e = El sin ut + E3 sin 3 cot -|- E^ sin Scoi -\- Ei sin 7 cat If a % pitch is used, the result.ant voltage generated in the coil will be _ , / Cr = l.lSEi sin ut + 1.73Eb sin (Scoi -^ 180°) -f 1.73^7 sin (7aj< ^480°) A ^ pitch would eliminate the sixth harmonic — this cannot appear even with full pitch — and will very nearly cut out the fifth and seventh. It will be shown later that there can be no third harmonic or multiples of the third harmonic between the terminals of a three-phase, F-connected generator. Therefore, by using a ^e pitch and Y connection there can be no third, ninth or fifteenth harmonics and only a small fifth and seventh between the line terminals. The first harmonic which can occur in any magnitude is the eleventh, and harmonics of as high order as this SYNCHRONOUS GENERATORS 45 seldom are present in sufficient magnitude to have much effect on the wave form. The magnitudes of the harmonics in fractional-pitch windings as compared with their magnitudes in a full-pitch winding having ■the same number of turns are given in Table II. Table II Pitch Harmonic 1 3 5 7 11 % 0.866 0.951 0.966 0.975 0.000 0.588 0.707 0.782 0.866 0.000 0.259 0.434 0.866 0.588 0.259 0.000 0.866 ^4 0.951 ^i 0.966 ^ 0.782 The Effect on Wave Form of Distributing a Winding. — When a winding is distributed, that is, when it occupies more than one slot per pole per phase, the electromotive forces generated in the turns of a single phase, which occupy different pairs of slots, will be out of phase. For the fundamental of the voltage wave, this difference in phase will be equal to the angle between the two pairs of slots occupied by the two groups of turns. For the third harmonic it will be three times this angle; for the fifth, five times; for the seventh, seven times, the angle, of course, being measured in electrical degrees. The general effect of distributing a winding is to smooth out the wave form by diminishing the amplitude of the harmonics with respect to the fundamental. This can be made clear by considering a specific case. Take, for example, a generator which has a distribution of flux in its air gap which gives an electro- motive force containing a third and a fifth harmonic in each turn of the armature winding. Let the equation of this electromotive force be e = E{sm ut + ^i sin 3cat + }i sin dwt) Let there be four turns per pole per phase. If all four turns are placed in a single pair of slots the resultant electromotive force generated in them will be e, = S(4 sin cot + 1.33 sin 3w« + 0.8 sin 5oot) 46 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY and the harmonics will have the following relative magnitudes : 1st :3rd :5th = 1:0.33:0.2 Suppose the four turns are distributed among four pairs of slots which are 15 degrees apart. This corresponds to the dis- tribution of the armature winding of a three-phase alternator having four slots per pole per phase'^and gives a phase spread of 60 degrees. Let ei, 62, 63 and 64 be the electromotive forces generated in the four turns referred to the center of the phase belt. Then ei = i;[sin(a)<-22.°5)+J^sin3(w<-22.°5)+>^sin5(w<-22.°5)] 62 = E[sm {o>t-7.°5)+M sin 3(a)i-7.°5)+J^ sin 5{cot-7.°5)] 63 = ^[sin icat+7.°5)+}4 sin 3(co<+7.°5)+K sin 5(w<+7.°5)] 64 = E[sm (co<+22.°5) + i'^sin3(co<+22.°5)+J.^sin5(w<+22°.5)] Adding these vectorially gives the resultant voltage e, equal to Br = E{3M sin'w<+0.869 sin 3&)f +0.164 sin 5w«) The relative magnitudes of the harmonics in this resultant wave are 1st :3rd :5th = 1:0.226:0.043 With all four turns in the same pair of slots the root-mean- square voltage is = 4.28^ \/2 With the turns distributed this voltage is P Er.r,...' = ;;^'V(3.84)2 + f0.869)2 + (0.16472 Distributing the winding has diminished the voltage by about 10 per cent. Therefore either 10 per cent, more turns or 10 per cent, more flux will be required in this particular case for the same voltage. The disadvantage of increasing the flux or the turns is usually more than balanced by the smoothing out of the wave form by diminishing the harmonics. The distribution of the armature copper loss is also improved. In the particular ex- SYNCHRONOUS GENERATORS 47 ample just given, distributing the winding reduced the third harmonic about 30 per cent, and the fifth about 79 per cent. Harmonics in Three-phase Generators. — There can be neither a third harmonic nor any multiple of the third harmonic in the voltages between the terminals of a three-phase generator, but such harmonics may exist between any one of the three terminals and the neutral point if the generator is F-connected. Let the phase voltages of a three-phase generator be given by ei = El sin wi + E^ sin 3wi -\- Es sin 5w< + Ei sin 7coi + . . . . 62 = Si sin (coi - 120°) + E^ sin3(wf - 120°) + Ei sin 5(co< - 120°) + E^ sin 7{ut - 120°) + . . . . ea = El sin {ut - 240°) + E3 sin 3(w« - 240°) -f Ei sin 5(ut - 240°) + Ei sin 7(«« - 240°) + . . . . The angular displacement between any harmonic of any one phase and the corresponding harmonic of phase one is given in Table III. Table III Phase Displacement in electrical degrees 1st 3rd 5th 7th 9th 1 2 120 3(120) = 360 = 5(120) = 600 = 240 7(120) = 840 = 120 9(120) = 1080 = 3 240 3(240) = 720 = 5(240) = ,. 1200 = 120 7(240) = 1680 = 240 9(240) = 2160 = Referring to Table III, it will be seen that all of the third har- monics are in phase. The ninth harmonics are also in phase. In fact, all multiples of the third harmonic will be in phase. The fifth harmonics are 120 degrees apart, but they occur in inverted order, that is in the order 1, 3, 2. The seventh harmonics are 12Q degrees apart and in natural order. In general, starting with the fifth harmonic and neglecting those harmonics which are in phase, the sequence in which the harmonics of any order occur in the three phases alternates from the order 1, 3, 2, to the order 1, 2, 3. 48 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Consider a F-connected generator. Fig. 31 represents'a space- phase diagram of the connections of the phases of a F-connected generator and a time-phase diagram of the voltages induced in them. The voltages across the three pairs of terminals 1-2, 2-3 and 3-1 are ei2 = Cio 4- eo2 = Cio — 620 623 = ^20 + ^03 = ^20 — 630 631 = 630 + 601 = 630 — 610 The voltage between any pair of terminals is, therefore, the vector difference of the phase voltages. Since the, third har- monics and all the multiples of the third harmonics are in phase, they will cancel in the differences. Therefore, there cannot be any third harmonic or any multiple of it in the line or terminal voltage of a three-phase F-connected alternator. The third har- monics and their multiples existing between the terminals, and neutral point, however, will be in phase. A study of the phase differences between the harmonics of the same order for the three phases will show that the voltages 612, 623 and 631 of a F-connected alternator when referred to Cio are: cot j- -5-) + (30°' cof ^-f-) +0+ . . . . 623 = \/3£/i sin {oit - 120° + 30°) + + V3£6 sin 5 (co< - 120° 1^ ^^ + 30° (I so \ ut - 120°^ ^j -1-0+ SYNCHRONOUS GENERATORS' 49 631 = \/3£i sin (cot - 240° + 30°) + -f co«-240°4-^j + aj«-240°--K^) +0+ . . . . Consider the conditions existing in a A-connected alterna- tor. The voltage acting around the closed delta is eio + 620 + 630. By referring to Table III it will be seen that the three compo- nents of the third harmonic voltage are in phase. They will, therefore, be short-circuited in the closed delta and cannot ap- pear between the terminals of the alternator. The ninth and all other multiples of the third harmonic will also be short-circuited in the delta. The vector sum of all other harmonics, including the fundamental, will be zero when taken around the closed delta. The three line or terminal voltages of a A-connected alternator are: ei2 = El sin oit -'t Q -\- Ef, sin bwt -}- S7 sin 7co< -|- -f . . . . en = El sin (ui - 120°) + + E^ sin 5(a)f - 120°) + El sin 7(£o< - 120°) + -f . . . . 631 = Ex sin (ojf - 240°) + -|- £^5 sin 5{U - 240°) + E7 sin 7{oit - 240°) -f -f- . . . . Although the terminal voltages of an alternator when con- nected in Y and in A contain the same harmonics in the same rela- tive magnitudes, the wave forms given by the two connections will be different, due to the phase displacement of 30 degrees which occurs in the harmonics of a F-connected alternator. The root-mean-square voltages given by the Y and A connec- tions will be in the ratio of VS to 1, but the maximum voltages will not be in this ratio since the phase relations between the harmonics are different for the two connections. The effective value of the circulatory current caused by the third harmonic and its multiples in the armature of a A-connected generator is 1 j/SEsy , (ZE^Y ^ . where the z's are the effective impedances of the armature per phase for the different harmonics. 50 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The effective reactance of the armature of an alternator for any harmonic will not be the effective or synchronous reactance of the armature for the fundamental multiplied by the order of the harmonic, but in general it will be considerably less than this on account of the difference between the armature reaction pro- duced by the harmonics and the fundamental. A cormection is objectionable for alternators unless their wave forms are free from third harmonics and their multiples. If third harmonics are present in any great magnitude, there will be a large short-circuit current in the closed delta formed by the armature winding. This current combined with the load cur- rent may cause dangerous heating. Most modern alternators are F-connected. The effect of the third harmonic in a A-con- nected generator is only one of several things which make Y connection preferable as a rule. CHAPTER IV Rating; Regulation; Magnetomotive Forces and Flttxes Concerned in the Operation of an Alternator; Arma- ture Reaction; Armature Reaction of an Alternator with Non-salient Poles; Armature Reaction of an Alternator with Salient Poles; Armature Leakage Reactance; Equivalent Leakage Reactance; Effective Resistance; Factors which Influence the Effect and Magnitude of Armature Reaction, Armature Leakage Reactance and Effective Resistance; Conditions for Best Regulation Rating. — The maximum output of any alternator is limited by its mechanical strength, by the temperature of its parts pro- duced by its losses, and by its voltage regulation. Usually the limit of output is fixed by the temperature. The maximum voltage any alternator can give continuously depends upon the permissible flux per pole. The armature cop- per loss limits the maximum safe current. The kilowatt output depends upon the voltage, the current, and the power factor, but the core and copper losses and, therefore, the temperatures of the parts of an alternator depend upon the voltage and cur- rent and are nearly independent of the power factor. For this reason, alternators are rated on their kilovolt-ampere output and not upon their kilowatt output. It is customary at present to rate alternators so that the maxi- mum rise in temperature of their parts above a specified ambient temperature, i.e., temperature of the surroundings, shall not ex- ceed a certain definite number of degrees after a full-load run of sufficient duration for constant temperature conditions to have been reached. In addition, generators are usually designed to carry a 25 per cent, overload for 1 hour immediately following the continuous full-load run without an additional rise in tem- perature of more than a specified number of degrees. The 61 52 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY ambient temperature of reference recommended by the American Institute of Electrical Engineers is 40°C. The permissible maxi- mum temperature rise in any part of an alternator depends upon the type of insulation used and upon the ambient temperature in which the alternator operates. It may be found from the limiting temperatures for different classes of insulation given on page 19 by subtracting the ambient temperature. There is a growing feeling among engineers that all electrical apparatus should have for its rating the maximum kilovolt- ampere output it is capable of giving continuously without in- jury, instead of a full-load rating with a provision for an over- load. Such maximum ratings are already in use for large turbo alternators. Regulation. — The regulation of an alternator is the percentage , rise in voltage, under the conditions of constant excitation and frequency, when the rated kilovolt-ampere load is removed. The change in voltage produced under this condition depends not only upon the magnitude of the load and the constants of the alternator, but also upon the power factor of the load. The regu- lation will be positive for both a non-inductive and an inductive load since both of these cause a rise in voltage when they are removed. A capacity load, on the other hand, may, if the angle of lead is sufficiently great, cause a fall in voltage instead of a rise. Under this condition the regulation will be negative. The inherent regulation is the regulation on full non-inductive load. The regulation of an alternator depends upon four factors, namely: I. Armature reaction. II. Armature reactance. III. Armature effective resistance. IV. The change in the pole leakage with change in load. Some of the four factors produce similar effects and for this reason they are combined in certain approximate methods for determining regulation. The relative magnitudes of the effects produced upon the terminal voltage of an alternator by these four factors depend not only upon the magnitudes of the factors, but also upon the power factor of the load. At 100 per cent, power factor with respect to the generated voltage, armature re- SYNCHRONOUS GENERATORS 53 action and reactance have a minimum effect upon the terminal voltage. Their maximum effect occurs at zero power factor. Just the opposite is true in regard to the effect produced by re- sistance. The actual magnitudes of reaction, reactance and re- sistance are fixed by the design and may be varied over quite wide limits, but considering merely the component change in voltage produced by each when acting separately, the magnitudes of their effects are usually in the order named. Magnetomotive Forces and Fluxes Concerned in the Operation of an Alternator. — There are two distinct magnetomotive forces and three component fluxes to be considered in the operation of any alternator. The two magnetomotive forces are: (a) the magnetomotive force of the impressed field; (b) the magneto- motive force due to the armature current, i.e., the armature reaction. Although both of these magnetomotive forces may be expressed either in ampere-turns per pole or per pair of poles, it is usually more convenient, especially when dealing with multi- polar alternators, to express them in ampere-turns per pole. The three component fluxes are : (a) the flux which is common or mutual to the armature and the field, this is the air-gap flux; (6) that portion of the total armature flux which links only with the armature inductors; and (c) the field leakage flux. This last is the portion of the field flux which passes between adjacent north and south poles without entering the armature. The ratio of the maximum flux in a pole to the portion of that flux which enters the armature is called the leakage coefficient or the leakage factor of the field. This coefficient varies from about 1.15 to 1.25 according to the design of the alternator. If the leakage coefficient were constant and independent of the load, the field leakage would produce no effect on the regulation of an alter- nator. The field leakage is inversely proportional to the reluct- ance of the path of the stray field and is directly proportional to the magnetic potential between the poles. The latter is made up of two parts: one, the drop in the magnetic potential necessary to force the flux through the armature and the air gap; the other, the opposing ampere-turns of armature reaction. Armature Reaction. — When a synchronous generator operates at no load, the only magnetomotive force acting is that of the field winding. The flux produced by this winding will depend 54 PRINCIPLES OF ALTERNATINO-CURRENT MACHINERY only upon the current it carries, the number of turns and their arrangement, and the total reluctance of the path through which the magnetomotive force acts. The distribution of the air-gap flux will depend chiefly upon the shape of the pole shoe, except in cases where the cyhndrical or drum type of field is used. In these latter, the distribution of the field winding will determine the distribution of the air-gap flux. When load is applied to an alternator, the magnetomotive force of the armature current will modify the flux produced by the field winding. The effect of the armature magnetomotive force, or armature reaction, will depend not only upon the arrangement of the armature winding, the current it carries and the reluctance of the magnetic circuit, but also upon the power factor of the load. Neglecting field distortion, the voltage generated in any coil or turn on the armature of a single-phase alternator will have its maximum value when the center of the coil lies midway between two adjacent poles. It will be zero when the center of the coil is directly opposite the center of a pole. If the power factor is zero with respect to the voltage produced by the.air-gap flux, the maximum current will occur when the voltage is zero or when the coil is directly opposite a pole. Under this condition the axis of the magnetic circuit for the armature reaction coincides with the axis of the magnetic circuit for the field winding, and the resultant magnetomotive force acting to produce the field flux will be the algebraic sum of the magnetomotive forces of armature reaction and field excitation. Under this condition the armature reaction will either strengthen or weaken the field without pro- ducing distortion. The armature reaction caused by a lagging current will oppose the magnetomotive force of the field winding and will weaken the field. A leading armature current strengthens the field. If, instead of the coil lying with its center opposite a pole when the current in it is a maximum, it lies with its center midway between two poles, it will cover half of two adjacent poles (a full-pitch winding is assumed) and will produce a demagnetizing action on half of one pole and a magnetizing action on half of the other. It follows that one-half of each pol^ is strengthened and the other half is weakened by the action of the armature SYNCHRONOUS GENERATORS 55 current. These two effects will be equal under the conditions assumed and the resultant action, therefore, produces a distor- tion in the flux distribution without changing the total strength of the field. The application of the cork-screw rule to the direc- tion of the current carried by the armature coils will show that the trailing pole tips are strengthened and the leading pole tips are weakened by a lagging armature current. The effect is merely a shift in the flux from the leading pole tip to the trailing pole tip. The condition just described, i.e., with the center of the armature coil midway between two poles when the current in it is a maximum, corresponds approximately to unit power factor with respect to the terminal voltage. Fig. 32. The approximate distributions of the flux at the instant when the armature current is a maximum for a reactive load of zero power factor and a power factor of unity are shown in Figs. 33 and 34 respectively. Fig. 32 shows the distribution at no load. In the preceding discussion only the instant when the current is a maximum was considered. While the field is moving through a distance corresponding to 360 electrical degrees, the current in any armature coil as ah, Fig. 34, will go through a complete cycle and consequently the value of the total flux from_a_Bole and its distribution will also go through a complete cycle. The average distribution of flux, however, will be about the same as when the current passes through its maximum value. Such a variation of the flux does not occur in the case of a polyphase alternator which carries a balanced load, since the armature reaction of such an alternator under such conditions is fixed in 56 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY magnitude and in direction with respect to the poles. The effect is the same as occurs in a single-phase alternator at the instant when the current passes through its maximum value. To sum up, the general effect of armature reaction is as follows: with a non-inductive load, it distorts the field without appreciably- changing the total field flux; with an inductive load of zero power factor, it weakens the field without distorting it; and with a load having a power factor between unity and zero, it both distorts the field and modifies its strength. In addition to the general distortion of the field which has been so far considered, there will be a local distortion in the neighbor- ^^ Direction of Kotatlon of Field < i i^^ Fig. 33. hood of each inductor. This distortion is limited to the region about the slots and to the air gap and does not extend to any depth into the pole faces. It is equivalent to a little ripple in the flux about each inductor and may be considered to be due to the superposition upon the main fleld of local fluxes which surround the armatiu-e inductors. These local fluxes are indi- cated by the dotted lines in Figs. 33 and 34. Although the local fluxes have no real existence except about the end connections of the coils, it is convenient to consider them separately as com- ponents of the main flux. They are alternating fluxes and are very nearly in time phase with the currents which cause them. They are the so-called leakage fluxes and give rise to a voltage of self-induction in the inductors with which they link. This voltage will alternate with the same frequency as the armature current and will lag 90 degrees behind that current. The re- actance corresponding to this voltage of self-induction is the so- SYNCHRONOUS GENERATORS 57 called leakage or slot reactance of an alternator. More will be said of this under reactance. A knowledge of armature reaction is necessary in order to pre- determine the regulation of an alternator and also to determine the number of field ampere-turns required at full load to main- tain the rated voltage at different power factors. In the case of alternators with salient or projecting poles, such as are illustrated in Figs. 32, 33 and 34, armature reaction produces a distortion of the air-gap flux except when the power factor is zero, a condi- tion which is impossible in practice and which is not even ap- proached under ordinary operating conditions. Fia.'34. The distortion of the air-gap flux which takes place, in an alternator with salient poles is caused almost entirely by the difference between the reluctance of the magnetic circuits for the armature reaction and the impressed field. Except when the power factor of the load is zero, the magnetomotive forces of the field and armature do not act along the same line. They are not in space phase and the axis of their resultant will not coincide with the axis of either. Since flux always distributes itself so as to follow the path of minimum reluctance, the flux caused by the combined action of the magnetomotive forces of the armature and field currents will still cling to the poles, but it will be crowded toward one side instead of being symmetrical about their axes. In the case of alternators with non-salient poles, however, the reluctance of the magnetic circuit for armature reaction is con- stant and independent of the power factor and is equal to the reluctance of the magnetic circuit for the impressed field. Under 58 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY this condition, there will be no distortion of the magnetic field under load provided the field and armature windings each give a sine distribution of magnetic potential in the air gap. This con- dition cannot be fulfilled exactly in practice. Armature Reaction of an Alternator vsrith Non-salient Poles. — The armature reaction of an alternator with non-salient poles will first be considered. A sinusoidal current wave and a distributed armature winding will be assumed. Under this condition, the space distribution of the magnetic potential in the air gap due to the armature current will be nearly sinusoidal and will be so as- sumed. The effect of the slots on the armature and the field core will be neg- lected. Their presence will in reality produce little ripples in the wave of flux distribution. Under the conditions assumed, the armature reaction of a single-phase alter- nator will be sinusoidal with respect to time and will oscillate along an axis which is fixed in space with respect to the arma- ture. Any simple oscillating vector which varies with time according to a sine law can be resolved into two oppositely rotating vectors, each with a maximum value equal to one-half of the maximum value of the vector they replace and having the same period. An in- spection of Fig. 35 should make this clear. The vertical dotted line on this figure represents the line along which the simple vector oscillates. A and B are the two oppositely rotating vec- tors. Their resultant, R, will be equal at every instant to the original vector and will lie along its axis. Consider the armature reaction of a single-phase alternator to be resolved into two oppositely revolving vectors. Both of these vectors will rotate at synchronous speed with respect to the armature, one right-handedly, the other left-handedly. One of these vectors will rotate in the same direction as the field and will be stationary with respect to it. Let N be the effective number of armature turns per pole and let Im be the maximum armature current. The value of the SYNCHRONOUS GENERATORS 59 component of armature reaction which is fixed in direction with respect to the field is ^iNIm per pole. Replacing 7„ by its root- mean-square value gives A = 0.707 N I (10) where I is the root-mean-square value of the current. The other component rotates at twice synchronous speed with respect to the poles and will set up in them a double-frequency or second- harmonic flux. This double-frequency component of the field flux in combination with the rotation of the field induces a third- harmonic voltage in the armature turns which will be present across the terminals of the alternator unless it is eliminated by the distribution, the pitch or the connections of the armature winding. The voltage generated in any armature turn is e = kcp sin ut where fc is a constant and

-0.5) ll Xe = 2Tr/(4.6 1 Z^) (logio j-, - 0.5.) 10-» ohms (24) This multiplied by the number of coils in series per phase will give the phase, end-connection reactance. Total Leakage Reactance. — The total phase reactance in ohms of a full-pitch winding having s straight-sided slots with two coil sides per slot is, from equations (23) and (24), Xa = 27r%Z"'{^[2.7d + 4«' + f + 2.9m> logic "^^ ' ^ ^ ] + 0.37z(logio J-, - 0.5) 1 10-' (25) The omission of the phase helt leakage seems justified as it is of minor importance with fractional-pitch windings and its value is quite uncertain with any type of winding. Equivalent Leakage Reactance. — For a given size and shape of slot and fixed coil pitch, the leakage flux per ampere per inductor per unit length of slot is nearly constant. This state- ment is also approximately correct when applied to the end connections. When dealing with a given type of armature stamping it is, therefore, permissible and often convenient to make use of an equivalent leakage flux which may be defined in the following manner: The equivalent leakage flux is that flux per ampere per unit length of embedded inductor which, if linked with all of the inductors in a slot, would produce a y ■ 78 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY reactance which would be equal to the actual slot and tooth- tip reactance plus one-half of the reactance of the end con- nections for the inductors in a pair of slots. The value of this ■equivalent leakage flux varies from 2.5 to 6 lines per ampere per centimeter length of embedded inductor. It depends mainly upon the shape and size of the slots. If (pe is the equivalent leakage flux per amper e per unit length of embedded inductor, the equivalent leakage flux per slot is l(PeZ where I and Z are, respectively, the length of the embedded inductors and the number of inductors in series per slot. The slot linkages due to this flux are l Fig. 50. E"a is equal to the vector sum of the terminal voltage and the resistance drop. To produce this voltage a magnetomotive force B" is required which is in quadrature with E"a and is equal to the field required to produce this voltage when the generator is on open-circuit. The fictitious reaction, —A', is found from measurements made with the alternator short-circuited. When the alternator is short-circuited, the terminal voltage becomes zero and the vector diagram reduces to the form shown in Fig. 50. .E"a = laTe IS Small. Consequently, R", the magnetomotive force corresponding to E"a, is also small and may be neglected in comparison with —A'. A' = V{Fy - {R"Y = F approximately. The. value of A' for the vector diagram given in Fig. 49 is, therefore, equal to the impressed field required to produce the armature current, la, when the generator is short-circuited. The SYNCHRONOUS GENERATORS 95 impressed field, F, under load conditions is the vector sum of B" and —A'. If E'a is the open-circuit voltage corresponding to the excitation F, the regulation is E'a — V — '^^ — 100 per cent. The value of —A' used in the calculation of the regulation by the magnetomotive-force method is found with the generator operating short-circuited and is, consequently, for low saturation as well as for low power factor. On low saturation, a smaller magnetomotive force is required to produce a given voltage than at normal operating saturation. Consequently, that part of —A' which replaces laXa will be too small for the condition of normal saturation. On account of the low power factor on short-circuit, the effect of armature reaction in the case of alter- nators with sahent poles will be a maximum and will be con- siderably higher than it would be if the power factor were more nearly that met under ordinary operating conditions. Due to this latter cause, the value of —A' will be larger than it should be. In the case of alternators with salient poles both these effects will be present and will tend to neutralize each other. In spite of this, however, the regulation of generators with sahent poles found by the magnetomotive-force method is usually lower than the regulation obtained from measurements made under actual load conditions. The effects produced, in the synchronous-impedance method, by low saturation and low power factor on the synchronous reactance both tend to make the value of the synchronous reactance too large. The synchronous-impedance method will, therefore, always give a poorer regulation than the actual. Data Necessary for the Application of the Synchronous- impedaace and the Magnetomotive-force Methods.-^In order to apply either the synchronous-impedance or the magneto- motive-force method, the following data are necessary: (a) The effective armature resistance per phase. (6) The open-circuit characteristic. (c) The short-circuit characteristic. No other information is required except the name-plate rating. Effective Resistance. — The effective resistance of an alternator 96 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY may be found by multiplying its ohmic resistance by a suitable constant which will depend upon the type and the design of the alternator, or it may be obtained by direct measurement by one of the approximate methods for measuring the effective re- sistance which will be given later. The ohmic armature resist- ance per phase of a three-phase alternator is not the same as the ohmic resistance between its terminals. For a F-connected alternator it is 3^, and for a A-connected alternator it is %, of the resistance between the terminals. The phase resistance and the resistance between the terminals are the same for either a single-phase or a two-phase alternator. Open-circuit Characteristic. — The open-circuit characteristic is a curve plotted for rated frequency with open-circuit voltages as ordinates and the corresponding field excitations as abscissae. Either terminal voltage or phase voltage may be plotted. The excitation may be expressed in amperes or in ampere-turns. Since with any fixed excitation the open-circuit voltage of a generator varies directly as the speed, it is possible to apply a correction to the measured voltages in case the frequency can- not be maintained exactly constant. The open-circuit char- acteristic should always extend from zero excitation up to the maximum excitation for which the alternator is designed. Short-circuit Characteristic. — The short-circuit characteristic shows the relation between the short-circuit armature current and the field excitation. This curve should always extend to at least one and one-half times the full-load current and as much further as is possible without overheating the alternator. The magnitude of the steady short-circuit current of an alternator at normal excitation depends upon its design and its size. This current will lie between one and one-half and five times the rated full-load current. It is limited by the syn- chronous impedance. The instantaneous rush of current, which takes place at the instant of short-circuit, is limited by the resistance and by the leakage reactance of the alternator. This current rush may be twenty or even thirty times as large as the normal full-load current (Chapter VIIT). Measurements for a short-circuit characteristic should be made at rated frequency, but a considerable variation in the frequency will produce a relatively small effect on the armature SYNCHRONOUS GENERATORS 97 current. Both the voltage induced in the armature and the synchronous reactance vary as the frequency. Therefore, if it were not for the armature resistance, which is always small compared with the synchronous reactance, a change in the fre- quency would have httle or no effect on the short-circuit current. ^ ' •p 2^6000 1000 5000 / / / / / / *- V / s / / 800 WOO 6 f / f # c. ff 600 3000 ? 7 , a 40Q|S 200 || /' f / / / 1 hree- Phas !, / / 5 MO-k' Uter ■Com latoi lecte 1 i I / 660 3 Vol s, 24 )tev per. min. 40 60 80 100 120 140 160 180 200 220 240 260 Eieia Current in Amperes, Fig. 51. Short-circuit characteristics are usually straight lines over the range of saturation through which it is possible to carry them. Although the impressed field may be large, the resultant field, which determines the degree of saturation, is small on account of the large armature reaction caused by the relatively large short- circuit armature current. The effect of the armature reaction of alternators with salient poles will, moreover, be a maximum 7 98 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY on account of the large angle of lag between the current and the generated voltage. The data for the open- and short-circuit characteristics of the 5000-kv-a., 60-cycle, 6600- volt, F-connected alternator men- tioned on page 87 are given in Table IV. The two characteristics are plotted in Fig. 51. Table IV Field current Open-circuit terminal voltage Short-circuit line current 100 150 200 250 4800 6500 7400 7900 680 1020 Examples of the Calculation of the Regulation by the Syn- chronous-impedance and the Magnetomotive -force Methods. — The regulation of the 5000-kv-a., F-connected generator will be calculated by both methods. The constants of this generator are given on page 87. Its open- and short-circuit characteristics are plotted in Fig. 51. Synchronous-impedance Method. — It is first necessary to find the synchronous reactance. From Fig. 51 take the short- circuit current and the open-circuit voltage corresponding to the largest field excitation used on short-circuit. The current and voltage, for an excitation of 160 amp., are 1085 amp. and 6720 volts respectively. The generator is F-connected. Therefore, the line voltage must be divided by -\/3 to get the phase voltage. The line current is the phase current. V3 6720 = 3.58 ohms. 1085 X, = Za = 3.58 ohms approximately. The regulation will be calculated for a power factor of 0.8 lagging. From Fig. 47, page 91, E'a = V + la (cos e - j sin e){re + jx,) = 6600 _^ 437 ^Qg _ jo.6) (0.069 + J3.58) V3 SYNCHRONOUS GENERATORS 99 = 4773 + il236 = V (4773) 2 + (1236)2 = 4931 volts. 4931 - ^ Regulation = ^^ 100 = 29.4 per cent. V3 The field excitation required for this load is the field current corresponding to a voltage of 4931 VS = 8541 on the open-circuit characteristic. This is beyond the range of the curve. Magnetomoiive-force Method. — Refer to Fig. 49. tJse V as an axis of reference. E"a = V + la (cos e - j sin 9)r, = ^ + 437(0.8 - j0.6)0.069 = 3835 - il8 = 3835 volts. The field current corresponding to a voltage of 3835 V'3 = 6642 on the open-circuit characteristic is 157 amp. This is R" on the vector diagram. The fictitious armature reaction, A', is the field current corresponding to an armature current of 437 amp. on the short-circuit characteristic. This field current is 64 amp. The angle /3 on the vector diagram is equal to 90° - a 18 sin a = cos a = 3835 3835 3835 The angle a is so small that it may be neglected and /3 taken as 90 degrees. F = A'{ - cos e +j sin 0) + Z?"(cos +j sin /3) = 64( - 0.8 + iO.6) + 157(0 + jl) = - 51.2 -I- jl95 = 202 amp. -A' 100 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The voltage on the open-circuit characteristic corresponding to thig is 7450. ^ , . 7450-6600 ,-_ , , • ^ Regulation = ^^ = 12.9 per cent. Potier Method. — The vector diagram given in Fig. 46 is known as the Potier Diagram. Although, as has already been pointed out, this diagram is correct only for an alternator with non-salient poles, it often is used in place of some of the more correct diagrams as an ap- proximate diagram for gene- rators with projecting poles. One of the more correct dia- grams is given later. The Potier method for determining the regulation of an alternator makes use of the vector diagram shown in Fig. 46. The important fea- ture, however, of this method is the manner of separating the armature reaction and the armature leakage re- actance. The terminal voltage of an alternator under load differs from its open-circuit voltage at the same excitation on ac- count of the change in the field caused by the armature reaction, and also on account of the drop in voltage through the armature produced by the leakage reactance and the armature effective re- sistance. The relative influence of the three factors depends upon the power factor of the load. With a reactive load at zero power factor, the decrease in the terminal voltage is due almost entirely to the armature reaction and the armature leakage re- actance. Under this condition, the effective resistance drop is in quadrature with the terminal voltage and has httle influence on the change in the terminal voltage caused by a change in load. This will be made clear by the vector diagram given in Fig. 52 which is for a reactive load of zero power factor. A" Fig. 52. SYNCHRONOUS GENERATORS 101 The resultant field, R, is almost exactly equal to the algebraic difference between F and A, and the terminal voltage, V, is very nearly equal to the algebraic difference between Ea and laXa. Under these conditions, the armature reaction subtracts directly from the impressed field and the armature leakage- reactance drop subtracts directly from the generated voltage. /^ ^ ■* p B 7000 y ^ ^ S N M^ / / S S ^ D 6000 / / y ^' y / / / .f/ / 4 r 5 / 4000 4 J / 4 / d &' 3000 1 / , ^/§ / «5 ny" 1 / / i / / / / G hree .Phaa s. Ji, -V, / / ^ OO-ks ra. TJ-Com Alternatoi ecte 1 / \ z 660 )Vol ;s, 24 )rev. per: Qin. 20 Ifi OOC 80 100 120 140 100 180 200 220 240 260 Meld Currents In Amperes. Fig. 63. The armature resistance drop has no appreciable effect on the terminal voltage. It follows from this that if an open-circuit characteristic, OB, and a curve, CD, showing the variation in the terminal voltage with excitation for the condition of constant armature current at a reactive power factor of zero, be plotted as is shown in Fig. 53 the two curves will be so related that any 102 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY two points as E and F, which correspond to the same degree of saturation and consequently to the same generated voltage, will be displaced from one another horizontally by an amount equal to the armature reaction and vertically by an amount equal to the leakage-reactance drop. GF represents the armature reaction in equivalent field am- peres, provided the excitation is plotted in amperes, and GE represents the leakage-reactance drop in volts. Let the curve CD be for an armature current I'. Then the armature reaction for any current, la, will be A. = la ~j7 ^^ laK and the armature leakage-reactance voltage for the same current will be 1 - T ^ EG _ J, ■— Xa is the armature leakage reactance. It has been shown experimentally that the open-circuit curve and the load characteristic at zero power factor, as curve CD is called, are sensibly the same shape and may be made to coin- cide if superposed.* This would be expected, since, at zero power factor, the effect of a fixed armature reaction should be independent of the degree of saturation of the armature and field as the axis along which it acts is fixed and coincides with the axis of the field poles. The leakage reactance of alternators with open slots, as are ordinarily used, should be nearly in- dependent of the degree of saturation of the armature teeth. In order to make use of the Potier method, it is necessary to find some means of locating two points, one on each curve, corresponding to the same generated voltage and the same resultant field. There are two ways by which this can be done. First Method. — Make a tracing of the open-circuit characteristic and the co-ordinate axes and mark some point, such as E Fig. 53, which is well up on the bend of the characteristic, on both the open-circuit characteristic and its tracing. Lay the tracing 'L'ficlairage Electrique, Vol. XXIV, p. 133. SYNCHRONOUS GENERATORS 103 on the plot. Then, keeping the axes on the tracing parallel with the axes on the plot, slide the tracing about until the traced curve coincides with the load characteristic CD. Then prick the point E through on to the load characteristic. By drawing the right-angle triangle EGF with its base parallel to the axis of abscissEe, the armature reaction and the leakage-reactance drop may be determined. The complete load characteristic is not necessary. Two points on this curve are sufficient, provided one of them, F, is well up on the bend of the curve. The other point is • preferably the point C. This latter point corresponds to the condition of short-circuit. The tracing is made as before but the point E is left off. The tracing is now moved parallel to itself until it touches the two points C and F. By transferring the point F to the tracing and then superposing the tracing on the open- circuit curve, the point E may be located. Second Method. — Since the two curves, Fig. 53, are parallel, the small right-angle triangle EGF will fit anywhere between them. Let it be moved down until its base lies on the line OC. It is shown dotted in this position. A new triangle QIC is formed with the lower part of the open-circuit characteristic. This new triangle has a definite base OC. From the point F draw a line FJ parallel and equal to OC. Through J draw another line parallel to the lower part of the open-circuit characteristic. The inter- section of this latter line with the open-circuit curve will locate the point E of the desired triangle. It will be seen that, unless the point E is taken well up on the bending part of the curve, the line JE will be nearly parallel to the open-circuit character- istic and the intersection between JE and the open-circuit char- acteristic will not be at all definite. The Potier method for determining experimentally the arma- ture reaction and armature leakage reactance of an alternator determines these quantities under approximately normal satura- tion but at a power factor which is very much below that met in practice. For this reason, the value of the armature reaction obtained will be too large in the case of alternators with salient poles. In practice it is impossible to obtain a load of zero power factor for determining a point as F on the load characteristic. 104 PRINCIPLES OF ALTERNATINO-CURfiENT MACHINERY but power factors sufficiently low may be obtained by using an under-excited synchronous motor operated at no load. American Institute Method.— The 1907 Standardization Rules of the American Institute of Electrical Engineers recommend the use of the magnetomotive-force method for calculating the regulation of an alternator. The revised Standardization Rules which were adopted by the American Institute of Electrical Engineers in 1914 recommend a modification of the synchronous- impedance method. The only essential difference between this modified method and the regular synchronous-impedance method is that the synchronous inipedance used is obtained at normal saturation instead of at low saturation. It is found from one point on the zero-power-factor load characteristic and one point on the open-circuit characteristic. Let the point F, Fig. 53, be a point at normal voltage on the zero-power-factor load characteristic for the armature current at which the regulation is desired. From F draw a vertical line intersecting the open-circuit curve at K. The distance FK represents the total change in voltage, at zero power factor, caused by armature reaction and the armature leakage-reactance drop. It is the synchronous-impedance drop. Since the curves shown on Fig. 53 are for a F-connected alternator and are plotted in terms of the voltage between lines, the synchronous impedance per phase is FK ^ — 7=— : ohms V3I where I is the line current. If the machine had been A-connected, the line current, I, would have had to have been divided by VS instead of the voltage FK. The distance FK may be divided into its two components by drawing a horizontal line through E intersecting FK at N. FN is then the leakage-reactance drop. NK is the drop in voltage which replaces the effect of the arma- ture reaction. Having found x^, the regulation may be calculated in the usual way. The American Institute Method of getting synchronous react- ance avoids the chief source of error in the value of synchronous reactance calculated from short-circuit data, namely, the error due to low saturation. The error due to low power factor which was mentioned in the discussion of the regular synchronous- SYNCHRONOUS GENERATORS 105 impedance method is still present unless the machine has non- salient poles. Example of the Calculation of the Regulation by the American Institute Method. — Referring to Fig. 53, FK 1300 •v/3 / V3 438 = 1.71 ohms. AArtO E'a = -j^ + 437(0.8 - jO.6) (0.069 + jl.71) = 4282 + j580 = 4321 volts. 4321 - ^ Regulation = ■ Karif) 1^0 = 13.4 per cent. V3" Value of A' of the Magnetomotive -force Method for Normal Saturation. — The fictitious magnetomotive force A' used in the magnetomotive-force method may be obtained at normal satu- ration from two points, one on the zero-power-factor curve and one on the open-circuit characteristic. If the voltage is to be kept constant when a zero-power-factor load is applied to a generator, the field current must be increased to balance the demagnetizing effect of armature reaction and to produce the increase in the generated voltage required to balance the leakage-reactance drop. MF, Fig. 53, represents this increase in field excitation. GF is the part of this increase required to balance the effect of armature reaction. MG is the part required to cause the increase in generated voltage needed to balance the leakage-reactance drop, FN. When A' is obtained from a short-circuit test, the magneto- motive-force method is an optimistic method since it gives a regulation which is usually better than that found from a load test. When, however. A' is found from a test made with the generator on a highly inductive load, the magnetomotive-force method becomes a pessimistic method if the generator has salient poles. The part MG of MF = A' on Fig. 53 which replaces the leakage-reactance drop is correct since it is for normal saturation. The other part GF, which is the field current 106 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY required to balance armature reaction is too large, when the gen- erator has salient poles, since with such a generator armature reaction has its maximum effect in modifying the field at zero power factor and has a greater effect than it does at any normal operating power factor. As a result MF = A' is too large for ordinary power factors. Example of the Calculation of the Regulation by the Magneto- motive-force Method Using the Value of A' Obtained from a Zero -power-factor Test. — From Fig. 53, A' corresponding to 6600 volts is 96 amp. From page 99, E"a = 3835. F = A'{ — cos e + j sin B) + i2"(cos a + j sin a) = 96( - 0.8 + jO.6) + 157 (0 + jl) = -76.8+i214.6 = 228 amp. From the open-circuit curve (Fig. 53, page 101), E'a = 7700 volts. Regulation = w^^ 100 = 16.7 per cent. Blondel Two-reaction Method for Determining the Regula- tion of an Alternator. — The armature reaction of an alternator with non-salient poles shifts the axis of the field flux and modifies the field strength without distorting it to any extent. In a generator with salient poles, however, the armature reaction not only modifies the field strength, but, except in the case of zero power factor with respect to the excitation voltage, it also distorts the field by crowding the fiux toward one pole tip and away from the other. The effects of the armature reaction of alternators with salient poles have already been discussed and are shown in Figs. 33, 34, 37, 38 and 39, Chapter IV. If the armature current of an alternator is in phase with its excitation voltage, i.e., with the voltage which would be produced on open-circuit by the impressed field, the armature reaction caused by that current merely distorts the field without modifying its strength. On the other hand, if the power factor is zero with respect to the excitation voltage, the armature reaction will modify the strength of the field without producing distortion. It is, therefore, convenient to resolve the armature reaction of SYNCHRONOUS GENERATORS 107 an alternator with salient poles into two quadrature components: one producing only distortion, the other producing only a change in the field strength. To take account of the two effects of the armature reaction of alternators with salient poles, Blondel suggested the two-reaction theory. "^ Since the magnetomotive force of armature reaction is in phase with the armature current, resolving it into the two components just mentioned is equivalent to considering the armature current to be resolved into two quad- rature components. If /„ is the armature current, the two com- ponents into which it should be resolved are la sin

given by equation (34). The vector diagram of the two-reaction method is given in Fig. 58. The generated voltage, E^, is found in the usual manner by adding the resistance and the leakage-reactance drops to the ter- 114 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY minal voltage V. In the two-reaction method of Blondel, the electromotive force, Ea, is considered to be the resultant of two quadrature components: one, Ec, induced by the flux produced by the transverse component of the armature reaction, and the other, Eo, induced by the flux from the main poles. For the present, assume that the angles made by these com- ponents with Ea are known. Ea may then be resolved into its two components Eo and Ec- Having obtained Ed, the excitation, FEj,, required on the main poles to produce this voltage may be found from the open-circuit characteristic by looking up the field magnetomotive force on that curve corresponding to the voltage Ed. The real excitation under load must be greater than this (an inductive load assumed) to balance the direct component, A'd (equation 34), of the armature reaction. The cross component, A'c (equation 39), of the armature reaction merely distorts the field without altering its strength. There- fore, the impressed field is F = Fe^ + A'd where F, Fed and A'd are considered in a purely algebraic sense. If E'a is the voltage on the open-circuit curve corresponding to the field F, the regulation is E'a — V — " y 100 per cent. The angle /3, which it is necessary to know in order to divide Ea into its two components, Ed and Ec, may be found in the following manner. Find the volts generated per ampere turn on the lower part of the open-circuit characteristic. Call this voltage v. For reasons already given, v will be assumed constant. Calculate ■^'c by equa^tion (39). Then Ec = vA'c - ._, ^ „ 2.22 (a 1 . 0x1 = V OAbhla Z cos ip -r T Sm -r" [' = QIa cos tp = QIa cos ((3 -I- 9') (41) where Q = vhZKc From Fig. 58, Ec = Ea sin /3 (42) SYNCHRONOUS GENERATORS 115 Combining equations (41) and (42) gives Ea sin |3 = QIa cos (/? + d') and Qlg cos 9' *^° ^ = E. + QLsme' (*^) e' = e + a (Fig. 58). Therefore, j3 can be found for aay given armature current, 7a, and load power-factor angle 6. Example of the Calculation of Regulation by the Two- reaction Method. — The regulation of the 5000-kv-a., 6600- volt, three-phase, F-connected generator which has already been used wiU be calculated. The rating and constants of this generator are given on page 87. The ratio of the pole arc to the pole pitch is 0.768. A full kv-a. load at 0.8 (lagging) power factor will be assumed. Refer to Fig. 58. The generated voltage was found, in the calculation of the regulation by the general method, to be Ea = 3900 + J70.3 = 3901 volts sin a = ^1 = 0.0180 a = 1° 1' cos d = 0.8 e = 36° 52' and and e' = + a = 37° 53' sin e' = 0.614 cos 6' = 0.789 sin (0.768 9 Kd = 0.45 ^ = 0.349 0.768 1 Kr = -^io.768 - - sin (0.7687r) [ = 0.160 TrKf I T From equation (43) QIa cos 6' tan ^ - £__ ^ Qi^ sin q> Q = vkiZKc 116 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY From the lower part of the open-circuit curve of this generator, • m , 1 TTr 4800 the data for which are given in Table 1 V, w = ,- Vo X 100 X 67.5 = 0.41 volt per ampere-turn per pole. Q is then equal to Q = 0.41 X 0.96 X 24 X 0.160 = 1.51 and + « 1-51 X 437 X 0.789 *^° ^ ^3901 + 1.51X437X0.614 = ^'^^^^ p = Q° 53' Ed = Ea cos /3 = 3901 X 0.993 = 3880 The field excitation corresponding to a voltage 3880 \/3 on the open-circuit curve of this generator, which is plotted in Fig. 51, page 97, is 160 amp. ^ = + a + /3 = 44° 46' sin ^ = 0.704 A'd — hIaZ sin (pKd = 0.96 X 437 X 24 X 0.704 X 0.349 = 2474 ampere-turns per pole. The impressed field, F, in amperes is 2474 160 + 67:5 = 197 The open-circuit voltage corresponding to this is 7360 between terminals. The regulation is, therefore, 7360-6600 ..^ ,_ — Rfin?) — ~ P®'" ^^'^''• The values of the regulation of the 5000-kv-a., 6600-volt, three-phase alternator calculated by the different methods given in this chapter are brought together in Table VI for comparison. SYNCHRONOUS GENERATORS Table VI 117 Method Per cent. regulation at unit power factor ' Per cent. regulation at 0.8 power factor General 3.0 9.4 4.6 2.7 1 9.1 3.0 4.1 11.4 Synchronous-impedance, using short-cir- 29.4 Magnetomotive-force, using short-circuit 12.9 Synchronous-impedance, using zero-power- 13.4 Magnetomotive-force, using zero-power- 16.7 Blondel double-reaction 11.5 Bv measurement h^^ ^>?t iS-t CHAPTER VI Short-circuit Method for Determining Leakage React- ance; Zero-power-factor Method for Determining Leakage Reactance; Potibr Triangle Method for Determining Leakage Reactance; Determination of Leakage Reactance from Measurements made with Field Structure Removed; Determination of Effect- ive Resistance with Field Structure Removed Short-circuit Method for Determining Leakage Reactance. — Short-circuit the armature and measure the phase current and impressed field at rated frequency for about full-load current. The vector diagram for a short-circuited alternator is shown in Fig. 59. E'a is the voltage on open circuit which corresponds to the impressed field F. i22 = i?'2 _|_ ^2 _ 2FA cos j3 P = a . „ . laTe Sin j8 = sm a = -^^^ ■H a (44) The armature reaction, A, may be calculated from equation 118 SYNCHRONOUS GENERATORS 119 (10), page 59, but it is better to calculate it from equation (34), page 109, which gives the direct component of the armature reaction used in the double-reaction method for determining the regulation of an alternator. On short-circuit the angle of lag be- tween the phase current and the excitation voltage is very large and, in consequence of this, the distorting component of the arma- ture reaction is very small and can be neglected. When equation (34), is applied to a generator having non-salient poles, the ratio of pole arc to pole pitch, i.e., t in equation (34), is determined by the arrangement of the field winding. By substituting the numerical values of F, A and ^ in equation (44) the resultant field R may be found. The angle /S is small and usually may be neglected. R = F — A approximately. Let Ea be the voltage on the open-circuit curve corresponding to R. Then ''^=yl(fy-''' The effective resistance, re, can be found by one of the methods which will be given later. The chief objections, to the short-circuit method for determin- ing the leakage reactance are the low degree of saturation and the low power factor for which the reaction is obtained. The objections are not of so great importance as might at first seem, since the reactance of ordinary alternators with open slots is not greatly affected by the degree of saturation of the armature teeth. Zero-power-factor Method for Determining Leakage React- ance. — When an alternator is operated on a reactive load at zero power factor, the axis of the armature-reaction magneto- motive force very nearly coincides with the axis of the impressed field and the two magnetomotive forces may be subtracted directly to give the resultant field. This has already been referred to in the Potior method for separating the effects of armature reac- tion and armature leakage reactance. Referring to Fig. 52, page 100, which is the. vector diagram of an alternator supplying a highly inductive load, it will be seen that the algebraic relation R = F - A 120 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY is very nearly correct. The armature reaction, A, as in the case of the short-circuit method for determining leakage reactance, is best found from equation (34), page 109. Ea is the voltage generated by the resultant field, R, and is equal to the voltage corresponding to an excitation, R, on the open- circuit characteristic. Again referring to Fig. 52, it will be seen that the following algebraic relation is very nearly correct: Ea — V = laXa from which Ea-V Xa — /. The highly inductive load required for the zero-power-factor method of determining the leakage reactance may be obtained by using as a load for the alternator an under-excited synchronous motor operating without load. The zero-power-factor method of determining the leakage reactance is not so simple to apply as the short-circuit method, but it has the advantage of giving the reactance for about normal saturation. The effect of the low power factor under which the reactance is obtained will tend to make the measured value of the reactance of alternators with salient poles slightly larger than it would be under ordinary operating power factors. If the equivalent leakage flux is desired, it can be found by making use of equation (26), page 78. Potier Triangle Method for Determining Leakage React- ance. — This method has already been given in Chapter V. The Determination of Leakage Reactance from Measure- ments Made with the Field Structure Removed. — An approxi- mate value of the leakage reactance of the armature of g.n alternator may be obtained by removing the field structure and measuring the voltage, V, required to send about full-load current through the armature. The ratio of this voltage to the current will be approximately equal to the armature impedance. _ V_ ■*■ a Xa = -s/27^^2 If r^ is known, Xa may be found. SYNCHRONOUS GENERATORS 121 The method just outUned for determining the leakage reactance of an alternator assumes that the leakage flux of the armature for a fixed armature current is the same whether the field structure is in place or removed. This assumption is probably not far from correct in many cases, since the only part of the leakage which would be affected materially by the removal of the field is the tooth-tip leakage. In addition to the leakage fiux, a second flux is caused by armature reaction which passes between the poles produced on the armature by the armature current. The voltage induced in the armature inductors by this second flux is not a part of the leakage-reactance voltage and should not be included in it. With the field structure in place, this flux combines with the flux caused by the impressed field to pro- duce the resultant field and it has nothing to do with the voltage drop through the armature. Although the voltage induced by this flux is included in the value of Xa obtained by the method just described, the error introduced by it in the measured value of Xa is probably not large, since the armature-reaction flux will be small when the field structure is removed on account of the high reluctance of its magnetic circuit under this condition. With the field structure in place, the effect of this flux would be very large. The Determination of the Effective Resistance with the Field Structure Removed. — If the power consumed by the armature is measured when the field structure is removed and a current, la, passed through it, the effective resistance may be found by dividing the power, P, per phase by the square of the phase current, h, _ L '■« — T 2 la This assumes that the armature-reaction flux is negligible so that the core loss is entirely due to the leakage flux. It also assumes that the core loss produced by a given change in flux is inde- pendent of whether that flux acts alone or in conjunction with another flux. When the field structure is removed, the core loss in the teeth is that caused by the leakage flux. This is the only flux which exists. Under this condition, all of the core loss in the teeth is effective in increasing the apparent resistance. Under operating conditions, the core loss in the 122 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY teeth is due to the resultant variation of the flux in the teeth caused by the leakage flux superposed upon the flux from the field poles. Only that part of this loss which is due to the leakage flux contributes to the loss caused by the effective resistance. The increase in the core loss caused by the superposition of the leakage flux can be the same as the core loss produced by the leakage flux when acting alone only when the core loss varies as the first power of the flux density. It actually varies between the 1.6 and 2 powers. Moreover, the superposition of the two fluxes not only changes the magnitude of the tooth flux but changes its distribution as well. A change in the distribution of a flux will alter the core loss produced by it even though the total flux remains unaltered. A second method for determining the effective armature resistance which can be used under certain conditions will be taken up after the discussion of the losses in an alternator. CHAPTER VII Losses; Measukement of the Losses by the Use of a Motor; Measurement of Effective Resistance; Retardation Method of Determining the Losses; Efficiency Losses. — With the exception of the commutator brush- friction loss, an alternator has the same losses as a direct-current generator, and in addition it has certain load losses which are not present in a direct-current machine. The losses in an alternator may be divided into two general groups, namely : the open-circuit losses and the load losses. The open-circuit losses are those which are present at no load. They are all also present under load, but under load conditions some of them are modified. The load losses are those which are caused either directly or indirectly by the armature current. The open-circuit losses may be divided into : (a) Bearing friction. ' (Jb) Brush friction. ■ ' (c) Windage loss. ' (d) Hysteresis and eddy-current losses caused by the resultant field. (e) Excitation loss. The load losses may be divided into two groups : (/) Armature copper loss due to the ohmic resistance of the armature winding. {g) Local core and eddy-current losses caused directly or indirectly by the armature current. (a) Bearing Friction. — The bearing-friction loss is proportional to the length and diameter of the bearing and to the three-halves power of the linear velocity of the shaft. It depends 'upon many factors such as the condition of the bearings, lubrication, etc., and it varies with the load, especially if the generator is belt-driven. The loss caused by the bearing friction is small and for this reason it is usually assumed to be constant. 123 124 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY (6) Brush Friction. — The brush-friction loss is caused by the brushes for the field excitation. On account of- the few brushes required and the low rubbing velocity of the slip rings against these brushes, this loss is very small. (c) Windage Loss. — The windage loss is iiot great except in the case of turbo alternators. It cannot be calculated. All of the friction and windage losses are generally grouped together and determined experimentally or estimated from experimental data obtained from measurements made on similar machines. (d) Hysteresis and Eddy-current Losses Caused by the Re- sultant Field. — The hysteresis and eddy-current losses caused by the resultant field include all eddy-current and hysteresis losses which are not directly or indirectly due to the armature current. Besides the ordinary eddy-current and hysteresis losses in the armature, there are certain additional eddy-current and hysteresis losses, namely: 1. The eddy-current losses in the armature end plates and bolts and in the frame due to leakage flux which gets into these parts. 2. The pole-face losses caused by the movement of the arma- ture slots by the pole faces. When solid poles are used, as in the case of some large turbo generators, these losses will increase very rapidly as the ratio of the width of the slot opening to the length of the air gap increases. Fig. 60 shows the distribution of the flux across the pole face, at one particular instant, in the case of an alternator with a Me-in. air gap and armature slots 1 in. wide. 3. The eddy-current losses in the armature conductors caused by the field. The flux entering a slot is not constant but varies with the position of the slot with respect to a pole. It is a maximum when the slot is opposite the center of a pole and a minimum when the slot lies midway between two poles. Fig. 61 shows the approximate direction of the flux lines in the slots and air gap of an alternator at no load. The number of these lines per inch represents in a very crude way the intensity of the field. The variation in the flux entering a slot will set up eddy currents in the inductors. The voltages producing these eddy currents will be different on the two sides of the slots, and will be greater at the top of the slots than at the bottom. Therefore, SYNCHRONOUS GENERATORS 125 to prevent eddy-current losses due to these differences in voltage, it is necessary to laminate the armature inductors both horizon- 1 1 1 f— -1 ^ irt I — J — r~-4 . H- a ■§ I 1 — f — f— hUT m . 1 -ir/^ ^Pc^^^ S 3 1 tW-, a N ^^^Lr^f^ S s r"~Ty--f. >j '//"/^ 600 00 //r^ 450 00 -TtH-^UTTyhM 400 00, XrrTM^rr^ JL350 OOJ^ — , nW-tefySt \^3rAvAjJtV^"TrnP V ' 30O 00 -^R=?v-f-44.nSU -.250. 00 /r--i 200 oo n^^ 150 00 100 00 iAtO* -Vwrr^S- ^ 50 00 \r~-/r/ Flv I Dist ibiitiof iiil i A \ ^V'\\ V-A — \ Ttbe Air "& a rolst le^ac lOf thl ^ple / 1 IJ~~~L,^I 1 >— ' ' \\\\\XX\Vrrr\ 1 1 ^ A^ Fig. 60. L_I Fig. 61. tally and vertically. It is not at all important that the lamina- tions should extend to the bottom of the slots, since the flux 126 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY entering a slot never penetrates to more than one-third to one- half the depth of the slot. The loss due to these eddy currents is nearly constant and independent of the load, and for this reason At is usually included in the core loss. It is not necessary to laminate the armature inductors except M^hen their cross-section is large as in the case of bar windings. Even when bar windings ate used, the inductors are laminated only in one direction, namely, vertically. (e) Excitation Loss. — The excitation loss is the copper loss in the field circuit and is equal to the field current multiplied by the voltage across the field circuit. The loss in the field rheostat is included in the excitation loss. The excitation loss varies both with the load and with the power factor. It will be greatest for inductive loads. (/) Armature Copper Loss Due to Ohmic Resistance. — The armature copper loss is the ordinary la^r^ loss and may be com- puted easily from the length, the cross-section and the specific resistance of the armature conductors. (g) Local Core Losses Caused by Leakage Flux. — In addition to the ordinary copper loss in the armature inductors, there are eddy-current losses in these inductors and hysteresis and eddy- current losses in the pole faces and armature teeth which are produced by the leakage flux set up by the armature current. To prevent the eddy-current losses in the armature inductors due to the leakage flux, it is necessary to laminate the armature inductors horizontally. The armature current also causes some eddy-current losses in the end connections and any adjacent metal. All the eddy-current and hysteresis losses which are directly due to the armature current produce an effect which is equivalent to an apparent increase in the armature resistance, and may be taken into account by using the so-called effective resistance in place of the ohmic when finding the armature copper loss. Measurement of the Losses by the Use of a Motor. — The open-circuit losses of an alternator, (a) to (cZ) inclusive, may be determined by driving the alternator on open circuit by a shunt motor. The open-circuit losses corresponding to any excitation are equal to the input to the armature of the motor minus the belt loss, the armature copper loss and the stray power SYNCHRONOUS OENERATORS 127 of the motor. The input to the alternator when its field circuit is open is its friction and windage loss. The difference between the open-circuit losses and the friction and windage losses is known as the open-circuit core loss. It is customary to plot this loss against field excitation expressed either in amperes or in ampere-turns. The load losses may be obtained by finding the power required to drive the alternator on short-circuit. All phases should be short-circuited. This power less the friction and windage losses is the load loss. The difference between the load loss and the short-circuit copper loss is known as the stray load losses or short-circuit core, loss. These losses depend upon the armature current and should, therefore, be plotted against that current. The stray load losses include all losses due to the armature leak- age flux and a small core loss due to the resultant field. The stray load losses under normal operating conditions are usually less than the stray load losses determined on short-circuit for the same armature current. The difference between these losses under the two conditions depends upon many factors; it is greatest in high- speed turbo alternators with solid cylindrical field structures. Although the stray load losses measured on short-circuit are greater than under operating conditions, the revised Standardiza- tion Rules (1914) of the American Institute of Electrical Engineers recommends the use of the stray losses measured in that way in calculating the efficiency of polyphase synchronous generators and motors. Measurement of Effective Resistance. — If the local losses produced by a fixed armature current are assumed to be the same on short-circuit as under normal conditions, the effective resistance of an alternator may be found by dividing the total losses produced by the armature current when the alternator is short-circuited by the number of phases and the square of the armature phase current. The losses caused by the armature cur- rent can be found by subtracting the core loss corresponding to the resultant field from the total short-circuit losses exclusive of friction and windage. This method of determining the effective resistance is not very reliable since, with the low field intensity used on short-circuit, the load losses are usually greatly exag- gerated. It is, moreover, subject to most of the errors of the 128 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY method for measuring effective resistance with the field structure removed (see page 121). Retardation Method of Determining the Losses. — It is often impossible or impracticable, when dealing with large machines, to drive them by motors to determine their losses. In the case of turbo generators there is often no projecting shaft to which a motor may be attached or belted. Under this condition the retardation method of determining the losses is the only one which can be used. The kinetic energy of any rotating body is W = Mw^l (45) where W, o and | are, respectively, the kinetic energy, the angular velocity and the moment of inertia of the rotating part. Differentiating equation (45) with respect to co gives dW , dw The differential of energy with respect to time is power, and the rate of change of angular velocity, i.e., -n, is angular accelera- tion. Replacing -rr by P, power, and —tt by a, i.e., by angular acceleration, gives equation (46). P = Ma. (46) The power, therefore, causing any change in the angular velocity of a rotating body is equal to the moment of inertia of the body multiplied by its angular velocity and by its angular acceleration at the instant considered. If the rotating body is coming to rest, the acceleration will be negative and is called retardation. The formula P = Icoa may be applied to a motor or a gen- erator to determine the losses, provided the moment of inertia of its rotating part can be found. There are several methods by which the moment of inertia may be determined. One of these is more satisfactory than the others and alone will be given. SYNCHRONOUS GENERATORS 129 If any alternator is brought- up above its synchronous speed with armature circuit open and its field circuit closed and is then allowed to come to rest, the retarding power causing it to slow down is its friction and windage and open-circuit core loss. If the angular retardation, i.e., a, is measured at the instant the generator passes through synchronous speed, the friction and windage loss plus the open-circuit core loss corresponding to the excitation used may be calculated from formula (46), provided the moment of inertia is known. If the generator comes to rest without field excitation, the formula will give the friction and windage losses alone. The chief source of error in the application of the retardation method lies in the determination of the retardation a. In order to find a, it is necessary to take readings for a speed-time curve as the generator slows down. Some form of direct- reading tachometer will be necessary for this. The interval required between the successive readings for the speed-time curve will depend upon the size and speed of the generator being tested, and will vary from 5 seconds for very small machines to as many minutes in the case of the largest turbo alternators. A speed- time curve is plotted in Fig. 62. If a line he is drawn tangent to the curve at a, which is the point of rated speed, the retardation a will be ed The simplest and most satisfactory method of finding the moment of inertia is first to measure the open-circuit losses at rated frequency and with some definite field excitation. This can be done by operating the machine as a synchronous motor and adjusting the excitation for unit power factor (Synchronous Motors, page 297). The power input to the armature under this condition is equal to the sum of the friction and windage losses, the core loss corresponding to the excitation used and a very small armature copper loss, which can usually be neglected if the power factor is properly ad j usted . Having determined the losses, the speed of the generator is increased 10 or 15 per cent, by in- creasing the frequency of the circuit from which it is operated or by any other convenient means. The generator is then allowed 9 130 PRINCIPLES OF ALTERNATINO-CURBENT MACHINERY to come to rest with its field circuit still closed and its excitation unaltered, and readings are taken for a speed-time curve as the generator slows down. By substituting in formula (46) the friction and windage and core losses as measured at synchronous speed and the values of w and a also at synchronous speed, the moment of inertia may be found. eou 500 \ a dOO 1 \ \ 300 f 1 1 1 a \ 1 1 — t N SCO 1 1 1 u \\ K N UK) 1 1 \ •^ 1 1 \ 1 e! \d 10 SO 30 40 50 80 Xime Fig. 62. 70 80 90 100 Having determined the moment of inertia, the friction and windage losses may be found by taking measurements for a speed- time curve while the generator comes to rest without field excita- tion. The friction and windage and core loss corresponding to different field excitations may be found by allowing the generator to come to rest with different field excitations. Knowing the friction and windage losses, the open-circuit core losses corre- sponding to these field excitations may be found. It is also possible to get the short-circuit losses by letting the generator come to rest with its armature short-circuited and with SYNCHRONOUS GENERATORS 131 a field excitation which will produce the desired short-circuit armature current at synchronous speed. The power found under this condition, minus the friction and windage losses and the Ih losses in the armature due to its ohmic resistance, is the short- circuit core loss corresponding to the current in the armature when the generator passed through synchronous speed. The armature current will remain very; nearly constant over a wide range of speed. The reason for this has already been given under the discussion of the short-circuit characteristic. Formula (46) will give the power in watts, provided the second member of the formula is multiplied by 10""' and | is expressed in c.g.s. units. The angular velocity, co, and the angular re- tardation, a, are expressed in radians per second and radians per second per second, respectively. As the method just out- lined is purely a substitution method, the units in which P, |, ai and a are expressed are of no consequence. Efficiency. — The efficiency of any piece of apparatus is equal to the ratio of its output to its output plus its losses. Efficiency = — : — t^^ (47) •' output -f- losses ^ ' If the losses corresponding to any given output are known, the efficiency corresponding to that output can easily be calculated by means of equation (47). For a three-phase alternator operating under a balanced load, equation (47) may be written V^VKp.f.) Efficiency = ^3^,^^.^.) + p^ + 3,^.,^ + p ^^^ ^j^y^ (48) where the letters have the following significance : V = Terminal voltage. I = Line current. la = Phase current. Pc = Open-circuit core loss. Te = Effective resistance of the armature per phase. If = Field current. Vf = Voltage across field including the field rheostat. p.f. = Power factor. Pf+v, = Friction and windage loss. 132 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The field current corresponding to any load may be found by any of the methods already described for determining the regula- tion. It is best to use the American Institute method. The proper value of the core loss, Pc, is that value, on the curve of open-circuit core loss, which corresponds to a voltage equal to the phase terminal voltage plus the armature-resistance and leak- age-reactance drops. The American Institute of Electrical Engineers recommends that the efficiency of an alternator be calculated by dividing its output by its output plus its losses where the losses are: open-circuit core loss, the copper loss in the field, the friction and windage losses, the armature ohmic copper loss and the stray load losses. Since there is no generally accepted method of determining the armature leakage reactance, it is recommended that the core loss be taken for a voltage corresponding to the terminal voltage plus the arma- ture-resistance drop. It is further recommended that the effi- ciency be referred to a temperature of 75°C. CHAPTER VIII Transient Short-circuit Current Transient Short-circuit Current. — The short-circuit current of an alternator under steady conditions is limited by the syn- chronous impedance of the armature and is determined by the open-circuit voltage corresponding to the field excitation and the synchronous impedance. The maintained short-circuit current under conditions of normal excitation is from one and a half to three or four times full-load current, depending upon the type of alternator. The lower limit applies to large, modern turbo alternators. Since the effective resistance is small compared with the syn- chronous reactance, equation (49) may be written The synchronous reactance, Xs, is made up of two parts: one the leakage reactance, Xa, the other a fictitious reactance, Xa., which replaces the effect of armature reaction on the voltage of the machine. The leakage reactance, Xa,-,iB a real reactance and is instantaneous in its action. The fictitious reactance, Xa, is not a true reactance. It is a term which replaces a magneto- motive force and is not instantaneous in its action chiefly on account of the mutual induction between the armature and field windings and the hysteresis and eddy currents in the poles. At the instant of short-circuit, the armature current is limited only by the effective resistance and leakage reactance. It is approximately equal to The ratio of the initial short-circuit current to the maintained 133 134 PRINCIPLES OF ALTERNATI NO-CURRENT MACHINERY short-circuit current is approximately equal to the ratio of the synchronous reactance to the leakage reactance. This ratio is very large for large turbo machines which, due to their design, have large armature reaction and low leakage reactance. Unless limited, the instantaneous short-circuit current of such machines may be twenty or even thirty times the full-load current. This is the reason for the use of current-limiting reactances in series with large generators. Such reactances were mentioned in Chapter VI, page 167. Since the force acting between two conductors varies as the product of the currents they carry, the forces produced on the end connections of an armature winding by the first rush of current on short-circuit are enormous. To successfully resist these forces, the end connections must be very strongly braced. The necessity for this bracing has been mentioned (page 37). One very satisfactory type of bracing is shown in Fig. 29, page 39. The end connections of the earher turbo alternators were not sufficiently braced to withstand short-circuits, and were fre- quently badly injured by a severe short-circuit. The actual magnitude of the initial rush of current on short- circuit depends upon the particular part of the voltage cycle at which the short-circuit occurs. It is, therefore, not the same for all phases of the alternator. The transient conditions existing in an alternator between the instant of short-circuit and the time when the short-circuit current reaches its final value are complicated and not well understood. They depend upon many factors, among which the mutual induction between the armature and field windings and the field leakage are very important. Due to the mutual induction between the armature and field^ windings, there is an increase in the field current when an . alternator is short-circuited. This increase gradually diminishfes and becomes zero when the armature current reaches its final value. Superposed on this transient increase in field current is a periodic variation in its strength which has the same frequency as the armature current. There is also an alternating voltage induced in the field winding which may be large, if the field reactance is high. The initial rush of current on short-circuit depends, to a large SYNCHRONOUS GENERATORS 135 extent, upoathe mutual induction between the armature and field and upon the field reactance. It is greatest in machines having large mutual induction between armature and field windings and low field reactance. Fig. 63 shows oscillograms of the armature currents, field current, and field voltage of a 9375-kv-a., 7200-volt, three-phase 06C1LLOC1RAPH CUJIVE6 SHOWJNG TRANSIENT 6H0RT-CIBCUIT CONHITIONS IN A fl375,KV.~J\, 60 CYCLE,. S-PHASE, 4-POLE, 7200-VOLT TURBO ALTEBNATOR. SHORT-OIRCUIT MADE-ATV4 RATED VOtTAGB WITH FIELD SHUNTEDWITH A dON^lNDUOTIVE RESrSTANC6 Field. Current Fig. 63. turbo alternator short-circuited at )^ rated voltage. During the short-circuit the field winding was shunted by a non-induct- ive resistance to protect it from injury. For this reason, the oscillograph curve of field voltage shown in Fig. 63 shows no rise in voltage. CHAPTER IX Conditions and Methods for Making Heating Tests of Alternators without Applying Load Conditions for Making Heating Tests. — In order to determine the actual temperature rise in the different parts of an alternator, it is necessary to 'run the alternator under normal load conditions until steady temperatures have been attained. Such a test would consume a large amount of power and would be very expensive. Moreover, there are few if any manufacturing companies which have sufficient power available to run at full load generators as large as are now being built. To meet these conditions and to obviate the necessity for actually loading an alternator in order to determine its temperature rise under normal rated load, certain methods have been devised by means of which a heat run may be carried out without a large expenditure of power. None of these methods reproduce the conditions of actual load, but some reproduce them much more closely than others. The chief methods of msLking heating tests without actually applying load are: (a) The zero-power-factor method. (&) Operating the generator short-circuited with 25 per cent, over full-load current and measuring the temperature, then re- peating the test with the generator on open circuit with 25 per cent, over rated voltage. (c) Hobart and Punga method using alternate periods of open and short-circuit. (d) Goldsmith method using direct current in the armature. (e) Mordey method and a modification of it. (a) Zero-power-factor Method. — The alternator, for this method of making a heat run, is operated at no load as an over-excited synchronous motor at rated voltage and frequency, with its field excitation adjusted so as to cause full-load current to exist in its armature. Under these conditions, the power factor 136 SYNCHRONOUS GENERATORS 137 will be very low and little actual power will be required. It is necessary, however, in order to carry out this test, to have a power plant which has a kilovolt-ampere capacity at least equal to the rated kilovolt-ampere capacity of the alternator being tested. The armature copper loss will be normal, but the field copper loss will be considerably too high. The core loss will also be somewhat too high on account of the over-excitation. To correct for the abnormal field heating, it is customary to multiply the field temperature rise obtained from the test by the ratio of the field loss under normal load conditions to the field loss during the test. The zero-power-factor method of making a heat run is the method usually selected when sufiicient kilovolt-ampere capacity is available. The test appears to be the most satisfactory of those mentioned. A modification of the zero-power-factor method consists of operating the alternator to be tested 6n alternate periods of over- and under-excitation. By properly adjusting the relative lengths of the two periods, the average field copper loss can be made equal to its normal full-load value. Under these conditions the core loss will also be very nearly normal. If two similar alternators are to be tested, both the heating and the losses may be obtained. One is driven as a generator and in turn drives the other as a synchronous motor. By properly adjusting the field excitation of both and the speed at which the first alternator is driven, the voltage, the current and the frequency of the two machines may be made equal to their normal full-load values. The field copper loss of one alternator will be larger, that of the other smaller, than under normal operating conditions. Correction for the field heating caused by this may be applied by the method, already indicated. The power required to drive the machine which operates as a genera- tor is the total losses of both alternators, exclusive of the field copper losses. One-half of this will be very nearly equal to the sum of the rotation and load losses of one alternator under the conditions of norrrial full load. (b) Separate Open-circuit and Short-circuit Tests at Respectively 25 Per Cent, over Voltage and 25 Per Cent, over Full-load Current. 138 PRINCIPLES OF ALTERNATINQ-CVRRENT MACHINERY The alternator, for this method of testing, is run at rated fre- quency on open circuit at 25 per cent, over its rated voltage until the temperatures of its parts become constant. It is then allowed to cool down. When cool, the test is repeated with the alternator short-circxiited and with its field excitation adjusted to give 25 per cent, over full-load current in its armature. The 25 per cent, over full-load current is a very crude attempt to produce the same heating in the armature conductors and in the armature teeth as occurs under normal full-load operating conditions. When a generator is short-circuited, the impressed field must be less than normal and as a result the core loss will be smaller than under full-load conditions. In consequence of this, the temperature of the iron as a whole will be less than under normal load and the loss of heat from the conductors will be greater than it should be. Moreover, in certain generators under load, some parts of the iron may be hotter than the adjacent parts of the embedded armature conductors. Under this condition, heat will pass from the iron to the conductors. The " factors which determine the temperature of the armature con- ductors and teeth of an alternator are altogether too complex to even be approximated by merely operating the alternator short-circuited at 25 per cent, over full-load current. Twenty- five per cent, over voltage is used in the open-circuit test to get approximately the same core temperature as at full load, but the conditions which determine core temperature under load are too complex to be reproduced in this way. The temperatures obtained from the separate open-circuit and short-circuit tests are unsatisfactory at the best and can be con- sidered only as guides for estimating the probable temperatures which would be reached under normal full-load conditions. (c) Hobart and Punga Method. — In the Hobart and Punga method of making a heat test of an alternator, alternate periods of open-circuit and short-circuit are used. The lengths of these periods as well as the voltage and current employed are adjusted so that the average losses throughout a complete cycle, consisting of an open- and a short-circuit period, are equal to the losses under normal load conditions. When an alternator is operated on short-circuit, the losses are: friction and windage, armature copper loss and core loss. On SYNCHRONOUS GENERATORS 139 open circuit, the losses are: core loss and friction and windage. The friction and windage losses need not be considered since they are nearly independent of the armature current and excitation/ Let the duration of a complete cycle consisting of an open- circuit and a short-circuit period be unity, and let x be the frac- tion of this period during which the alternator is short-circuited. Let I be the full-load armature current and let Pc be the normal full-load core loss. If the armature current on short-circuit is ~7=> and on open circuit the field current is adjusted to cause a core loss equal to Pc z — ^—, the average armature copper and core losses over the two X X periods will be the same as under load conditions. If lo and /« are the field currents for the periods of open-circuit and short- circuit, respectively, the average field loss will be 7.2x + 7„2(1 - x) = Ie,\ leq. may be called, for want of a better name, the equivalent field current. It is the constant field current which would pro- duce the same heating as the average heating caused by I, and /„. In so far as the average armature copper loss and the average core loss are concerned, x may have any value, being limited only by the safe limits of short-circuit current and open-circuit excitation. leg. depends upon the value chosen for x. By giving x the proper value, it should be possible to make the equivalent field current equal to the field current under load conditions. If this is done, the average losses will be the same as the losses under normal full-load conditions. The limits of possible short-circuit current and open-circuit voltage often make it impossible to use a value of X which will make the equivalent field excitation loss normal. (d) Goldsmith Method. — For this method of making a heat run, the alternator is operated at normal full-load excitation in order that the iron loss caused by the field and the field copper loss shall be the same as those occurring at full load. The armature copper loss is supplied by sending a direct current through the armature equal to the full-load armature current. The connections for supplying the direct current to the armature must be made in such a way as to prevent the high 140 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY alternating voltage, which will be induced in each phase of the armature, reaching the source from which the direct current is taken. This can be accomplished in several ways. If the alternator is three-phase A-connected, one corner of the delta may be opened and the direct current introduced at this point. If the alternator is F-connected, it may be reconnected in delta and then treated like a A-connected machine. If the alter- nator is single-phase, the armature winding may be divided into two equal groups of coils which may be connected in opposition. Each of the two phases of a two-phase alternator may be treated like the one phase of a single-phase machine. The objection to the Goldsmith method of making a heat run is that the ordinary load core losses are not present, and in their place there are other losses, produced mainly in the pole faces by the magnetic poles of the armature, which are caused by the direct current. These magnetic poles instead of being fixed with respect to the field poles, as they are when produced by the ordinary armature reaction of a polyphase alternator, revolve at synchronous speed. (e) Mordey Method and a Modification of It. — In the Mordey method, the armature winding, or in the case of a polyphase alternator each phase of the armature winding, is divided into two unequal parts which are connected in series so that their electromotive forces oppose each other. The winding is then short-circuited through a suitable adjustable reactance coil. The alternator is driven at its rated frequency with the field excitation adjusted so that the core loss is the same as under full-load conditions. Full-load current in the armature is obtained by adjusting the reactance coil in series with the arma- ture winding. Instead of dividing the armature into two unequal parts, the field may be similarly divided and connected so that two opposing but unequal electromotive forces are induced in each phase of the armature. Neither of these two methods can be applied to modern al- ternators owing to the severe mechanical vibration which results from their use. Behrend's modification of the Mordey method consists of dividing the field into two equal parts and varying the excita- SYNCHRONOUS GENERATORS 141 tions of these independently until full-load current exists in the armature which is short-circuited. This modification of the Mordey method does away to a considerable extent with the vibration and makes it possible to apply the method in many cases to modern slow-speed alternators. CHAPTER X Calculation of Ohmic Resistance, Armature Leakage Re- actance, Armature Reaction, Air-gap Flux per Pole, Average Flux Density in the Air Gap and Average Apparent Flux Density in the Armature Teeth from the Dimensions of an Alternator; Calculation of Leakage Reactance and Armature Reaction from AN Open-circuit Saturation Curve and a Saturation Curve for Full-load Current at Zero Power Factor; Calculation of Equivalent Leakage Flux per Unit Length of Embedded Inductor and Effective Resist- ance FROM Test Data; Calculation of Regulation, Field Excitation and Efficiency for Full-load Kv-a. AT 0.8 Power Factor by the A. I. E. E. Method Alternator. — The calculations will be made for a 1000-kv-a., three-phase, 60-cycle, 32-pole, 225-rev. per min., 2400- (hne) volt, F-connected alternator. The principal dimensions of this alternator are : Number of slots 192 Size of slots 0.85 by 2.6 in. Width of tooth at bottom 1.125 in. Width of tooth at tip 1.04 in. Diameter of armature at air gap 115.5 in. Effective radial length of armature core 9 J| in. Mean radial depth of air gap % in. Armature coils lie in slots 1 and 5 Armature turns per phase 192 Inductors per slot 6 Each inductor consists of two bars in parallel, each bar ' 0.27 by 0.283 in. Length of embedded armature inductor 9J^ in. Length of end connections per coil on one side of armature 16.6 in. Number of poles 32 Number of turns per pole 65 Ratio of pole arc to pole pitch 0.72 Pole pitch measured on armature bore 11.4 in. Friction and windage loss 10 kw. 142 •'.-atk. SYNCHBONOUS GENERATORS 143 The test characteristics of the alternator are shown in Fig. 64. Figs. 65 and 66 show, respectively, the arrangement of the arma- ture winding and a slot. The cross-hatched rectangles in Fig. 66 represent the inductors. Each inductor consists of two bars in parallel as indicated. 40 C 60 80 100 120 140 160 180 200 220 240 Field Excitation in Amperes. Fig. 64. Ohmic Resistance of Armature from Dimensions of Alter- nator. — Length of two inductors 18.25 in. Length of two end connections 33.2 m. Length per turn 51.5 in. Length of conductor per phase =61.5 X 192 9890 in. Cross-section of conductor = 0.54 X 0.283. . .' 0.153 sq. in. 144 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The specific resistance of copper at 20°C. per centimeter cube is 1.72 X lO"* ohms. Armature resistance per phase at 20°C. 9890 X 2.54 X 1.72 X IQ-^ „ „ .„_ , = 0.153 X (2.54)^ = O-O^^^ °*^^- Armature resistance per phase at 25°C. = 0.0437(1 + 5 X 0.00385) = 0.0445 ohm. The measured resistance per phase was (see plot, Fig. 65) -^-x — = 0.0463 ohm per phase. Armatixre Leakage Reactance from Dimensions of Alter- nator.— Referring to equations (16), (17), (20), (21) and (24), I* — rr — ■^ — r~^i* — :"; n^o- K-30 ->K— S0^<-3O°-»j<-S0->|<-3O->f«-3O^ 120- >1 A = —;^ j-g- + (« + + 0.73m) logio ~ 1 (4) (3.14) (9.13) (9) 0.85 2.54 [l.267 + 0.55 = 3080 X 2.24 = 6900 B=i^^|^+r + 0.73z. logic + 0.73(0.85) logic ■ ■KW" + w 3.14(1.04) + 0.85 0.85 ]1 C = w 12 ■ ' ' ' ' ' °'" w = 3080[0.475 + 0.40 +0.424] = 4010 g + i' + 0.73w log w w = 3080[0.317 + 0.40 + 0.424] = 3510 D = B = 4010 I Xe = 2-KJ -^^MZ^ (logic ^ - 0.5) f 10-» 2x/ J4.6 X 33.2 X 2.54 X 9 (logic |^ - O.5) jlO" 27r/{ 3450} 10-8 SYNCHRONOUS GENERATORS 145 Since the alternator has a three-phase winding, the self- and mutual induction of each coil side are 60 degrees out of phase. Consider the coil of phase 1 which is in slots 3 and 7, Fig. 65. This has the back side of a coil in phase 3 with it in slot 3 and the front side of a coil in phase 2 with it in slot 7. The mutual induction produced on phase 1 in slot 3 by phase 3 is 60 degrees behind the self-induction of phase 1. The mutual induction produced on phase 1 in slot 7 by phase 2 is 60 degrees ahead of the self-induction of phase 1. Therefore, if s is the number of slots in series per 'phase, the leakage reactance of phase 1 is Xa = 27r/s| C -I- A -H (I> + B) cos 60°} lO"' + sx, = 2ir/s(3510 + 6900 +(4010+4010) M} 10- '+2Tr/s{ 3450} 10-' = 0.432 ohm. Armature Reaction .from Dimensions of Alternator, — The order in which the inductors of an armature winding are con- nected in series does not influence the voltage induced in the winding or the armature reaction it produces, provided the direction of current flow through the inductors is not changed. The volt- age across the terminals of any phase is equal to the vector sum of the volt- ages induced in all the inductors of the phase and it is entirely independent of the order in which the component volt- ages are taken in making the vector summation. The actual winding of the generator which is shown in Fig. 65 may be re- placed, so far as voltage and armature reaction are concerned, by the equiva- lent winding shown in Fig. 67. To avoid confusion, the end connections of only one phase are indicated in this figure. The second winding differs from the first only in the order in which the end connections are made. The equivalent winding is a full-pitch winding con- taining two groups of coils which are. slipped by each other by 10 Fig. 66. 146 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY an angle which is equal to 180 — 120 = 60 degrees, that is, by an angle equal to the pitch deficiency. Each group of coils has a phase spread equal to the phase spread of the original wind- ing. In general, any fractional-pitch winding may be replaced by two full-pitch windings which have a phase spread equal to the phase spread of the original winding and which are slipped by each other by an angle equal to the pitch deficiency. The voltages induced in the two windings are, consequently, out of phase by an angle equal to the pitch deficiency measured in degrees. For purposes of calculation it is often convenient to replace a fractional-pitch winding by its equivalent full-pitch winding. The armature reaction of the 1000-kv-a. generator will be calculated from the equivalent winding shown in Fig. 67 by find- ing the reaction of each of the two groups of full-pitch coils and then adding these reactions vectorially. The reactions of the two groups of coils will, of course, be equal. Each group of the Fig. 67. full-pitch coils will contain on&-half the total number of series armature turns. A = 0.707 — ^ — j 2 cos ^ [ampere-turns per pole where N, la, h, p, and p are, respectively, the total number of armature turns in series, the phase current, the breadth factor, the number of poles and the coil pitch. The equivalent winding as well as the original winding has two slots per pole in each group of coils. From Table I, page 41, h is 0.966. /'192 X 6\ (^^>240.5 )(0.966) 2 X 32 = 2560 ampere-turns per pole. A = 0.707 2X"32 ^ "^°^ ^^° SYNCHRONOUS GENERATORS 147 Air-gap Flux per Pole, Average Flux Density in the Air Gap and Average Apparent Flux Density in the Armature Teeth at No Load for a Terminal Voltage of 2400 Volts, from Dimensions of the Alternator. — The equivalent winding given in Fig. 67 will be used. The harmonics in the air-gap flux will be neglected. E = 4.44 I^^f The effective or root-mean-square voltage in volts will be = 4.44/iVi«,,aO-s (50) 164 STATIC TRANSFORMERS 165 If the voltage is not a sine wave, expression (50) becomes El = 4(form factor)/iVi,p™10-8 (51) Transformer on Open Circuit. — When an alternating potential is impressed on an inductive circuit, the current will increase until the total voltage drop around the circuit is zero. Under this condition the total voltage drop due to induction plus, vectori- ally, the resistance drop in the circuit will be equal and opposite to the impressed voltage. F = -E + Ir V and E are, respectively, the impressed voltage and the total voltage induced in the circuit by the flux linking with it. The diagram of connections for such a circuit when it contains iron and its vector diagram are shown in Figs. 82 and 83 respectively. The conditions shown in these figures cor- respond exactly to those existing in a trans- former with the secondary circuit open. Referring to Fig. 83, Ei is the voltage induced by the flux linking- with the wind- ing. To induce this voltage, a flux

J (73) Equation (73) is an equation of the first degree with respect to p f and if plotted with —r^ as ordinates and / as abscissae will give 216 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY a straight line. The intercept of this line on the axis of ordinates will be Kk(S,m'^-^. Kef(S>m^ is equal to the ordinate at a point on the line for a frequency / minus the intercept oa. Equation (73) is plotted in Fig. 106. Referring to Fig. 106, oa and cd both multiplied by the fre- quency / are, respectively, the hysteresis and the eddy-current losses corresponding to that frequency. Measurement of Equivalent Resistance. — The equivalent re- sistance of a transformer may be calculated from the ohmic resist- ance of its primary and secondary windings, but it is sometimes better to measure it directly in order to include all local eddy- current losses or hysteresis losses which are produced in the conductors or in the iron core by the currents in the primary Erequency Fig. 106. and the secondary windings. The equivalent resistance, in- cluding these local losses, may be obtained from measurements made with the transformer short-circuited. The vector diagram of a short-circuited transformer is shown in Fig. 107. The flux in a short-circuited transformer is merely that re- quired to produce a voltage equal to the impedance drop in the secondary (Fig. 107). The secondary impedance drop will be approximately equal to one-half of the total impedance drop in the transformer. This total impedance drop is equal to the impressed voltage and, as a rule, it will not be over 4 or 5 per cent, of the rated voltage of the transformer even with full- load current in the short-circuited winding. The secondary induced voltage is only half of this or from 2 to 2}4 per cent. STATIC TRANSFORMERS 217 of the rated voltage. Since the flux is proportional to the induced voltage and since the core loss produced by a flux varies between the 1.6 and 2 power of the flux, the core loss in a short-circuited transformer is entirely negligible in comparison with the copper loss. The input to a short-circuited transformer will, therefore, be equal to the total copper loss corresponding to the short- circuit current plus all local losses that are produced by the short-circuit current. If P and I are, respectively, the input and the short-circuit current both measured on the side of the transformer to which the power is supplied, the equivalent re- P sistance referred to that side is j^- ^^~^^~r^~^ r. ^i Hi y Fig. 107. Measurement of Equivalent Reactance, Short-circuit Method. — When a transformer is short-circuited Vi = alaZi + IiZi Ze is the equivalent impedance and is referred to the primary side since /i is the primary current. _Zi Ze - j^ and If the primary and the secondary leakage reactances Xi and x^ are assumed to be proportional to the square of the number of turns in the two windings, the equivalent reactance may be divided into its two component parts. Although the short-circuit method of determining the leakage reactance of a transformer necessitates the use of very low saturation, the value of the reactance given by it will differ only 218 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY slightly from the value corresponding to normal saturation since the reluctance of the path of the leakage flux in most transformers is nearly independent of the saturation of the iron core. Measurement of Equivalent Reactance, Highly Inductive- load Method. — The simplified vector diagram of a transformer delivering a highly inductive load is shown in Fig. 108. Every- thing on the diagram is referred to the secondary winding. The equivalent reactance may be calculated from the follow- ing equation which is approximately trUe when applied to a transformer which carries a very highly inductive load (Fig. 108). a — CCfi The advantage of this method is that it gives a value of reactance which corresponds to very nearly normal saturation of the trans- hrrC li xe a ily. Fig. 108. former core. The disadvantage is that it necessitates the sub- traction of two voltages, — and V2, which are very nearly equal, and any error in the determination of either will be very much exaggerated in their difference. Opposition Method of Testing Transformers. — The limit of the output of a transformer is determined by the rise in temperature of its parts and by its regulation. Of the two, the temperature rise is by far the more important in most cases. The methods for determining the regulation of a transformer have already been given. In order to obtain the increase in temperature of a transformer under load, it is necessary to operate it under conditions which produce normal full-load STATIC TRANSFORMERS 219 heating for a sufficient length of time for the temperature of its parts to become constant. This will require from 2 to 3 hours for small transformers to 24 hours or longer for very large trans- formers. When merely the ultimate temperatures are desired, the time required to make a heat run may be reduced considerably by accelerating the heating during the first part of the test by operating at overload. Small transformers may be tested by applying an actual load, but when large transformers have to be tested, the cost of the power required for loading becomes prohibitive. In such cases, the opposition method may be used, provided two similar trans- formers are available. A modification of this method may be applied to a single transformer if it has two primary and two Fig. 109. secondary windings. The opposition method is equally ap- plicable to small transformers as to large and it is in very general use. It requires merely enough power to supply the core and copper losses of the two transformers being tested. For the opposition method, the primary windings of the two transformers are connected in parallel to mains of the proper voltage and frequency. The secondary windings are then connected in series with their voltages opposing. Fig. 109 gives the proper connections. A and A' represent the primary and secondary windings, respectively, of one transformer; B and B' are the corresponding windings of the other. If the secondary windings are opposed with respect to the series circuit, they are virtually on open circuit so far as their 220 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY primaries are concerned, and no current will flow in them when the primaries are excited. So far as the secondaries are con- cerned, the primaries are virtually short-circuited with respect to any current which is sent through the secondaries. The correctness of these two statements will be made clear by referring to Fig. 109. The plus and minus signs on this figure merely indicate the polarity of the transformer windings at some particular instant. The arrows show the direction of the current which would be produced at some instant by inserting an alternating electromotive force anywhere in the secondary circuit, as at e. By following through the circuit in the direction of the arrows, it will be seen that the transformers are short- circuited so far as the electromotive force inserted at e is. concerned. The secondary voltages are in opposition when con- sidered with respect to the electromotive force impressed on the primaries Therefore, if the rated voltage is applied to the primary windings, the transformers will be operating under normal conditions so far as core loss is concerned. If, at the same time, the voltage inserted at e is adjusted so that full-load current exists in the secondaries, full-load current will also exist by induction in the primary windings and the transformers will be operating under conditions of full load so far as the copper loss is concerned. The only power required under these conditions is that neces- sary to supply the core loss, which is measured by a wattmeter placed in the primary circuit at Wi, and the power required to supply the copper loss. This latter will be measured by a wattmeter at W2 with its potential coil connected about the source of electromotive force at e. One-half of the reading of the wattmeter at w^ divided by the square of the current measured by an ammeter in series with it will be the equivalent resistance of one transformer. A voltmeter connected about the source of electromotive force at e will record twice the equiva- lent impedance drop in one transformer. The reading of this instrument divided by twice the current in the circuit, given by an ammeter placed at aa, will 'be the equivalent impedance of one transformer. An ammeter placed at ai, in the primary circuit, will record twice the no-load current of one transformer. The temperature rise may be obtained both by thermometers STATIC TRANSFORMERS 221 and from resistance measurements. The resistances for the calculation of the temperature rise may either be obtained from measurements made by any suitable method at the beginning and at the end of the run or from the readings of the wattmeter and the ammeter placed at Wi and a^, respectively. The best way to obtain the voltage required at e is to insert the secondary of a suitable transformer at that point. The voltage may be varied by a resistance in series with the primary of this auxiliary transformer. If the core losses are put in on the low-voltage side of the transformers and the voltage at e is obtained from a third trans- former, all necessity for handling high-voltage circuits when adjusting for load conditions is avoided. CHAPTER XVIII Current Transformer; Potential Transformer; Constant- current Transformer; Auto-transformer; Induction Regulator Current Transformer. — Current transformers are used with alternating-current instruments and serve the same purpose as shunts with direct-current instruments. When a current transformer is used, its primary winding is placed in the line carrying the current to be measured and its secondary is short- circuited through the instrument which is to measure the current. Current transformers serve the double purpose of increasing the current range of an instrument and insulating it from the line. The ratio of the secondary current in any transformer to the load component of the primary current is constant and is fixed by the ratio of the turns on the primary and the secondary windings. The two currents are exactly opposite in phase. The total primary current and the secondary current are not exactly opposite in phase, neither is their ratio exactly constant. Both their phase relation and their ratio varies on account of the magnetizing current in the primary and the component current in the primary which is required to supply the core losses. When the secondary winding is closed through a very low impedance, such as an ammeter or the current coil of a wattmeter, the secondary i'ziduced voltage becomes very small and is equal to the impedance drop in the instrument plus the impedance drop in the secondary of the transformer. The mutual flux required to produce this small induced voltage will be corre- spondingly small and, since it is the mutual flux which deter- mines the magnetizing current and the component current supplying the core losses, these two components of the primary current will be small. Under normal conditions, i.e., with the secondary winding short-circuited through an instrument, 222 STATIC TRANSFORMERS 223 neither of these two components of the primary current should be more than a fraction of a per cent, of the rated current of the transformer. The voltage drop across the primary winding will, of course, be merely the equivalent impedance drop in the transformer plus the impedance drop in the instrument, both referred to the primary winding. Although the induced voltage in the current transformer and, therefore, the mutual flux are both directly proportional to the secondary current, assuming the impedance of the transformer and the instrument are constant, the small exciting current will not be exactly proportional to or make a constant angle with the induced voltage, sinc6 neither component of this current varies as the first power of the mutual flux. The magnitudes of both components of the exciting current will depend upon the degree of saturation of the iron core of the transformer. For this reason, direct current should not be put through a current transformer unless the precaution is after- ward taken to thoroughly demagnetize the core. For the same reason, the secondary winding should not be opened while the primary carries current. Passing either direct current through the windings of a current transformer or opening its secondary circuit while its primary winding carries current will change its ratio of transformation. The winding with the fewer turns is the one placed in the line; therefore, if the secondary winding is opened, the current transformer becomes a step-up transformer and a voltage both dangerous to life and to the insulation of the transformer may be induced in its windings. This voltage is limited by the saturation of the core. It will be very much less than the voltage- of the circuit in which the transformer is placed multiplied by ratio of turns. If the secondary in any way should be accidentally opened, the core should be completely demagnetized before putting the transformer back in service. A current transformer should be insulated for the full voltage of the line on which it is to be used and should be operated with its secondary winding and also its case solidly grounded. On account of the effect of the exciting component of the primary current upon the ratio of the primary and secondary currents and upon the phase relation between them, the excit- ing currents of current transformers must be made small by 224 PRINCIPLES OF ALTERN ATING-CURRENT MACHINERY designing such transformers to operate at relatively low flux densities. The windings must also be arranged for minimum leakage since any increase in the leakage reactance will increase the mutual flux and, therefore, both components of the exciting current. From what precedes, it is obvious that current transformers should be calibrated with the instruments with which they are to be used, as well as at the currents to be measured. For power measurements where accuracy is essential, it is often necessary to apply corrections for the phase displacement be- tween the primary and secondary currents caused by the excit- ing current. Fig. 110 will apply to a current transformer if x^ and r^ ss ^)| are considered to include the reactance and resistance of the instrument with which the transformer is used. Cm-rent transformers are made for two classes of work, namely: for use with instruments, and for operating protective and regulating devices such as automatic oil switches. For the second class of service great accuracy or constancy of trans- formation ratio with change in load is not required, but great reliability is of prime importance. Current transformers, in the case of high-voltage power stations, form an extremely important part of the auxiliary apparatus and require no small amount of space. They range in weight from 40 to 50 lb. for very low voltages to as much as 4000 lb. for 110,000 volts, and in height from 6 or 8 in. to 8 ft. and a diameter of 3 ft. A current transformer for a 66,000-volt circuit is shown in Fig. 111. STATIC TRANSFORMERS 225 Potential Transformer. — Potential transformers are used to increase the range of alternating current voltmeters and watt- meters and at the same time to insulate them from the line voltage. They do not differ from ordinary transformers except in detail of design. The ratio of the terminal voltages of an ordinary transformer does not change by more than a few per cent, from no load to full Fig. 111. load and the voltages would be in opposition if it were not for the resistance and the reactance drops. By designing a potential transformer with low resistance and reactance, the change in phase and in magnitude of the terminal voltages may be made small. The phase relation is of importance only when potential transformers are used in connection with wattmeters. Since the magnetizing current and the current supplying the core losses are important parts of the primary current, these com- 15 226 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY ponent currents should be kept small. The influence of the resistance and the leakage reactance of the windings is far more important in a potential transformer than in a current trans- former, since these factors affect both the ratio of transformation and the phase relation between the primary and secondary terminal voltages directly. The exciting current of a properly designed potential transformer should have relatively little influence on either the ratio of transformation or the phase relation between .the termi- nal voltages. When potential trans- formers are used for accurate power measurements, correction for the phase displacement between the primary and secondary voltages caused by the re- sistances and leakage reactances may have to be applied. Potential trans- formers as well as current trans- formers should always be calibrated. The space required for high-volt- age potential transformers, and their weights are somewhat greater than the space and weights of current transformers for the same line voltage. A 110,bOO-volt potential transformer is shown in Fig. 112. Constant-current Transformer. — When arc or incandescent lamps are operated in series, as is almost uni- versally done when they are used for street lighting, they must all have the same current rating and must be operated from a circuit which carries a constant current and which varies its voltage with the number of lamps in use. Except in some of the older central stations, where there may still be some Brush arc-light generators, constant- current or "tub" transformers are now almost universally em- ployed for such circuits. Since all modern arc lamps are of the luminous or flame type and require unidirectional current for their operation, the constant-current transformer would be of STATIC TRANSFORMERS 227 little use if it were not for the mercury arc rectifier. The constant- current transformer, however, with a mercury arc rectifier and suitable reactances to smooth out the current wave, forms a very satisfactory source of power for constant-current circuits feeding modern arcs. They are extensively used with rectifiers and form an important part of the auxihary apparatus of all central stations supplying power for street lighting. If a transformer of the ordinary type is designed with very high -leakage reactance, it wiU have a very drooping voltage characteristic and it may even be short-circuited without pro- ducing excessive current. A core-type transformer which has its primary and secondary windings on opposite sides of a core which is designed to give excessive leakage will have a characteris- tic of this kind. A transformer which is designed in this way, Fig. 113. if operated on the drooping part of its characteristic, will give a considerable range of voltage at sensibly constant current. The characteristic of a transformer which has excessive magnetic leakage is shown in Fig. 113. Between a and h on the characteristic there is a large change in voltage with a comparatively smaU variation in current. If the leakage reactance can be increased automatically as the current tends to increase, the transformer may be made to regulate for constant current throughout any desired range of load. The necessary automatic increase in the reactance is obtained in the constant-current transformer by arranging the primary and the secondary windings so that they may move relatively to one another. The increase in the repulsion between the two 228 PRINCIPLES OF ALTERNATINO-CURBENT MACHINERY windings produced by an increase in the current, causes them to move apart and increase the cross-section of the path for magnetic leakage and thus increase the reactance. The simple arrangement by which' this is usually accomplished is shown in Fig. 114. CCC is the iron core which should be long and should operate at relatively high density. A and B, respectively, are the primary and secondary windings. The secondary winding, B, is movable and is supported from an arm pivoted at D. A weight W, which is hung from the sector S attached to the Fig. 114. swinging arm, partially counterbalances the weight of the secondary winding. Due to the force of repulsion between the two windings caused by the primary and secondary currents, the winding B will move away from A until this force of repulsion is just equal to the unbalanced weight of the arm and the coil. If the impedance of the external circuit is diminished, the current will increase and the winding B will move farther away from A increasing the reactance and diminishing the current. By properly adjust- ing the counter-weight, W, and the shape of the sectors and angle at which they are set, the transformer may be made to regulate for very nearly constant current over any desired range of load, provided the core is long enough to allow the windings STATIC TRANSFORMERS 229 to get far enough away from one another at no load, i.e., short- circuit on the secondary. The maximum load iS that at which the windings come in contact. The conditions under which constant-current circuits are Fig. 115. operated seldom require constant-current regulation from full load to no load; consequently most constant-current transformers are designed for a limited range of regulation. This range is usually from full load to about one-half or one-quarter load. Since the secondary current in a properly adjusted constant- FiG. 116. current transformer is constant, the load component of the primary current will also be constant. If it were not for the variation in the exciting current, the whole primary current would be constant. Therefore, the primary winding will oper- ate, at a constant voltage and very nearly constant current, and 230 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the entire change in input will be caused by a change in the primary power factor. The secondary winding will deliver power at constant current and variable voltage and at a power factor which is determined by the constants of the load. The method by which a constant-current transformer regulates for constant current should be made clear by inspecting Figs. 115 and 116. Fig. 115 is for no load, i.e., short-circuit; Fig. 116 is for a large inductive load. The entire regulation is due to the change in the leakage-reactance drop with the change in load. When constant-current transformers are designed for more than 50 lights, the middle point of the secondary circuit feed- ing the lamps is sometimes looped back to the transformer giving in effect two independent circuits. No change in the trans- FiQ. 117. former is required for this arrangement of secondary circuit. The connections for the two circuits are shown in Fig. 117. The two circuits I and II are brought back to the transformer and grounded at E. Either of these two circuits may be short- circuited and cut out by the switches si and S2 and the remain- ing circuit operated alone. A constant-current transformer is started with the second- ary winding lifted to its highest position and with the load short-circuited. After the primary circuit has been closed, the short-circuit switch on the load is opened and the secondary winding released and allowed to take up the position correspond- ing to the load on the transformer. Constant-current transformers are extensively used with mercury-arc rectifiers to supply arc lights requiring unidirec- tional current. STATIC TRANSFORMERS 231 Auto-transformer. — In addition to the regular type of trans- former in which the primary and secondary windings are entirely independent, there is another type known as the auto-transformer or compensator which has a single continuous winding, a portion of which may be considered to serve both as primary and second- ary. The size of the wire used for the continuous winding will not be the same throughout unless the ratio of transformation is such that its two parts carry the same current. The arrangement of the auto-transformer should be made clear by Fig. 118. If used as a step-down transformer, all the turns between a and c will serve as the primary winding. Some of these, namely, those between b and c, will also serve as secondary. If the transformer is used to raise the voltage, all the turns will act as a secondary winding but only those between b and c will serve as a primary. Some of the turns on an auto-transformer may be considered to serve the double purpose of primary and secondary windings. Since a „ -.„ part of the winding on an auto-transformer serves for both primary and secondary, an auto-transformer will require less material and will therefore be cheaper than an ordinary transformer of the same output and efficiency. The saving, however, is large only when the ratio of transformation is near unity. Since the primary and secondary windings of an auto-transformer are in electrical connection, the use of auto- transformers for high ratios of transformation is limited to those places where electrical connection between the low-voltage wind- ing and a high-potential circuit is not objectionable^ \ Since all the turns on the auto-transformer between a and c link the same mutual flux, the voltage induced per turn will be the same throughout the winding. Therefore, if Nac and Nhe are, respectively, the turns on the winding between ac and be, the ratio of transformation will be iV ac If the secondary circuit is closed, a current Im will flow to the load. In the case of the ordinary transformer, this current 232 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY would flow in an independent secondary winding haying Ni^ turns and would exert a demagnetizing action equal to IbdNte. This demagnetizing action would be balanced by an increase in the primary current which would also flow through an inde- pendent primary winding having Nac turns. These two currents would produce equal and opposite magnetizing effects and f hcNao = ImN^c* (74) In the case of the auto-transformer, the secondary turns and a part of the primary turns are combined. These combined turns will carry a current which will be equal to the vector sum between lu and /«. The two components lu and lac may, however, still be considered to exert the same component magnetizing effects as when they existed in separate windings. Considering lac and 7m as conaponents, equation (74) is equally true for the auto-transformer. The current lac is the load component of the primary current and corresponds to the component Z'l on the vector diagram of the regular transformer. In addition to the component current, lac, all turns between a and c will carry a small component 7„ of the primary current which supplies the core loss and produces the mutual flux. Since the actual current (neglecting the exciting current), which exists in the turns between h and c is the vector sum of 7m and lac - Icb = 7m )f lac (75) Replacing 7m in equation (74) by its value in equation (75) gives - lac Nac = {Lb - Iac)Nic (76) ■*-cb -^V (£c , - ^ ■*■ ac -^V be The currents lac and lab are the same; therefore, Icb I ab = a - 1 (77) The load currents carried by the two parts of an auto-trans- former are, therefore, in the ratio a — 1, where a is the ratio of transformation of the auto-transformer as a whole, or the ratio of transformation between the portions ac and 6c. * The order of the subscripts on the currents and voltages indicates their direction. Eia = E.a- -Ecb Eba Ecb Eca — Ech Ecb = STATIC TRANSFORMERS 233 Let Eba, Ecb and Eca be the voltages induced by the mutual flux in the turns between ab, be and ac respectively. Then and K._ E.. — E.^ 1 Therefore, the ratio of the voltages and the ratio of the load currents in the turns between a and b and between b and c are the same as if the turns Nab and Nbc formed the primary and secondary windings of an ordinary transformer having a ratio of transformation of a — 1. In the case of a step-down auto-transformer, the current going to the load. may be considered to be made up of two parts: one supplied directly from the line through the coils Nab without transformation, and the other supplied by transformer action in the coils Nic- These two component currents will be in phase with respect to the load and in opposition in so far as their magnietic action on the transformer core is concerned. If the auto-transformer is used to step up the voltage, the voltage on the secondary or load side will be made up of two parts: one due to the transformer action in the coils Nab, and the other the voltage impressed across the primary winding Nbc. These two voltages will be very nearly in conjunction with respect to the load. The gain in output of the auto-transformer over the ordinary transformer is due to the fact that only a portion of the power delivered by it is transformed. A portion is always obtained directly from the line without transformation. If the exciting current taken by an auto-transformer is neg- lected, the solution of the vector diagram becomes simple. Consider a step-down auto-transformer having a ratio of transformation equal to a. Since the ratio of the load currents and the ratio of the induced voltages in the coils Nab and Nbc, Fig. 118, are the same as they would be in an ordinary trans- former with independent primary and secondary windings having a ratio of transformation equal to a — 1, the voltage across the coil Nab may be found by considering Ndb to be the primary and Nbc the secondary of an ordinary transformer having a — 1 for a ratio of transformation. The voltage impressed across 234 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Nac, i.e., the real primary voltage of the auto-transformer, will be the vector sum of Vab and Vhc. The vector diagram of an auto-transformer, neglecting the exciting current, is shown in Fig. 119. The regulation is V ■ — Vhc a 'be Ecb Fig. 119. In Fig. 119, Ibd is the current going to the load. The current in the winding Nu must, of course, be used for finding the ' impedance drop in Nic. This current is Icb = Ibd + lab J Ibd iab = —- a Icb = ■ Ibd or smce Ibd a — 1 ~7 ~ ■^''^ — Z — a a The vector diagram of the auto-transformer may be simplified by combining the resistances nc and Tab into a single equivalent resistance, and the reactances Xbc and Xab into an equivalent reactance. Te = Tab + nda — ly and Xe = Xab + Xbc{a — 1)'' The simplified diagram of an auto-transformer with all vectors referred to the winding Nab is given in Fig. 120. Fa. = Vbc + Vai = Vhc + (a — l)Vhc + lahiVe + jXe) = aVbc + -^{re -\- 3^e) STATIC TRANSFORMERS 235 The resistance r^ and the reactance Xe may be found by any of the methods used for determining the equivalent resistance and the equivalent reactance of an ordinary transformer, by merely treating Ndb and Ntc as the primary and the secondary windings, respectively, of an ordinary transformer. The electrical con- nection between these two coils will not influence the measure- ments. The core loss may be found by applying the proper voltage across any two terminals. Fig. 120. Relative Outputs of the Auto-transformer and the Regular Transformer. — Since a portion of the turns of an auto-trans- former serve the double function of primary and secondary windings, less copper will be reqjH It should be noted that the division of load between trans- formers which have dissimilar ratios of transformation depends upon the load carried by the system. From what precedes, it should be clear that transformers which are to be operated in parallel should have: (a) Equal voltage ratings. (6) Equal ratios of transformation. (c) Equivalent impedances which are inversely proportional to their current ratings. (d) Ratios of equivalent resistance to equivalent reactance which are equal. These four conditions are stated in the order of their relative importance. That the transformers should have the same voltage rating needs no explanation. If their voltage ratings are not the same, some will be operating on a higher voltage than that for which they are designed, and some on a lower. If the ratios of transformation are not the same, there will be currents in the transformers, in addition to the exciting currents, when the load on the system is zero. The magnitude of these currents will depend upon the differences between the ratios of STATIC TRANSFORMERS 251 transformation, and they cannot be eliminated without re- designing the transformers. If the impedances are not inversely proportional to the current outputs which produce the maximum safe temperature rises in the transformers, the transformers will not divide the load properly and some will become overheated while others are below their safe temperatures, unless the system is operated at less than its total rated capacity. If the ratios of equivalent resistance to equivalent reactance are not the same for all of the transformers, the currents delivered by them will not be in phase with each other or with the load current and the transformers will be carrying kilowatt loads which are not proportional to their current loads. As a result, the copper loss for a given load on the system in all of the trans- formers and in the system as a whole will be greater than it would be if all of the currents were in phase. In other words, the maximum safe kilowatt output of the system will be diminished. The last two faults, i.e., impedances not in the proper ratio and unequal ratios of resistance to reactance, may be corrected by inserting the proper amount of resistance, or reactance or both, on either the primary or the secondary sides of the transformers. CHAPTER XX Transformer Connections for Three-phase Circuits USING Three Transformers; Three-phase Transforma- tion WITH Two Transformers; Three- to Four-phase Transformation and Vice Versa; Three- to Six-phase Transformation; Two- or Four-phase to Six-phase Transformation; Three- to Twelve-phase Transfor- mation Transformer Connections for Three-phase Circuits using Three Transformers. — ^A and Y Connections. — When three single- phase transformers are used in connection with three-phase cir- cuits, they may be grouped in any one of the following ways: 1. Primaries in A, secondaries in A. 2. Primaries in Y, secondaries in Y. 3. Primaries in A, secondaries in F. 4. Primaries in Y, secondaries in A. Any one of these arrangements is symmetrical and will, therefore, give balanced secondary voltages on balanced loads, provided the primary impressed voltages are balanced. It is best not to use the Y connection without a neutral on the primary side except for balanced loads. If the load is much un- balanced, this connection will give unbalanced Une voltages on the secondary side. In any three-phase system, the vector sum of the currents at the neutral point must be zero and the voltages between the lines and neutral must change in such a way that this condition will be fulfilled. This unbalancing of voltages or shift of neutral will not occur if the neutral of the transformers and the neutral of the source of power are interconnected, since under this condition any one transformer can receive power entirely independently of any other. If a single-phase load is applied between the line and neutral of a group of transformers which are connected in double Y and which have no neutral connection on their primary side, only a small current can be obtained even if the impedance of the load 252 STATIC TRANSFORMERS 253 be reduced to zero. All of the current on the primary side of the loaded traiisformer must come through the primaries of the other two transformers which are on open circuit. Since these transformers are on open circuit, all of the current on their primary sides will be exciting current. It follows, therefore, that the only current that can be obtained from the loaded trans- former is a current which is equal, assuming a ratio of transfor- A^i No Load E^P, Secondaries E,=K Fig. 128. mation of 1, to the vector sum of the exciting currents of the other two transformers. If the impedance of the load be reduced to short-circuit, the only voltage across the primary of the loaded transformer will be the equivalent impedance drop in that transformer for a current which is much smaller than full-load current. As a result, the neutral point of the transformers on their primary side will shift until it almost coincides with the line to which the loaded transformer is connected. This puts 254 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the other two transformers very nearly across line voltage or across a voltage which is very nearly \/3 times the voltage for which they are designed. This, of course, will very much increase their exciting currents, but even a considerable increase in the exciting currents will allow only a small percentage of full- load current to flow in the loaded transformer. Even a slight unbalancing of load on the secondaries will produce a bad un- balancing of the secondary phase voltages. If the normal exciting currents of three transformers which are connected in Y on both their primary and their secondary sides are unequal, the secondary voltages to neutral will be unbalanced at no load as well as under load. If the secondaries are in delta, a small current will circulate in the closed delta. This will act as a magnetizing current and will very nearly restore the balance of the voltages. The effect of a single-phase load applied to a double- F-con- nected group of transformers, which have no primary neutral connection, is shown in Fig. 128. In order to make the diagram clearer, voltage drops are used on the primary and the secondary sides. This makes corresponding vectors for currents and voltages on the diagrams for the primary and the secondary sides in phase. The subscripts 1, 2 and 3 indicate the phases. A current vector with a prime represents a primary load component. 7'i on the left-hand side of the figure is the load component of the primary current for phase 1. 7i on the right-hand side of the figure is the corresponding secondary current of phase 1. Table XVI Connection Primary voltage Secondary voltage Primary Secondary Between lines To neutral Between lines To neutral A A 1 1 a A Y 1 V3 a 1 a 1 1 1 Y Y 1 V3 a aVS 1 1 Y A 1 V3 ay/S STATIC TRANSFORMERS 255 The effect of an unbalanced load on transformers which are connected in F — A, and which have no primary neutral con- nection, will be similar to that produced with the Y — Y con- nection but is less exaggerated since even with a single-phase load all of the secondaries will carry some current. Table XVI shows the voltage which will be given by the differ- ent three-phase transformer connections indicated on page 252. The principal advantages of the different connections given in Table XVI are: A A. If one transformer is damaged, the system may still be operated at about 58 per cent, of its normal capacity with the remaining two transformers connected in open A or V. AY. This gives a higher secondary hne voltage for trans- mission purposes than the other connections without increasing the strain on the insulation of the transformers. YY. This permits grounding the neutral points of both the primary and the secondary three-phase circuits. FA. This permits the primary neutral to be grounded. If the F — A connection is used for transmission purposes, the secondaries may be reconnected in F if at any time it becomes desirable to raise the transmission voltage in order to increase the capacity of the line. F connection of secondaries permits the use of a four-wire distributing system. This is sometimes desirable for Hghting. Method of Testing for Proper Connections. — When three single- phase transformers are to be connected in three-phase, their primary windings may be connected at random since the trans- formers have entirely independent magnetic circuits and the phase relations between their voltages depend merely upon the way in which they are connected together and to the line. After the primary windings have once been connected, the secondary voltages are fixed. The proper connections for the secondaries must, therefore, be tested out with a voltmeter or by other means. To connect the secondary windings in F, connect one terminal of each of two secondaries together and then put a voltmeter across the remaining two free terminals. The voltage across these will either be equal to the voltage of one secondary or to s/S times that voltage. It should be \/3 times that voltage. 256 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Fig. 129. If it is not, reverse the connections of either of the two second- aries. When the two secondaries have been connected properly, connect one end of the remaining secondary to the common junction of the other two. The voltage between the free terminal of this last secondary and the free terminal of either of the other two should be ■\/3 times the voltage of one secondary. If it is not, reverse the con- nections of the last secondary. The method of testing for the proper con- nections for putting the secondaries in A is similar to the method of testing for putting them in Y. For the A connection, connect one end of each of two secondaries together. The voltage across the free ends should be the same as the voltage of one secondary. If it is not, reverse one of the secondaries. Then connect one end of the remaining secondary to one of the free ends of the other two. The voltage across the remaining gap will be either zero or twice the voltage of one winding. 1 If it is double the voltage, reverse the con- nections. When it is zero, the remaining gap may be closed and the secondaries will be in A. If this gap is closed when the last secondary is connected reversed, the transformers will be virtually short-circuited. Twice the volt- age of one winding will act on an impedance which is equal to three times the impedance of a single winding. The current under this condition will be ^ what would flow if a single trans- former were short-circuited. Let Fig. 129 represent the secondary windings and also a vector diagram of the secondary voltages. If a and b are connected together, the voltage across the free ends or across a'b' will be Va'a + Vw This voltage will be ' This assumes there are no third harmonics in the secondary voltages (see page 278). JlVaa' STATIC TRANSFORMERS 257 equal to either V^'a or Vn multiplied by ^/3 and will lag behind the voltage Vw by 30 degrees. This is the correct connection of the windings aa' and 66' for Y. If 6' is connected to a, the voltage across the free ends or across a'b will be Va'a + Vvi. This will be equal to either 7„„/ or Vw, and will lead Vaa' by 120 degrees. This is the correct connection for A. The third winding should have c' connected to 6 of the second if A con- nection is desired. The vector diagram for A connection is shown in Fig. 130. The connections are a to 6', 6 to c' and c to a'. The vector sum of the three voltages Vvb, Vc'c and Va-a which act around the closed A is zero. If c is connected to 6 the resultant voltage in the three windings will be (Fj/j + Va'a) + F„' = 2F„,. Three-phase Transformation with Two Transformers. — Three-phase transformation may be obtained with only two single-phase transformers by connecting them either in open delta or F or in T. I" Both of these connections are unsym- Jf* metrical and will, therefore, give slightly / i\ unbalanced voltages under load. The / I \ amount of this unbalancing is, however, , / ^^-^^ \ small under ordinary conditions, especi- S^— ^"^"a ally with T connection. i^ ^^^^ Open-delta Connection. — The open- pj^ ^3^ delta or F connection is the same as the delta connection with one transformer removed. Therefore, when similar transformers are used, the voltages given by the delta and F connections are the same, and their outputs will be proportional to their line currents. Let I be the maximum current output per transformer. The current out- put per line of the A connection is -v/3/. The current out- put of the F connection is equal to the current output of one transformer or equal to /. Therefore, the output of the open delta will be —7= or 58 per cent, of the output of the delta. The actual transformer capacity of the open delta is two-thirds of that of the delta, but all of this cannot be utilized on account of the power factors at which the transformers of the open delta operate 17 258 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY as compared with the power factor of the load. With a non- inductive balanced load, each transformer of the delta system carries one-third of the total load at unit power factor. Under the same conditions, each transformer of the open-delta system carries one-half of the load at a power factor of -g- = 0.866. Multiplying 0.866 by ^ gives 0.58, which is the capacity of the open delta as compared with the delta. The transformers of the open delta will not carry equal watt loads except when the power factor of the three-phase load is unity. The current loads will, how- ever, be equal whenever the three-phase load is balanced. Fig. 132. The output of a system made up of two groups of transformers in parallel, one group consisting of two transformers in open delta or V, while the other consists of three transformers in A, is only 33)^ per cent, greater than the output of the A-connected group alone and not 58 per cent, greater as might be expected (see page 286). Fig. 132 is a vector diagram for two transformers connected in open delta or V. The lettering on this diagram corresponds to the lettering on the diagram of connections shown in Fig. 131. Equivalent resistances and equivalent reactances are used. Single and double primes indicate, respectively, primary and secondary values. STATIC TRANSFORMERS 259 The transformers forming the open delta are ca and be. Trans- former ca carries the current ha'- Transformer be carries the current Iw The voltage across the open part of the delta, i.e., across ab, will be equal to the vector sum of Vac and Vob- If is the angle of lag for the load, the current in the lines will lag behind the Y voltage of the system by an angle 6. To simplify the construction of the vector diagram, let be the angle of lag of the secondary current with respect to the primary voltage referred to the secondary, and assume the current load to be balanced with respect to the primary voltage. The current laa', Fig. 131, will lag behind V'ca by an angle 6 — 30 degrees. On Fig. 132, d is 30 degrees. Referring to Fig. 132, V'ca, V'ab and V'so are the three primary voltages referred to the secondary windings. V'ca, V'ai, and V'bc are the three corresponding secondary voltages. V'db is the voltage across the open side of the delta and is the vector sum of the voltages produced by the two transformers ca and be. y ab ^^ ' ac I" ' cb It will be seen from Fig. 132 that the secondary voltages cannot be exactly balanced for a balanced load. The unbalanc- ing on the diagram is very much ^ greater than will be found in practice on account of the exaggerated impe- dance drops. T Connectim. — Two transformers / with the same current ratings but / with different voltage ratings are used. / One transformer, which is called the / 'teaser," is connected to the middle c^ \ \ \ \ \ \ d \ of the other as is indicated in Fig. pjg, 133 133. Both the primary and the sec- ondary windings are connected in the same way. Fig. 133 will serve either for the diagram of connections or for the vector diagram of the voltages. The teaser transformer is represented by ad on the diagram. The second transformer is indicated by the line cb. The three- phase voltages are impressed across the terminals, a, b and c. The secondaries being similarly connected will supply three- 260 PRINCIPLES OF ALT ERN AT I NO-CURRENT MACHINERY phase power at a voltage which, except for the impedance drops, will be equal to the impressed voltage divided by the ratio of transformation. If the impressed voltages are balanced, the primary voltages Vdb, Vic and Vca will be equal and each equal to 2Vcd- The voltages Vda and Vdo and also Vda and Vdb will be in quadrature. The angle acd is 60 degrees; therefore, •J^ = sin 60 = ^ = 0.866 y ca ^ The teaser transformer, therefore, should be wound for a voltage which is 86.6 per cent, of the voltage of the line or of the main transformer. Usually the teaser transformer is wound for the same voltage as the main transformer but is provided with a tap for 86.6 per cent, of full voltage. A neutral point may be obtained from the T connection by bringing out a tap from the teaser transformer at a distance from a equal to two-thirds of the distance between a and d. V da 7? - 73 If w. Fig. 133, is the neutral point of the three-phase system, Vna^A^ Vca V3 but therefore, y na — /- ' ca F,a = 0.866 Fea =\-V c. — ^Ya = 2/^F The T system is unsymmetrical and, therefore, cannot give per- fectly balanced secondary voltages under load conditions. It is, however, perfectly satisfactory and gives less unbalancing than the open delta. STATIC TRANSFORMERS 261 Two exactly similar transformers can be used for the T con- nection with fair results, but this is not advisable except for temporary work or in an emergency. If the two transformers are similar, the one which is used for the teaser will have more turns than it should for the voltage impressed upon it and the impedance drop will be unnecessarily large. Fig. 134 is a vector diagram for the T connection. The load is assumed to be balanced with respect to the primary voltage. The angle of lag, fl, is 30 degrees with respect to the primary voltage. All vectors are referred to the primary. The voltage V'da is in phase with the Y voltage of the system. The transformer da carries line current. Therefore, the power V^=Vcd+V^ Fig. 134. factor for the transformer da is the same as the power factor of the three-phase load. The two halves of the secondary coil of the main transformer carry currents which are out of phase. Therefore, in order to find the voltage across the secondary, the transformer be must be treated like a transformer with two secondary windings which are independently loaded. By inspection of Fig. 134, it will be seen that the T system is not symmetrical and cannot, there- fore, give exactly balanced secondary voltages under load. The capacity of the T system for three-phase transformation is somewhat less than the sum of the capacities of the two trans- formers used. Take, for example, the case of a load at unit power factor and 262 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY assume that the teaser transformer, i.e., transformer da, Fig. 133, is wound for the correct voltage. Let the Une current and line voltage of the three-phase system be / and V respectively. The transformer da has a voltage equal to 0.866 F and works at the power factor of the load. Its output is therefore 0.866F/. The two halves of the secondary of the other transformer carry the current 7 at a power factor equal to cos 30° = 0.866. Its output is, therefore, VI cos 30° = 0.866F7 The total output of the system is 2(0.86677) The total rated capacity of the two transformers is 0.866 F7 + 77 = 1.86677 Comparing the actual output with the rated capacity gives 1.73277 1.86677 0.928 as the fraction of the total transformer rating which is available for the three-phase output. If the transformer da is wound for the same voltage as the transformer be but has a voltage tap for 86.6 per cent, of full voltage, 86.6 per cent, of the rating of this transformer will be utilized. In this case the output of the T system will be 86.6 per cent, of the total transformer rating, or will be the same as the three-phase output of the same two transformers when connected in 7 or open delta. Three- to Four-phase Transformation or Vice Versa. — Trans- formation from three- to four-phase or vice versa is easily accom- plished by means of the Scott- or T-transformer connections. Referring to Fig. 133 it will be seen that the voltages across the primary terminals of each of the two transformers are in quadrature and are in the ratio of 1 to 0.866. The secondary voltages will also be" in quadrature and in this same ratio. A symmetrical four-phase system may be obtained on the sec- ondary side by connecting the secondary windings together at STATIC TRANSFORMERS 263 their middle points and adjusting the turns on the two secondary windings so that the voltages of both are equal. This can be accomplished by making the ratios of transformation of the two transformers ad and be equal to „ r,„„ and - respectively. In order to have the two transformers interchangeable, both are usually provided with taps on their primary side for 0.866 per cent, of full voltage, but the tap on only one transformer is used. The Scott connection for three- to four-phase transformation is shown in Fig. 135. The point n of the common connection is the neutral point of the f our-phdse side. The secondaries may be considered to give either a four-phase or a two-phase system. The four-phase -n- •> Fig. 135. a C n 1 1 1 1 J 'L d ■^ tT voltages are na', nV, nd and nd' with ?i as a neutral point. The two-phase voltages are a'd' and h'c' . To transform from two-phase to three-phase, it is merely necessary to consider a', V, d' and c', Fig. 135, as the primary terminals and a, b and c as the secondary terminals. If the Scott connection is used to transform from two-phase to three-phase, one of the three-phase voltages, i.e., Veb is derived directly from one transformer. The other two three-phase voltages are equal to the vector sum of two quadrature voltages one of which is derived from each of the two transformers. Therefore, neglecting the insignificant effect of the impedance drops in the transformers, the wave form of the voltage Vic will be the same as the wave form of the'Voltage impressed on the two-phase side. The other two three-phase voltages, how- ever, will not be of the same wave form as the two-phase voltage 264 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY or of like form except when the two-phase voltage is sinusoidal, since the harmonics in the resultant of the two voltages which are out of phase are different in phase from the harmonics in the components. If power is put in on the three-phase side, one of the two-phase voltages, i.e., dh' , will be of the same wave form as the three- phase voltages. The other, however, except when the im- pressed voltage is sinusoidal, will be either more or less peaked than the impressed voltage. Whether it is more or less peaked will depend upon the harmonics present and their phase relations. Let the two two-phase voltages. Fig. 135, be evc' — El sin (co< -|- ai) + Es sin {Zoit + ag) + ^6 sin (5w< -I- as) + E7 sin (lut + a?) ed'a' = El sin {U + on - 90°) -t- Ei sin {SU + as + 90°) + Ei sin (5a)« + a^- 90°) + Ev sin (7ut + a, -f- 90°) These two voltages are alike in wave form but differ by 90 de- grees in phase. Assume a ratio of transformation of unity between the three- phase and four-phase voltages. Then, remembering that the secondary voltage of a transformer is opposite in phase to the primary voltage, — Bab = 0.866ea'd' 4-0.5 ec'b' — Cca = 0.866edv + 0.5 Cc'v Referred to 6c as an axis, the three-phase voltages are: — Che = E-i. sin (cot -|- ai) 4- Ei sin (Swi -\- as) + Ei sin (5w« + ae) + E^ sin (7w< + «?) (83) -eca= El sin (cot + ai - 120°) + Es sin (3co« + as - 240°) + Et sin (5coi + a^ - 120°) -|- Ej sin (7coi + a, - 240°) (84) -e„6= El sin (coJ + ai - 240°) + Es sin (3cot + a^ - 120°) + Ei sin (5cof -f as - 240°) -|- E^ sin (7coi + aj - 120°) (85) It will be seen from equations (83), (84) and (85) that the wave forms of the voltages .Vab and Vca are different from the wave form of the voltage Vbc- All three of the three-phase voltages contain third harmonics which differ by 120 degrees in phase. Except when three-phase voltages are obtained from Scott- STATIC TRANSFORMERS 265 connected transformers or some other unsymmetrical system, they cannot contain third harmonics (see page 47). The wave forms of the three-phase voltages are plotted in Fig. 136 for the case where the two-phase voltages contain 30 per cent, third harmonics. The angles ai and aa are assumed Fig. 136. to be 0° and 180° respectively. The fundamentals and the third harmonics of each wave are shown dotted. Three- to Six-phase Transformation. — Double A and Double Y. — A six-phase system may be derived from any three-phase system by the use of three single-phase transformers which are each provided with two independent secondary windings. The 266 PRINCIPLES OF ALTERNATINO-CVRRENT MACHINERY primaries should be connected for three-phase in either Y or A. The two sets of secondaries are connected to form two independ- ent three-phase systems with the connections of one set of sec- ondaries reversed with respect to the connections of the other. Fig. 137. The phase relations of the six secondary voltages are shown by Fig. 137. Reversing one group of secondaries gives the phase relations shown by Fig. 138. Secondaries In Double A I 11 m IV T VI Fig. 139. The two groups of secondaries may be connected in A or in F giving what is known as the double-A or the double- F connection respectively. In case of either the double-F or the double-A STATIC TRANSFORMERS 267 connection, one-half of the power deUvered by the transformers will be supplied by each group of secondaries at the three- phase voltage. The connections with the secondaries in double A and with the primaries in Y are shown in Fig. 139. The con- nections are shown diagrammatically at the left of the figure. The actual connections are shown at the right. Fig. 140 shows the diagrammatic and the actual connections for the double Y. The two deltas forming the double A have no electrical con- nection and therefore cannot be considered to form a true six-phase system. When, however, they are connected to the armature of a motor or a synchronous converter, the electrical PrImaiUea Secondaries In Double Y 1 I II in IV V VI Fig. 140. connection between the two deltas is established and the effect is the same as if six-phase power was being fed to the machine. The two F's forming the double Y may be interconnected at their neutral points, n and n', and form, under this condition, a true star six-phase system. Diametrical Connection. — Three single-phrase transformers with single secondaries may be used to supply six-phase power to a rotary converter or motor by making use of what is known as the diametrical connection for the secondaries. The diametrical connection is probably more used than either the double A or the double Y. The double Y is always used when a neutral point is desired for grounding or for the neutral wire of a three- 268 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY wire direct-current system which receives power from a six- phase rotary converter. The diagram of connections for the diametrical connection of transformers to feed six-phase power is shown in Fig. 141. The hexagon at the bottom represents the armature which is to receive six-phase power. If taps are brought out from the middle points of each of the three secondaries and these taps are interconnected, the dia- metrical connection becomes the double Y. Primaries inY 1 Secondaries Diametrical Fig. 141. Two- or Four-phase to Six-phase Transformation. — Two- or four-phase to six-phase transformation may be accomplished by use of double-T connection on the secondary side of Scott transformers. The connections for this are shown in Fig. 142. The ratio between the primary and secondary voltages should be the same as for the Scott transformers. If the primaries are also connected in T, the Scott transformers may be used to transform from three- to six-phase. The chief use of three- to six-phase transformation is in con- nection with rotary converters which are more efficient and give a greater output for the same copper loss when connected for STATIC TRANSFORMERS 269 six-phase than when connected for three-phase. All rotary converters of more than a few hundred kilowatts capacity are tapped for six-phase and are operated through transformers from three-phase mains. A rotary converter connected for twelve phases will give a larger output than when connected for three phases, and in addition it possesses certain other marked advantages, the prin- cipal among which is the much more uniform distribution of armature copper loss. Twelve-phase converters are not at present built, but it is quite possible that with the growing demand for larger units they may come into use. For this reason the transformer connections for changing from three- to twelve-phase will be given. One of these, namely, the double- chord connection, is very simple. mi Prim ^ai k.K Vo2 Secondaries IV," 4« Fig. 142. Three- to Twelve-phase Transformation. — There are 30 degrees difference in phase between corresponding Y and A voltages of a three-phase system. Therefore, two groups of transformers connected for three- to six-phase transformation will have their corresponding six-phase voltages 30 degrees apart, provided the primaries of one group are connected in A and the primaries of the other are connected in Y. If the ratios of transformation of the A- and F-connected groups of transformers are in the ratio of a , . . a to ^~7^' the six-phase voltages of both groups will be equal in magnitude and they may be interconnected to give either a star or a mesh twelve-phase system. The diagram of connections and the phase relations of the primary voltages are shown in Fig. 143. Fig. 144 gives the secondary connections and vectors of the twelve-phase star connection. To simplify the reference to 270 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Fig. 143, the secondary voltages are assumed to be in phase with the primary voltages instead of in opposition to them. The connections shown in Fig. 144 require six single-phase Connections Fig. 143. transformers or two three-phase transformers with two different ratios of transformation. The complication of such connections would as a rule offset any gain that might be derived from their use. "k \ 00. /" \^ \ / / XV^o ^bo* y Y^ * OQ "/ ' ^ \^ -^Kb (US V ao Fig. 144. The equivalent of twelve phases may be obtained for any mesh- connected twelve-phase system by the use of a very simple double-chord connection which requires only three single- STATIC TRANSFORMERS 271 phase transformers or one three-phase transformer. Each transformer, or phase in the case of the three-phase transformer, must have two similar secondary windings. All secondaries will be wound for the same voltage and the same current. The chord connection can be used to supply twelve-phase power from a three-phase system. 4 u Connections Fig. 145. Fig. 145 shows the vector diagram of the voltages and the connections of the twelve-phase double-chord connection. The chord voltages are approximately 96.5 per cent, of the diametrical voltage. CHAPTER XXI Three-phase Transformers; Third Harmonics in the Ex- citing Currents and in the Induced Voltages of Y- AND A-CONNECTED TRANSFORMERS; ADVANTAGES AND DIS- ADVANTAGES OF Three-phase Transformers; Parallel Operation of Three-phase Transformers or Three- phase Groups of Single-phase Transformers; V- and A-CONNECTED TRANSFORMERS IN PARALLEL Three-phase Transformers. — A considerable saving in material and therefore in the cost of transformers required for three-phase circuits may be effected by combining their magnetic circuits. Core Type. — For example, consider the case of three similar single-phase core-type transformers which are to be used on a Fig. 146. Fig. 147. three-phase circuit. If both windings on each transformer are placed on one side of the core and the opposite sides of the iron cores are butted together as shown in Fig. 146, the component fluxes in the three sides which are placed together will be 120 degrees apart in time phase and their resultant will be zero. The 272 STATIC TRANSFORMERS 273 common portion of the iron core may, therefore, be removed without affecting the operation of the transformers. The core type of three-phase transformer as actually built has the three parts of the core which carry the windings in 'one plane as shown in Fig. 147. This arrangement is derived from that shown in Fig. 146 by removing the p'arts of the cores which butt together and then contracting the horizontal portions of the Fig. 148. core of one phase and bending the corresponding parts of the cores of the other two phases until the three windings lie in the same plane. Any one leg of the iron core will carry a flux which is the resultant of the fluxes in the other two; consequently, the re- luctances of the magnetic circuits for the fluxes of phases X ^^d 3, Fig. 147, will be slightly greater than the reluctance of the magnetic circuit for the flux of phase 2. The only effect of this 18 274 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY will be a slight unbalancing of the magnetizing currents. This will have little influence upon the operation of the transformer. The yokes between the portions of the iron core which are surrounded by the windings form a Y coupling for the three magnetic circuits of the three-phase transformer shown in Fig. 147. They, therefore, carry the same flux, neglecting the leakage fluxes, and should have the same cross-section as the portions of the iron core surrounded by the windings. The yokes may be arranged in A, but this arrangement is more ex- pensive to construct and requires more space and possesses no particular advantage, and is not used. A three-phase core-type transformer is shown in Fig. 148. Shell Type. — When the three-phase transformer is of the shell type, the windings are embedded in the iron core instead of III d e P / 9 1 3 k — h ;§[|; Si IffM II a b c i tm in m ■I.. I m n Fig. 149. surrounding the iron core as in the core type. The usual ar- rangement of a shell-type, three-phase transformer is shown in Fig. 149, which gives two sectional views. The three groups of coils are 1-1, 2-2 and 3-3. The resultant magnetomotive force producing flux along the whole length of the core, i.e., along the line a, b, c, is the vec- tor sum of the three magnetomotive forces due to the three, groups of windings. If the three groups of coils are connected in the same relative direction, the magnetomotive forces produced by them will be 120 degrees apart and their vector sum taken along abc will be zero. The flux passing between the two pairs of adjacent windings 1 and 2 and 2 and 3 in the spaces d and e and / and g, respectively, will be equal to one-half of the vector difference of two fluxes which are equal but 120 degrees apart. STATIC TRANSFORMERS 275 The fluxes in the spaces d and e and / and g, therefore, are equal to Ka/S = 0.866 of the flux linking a sifigle phase. The magnetomotive forces acting to produce fluxes between any pair of coils are in parallel instead of in series as in the case of the core type of transformer. The magnetic circuits of the three phases of a shell-type transformer are, therefore, much more independent of one another than the magnetic circuits of a transformer of the core type. If the flux is prevented, in any way, from passing through the windings of any one phase of a shell type of transformer, there will still be magnetic circuits for the fluxes of the other two phases and they may be operated in open delta. Two windings of a core-type transformer cannot be operated in open delta if the flux is prevented from passing through the core of the third phase, since in this case both of the active windings would have to carry the same flux instead of fluxes 120 degrees apart, as they should. The action of a shell- type transformer under the preceding conditions is of some importance since it permits such a transformer to be operated temporarily with one winding out. If one winding of a shell type becomes injured in any way, the remaining two windings may be operated in open delta giving 58 per cent, of the normal capacity of the transformer, provided the injured winding is disconnected and either its primary or its secondary winding, or preferably both, are short-circuited. If the injured phase is short-circuited, any current which flows in it will have no circuit upon which to react and will, therefore, be all magnetizing current. As a result of this, any flux which tends to pass through the injured phase will be forced back and only a very small current will flow in the short-circuited phase. The voltage in- duced in this phase will merely be equal to the impedance drop due to this small current. If one phase of a core-type trans- former is short-circuited, the remaining two cannot be operated in open delta since their magnetic circuits would be in series. Some iron may be saved in the construction of a shell-type transformer by reversing the connections of the middle phase, that is, by reversing the connections of the windings of phase 2, Fig. 148. If the connections of the middle coil are reversed, the fluxes carried by the portion of the core between the coils, i.e., by the portions d, e, f and g, will as before be equal to one-half 276 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY of the vector difference of the fluxes Unking two adjacent wind- ings, but in this case the fluxes threading two adjacent windings are 60 degrees apart instead of 120 degrees. Their vector difference will, therefore, be numerically equal to either flux, and the parts d, e, f and g of the core will carry fluxes which are equal to one-half of the flux through any one coil instead of 0.866 of this flux, as was the case when the windings of all phases were connected similarly. When the middle phase is reversed the cross-section of the magnetic circuit throughout the trans- former should be the same. It should be remembered that Fig. 150. certain portions of the magnetic circuit consist of two parallel paths. With the middle coil reversed the cross-sections of a, b, c, h-\- i,d + e,f + g, j + k, o -\-l,p -\-m, and q-\-n should be equal. If the phases are all connected similarly, the cross-sections of d + e and f + g must be s/Z = 1.73 times the cross-section of the other parts of the magnetic circuit. The actual appearance of a three-phase shell-type trans- former is shown in Fig. 150. Third Harmonics in the Exciting Currents and in the Induced Voltages of Y- and A-connected Transformers. — Due to the STATIC TRANSFORMERS 111 variation in the permeability of the core of a transformer with varying flux density as well as to hysteresis, the wave form of the magnetizing current of a transformer will be different from the wave form of the impressed voltage. If the impressed voltage is sinusoidal, the magnetizing current will not be sinusoidal but will contain harmonics, the most prominent among which is the third. Consider first three single-phase transformers connected for three-phase. If sinusoidal electromotive forces are impressed on three single-phase transformers which have their primary windings connected in Y with a neutral, there will be a third harmonic in the magnetizing current of each phase. These harmonic component currents will flow over the three lines and return on the neutral, where they will all be in conjunction and will add directly giving a third-harmonic current in the neutral equal to three times the third-harmonic current in each line. Under this condition the voltage induced in each transformer will be of the same wave form as the impressed voltage, except as its shape may be very slightly modified by the impedance drop in the primary winding. If the neutral connection is broken, there caSTBe^no third harmonic in the magnetizing current and the flux in each transformer will then be so modified that a third- harmonic voltage will be induced in each winding. This har- monic voltage cannot appear between any pair of lines, since the third harmonics in the two phases between any pair of lines will be in opposition and will, therefore, neutralize. There will be third-harmonic voltages indu ced in the secondary windings. If these are connected in Y this harmonic voltage will appear be- tween the lines and neutral, but it cannot appear between any pair of lines. If the secondaries are in A, the third-harmonic voltages in- duced in them will be in conjunction in the closed A and will cause a third-harmonic current to circulate in the delta. This current has no electric circuit upon which it can react. It will, therefore, act as a third-harmonic magnetizing current- for the core and will suppress the third harmonic in the induced voltage. The third-harmonic current in the closed A will not be large compared with the rated current of the transformers, since it can be no larger than the third-harmonic components which 278 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY would exist in the exciting currents of the transformers if they were excited, on the side in which it occurs, from a single-phase line. It may, however, in extreme cases be equal to 30 or even 50 per cent, of the fundamental of the normal exciting currents. Its magnitude will depend very largely upon the magnetic den- sity at which the cores of the transformers are operated. It will increase rapidly with the magnetic density. If the A in which this third-harmonic current flows is opened, a large third-harmonic voltage will appear across the gap. Since, with respect to one another, the third harmonics in the three transformers are 3 X 120 = 360 degrees apart in phase, this voltage will be equal to three times the third-harmonic voltage the third-harmonic current in the closed A produces in each transformer. The third-harmonic voltage in each trans- former may be as great as 40 or even 50 per cent, of the rated voltages of the transformers if the cores are operated at high magnetic density, and will usually be as much as 25 per cent. If the transformers have both primaries and secondaries Y- connected, the third-harmonic voltage will not appear between the mains, but will appear between the mains and neutral. If it were 50 per cent, of the fundamental, the root-mean-square value of the resultant voltage to neutral would be -\/(50)^ -f- (100)'' = 112 per cent, of the rated voltage. The increase in the maxi- mum voltage of the wave would be much more than 12 per cent. The effect of this increase in voltage is not only to give an ab- normal ratio of transformation, but also to increase the insulation strain in the transformers. An increase of 10 or 12 per cent, in voltage, with a much greater corresponding increase in the maximum voltage, is of importance in very high-voltage trans- formers where the factor of safety of the insulation may not be much over 2. If the primaries are in A, each phase may be considered to receive power independently of the others and the required third harmonic in the magnetizing currents may be considered to come in over the lines. A little thought, however, will show that the third-harmonic component currents which come in over any one line for the two phases connected to that line will neutralize. The result is, there will be merely a third-harmonic current circulating in the closed A formed by the primary windings. STATIC TRANSFORMERS 279 The effect of this current will be the same as the effect of the third-harmonic current which existed in the secondary windings when they were in A with the primaries in Y without neutral connection. What has been said about third harmonics in single-phase transformers connected for three-phase transformation applies ^ equally well to three-phase shell-type transformers, but does not apply to the three-phase core , type.. The portions of the core about which the windings of a three-phase core-type transformer are placed are joined in Y without a common return correspond- ing to the neutral wire of a F-connected electric circuit. This should be made clear by referring to Fig. 146, page 272, remember- ing, however, that the central portion of the core shown in this figure, i.e., the portion made by the three sides which are butted together, is left out in a three-phase transformer. A little thought will show that the two third-harmonic fluxes in any magnetic circuit which includes two of the upright portions of the core are in time-phase opposition and cancel. There can be, therefore, no third harmonic in the mutual flux of a three- phase core-type transformer with balanced impressed voltages, but there may be a third-harmonic leakage flux between any two upright legs of the core. This leakage flux will be small compared with the mutual flux on account of the high reluctance of its path. There can be no third-harmonic voltages in the windings of a core-type three-phase transformer with symmetrical mag- netic circuits under the condition of balanced impressed voltages except those due to the third-harmonic leakage fluxes. These latter should be very small. Neither can there be any third- harmonic components in the magnetizing currents in any of windings no matter how connected. In what follows balanced impressed voltages will be assumed and the effect of the leakage fluxes will be neglected. Sinusoidal impressed voltage will also be assumed. Assume the primaries are connected in Y with neutral. Let the secondaries be open. Under this condition the primary windings receive power inde- pendently of one another. The neutral may be considered to carry the combined third-harmonic currents for the three phases provided such currents exist. Refer to Fig. 146. Consider the common central leg of the 280 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY core removed. There will then be no common return path for the third-harmonic flux. Let the instantaneous values of the magnetotnotive forces of the three phases at any instant due to the magnetizing currents be JFi, JFa and 3^3 and let (Ri, (Rj and OI3 be the corresponding instantaneous values of the teluctances of the three magnetic circuits up to their common junctions. Let / / A / / ■ a 6D / / o^ f o^ r o .&' / / /- ■■ f J ^^ /^ I 50 / ^ ^ s 1 / /^ 1 40 1^20 o > 0.0 p 30 ( A rape :e-T ami per Inc 1. I 20 ) ] i i 4 . ) ( ) ' i i ' ) 1 1 1 1 i 1 i 1 1 1 5 1 6 1 < If Short-Ciroult Current on "Low-'Voltage Side. 10 20 30 .40 50 60 70 80 90 100 110 120 130 140 150 Fig. 155. Neglecting the primary impedance drop and assuming a sinu- soidal voltage i X 9K) + (4^ + 7M)1%6 = 73.5 sq. in. The space factor for the core is 0.9. 4 540 000 Flux density = ' y'yo g = 68,700 lines per square inch. Primary and Secondary Leakage Reactances. — From equa- tions (58) and (59), page 189. X2= 27r/ < — j^ — 1^-^ + -^ + }4{di + d2 + }4d3)d3J > 10 ' „ J 8Tmi^ [ {di+d2+ds)di ,di\ ,^,^ , _, , i/.j n^ 1 1 ,»-» xi = 2x/ 1 — ^ — ^- g [—^+}4idi+d2+}id3)d2^ | 10 » From Fig. 154. di = 5.765 in. da = 0.345 in. dg = 1.003 in. di = 0.500 in. Li = 33^6 in. L2 = 331^6 in. There are two low-voltage and two high-voltage coils in series on the transformer. All dimensions must be reduced to centimeters. ». - 2 X 2.54{2,60^' fWIX0J4_5 ^ (0^. + M (5.77 + 0.345 + ^^) 1.003] j io-» = 0.162 ohm. .. = 2 X 2.54{2.60^gf ^ (5.77 + 0.345 + 1.003)0.500 . + (^ + i'(5.77 + 0.345 + if?) 1.003] ) 10- = 4.28 ohms. Equivalent Reactance. — The equivalent leakage reactance re- ferred to the low voltage side is Xe = 0.162 + 4.28 (4^)' = 0.349 ohm. 292 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Resistance of Low-voltage Winding. — Mean length of turn of the low-voltage winding (Fig. 154) is 27r(5.765 + ^^) = 37.3 in. Cross-section of copper conductor = 3(0.345 X 0.100) = 0.1035 sq. in. The specific resistance of copper at 25°C. is 10.42 ohms. 37.3 X 190 TT 0.1035 4(1000)^12 0.0467 ohm. 10.42 Mean length of turn of high-voltage winding (Fig. 154) is 0.500\ 2.(7.11+^) = 46.3 in. Cross-section of copper conductor = 0.500 X 0.045 = 0.0225 sq. in. ,-, _ ^6-3 X 910 X ' 0.0225 4(1000)212 ^" = 1.28 ohms. The calculated resistances do not include the resistances of the leads or the effect on the resistance of bending the copper when forming the coils. Core Loss. — Allowing a space factor of 0.9 for the core, the volume of steel contained in it is 2{(52M X 73.4) + (6K X 73.5)} 0.9 = 7780 cu. in. The flux density was found to be 68,700 lines per square inch. The loss per pound at this density is (Fig. 155) 1.055 watts. The density of the silicon steel of the core is 0.26 lb. per cubic inch. Total core loss = 7780 X 0.26 X 1.055. = 2140 watts. Component of No-load Current Supplying Core Loss. — This is the current marked J^ + e on the transformer vector diagrams. Assuming a sine wave of voltage and current 7 2140 . _„ Ih + ,= 2300 = 0.93 amp. This current is on the low-voltage side. STATIC TRANSFORMERS 293 Magnetizing Current. — The magnetizing current can be found from equation (52), page 166, but it is simpler to get it from a magnetization curve plotted with flux densities against ampere-turns per unit length of core. Such a curve is given on Fig. 155. The maximum flux density in the core was found to be 68,700 lines per square inch. From Fig. 155, 6.35 ampere-turns per inch of length of the iron core are required at that density. The 6.35 is the maximum value of the ampere-turns. The approximate mean length of the core is 2((6K + 9M) + (35 + 8^^)} = 119M in. The root-mean-square ampere-turns for the iron of the core are -^119>i^ X 6.35 = 535 The lap joints at the corners of the core must be figured as small air gaps. Each joint of ordinary transformer cores is equivalent to an air gap of about 0.002 in. According to this assumption, the ampere-turns for each joint may be found from equation (52), page 166. 0.4iriV/ Since n for air is one, this equation may be written NI = 0.00044(33™ where (&m is the maximum flux density in lines per square inch and NI is the root-mean-square ampere-turns required per joint. NI = (0.00044)68,700 = 30 ampere-turns. The total ampere-turns are, therefore 535 -I- 4 X 30 = 655 The magnetizing current measured on the low-voltage side is T 655 „.. " "^ l90 "^ ^™^' The large increase in magnetizing current produced by a moder- ate increase in the voltage impressed on a transformer may be 294 PRINCIPLES OF ALTEBNATINO-CURRENT MACHINERY seen by referring to Fig. 155. At an impressed voltage of 2300, the flux density in the core was found to be 68,700 hnes per square inch. At a voltage 25 per cent, greater than 2300, the flux density would be 68,700 X 1.25 = 85,900. The ampere-turns per inch of length of core corresponding to this density are, by extrapolation on the plot, approximately 23. This calls for an increase in the magnetizing current for the iron of the core of 23 1 — w-oK = 2.6 or 260 per cent. No-load Current. — The no-load current on the low-voltage side is J-n = Ih + e \ Jlif = 0.93 + i3.45 = 3.57 amp. Equivalent Resistance From Test Data. — The short-circuit loss at 130.4 amp. from the plot. Fig. 155, is 2020 watts. There- fore, Te referred to the low-voltage side is 2020 ^,.„ , ^' ^ (130.4)'' ^ ^^^^ °^^- The equivalent resistance at 25 degrees calculated from the measured resistance is Te = 0.0495 + 1.31 74-^ = 0.107 ohm. This last value of re does not include certain eddy-current and hysteresis losses which are caused by the leakage flux. For this reason it should be slightly smaller than the value calculated from the short-circuit data. Equivalent Reactance from Test Data. — The full-load current on the low-voltage side is 300,000 = 130.4 amp . 155, th to 130.4 amp. is 53.5 2300 From the plot, Fig. 155, the impedance voltage corresponding Ze = jgQ^ = 0.41 ohm. Xe = V (0.41)2 - (0.119)2 = 0.39 ohm. This is referred to the low-voltage side. STATIC TRANSFORMERS 295 The equivalent reactance calculated from the dimensions of the windings was 0.35 ohm. Regulation. — The regulation will be calculated for a full kilo- volt-ampere load of 0.8 power factor and a temperature of 75°C. using the values' of Xe and r^ obtained from the test data. re = 0.119 ohm at 25°C., Xe = 0.41 ohm. Te at 75°C. = 0.119 (l-t- 50 X 0.00385) = 0.142 ohm. From equation (64), page 198. — = Fa + /2(cos 02 — j sin 62) {ve + jx^) = 2300 + 130.4(0.8 - jO.6) (0.142 +J0.41) = 2345.3 + J29.6 = 2345. 2345 2300 Regulation = Horjo ^^^ ^ ^'^^ P^'^ *'®'^*' Efficiency. — The efficiency will be calculated at 0.8 power factor. From equation (68), page 207, the efficiency is VJi cos Bj 72/2 cos 02 + P. + /2V, The core loss corresponding to V2 = 2300 volts has already been found to be 2140 watts. From the plot li^Ve corresponding to the full-load current, Zz = 130.4, is 2020 watts. This is at 25°C. At 75°C., it is 2020 X (1 + 50 X 0.00385) = 2409 watts. 300,000 X 0.8 300,000 X 0.8 + 2140 + 2409 100 = 98.14 per cent. SYNCHRONOUS MOTORS CHAPTER XXIII Construction; General Characteristics; Power Factor; V-CuRVEs; Methods of Starting; Explanation of the Operation of a Synchronous Motor Construction. — Synchronous motors are always built with saUent poles. In other respects there is no essential difference between their construction and the construction of a synchronous generator. The only differences which exist do not involve prin- ciples of design, and are merely to better adapt the machines to the particular purpose for which they are to be used.' The chief differences are in the relative amounts of armature reaction and in the damping devices. Any synchronous generator will oper- ate as a synchronous motor and, vice versa, any synchronous motor will operate as a synchronous generator, but, as a rule, a synchron- ous motor will have a more effective damping device to prevent hunting than is necessary for a synchronous generator and its 'armature reaction will be larger than is desirable for a generator. General Characteristics. — A synchronous motor will operate at only one speed, i.e., at synchronous speed. This speed depends solely upon the number of poles for which the motor is built and upon the frequency of the circuit from which it is operated. The speed is entirely independent of the load. A change in load is . accompanied by a change in phase and in the instantaneous speed, but not by a change in the average speed. If, due to excessive load or any other cause, the average speed differs from synchron- ous speed, the average torque developed becomes zero and the motor comes to rest. A synchronous motor as such has abso- lutely no starting torque. Power Factor. — The power factor of a synchronous motor operating from constant-potential mains is fixed by its field ex- citation and by the load it carries. At any given load the power factor may be varied over wide limits by altering the field excita- 297 298 PRINCIPLES OF ALTERNATINO-CURRENT MACHINERY tion. A motor is said to be over- or under-excited according as its excitation is greater or less than normal. Normal excita- tion is that which produces unity power factor. Ovtir excitation produces condensive action and causes a motor to take a leading current. An under-excited synchronous motor will take a lagging current. The field current which produces normal excitation depends upon the load and in general, except at very small loads, it increases with the load. V-Curves. — Since it is possible to operate a synchronous iriotor at different power factors, curves may be plotted showing the iOOO relation between the armature or line current and the excitation for different constant loads. Such curves are called V-curves on account of their shape. Lines drawn through points of equal power factor on the V-curves are called compounding curves. Fig. 156 shows three V-curves and three compounding curves of a synchronous motor. Curves I, II and III are the V-curves for three different loads, and A, B and C are compounding curves. B is the compounding SYNCHRONOUS MOTORS 299 curve for unity power factor and gives the normal excitation for different loads. Methods of Starting. — Since synchronous motors have no starting torque, some auxiliary device must be used to bring them up to speed. Polyphase synchronous motors may be brought up to speed by the induction-motor action produced in their damping windings and by the hysteresis and eddy currents in the pole faces. The field winding is usually open while the motor is being started in this way, but in some cases it is short- circuited. The damping winding usually consists of copper bars which pass through the pole faces near their surface. The ends qf these bars are connected together by copper or brass straps. If the synchronous motor is provided with an exciter which is mounted on its shaft, this exciter may be used as a direct-current motor to bring the synchronous motor up to speed. A small in- dxxction motor mounted directly on the shaft of the synchronous motor is occasionally used for starting. In this case the in- duction motor must have fewer poles — ^usually two Jess — than the synchronous motor in order that it may bring the syn- chronous motor up to synchronous speed. Explanation of the Operation of a Synchronous Motor. — A single-phase motor having a concentrated winding will be con- sidered in order to simplify the explanation. Let the squares marked N and S in Pig. 157 represent the ends of the pole faces and let the rectangle represent the armature winding. The electromotive force induced in the armature winding will be zero for the position of the coil shown. Let the direction of rotation of the motor be such that the armature moves from left to right relatively to the poles. Call an electromotive force positive when it acts in a clockwise direction. Assume the armature to be driven at a uniform speed. The electromotive force generated in the coU while it passes across the pole faces is plotted on the reference line AB in Fig. 157. Now let the armature circuit be closed through a load of such constants that the current in the coil is in phase with the gen- erated electromotive force. This current is marked /. While the coil moves from a to b, the face of the coU toward the poles will be south. There is, therefore, a force of attraction between it and the pole a, and a force of repulsion between it and the 300 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY pole b. That is, during the movement from a to b, there is a torque which opposes the motion of the coil. The power de- veloped at any instant is equal to the product of the instan- taneous values of the current and the voltage. Since the speed is constant, the torque is also proportional to this product. WhUe the coil moves from b to c the current and the induced electromotive force both reverse. Their product is still positive and the sign of the torque remains unchanged. The torque curve is marked T on the figure. The torque is intermittent but is always positive and since it opposes the motion of the coil, it corresponds to generator action. (The torque of a polyphase generator is the algebraic sum of the torques developed by all Xs/ Fig. 157. phases and is constant if the current and voltage are both sine waves and the impressed voltages are balanced.) If the load on the generator is such that the current is not in phase with the generated voltage, the torque curve will have positive and negative loops. The average torque will be pro- portional to the difference between the areas enclosed by these loops. It will be positive for any angle of lag or lead which is less than 90 degrees. A study of Fig. 157 will show this. This study will also show that a lagging current in the case of a generator will produce a demagnetizing action on the poles and that a leading current will produce the opposite effect. Suppose that while the generator is running with the current SYNCHRONOUS MOTORS 301 and the voltage in phase, the current is reversed in some way. This condition is represented in Fig. 158. Fig. 158. The current and the voltage are now exactly 180 degrees apart and their product, which is proportional to the torque, is negative and corresponds to motor action. b 1 1 1 ^^— — ^ Fig. 159. The current in the coil while it passes from a to 6 is in a clock- wise direction and causes the face of the coil toward the poles to be a north pole. There is, therefore, a force of repulsion between 302 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the coil and the pole a and a force of attraction between the coil and the pole b. The resultant of these two forces assists the motion of the coil and produces motor action. The conditions existing with a leading current are shown in Fig. 159. The torque in this case has positive and negative loops. For angles of lead between zero and 90 degrees, the negative loops are larger than the positive ones and there is a resultant motor torque. The conditions for a lagging current are shown in Fig. 160. The effect produced on a motor by a lagging or a leading current is just the opposite to that produced by these currents on a generator. The effect of armature reaction depends upon Fig. 160. the phase relation between the current and the generated voltage. Therefore, since the current of a motor and the current of a generator are nearly opposite in phase with respect to the generated voltage, the effect produced on the field by a leading or a lagging current in a motor is just opposite to the effect produced by similar currents in a generator. A leading current in a motor demagnetizes and a lagging current magnetizes the field. This can easily be seen by referring to Figs. 159 and 160. Consider the case of the lagging current shown in Fig. 160. When the coil is over the pole b it is still carrying a positive or clockwise current. This current, according to the cork-screw rule, will cause the face of the coil which is toward the pole b SYNCHRONOUS MOTORS 303 to be a north pole. The magnetomotive force of the coil, there- fore, is in the same direction as the magnetomotive force of the field excitation. At constant output, the effect of a change of field excitation is to alter the armature current and hence to change the power factor. A synchronous motor, unlike a direct-current motor, may be operated with a generated voltage which is considerably greater than the impressed voltage. If it were possible to build a motor without reactance it would not operate except with a generated voltage less than the impressed voltage and even under this condition it would be very unstable. CHAPTER XXIV Vectob Diagram; Magnetomotive-force and Synchronous- impedance Diagrams; Change in Normal Excitation WITH Change of Load; Effect of Change in Load and Field Excitation Vector Diagram. — The same notation will be used as was adopted for the generator. For generator action, there must be a component of the armature current in phase with the generated voltage. For motor action, there must be a com- ponent of this current opposite in phase to the armature voltage. The vector diagrams of a synchronous motor and of a synchronous generator are similar. They differ only in the relative positions of the vectors of generated voltage and current and those vectors —r Fig. 161. whici depend upon the current. The vector diagram of a synchronous motor is shown in Fig. 161. Compare this diagram with the vector diagram of a generator shown in Fig. 46, page 86. V is the rise in voltage through the motor. To get the internal or generated voltage, Ea, the resistance and reactance drop>s must be added to the voltage V. This corresponds exactly to what was done in the case of the generator. The resultant field, R, leads the generated voltage, Ea, by 90 degrees. . The vector sum of R and the magnetomotive force, —A, which is required to balance the armature reaction, is equal to the impressed field 304 SYNCHRONOUS MOTORS 305 F. — F on the diagram is the voltage drop across the motor terminals. The cosine of the angle between this voltage drop and the current, h, is the power factor of the motor. Refer all vectors to V as an axis. la = —h (cos 6 — j sin d) Ea=V + Ia (re + jXa) = V — Ja(cos — j sin d) (re + jXa) R is found from the open-circuit characteristic and corresponds to the voltage Ea on that curve. c R = R (sin A + j cos A) —A = i4(cos 6 — j sin 8) Fig. 162. The impressed field is the resultant of R and — A and is equal to F = R (sin A + i cos A) + A (cos 8 - j sin 6) The electromagnetic power developed by the motor is equal to the current multiplied by the energy component of the gener- ated voltage taken with respect to the current. This is equal to hEa cos (0 - A) = hVC' + D^ cos (8 - A) 20 306 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The electromagnetic power is the total internal power developed by the motor. It is equal to the external load plus all rotational losses. These latter include friction and windage and all eddy- current and hysteresis losses due to rotation. The Magnetomotive-force and the Ssmchronous-impedance Diagrams. — Either the magnetomotive-force or the synchronous- impedance diagram may be applied to the motor. These two diagrams are shown in Figs. 162 and 163 respectively. The internal power developed by a motor is always equal to the energy component of the generated voltage with respect to the current multiplied by the current. It is immaterial whether the voltage generated by the resultant field, by the impressed field or by B" on the magnetomotive-force diagram is used, since Fig. 163. the energy components of all three of these with respect to the current are the same. Change in Normal Excitation with Change of Load. — Under the heading "Power Factor" on page 297, the statement is made that the field current which produces normal excitation increases with the load except for small loads. The reason it should decrease with small loads may be seen from the vector diagram of the synchronous motor. Fig. 164 is the vector diagram of a synchronous motor for unit power factor. Assuming that the synchronous reactance and effective re- sistance remain constant, the line laZs on the diagram will make a constant angle with the vector, V, which represents the rise in voltage across the motor terminals. At light loads Ea and V nearly coincide. As the load is increased, the power factor SYNCHRONOUS MOTORS 307 remaining unity, the extremity of the vector EJ travels out along the line ab and will decrease in length until it reaches the position where it is perpendicular to db. Beyond this position it will increase in length. Since Ea is the excitation voltage, i.e., the voltage which the impressed field would produce on open circuit, the impressed field will vary in a similar manner. The bottom of a compounding curve of a synchronous motor wUl, therefore, be inclined slightly toward low excitation. The point of ex- citation at which it commences to slope toward higher excita- tion will be where the vector Ea becomes perpendicular to laZa- This excitation will depend upon the ratio of Xs to re. It will usually be well down on the compounding curve and in some cases it may be too far down to show at all. If the motor had no reactance, the field excitation for unity power factor wotild Fig. 164. decrease continuously with increasing load. On the other hand, if the motor had no resistance, the field excitation under similar conditions would increase continuously. Effect of Change in Load and Field Excitation. — The current V — Ea taken by a direct-current motor is equal to = la. If fa load is applied the motor slows down and decreases Ea. It will continue to decrease its speed until the current has increased sufficiently to carry the load. The theoretical limit of load can be readily shown to be reached when Ea — la^a — o^- Beyond this limit the decrease in Ea will more than balance the increase in /„. If the field excitation is increased, Ea increases, decreas- ing V — Ea and consequently the current. The power developed by the motor is now too small to carry the load and it will start to slow down. It will continue to slow down until the effect on 308 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Ea of the decrease in speed balances the effect of the increase in the excitation. The current will then have increased to nearly its original value. A direct-current motor adjusts itself to a change in load or in its excitation by changing its speed. A synchronous motor must run at synchronous speed. It carmot change its average speed to accommodate itself to a change in load or in excitation. The current taken by a synchronous motor is equal to _ V -EJ °~ z. For any given excitation, EJ is fixed, but its phase relation with respect to V may change and alter the current. A synchronous motor accommodates itself to a change in load by changing the phase of its generated voltage with respect to the voltage im- pressed across its terminals. Its average speed does not alter but its instantaneous speed changes long enough to permit the required change in phase to take place. If load is applied, it starts to slow down and will continue to slow down until sufficient change in phase has been produced. If the motor is not properly dampened, it may over-run and develop too much power. It will then speed up and may again over-run. It will now be developing too little power and the action will be repeated. This is called hunting. Hunting will be taken up later some- what in detail. If the field excitation is altered, EJ and the power developed will change. The motor will then immediately alter its phase until equilibrium has been re-established. In general, an increase in the load carried by a synchronous motor will cause it to increase its lag, and a decrease in load will cause a decrease in lag. An increase in field excitation will cause the lag to decrease, and a decrease in the field excitation will cause the lag to increase. CHAPTER XXV Maximum and Minimum Motor Excitation for Fixed Motor Power and Fixed Impressed Voltage; Maximum Motor Power with Fixed E^', V, Re and x,; Maximum Possible Motor Excitation with Fixed Impressed Voltage and Fixed Resistance and Reactance; Maximum Motor Activity with Fixed Impressed Voltage and Fixed Reactance and Resistance Maximum and Minimum Motor Excitation for Fixed Motor Power and Fixed Impressed Voltage. — The synchronous re- actance and effective resistance will be assumed constant. Refer all vectors to V as an axis. Small letters will be used 1^' ! ^"""^^^^ k- 6 J r-:^^ -i ^IG. 165. NjA r to represent components. A prime on a small letter indicates that it is a component which is in quadrature with V, the axis of reference. The synchronous reactance and effective resistance win be represented, respectively, by x, and Ve. Pm will represent the internal motor power. The synchronous impedance vector diagram is shown in Fig. 165. Referring to Fig. 165. V = v+30 EJ = e- je' (88) Ia= -i + ji' 309 310 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The resultant voltage causing the current is equal to the imped- ance drop and is equal to E„ = EJ - V* I =^ ^ e — je' — V "~ re+ jx. Rationalized, this becomes (ne - TeV — x,e') + j {x,v -Xee-Tje^ The_power in any circuit is equal to the product of thexeal_ parts of the current^nd the.¥oltage plus the product of-the, imaginary parts of the current and the voltage. p _ e(r^e — VeV — x,e') — e'jXsV — XsC — r^e') ^^ ~ u^^x^ (90) n^ + x,^ nEa'^ - v(ree + x,e') Te^ + a;/ For the condition of fixed motor power, the numerator of the expression for the motor power, i.e., the numerator of equation (90) must be constant. Replace e' in the numerator of equation (90) by its value from equation (88). e' = MSa'^ - e^ Putting this in equation (90) gives r,Ea'^ - vine + x,V Eg'^ - e") ^^ " re^ + a;/ ^^^' Since the motor power is constant, the differential of Pm with respect to e will be zero. „ ^,dEg' ^^ de ' ^ 2reEa —j Vre VXs — ;; — „ = de ^Ea"" - e^ dEg' de ■ vXsEg' 1 f x,e 2reEg' , „ — V \ re . „ = * The voltage, V, on the diagram is the rise in voltage through the motor. The current is in the general direction of the voltage drop, i.e., — V. SYNCHRONOUS MOTORS 311 x,e dEa' ^ " I ""^ VEg'' - e^ , ^ ^ Vej^' & 2 vlve . = 1 VEJ^ - e^ I VTey/Ea'^ — e^ = x,ev re^Ea'^ - re^'e^ = x,^e^ (92) Substituting the value of e from equation (92) in equation (91) and replacing {r^^ + a;,^) by Zj" gives and = VeEJ' Z9 Zg = reEa'^ - VZ.Ea' Ea' = ^ j f + V4P„r. + z;2 but y = F, therefore, i/a'=^(F+ V4P.r-. + F=') (93) When substituting the numerical value of the motor power in equation (93), it must be remembered that motor power is negative according to the direction of the vectors for motor cur- rent and voltage given on Fig. 165. The maximum possible motor excitation is when the motor power is zero; therefore, from equation (93) Vz, maximum Ea = — ' The minimum excitation is zero. Maximum Motor Power with Fixed Ea, V, r^ and a;». — The motor power must be negative. Therefore, since the first term of equation (90) is constant, the second must be negative for 312 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY maximum motor power. For a maximum motor power ne + x^e' must be a maximum. ^ (r-ee + x,e') = r, + x, -^ = (94) -Ba" = e' + e'2 = constant ^=2. + 2e4' = (95) Combining equations (94) and (95) gives -J- (ne + Xse) = Ve — Xs—, = (96) That is, with the impressed voltage constant and the motor voltage, Ea', as well as the resistance and the reactance fixed, the maximum motor power will occur when the angle of lag of Ea X behind V is equal to tan~' — • Putting the value of e' from equation (96) in equation (90) gives TeEa" - i;jr,e+^% j maximum Pm = ^ But V = V, therefore, maximum /*„ = — ^ (97) Ea"- = e^ + e'2 and from equation (96) / Xs e' = e — Te therefore, e^^EJ^-e^i^^y EJn e = z. Substituting this value of e in equation (97) gives p Ea'he VEJ ,„-, maximum Pm = — ^ (98) SYNCHRONOUS MOTORS 313 A synchronous motor cannot operate with an excitation voltage greater than the impressed voltage unless it has reactance. This follows from equation (98). If the reactance is zero, equation (98) becomes „ Ea'' ye: maximum fm — ■■ — re Te For any value of E^ greater than Y , Pm will be positive and will represent generator action. Maximum Possible Motor Excitation with Fixed Impressed Voltage and Fixed Resistance and Reactance. — In order that the machine shall run as a motor Pm must be negative. The limiting value of EJ will be that value which makes Pm zero. Ea'he YEa' Maximum Pm = " 2 " ~ Zg Zg 2-3 Zg The maximum possible motor voltage is equal to the impressed voltage multipUed by the ratio of the impedance to the resistance. Maximum Motor Activity with Fixed Impressed Voltage and Fixed Reactance and Resistance. — dPm f. d Ea"r. dEa' ~ dEa' [ Zg^ Ea'v Zs 2Ea're V Zs^ Zs , vz, Vz, The maximum motor power, therefore, occurs when the motor voltage has one-half of its maximum possible value. This corresponds to Jacobi's law for a direct-current motor operating with a constant field. CHAPTER XXVI Hunting; Damping; Stability; Methods of Starting Synchronous Motors Hunting. — All synchronous machines in which a change in load is accompanied by a change in phase are subject to hunt- ing. Consider the case of a synchronous motor operating under constant excitation and load. Under this condition there will be a perfectly definite phase angle between the impressed and excitation voltages. Suppose the load it carries is increased. The motor will now be developing less power than is demanded by the load and, as a result, it will immediately start to slow down. It will continue to slow down, thereby changing its phase, until the phase displacement between its impressed and excitation voltages corresponds to that required for the load. This slowing down may last several cycles, but unless the load exceeds the maximum load the motor can carry, the change in speed will not last long enough to produce more than a moder- ate change in phase. This change in phase can never equal 90 electrical degrees, unless the excitation is changed. If the change in phase should exceed that which corresponds to the maximum load the motor will carry, the motor will "break down," i.e., fall out of synchronism and come to rest. While the motor is slowing down, the increase in load is being suppUed by the change in the kinetic energy of the moving part of the motor. At the instant the motor passes through the phase displacement corre- sponding to the load, the electromagnetic power developed will be equal to the entire load plus the rotational losses of the motor. Due to the inertia of the rotor it will not stop changing its speed at this instant, but will over-run. It will now be developing more power than is required for the load and the rotational losses, and it will start to speed up. If it again over-runs it will be developing too little power and it will immediately start to slow down. This action is called hunting. It is equivalent to an oscillation in speed which is superposed on a uniform speed of 314 SYNCHRONOUS MOTORS 315 rotation. A slight amount of hunting must always take place when the load on a synchronous motor is changed but, with a properly designed motor operating under good conditions, it should be small and not noticeable. In the case of a poorly designed motor or a motor operating under bad conditions, hunting may become excessive. The effect of hunting will be made clear by a vector diagram. Fig. 166 is a vector diagram of a synchronous motor to which the drop in voltage, — V, across the terminals has been added. The resultant voltage, Eo = hza, which causes the current, la, in the circuit, is equal to the vector sum of — F and Ea'. The current, /„, is equal to this voltage divided by the syn- chronous impedance of the motor and lags by an angle p = tan"' — ° behind the voltage Eo. This angle, p, would be con- stant if Xs and r^ were constant. If hunting takes place, the extremity of the vector EJ will oscillate on the arc of a circle about its mean position. At the same time Eo and also /„ will change in magnitude and in phase. The effect of hunting on Fig. 166 is shown in Fig. 167. The full lines on this figure represent the stable condition and the dotted and dashed lines represent the. two extreme displace- ments due to hunting. The position of the vectors, — V and V, representing the impressed voltage, does not change. The vector Ea is assumed to oscillate from a to b. The re- sultant voltage Eo oscillates from c to d and at the same time changes its magnitude. The current, /„, is proportional to Eo at every instant and will swing through an angle equal to the angle through which Eo moves. The minimum power is de- veloped when Ea' is ahead of its mean position and has its greatest displacement. This power is equal to the projection of the 316 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY motor voltage, Oa, on the current, Og, multiplied by that cur- rent. The maximum power is developed when the motor has its extreme displacement in the direction of lag. This is equal to the product of the current, Of, and the projection of the motor voltage, Ob, on that current. It will be seen that there may be a large variation in the power developed if hunting occurs. The rotating part of the motor acts like a torsional pendulum where the change in the couple producing rotation corresponds to the torsional couple in the fiber or supporting wire of the pen- dulum. In the case of the motor, the change in the couple is caused by the displacement of the rotor from its mean position. The moment of inertia of the rotor corresponds to the moment of inertia of the mass of the pendulum. -^4 '#' <.^) where f, l^md^ and M are respectively, the time of an oscillation, the moment of inertia of the rotor and the restoring couple per unit of angular displacement from the mean position. From equation (90), page 310, the electromagnetic power de- veloped by a motor is TeEa' — v{ree + x,e') Pn. = z. Replacing vhy V and e and e' by their values in terms of the angle, a, between EJ and V (Fig. 165, page 309). gives, _ TeEg'^ — VEg'ire cos a + x, sin a) i^m — 2 (100) SYNCHRONOUS MOTORS 317 To make equation (100) apply to a polyphase motor, it must be multiplied by the number of phases, n. If p and / are, respectively, the number of poles on the motor and the frequency, the electromagnetic torque developed by a polyphase motor is T-r,^P - - P f ^^^"'^ ~ ^^"'(^^ cos a + X, sin g) ] M in equation (99) is equal to the differential of T with respect to a', where a' is the displacement in space radians of Ea from V. The angle a in equation (101) is in electrical radians. Therefore, since a = ^a' dT npWE,' . M = j—7 = p . „ (r-e sin a — Xs cos a) da oTjZis The moment, M, is negative since, according to the convention adopted for motor power, motor power is negative. Before sub- stituting M in equation (99) its sign should be reversed in order to make it positive and avoid an imaginary value for the time of oscillation, t. Substituting this value of — M in equation (99) gives for the period of hunting in seconds of a polyphase synchronous motor. \npWEa {Xs cos a — re sm a) V, E'a, 2s, Xs and r^ in equation (102) are per phase and are in c.g.s units. If practical units are to be used in place of c.g.s. units, the expression under the square-root sign must be multi- plied by 10-^ Equation (102) is only approximate as it neglects the effect of damping due to currents induced by the hunting in the field winding, in the pole faces and in the damping bridges with which all synchronous motors are provided. One effect of these in- duced currents is to diminish the apparent reactance of the motor. If the free period of the hunting as given by equation (102) coincides or nearly coincides with any periodic variation in the 318 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY load or in the frequency of the power supplied to the motor, the effect will be cumulative and violent hunting will occur, which, unless damped out, will probably cause the motor to swing beyond the phase displacement corresponding to the maximum power and to drop out of synchronism and come to rest. The maximum possible phase displacement at which the motor can X, operate has already been shown to occur when tan a = -r (equation 96). This angle for maximum power may also be derived by differentiating equation (100) with respect to a and equating the differential to zero. dPm VEa , ■ \ t\ — = — - — — (x cos a — re sin a) = Xs cos a = re sin a . Xg tan a = — re It will be seen from equation (102) that the period of oscilla- tion of a synchronous motor about its mean angular position depends upon the excitation voltage, Ea, and the phase dis- placement, a, of this voltage from the voltage impressed on the motor. Consequently, the period depends upon the excitation of the motor and the load it carries. Therefore, if there is any periodic variation in the load, in the impressed voltage or in the excitation, hunting may occur at some load or at some excitation and not at others. Damping. — There are two ways by which hunting may be diminished. One of these is to increase the moment of inertia of the rotor by increasing its mass by adding a flywheel. This method is applicable to either single or polyphase motors. The other consists of using a short-circuited low-resistance winding, an amortisseur or damping winding or damper as it is called, placed in the pole faces. When an amortisseur winding is used on a single-phase motor, double-frequency currents are induced in it by the double-frequency flux variation produced by arma- ture reaction in the poles (see page 59, under Synchronous Generators). This double-frequency current increases the cop- per loss in the damper and tends to damp out the flux variation which causes it. Its existence is not dependent upon hunting. SYNCHRONOUS MOTORS 319 Adding a flywheel may decrease the tendency to hunt by making the free period of oscillation of the motor lower than any which is likely to start hunting. This is an effective way to diminish the tendency to hunt but it does not diminish this tendency by real damping action. Flywheels are not used to decrease hunting, mainly on account of their weight, except in special cases. An amortisseur winding or damper exerts a real damping action. Such windings are universally employed on polyphase synchronous motors. Besides effectively diminish- ing hunting, they very greatly increase the starting torque of a synchronous motor when it is started as an induction motor. An amortisseur winding or damper usually takes the form of copper grids placed in the pole faces and copper bridges between Fig. 168. the poles. The grids are usually made by placing copper bars in the pole faces near their surfaces and then short-circuiting these bars by bolting or welding them to end straps of brass or copper. An amortisseur winding is shown in Fig. 168. For the most effective damping, the damper should have as low a resistance as possible, but if this winding is to be used for starting the motor, the resistance which gives the best damping action may be too low to give the best starting torque. The armature reaction of a polyphase synchronous motor, which operates under steady conditions is fixed in space phase with respect to the poles. Under this condition, the resultant flux is also fixed with respect to the poles and the damper is 320 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY inactive and produces no effect whatsoever upon the operation of the motor. When, however, hunting starts, the armature reaction is no longer constant or fixed in space phase with respect to the poles, but sweeps back and forth across them with a period equal to the period of oscillation of the rotor. This causes the resultant flux to cut the damper and to induce currents in it. Eddy-current and hysteresis losses will be produced ill the pole faces which will assist, to a slight extent, in damping out the hunting. The main damping action is due to the cur- rents induced in the damper which are in such a direction as to oppose the change in the angular velocity of the rotor which produces them. The reactance of the damping winding for the period of the current induced in it by hunting is very small. Assuming it to be zero, the damping action would be a maximum when the rotor swings through its mean position, and zero when it has its extreme displacement. The effect of damping pro- duced in this way is much the same as the damping produced by a viscous fluid on a torsional pendulum. The braking action produced by a damper is only in part due to the energj' dissi- pated in copper loss in the damping winding. On account of the reaction between the currents in the damper and the arma- ture winding, energy will be returned to the line while the rotor is accelerating and taken from the line while the rotor is retarding. Due to the reaction of the currents induced in the damper and in the field windings on the armature winding, the apparent re- actance of the armature will be slightly diminished when hunting starts. This reaction is similar to the reaction existing between the primary and secondary windings of a transformer when the secondary is loaded. This decrease in the apparent reactance of the motor will slightly decrease the free period of oscillation of the rotor, according to equation (102). If an amortisseur winding or damper is used on a single-phase synchronous motor, there will be currents induced in it even when the motor is entirely free from hunting. These currents are caused by the armature reaction which, in a single-phase syn- chronous motor, is neither constant in space phase nor in mag- nitude. The main- effect of a short-circuited winding on a single-phase synchronous machine is to damp out any harmonics SYNCHRONOUS MOTORS 321 there may be in its wave form. The copper loss of such a wind- ing will considerably lower the efficiency. The armature reaction of the single-phase motor may be re- solved into two revolving vectors, rotating in opposite directions. One of these will be stationary with respect to the rotor, while the other will revolve at twice synchronous speed with respect to it. The component which is fixed with respect to the rotor will produce no effect on the damper; the other, however, will produce double-frequency currents in it. These currents will not be very large on account of the relatively high reactance of the damper to the double-frequency currents. The width of the air gap and the magnitude of the leakage reactance have a large influence on the stiffness of coupling of a motor. By stiffness of coupling is meant the tendency of a motor to follow every irregularity in the speed of the generators from which it is operated. The degree of stiffness of coupling depends upon the change in power produced by a given change in phase between the impressed voltage and the voltage cor- responding to the field excitation. This change in power is determined mainly by the change in the armature current caused by the change in phase. Since I _ y -Eg' Zs this change in current is fixed by the synchronous impedance, z„ of the armature. The magnitude of the part of the syn- chronous reactance which replaces the effect of armature re- action depends upon the effect produced by armature reaction on the field strength. This is greatest when the air gap is small. A small air gap, therefore, makes the synchronous reactance large and, conversely, a large air gap makes the synchronous reactance small. Therefore, a large air gap and low leakage reactance will give a stiff coupling, i.e., the motor will tend to follow every irregularity in the speed of the generator. A large air gap and large leakage reactance will produce what is known as a soft coupling. With too stiff a coupling, a motor will tend to follow the generator too closely and will be subjected to shocks and strains of considerable magnitude whenever irregularities occur in the load, excitation or speed of the system. With too 21 322 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY soft a coupling, there will not be sufficient stability and there will be danger of a motor dropping out of step when any sudden change occurs in the system. A compromise between the two extreme conditions must be made. An objection to a soft coup- ling is that, under the condition of constant excitation, there will be a large variation in the power factor from no load to full load. An inspection of the vector diagram given in Fig. 164, page 307, should make this clear. What has just been said in regard to stiffness of coupling neglects the effect of the damper and any damping action that may be produced by eddy-current or hysteresis losses in the pole faces. The damping action of pole-face losses and of a damping winding increases as the width of the air gap is decreased since the smaller the air gap the larger is the effect of armature reac- tion and the greater is the magnitude of the current induced by it in the damper and pole faces when there is a change in phase between the impressed and excitation voltages. Stability. — According to equation (98), the maximum electro- magnetic power developed by a synchronous motor operating with fixed excitation and fixed impressed voltage may be written maximum P„ = j^^^^ - ^y^T^, (103) To find the value of a;, which will make the power a maximum when Te, V and EJ are fixed, differentiate equation (103) with respect to x, and equate the differential to zero. d \ nEa'^ VEa' 1 dx, [ r/ + x,2 Vre= -f- x/ J - 2XsreEa'^ H Vr-e^ + X,^ 2XsVEa' ■2xsreEa'^ + Vre^ + x,^ XsVEJ V 4r/^<.'2 = yi{r.^ + x,^) If V and Ea are equal, \ V ) = TeVZ SYNCHRONOUS MOTORS 323 for a maximum motor power. This corresponds to a difference in phase between the impressed and excitation voltages of tan"i = \/3 or 60 degrees (equation 96, page 312). This is the phase displacement at which " breakdown" will occur. Since the field excitation of synchronous motors is usually adjusted to make them operate at unity power factor or with a leading current at full load, Ea will generally be at least equal' to V. Therefore, in order to get the maximum possible output from a motor under such conditions, the ratio — should be equal to or somewhat Te greater than 1.73. A motor can, of course, never be used at an output approaching its maximum, since, under this condition, any hunting would be likely to increase the phase displacement between V and Ea beyond its limiting value and cause the X motor to break down. Increasing the ratio — beyond the value which gives the maximum output, will decrease the maximum output, but it will at the same time increase the displacement at which the maxinaum output occurs. For maximum stability, the change in the power developed should be a maximum for a given change in phase. In other words, -^r^ should be a maximum. Differentiating the ex- aa pression for motor power given by equation (100), page 316 with respect to a gives dPm _ VEa'iVe slu a — X, COS a ) , , da " Te^ + x,2 — r^ might be called the stability factor. It will be seen from da equation (104) that this stability factor is directly proportional to Ea, the excitation voltage. An over-excited synchronous motor is, therefore, more stable than one operating under- excited. The maximum power occurs when tan a = -^. Sub- stituting the values of sin a and cos a corresponding to this in equation (104) makes -r^ zero as it should. For any value of tan a greater than — , —r^ becomes positive and, since motor power is negative according to the notation adopted, it represents 324 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY a decrease in motor power. If, therefore, tan a exceeds — , the motor will break down. The relative magnitudes of re and Xs which make the stability factor a maximum can be found by equating the differential of dPm -r~ with respect to x^ to zero. (to. d (dPml _ d I VEa'ire sin a — x, cos a) dx,\ da J dxs\ r^^ + Xs^ , 2x/ cos a — {re^ + Xs^) cos a — ^rg,, sin a _ Xs^ cos a — r^^ cos a — 2reXs sin a = - = tan a ± Vl + tan^ a (105) Te The minus sign before VI + tan^ a in equation (105) has no significance, since Xs cannot be negative.. The ratio of — , given Te by equation (105) for values of tan a equal to or greater than ~, is of no importance since it represents unstable operation. X The ratio of — for maximum stabihty is unity for a = 0, and ^e X increases with an increase in a. For a = 30 degrees, — for fe X maximum stability should be 1.7; for a = 45 degrees, — should be 2.4. The angles of breakdown corresponding to these are, respectively, 60 and 67 degrees. All that which has preceded on stability is only approximate, as it neglects the effect of the damping, and also the effect of the free period of oscillation of the motor as a torsional pendulum. X The usual value of the ratio of — for a synchronous motor is much greater than the values of the ratio deduced for maximum output and maximum stability. Moreover, the steadiness of operation of a motor, which shows a tendency to hunt, is often improved by adding reactance to its circuit. The effect of increasing the reactance is to increase the time of the free period of oscillation as a pendulum, and make it longer SYNCHRONOUS MOTORS 325 than the period of the disturbance which is caiising the hunting. It is usually more important to have the free period quite different from the. period of any disturbance which is likely to produce hunting. It is not particularly important to have the natural tendency to damp out oscillations a maximum since this is always supplemented by the strong damping action of the amor- tisseur winding. If the stability is too great, severe strains will be put on the motor when sudden disturbances occur in the system. In other words, the stiffness of coupling will be too great (see page 321). Methods of Starting Synchronous Motors. — A synchronous motor may be started by means of an auxiliary motor. When started in this way it is brought up to speed and synchronized like an alternating-current generator. When a synchronous motor is provided with an exciter which is mounted on its shaft, the exciter may be used as a starting motor. Synchronous motors which form one unit of a motor-generator set where the other unit is a direct-current generator are very often started by using the direct-current generator as a motor. A polyphase synchronous motor may be started as an induction motor by making use of its amortisseur or damper winding. The eddy-current and hysteresis losses in the pole faces produced by the revolving field set up by the armature reaction of a polyphase motor will produce a starting torque which may cause the motor to start even without the amortisseur winding. The starting torque produced without the amortisseur would be small and might not be sufficient alone to start the motor. Moreover, the current required would be excessive. Single- phase motors have no revolving field due to their armature reaction. Consequently, they cannot be started by means of an amortisseur winding. Single-phase motors are of little practical importance and are seldom used and then only in very small sizes. If a synchronous motor is to be brought up to speed as an induction motor, care must be taken to design it in such a way that the reluctance of the air gap under the poles is constant for any position of the poles with respect to the armature. If the reluctance of the air gap under the poles varies with their position, the motor will tend to lock in the position of minimum 326 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY reluctance when the stator is excited, and a large torque will be required to move it from this locked position. The question of whether the reluctance of the air gap over the poles varies with the position of the rotor, depends upon the spacing of the armature slots. Fig. 169 shows a spacing for which the air-gap reluctance is not constant. The left-hand of the figure shows a field pole in the position which makes the reluctance a maximum. The position for minimum reluctance is shown in the right-hand half of the figure. Fig. 169. The armature reaction of a polyphase synchronous motor operating at synchronous speed is constant and fixed in space phase with respect to the poles and produces no effect on the damping winding, except when there is hunting. When the rotor is at rest or revolving at any speed below synchronism, there is relative motion between the field produced by the arma- ture reaction and the poles, which causes currents to be induced in the damper. These currents produce the same effect. as the currents induced in the squirrel-cage winding of an induction motor and will cause the motor to speed up. A synchronous motor can never reach synchronous speed under the action of the currents induced in its damper alone, but, if the damping winding is properly designed, the motor may reach a speed which is near enough to synchronous speed to pull into step before the field is excited, provided the motor has salient poles as all synchronous motors do. The lagging component of the starting current will usually produce sufficient field excitation to cause the motor to pull into step. When the motor has reached synchronous speed, the excita- tion is entirely due to the armature reaction. If now when the field is closed it happens to oppose the polarity produced by armature reaction, the motor will slip 180 degrees and will only SYNCHRONOUS MOTORS 327 be pulled into step at the expense of a large rush of current. To avoid this current rush, it is best to excite the field through a large resistance just before synchronous speed is reached. This will cause the motor to pull into step with the correct polarity. The starting torque of an induction motor depends upon the resistance and the reactance of its short-circuited rotor winding. For maximum starting torque, the resistance should be equal to the reactance measured at the impressed frequency. The difference between the actual speed of an induction motor and its synchronous speed, i.e., its slip, is directly proportional to the resistance of its rotor winding. If it were possible to make the resistance of the rotor winding zero, an induction motor would operate at synchronous speed at all loads. The require- ments for small slip and large starting torque are opposite so far as the resistance of the rotor is concerned. Since a syn- chronous motor starts as an induction motor, to pull into step easily, it should have a damping winding of very low resistance, but in order to start readily, especially under load, the resistance of its damper should be high. For maximum starting effort the resistance should be equal to the reactance. The conditions for good starting torque are incompatable with pulling into step readily and a compromise is, therefore, necessary. The frequency of the current in the damper is very low when synchronous speed has nearly been reached and the local core loss produced by the current in the winding and also the skin effect become very small. The ohmic and effective resistances under this condition are very nearly equal. At the instant of starting, however, the current in the damper is of the same frequency as the voltage impressed on the motor. The local losses produced by this current as well as the skin effect may make the apparent resistance of the damper considerably greater than its ohmic resistance. By making use of this dif- ference between the ohmic and effective resistances, it i.s possible to design a damper which will start a synchronous motor under load. The chief objections to starting a synchronous motor by the use of a damping winding are the large current required and the high voltage induced in the field winding. The large cur- rent, which is a lagging current, may seriously disturb the voltage regulation of the system. 328 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The high voltage induced in the field winding during starting is caused by the armature reaction flux sweeping across the pole faces. This voltage is a maximum at the instant of starting and zero when synchronous speed is reached. To keep this voltage as low as possible, the voltage for the field excitation of a self-starting synchronous motor should be low in order to permit a small number of field turns. A switch may be provided to sectionalize the field winding during starting though this is seldom done on account of constructional difficulties. The voltage strain on the field insulation is then limited to that generated in a single section instead of that generated in the entire field winding. Extra insulation must always be provided on the fields of self-starting synchronous' motors. The presence of the damping winding considerably reduces the voltage which would otherwise be induced in the field winding by the reaction of the currents induced in it. Short-circuiting the field winding wiU also reduce the voltage induced in it during starting and at the same time slightly increase the starting torque. The increase in the starting torque produced in this way is small on account of the high reactance of the field winding. When synchronous motors are started as induction motors, the voltage impressed on them should be reduced while they are coming up to speed. This reduced voltage may be obtained by using a starting compensator or from taps on the secondary windings of the transformers supplying the motors, in case trans- formers are used. Transformers are seldom used with syn- chronous motors unless the voltage of the line from which they are operated exceeds 13,500. Above that voltage, it is more economical to use transformers than to insulate a motor for full line voltage. To bring a synchronous motor, which has a damping winding, up to speed, its field is opened or in some cases short-circuited. About one-half normal voltage is then applied to its terminals. It should start slowly and if the damper has been properly de- signed it should speed up with increasing acceleration. The time required to come up to speed will depend upon the fraction of full voltage applied, the size of the motor, and its design. It should not exceed a minute or a minute and a half, for moderate sized motors. When nearly the maximum speed has been at- SYNCHRONOUS MOTORS 329 tained — this can be told by the sound — the field circuit should be closed through a moderate amount of resistance and full voltage applied to the motor. The field should then be adjusted to make the motor operate at the desired power factor. Slight over-excitation is more desirable than under-excitation since it will make the motor take a leading current and in a measure compensate for the reactive components of the currents taken by other loads on the line. Moreover, a slightly over-excited synchronous motor is more stable than one which is under- excited. If the motor is to operate with fixed excitation, the field should be adjusted initially to make the motor operate at approximately unit power factor at its average load unless the conditions under which it is to operate make some other power factor more desirable. When a synchronous motor is to be started by the induction- motor action of its damping coils or by the torque produced by hysteresis and eddy-current losses in the pole faces, a short air gap is desirable in order to keep the starting current small. A short air gap will give rise to a stiff coupling which may be un- desirable. The selection of the best length of air gap for any motor is a compromise and depends upon the particular service demanded. CHAPTER XXVII Circle Diagram of the Synchronous Motor; Proof op the Diagram; Construction op the Diagram; Limiting Oper- ating Conditions; Some Uses op the Circle Diagram Circle Diagram of the Synchronous Motor. — Circle diagrams were first applied to synchronous machines by Andr6 Blondel.^ Although such diagrams assist in determining the general oper- ating characteristics of a motor, they cannot be used for prede- termining these- characteristics with accuracy, since all circle diagrams are based upon the assumption of constant resistance and constant synchronous reactance. Blondel's original circle diagram of the synchronous motor and generator was a diagram of voltages. The circle diagram of currents, which is in reality merely a modification of the voltage diagram, is, in some respects, more convenient for the motor than the diagram of voltages and alone will be given. Proof of the Diagram. — ^Let Fig. 170 be the vector diagram of a synchronous motor on which V, without the minus sign, will be used to represent the drop in voltage through the motor. This diagram is similar to the one given in Fig. 166, page 315 ro- tated through 90 degrees. Let P, Pm and Eo = laZs represent respectively, the power input, the internal power developed, and the resultant voltage forcing the current through the cir- cuit. Everything on the diagram, as on all previous diagrams, is per phase. Take F as a fixed reference line. This will be drawn vertically for convenience. and according to the assumptions made in regard to Xs and r^, tan A = — is constant. ' Moteurs Synchrones a Currents Alternatifs, by Andr6 Blondel, also L'Industrie Electrique, February, 1895. ' 330 SYNCHRONOUS MOTORS 331 If the motor excitation remains constant, ab = EJ will be constant and the extremity of the vector Eo will swing on the arc of a circle, bee, as the current varies. Since the current is proportional to and makes a constant angle with E„, the ex- tremity of the current vector Oh will also swing on the arc of a circle HIa. Fig. 170. If the motor excitation is decreased, Ea will decreasfe and the point Eo will approach the point V. It will coincide with V, which is the center of the voltage circle bcfi, when the excitation is zero. At the same time la will approach the center of its circle, and will coincide with this center when Eo coincides with V. When Eo and V coincide, Ola will he along the diameter of the circle HI a- The radius, OC, of this current circle will, therefore, make an angle tan"^ — with OV and will be equal to the voltage, OV, impressed on the motor divided by the synchronous impedance. OV DC = ~ (106) 332 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY C will be the center of a system of concentric circles cor- responding to different motor excitations. These circles are the motor excitation circles of the circle diagram. If the excitation is constant, h will travel along the arc of the circle HIa and when Eo coincides with c, h will coincide with H, and will be equal to HC = OC-OH = ^-^ =^' ^8 ^s *a That is, the radius of any motor excitation circle is equal to the corresponding excitation voltage divided by the synchron- ous impedance. Take any point, such as d, on the line OV which represents the impressed voltage. (dl„)2 = my + {OlaY - 2{0d){0Ia) COS e and my - {dlaY = 2{0d){0Ia) COS B - {OhV If Od is made equal to OV _ impressed voltage 2re twice the effective resistance and {Ody - (dhy = v-^-'' my - {dhy = ^ (lo?) where P and P™ are, respectively, the input and the internal output of the motor. If the motor power, P™, is fixed, equation (107) is the equa- tion of a circle having a radius equal to dl^ and a center at a dis- V tance, Od = k~, above the poiflt 0. Hence, for any fixed motor power the extremity of the current vector 01^ must be on a circle drawn about the point d as a center. The point d will be the center of a system of power circles corresponding to different motor powers. SYNCHRONOUS MOTORS 333 V Substituting Od = k- in equation (107) gives © I e H (108) Equation (108) gives the radius of any power circle in terms of the motor power, P^, the impressed voltage, V, and the resistance of the motor, re. Od and Cd are equal. This can be proved by showing that the apex of the isosceles triangle, having OC for a base and OV for the direction of one side, coincides with the point d. From the construction of Fig. 170, the angle COV is equal to the angle A. The length of the side of the isosceles triangle is, therefore, OC 1 ^gcz. 2 cos A 2 n OV From equation (106) OC = — ■ Substituting this in the preceding equation gives for the length OV of the side of the triangle -x— This is equal to the distance Od on the diagram. The radius of the circle of zero power may be found by putting Pm = in equation (108). Making this substitution gives dl^ V . . = ^z— as the radius of the circle of zero power. Since dO is equal V to 75— and dO and dC are equal, this circle passes through the two points and C. The circle of zero power, therefore, passes through the center, C, of the system of excitation circles. Construction of the Diagram. — Choose a suitable current scale. This scale will be used for all lines on the diagram. Everything will be per phase. Refer to Fig. 171. Lay off the X line OC making an angle tan"' — - with the line Od. Ve Power Circles. — The circle, OCD, of zero power is fixed by the points and C, and the direction of its diameter, Od. A more 334 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY accurate method of determining this circle is to find the position of its center, d. V Od = 2re The diameter of the circle of maximum power is zero. The radii of other power circles are found from equation (108) which gives for the radius of any power circle It is usually convenient to have the power circles represent definite electromagnetic outputs, as for example, 100, 200, 300, 400, etc., kUowatts. Three power circles are shown on Fig. 171. Excitation Circles. — A series of concentric circles representing different motor excitation voltages may be drawn with C as a center. It is sometimes convenient to draw these circles to represent different per cents, of the excitation voltage which SYNCHRONOUS MOTORS 335 makes EJ equal to V. One hundred per cent, excitation is rep- resented on the diagram by OC, which is equal to — . Three excitation circles are shown on the figure. A line drawn from C to la represents the motor excitation, corresponding to the current 7„ and the power P^, both in magnitude and in phase, and the angle laCO this line makes with the line OC is the phase angle between Ea and — F on Fig. 170. Current Circles. — A series of current circles may also be drawn with as a center. Only one of these, mlaU, is shown. For any fixed electromagnetic motor power, such as is represented by the power circle marked Pm, there can be two motor excita- tions corresponding to the current /„. The two corresponding excitation circles are fixed by the intersection of the current circle, mlaU, with the power circle Pm- Limiting Operating Conditions. — Maximum and Minimum Excitation for Fixed Motor Power. — The maximum and minimum excitations at which the motor can develop the power Pm, are re- spectively, Cg and Cf. The excitation circles corresponding to these are not shown. The points g and / are points of tangency between the motor power circle and the two excitation circles. The currents corresponding to these excitations are Og and Of. The former leads, the later lags. Minimun Power Factor. — The power factor for any condition is equal to the cosine of the angle made by the current line with the line Od. All currents to the right of Od are lagging. All those to the left of Od are leading. The minimum power factor for any load occurs when the current line is tangent to the power circle for the given load. If the point of tangency for lagging current lies above the line CD, it represents an unstable condition. If this case the minimum power facter for lagging current for stable operation will occur when the extremity of the current line lies on CD. The Maximum Possible Motor Excitation. — The maximum pos- sible motor excitation is CD, where D is the point of tangency of the circle of zero power with a motor excitation circle. CD is the diameter of the circle of zero power. This diameter is V equal to — laid off to the scale of currents. The maximum ex- re citation in volts is {CD)Za. 336 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The Maximum Possible Motor Power. — The diameter of the circle of maximum motor power is zero. The radius of any power circle according to equation (108) is If this equation is to be equal to zero, V^ must be equal to 4rcPm, and P =^ ^'" 4?-, V . . The excitation corresponding to this is Cd = „— which is equal to one-half of the maximum possible excitation. Siability. — All currents lying above the line CD represent unstable conditions of operation. Any increase in load on a motor will cause it to start to slow down, that is, to cause the lag of the motor voltage to increase. If the excitation is fixed, this increase in lag will produce an increase in the current (Fig. 171). With fixed excitation, any increase in the current beyond the line CD (Fig. 171) will cause the extremity of the current line to move to a power circle of larger radius and conse- quently of smaller power. Some Uses of the Circle Diagram. — Besides being of use for determining the approximate operating characteristics of a syn- chronous motor, the circle diagram is very convenient in help- ing to explain simply certain peculiarities in the characteristics of such a motor. For example: the possible range of under- excitation with fixed motor power is much less than the range of over-excitation for the same power. Referring to Fig. 171, let the constant motor power be Pm. The range of under- excitation is confined to excitation circles which cut the power circle, Pm, between a and /, but the range of possible over-excita- tion includes excitation circles which cut the power circle be- tween a and g. This shows why F-curves always extend further on the side corresponding to over-excitation than on the other side. The complete F-curve of a synchronous motor calculated from the circle diagram is plotted in Fig. 172. The dotted part of this curve corresponds to the part of the circle diagram beyond CD and represents unstable conditions. SYNCHRONOUS MOTORS 337 That the compounding curve for unity power factor should at first bend toward lower excitations as the output is increased can easily be seen from the circle diagram. Why the compounding curve for unity power factor should bend in this way was explained in Chap. XXIV. The middle power circle on Fig. 171 corresponds to the power for which the excitation is a rfiinimum for unity power factor. For this power, the excitation circle for unity power factor is tangent to the line ■ , . C .y ,.'-' ^' \ ] 160 ^140 y ^ 1 j • 1 / / /I S 5 100 / / / ^ 80 60 / \ / \ V y 20 JSOO 2000 3000 Szcitaticm. Fig. 172. Od. For powers either greater or less than this, the excitation for unity power factor is greater. The power for which the ex- citation is a minimum depends upon the angle COd. This is tan~^ — . For most motors the ratio of — is so large that the output at which the bend in the compounding curve for unity power factor occurs is not much above, or even may be less than, the electromagnetic output on the diagram which corre- sponds to no load on the motor. CHAPTER XXVIII Losses and Efficiency; Advantages and Disadvantages; Uses Losses and Efficiency. — A synchronous motor does not differ essentially from a synchronous alternator. Consequently, the losses in the two machines are identical and they may be deter- mined in exactly the same manner. The same methods may be employed for calculating the efficiency of a synchronous motor and a synchronous generator. The losses and the method of calculating the efficiency of a generator are discussed in Chapter VII, page 123, under "Synchronous Generators." Advantages and Disadvantages. — The chief advantage of the synchronous motor is its ability to operate at different power factors and the ease with which its power factor may be adjusted. Its comparative simpUcity and the possibility of winding it economically for high voltages, thus doing away with the neces- sity for transformers, are advantages, but these are also pos- sessed by the induction motor. Its invariable speed under vary- ing load is also an advantage under certain conditions. Its main disadvantages are its tendency to hunt and its lack of any inherent starting torque. In the case of polyphase motors, neither of these objections are serious and they are of little consequence when a motor is provided with a properly designed damper and operates under reasonably good conditions. The lack of good starting torque limits the use of synchronous motors to places where frequent starting is unnecessary? They are seldom built in small sizes, that is under 100 or 200 kw. Synchronous motors are less sensitive to variations in the voltage impressed upon them than induction motors. This is an advantage in some cases as it enables them to carry their loads without getting out of synchronism during periods of reduced voltage caused by some temporary trouble on the line or in the power house. A synchronous motor which has the same breakdown torque as an induction motor will continue 338 SYNCHRONOUS MOTORS , 339 to carry its full load at a voltage which is considerably lower than that which would cause the induction motor to break down. The maximum output of an induction motor varies as the square of the impressed voltage, while the maximum output of a syn- chronous motor having usual constants operating with con- stant excitation varies nearly as the first power of the impressed voltage (equation 98). If a synchronous motor and an in- duction motor each having a maximum output equal to twice their rated outputs were put on half voltage, the synchronous motor could still develop its full load without getting out of synchronism. The induction motor, however, would break down at one-half its rated output. Uses. — The principal uses of synchronous motors are in con- nection with motor generators, including frequency changers, and as synchronous condensers. They are not very satisfactory for ordinary power work mainly on account of their lack of good starting characteristics. The chief reason for their use in con- nection with motor-generator sets is the possibility of varying their power factor to control the wattless current taken from the supply system. By operating the synchronous motors slightly over-excited, the reactive current taken by transformers or by inductive loads connected to the system can be in part or wholly neutralized. If the distributing system of a power plant sup- plying a part of its power through synchronous motor-generators is properly laid out, unity power factor may be maintained at the station. The constant-speed feature of synchronous motors makes theni particularly adapted for use in frequency chaijgers. Synchronous motors are frequently used without load in con- nection with transmission systems to control the power factor and to better the voltage regulation. When used in this way, they are called synchronous condensers. Although synchronous motors, if over-excited, take a leading current like a condenser, they do not behave like a condenser in other respects. The wattless current taken by a condenser is directly proportional to the voltage. The wattless current taken by an over-excited synchronous motor operating with fixed excitation decreases with an increase in voltage and becomes zero at a certain voltage. A further increase in the impressed voltage will cause the wattless 340 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY component of the current to reverse and become a lagging component. If a synchronous motor is to be used as a synchronous con- denser to control the voltage of a line, the line must contain reactance. This reactance may be the natural reactance of the line if the line is of sufficient length or it may be inserted arti- ficially. The entire control of the voltage is due to the voltage drop in the reactance. The motor merely serves as a means of making the current through the reactance lead or lag. A leading current through a reactance will cause a rise in voltage. A lagging current will produce the opposite effect. A synchronous motor operating with constant excitation tends to automatically maintain constant voltage across its terminals provided there is reactance in the power mains supplying it. For example, suppose a synchronous motor were operating with normal excitation, i.e., the excitation which produces unit power factor, and the line voltage drops. The excitation of the motor will now be higher than normal for the reduced impressed voltage, and the motor will take a leading current. This lead- ing current will cause a rise in voltage through the line reactance which will tend to restore the voltage at the motor. If the voltage at the motor rises, the motor will take a lagging current which will produce a drop in voltage in the reactance of the line which will tend to offset the change in voltage. The ten- dency of a synchronous motor to maintain constant voltage at its terminals does not depend upon its initial excitation. Normal excitation was chosen merely to simplify the explanation. A polyphase synchronous motor floated on a circuit carrying an unbalanced load tends to restore balanced conditions both in regard to current and voltage. If the system is badly out of balance, the synchronous motor may take power from the phases with high voltage and deliver power to the phase or phases with low voltage. PARALLEL OPERATION OF ALTERNATORS CHAPTER XXIX General Statements; Batteries and Direct-current Gen- erators IN Parallel; Alternators in Parallel; Synchronizing Action; Two Equal Alternators; Synchronizing Current; Reactance is Necessary FOR Parallel Operation; Constants of Generators for Parallel Operation need not be Inversely Proportional to Their Ratings General Statements. — Since the terminals of all generators operating in parallel are connected to common busbars, the terminal voltages of all generators so operating must be equal if measured at the point at which they are paralleled. The load carried by the individual generator and the phase relation between its armature current and generated voltage must adjust themselves to maintain equal terminal potentials. If the impedance in the cables between the generators and the bus- bars or the point at which the generators are put in parallel is zero, the actual potentials at the generator terminals will be equal. In all that follows, unless otherwise stated, the words terminal voltage when applied to a generator will mean the voltage of the generator measured at the point of paralleling and the constants of the generators will include the constants of the line or leads up to this point. The terminal voltage at the point of paralleling will be equal to the actual terminal voltage of the generator minus the drop in the cables between it and the point of paralleling. In the case of alternators, not only must the terminal voltages as measured by a voltmeter be equal, but they must be equal at every instant. In other words, alternators operating in parallel must be in synchronism and their terminal voltages must also be in phase with respect to the load and must so remain. Fortunately, the natural reactions which result from a departure from synchronism are such as to re-establish it. 341 342 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Unless mechanically coupled, alternators cannot ordinarily be operated in series, their natural stable condition being in parallel. If one of two alternators in parallel leads its proper phase with respect to the other, more load is automatically thrown upon it. At the same time the other generator, lagging its proper phase, is relieved of some of its load. The result is that the generator which is leading slows down and the generator lagging speeds up until the proper phase relation is restored. This shift of load between two or more generators which are in parallel is equivalent to a transfer of energy from one to the other. Although it is sometimes convenient to consider the shift of load as an inter- change of energy, in reality no actual transfer of energy takes place except when the load on the system is zero or when the load on a generator is less than the change in its load required to restore synchronism. Batteries and Direct-current Generators in Parallel. — Con- sider first a battery consisting of a number of cells connected in parallel. Let E, I and R with subscripts 1, 2, 3, etc., indicate, respectively, the internal voltage, the current and the resistance of the cells. Let V be their common terminal voltage. The currents delivered by the individual cells must be such as to make all terminal voltages equal. V = El- LRi V = Ei — I2R2 etc. etc. and T El — V I Ri E2-V 2 Xt2 etc. etc. The total current supplied by the batteries in parallel is E -V lo = I1 + h + etc. = S - V = R ^R 4' 72 lo 1 R 4 PARALLEL OPERATION OF ALTERNATORS 343 Substitutihg this value of V in the expression for the current in battery No. 1 gives ^'^R ^R If the internal voltages of the batteries are all equal lo 1 7i = R^ 1 Therefore, if all the internal voltages are equal, the total current carried by the system will be divided anaong the cells in inverse proportion to their internal resistances, and no cur- rent will flow in the cells when the external load is zero. If the resistances are all equal, the currents in the cells will also be equal. If the internal voltages are not all equal, the currents carried by the cells will not be inversely proportional to their resistances and a current will flow in some of the cells when the external load is zero. The cells with low voltage may have current flow through them against their internal voltages, and at some loads certain of the cells may deliver no current. This latter condition will occur whenever the internal voltage of a cell is equal to the common terminal voltage of the system. When it is greater than the common terminal voltage, the cell will deliver current and when it is less the cell will take current from the system. This last condition corresponds to motor action. The former corresponds to generator action. The pre- ceding statements apply equally well to direct-current generators provided R is used as the total resistance of the armature circuit. This includes the resistance of a series field when the generators are compounded. In the case of compound generators, the equalizer is assumed to carry no current. Alternators in Parallel. — The method which has just been applied to batteries in parallel for determining the currents delivered by the individual cells, may be applied to alternators, but when so applied all currents and voltages must be taken in a vector sense and the resistances must be replaced by impedances. 344 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Consider a number of alternators which are in parallel. Let E, V and 2 be, respectively, the generated voltage, the terminal voltage, and the impedance of the alternators. E will be either the voltage corresponding to excitation or to the resultant field according as z is the synchronous impedance or the armature- leakage impedance. Subscripts will be used to distinguish the different alternators when such distinction is necessary. It must be remembered that all of the expressions which follow are to be taken in a vector sense and that sine waves are assumed. The effect of harmonics will be considered later. The current delivered by any alternator is I = ^^^ (109) z and the total current 7„ taken by the load is E — V z z But since all the terminal voltages, V, are equal and in phase, /„ = 2 - - ys i z z where Yo is the resultant admittance of the armatures in parallel. Therefore, V = £ : (110) Yo Hence, J El z lo zi Y^i Y„zi h = y,E, - p XiEy) + lop (HI) Fo ^ "' ' °Yo where yi is the admittance corresponding to Zi. PARALLEL OPERATION OF ALTERNATORS 345 If the voltages, E, are all equal and in phase, then and T. — T h = lo p (112) Hence when all generated voltages are equal and in phase, the currents carried by the individual alternators are directly pro- portional to their admittances or inversely proportional to their impedances. The vector sum, but not the algebraic sum, of these currents is equal to the total current delivered by the system. If, under this condition — = — , etc., Yq will be equal X\ X2 to the algebraic sum of yi, 2/2, 2/3, etc., and the total current de- livered by the system will be divided among the generators in direct proportion to their admittances. All of the indi- vidual armature currents will be in phase with the load cur- rent and their algebraic sum will be equal to the total current delivered by the system. The condition of the generated voltages being equal and in phase corresponds to the condition existing when transformers with equal ratios of transformation and negligible exciting currents are paralleled. If the con- stants are not in the relation indicated above, it is still possible to have the component currents in phase and inversely propor- tional to their impedances or in any other proportion but in this case Ei and E2 will neither be equal nor in phase. Since the vector sum of the first two terms of equation (111) is the component current carried by the armature of an alter- nator due to its voltage being out of phase or not equal to the generated voltages of the other alternators, the first two terms of equation (111) may be considered to be a current interchange or a circulatory current between one alternator and the others. This circulatory current never gets to the load. It may or may not be wattless with respect to the terminal voltage, depend- ing upon the conditions which cause it. This interchange current is not a separate current. It is merely one of the com- ponents into which the armature current may be resolved under certain conditions. Its presence may or may not be desirable. 346 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Two Equal Alternators. — Consider two equal alternators, i.e., alternators with equal constants and the same rating. For two alternators equation (111) reduces to If 7, = y,E^ - |1 (E,yr + E^y,) + h y^ , ri r-2 yr = y, and - = ^ /i = y,E^ - Yy^{E^ + E2) + loY^ h = ViE, - I {E, + E,) + 1° (E, (113) jhi — Ei\ 2 / ' 2 (114) The current carried by each alternator under these conditions is equal, vectorially, to one-half the load current plus a com- ponent current Ji = yA ^ „ — ^j which circulates between the two armatures. The conception of a circulatory current is apt to mislead except with two identical alternators. Synchronizing Action, Two Equal Alternators. — The circulatory current is equal to yi ^ (equation 114). This current may Fig. 173. be caused by one of two things: by Ei and E^ being out of phase; by El and E^, differing in magnitude. The effect of the cir- culatory current will be different in the two cases. When it is produced by a difference in phase, it produces synchronizing action. When resulting from an inequality in the voltages, it merely equalizes the terminal voltages, mainly by its effect on armature reaction. If two equal alternators which have equal PARALLEL OPERATION OF ALTERNATORS 347 excitations and carry equal loads are in parallel, there will be no interchange current between them unless they are displaced in phase from exact synchronism. If they become so displaced, an apparent interchange of energy will take place between them which will tend to restore synchronism. The natural tendency of two alternators, which are in parallel, to remain in synchronism, will be made clear by the vector diagrams shown in Figs. 173 and 174. These diagrams are for equal alternators with equal excitations and equal loads. Both diagrams are drawn with respect to the series circuit consisting of the two armatures. The terminal voltages which are equal and in phase when con- sidered 'with respect to the parallel circuit are opposite in phase when considered with respect to their own series circuit. Equa- FiG. 174. tion (114) applies to the parallel circuit. To make it apply to the series circuit, the sign of Ez must be changed. Fig. 173 represents the condition before either generator has become displaced. Fig. 174 represents the condition when the generators are slightly out of phase. V, E and I are, respectively, the terminal voltage, the excitation voltage and the load component of the armature current. The subscripts 1 and 2 refer to generators 1 and 2, respectively. Since Fig. 173 represents the condition of synchronism and equal excitations, Ei and E2 are equal and opposite and their resultant is zero. The total load current delivered by the system is 7i minus 1 2 when referred to the voltage Vi. When referred to the voltage V2, it is h minus h. In Fig. 174, Ei and Ei are still equal but they are not exactly in opposition since the generators are assumed to be slightly out of phase. 348 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Since Ei and E2 are not in opposition, their resultant J57o = Ei + E2 (Fig. 174) is not zero. The resultant voltage Eo acts through the impedances of the two armatures and produces a component current J Eq ^' ~ 2z in the series circuit consisting of the two armatures. The synchronous impedance of each armature is z = r + jx. This component current or circulatory current, as it is called, lags behind Eo by an angle tan~^ — ■ Since the ratio of the synchronous reactance to the effective resistance of a synchron- ous generator is usually large, this angle is also generally large, equal to at least 85 degrees. The current. It, has a component which is in phase with Ei and a component which is in opposition to E^. It, therefore, produces generator power with respect to machine No. 1 and motor power with respect to machine No. 2. The power output of an engine or turbine depends upon its mean speed. Since the mean speed of the alternators does not change when they are displaced in phase, the power they receive from their prime movers is not altered by a change in phase. The effect of the interchange of current due to the phase displacement is, there- fore, to slow machine No. 1 which leads and accelerate machine No. 2 which lags. In other words, the circulatory current tends to bring the rotors of the generators into synchronism, i.e., into the phase position which puts Ei and E2 in opposition as shown in Fig. 173. As a result of the circulatory current, /<, there is an apparent transfer of energy from one generator to the other. This apparent transfer of energy between two generators which are in parallel is the synchronizing action which makes the parallel operation of alternators possible. The current carried by the armature of either alternator is the vector sum of the circulatory current and the component of the load current the alternator would carry if there were no circulatory current. Referring to Fig. 174, generator No. 1 has an armature current, /aj, which is equal to the vector sum of /i PARALLEL OPERATION OF ALTERNATORS 349 and li. The armature current of generator No. 2 is the resultant of I2 and li. This is /aj- The only actual currents are armature currents lai and 1^2- Ii, h and li exist merely as components. The current delivered by the system is still the difference between /i and 1 2. This is equal to the vector difference between the armature currents lai and la^. Synchronizing Current. — The change in the electrical output of the generators when they are displaced in phase is in part due to the power developed by the interchange current, li, considered with respect to the generated voltages Ei and E2, and in part due to the change caused by It in the phase and mag- nitude of the generated voltages with respect to the currents 7i and I2. Referring to Figs. 173 and 174, the change in the power developed by generator No. 1 due to a phase displace- ment such as is indicated in Fig. 174 is EJai cos a'l — Eili cos ai Assuming that the terminal voltage, V, does not change, 7i will not change and EJai cos a'l — Eili cos ai = EJi (cos ffi — cos ai) -|- Exii cos pi The greater part of the synchronizing power is caused by 7< directly and, for this reason, li is sometimes called the syn- chronizing current. Reactance is Necessary for Parallel Operation, — Considering the part of the synchronizing action which is due to 7, — the only part which can exist when there is no load on the system — it will be seen by referring to Fig. 174, that this part can be present only when li lags behind Eg. If E\ and E2 are equal, as was assumed, and 7,- is in phase with Eo, it would have equal positive projections on E\ and E2 and would produce an equal generator effect on each alternator. Under this condition, it would have no tendency to restore synchronism. Under certain special conditions, there still may be a slight synchronizing action due to the change in the phases of E\ and E2 with respect to 7i and I2, respectively, but this action does not always exist, can never exist at no load and is always too small alone to make the parallel operation of alternators possible. The synchronizing 350 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY action of li is dependent upon its lag behind Eo. Reactance is, therefore, absolutely necessary for the parallel operation of alternators. By putting capacity between two alternators which are connected in parallel, the circulatory current, li, can be made to lead Eo. Under this condition, the action of /; is to bring the voltages Ei and Ei into conjunction on their series circuit- and to bring the generators into conjunction on the series circuit. Although this condition may be produced experimentally, it is of no importance practically. Generators cannot be built with- out reactance. The natural tendency is, therefore, for all generators which are connected together to assume the proper phase relation for parallel operation. Their stability will depend upon the amount of reactance in their armature circuits and to some extent upon the constants of the load. The Constants of Generators for Parallel Operation need not be Inversely Proportional to Their Ratings. — When trans- formers having equal ratios of transformation are operated in parallel, all of their primary voltages must be equal and in phase. All of their secondary voltages must also be equal and in phase. The load they carry and the phase relations between the currents they deliver depend solely upon their constants. For this reason it is important that transformers which are to operate in parallel should have constants approximately in- versely proportional to their ratings. The conditions for suc- cessful parallel operation of alternators are not nearly so rigid since the excitation voltages, corresponding to the primary vol- tages in the case of transformers, do not have to be equal or in phase, fli the constants of the alternators are not in the inverse ratios of their ratings it makes Uttle difference since the load may still be divided between alternators in any desired way and their armature currents brought into phase. This may be accomplished by properly adjusting the power they re- ceive from their prime movers and also varying their field ex- citations. If two alternators having dissimilar constants have been made to share the load properly and their armature cur- rents have been brought into phase, there will be a circulatory current according to equation (113), page 446. A circulatory current under these conditions is highly desirable. For this reason, too much emphasis should not be placed on the exist- PARALLEL OPERATION OF ALTERNATORS 351 ence of a circulatory or interchange current. Such a current is very desirable except when the constants of the alternators operating are exactly inversely proportional to their ratings. It is what makes possible the successful parallel operation of alternators of totally different design. Modifying equation (111), pag^ 344 so as to make it apply to two alternators, gives for the current in the armature of alter- nator No. 1 h = y,E, - 1^ {E^y^ + E^) + h^^ (115) A similar expression applies to alternator No. 2. If the ratio of the loads to be carried by the two alternators is A, then for ideal conditions 7i should.be equal to I^A and in phase with it. The value of Ex in terms of E^ which will give this condition may be found by equating 7i and I^A and replacing Jj and I^ by their values as given by equation 115. This gives ^' = y.{Yo + Ay, - 2,0 (116) Although equation (116) looks somewhat formidable, it shows that for definite constants and any desired ratio, A, between the loads, there is a definite relation between Ei and E, in phase and in magnitude which will not only make the ratio of the loads equal to A but will also bring 7i and 1% into phase. In equation (116), El is referred to the same axis as that to which 7„ is re- ferred, but beyond showing that a definite relation exists between El and E-^ which will make the alternators divide the load properly, the equation is of no practical value. In order to get Ei in terms of E2 from equation (116) the y's would, of course, have to be replaced by their components, g — jh, and 7„ would have to be expressed as a vector referred to sonie axis such as V. It will be shown that changing the amount of power given by a prime mover to an alternator which operates in parallel with others changes its output and also the phase but not the magnitude of its excitation voltage. Changing the field ex- citation alters the magnitude of the excitation voltage and also changes its phase but does not appreciably alter the output. Therefore, by adjusting the power given by prime movers to alternators which are in parallel and at the same time adjust- 352 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY ing their field excitations the alternators may be made to divide the load properly, provided any reasonable relation exists be- tween their constants. That it is not necessary, although desirable, for alternators which are to operate in parallel to have constants approximately inversely proportional to their ratings, should be made clear by Fig. 175. Fig. 175 is drawn for two alternators considered with respect to their parallel circuit. lai and la^ are the two armature currents. These are assumed to have been brought into phase by adjusting the field excitations of the alternators and to have been made proportional to the ratings of the alternators by adjusting the amount of power given to each alternator by its prime mover. Fig. 175. Adding the drops due to the currents la^ and /„2 to the terminal voltage gives the excitation voltages Ei and E^. Since Ei and J?2 are not equal, each machine will have a circulatory current according to equation (115). If the excitations are adjusted to give the voltages Ei and E^, the condition represented by the vector diagram shown in Fig. 175 will be fulfilled. With any division of load the armature currents may be brought into phase with the load current by properly adjusting the field ex- citations. Although there is an unbalanced voltage Ei — Ei acting in the series circuit consisting of the two armatures, this voltage /will not cause any currents in the armatures other than those already existing, since the voltage Ei — Ez = Ed. is just balanced by the impedance drops which are already present in the armatures. The rise in voltage represented by Ei — E^ = Ecb is just balanced by the fall of voltage Eia plus the rise Eac. CHAPTER XXX Synchronizing Action of Two Identical Alternators; Effect of Paralleling Two Alternators through Transmission Lines of High Impedance; the Relation between r AND X for Maximum Synchronizing Action Synchronizing Action of Two Identical Alternators. — Consider the case of two identical alternators, that is, of two alternators which have equal electrical and mechanical constants. Assume that the governors of the prime movers which drive the al- ternators are sluggish and do not respond to changes in the Fig. 176. angular velocity of the prime movers which are caused by hunt- ing. Under these conditions when hunting occurs, it will not be influenced by any changes in the power developed by either prime mover. When one generator is ahead of its mean phase position the other generator will be behind its mean phase posi- tion. In order to simplify the discussion the only case which will be considered is where the alternators have equal excitations and share the load equally when there is no hunting. The excitation voltages will be assumed equal. Fig. 176 is the vector diagram of the alternators drawn with respect to their 23 353 354 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY parallel circuit and represents the conditions which exist at some instant when their excitation voltages have been displaced in phase from each other by an angle, a. I is the component of the armature current of each generator which is in phase with the load current. Since the generators are equal, the current /, according to equation (114), is equal to one-half of the current /„ taken by the load. The excitation voltages El and E2 are assumed equal and constant. When hunting occurs these voltages remain unchanged in magnitude but swing in opposite directions. They are shown in Fig. 176 displaced by an angle a. When there is no hunting they coin- cide, li, the circulatory current for generator No. 1, is equal to — ^ (equation 114). The circulatory current for gen- erator No. 2 is equal to — ^ and is opposite to /,•. The latter is not shown on the diagram. Let Pi and Pa be the powers developed by generators No. 1 and No. 2, respectively, when they are displaced in phase by an angle a and let /j and li be the armature currents under this condition. P. = hEi cos <; = (1° + ^^)ei cos «;; = (7 + Ii)Ei cos e f; El sm / \ /ii sm s / s = Z^,eos(,+ |)+^^,eos(|-|-,) = IEiCos,[e -\-^\ H = sin OL Ot = lEi cos d cos 2 — IE\ sin 6 sin ^ El-' sin I ^ Ei^ sin ^ + sin ^ cos 7 -I- cos ^ sin 7 z I z 2 Ot (X = lEi cos d cos 2 — I El sin d sin -^ Ei^ sin I Ei^ sin ^ I ^ r ■ a , 2 X a + sm - + cos ~ z z 2 z z 2 PARALLEL OPERATION OF ALTERNATORS 355 = lEi cos cos I — lEi sin S sin | + lih + —^ sin 2 cos 2 (117) Pa = lEi cos 9 cos I + 7£2 sin e sin ^ + li^r ^ sm 2 cos 2 (118) The term ZjV in equations (117) and (118) does not necessarily represent the copper loss caused by the circulatory current. The copper loss caused by this current is equal to (I + 7,)V - Pr. This is not equal to li^r except when I and li are in quadrature. Since Ei is the voltage causing the current (7 + li) in generator No. 1 and }4Eo = M(-2?o — -Ea) is. the voltage absorbed in the impedance drop caused by the circulatory current, h, the dif- ference between Ei and }4Eo or Eoa must be the voltage causing the current 7. If there were no hunting, IE cos 8 would be the power developed by each generator. E = Ei = E2 according to the assumed conditions. The change in the power developed by each generator which is caused by any change in their phase displacement, a, may be found by subtracting EJ cos 6 and E2I cos d, from equations (117) and (118), respectively. Making this subtraction gives equations (119) and (120) for the change in the power developed by generators No. 1 and No. 2, respec- tively. In (119) and (120) the subscripts have been dropped from the E's since, according to the assumed conditions, Ei = E^. IE cos eicos ^ — 1) — IE sin 5 sin | + hh + -^ sm 2 cos 2 (119) IE cos eicos ^ — 1) + IE sin 9 sin | + 7iV ^sm2COS2 (120) The first terms and the third terms of expressions (119) and (120) are equal both in magnitude and in sign. Therefore, if the moments of inertia of the generators are equal, these terms 356 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY tend to produce equal retarding effects on the angular velocity of the generators and, consequently, cannot influence the relative phase displacement of the generators. Hence, they cannot cause any synchronizing action. The effect of the first and third terms of (119) and (120) is to cause a slight variation in the angular velocity of the entire system. The period of this variation in the angular velocity is, of course, the same as the period of the hunting. Due to this variation in the angular velocity of the system, the terminal voltage of the system will swing in phase when hunting occurs. This will not influence the hunting between the alternators which are in parallel, but it will tend to start hunting between the alternators and any synchronous apparatus they supply. Since the first and third terms of (119) and (120) do not affect the synchronizing action of the two alternators, it follows that the second and fourth terms must represent the power acting on each alternator to hold it in synchronism with the other. The synchronizing power, P^, acting on each of two equal alternators is, therefore P, = sj^cos| - /sinej sin| (121) p p which is equal to „ , i.e., to one-half of the difference between the powers developed by the alternators when they are displaced. ^ fV fY fY The terms E^-i cos-^ sin -^ and El sin d sin -x in equation (121) represent the synchronizing power due, respectively, to the circulatory current and to the change in the phase angle between E and /. Equation (121) shows that with constant excitation voltage, i.e., with constant excitation, the synchronizing power and, therefore, the stability of two alternators which are in parallel is greatest when sin Q is negative, that is, the stability is greatest with capacity loads. Generators do not operate as a rule with constant excitation but with constant terminal voltage. When the terminal voltage is kept constant the synchronizing power, Ps, corresponding to a given phase displacement, a, is greatest for inductive loads. PARALLEL OPERATION OF ALTERNATORS 357 If R and X are, respectively, the resistance and the reactance of the load jb cos 75 I = I'^Z = ^^ ° \/{2R + r)^+ (2X + a;)=' and sm e = , -s/(2/i + r)2 + {2X + xY Substituting these values in equation (121) gives _ ( X (2X + x) 1 • ^ ^ " ~ [z^ ~ {2R + ry + (2X + xYl ^^^ 2 ^°^ 2 E^ ( X 2X + a; 1 = T 17^ ~ (2R + r)''+(2X + a;)4 ^'"^ " "^^^^^ If a short-circuit should occur at the generators, both R and X would be zero and the synchronizing power would reduce to E^fx X ] . "« = "S" i ~i i [ sin a = Under this condition the synchronizing power would be zero and the generators would fall out of step. A short-circuit on the feeders outside of a generating station may decrease R and X sufficiently to so much reduce the synchronizing power as to cause the generators to drop out of step. To get the synchronizing power in terms of the terminal vol- tage, V, under steady operating conditions, replace E in equation (122) by V. When hunting occurs, V will vary in direction and magnitude but E will be constant in magnitude. E will be equal to the terminal voltage, V, before hunting starts plus the impedance drop due to the current, /, carried by either generator under steady conditions. E =V + Iir+ jx) V = I{2R+ j2X) = 2IVr^ + X^ E = lV{2R + r)2 + {2X + xY ^ ' l{2R + ry + {2X + xY -u R^-irX^ 358' PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Putting this value of E in equation (122) gives [(2R + rY + (2Z + xy] - [4 (fl2 + Z2) + (r^ + x^) + 4(fir + Xx)] 7^ fa; 8(/?2 + Z^) 1 3^ -(2Z+a;)| sin a = 4^2(2;!+ j^2) { 2(fi'+ X=')a;+(x2-r2) Z + 2ii;ra;)sina (123) 7^ f a; , (x^ - r') ^ . fira: ] . . „ .. " 2 U2+ 2^2 (i22 + X=')+g2(222 + X2) pma ix^4j It should be remembered that 7 in equations (123) and (124) is the terminal voltage of the generators before hunting starts. This is constant. The actual terminal voltage, as has already been pointed out, varies both in phase and magnitude when hunting occurs. The first term of equation (124) represents the synchronizing power due to the circulatory current. The second and third terms represent the synchronizing power due to the reactive and energy components of the load current, respectively. These last two terms become zero when there is no load on the system. The first term of equation (124) is the most important. The relative importance of the second two depends upop the power factor of the load. The second dis- appears when the load is non-inductive. It follows from equation (124) that for any fixed terminal voltage, 7, Ps will be greater when X is positive than when negative, provided x is greater than r. Therefore, since the synchronous reactance, x, of alternators is always greater than their resistance, two equal alternators when operated at con- **stant terminal voltage will be more stable on inductive loads than on capacity loads. This statement also holds when the alter- nators are not equal. For any fixed ratio of r to x, the synchronizing power will de- crease with an increase in either r or z. This may be shown by replacing x in equation (124) by kr where A; is a constant. Mak- ing this substitution gives ^' ~ 2\r\k^ + V^ 2{k^+l) (R'^+X^) ^ k R ] (F+l)(K2+z=)J sin a PARALLEL OPERATION OF ALTERNATORS 359 Effect of Paralleling Two Alternators through Transmission Lines of High Impedance. — When alternators are paralleled through transmission lines, the resistance and the reactance of the lines add directly to the constants of the machines. The easiest way to determine the effect of paralleling through lines of considerable impedance, is to substitute numerical values in equation (124). V in this equation is the potential at the point of paralleling. Assume a 2 per cent, copper loss in each generator at full-load current. According to this assumption Z, of the load, will be 25r. Let x be 20r. Then, if the generators are paralleled with no impedance between them on a full kilovolt- ampere load of 0.8 power factor Ps=-Yr (0-063) sin a Let them be paralleled through lines having a 15 per cent, copper loss at full load. The line resistance now will be 7.5r. If the line reactance is equal to the line resistance 72 P, = 2^ (0.052) sin a which shows a decrease of about 18 per cent, due to the effect of the line. Unless the excitation of the generators is increased, the potential, V, at the point at which the generators are paral- leled will be decreased by the line drop. Any decrease in V will have a marked effect on Ps, since Ps varies as V^. The effect of a given line impedance depends upon the ratio of its component parts as well as upon the load power factor. Any increase in line resistance, when the load is either inductive or non-inductive, will decrease the synchronizing action under most conditions. The effect of an increase in the line resistance is most marked where the power factor of the load is low. Anything which affects the synchronizing power 'will also affect the period of hunting but in an opposite manner (equation 130, page 362). Due to the decrease in the synchronizing action when generators are paralleled through lines of consider- able impedance as well as to the change in the period of hunt- ing, generators paralleled through transmission lines of high impedance may show a tendency to hunt. This, however, does not often occur. 360 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The Relation between r and x for Maximun Synchronizing Action. — The synchronizing power will be a maximum when the term of equation (124), X x^ — r^ X - Rrx z^ ^ 23^ R^ + X^^ z^{R^ + X^) is a maximum. Differentiating this term with respect to x and equating the differential to zero gives Xr X = r\ i22 + Z2 + Rr +Vl+ U + g + «j '|<'25) for the maximum synchronizing action corresponding to any fixed armature resistance, r. If the load is zero, equation (125) reduces to a; = r. For any reasonable values of X and R as compared with r, x, according to equation (125), should be very nearly equal to r for maximum synchronizing action. For a power factor of 80 per cent, and a copper loss in the generators of 2 per cent, at full load, equation (125) becomes 'x = 1.02 r. This relation between r and x is of little importance since for best operation other conditions than having Pj a maximum for a given phase displacement, a, call for a ratio of a; to r which is very much larger than unity. These other considerations are: the stiffness of coupling, the period of oscillation as a torsional pendulum and the short-circuit current. The resistance, r, must be made as small as possible in order to keep down the copper loss. If x could be made as small as r, the stiffness of coupling between the two alternators would be too great and they would, in consequence, be subjected to very severe strains whenever hunting started or when they were synchronized slightly out of phase. The circulatory current would also be very large even for slight phase displacements. The ratio of the synchronous reactance of ordinary alternators to their effective resistance is always greater than 10 and more often greater than 25. CHAPTER XXXI Period of Phase Swinging or Hunting; Damping; Irregu- larity OF Engine Torque during Each Revolution and Its Effect on Parallel Operation of Alternators; Governors Period of Phase Swinging or Hunting. — If one of two equal alternators which are operating in parallel momentarily changes its angular velocity, synchronizing power will be developed between the two machines according to equation (124), page 358, which will tend to restore them to their proper phase relation. This will cause the machine which lags to speed up and the machine which leads to slow down. Due, however, to the inertia of their moving parts, the generators will swing past the position ' of no synchronizing action. The synchronizing power will then reverse and tend to pull the generators together again. This action is the same as the hunting which takes place with a synchronous motor. It would continue indefinitely if it were not for the damping action of the losses produced by the hunting in the pole faces and in the dampers in case dampers are used. The period of hunting can be found in the same way as it was found for a synchronous motor (page 317). If p, / and n are, respectively, the number of poles, the frequency and the number of phases, the synchronizing torque is r.=^ (126) Substituting P, from equation (124), page 358, in equation (126) and dividing by — gives the synchronizing torque developed per unit of space angular displacement of the generators from their mean position. ■pT, ,., nVv^lx ,x^ -r"" X , Rrx \ sin a , „_. where Z^ = W + X^. 361 362 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Sin oi Since for small angles is nearly equal to unity, equation a (127) may be written ^^nV^lxx^^-r^XRrx] .(128) The time of oscillation of the rotor of an alternator as a torsional pendulum about its mean angular position is I = 2,^^' (129) where M and Smd^ are respectively, the synchronizing torque per unit angle of phase displacement, and the resultant moment of inertia of the rotor of the generator and the prime mover. Substituting the value of M from equation (128) in equation (129) gives for the approximate time of oscillation SirfLmd^ t -^■^ I ^ X , x^-r^ X , Rrx] (130) rvp^ Z2+ 2^2 Z2+22Z2J Equation (130) shows that the period is inversely proportional to the terminal voltage, V, of the system before hunting starts. It is also proportional to the square root of the moment of inertia of the moving parts of the generator and its prime mover. The time of oscillation will increase nearly as the square root of the synchronous reactance of the generators. Putting react- ance in series with the generators increases their period of hunting. The period is also affected by the load and its power factor. With fixed terminal voltage, the period increases with an increase in the load. When the load is zero, equation (130) reduces to ^^2 118^^ (131) Damping. — Synchronous generators which are to be operated in parallel require a certain amount of damping, but as a rule the damping does not need to be so great as for synchronous motors. Generators which are driven by internal-combustion engines are an exception to this rule. Such generators require strong damping. The most common form of damping device PARALLEL OPERATION OF ALTERNATORS 363 is an amortisseur. An amortisseur for an alternator as a rule, need not be as effective as for a motor, since for a motor it must serve not only for damping out hunting but also as a starting device. The damping action of the hysteresis losses and the eddy currents produced in the pole faces by hunting is often sufficient for alternators. If the ordinary pole-face losses due to the armature slots are eliminated or largely diminished by getting the effect of closed slots by the use of magnetic wedges for holding the armature winding in place, soUd poles may be used. Under these conditions, if hunting occurs, large eddy currents will be set up in the solid pole faces by the oscillating armature-reaction field. Very strong damping action may be obtained in this way. It may, in fact, be made large enough to permit its use for damping and for starting synchronous motors as well. Irregularity of Engine Torque during Each Revolution and Its Effect on Parallel Operation of Alternators. — The turning Fig. 177. monient or torque developed by any reciprocating steam engine or internal-combustion engine is not uniform but goes through a definite cycle in each engine revolution. The form of this cycle depends upon the type of engine and the load it carries. The torque curve of a double-acting single-cylinder engine or of a tandem-compound engine has two maximum and two zero points in each revolution. The torque of a cross-compound engine with 90-degree cranks has four maximum and four minimum points in each revolution but it never falls to zero. Typical torque curves for large slow-speed engines with Corliss valves are shown in Fig. 177. Curve / is for a single-cylinder 364 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY or a tandem-compound engine. Curve /// is for a cross-com- pound engine with 90 degree cranks. For this curve the power is assumed to be divided equally between the two cylinders. Curve III is obtained by adding the ordinates of curve / to the ordinates of a similar curve, //, which is displaced from curve I, by 90 degrees. The torque curve of a high-speed engine will have the same number of maximum and minimum points as the torque curve of a slow-speed engine. Its shape will be different on ac- count of the greater effect of the inertia of the reciprocating parts. The frequency of the variation in the torque of a cross-com- pound engine will be twice the frequency of the variation in the torque of a single-cylinder engine or a tandem-compound engine having the same speed, and for the same average turning moment the magnitude of the variation will, as a rule, be less than one-half as great. A cross-compound engine may have three distinct frequencies of torque variation, namely: (a) A frequency of one per revolution caused by one end of one cylinder receiving more steam than the other. {b) A frequency of two per revolution caused by the work being unevenly divided between the cylinders. (c) A frequency of four per revolution caused by the combined double-frequency torque waves of the two cylinders. The last of these three frequencies, i.e., four per revolution, is by far the most important. In designing a flywheel for an engine, the maximum variation of the torque must first be found and then the flywheel must be so designed that its moment of inertia when combined with the moment of inertia of the rotor of the generator will limit the variation in the angular velocity during a revolution to some definite prescribed value. The permissible variation produced by irregularities in the engine torque in the angular velocity of alternators which are to operate in parallel depends upon the frequency of the alter- nators and the ratio of their short-circuit and full-load currents. Under ordinary conditions, this variation should not be allowed to cause a displacement in the excitation voltage of any alter- nator from its mean position of much more than 1^ electrical degrees. This corresponds to space degrees where p is PARALLEL OPERATION 0F_ ALTERNATORS 365 the number of poles. The permissible variation in the angular velocity of multipolar generators is very small. For this reason, engines which drive multipolar alternators must have large flywheel action. The effect of a displacement can readily be seen by referring to equation (124) page 358. For most alternators the ratio of their synchronous reactance to their synchronous impedance is very nearly unity. Making this assumption and neglecting the effect of the load, equation (124) may be written, for small values of the angle, a, in the following approximate form, 72 ^' = T.« V . Under ordinary conditions — is nearly equal to the short- z circuit current, /«, at full-load voltage. Making this assump- tion gives Pa = y^VIscCt, approximately (132) Suppose that two similar alternators are paralleled which have short-circuit currents that are equal to three times their full-load currents, and also suppose that the maximum displace- ment caused by the engines is l]/i electrical degrees. If it happens that the alternators are synchronized in such a way that the engines produce maximum displacements in opposite direc- tions at the same instant, a. will be 2 (134) = 2.5 electrical degrees. According to these assumptions P, = K7(3/) (2.5) ^ = 0.065F7 where I is the full-load current. The synchronizing power under this condition is 63^ per cent, of the rated output of each generator. The smaller the short-circuit current, the larger the permissible variation in angular velocity. The actual variation in the angular velocity, and consequently in the phase displacement produced between the excitation voltages of generators which are in parallel, depends not only upon the magnitudes of the variation in the torques of the engines and the moment of inertias of the flywheels and alter- nators, but also upon the synchronizing torque of the generators and their damping. The synchronizing torque of an alternator 366 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY is nearly proportional to its phase displacement and directly opposes the displacement. On the other hand, the damping action of eddy currents and hysteresis caused by any oscillation of the rotor about its mean angular position as well as the damp- ing action of a damping winding if one is used, is nearly in quadrature with the displacement, and, therefore, nearly in quadrature with the variation in the engine torque. The curve representing the torque of a reciprocating engine during a revolution may be resolved into a straight line, which Angular Displacement frooLMean Position. Synchronizing Torque te=^+fc sla (J)t h tt — „ sin U)t 2 mr^ w'=K'+fat dt ~K''~A. COS flit d'=f—A COS tot dt =B sta (Dt = -C eln 2d' = -C sln(2B sin o)t) CErom-Equatlon 127) t^—Du>'=—G cos (ot represents the mean torque, and an irregular curve which shows the variation of the actual torque from this mean. If the irregular curve is resolved into a fundamental and a series of harmonics and then the latter are neglected, the resulting ap- proximate torque curve becomes a straight line upon which a sine curve is superposed. In a good many cases, this substitu- tion may be made when studying the effect of thp variation of the engine torque on parallel operation, but when the harmonics are large, as, for example, when the prime movers are internal- combustion engines, the harmonics must be considered. Fig. PARALLEL OPERATION OF ALTERNATORS 367 178 shows the curve of the engine torque, neglecting all har- monics, and the corresponding curves of the angular accelera- tion of the engine, the angular velocity of the engine, the angu- lar displacement of the generator from its mean position, the synchronizing torque, and the damping torque. These curves are plotted one over the other with a common scale of time in order that the phase relations between them may be seen. Equal machines are assumed in the case of the curve of synchro- nizing torques. In addition to keeping within certain limits the phase dis- placement due to the irregularities in the engine torque, it is necessary to make the time of free oscillation of the generator and its flywheel as a torsional pendulum different from any frequency in the torque produced by the engine. If the natural frequency of the oscillation of the generator and its flywheel should coincide with the frequency of any of the engine impulses, violent hunting would occur which would make parallel operation impossible and which might even make it impossible to hold the generators in synchronism. To prevent this resonance between the frequency of the engine torque and the time of oscillation of a generator as a torsional pendulum, care should be taken .when designing a flywheel to make the natural frequency of the system at least 20 per cent, lower than the lowest frequency of the impulses from the engine. Hunting may be caused by any periodic variation in the circuit fed by the alternators as, for example, a periodic varia- tion in the load, but it is seldom that the frequency of the variations in a load will coincide with the natural frequency of the alternators. Turbines for both water and steam have uniform turning moments. Turbo-driven generators are, therefore, free from hunting caused by their prime movers. In case alternators which are driven by internal-combustion engines are to be paralleled, it is necessary to provide them with massive flywheels and in addition to use damping grids. Governors. — Hunting may be caused by improperly designed governors. Governors for engines which are to drive alternators must not be too sensitive and must be sufiiciently damped to prevent over-running. If wi and wj represent, respectively, the 368 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY maximum and the minimum angular velocity of an engine during a revolution, the mean angular velocity is 0)1 + 0)2 The variation in speed referred to the mean speed is 0)1 — 0)2 '2 give, respectively, the maximum and the minimum speeds for different loads at which the engine PARALLEL OPERATION OF ALTERNATORS 369 can operate without action of its governor. If the engine is a reciprocating engine, its speed will vary by an amount equal to 3^0- Wm above and below its mean speed. Therefore,, in order that its speed shall lie within the limits of maximum and mini- mum speed fixed by the lines marked o'l and w'2, Fig. 179, its average speed during a revolution must always be less than the speed represented by the line w'l and greater than the speed represented by the line ca'i by an amount equal to J'^crtom. Sub- tracting y^(T(i3m from the ordinates of the curve marked co'i and adding it to the ordinates of the curve marked co'2 gives curves + Aco)< be the voltage of the generator which is being synchronized, then / Bl = eb + eg = Em sin o>t + E'm sin (co + Ac / 1 / i /' \ 1 / ^4 1 / 1/ F] G. 196. Replacing the coil current in equation (137) by the total line current, lac, gives lac _ . ■IT 1 Vdc Idc n n{p.f.) iv) Vac is very nearly equal to the ratio of the induced voltages and we may write as an approximation lac „ . /^\ 1 1 2V2 = 2 sin Idc © n(.p.f.) iv) 1 . TT •v/2 n n{p.f.) M (138) This shows that the line currents in converters are inversely proportional to the number of slip rings or inversely proportional to the number of phases except for single phase. Table XVIII gives the ratio of the currents on the two sides of converters 402 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY with different numbers of phases. One hundred per cent, efficiency and unity power factor are assumed. Table XVIII No. of taps per pair of poles No. of phases per pair of poles 2 1 1.41 3 3 0.943 4 4 0.707 6 6 0.471 12 12 0.236 It will be seen from Table XVIII that a three-phase converter having an efficiency of 94.3 per cent, and operating at 100 per cent, power factor has equal a.c. and d.c. currents. CHAPTER XXXVII CoppEB Losses of a Rotary Converter; Inductor Heating; Inductor Heating of an n-PnASE Converter with a Uniformly Distributed Armature Winding; Relative Outputs of a Converter Operated as a Converter and AS A Generator; Efficiency Copper Losses of a Rotary Converter. — The output of all commutating machines is limited by commutation and by the heating produced by the losses. A large part of the difficulties of commutation in direct-current motors and generators is due to the field distortion produced by armature reaction. Poly- phase rotary converters are almost entirely free from this field distortion. Since motor and generator currents are opposite when considered with respect to the generated voltage, the currents carried by the armature inductors of a rotary converter will be the difference between the alternating-current and direct- current components. The average copper loss produced by the resultant current carried by the inductors is less than would be produced by either component alone, except for a single-phase converter. The average copper loss is not the same in all the armature inductors of a rotary converter, but varies with the position of the inductors with respect to the taps. The difference between the copper loss in the hottest and coldest inductor depends upon the number of phases for which the converter is tapped and upon the power factor at which it operates. This difference decreases as the number of phases is increased and as the power factor is raised. ^^^ Inductor Heating. — ^Let Fig. 197, represent the armature of a two-pole rotary converter. The direct-current brushes are dA. t\ and U are two tap inductors. U is the inductor midway between the two tap inductors t\ and U. The electromotive force induced in the phase between t\ and 403 404 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY ti is a maximum when the axis of the field bisects the angle sub- tended by the tap inductors h and h. This occurs when t„ lies on the field axis. The alternating current in all inductors between ii and <2 is the same at any instant, but it varies as the armature revolves. The phase of the voltage generated in the winding between inductors h and h is the same as the phase of the voltage gen- erated in the inductor to, which is midway between the two taps ti and fz. Therefore, for unity power factor with respect to the voltage generated in the winding between ti and <2, the current will be a maximum when f„ lies on the axis, R, of the field. The alternating current in all inductors between h and U will be zero when the current in <„ is zero. At unity power factor this will occur when to is under a direct-current brush. Fig. 198. The direct current in all inductors on the armature is the same in magnitude, but reverses in direction in each inductor as it passes under a direct-current brush. At unity power factor, the direct and alternating currents in the inductor to will reverse at the same instant. The two currents must be opposite in phase since one represents motor action and the other generator action. Neglecting the effect of the coils short- circuited by the direct-current brushes, the direct-current wave must be rectangular. The dotted lines in Fig. 198 show the direct and alternating currents carried by the inductor to when the power factor with respect to the generated voltage is unity. The full line shows the resultant current. The direct current in inductor ii reverses when ti passes under a direct-current brush, but the alternating current, assuming unity power factor, does not reverse until to passes under the SYNCHRONOUS CONVERTERS 405 brush. For a three-phase converter, this will occur 60 degrees later. Fig. 199 shows the resultant and component currents carried by ti at unity power factor in a three-phase converter. It can readily be seen from Figs. 198 and 199 that the root- mean-square currents in inductors, to and ti, are not the same. If the current lags behind the generated voltage, the alter- nating current does not reverse when f„ passes under a brush but reverses later. Considering the inductor to, the alternating current reverses later than the direct current and the angle of ' lag between the reversal of the two currents is the same as the angle of lag between the alternating current and the alternating Fig. 199. electromotive force in the inductor to- If the angle of lag were 60 degrees, the current relations for to would be the same as those for ti shown in Fig. 199, i.e., they would be the same as those existing at unity power factor in an inductor 60 degrees ahead of to. In general, the current relations produced in any inductor by a lagging current are the same as those which exist at unity power factor in an inductor which is ahead of the one considered by an angle equal to the angle of lag of the current behind the voltage. For leading currem, they would be the same as those in an inductor behind the one considered by an angle equal to the angle of lead between the current and voltage. 406 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Inductor Heating of an n-Phase Converter with a Uniformly Distributed Armature Winding. — Referring to Fig. 200, h and ti are taps. Co is a point on the armature midway between the two taps U and U. 2 a is the phase spread and is equal to 2 - where n is the number of taps. When there are more than two poles, n is the number of taps per pair of poles. The resultant current in any in- ductor such as Ci will be VS/'a^sin (A -/3 - fl) (139) where I'ac is the coil value of the alter- nating current, I'dc the direct current delivered per brush and B the angle of lag between the alternating current and the generated voltage in the coil Co. Each path between any pair of Fig. 200. brushes carries- of the total direct V current or one-half of the current delivered per brush, p being the number of poles. The average heating in the inductor ci during a cycle is proportional to the mean-square current or to T\ = ^ C/[^/2Z'„, sin (A _ /3 - e) - ^^]'dA (140) Replacing I'ac by its value in terms of I'i^ from equation (137), page 400, remembering that the current 7'^^ per brush is equal to the total d.c. current, Idc, divided by the number of pairs of poles gives 1 v/2 r = r M. ac ■*- t Pc he" 4 4Tr Ja = o n 4 sin (A — /? — e) _ (P-f-) (v)nsia^ dA n + 1 - 16 cos (|8 + 0) (p./.)(7))7rnsin- n (141) SYNCHRONOUS CONVERTERS 407 Since the first term in equation (141) is constant, the copper loss in any inductor, such as Ci, Fig. 200, will be a maximum when the last term has either its minimum positive or its maxi- mum negative value. It will be negative whenever /3 + 5 is greater than either + 90 degrees. It is obvious that under ordinary conditions the maximum copper loss will always occur at one of the tap inductors of each phase. At unit power factor the copper loss in all tap inductors will be the same. Under this condition, the minimum copper loss will occur in inductors midway between taps. Except in the case of single-phase converters, and these are never used in practice, the last term of equation (141) is not likely to be negative under commercial operating conditions since converters are never operated at low power factor. The power factor of a converter under load conditions is seldom allowed to get as low as 0.9. The ratio of the maximum to the minimum inductor heating in three-, four-, six- and twelve-phase converters for unity power factor and for a 90 per cent, power factor, for both lagging and leading current, are given in Table XIX. One hundred ■ per cent, efficiency of conversion is assumed. Table XIX Hatio of maximum to minimum inductor heating Number of phases Power factor = 1 Lagging-current, power factor = 0.9 factor = 0.9 1 6.6 7.4 7.4 3 5.3 8.1 8.1 4 3.6 6.8 6.8 6 2.2 4.9 4.9 12 1.3 2.8 2.8 The ratio of the temperatures of the hottest and coldest in- ductors will be much less than the ratio of the copper losses given in Table XIX on account of the tendency for the temperature of the inductors to become equalized by heat conduction through the end connections and across the armature teeth. The copper loss in inductors at different points on the armature of a converter is plotted in Fig. 201. All four curves are for the 408 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY same converter operated at a fixed total armature copper loss. The efficiencies and relative outputs for the conditions shown are given on the plots. 1 Three Phase (Tap InduotorB at / FoIntB niDrkedTon axle ol abBoisBse, Dotted Una Unl^ Power Factor Full Line 0.0 Power Factor Dashed Line Avetage loduotoi Copper LosB Total Inductor Copper Lohb Conatant. At Unit; Power Factoi Effioienc; = 93.7^ At Dnltj Power Factor Output = 1-0 At 0.0 Power Faotor Output =0.81 y / b / / f \ / / / i / \ / / / f / \ & 3 / / / \ / / \ s f \ y V / \ 1 \, N. y ^ \ ^\ ^-.^ ^^ ,^' ~^ -^ a i 5 © 10 50 60 70 80 90 100 110 120 130 UO 160 160 170 180 Klectrical Degrrees. Six Pha-^e Tap InduotorB at Points marked T Total Induutor Copper La ox axla of abaolBsae. B Conatant. f A Dotted Line Unity Power Factor Full Line 0.9 Power Factor / Daahed Line ATerage Induotor Copper Loib At 0.9 Power Factor Efficiency =94.3i6 Output = 1.09 A /■ Effiolanoy = 95.6^ / A Oo / i / \ / > \ / / '\, S / / i. / \ \, / / s / y / \ v / ^ X y- \/- \, V^ v^ ^' -V v^ . . — I ? (5 D q !) 10 20 30 10 SO 70 80 90 100 110 120 130 Electrical Degrees. 110 laO IGO 170 180 Fig. 201. Relative Outputs of a Converter Operated as a Converter and as a Generator. — The ratio of the copper loss in the armature of an n-phase converter to the copper loss in the same machine when operated as a direct-current generator is given by the ratio of average mean-square current carried by an armature inductor under the two conditions for the same direct-current output. This ratio is given by the following expression. Note. — The current per inductor of the direct-current gen- erator is lie _ I dc p - 2 SYNCHRONOUS CONVERTERS 409 "-Ct +1- 16 cos (jS + e) H = +1 - n {p.f.)r]Tn sin^ d0 (142) {v-f-Yn^n^ sin^- {p.f.)rrK^ sin- 8 {p-f-Yv^n^ sin^ + 1 26 (143) The ratio of the outputs for the same copper loss in the arma- ture is reciprocal of the square root of the ratio of copper losses for the same output. Therefore, Output of n-phase rotary 1 (144) Output of a direct-current generator The outputs of a converter compared with the output of the same machine as a direct-current generator are given by Table XX. Table XX Relative outputs a.c. to d.c aBSuming 100 per cent, efficiency Number of phases Unit power factor 90 per cent, power factor 1 0.85 0.74 3 1.33 1.09 4 1.65 1.28 6 1.93 1.45 12 2.18 1.58 OC 2.29 1.62 The gain in output by increasing the number of phases de- creases rapidly as the power factor decreases. If converters were operated at low power factors, little would be gained by increasing the number of phases. It is seldom, however, that the power factor of a converter in commercial operation will be as low as 0.9. The decrease in output with power factor for three- and six- 410 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY phase converters is shown by Table XXI in which the output of the converters as direct-current generators is taken as unity. Table XXI Power factor in per Relative outputs, a.o. to d.c. assuming 100 per cent., efficiency cent. three-phase six-phase twelve-phase 100 1.34 1.93 2.18 95 1.20 1.65 1.83 90 1.09 1.45 1.58 85 0.99 1.28 1.38 80 0.90 1.14 1.22 1.7 1.6 fl.5 O 1.1 1.3 1.2 1.1 1.0 0.0 \, \ \ \ \ \ \ s \^ \ s ^t ^ k ^ \ s. * s \ \ ^v. -v'-A « '\ V ^^ ^» \ "^ "^ -^ 100 98 96 91 92 90 Power Factor Fig. 202. The results given in Table XXI are shown plotted in Fig. 202. The outputs at different power factors expressed in per cent, of the output at unity power factor are plotted in Fig. 203. From Fig. 203 it will be seen that, for a fixed armature copper loss, the percentage decrease in output produced by a decrease in power factor increases slightly as the number of phases is increased. The difference between the temperature of the hottest and the coldest inductors in the armature of a rotary converter will be less than the difference between the copper losses in these SYNCHRONOUS CONVERTERS 411 inductors on account of the equalization of temperature by- conduction through the end connections and armature core. In spite of this tendency to- equalization, there will still be con- siderable difference between the temperature of the hottest and coldest inductors under operating conditions. This difference will have to be considered when determining the proper rating for a converter. Since the difference in temperature decreases with increasing number of phases, converters can more safely be given ratings which are determined by their average inductor heating as the number of phases is increased. For this reason the actual gain in output by increasing the number of phases is greater than that indicated in Table XX. It is possible to d) OO v N \: \ N ^ s. \ k ^ N*J T<^ ^ ^ ^ 3 \ \ "-^ h •\, \ \J ^ s. V < 100 98 9S 91 Power Eactor Fig. 203. equalize in some degree the slot heating by making use of frac- tional-pitch windings. Pitches which differ much from full pitch cannot, however, be used on account of commutation difficulties. By the choice of a proper pitch, it is possible to make the slot heating of a twelve-phase converter at unit power factor almost uniform. Efficiency. — Since the output of a rotary converter for given losses, is greater than the output of the same machine operated as a generator, it follows that the efficiency of a machine when operated as a converter is greater than when operated as a generator. Table XXII gives the armature efficiencies and outputs at 412 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY unity power factor of a rotary converter, when operated with different numbers of phases and as a direct-current generator. These efficiencies and outputs neglect the increase in commutator friction losses and commutator losses as the number of phases is increased. This increase will be to some slight extent offset by the decrease in the local core losses in the armature and pole faces. No account is taken of the effect on the outputs of the uneven distribution of the armature copper loss. The output and the efficiency for the direct-current generator are taken as 100 per cent, and 92 per cent., respectively. Table XXII Machine Output in per cent. Efficiency in per cent. Direct-current generator 100 133 165 193 218 92.0 Three-phase rotary converter Four-phase rotary converter .... 93.7 94 7 Six-phase rotary converter 95 6 Twelve-phase rotary converter 96.0 The increase in the efficiencies of converters of the same rated output would not be so great as the increase shown by Table XXII. For Example. — Compare the efficiency of a 500-kw. six-phase converter with a 500-kw. generator of the same speed. The converter would have an output as a generator of Yn^ 500 or of about 250 kw. A 500-kw. generator would have an armature efficiency of about 95.2 per cent. The armature efficiency of a 250-kw. generator should be about 90.5 per cent. Assuming that the 500-kw. converter, when operating as a generator, has an armature efficiency of 90.5 per cent., its armature losses as a converter would be r^o 4.9 per cent. Its armature efficiency as a converter would be 95.1 per cent, or substantially the same as the efficiency of the 500-kw. generator. Although the efficiency of a converter may not be greater than the efficiency of a generator of the same rating, the efficiency of the converter will be greater than the overall efficiency of the generator and motor required to drive it. SYNCHRONOUS CONVERTERS 413 Whether the cost of a converter per kilowatt of rating is de- creased by increasing the number of phases, depends upon the relative cost of the labor and the material used in its construc- tion. The ratio between labor and material costs increases rapidly as the output is decreased and below outputs of 100 or 200 kw., the cost of adding extra slip rings and increasing the overall length to provide for these usually more than offsets the saving in material in other parts of the converter. Twelve- phase converters will probably not be economical to construct except in very large sizes. In addition to the expense of adding extra slip rings, there will also be a slight increase in the cost of the transformers in some cases. Transformers for three-phase y or A connection and six-phase diametrical connection ought to cost substantially the same, as they differ only in the magnitude of their secondary voltages. Transformers for double A and double Y and for twelve-phase connection require two secondary coils and would, therefore, be slightly more expensive than trans- formers for three-phase or six-phase diametrical connection. The difference in cost, however, would be small. CHAPTER XXXVIII Armature Reaction; Commutating Poles; Hunting; Methods of Starting Converters Armature Reaction. — For convenience in considering the arma- ture reaction of a rotary converter, let the armature current be divided into four components, namely: (a) The direct current. (6) The component, I,, of the alternating current, which is in quadrature with the generated voltage. (c) The component, Ii, of the alternating current which is opposite in phase to the generated voltage and which supplies the rotational losses. (d) The remainder, le, of the alternating current. This is opposite in phase to the generated voltage and is the component which is effective in producing the direct-cur- rent output. le is the alternating current the converter would carry at unity power factor if the efficiency were 100 per cent. Assume a converter with p poles and N uniformly distributed armature turns. The turns per pole and the direct current N Id I'dc, per conductor, will be, respectively, — and — °-. The V P ampere-turns per pole per elementary angle dtp on the armature are Ido N d

600 ! 500 S 450 in a I 400 u S '^ 350 60 300 -2 *. 250 d O 200 B 150 100 .60 E 1( ) 15 20 2S 30 ^ ^ ^ / y ^ ^ / / / ^\ V Is • ormal )ix'ect-Ourreat Voltage PI a'^ f Ci -iVfl.fl^ .^ ^SSi ^ ii ^ o5^ ■i/ .' f z* / f / ' 1 1 "^ Y / / . / / ' 10 600 V Rota OKw., )lt8 D.C y Conv ;o cycle .,1667 A !rter 1 12 PoU nperes D.C. - 1 / / 1 // f 10 12 14 Field Amperes, Fig. 211. 16 18 20 22 producing distortion. The ampere-turns corresponding to it, therefore, add directly to or subtract directly from the SYNCHRONOUS CONVERTERS 439 excitation of the shunt and series fields. In the case of a con- verter delivering direct current, a reactive lagging component of the alternating current strengthens the field. The reactive component of a leading current weakens the field. The resultant or net ampere-turns of excitation for any terminal voltage under load conditions are approximately equal to the ampere-turns necessary to produce the required voltage when the converter is driven at no load as a generator. The efficiency of a rotary converter operating at a power factor in the neighborhood of unity is always high at full load. For a large converter it is usually 95 per cent, or better. On account of this high operating efiiciency, it is usually close enough to assume the efiiciency to be 95 per cent, when calculating the armature reaction caused by the reactive component of the alternating current. The field excitation for the 1000-kw. converter will be cal- culated for a full direct-current load and a power factor of 0.95 with a leading current. The coil alternating current, I'ac, may be found from equation (137), page 400. The coil current is the same as the inductor current. 2 Vdc = /, dc Pnip.f.)v Vac J 1000 X 1000 Idc = gQQ = 1667 amp. Assuming the efficiency and power factor each to be 0.95 ^'- = 1667 12 X 6 X 0.95 X 0.95 ^ = '^' "'"P" sin - n The reactive component of this current is h = USVl - (0.95)2 = 45.3 amp. The armature reaction, Ax, per pole for this current may be found from equation (10), page 59, provided the breadth factor is added to the equation where kb is the breadth factor. 440 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The phase spread of a six-phase converter is 60 degrees or one-third of the pole pitch. The converter has 180 slots and 12 180 poles, or ^-TTs = 5 slots per phase per pair of poles^ '^ ' o X From Table I, page 41, the breadth factor for a spread of 60 degrees and four slots per phase is 0.958. For five slots per phase it would be about 0.957. 1 Sin V fi A. = 0.707 X 0.957 X ^ x 12 ^^"^ = 1380 ampere-turns per pole. These are demagnetizing ampere-turns since a leading current was assumed. The ampere-turns per pole due to the series field are 1667 X 2 = 3334 The field current required for 600 volts when the converter is driven at no load as a direct-current generator is 9.25 (open- circuit saturation curve, Fig. 211). This corresponds to 9.25 X 864 = 7990 ampere-turns per pole. The shunt excitation required under full-load conditions at a power factor and efficiency each of 0.95 and with a leading current is 7990 + 1380 - 3330 = 6040 ampere-turns. This corresponds to a shunt-field current of 6040 . __ gg^ = 6.99 amp. Efficiency. — The efficiency is IdcVdc '' ~ licVac + Hlic^nc + LK.Vic + /cV. -f P. + (i? + TF) where lie = Direct current. Vie = Direct-current voltage. r^c = Armature resistance between direct-current terminals. I,h, = Shunt-field current. SYNCHRONOUS CONVERTERS 441 • Ic = Compound-field current. Tc =5 Resistance of compound winding. Pc = Core loss. F + W = Friction and windage loss. The armature copper loss may be found by multiplying the copper loss corresponding to the direct-current component of the armature current by the ratio of the copper loss of the converter as a converter to its copper loss at the same output as a direct- current generator. This ratio, H, may be found from equation (143), page 409, H = '- +l-i^ For a power factor of 0.95 and an assumed efficiency of 0.95 fl- ^ 8 16 (0.95)2(0.95)2(6)^0.5)2 ^ (3.142)2(0.95) = 0.385 The armature resistance at 75°C. between direct-current terminals is 0.00589 (1 + 50 X 0.00385) = 0.00702 ohm. The armature copper loss is /■fc2 Xr^X 0.385 = (1667)2 X 0.00702 X 0.385 = 7510 watts. The ohmic resistance is used in finding the armature copper loss. This loss is small and the error introduced by using ohmic resistance in place of effective is not great. Since the armature inductors of a converter carry differently shaped current waves, the ratio of ohmic to effective resistance would not be the same for all inductors. It would also change with power factor. The shunt-field loss including the loss in the field rheostat is equal to the shunt-field current multiplied by the voltage across the direct-current brushes. This voltage is equal to the terminal voltage plus the drop in the series field. The drop in the series field will be neglected. The shunt-field loss is, therefore, 6.99 X 600 ^ 4194 watts. 442 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The resistance of the whole series field at 75°C. is • .0.000610 X (1 + 50 X 0.00385) = 0.000728 ohm. The series-field Ipss is (1667)2 0.000728 = 2025 watts. The core loss from Fig. 211 corresponding to a direct-current voltage of 600 is 14,700 watts. The efiiciency = 1000 '' ~ 1000 + 7.5 + 5.0 + 2.0 + 14.7 -|- 8.1 ~ ^*'"* ^^^ POLYPHASE INDUCTION MOTORS CHAPTER XLIII Asynchronous Machines; Polyphase Indtjction Motor; Operation of the Polyphase Induction Motor; Slip; Revolving Magnetic Field; Rotor Blocked; Rotor Free; Load is Equistalent to a Non-inductive Re- sistance ON a Transformer; Transformer Diagram of a Polyphase Induction Motor; Equivalent Circuit of a Polyphase Induction Motor Asynchronous Machines. — Up to this point, only machines which operate at synchronous speed have been considered. There is, however, another"^tess known as asynchronous machines. As their name implies, these do not operate at synchronous speed. Their speed varies with the load and may or may not be influenced by the frequency of the circuit to which they are connected. For motors of the series or repulsion types the speed is not so influenced. One distinguishing feature of all com- mercial synchronous~madiines is that they require a field of constant polarity excited b^t^^rect current. Such a field does not exist in an asynchronous machine. Both parts of an asyn- chronous machine, L^Ats armature and field, carry alternating current and are either connected in series, as in the series motor, or are inductively related, as in the induction motor. The induction motor and generator, the series and repulsion motors and all forms of alternating-current commutator motors are included in the general class known as asynchronous machines. The induction motor is probably the most important and most widely used type of asynchronous motor. It has essentially the same speed and torque characteristics as a direct-current shunt motor and is suitable for the same kind of work. Its ruggedness 443 444 PRINCIPLES OP ALTERNATING-CURRENT MACHINERY and ability to stand abuse make it a particularly desirable tjfpe of industrial motor. Polyphase Induction Motor. — The induction motor differs from the synchronous motor in that the current in its armature, which is usually the revolving part, is produced by electromagnetic induction while in the synchronous motor it is produced by con- duction. The polyphase induction motor is exactly equivalent to a static transformer on a non-inductive load. It is a trans- former with a secondary which is capable of rotating with respect to the primary. Although the secondary is usually the rotating part, the motor will operate equally well if the secondary is fixed and the primary revolves. In what follows, the primary will be assumed stationary and will be referred to as the primary, the stator or the field. The secondary, which in this case will rotate, will be called the secondary, the rotor, or the armature. The terms primary and secondary are perfectly definite, meaning respectively the part which receives power directly from the mains and the part in which the current is produced by electro- magnetic induction. The terms stator and rotor are not so definite, since their significance is not determined by the electrical connections, but merely by the particular part which is stationary. Operation of the Polyphase Induction Motor. — The stator winding of a polyphase induction motor is similar to the armature winding of a polyphase alternator. This winding produces a rotating magnetic field which corresponds to the armature reac- tion of the alternator. As with the armature reaction of an alternator, the fundamental of this field revolves at synchronous speed with respect to the stator. With respect to the rotor it revolves at a speed which is the difference between the synchron- ous speed and the speed of the rotor. This difference is known as the slip. A portion of the stator of an induction motor with a few coils in place is shown in Fig. 212. The rotor winding will have as many poles as the stator and will have currents induced in it by the revolving magnetic field. These currents will cause the rotor to revolve in the same direc- tion as the magnetic field set up by the stator. If it were not for rotational losses, synchronous speed, would be reached at no load. Under load conditions, the difference between the speeds of the magnetic field and of the rotor will be just sufficient to POLYPHASE INDUCTION MOTORS 445 cause enough current to be induced in the rotor to produce the torque required for the load and to overcome the rotational losses. The speed of the revolving magnetic field depends upon the frequency and the number of poles for which the motor is wound. It is entirely independent of the number of phases. The only condition which must be fulfilled in regard to the number of Fig. 212. phases is, that the space relations of the windings for the different phases in electrical degrees must be the same as the time-phase relations between the currents they carry. Thus for a three- phase winding, they must be 120 electrical degrees apart. For a four-phase winding, they must be 90 electrical degrees apart. Slip. — If /i and p are, respectively, the impressed frequency and the number of poles, the speed of the revolving magnetic ; 446 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY field and also the synchronous speed of the motor in revolutions per minute is m = ^ 60 (149) P The actual speed of the rotor will be less than this and is 712 = ni(l - s) (150) where s is the slip expressed as a fraction of synchronous speed. Revolving Magnetic Field.^ — Assume the rotor to be on open circuit. This corresponds to the condition in a static trans- former when the secondary is open. The only magnetomotive forces acting in this case are the magnetomotive forces produced by the primary windings. The primary winding of an induction motor is distributed and is similar to the armature winding of an alternator having the same number of phases and poles. At any instant, the space distribution of the flux caused by any one phase will be determined by the distribution of the winding. The air gap of an induction motor is uniform and, except for the presence of the slots, does not affect the flux dis- tribution. The space distribution of the flux set up by the stator will be more nearly sinusoidal as the number of slots per phase is increased. This distribution may be found by the method indicated on page 84 in the section on Synchronous Gen- erators. The time variation of the air-gap flux due to any one phase may or may not be sinusoidal depending upon the wave form of the impressed voltage. If the space distribution of the flux produced by each stator phase is sinusoidal, the fundamentals of the time variation of the air-gap flux for all phases combined will produce a revolving magnetic field revolving at synchronous speed, constant in value and sinusoidal in its space distribution. The flux due to any one phase is oscillatory. As in the trans- former, it induces a voltage which is equal to the voltage im- pressed on the phase less the impedance drop due to the resistance and leakage reactance of the primary winding. Except as this induced voltage is influenced by the impedance drop, it will be of the same wave form as the impressed voltage and the magnetizing current must adjust itself to meet this condition. If the im- pressed voltage is sinusoidal the induced voltage will be very POLYPHASE INDUCTION MOTORS 447 nearly sinusoidal, since the impedance drop is small. If the impressed voltage contains harmonics, the induced voltage will contain the same harmonics for the same reason. Fig. 213 shows the developed stater of a three-phase induction motor. The dots represent inductors and the numbers indicate the phases to which the inductors belong. The full line, the dotted line and the dot-and-dash line show, respectively, the fundamentals of the space distribution of the fluxes produced in the air gap by phases 1, 2 and 3 at the instant when the current in each phase has its maximum positive value. Consider a point 6, situated a electrical degrees from the beginning of phase 1. The flux density, (Zi, at this point is (Rb = (Bi sin a + (&2 sin (a - 120) + (B3 sin (a - 240) (151) Fia. 213. where (Bi, (B2 and (B3 are the flux densities at the centers of phases 1, 2 and 3, respectively, at the instant considered. If only the fundamental of the time variation of the flux is considered, equation (151), may be written (Sb = (S.m {sin a sin wt + sin (a - 120) sin (cot - 120) -I- sin (a - 240) sin (wt - 240) } = %(S>m cos {a — cat) = % (B™ sin {cct + l- a) (152) Equation (152) shows that the flux density at any point such as b is sinusoidal with respect to time. It also shows that at any given time, i.e., for any fixed value of t, the space distribution of the air-gap flux is also sinusoidal. 448 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY If a, equation (152), equals ut (Rb = /i (S>m Sin 2 or the magnetic field travels about the air gap at synchronous speed and has a constant value. If w is the thickness of the stator core and A is the pole pitch in centimeters, the fluxes T + 120° ^2 =%l«(Bm— I sin(a)< + o - «) da = 3u)(B„ - sin (co< - 120°) (154) TT .■>,r + 240<' ^Js =Hw(&,n — I sin (a)< + 2 — a) (ia — I sir TT I 1/240" = 3u)(B,„ - sin (oji - 240°) (155) It may be seen from equations (153), (154) and (155) that the total flux through each phase is sinusoidal with respect to time and that the total fluxes linking the phases differ in time phase by 120 degrees. Rotor Blocked. — The fundamental of the flux due to each phase induces a sinusoidal electromotive force in the rotor. If the rotor circuits are closed, polyphase currents will be induced in them of the same frequency as the primary currents provided the rotor is blocked. In this case the conditions are those of a short-circuited transformer. The ciu*rents in the rotor have the same frequency as the primary or stator currents and react on the stator in exactly the same way as the secondary current of a static transformer reacts on the primary. They cause an equiva- lent load-component current in the stator windings. This load- component current and the secondary current are opposite in phase and their ratio is equal to the ratio of the effective turns POLYPHASE INDUCTION MOTORS 449 per phase in the rotor windings to the effective turns per phase in the stator windings/ If Ni and iV2 are the effective turns per phase in the stator and rotor windings, respectively, and I'l and I2 the load com- ponent of the stator and rotor current, respectively, h ~ Ni~ a where a is the ratio of transformation. The rotor current will lag behind E2, the induced voltage in r2 the rotor winding, by an angle whose cosine equals / ^ = where rz and X2 are the rotor resistance and the rotor leakage reactance, per phase, at stator frequency. The vector diagram for a polyphase induction motor with rotor blocked is ex- actly the same as that for a short-circuited transformer. The magnetizing component of the stator current and the stator and the rotor reactances, xi and X2, are larger for the motor due to the air gap between stator and rotor windings. The rotor current, considered with respect to the revolving magnetic field, produces a torque which acts in the direction of rotation of the magnetic field. If the rotor is free to revolve, it will speed up. Rotor Free. — When the rotor is blocked, the speed of the stator field with respect to the rotor inductors is proportional to the primary frequency. When the rotor revolves, the speed of the stator field with respect to the rotor inductors is equal to the difference between the speed of the field in space and the rotor speed. This relative speed is ris = ni — Hi where ni and ^2 are the speeds of the stator field and the rotor, respectively. Replacing m and ni by their values from equations (149) and (150), page 446, n, = — 60s P The frequency of the rotor currents corresponding to the speed n» is /. =/lS 29 450 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The rotor currents at a frequency /is produce a rotating mag- netomotive force in the rotor. This revolves at a speed n, = ^ 60s V with respect to the rotor. Since this magnetomotive force is revolving in the same direc- tion as the rotor, its speed with respect to the stator is equal to its speed with respect to the rotor plus the speed of the rotor itself, or to n,-^n2= '^ 60s -]- ^ 60 (1 - s) = ?^60 V Its speed with respect to the stator is the same as the speed of the stator field in space. The frequency of the rotor current If^. h^i I,N^ Fig. 214. considered with respect to the stator current is the impressed frequency, /i, of the circuit. The rotor current, therefore, re- acts on the stator current at this frequency whatever the speed of the rotor. The resultant magnetomotive force causing the aii'-gap flux is equal to the vector sum of the magnetomotive forces of the stator and rotor currents. The condition is the same as in a static transformer. Fig. 214 shows the relation existing between these magnetomotive forces. The letters on Fig. 214 have the following significance: ' 7i = Stator current. h = Rotor current. Ni = Number of effective turns per phase on the stator. Ni = Number of effective turns per phase on the rotor. POLYPHASE INDUCTION MOTORS 451 The flux, tp, corresponding to the magnetizing component, I^, of the stator current, 7i, induces a voltage Ei in the stator aild a voltage EiS in the rotor. E2S has a frequency of fis with respect to the rotor but a frequency of /i with respect to the stator. The secondary current, h, corresponding to the voltage E2S is where X2 is measured at primary frequency. At a frequency /2= fis, the secondary reactance is xzs. /2 may be equally expressed by h = , . ^: (157) n/(?) 2 + X2' This form will be used later with the vector diagram. E^, — and o Xi in equation (157) are all referred to the stator and when so referred are at stator or impressed frequency. li is also at stator frequency when so referred. The resistance, —, is the apparent resistance of the rotor when referred to the stator. Ei is the voltage which would be induced in the-rotor by the flux

x EJl'-s-) — >l Fig. 215. Fig. 215. Everything on this diagram is per phase and is referred to the stator. The relative positions of the vectors on the secondary side of the diagram may be changed to correspond to their usual positions on the ordinary transformer diagram as indicated in Fig. 216. hR corresponds to the potential difference, V2, at the secondary terminals of a transformer. Equivalent Circuit of a Poljrphase Induction Motor. — The con- ditions of the vector diagram are exactly those of the circuit shown in Fig. 217. This diagram shows what is known as the equivalent circuit of the induction motor. This circuit is in reality the POLYPHASE INDUCTION MOTORS 453 equivalent circuit of a transformer which supplies power to a non-inductive load, R. Everything in the equivalent circuit is referred to the primary or stator. For example, r^ on Fig. 217 is the actual secondary resistance multiplied by the ratio of trans- formation squared where the ratio of transformation is obtained with the rotor blocked. The susceptance and conductance 6„ and g„ are such that In = Ei{gn — jbn) With the ordinary transformer, little error is introduced into calculations based on the equivalent circuit if the portion of the circuit represented by 6„ and gn be placed directly across the impressed voltage. When this change is made in the equivalent diagram of an induction motor, the error introduced is much greater, since the exciting current, /„, of an induction motor is large com- pared with the load component, I'l, of the stator current. The reactance, Xi, of an induction motor is also much larger than the reactance of the primary winding of a transformer, chiefly on account of the air gap. The approximate equiva- lent circuit of the induction motor is given in Fig. 218. The —E-^- FiG. 216. I I A Fig. 217. use of this circuit will generally introduce a nearly constant error of about 5 per cent, in the induced voltages Ei and E2 between no load and full load. The power and the torque corresponding to any given slip vary as the square of E2, and the error in these quantities introduced by the use of the approxi- mate circuit may, therefore, be as high as 10 per cent. 454 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY. Since everything on Fig. 218 is referred to the primary, Z'l = Zj and Ii = In -\- h vectorially. The use of the true equivalent circuit for purposes of calcula- tion can be considerably simplified by dividing the impedance drop in the primary into two components, one, produced by the exciting current, In, and the other, by the load component, I'l. Under ordinary conditions, the drop due to the exciting current will subtract almost directly from the impressed voltage. It may Fia. 218. be assumed constant without introducing any great error in the value of the induced voltage, Ei. According to this assumption ^1 = Fi - ZnVnM^ - I'liri + jxi) = V\-I\in+jxO where F'l is a constant voltage obtained by subtracting In -v/j"!" + Xi^ directly from Vi. The error in Ei produced by this assumption ought not to exceed 2 per cent, from the condition of no load to that where the rotor is blocked. CHAPTER XLIV Ettect of Harmonics in the Space Distribution of the Air-gap Flux Effect of Harmonics in the Space Distribution of the Air-gap Flux. — Thus far only the fundamental of the space distribution of the flux due to each phase has been considered. The harmonics in the time variation of the air-gap flux were also neglected. This was equivalent to assuming that both the space distribution and the time variation of the air-gap flux were sinusoidal. The voltage induced by the air-gap flux in the primary winding must be equal at every inStant to the primary impressed voltage minus the primary leakage impedance drop. The wave shape of the air-gap flux must adjust itself to meet this condition. It follows, that if the impressed voltage is sinusoidal, the time variation of the air-gap flux will also be sinusoidal except in so far as it may be slightly affected by the small resistance and leakage drops in the primary windings. The space distribution of the flux cannot be exactly sinusoidal with any possible distribution of the primary winding, but it approaches this form as the number of slots per phase and the number of phases are increased. The presence of the stator and the rotor slots will introduce small harmonics into both the time variation and the space distribution of the air-gap flux but these will have relatively little effect. For the present neglect all harmonics in the time variation of the air-gap. flux. Under this condition, all of the harmonics in the space distribution of the flux, when considered with respect to any phase of the stator winding, have fundamental frequency with respect to time. They can, therefore, induce only electro- motive forces of fundamental frequency in the stator winding. The fundamental of the space distribution of the flux induces currents in the rotor which react to diminish the flux that pro- duces them. In a similar way, the harmonics in the space 455 456 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY distribution of the flux induce currents in the rotor which also react to diminish the harmonics in the flux causing them. These currents will not be true harmonics of the rotor currentj since the ratios of their frequencies to the frequency of the fundamental of the rotor current cannot, under ordinary conditions, be integers. All the odd harmonics with some exceptions occur in the space distribution of the air-gap flux. In a three-phase winding, the third harmonic in the three phases cancel. In a six-phase wind- ing, the third and fifth harmonics cancel. In general, the possible harmonics may be expressed by p = 2xm ± 1 (160) where p is the order of the harmonic, m the number of phases and X any integer. In the case of a three-phase winding, the first harmonic which can occur is the fifth. The rotating field due to the fifth har- monic turns in the opposite direction to the field produced by the fundamental. The field due to the seventh harmonic turns in the same direction as that due to the fundamental.' In general, the fields caused by harmonics of the order p = {2xm + 1) turn in the same direction as the field due to the fundamental. The fields due to harmonics of the order p = {2xm — 1) turn in the opposite direction to the field due to the funda- mental. The speed of these fields is 111 rii ■ , ^ "^ = 7 = 2^^^^^l (161) where rii = is the speed of the field produced by the fundamentals of the stator flux.^ ' Section on Synchronous Generators, page 47. " The number of poles produced by any harmonic in the space distribution of the flux is equal to the number of poles produced by the fundamental multiplied by the order of the harmonic. The frequency of the harmonic and fundamental are the same. POLYPHASE INDUCTION MOTORS 457 The harmonics in the stator field induce electromotive forces in the rotor. The frequencies of the electromotive forces cor- responding to the harmonics of the order {2xm + 1) are , _ (2xm + l)p , /r,(2lm+l) — 2(60) U'-i^xm+l) — W2J where /r,(2im + i) is the frequency of the harmonic induced in the rotor by the {2xm + l)th harmonic of the primary field and n2 is the actual rotor speed. n(^2xm+i) is the speed with respect to the stator of the rotating field due to the (2xm + l)th harmonic. Replacing n(2im+i) = n,, by its value from equation (161) gives V 1 In a similar way, the harmonics of the order (2xm — 1) induce electromotive forces in the rotor of frequencies P 1 These harmonics rotate in a direction which is opposite to that in which the rotor turns. Remembering that p = {2xm + 1), equa- tions (162) and (163) may be combined into a general equation which is pi _ The slip of the rotor with respect to any harmonic in the stator of the order p is Replacing n^ by its value from equation (161) gives (165) Replacing nz in equation (165) by wi(l — s), where s is the slip of the rotor with respect to the fundamental of the flux, gives s, = 1 + p(l - s) (166) '[w(2im + i) — rii] is the slip and {2xm + l)p is the number of poles corresponding to the {2xm. + l)th harmonic. 458 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY From equation (166), it is obvious that the slip of the rotor with respect to the fifth harmonic, which turns in the opposite direc- tion to the fundamental in the case of a three-phase winding, is SB = 1 + 5(1 - s) = 6 - 5s If the rotor is driven at synchronous speed with respect to the fundamental of the stator field, s will be zero. Under this condition, the rotor slip will be six times the speed of the field due to the fifth harmonic. The ratio of the frequency of the rotor electromotive force, j&2,p, caused by any harmonic of the order p to the frequency of the electromotive force induced in the rotor by the fundamental is (equation 164) Jr,l /r,l Replacing (rii + pn^) by its value from equation (165) and P 1 remembering that o" «fj ^i = /i ^^'^ /'■.i ~ ®/i gives /r,p _^ Sftfl fr.l Sfl By substituting the value of s^ from equation (166) this becomes / ^ L+_(i_-_^p ^jg^^ /r,l S This ratio will be an integer only in exceptional cases. No simple relation exists between the frequencies of the currents induced in the rotor by the fundamental of the stator field and by the harmonics of that field, even when the time variation of the stator flux due to any phase is assumed sinusoidal. The rotor current, therefore, cannot be resolved into a fundamental and a series of harmonics and is not a periodic current in the ordinary understanding of the term. If the time variation of the stator flux is not sinusoidal, the flux may be resolved into a fundamental and a series of harmonics. The effect of these harmonics in producing currents in the rotor will be similar to the effect of the harmonics in the space POLYPHASE INDUCTION MOTORS 459 distribution of the stator or air-gap flux. The harmonics in the time variation of the flux induce currents in the rotor, but the frequencies of these currents do not have integral relations among themselves and their relative phases will be continually changing. The relation between the frequencies of the currents caused by the harmonics and by the fundamental of the time variation of the flux are the same as given by equation (167). It is, therefore, useless to attempt to represent the secondary current by any definite curve since its wave form changes from instant to instant. If the wave form of the current in the rotor were obtained by a contact method, the instantaneous values of only that part due to the fundamental of the flux would be constant for any setting of the contact device, and alone would be recorded. The parts due to the harmonics would vary progressively from instant to instant, since the contact device would close the circuit at progressively different points on their waves. Their average over any reasonable length of time would, therefore, be zero. Certain of the harmonics in the air-gap flux will tend to dimin- ish slightly the torque developed by the motor. The air-gap flux caused by harmonics of the order p = {2xm — 1) in the space distribution of the flux of each phase rotates in the opposite direc- tion to the flux due to the fundamentals. The torque produced by these harmonics will be in the direction of their motion and will, consequently, oppose the main torque of the motor. In the case of a three-phase motor, the harmonics in the air-gap flux which can produce this diminution in torque are the 5th, 11th, 17th, etc. Due to the large slip and the high rotor reactance with respect to these harmonics, their effect on the torque developed by the motor will be small. In what follows only the fundamental of the revolving field due to the stator windings will be considered. CHAPTER XLV Analysis of the Vector Diagram; Internal Torque; Maxi- MTTM Internal Torque and the Slip Corresponding Thereto; Effect of Reactance, Resistance, Impressed Voltage and Frequency on the Breakdown Torque and Breakdown Slip; Speed-torque Curve; Stability; Start- ing Torque; Fractional-fitch Windings; Effect of Shape of Rotor Slots on Starting Torque and Slip Analysis of the Vector Diagram. — Refer to the vector diagram of the induction motor, Fig. 215, page 452. The power input to the motor or the stator power per phase is Pi = 7i/i cos e]^ (168) Resolving the impressed voltage, Vi, into its components) Pi = (El + Iixi + hn) 7i cos el' = EJi cos ef/ + (/ixi)7i cos 2 + iliri)li cos = Eili cos dj' + + stator copper loss. The expression for Pi may be further expanded by replacing h by its components. Pi = Eiil'i + I^ + Ih + e) cos dj^' + + stator copper loss. = EJ'i cos df'\ + Eil^ cos I -f- Eih+e cos + -t- stator copper loss. = Eil'i cos 9j>\ -|- 4- core loss + -|- stator copper loss. Eil'i cos 9i'\ is the power transferred across the air gap to the rotor by electromagnetic induction and is the total rotor power, P'a. P'i = EJ'i cos e^>\ = Eih cos ef,' (169) 460 POLYPHASE INDUCTION MOTORS 461 If E2 is resolved into its components, the expression for P'2 becomes P'i = [hr2 + I7.X2S + £^2(1 - s)]li cos N X \ ^^ ^ ^ N| \ / \ \ \^ ^^ y X \ V \'(» *\ >^ y \ N^* \ p 1 ^ N W r \ \ ^ '% \ \ Thr !6-Pha eindu itionK otor X N, \\ li is the in the rOtal re rotor 1 ristanc ircuit. ) s ^ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Slip Fig. 219. its value from equation (174), page 462, and reniembering that s = 1 at starting gives (180) ^" = iiF' That is, the starting torque is proportional to the copper loss in the secondary or rotor circuit. This torque for any fixed im- pressed voltage may be increased up to a certain maximum value by increasing the resistance r2 of the rotor circuit. It makes no difference whether the increase in resistance is obtained by actu- ally increasing the rotor resistance or by putting external resist- so 466 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY ance in series with the rotor windings. The starting torque will be a maximum when the constants of the motor are such as to make the slip unity in equation (177), page 463. From equation (177) for maximum torque at starting r^' = n^ + (xi + x^y (181) By properly adjusting r^ the maximum torque may be made to occtu: at starting, but for this value of resistance, the slip under normal running conditions will be large and the efficiency low. One curve on Fig. 219, page 465, is drawn for that value of r^ which gives maximum torque at starting. It will be seen from Fig. 219, that the portions of the speed- torque curves between maximum torque and synchronous speed are approximately straight lines. Therefore, when the maximum torque is made to occur at starting by increasing the rotor resistance, the slip at which full-load torque occurs is approxi- mately equal to the ratio of full-load torque to maximum torque. Under this condition both, the speed regulation and the efficiency are very poor. For best running condition, r^ should be as small as possible. For best starting torque, it should be large. In any motor, a compromise must be made between these two requirements. By proper design, it is possible to obtain good speed regulation with sufficiently satisfactory starting torque. When large start- ing torque is required, r^ must temporarily be increased by inserting resistance in the rotor circuit. This resistance is cut out when the rotor is up to speed. Fractional-pitch Windings. — In order to obtain good operat- ing characteristics, it is desirable to make the reactances of induction motors low, equations (176), (179) and (180). For this reason, fractional-pitch windings generally are used for both the stator and rotor windings. Fractional-pitch windings reduce the amount of copper required, due to the shortened end connections and consequently decrease the reactance^ and resistance. By distributing the coils of the rotor and stator in a greater number of slots, the effect of more slots per phase is obtained. ^ Section on Synchronous Generators, page 80. POLYPHASE INDUCTION MOTORS 467 Effect of Shape of Rotor Slots on Starting Torque and Slip. — By proper shaping of the rotor slots and also of the inductors much can be accomplished in increasing the starting torque without sacrificing good speed regulation. If deep, narrow rotor slots with low-resistance inductors and end connections are used, the rotor resistance at standstill may be made several times greater than its resistance under normal running conditions. The increase in the apparent resistance at standstill is due in part to the local losses set up by the slot leakage, but the chief cause of the increase is the tendency of the slot-leakage flux to force the current toward the top of the inductors. If the inductors are considered to be divided into horizontal elements similar to the elements, dx and dy, shown in Fig. 41, page 67, the linkages with these elements due to the slot leak- age wUl increase in passing from the top to the bottom of an in- ductor, causing the leakage reactance of the lower elements to be higher than the leakage reactance of those above. As a result, the current will not be distributed uniformly over the cross-section of the inductors but will be forced toward their upper portions producing an apparent increase in their resist- ance. The effect is the same as the ordinary skin effect of circular wires but is much more marked for the motor. The reactance of these elements, and consequently the apparent increase in the resistance, is dependent upon the frequency. At starting the frequency of the rotor current is fi. At any slip, s, it reduces to /is, and at full load it has from 2 to 10 per cent, of its starting value according to the size and type of the motor. Due to the decrease in the local losses and in the skin effect with decreasing frequency, the resistance of the rotor when running may be much less than at starting. CHAPTER XLVI Rotors, Number of Rotor and Stator Slots, Air Gap; Coil- wound Rotors; Squirrel-cage Rotors; Advantages and Disadvantages op the Two Types of Rotors Rotors, Number of Rotor and Stator Slots, Air Gap. — ^Two distinct types of rotors are used in induction motors, the coil- wound and the squirrel-cage. Each of these possesses certain distinct advantages. Both have slots which are usually par- tially closed. Very open slots are undesirable as they would materially increase the effective length of the air gap. This would increase the magnetizing current and hence decrease the power factor. On the other hand, completely closed slots are usually undesirable as they would decrease the reluctance of the path of the leakage flux and consequently increase the stator and rotor reactances thus decreasing the maximum torque de- veloped by the motor. Magnetic wedges are sometimes used to hold the coils in the slots. Such wedges give the effect of closed or partially closed slots and decrease the effective length of the air gap. The stator shown in Fig. 212, page 445, has such wedges. Induction motors always have very short air gaps. For this reason, they should be provided with such bearings as will minimize the effect of wear and the danger of the rotor striking the stator. The number of slots in the rotor and stator must not be the same. In order to prevent a periodic variation in the reluctance of the magnetic circuit of the motor the ratio of these numbers must not be an integer. Moreover, if the rotor and stator had the same number of slots, there would be a tendency for the rotor at starting to lock in the position which makes the reluctance of the magnetic circuit a minimum. Coil-wound Rotors. — The windings of coil-wound rotors are similar to those of alternators. They must be arranged for the same number of poles as the stator, but the number of 468 POLYPHASE INDUCTION MOTORS 469 phases need not be the same, although in practice it usually is so. Either mesh or star connection may be used, the rotors of thiee-phase motors being either A- or F-connected. It is customary to use Y connection, not only for the rotor but also for the stator, as it gives a better slot factor than the A connec- tion. ^ The terminals of the rotor winding are brought out to slip rings mounted on the shaft. These shp rings may be short- circuited for normal running conditions and connected through suitable resistances for starting or varying the speed. Since the current in the rotor is obtained entirely by induction, the opera- tion of the motor is not influenced by the voltage for which the Fig. 220. rotor is wound. The best voltage for a rotor is usually that which makes the cost of construction a minimum. A coil-wound rotor with a part of the winding in place but without the slip rings is shown in Fig. 220. Squirrel-cage Rotors. — The windings of sqtiirrel-eage rotors consist of solid copper inductors of either circular or rectangular cross-section, placed in the rotor slots with or without insulation and then short-circuited, by copper end rings or straps to which the inductors are bolted, soldered or welded. Since low resist- ance is desirable, it is' best to solder the short-circuiting end rings to the bars even if they are also bolted. The inductors of most squirrel-cage rotors are now electrically welded to the end rings. One type of squirrel-cage rotor is indicated in Fig. 221. ' Page 35, Synchronous Generators. 470 PRINCIPLES OF ALT ERN AT I NO-CURRENT MACHINERY Advantages and Disadvantages of the Two Types of Rotor. — The chief advantage possessed by the coil-wound rotor is the possibility it offers of having its apparent resistance varied by inserting resistances between its slip rings. This variation in resistance may be used to increase the starting torque or to vary the speed. The chief disadvantages of this type of rotor are its higher cost, slightly higher resistance^ and less ruggedness 07' 31." xi ^A — " "Mil ■''" ^^^ Fig. 221. than the squirrel-cage type. Squirrel-cage rotors are extremely rugged and have very low resistances. Consequently, they de- velop low starting torque but have good speed regulation. The starting current taken by a motor having a squirrel-cage rotor is large and the power factor at starting is low. 1 This resistance is referred to the primary. Its actual resistance is necessarily many times that of the squirrel-cage type. CHAPTER XL VII Methods of Starting Polyphase Induction Motobs; Meth- ods OP Varying the Speed op Polyphase Induction Motors; Division op Power Developed by Motors in Concatenation; Losses in Motors in Concatenation Methods of Starting Pol3rphase Induction Motors. — Referring to the equivalent circuit of the polyphase induction motor, Fig. 217, page 453, the stator current is Ii = h + h where 7„ and I2 are considered as vectors. At starting, full voltage being applied and no resistance added to the rotor circuit, the current taken by an induction motor is from five to eight times the full-load current. If the stator and rotor constants are assumed to be approximately equal when referred to either the stator or rotor, the magnetizing current when the slip is unity will be only about half as large as it is when the motor is running under normal conditions, same impressed vol- tage being assumed. Consequently, the starting current taken by a motor which has no resistance added to its rotor circuit may be considered to be approximately equal to the secondary current referred to the primary. From Fig. 217, neglecting the divided circuit and making s = 1, ( V (ri + rsy + {xi + X2r] The approximate power factor corresponding to this is P--^-"- = /, , ""iV") , ,2 (183) Vin -frz)^ + {xi+ X2r ■ The reactances are usually from three to four times the re- sistances. Therefore, the power factor at starting is low if no resistance be added to the rotor circuit. It must be remembered that equations (182) and (183) can be applied only when no resist- ance is added to the rotor circuit as under this condition alone is the magnetizing current negligible. 471 472 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The starting torque, equation (180), page 465, has already been found to be I St ^ ~A TJ- 2^2 47r/i The starting torque is, therefore, proportional to the copper loss in the rotor circuit. It may be increased to the maximum torque of the motor by increasing the resistance of the rotor circuit. Small motors may be started by connecting them directly to the line, but when started in this way, they take a very large current at low power factor. The magnitude of the current taken by a motor larger than a few horsepower prevents the use of this method for starting large motors. It is seldom employed even in the case of small motors. There are two methods for starting polyphase induction motors without taking excessive current from the line: by reducing the impressed voltage, by inserting resistance in the rotor circuit. Motors with squirrel-cage armatures must be started at reduced voltage. Motors with coil-wound armatures may be started by either reducing the voltage or by inserting resistance, although the latter is usually employed. There would be no object in using a coil-wound armature except for increasing the starting torque or for varying the speed. The reduced voltage for starting is usually obtained by means of a compensator giving from one-half to one-third normal voltage. The motor is brought up to speed on this reduced voltage and then thrown oh full line voltage. For starting motors with coil-wound armatures, drum-type con- trollers, similar to those employed for varying the speed of direct-current series motors, are generally used. The first position of the handle on these controllers puts the stator across full line voltage and closes the rotor circuit through resistance. Successive positions of the controller handle reduce the re- sistance and finally the rotor is short-circuited. The resistance units are usually of the grid type and are external to the con- troller. When this method of starting is employed, motors may be brought up to speed under any load which requires a torque not exceeding the maximum torque of the motor. The current required to develop a given torque when starting with POLYPHASE INDUCTION MOTORS 473 resistance in the rotor circuit is the same as that required to develop the same torque under running conditions. The torque per ampere is a characteristic constant of the induction motor when operating on the stable part of its speed-torque curve, i.e., on the part between synchronous speed and maximum torque. If full-load torque is required, the current will be equal to the normal full-load current of the motor. Equation (176), page 462, for the torque may be written T, 4x/i s From Fig. 218, page 454, (184) h = Vy ri + n gn + (n +7)'+ (a;i + x,y -J bn + Xl + X2 (185) (n + ^) +ixi,+ X2y = V,{G-jB) ' (186) At starting, S = 1. Therefore, if r2 plus the resistance, r'2, inserted in the rotor circuit at starting is made equal to r2 divided by the slip at full load, both the torque developed by the motor and the current it takes will be the same as at full load. From equation (186) the approximate power factor is f ^ This will also be the same as at full load when r^ + r\ is made equal to — where s is the slip at full load. If it is desired to have the motor develop its maximum torque at starting, ri must be made equal to y/r^ H- (xi + xi)"^, equation (181), page 466. When resistance is inserted in the armature during starting only, it is sometimes placed inside the rotor and arranged to be cut out or in by means of a sliding rod passing through the 474 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY hollow shaft. The objection to this arrangement is the danger of over-heating the starting resistance either by leaving it in circuit too long or by trying to start the motor under too great a load. Although much can be accomplished in increasing the start- ing torque of motors with squirrel-cage armatures by properly shaping the rotor slots and inductors, motors with coil-wound rotors and slip rings should be employed when high starting torque is desired. It is possible to design motors with squirrel- cage armatures which will give full-load torque at starting and will also have fairly satisfactory speed regulation. Such motors require several times the normal full-load current to de- velop this torque at starting since at starting, they are not on the part of the speed-torque curve where the torque per ampere 'is approximately constant. Methods of Varying the Speed of Polyphase Induction Motors. — There are four ways by which the speed of a polyphase induction motor may be changed : (a) By inserting resistance in the rotor circuit. (b) By using a stator winding which can be connected for different numbers of poles. (c) By varying the frequency. (d) By concatenation or series connection for two or more motors. (o) By Resistance. — This method of controlling the speed of an induction motor requires a coil-wound rotor with slip rings. Rotors are usually star-connected. For normal speed, the slip rings are short-circuited. During starting and also when the speed is to be reduced below normal speed, the slip rings are connected through suitable resistances. The slip of an induction motor may be found by + y\[n - ^J - W + {xr ^ x^Y]]^ (187) where K = -T~r> equation (176), page 462. For a given impressed voltage and frequency, if is a constant. POLYPHASE INDUCTION MOTORS 475 Therefore, for any fixed internal torque, T2, and constant impressed voltage and frequency, the slip of an induction motor varies directly as the rotor resistance, r2. Consequently, the speed of an induction motor may be varied by inserting resistance in its rotor circuit. According to equation (169); page 460, the power transferred across the air gap to the rotor is P'2 = B2I2 cos ef,' But T -Egg J aE2 ^2 I2 = — / :=: and cos Oj' = — , ^= V J'a^ + X2^s^ Wr^^ + x-^^s^ Hence and p, _ /2V2 " 2 = — : — « = ¥-' (188) The slip is equal to the ratio of the copper loss in the rotor circuit to the total power received by the rotor from the stator, or the loss of power in the rotor circuit is proportional to the slip. If the slip is 25 per cent., the electrical efficiency of the rotor is 75 per cent. If the slip is 50 per cent., the rotor efficiency is 50 per cent. If the slip is increased to 75 per cent., the efficiency is reduced to 25 per cent. The percentage decrease in the rotor efficiency is proportional to the slip. Although the resistance niethod of controlling the speed is simple and often convenient, it is not economical and the drop in speed obtained by means of it is dependent upon the load. A motor, delivering full-load torque, which has its speed decreased to 50 per cent, of its synchronous speed by adding resistance to the rotor circuit, will speed up to nearly normal speed when the load is removed. The speed regulation of a motor, with resistance added to its rotor circuit, is poor. It has already been shown that the maximum internal torque developed by a polyphase induction motor is independent of the resistance of the rotor circuit. Adding resistance changes the slip at which this maximum torque occurs and at the same time lowers the efficiency. Adding resistance to the rotor circuit 476 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY of a polyphase induction motor has much the same effect as adding resistance to the armature circuit of a direct-current shunt motor. (6) By Changing Poles. — The speed of an induction motor is proportional to the frequency and inversely proportional to the number of poles for which the stator is wound. Therefore, if induction motors which operate at the same frequency are to run at different synchronous speeds, they must have different numbers of poles. Induction-motor windings may be arranged to be connected for two different numbers of poles which are in the ratio of 2 : 1. By the use of two independent windings four speeds may be obtained. Unless squirrel-cage rotors are used with such motors, the general arrangement of the rotor winding must be similar to that of the stator and its connections must be changed whenever the connections of the stator are changed in order that the rotor and stator shall have the same number of poles. On account of the additional slip rings and extra complication involved in arranging the rotor windings for pole changing, squirrel-cage rotors are generally used for multi- speed motors unless speeds are required intermediate to those obtained by changing the number of poles. Multispeed induction motors are used to some extent on electric locomotives. The locomotives on some of the Italian State Railroads and on the Norfolk & Western Railroad in this country are of this type. Small multispeed motors for driving machine tools may be obtained in sizes up to 10 or 15 hp. from several companies manufacturing electrical machinery. The difficulties in the design of a satisfactory multispeed motor are due to the change in the effective number of turns per phase, and consequently in the flux density, and to the change in the coil pitch when the connections are altered to change the number of poles. There are several practical ways to change the number of poles, ^ but all of these, if the voltage is kept constant, involve a change in the flux density and magnetizing current which may, in some cases, be as high as 100 per cent., and a change in the blocked current which is even greater. As a result the power and breakdown torque may be quite different for the two connections. The design of multispeed motors, 1 Die Wechselstromtechnik, E. Arnold, Vol. Ill, Chap. VII. POLYPHASE INDUCTION MOTORS 477 consequently, must be more or less of a compromise between the designs which would give the best operating conditions at either speed. In the practical design of two-speed induction motors, the speed ratio with a single winding is two to one. In these motors the coils are of such a width as to give full pitch for the connec- tion producing the greater number of poles. Consequently, when connected for the smaller number of poles the pitch is one- half. The connections are so made that half of the poles are consequent poles when connected for the greater number of poles. The smaller number of poles is obtained by conducting the current to the center points of the windings of each phase. To keep the flux density somewhere nearly constant, a change may be made from delta to Y or from series to parallel connection. If, for example, for the larger number of poles, the connections are series delta and for the smaller number parallel Y, the flux densities will not be seriously different for the two speeds. (c) By Varying the Frequency. — The speed of an induction motor is directly proportional to the frequency impressed on the stator. By varying this frequency, the speed may be changed. This method of varying the speed has the objection of requiring a separate generator for each motor, and for this reason it is applicable only in special cases. Since an induction motor is in reality a transformer, the flux at any fixed voltage will vary inversely as the applied frequency. In order to prevent this change in flux density when the fre- quency is lowered with its attendant increase in core loss, mag- netizing current and magnetic leakage, the voltage impressed on the motor must be varied in proportion to the frequency. This does not involve any difficulty, since the voltage of a generator varies in direct proportion to the frequency provided the excita- tion is kept constant. If the ratio of the frequency to the impressed voltage is kept constant, the torque at any given slip will vary in direct proportion to the voltage or the speed, equation (176), page 462. (d) By Concatenation. — Concatenation, tandem or series connection for induction motors gives much the same effect as the series connection for direct-current series motors. In both cases, if the current taken from the mains is equal to the 478 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY full-load current of one motor, approximately twice the full-load torque of one motor at approximately one-half full-load speed results. Motors which are to be connected in concatenation should have wound rotors and their ratios of transformation should preferably be unity. The rotors must be rigidly coupled. The stator of one motor is connected to the mains and its rotor is connected to the stator of the second motor. The rotor of the second motor is either short-circuited or connected through resistance. The resistance is used either during starting or when intermediate speeds are required. Even if the ratios of transformation are not unity, the motors may still be operated in concatenation provided they have equal ratios of transforma- tion. The rotors must be electrically as well as rigidly coupled. The primary of one must be connected to the mains and the "^.p rimary o f the other must be short-circuited. Let pi and p^ be the number of poles and let Si and Si be the slips for the two motors respectively. If /i is the frequency of the voltage impressed on the first motor, the frequency, /a, of the current in the primary of the second motor is h = /isi Synchronous speed for motor No. 2 is, therefore, 2/iSi Its actual speed is — (1 - ..) The speed of motor No. 1 is Pi Since both motors are rigidly coupled they must run at the same speed, hence P2 Pi ^ and Sl P2 Pi — S2P1 + Pi POLYPHASE INDUCTION MOTORS 479 As the rotor of the second motor is short-circuited, S2 will be small. Therefore, the term s^pi may be neglected, giving approximately. (189) The speed of the system is the same as the speed of the first motor or Pi Pi V P1 + P2/ Pi P1-I-P2 If Pi and Pi are equal, the speed of the system will be equal to one-half of the normal speed of either motor. The use of two similar motors both in parallel and in concatenation, gives two efficient running speeds, viz., full speed with two motors in parallel, and half speed with the motors in concatenation. When the motors are in concatenation, other speeds may be obtained by the use of resistance in the rotor of the second motor. When the motors are in parallel, other speeds may be obtained by the use of resistance in both rotors. The use of two similar motors gives essentially a constant-torque system since approximately twice the full-load torque of one motor can be obtained at any speed and this without exceeding full-load current in either motor. If motors having different numbers of poles are used, three different running speeds may be obtained, but in this case, two of these speeds make use of but one motor at a time. The full torque of the system is available only when the motors are in concatenation. The three speeds are obtained by the use of (a) Motor No. 1 alone. (6) Motor No. 2 alone. (c) Motors No. 1 and No. 2 in concatenation. For example: let the motors have eight and twelve poles, respectively, and let the frequency be 25 cycles. Then, pi = 8, P2 = 12 and /t = 25. The speeds obtainable in revolutions per minute are, (a) No. 1 alone 2(26) speed = — ^ — 60 = 375 rev. per mm. o (b) No. 2 alone 480 PRINCIPLES OF ALT ERN AT I NO-CURRENT MACHINERY 2(25) speed = ^g 60 = 250 rev. per min. (c) No. 1 and No. 2 in concatenation. With No. 1 connected to the mains speed = „ 60 ( o , lo ) ^ -^^^ ^®^" P®'' ™'^'^' With No. 2 connected to the mains 2(25) „„/ 12 \ ,,„ speed = -~- 60 L ,, ) = 150 rev. per mm. In concatenation, it makes no difference so far as the speed of the system is concerned which motor is connected to the mains. Division of Power Developed by Motors in Concatenation. — The complete expression for the division of the power developed by motors in concatenation is very complicated. When the magnetizing currents and the impedance drops are neglected, however, the expression becomes simple. Neglecting the magnetizing current will produce considerable error yet expressions deduced under this assumption are of value, and may be considered as first approximations. If the magnetizing current of the second motor is neglected, the effect of this motor on the first is very nearly the same as if a non-in- ductive resistance were added to the rotor of the first motor. The actual effect of the second motor on the first is the same as adding an impedance to the rotor of the first motor. The ratio of the resistance of this impedance to the impedance itself is equal to the power factor of the second motor. This may be from 0.85 to 0.92 at full-load current. The effect of adding a non-inductive resistance to the rotor of an induction motor is to change the slip for a given current without altering the internal torque. The effect of adding impedance is to change not only the slip but the torque also. Single and double primes added to the letters for voltage and current will refer to the first and second motor, respectively. The mechanical power developed by the first motor is (equa- tion 170, page 461). E\ (1 - si) Z'2 cos e , 1 2 POLYPHASE INDUCTION MOTORS 481 That developed by the second is E". (l-S2)Z"2Cos/7'~"^ J- 2 Since the magnetizing currents and drops are neglected, the two currents, I'i and I''^, will be equal. The two power factors, cos 6„ and cos 9„, 1 will I't J i also be equal. Therefore, Power of No. 1 _ E'2{\ - sQ Power of No. 2 ~ £?"2(1 - Sj) The slip of an induction motor is equal to the ratio of the copper loss in the rotor circuit to the power received by the rotor from the stator. Since the drops are to be neglected, the copper loss in the rotor of the second motor will be zero. The slip, S2, of this motor is, therefore, zero. The second motor is connected to the rotor of the first. Since magnetizing currents and drops are neglected, its effect on that motor is like a non- inductive resistance. The slip, s\, of the first motor, therefore, cannot be zero. Power of No. 1 ^ g%(l - Si) Power of No. 2 " E"^ With the drops in the second motor neglected, E"i = E'iSx. Therefore, Power of No. 1 1 — Si nan Power of No. 2 Si Pi + Vi The slip of the system is Si = . . Substituting this in equation (191) gives Power of No. 1 Power of No. 2 P2 Pl + P2 Pa = 'Sl Pi (192) Pl + 'P2 The division of power between the two motors is approxi- mately proportional, therefore, to the ratio of the numbers of poles. Since the first motor has line voltage and line frequency im- pressed on it, its flux is normal. The second motor receives a 31 482 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY voltage E'iSi at a frequency /a = fiSi. Since the voltage and frequency impressed on the second motor are reduced in the same proportion, its flux is also normal. When the magnetizing currents are neglected, both rotors carry the same currents and at full-load current for the system each motor will develop its normal full-load torque. Losses in Motors in Concatenation. — Motors with the Same Number of Poles. Conditions in the Motors. Current. — The current in the first motor is normal. Neglect- ing the exciting current of the first motor, and assuming a ratio of transformation of unity, the current in the second motor is also normal. Voltage and Frequency. — ^The voltage and frequency im- pressed on the first motor are normal, but the second motor receives only half voltage at half frequency. Flux. — The flux of both motors is normal since the first re- ceives normal voltage at normal frequency and the second re- ceives half normal voltage at half normal frequency. Speed. — The speed of the motors is one-half normal speed. Torque. — Since each motor has normal current (exciting cur- rents are neglected) and normal flux, the torque of each will be normal. Copper Losses. — Each motor carries full-load current and will have normal full-load copper loss. Core Losses. — The core loss in the stator of motor No. 1 is normal. The core loss in the rotor of this motor is greater than normal on account of the large slip (50 per cent.). The frequency and voltage impressed on motor No. 2 are each one-half normal. The flux is normal. This motor runs at normal speed, i.e., with small slip, for the frequency which is impressed on it. The core loss in this motor will be less than normal on account of the low frequency. Power. — Each motor. develops full-load torque at half speed. The output of each is, therefore, one-half its full-load output. Motors with the Same Number of Poles. Conditions in the System. Power. — The power will be the full-load power of one motor. The first motor converts one-half the power it receives into mechanical work and transforms the other half into electrical POLYPHASE INDUCTION MOTORS 483 power at one-half normal voltage and one-half normal frequency. This electrical power is transformed into mechanical power by the second motor. This statement neglects the losses in the system. Torque. — The torque will be twice the full-load torque of a single motor. Losses. — The copper losses will be the full-load copper losses of both motors. The core losses will be somewhat less than the full-load core losses of both motors. Efficiency. — The efficiency will be less than the full-load efficiency of one motor. If the full-load efficiency of each motor under normal conditions is 90 per cent., the total losses will be approximately 20 per cent., the efficiency of the system will be approximately 80 per cent. Motors with Different Numbers of Poles. The conditions existing when the motors have different numbers of poles may be analyzed by following the method used for motors with the same number of poles. It must not be forgotten that what has preceded in regard to conditions existing in motors when in concatenation has neg- lected an important factor, the magnetizing current, and cannot, therefore, be considered as more than an approximation to actual operating conditions. CHAPTER XL VIII Calculation of the Pebpormance of an Induction Motor FROM Its Equivalent Circuit; Determination op the Constants for the Equivalent Circuit Calcixlation of the Performance of an Induction Motor from Its Equivalent Circuit. — The equivalent circuit of the induction motor is again shown in Fig. 222. The same notation will be used as in the vector diagram of Fig. 215, page 452. I„ = Ih + e + jlp, the exciting current, is not the no-load current as in a transformer. The no-load current of an induction motor is equal to 7„ plus a component which supplies the no-load copper and friction and windage losses. The letters Qn and &„ are yoM^vwv^ Fig. 222. the conductance and the susceptance, which, at normal fre- quency, take, respectively, the currents h + e and 7^ at a voltage equal to Ei. Everything will be referred to the stator and will be per phase unless otherwise stated. h = h + c +jl^ =.Ex {gn - jbn) The apparent resistance of the rotor circuit, including the load, is R + Ti = ra — ; H ?-2 =- 484 POLYPHASE INDUCTION MOTORS 485 The apparent conductance, gfa, of the rotor and load is The apparent susceptance, 62, of the rotor and load is X2 b,= (?) + X2' X2S''' rz^ + X2^ The resultant conductance and susceptance, gab and 606, of the portion to the right of the points a and & of the equivalent circuit shown in Fig. 222 are gab = gn + gz hah = hn-\- &2 Fi = J5?i + Zi(ri+ja;i) /i = Slight - jbab) (193) Fi = E,[l + (gab- jbab) in +jx,)] = El[l + {gabTl + babXl) - JibabVl - gabXl)] (194) = Ei{G - jB) (195) Both G and B depend on the load. JFrom equation (195) 7i Si = G-jB El = 77p=T^ numerically, (196) The power given to the rotor, or the synchronous power is P'2 = Ei'g, (197) This is the power which is transferred across the air gap 486 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY to the rotor and represents the internal power the motor would develop if it were to run at synchronous speed: The actual internal power developed by the rotor at the speed — ^ (1 — s) is P2 = E^'g2{l - s) (198) The power Pp developed at the pulley is Pp = £1^32(1 — s) — (friction and windage loss) (199) The torque at the pulley is Tp = —Kf^^ (200) 2x ^ (1 - s) Assuming the friction and windage loss to be constant (friction and windage loss) P From equations (193) and (195), the stator input, Pi, is Pi = {EiG){Eigai) + iEiB)(EJ,^) = Ei\Ggai + Bhab) (201) The stator power factor is The efficiency is P. All of the preceding equations are in c.g.s. units. If the quantities are expressed in practical units, the equations become Stator phase voltage, Vi, in volts = Eiy/G^-\-B^ (202) Stator phase current, Ji, in amperes = Ei-y/^J~+hJ^ ■ (203) Stator power, Pi, per phase in watts = Ei^ (Gga + Bbai) (204) POLYPHASE INDUCTION MOTORS 487 Stator power factor = ^^-(gg;; + ^M (205) Pulley output, Pp, per phase in horsepower ==2« [Ei^Qi (1 — s) — (friction and windage loss in watts) }^ (206) Torque at pulley, Tp, per phase in pound feet 550 / 1 \ f (friction and windage loss in watts) 1 ,„„_. = 7^1 1746/ 1^^ ^^ T^s 1 (^^^^ V Rotor phase current, I2, in amperes Eis ■\/r2^ + Xi^s^ Slip in per cent. = ^100=^J^100 f208) where P'2 is the power in watts per phase transferred across the air gap to the rotor. If the constants of a motor are known, the performance may be calculated for any assumed slip from equations (202) to (208), inclusive. Detennination of the Constants for the Equivalent Circuit. — Let Pn be the no-load input less friction and windage and primary copper losses. The rotor copper loss at no load may be neglected. Pn should be measured at a voltage E., but a voltage, equal to Fi may be used without producing any great error. If necessary a correction may be made for the voltage, by assuming P„ to vary as the square of the voltage. Let Z'„ be the total measured no-load current. Then iA + e — Y^ /p = /'»\/l — (no-load power factor)^ ih+e gn and Fi On - y^ ' The friction and windage loss should be measured at a speed corre- sponding to (1 — s). It is, however, sufficiently accurate to measure it at the no-load speed and assume it to remain constant. 488 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY On account of the small slip of an induction motor under ordinary operating conditions, the frequency, /2 = /is, of the current in the rotor is low. The effective and the ohmic re- sistances of the rotor will be nearly the same since the core loss due to the rotor leakage flux will be negligible on account of the low frequency The ohmic resistance should be used for ra in the formula. If the motor has a wound rotor, the ohmic resistance of its rotor may. be measured directly. It must be referred to the primary as in a transformer before it can be used in the equa- tions. Correction will usually have to be made for the stator impedance drop when finding the ratio of transformation. A correction must also be applied in case the rotor and , stator windings are not of the same type, i.e., both A or both Y. There is no satisfactory way of measuring the ohmic* resistance of a squirrel-cage rotor. It is possible to calculate its value from the dimensions of the rotor but this is a complicated process. The effective resistance should be used for the stator resist- ance, ri, since the core loss due to the stator leakage flux must be included on account of full frequency being impressed on the stator. If the motor has a wound rotor and the ratio of the ohmic to the effective resistance is assumed to be the same for both stator and rotor windings, the effective resistance of the stator and of the rotor may be found by measuring the equivalent re- sistance of the whole motor at approximately full-load current with its rotor blocked and then dividing this resistance into two parts which are in the ratio of the stator and rotor ohmic resistances. The power input to the stator with blocked rotor is the total copper loss in the motor plus a core loss due to the rotating magnetic field of the stator. This core loss will not be large compared with the copper loss; It will be approximately equal to the input to the stator minus the stator copper loss' with the rotor blocked and on open circuit and with an im- pressed voltage equal to one-half the voltage used for the usual blocked run. If Pi, is the power input at frequency /i with the rotor blocked, ' This is a small correction and for this reason ohmic resistance may, if necessary, be used in computing it. POLYPHASE INDUCTION MOTORS 489 and Vt and lb are the corresponding impressed voltage and stator current, the equivalent reactance of the entire motor at primary frequency is Xe = "f"^! ~ {blocked power factor)^ lb As there is no way of determining exactly how Xe divides between the rotor and stator, it is customary to assume it divides equally between them. This assumption is not correct in many cases but it does not affect the performance of the motor so far as torque and output are concerned, as may be seen by referring to equation (176), page 462. This same equation also shows that the effect of the rotor resistance is much greater than that of the stator. If only an approximate value of the primary resistance is used, little error will be introduced in the calculated torque and output. When the rotor is blocked, the conditions are the same as in a short-circuited transformer except that a much greater voltage must be impressed to give any fixed percentage of full- load current in the motor on account of the presence of the air gap. The leakage reactances are also much higher for the motor. The no-load current of an induction motor is usually between 30 and 50 per cent, of the full-load current. The equivalent impedance drop at full-load current is usually between 15 and 20 per cent, of the rated voltage. CHAPTER XLIX Circle Diagram of the Polyphase Induction Motor; Scales; Maximum Power, Power Factor and Torque; Determination op the Circle Diagram Circle Diagram of the Polyphase Induction Motor. — The circle diagram was first applied to the induction motor by- Alexander Heyland in 1894.' Many modified forms of this diagram have since appeared. One of the simplest of these, in construction and use will be given. Although certain ap- proximations are made in the construction of this diagram, the results obtained by it are, as a rule, quite satisfactory. This diagram like all other circle diagrams of the polyphase induction motor, may be constructed from two sets of readings which may be obtained quickly and without the use of special apparatus. These readings are taken under conditions which correspond to those existing in a transformer on open circuit and on short-circuit, giving current, voltage and power with the motor operating at no load and again with blocked rotor. In addition the rotor or stator resistance is required. Reference will be made to the approximate equivalent circuit shown in Fig. 218, page 454. ' V(r-i + r2 + E)==+(xi + a;2)^ The sine of the angle of lag between /2 and Fi is Xl -\-X2 sin 02 ■Vin + n + RY + ix^ + x^y Hence, ^^ = ^, «in ^^ (209) If X\ and Xi are assumed to be constant, this is the polar equation 1 Electrotechnisohe Zeitschrift, Vol. XLI, p. 561, 1894. 490 POLYPHASE INDUCTION MOTORS 491 of a circle with Fi Xi + Xi as diameter. This circle is plotted in Fig. 223 with AB = Vt as diameter. i.Xi + X2) AI2 is the rotor current to any suitable scale. To this same scale AB is the impressed voltage divided by the total motor reactance, i.e., by xi + X2, Xt being referred to the stator. To obtain the stator current, 7i, the current I„ = h+e+jlr. must be added to I2. Continue ViA to D and draw OD per- pendicular to AVi. Make AD and OD equal to Ih+e and I^, respectively. Let Oa be a line drawn parallel to A Vi. Then OA is the current !„ and 01 2 is the stator current. 61 is the stator power-factor angle. I2C' — Ii cos di and is the energy component of the stator current. If Vi is constant, I2C' represents the input to the motor. To the same scale AD represents the core loss. The further construction of the diagram can be considerably simplified by making an approximation, which wUl have little effect on the results, except at small loads. In the equivalent circuit. Fig. 218, the branch marked jTn takes a current equal to Ih+e- The friction and windage losses are supphed by the secon- dary current I2. Let the current I2 be decreased by an amount equal to the energy component of the current supplying the fric- tion and windage losses, and let this amount be added to the current Zt + e to give I'h + e, which is then the friction-and-windage 492 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY and core-loss current. Let I h + e also include the no-load primary copper loss. If these changes are made on the circle diagram shown in Fig. 223, AD to the proper scale becomes the core loss in the stator at no load, plus the no-load stator copper loss and the no-load friction and windage losses. OA will be the no- load current, I2C will be the motor output plus the secondary- copper loss and the increase in the primary copper loss caused by the load. As the motor is loaded, the true stator core loss will decrease slightly on \ account of a slight decrease in the value of Ei. The decrease in E^ with load is neglected in the approximate equivalent circuit. The rotor core loss will increase but this increase will be small and will tend to balance the decrease in the stator core loss. Little error is introduced by assuming the total core loss to remain constant and letting any increase in the rotor core loss as the motor is loaded be included in AD to balance the decrease in the stator core loss. As the motor is loaded, I^ will travel toward the point B on the circle and will reach some point, such as R, when the rotor has come to rest. This is the condition existing when the rotor is blocked under full voltage. Under this condition, OR is the primary current and, since the output is now zero, Rr must be the secondary copper loss plus the increase in the primary copper loss caused by the load. Let d divide Rr into two parts such that Rd is the rotor copper loss and dr is the increase in the primary copper loss caused by the load. Join the points d and A. Then ef is the rotor copper loss and fC is the increase in the primary copper loss due to the current AI^. AI2 represents the increase in the primary current caused by the. load. It is also the secondary current. Ce, ef and fC are, respectively, the total copper loss, the copper loss in the rotor and the increase in the copper loss in the stator produced by the load, i.e., by the current AI2. This may be shown as follows: Ce _ AC _ Ah cos BAI2 Rr~ Ar ~ AR cos BAR ^AB {Auy AR {ARY ^^AB POLYPHASE INDUCTION MOTORS 493 Since Ce and Rr are in the ratio of the square of the currents, AI2 and AR, Ce must be the sum of the rotor copper loss and the increase in the stator copper loss produced by the load, or by the load current AI2. In a similar way, it may be shown that fC is the stator copper loss due to AI2. From this it follows that ef must be the rotor copper loss. The slip of a motor is equal to the rotor copper loss divided by the power transferred across the air gap and is therefore ef Y^ See equation (208), page 487. The power given to the rotor is transferred at synchronous speed. This power divided by 2x times the synchronous speed is the torque at which the power is transferred. Since action and reaction between rotor and stator must be equal, this torque must also be the rotor torque. Therefore, since /a/ is the power given to the rotor less friction and windage losses, /a/ divided by 2t times the synchronous speed must be the pulley torque. The rotor losses affect only the speed and do not affect the torque at any given current. The following quantities may now be obtained from the diagram by applying the proper scales. Stator current Oh Stator power Stator power factor No-load current No-load losses No-load power factor Pulley output hC cos di OA AD cos 9n Power transferred across the air gap hf 494 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY Torque is also proportional to hf Slip hf Efficiency , he Scales. — The current scale is arbitrarily assumed. The power scale in watts is the current scale multiplied by the voltage, i.e., if the current scale is 5 amp. per inch and the phase voltage is 2200, the power scale is 11,000 watts per inch. The power scale in horsepower is equal to the watt scale divided by 746. The torque scale in pound feet is equal to the watt scale multi- plied by -'.„ ^ — ' where n is the synchronous speed in revolu- tions per minute. The slip is given by the ratio of the lengths of two lines and hence does not involve a scale. Maximum Power, Power Factor and Torque. — The maximum power will occur at that current -which makes the distance I^e on the diagram a maximum. To determine the maximum power it is necessary to draw a tangent to the circle parallel to the line AR. The point of tangency represents the position of the end of the primary current line, OI2, when the power is a maximum. The easiest way to determine the point of tangency is to erect a perpendicular bisector to the chord AB. The current 01 ^ on Fig. 223 is drawn for the condition of maximum power. On Fig. 223, I^ is the maximum power output. The maximum power factor will occur when the primary current line, OI1, becomes tangent to the circle. The maximum torque may be found by drawing a tangent to the circle parallel to the hne Ad. The point of tangency in this case locates the extremity of the current line under the condition of maximum torque. An inspection of the diagram will show that the motor will develop its maximum power output before it develops maximum torque. A properly designed motor under ordinary operating conditions should work on the part of the diagram considerably to the left of the extremity of the current line O/2 for maximum POLYPHASE INDUCTION MOTORS 495 power shown on Fig. 223. The breakdown or maximum torque of a properly designed motor is seldom less than twice full-load torque. Determiaation of the Circle Diagram. — The circle diagram is determined from two sets of measurements, one obtained with the rotor blocked and the other with the motor running at no load. The readings which are required under each of these conditions are: power input, current and impressed voltage, all under conditions of normal frequency. The no-load run should be made at rated voltage, but it is seldom safe to apply rated voltage to the motor when its rotor is blocked. Usually 40 to 60 per cent, of this voltage may be apphed with safety. Under the blocked condition the current varies nearly as the impressed voltage. This assumption is used in finding the current the motor would take if blocked and with rated voltage impressed. The power taken when the rotor is blocked varies nearly as the square of the voltage. In addition to the readings already mentioned, either the ohmic resistance of the rotor or the effec- tive resistance of the stator is necessary. If the motor has a wound rotor, it is an easy matter to measure the ohmic resistance of its stator and rotor. The rotor resistance as measured must be referred to the stator by multiplsdng it by the square of the ratio of transformation of the motor. The stator effective re- sistance may be obtained by multiplying its ohmic resistance by a suitable constant. This constant will depend upon the design of the machine. To construct the diagram, choose a suitable scale for the cur- rents. All the other scales depend upon this one. Take any line OC, Fig. 223, as a base line and erect a perpendicular at as a reference line from which to measure power factors. Every- thing on the diagram will be per phase. From lay off the blocked current, OR, corrected to rated voltage, and the no-load current, OA, making angles 9g and dn, respectively, with the volt- age reference line Oa. Through A, draw a line AB parallel to OC and drop a perpendicular, AD, from A to the base line. Both of the points A and R lie on the current circle. The diameter of this circle is on AB. A perpendicular erected at the middle point of a line connecting A and R will intersect the hne AB at the center of the circle. Draw Rr perpendicular to AB and 496 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY « locate the point d on this hne by either making dr equal to the effective resistance drop caused by the current ARin the stator or by making Bd equal to the ohmic drop in the rotor due to this same current. Joining R and d with A completes the diagram. The conditions corresponding to any desired current or output or torque may then at once be found. 100 800 32 000 9.0 lOO 28 000 S600 s ; 1 500 2 : ^400 S] 4.000 Pow'er Fi etor / '"^ / P^ -^ ~^ iigjc (- ^ / / f / jx V ) — 7 / •fA ^^ i-' N y / /- — ^. .>' t^' y 7 I / y r-'-. --, ^' >' / / .,'-- 1 ,-' ,-'' --- ^-, / / /. --'' A -'"' / K / / r. -- .-'' I / / /' \ ~~~~ — — Slip -,/' ^ \ - c^ ?^ 4> y ^ ^ / -^ Inductioh Motor. 500 Horse Power, ^ S-Phaae, 25 Cycles, 12 P^les, 2200 yolts. 400 600 800 1000 Output in Horse Power. Fig. 224. J.200 60 a u ■a 40 8 20 30 30 ^0| 50 fe 20 60-3 TOm 10 80 00 . JOO The complete characteristic curves of a three-phase, 25- cycle, 500-hp. induction motor calculated from a circle diagram are plotted in Fig. 224. The arrows on the curves indicate the direction the motor would pass over the curves in going from no load to the blocked condition. The portions of the curves beyond the point of maximum torque represent unstable conditions. These portions can, therefore, only be obtained by calculation. CHAPTER L Gbneral Characteristics of the Induction Generator; i^ Circle Diagram of the Induction Generator; Changes IN Power Produced by a Change in Slip; Power Factor OF THE Induction Generator; Phase Relation between Rotor Current Referred to the Stator and Rotor Induced Voltage, E^; Vector Diagram of the Induc- tion Generator; Voltage, Magnetizing Current and Function of Synchronous Apparatus in Parallel with an Induction Generator; Use of a Condenser instead of a synchronous generator in parallel WITH AN Induction Generator; Voltage, Frequency AND Load of the Induction Generator; Short-circuit Current of the Induction Generator; Hunting op the Induction Generator; Advantages and Disadvan- tages OF the Induction Generator; Use of the In- duction Generator General Characteristics of the Induction Generator. • — An induction generator does not .differ in its general construction from an induction motor. Whether an induction machine acts as generator or motor depends solely upon its slip. Below synchronous speed, it can operate only as motor, above synchron- ous speed it becomes a generator. The power factor at .which an induction generator operates is fixed by its slip and its constants, and not in any way by the load. The quadrature component of the current output is nearly constant for any fixed terminal voltage and frequency and always leads the voltage. The power factor of the induction generator is fixed by the machine and not by the load, and it is, therefore, necessary to operate such generators in parallel with synchronous machines. These synchronous machines serve not only to supply the quadrature lagging current demanded by the Toad, but in addition to supply sufficient quadrature lag- ging current to neutralize the quadrature leading component of 32 497 NER-i 498 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the current delivered by the induction generator. The induction generator depends upon its quadrature leading current for excitation and unless the combined connected load calls for this leading component, the induction generator will lose its excitation and hence its voltage. The synchronous machines which are in parallel with an induction generator determine its voltage and frequency. Its sli p fixes it s output. Circle Diagram of the Induction Generator. — The circle diagram of Fig. 223, page 491 may be apphed to the induction generator by merely completing the circle. All currents which lie below the base line OC represent generator action. Changes in Power Produced by a Change in Slip. — The following changes in power occur as the slip of an induction generator changes. (a) At synchronous speed the rotor current is zerc^ The current in the stator comes entirely from the synchronous machines, and is the exciting current, 7„, of the vector diagram of Fig. 215, page 452. The core losses are supplied by the synchronous generators. The mechanical power required to drive the rotor at synchronous speed is equal to the friction and windggeHlosses. " (6) Below synchronous speed there is rotor current. To balance the demagnetizing action of this current there must ■ be an equivalent component current in the stator circuit. Under this condition only motor power can be developed. (c) Above synchronous speed the current in the rotor reverses in direction as will also the component current in the stator required to balance the* demagnetizing action of this rotor current. At any speed above synchronism generator action exists, but power will not be delivered to the external circuit until the current in the stator, which balances the demagnet- izing effect of the rotor current, has a component equal and opposite to the current, h + e, required to supply the core loss. At the slip at which this particular condition occurs, the gen- erator supplies its own core loss. Its external output is zero. At larger slip, power will be delivered to the load. 1 There may be harmonic currents in the rotor due to harmonics in the air-gap flux, but these harmonics and their effect will be small. POLYPHASE INDUCTION MOTORS 499 Power Factor of the Induction Generator. — The only current which can produce generator power in an induction generator is that component of the primary current which is equal and opposite to the rotor current. A 1:1 ratio of transformation between the rotor and stator is assumed. The power factor of this current with respect to the generated voltage is fixed by the rotor constants and the slip. It is given by cos 0, = Since the slip is small, x^'s'^ is small compared with rz^ and cos 0j^ is nearly unity. The load component of the primary current, the I'l of the usual transformer diagram, is, therefore, nearly in phase with the primary induced voltage. Neglecting the magnetizing current and the phase displacement of the ter- minal voltage due to the resistance and reactance drops in the primary windings, the primary current will be very nearly in phase with the terminal voltage. This is the basis of the common but incorrect statement, that an induction generator can deliver power only at unity power factor. The magnetizing current is not negligible and the power factor in consequence of this may differ considerably from unity. The correct statement is that an induction generator can deliver power only at leading power factor. The power factor, in the case of large machines, usually is over 90 per cent, at full load, but at no load or small loads it may be quite low. The quadrature component of the current, mainly magnetizing, varies little with the load. Phase Relation Between Rotor Current Referred to the Stator and Rotor Induced Voltage, E2. — The current in the rotor of an induction machine is always given by the following expression " rz + JX2S ^ J- Rationalizing this by multiplying both the numerator and the denominator by r^ — jx^s gives ^~ ^ ^ Below synchronous speed s is positive and the expression for Iz 500 PRINCIPLES OF ALTERNATING-CURRE^NT MACHINERY takes the form I2 = A — jB which represents a lagging current with respect to E^s. Above synchronous speed the slip becomes negative and the real part of the expression (210) for the current reverses its sign while the sign of the imaginary part remains unchanged. Under this condition, the expression for the rotor current becomes I2 = — A — jB. This represents a leading current with respect to E^s which has reversed its sign with the change in the sign of the slip. The current h in the rotor cannot, of course, actually lead the voltage in the rotor which causes it, since the rotor circuit is inductive. It is only when this current is considered with re- spect to the stator that it has this apparent phase relation. The reason for the apparent change in phase is the reversal of the relative direction of mo- tion of the revolving magnetic synchronouB Speed field and the rotor when the slip 1 changes sign. This may be seen — • ♦ e^^loTC^i by referring to Fig. 225. Let Fia. 225. the magnetic field move to the left, as is shown by the arrow. Consider the voltage induced in any inductor, a, on the rotor. This voltage will have its maximum value whdn the inductor is in the strongest part of the stator field, that is, in the position a, in Fig. 225. Below synchronous speed the rotor moves to the right relatively to the field, and since the rotor circuit is inductive, the inductor will move to some position as h before the current in it reaches its maximum value. Above synchronous speed, the rotor moves faster than the field and will be moving to the left with respect to it. In this case the inductor a will move to some such position as b' before the current in it reaches its maximum value. In both cases the rotor when coiisidered with respect to the stator moves in the same direction as the field, that is, from right to left. Therefore, if the electromotive force and current in the rotor are observed from any fixed point on the stator, the electromotive force will be seen to pass through its maximum value before the POLYPHASE INDUCTION MOTORS 501 current when the rotor is below synchronous speed and after the current when the rotor is above synchronous speed. Vector Diagram of the Induction Generator. — The vector diagram of the induction generator is shown in Fig. 226. 7i is the total stator current and cos di is the stator power factor. It is the power factor which would be calculated from the readings of instruments placed in the mains leading from the generator. The angle di cannot under any conditions be an angle of lag. For fixed terminal voltage, Vi, current, /i, and frequency /, there can be but one value of di and this an angle of lead. The induction generator, therefore, is a machine which has its power factor fixed by its constants and not by the power factor of the load. c>^^ .^--A"- Ei-E-i, Fig. 226.' Voltage, Magnetizing Current and Function of Synchronous Apparatus in Parallel with an Induction Generator. — The voltage of the generator depends upon the magnetizing com- ponent, 1^, of the primary current, I\, and unless the load calls for a component equal to this, the generator will lose its voltage. The function of the synchronous apparatus which must be operated in parallel with an induction generator is to absorb this current 7^, or, more correctly stated, to adjust the power factor of the load on the induction generator to that at which it can deliver the required power. With respect to the synchronous apparatus in parallel with the induction generator, the magnetizing current, 7^, which 502 PRINCIPLES OF ALTEBNATINO-CVBBENT MACHINERY leads when referred to the terminal voltage of the induction generator, becomes a lagging current when referred to th^ terminal voltage of the synchronous apparatus. The syn- chronous apparatus must not only supply whatever lagging current is called for by the load but must also supply a lagging current equal to the leading magnetizing current of the induction generator. For this reason the use of induction generators is limited to systems which have inherently high power factor. The synchronous apparatus may be synchronous generators, synchronous motors, or rotary converters. Use of a Condenser instead of a Synchronous Generator in Parallel with an Induction Generator. — It is possible to operate an induction generator without synchronous apparatus in parallel with it provided a suitable condenser is connected across its terminals, but this method of operation is of no prac- tical importance on account of the size and cost of the con- denser which would be required, as well as on account of the very drooping voltage characteristics of such a system. More- over, the system would not be self-exciting and would, therefore, ■require an initial excitation from a synchronous generator. Voltage, Frequency and Load of the Induction Generator. — Case I. — In Parallel with a Synchronous Generator. — The vol- tage of the induction generator is equal to the voltage; impressed across its terminals by the synchronous generator to which it is connected. The magnetizing current automatically adjusts itself to give this voltage. The frequency is determined by the frequency of the magnetizing current and is the same as the frequency of the synchronous generator. The load is fixed by the rotor current which depends on the slip. Case II. — In Parallel with a Synchronous Motor or a Rotary Converter. — As in Case I, the voltage of the induction generator is determined by the terminal voltage of the synchronous motor or the converter. The initial excitation must come from a synchronous generator or from the synchronous motor or the converter driven as a generator. The frequency is fixed by the speed of the rotor and by the load. It is equal to p n 2 60(1 - s) where p and n are, respectively, the number of poles and the POLYPHASE INDUCTION MOTORS 503 speed in revolutions per minute. It should be remembered that s is negative for generator action. The induction generator will carry the entire load and in- addition will supply all the losses of the synchronous machine. The synchronous machine will supply sufficient quadrature current to adjust the power factor of the load on the system to that corresponding to the inherent power factor of the induction generator for that load. The slip is fixed by the load. The voltage regulation of the system is similar tp the voltage regulation of a synchronous generator. The voltage at any given load is fixed by the con- stants of the induction machine, the excitation of the synchronous machine and the power factor of the circuit external to the induction machine. Short-circuit Cun;ent of the Induction Generator. — Since an induction generator depends for its excitation upon the synchronous apparatus with which it is in parallel, the current it can supply on short-circuit depends upon the drop in voltage produced at the terminals of the synchronous apparatus by the short-circuit. On a short-circuit which drops the terminal voltage to zero, no current will be supplied by the induction generator. Very little current will be suppUed on partial short-circuit since the maximum power an induction machine can deliver at any fixed slip and frequency is proportional to the square of its terminal voltage, equation (176), page 462. The inability to back up a short-circuit considerably reduces the resulting damage and permits the use of smaller and less ex- pensive circuit-breakers than could safely be used if the whole capacity of the system were in synchronous generators. Hunting of the Induction Generator. — An induction generator is free from hunting since it does not operate at synchronous speed. Any change in load must be accompanied by an actual change in speed instead of by a small angular displacement as with a synchronous "generator. The irregularities in angular velocity of prime movers during a single revolution are so small as to produce only insignificant changes in load. Advantages and Disadvantages of the Induction Generator. — Most of the advantages and disadvantages possessed by an induction generator are obvious from what has already been said. In a few words, the advantages are: the ruggedness of 504 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the rotating part, its failure to back up a short-circuit, its freedom from hunting, the construction of its rotating parts makes it well suited to high speeds, it requires no synchronizing, its voltage and frequency are automatically controlled by the voltage and frequency of the synchronous machines which operate in parallel with it, it requires little attention. Its disadvantages are: its fixed power factor and the con- sequent necessity to operate synchronous apparatus in parallel with it, the additional quadrature lagging current and reduced power factor at which the synchronous generators in parallel with it must operate. Use of the Induction Generator. — The induction generator, on account of the leading component of the current it delivers, is suitable only for central stations operating at high power factor, such as those feeding substations containing synchronous appa- ratus. The induction generator is well suited for operation when driven by exhaust-steam turbines which receive steam from reciprocating engines directly connected to synchronous genera- tors. In such cases the induction generator and its corresponding synchronous generator are connected together electrically as a unit and are brought up to speed together. No governor is required on the low-pressure steam turbine but it should be provided with some form of speed-limiting device. CHAPTER LI Calculation of the Constants of a Three-phase Induction Motor for the Equivalent Circuit; Calculation of Output, Torque, Input, Efficiency, Stator Current AND Power Factor from Equivalent Circuit for a Given Slip Motor.— A 1000-hp., 2200-volt, three-phase, 25-cycle, 12-pole motor will be used. The motor is F-connected and has a phase- wound rotor. Test data obtained from no-load and blocked runs and the measured stator and rotor resistances are given below. Ohmic resistance at 25°C. of stator between terminals 0.130 ohm Ohmic resistance at 25°C. of rotor between terminals . 0772 ohm At no load | Stator voltage between terminals = 2200 volts Temperature of windings \ ■ Stator current per termi- 25°C. nal. = 75.1 amp. Rotor short-circuited j Total input to stator = 15.2 kw. With rotor blocked Stator voltage between terminals = 2200 volts Temperature of windings ■ Stator current per termi- 25°C. nal = 1960 amp. Rotor short-circuited ^ Total input to stator = 1960 kw. With rotor blocked, at Stator voltage between approximately full-load terminals = 290 volts current Temperature of windings Stator current per termi- 25°C. nal = 250 amp. Rotor short-circuited Total input to stator Stator voltage between = 38 kw. terminals = 2200 volts Rotor on open circuit ; Rotor voltage between terminals = 1500 volts Friction and windage loss at rated speed 3.1 kw. 505 506 PRINCIPLES OF ALTERNATINO-CURRENT MACHINERY Calculation of Constants for the Equivalent Circuit. — The equations given in Chap. XL VIII, page 484 will be used. All constants will be calculated per phase. A slip of 0.018 and a. temperature of 75°C. will be assumed. Ratio of transformation = ., -^„ = 1.467 1500 2200 Rated stator phase voltage = —-^ — 1270 volts. The equivalent resistance is where Pt and h are the stator input per phase and the stator phase current, respectively, with blocked rotor. As the core loss due to the leakage flux which is included in Pb does not vary as the square of the current, Ve cannot be constant. For this reason, Pb should be for about full-load current. The equivalent resistance per phase at 25°C. is 38 X 1000 „„„„ , '^ = 3 X (250)^ = O-^O^ °^"'- The ohmic resistance of the rotor per phase at 25°C. referred to the stator is 0772 rj = ^^^Y~ (1-467)2 = 0.0831 ohm. If the effective resistances of the stator and rotor are assumed to be in the same ratio as the ohmic resistances, the effective resistances of the stator and rotor may be found by dividing the equivalent resistance of the motor into two parts which are proportional to the ohmic resistances of the stator and rotor. The effective resistance of the stator per phase at 25°C. is .. = 0.203 0-««^ 0.065 + 0.0831 0.089 ohm. The ohmic resistance of the stator per phase at 75°C. is r-i = 0.065 (1 + 50 X 0.00385) = 0.0775 ohm. POLYPHASE INDUCTION MOTORS 507 The local core losses produced by the stator which are included in the effective resistance are not appreciably affected by the temperature.^ Therefore, the effective resistance of the stator at 75°C. is equal to its effective resistance at 25°C. minus its ohmic resistance at 25°C. plus its ohmic resistance at 75°C. The effective resistance per phase of the stator at 75°C. is, therefore, Tel = 0.089 - 0.065 + 0.0775 = 0.102 ohm. The ohmic resistance of the rotor per phase at 75°C. referred to the stator is ra = 0.0831 (1 + 50 X 0.00385) = 0.0992 ohm. The no-load copper loss in the stator at 25°C. is 3(75.1)2 0.089 j^ = 1.5 kw. Neglecting the rotor copper loss, the no-load core loss is F„ = 15.2 - 3.1 - 1.5 = 10.6 kw. where the 3.1 is the friction and windage loss. This core loss is for a voltage equal to the rated terminal voltage instead of for a voltage equal to the full-load induced voltage, El. The use of this value of the core loss will cause little error in the case of a motor as large as this. Ih + e — where P„ is the core loss per phase. J^(10.6) X 1000 „ „ , Ih + e = ^^ — j^To = 2.78 amp. per phase. No-load power factor is 15.2 X 1000 = 0.0531 V3 X 2200 X 75.1 If, = I'n s/l — {no-load power factor)^ 508 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY where /'„ is the no-load current of the motor at rated voltage with the rotor short-circuited. I^ = 75.1 Vl - (0.0531) = 75.0 amp. 9"= V, Ih + e 2.78 1270 = 0.00219 mho per phase. h = ^ 75 = 0.0590 mho per phase. 1270 The equivalent impedance per phase is equal to the ratio of the stator phase voltage to the stator phase current with the rotor blocked preferably at rated voltage. V^ Xe = -f- y/\ — {blocked power factor)^ J- b The blocked power factor at rated voltage is 1960 X 1000 p.f.b = — 7= = 0.262 ^ VS X 1960 X 2200 1270 Xe = Y^Vl-{0.262r = 0.625 ohm. Assuming xi = Xi when X2 is referred to the stator Xi =■ X2 = -^ = 0.313 ohm. For the assumed slip of 0.018 T tS 0.0992 X 0.018 (0.0992)2 _|. (0.313)2(0.018)2 = 0.1809 mho. POLYPHASE INDUCTION MOTORS 509 0.313 X (0.018)2 (0.0992)2 + (0.313)2(0.018)2 = 0.0103 mho. The constants just calculated are brought together in the fol- lowing table. Everything in this table is referred to the stator and is per phase. All resistances, susceptances and admittances are for 75°C. 7, = 1270 volts h + e = 2.78 amp. 7p = 75.0 amp. gn = 0.00219 mho hn = 0.0590 mho r.i (effective) = 0.102 ohm r2(ohmic) = 0.0992 ohm xi = 0.313 ohm X2 = 0.313 ohm g2 = 0.1809 mho for a slip of 0.018 62 = 0.0103 mho for a slip of 0.018 Qab = Qn + Qi — 0.183 mho for a slip of 0.018 6a!, = hn + hi = 0.0693 mho for a slip of 0.018 Output. — . ^ II ' V(?2 + 52 Where G = I -\- gob ri + bob xi = 1 + 0.183 X 0.102 + 0.0693 X 0.313 = 1.040 B = habri + gab xi = 0.0693 X 0.102 - 0.183 X 0.313 = 0.050 127 ^' ^ Vl:i.040)2+(0.650p = 1219 volts. Output of motor = SPp = SEi^gtil — s) - {friction + wind- age loss) 510 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY 3(1219)^X0.982X0.1809 = 787 kw. 787 = 746 = 1055 hp. Torque, — The torque of the motor is 3P„ 27r ^ (1 - s) p To express the torque in pound feet, Pp must be expressed in foot-pounds per second. _ 1055 X 550 2t ^3^(1 - 0.018) = 22,550 pound feet. Input. — The input to the stator per phase is Pi = EAGgai + BK„) = (1219)2(1.040 X 0.1831 + 0.050 X 0.0693) = 287.5 kw. Pi for the whole motor = 3 X 287.5 = 862.5 kw. The values of G and B were found in calculating £^i. Efficiency. — The efficiency of the motor is Output 787 ,„„ ^, „ = -J^^ = 862:5 1°^ = 91-2 P^'- «e^t- Stator Phase Current. — ^The stator phase current is II = Eligab — jbai) = 1219a/(0.1831)2 + (0.0693)2 = 238.9 amp. Power Factor.— The stator power factor is Pi 287.5 Wl = 1270 X 238.9 ^^^ = ^^''^ P^"" ««°*- SINGLE-PHASE INDUCTION MOTORS CHAPTER LIT Single-phase Induction Motor; Windings; Method of Fer- raris FOR Explaining the Operation of the Single- phase Induction Motor Single-phase Induction Motor. — The running characteristics of a single-phase induction motor are quite satisfactory, but the motor is not so good as a polyphase motor since it possesses no starting torque. It is also much heavier than a polyphase motor for the same speed and output. The greater weight for a given output is not an inherent peculiarity of the single-phase induction motor alone but is characteristic of any single-phase motor or. generator. A polyphase induction motor has a starting torque which may be increased up to a certain limiting value by putting resistance in the rotor circuit. No amount of resistance inserted in the rotor of a single-phase induction motor can give it an initial starting torque. It must be started by some form of auxiliary device and must attain considerable speed before it will develop sufficient torque to overcome its own friction and windage. The direction of its rotation depends merely upon the direction in which it is started. Once started, it will operate as well in one direction as in the other. This absence of a starting torque and the consequent necessity for some form of auxiliary starting de- vice, are the chief factors which limit the use and size of single- phase induction motors. Motors of this type are not often used in ratings over 10 or 15 hp. except in those cases where only single- phase power is available. Single-phase motors cost from 30 to 60 per cent, more than polyphase motors of the same rating and speed. I Windings. — The general features of construction of a single- phase motor are similar to those of a polyphase motor. The es- sential difference is in the windings. The stator of a single-phase 511 512 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY motor always has a distributed single-phase winding usually with fractional pitch. The rotor is generally of the squirrel-cage type except when the auxiliary starting torque is obtained by convert- ing the motor into a repulsion motor while coming up to speed. If the repulsion-motor action is not used for starting, the stator must have an auxiliary starting winding or its equivalent, in addition to the regular winding. By subdividing the main wind- ing it is possible to make a part of this serve as the auxiliary wind- ing. Method of Ferraris for Explaining the Operation of the Single- phase Induction Motor.- — Ferraris has given an ingenious and simple explanation of the operation of a single-phase induction motor, but as it omits several important fac- tors it cannot be used for analytical develop- ment. It serves a useful purpose, however, in bringing out certain pecuharities in the operating characteristics of the motor. Any simple harmonic vector may be re- solved into two oppositely rotating vectors, each of the same period as the given vector and of one-half its magnitude (Synchronous generators, page 58). Let the single-phase stator field of the induction motor be re- placed by two such revolving vectors (Fig. 227). Each of the revolving component fields I and II acting alone would give rise to a speed-torque curve similar to that of any polyphase motor. Such a curve is shown for slips between and 200 per cent, in Fig. 228. If a polyphase motor is driven backwards from rest, its torque decreases. Below standstill or 100 per cent, slip, its torque curve is similar to the portion of the curve between s = 100 and s = 200 in Fig. 228. Synchronous speed with respect to field No. I is 200 per cent. sHp with respect to field No. II and synchronous speed with re- spect to field No. II is 200 per cent. sUp with respect to field No. I. The torques produced by the two fields are oppositely directed since the fields rotate in opposite directions. The speed-torque curves for the two component revolving fields are shown in Fig. 227. SINGLE-PHASE INDUCTION MOTORS 513 Fig. 229, where the torque produced by field No. I is assumed positive. Fig. 229. The sum of the ordinates of the two torque curves gives the resultant torque curve of the motor. This curve of resultant torque is shown dotted in Fig. 229. When the motor is at rest 33 514 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY the resultant torque is zero, and it becomes zero again slightly below synchronous speed, for either direction of rotation. At other speeds, it has perfectly definite values. If the motor is started in either direction and is brought up to such a speed that the resultant torque is greater than that required for load plus losses, the motor will gain speed until the stable part of the speed- torque curve is reached corresponding to the direction of rotation in which the motor is started. Fig. 230. One peculiarity of the single-phase induction motor is that its internal torque becomes zero at a speed slightly below syn- chronous speed. A single-phase induction motor could never reach synchronous speed even if its rotational losses could be made zero. Although the torque of a single-phase induction motor becomes zero before synchronous speed is reached, it can be shown that the change in slip under load is less than the change in slip of a polyphase motor. The slip of a polyphase motor is proportional to the rotor copper loss (equation 188, page SINGLE-PHASE INDUCTION MOTORS 515 475). For small values of slip, the slip of a single-phase induc- tion motor is very nearly proportional to the square root of the rotor copper loss. Adding resistance to the rotor of a single-phase induction motor not only increases its slip but decreases its maximum internal torque as well. The maximum internal torque devel- oped by a polyphase motor is independent of rotor resistance (equation 178, page 463). Fig. 230 shows the effect on the torque of adding resistance to the rotor of a single-phase induction motor. Although the method of Ferraris just outlined serves to ex- plain the general action of the single-phase induction motor, a rigorous analysis must include the reaction of the rotor. This factor is neglected in the method of Ferraris. CHAPTER LIII Quadrature Field of the Single-phase Induction Motor; Revolving Field OF THE Single-phase Induction Motor; Explanation of the Operation of the Single-phase In- duction Motor; Comparison of the Losses in Single- phase AND Polyphase Induction Motors Quadrature Field of the Single-phase Induction Motor. — At any speed other than zero a single-phase induction motor has a revolving magnetic field, produced by two component fields which are in space quadrature and very nearly in time quadra- ture. One of these component fields is due to the stator winding and for any given impressed voltage would be constant were it not for the change in the stator impedance drop with change in load. The other component field is due to the -current in the rotor produced by its rotation in the stator field. This second or quadrature field varies in magnitude with the speed. It is zero at zero speed and would be equal to the stator field at synchronous speed, were it not for the resistance and leakage- reactance drops in the rotor winding. The trace of the extremity of the vector which represents the revolving field produced by these two component fields, in quadrature, is very nearly circular at synchronous speed. Both below and above synchronous speed it is elliptical, with the major axis of the ellipse along the stator field below synchronous speed, and at right angles to the stator field above synchronous speed. Fig. 231 represents diagrammatically a single-phase induction motor with squirrel-cage rotor. M is the stator winding, which is distributed in the actual motor. The axis, aa, of the stator field is vertical. The variation in the stator flux induces voltages in the inductors of the squirrel- cage rotor. These voltages act in opposite directions on oppo- site sides of the axis aa. So far as these voltages are concerned, 516 SINGLE-PHASE INDUCTION MOTORS 517 the armature inductors may be paired off to form a series of closed coils, as indicated by the horizontal lines of Fig. 231. The voltages induced in these coils by a variation in the stator field will set up currents in the coils and these currents will react on the stator winding just as the current in the secondary of a short-circuited static transformer reacts on the primary. So far as concerns the effect of these currents on the stator winding, the rotor winding may be replaced by a single concen- trated winding whose axis is coincident with the axis of the stator field and whose sides are 66. When the rotor turns, two compo- nent electromotive forces are induced in its inductors by the stator field. One is caused by the transformer action of the stator field and is the same as the electromotive force induced in the rotor when at rest. The other is induced by the movement of the rotor inductors through the stator field due to rotation. The first is a pure transformer voltage, the second a pure speed voltage. The electromotive forces induced in the in- ductors by the transformer action are in the same direction in all inductors on the same side of the axis, aa, of the stator field (Fig. 232). Therefore, the axis of the rotor for these voltages is vertical, and any current they cause will react on the stator field. The rotor, so far as the effect of this current is concerned, acts like the closed secondary of a static transformer. For this current the rotor winding has the same effect as an equivalent number of turns concentrated at 66. The application of the right-hand rule will show that the volt- ages induced in the rotor inductors by their movenient through the stator field will act in the same direction in all inductors above the horizontal axis (Fig. 233) and in the opposite direction in all inductors below that axis. The axis of the rotor for these voltages is horizontal, therefore, and any currents they cause will react on the stator along a horizontal axis. So far as con- FiG. 231. 518 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY cerns the effect of these currents on the stator, the rotor winding may be replaced by a single concentrated winding having its axis at right angles to the axis of the stator field and its sides at aa. Since the axis for the speed electromotive forces is horizontal, i.e., at right angles to the stator field, any currents these electro- motive forces may cause can have no electromagnetic reaction on the stator winding. The axes of the rotor, for the two component electromotive forces induced in it by the stator field are at right angles or in space quadrature. Both component electromotive forces are produced in the same inductors by the same flux. The trans- FiG. 232. Fig. 233. former voltage is in time quadrature with the stator flux. The speed voltage is produced by a movement of the inductors across the stator field at a speed which is constant for any fixed load. The speed voltage, therefore, must be directly proportional to the stator field at every instant. The two component voltages, therefore, are in time quadrature. That component of the rotor current which may be considered as due to the rotation of the rotor in the stator field gives rise to a field which has its axis along bb, that is, in space quadrature with the axis of the stator field. Since there is no winding on the stator or on any other part of the motor upon which this current can react electromagnetically, the rotor must act, so far as this current and, therefore, so far as the axis bb is concerned, SINGLE-PHASE INDUCTION MOTORS 519 like a reactance coil. It is equivalent to a single winding on a magnetic circuit with two air gaps. Its reactance for the axis bb must be high. The current producing the quadrature field, therefore, must lag nearly 90 degrees behind the speed voltage. Since the maximum of the speed voltage coincides in time with the maximum of the stator flux, it follows that the current pro- ducing the quadrature field, and hence the quadrature field itself, are both nearly in time quadrature with the stator field. The two component currents in the rotor will be considered separately and will be referred to the stator as was the rotor current of the polyphase induction motor. All voltages induced in the rotor will also be referred to the stator. The actual cur- rent in a rotor inductor is the vector sum of the two component currents. The component currents in the rotor when referred to the stator will be designated by the letter I with a subscript, a or b, to indicate along which axis they react. For example lb is the component current producing an armature magnetic axis which coincides with the axis bb. It is the component cur- rent producing the quadrature field. Three subscripts must be used with the component voltages: one, a or 6, to indicate the armature axis to which it belongs; a second, M or Q, to indicate whether it is produced by the main or stator field or by the quad- rature field; a third, T or S, to indicate whether a transformer or a speed voltage is intended. For example, Ei,ms is the speed voltage induced in the rotor by its rotation in the stator field M. It produces a component current lb in the armature which reacts along the axis bb. Let (Pm and between the brushes aa. AH four voltages are assumed to be referred to the stator. They have the same effect as the corresponding voltages of the single-phase induction motor when operating with a squirrel-cage rotor. The current la between the brushes aa produces the motor power. The current h between the brushes bb is the magne- tizing current for the quadrature field. The brushes aa may be called the power brushes since they carry the current which produces the motor power. The brushes bb carry the current for the quadrature field and may, for this reason, be called the field brushes. The actual current carried by the rotor inductors is either the vector sum or difference of the currents la and h according to which quarter of the rotor is considered. When these currents are used in the vector diagram, they are referred to the stator. A motor with a drum-wound rotor, short-circuited along two diameters. Fig. 237, operates exactly like a motor with a squirrel- cage rotor. The vector diagrams of the two types of motor are the same. The addition of the commutator and the short- circuited brushes, however, makes it possible to control the power factor of the motor and also to vary its speed over a considerable range, both above and below synchronism. As low as one-half synchronous speed may be obtained in practice. The motor may also be operated above synchronous speed. There is no object of using a drum- wound armature except to secure one ox both of these results. Power-factor Compensation. — The single-phase induction mo- tor may be compensated for power factor by inserting a voltage in the brush circuit bb to cause the resultant voltage Ea (Fig. 236, page 528) to rotate in the direction of lead until the current la, which lags behind the voltage Ea by a fixed angle, leads the trans- former voltage EaMT- The angle by which 7„ lags behind Ea is fixed by the leakage reactance and the resistance of the rotor. SINGLE-PHASE INDUCTION MOTORS 535 A voltage which is in phase with that induced in the rotor by the transformer action of the stator field will rotate the voltage Ea in the desired direction. With this voltage inserted between the brushes bb, the component of the stator current which balances the demagnetizing effect of the rotor current /„ will lead the volt- age induced in "the stator by the stator flux. By giving this component of the stator current sufiicient lead, the quadrature component may be made to neutralize the lagging magnetizing current carried by the stator. The voltage required for power- factor compensation may be obtained from a compensating wind- ing placed in the stator slots with the regular stator winding. The brushes bb instead of being short-circuited are connected to the terminals of this winding. The voltage induced in the com- pensating winding will be in phase with the voltage EaMT (Fig- 236, page 528) when considered with respect to the stator, but when considered with respect to the rotor, it may be either in phase with or in opposition to the voltage EaMT according to the way the terminals of the compensating winding are connected to the brushes bf). Reversing the connections of the compensating coil with respect to the brushes bb reverses the phase of the voltage inserted in the rotor with respect to the voltage EaUT ^^ the rotor. The compensating field should be connected so as to make the inserted voltage in phase with EaMT- Instead of hav- ing a separate winding for the compensation, the regular stator winding may be made to serve the purpose by bringing out two taps from suitable points. When this is done, the stator wind- ing acts both as the primary winding with respect to the arma- ture and as the compensating winding. Vector Diagrams of the Compensated Motor. — The vector diagram of the compensated motor is shown in Fig. 238. This is similar to the diagram in Fig. 236, page 528, except that the voltage inserted by the compensating coil has been added. Referring to Fig. 238, let Ebc be the voltage induced in the compensating winding. The voltage EbB causing the current in the armature between the brushes bb is the vector sum of EbMs and Ebc, Fig. 238. This is balanced by the voltage Ej,qt together with the leakage-reactance and resistance drops due to h in the rotor and in the compensating winding. The current h is proportional to E^r and lags behind that voltage by nearly 90 636 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY degrees just as it did behind the voltage E^ms i" t^e case of the nncompensated motor. The result of adding the voltage Ei,c to the circuit 6?> is to rotate the current h and with it the quadra- ture flux ipQ so that tpQ lags by more than 90 degrees behind the voltage E^ms- The resultant voltage Ea causing the current in the power circuit of the motor, now leads E^^f instead of lagging behind it as in the uncompensated motor (Fig. 236). As a result, the c-oss Fig. 238. current la (Fig. 238) leads the transformer voltage Eaur by ^-n amount which depends upon the magnitude of the voltage Ef,c in- serted in the brush circuit 66. The current 7a reacts on the stator and causes by transformer action an equivalent and opposite current J'l to flow in the stator winding. If /„ leads the voltage EaMT the equivalent stator current I\ will lead the component voltage — Ey which must be impressed on the stator to balance the voltage Ex = Eaur induced in the stator winding by the SINGLE-PHASE INDUCTION MOTORS 537 flux PQ = k'\E,MT{l -s)±e'] = k'E.Mrlil - s) ± p-— At synchronous speed both e' and s must be zero and N li'ni ^ v^ Fia. 256. Vector Diagram. — The vector diagram of a singly fed series motor is given in Fig. 257. The subscripts /, a, and c refer to the main field, the armature and the compensating field, respectively. 564 PRINCIPLES OF ALTERNATING-CURRENT MACHINERY The current I, which is the same in all windings of the motor, is resolved with respect to the main field into a magnetizing com- ponent, Jp, and a component, h + e, which supplies the iron losses caused by the field. Referring to Fig. 257, Ir/ is the drop in voltage due to the resistance of the main field winding. I^/, in quadrature with I^, is the actual reactance drop in the main field due to the flux for the flux, (p^, a component, Fig. 266. h + e, supplying the core loss due to ^t and a load component, /'i, which balances the demagnetizing action of the armature current. The flux T, and a transformer voltage due to (pg. Since vT and (ps are both in space quadrature and also nearly in time quadrature, the two voltages they produce in any given circuit on the arma- ture must be nearly in time phase oppposition. The letter E will be used with three subscripts to indicate a voltage induced in the armature between either set of brushes. A letter, a or b, will indicate which set of brushes is considered, >Si or T, will indicate the field which is producing voltage and s or t will indicate whether the voltage is a speed or a transformer voltage. Power-factor Compensation. — The power factor of the repul- sion motor is determined largely by the reactance of its torque field. If this can be compensated for, the power factor of the motor will be high. It will not be unity even in this case on account of the magnetizing current required for its transformer field. By over-compensating, it may be made unity, however. In the circuit formed by the series transformer S.T. and the arma- ture between the excitation brushes bb, there are three voltages, namely: the voltage E^Ta produced by the rotation of the arma- ture in the field (pr, the voltage Ej,si produced by the transformer action of the field g and with I. 'The component current la in the armature lags behind Ea, the resultant of EaTt and EaSs) by an angle which is determined by the resistance and leakage reactance of the armature between the brushes aa. A voltage — EaTt, equal and opposite to EaTt, must be im- pressed on the stator to balance the voltage induced by (p^,. Add- ing the stator leakage-impedance drop Izt, of the stator, to this voltage gives the voltage drop Vt across the stator. Thus far the diagram is exactly like that for the uncompensated motor. There are two voltages in the armature between the brushes bb which must now be considered. These are the transformer voltage Ejjst and the speed voltage EbT,. As perfect compensa- tion is assumed, these voltages are equal. E^st is 90 degrees behind tps, and according to the convention E^ts is opposite in time phase to ^j>. The resultant Eb of these two voltages is equal to the resistance and leakage-reactance drops through the armature between the brushes bb and is equal to the voltage which must be impressed across the brushes bb by the series transformer. The voltage drop across the primary of the series transformer is equal to —Eb plus the equivalent leakage-imped- ance drop Izfr in the transformer. Adding —Eb and this equiv- alent impedance drop to Vt gives the voltage V, which must SERIES AND REPULSION MOTORS 593 be impressed across the whole motor, including the series transformer. Comparing Figs. 266 and 269, pages 575 and 591 respectively, it will be seen that the power factor of the compensated motor is much higher than the power factor of the uncompensated motor. On Fig. 266, the reactance part of the impedance drop Izg corresponds nearly to the voltage — E^st on Fig. 269, and is 90 degrees ahead of the current. Ebst, Fig. 269, is the real reactance voltage rise due to the speed field flux and is 90 degrees behind the flux. It is the neutralization of this voltage by the speed voltage which improves the power factor. Compensation by this method brings the impressed voltage into phase with the stator current' by neutralizing a reactive drop. The power factor Of the trans- former formed by the stator and the armature considered with respect to the brushes aa is not changed. When compensation is effected by adding voltage to the armature between the brushes bb, as. was done in the single-phase commutator-type induction motor, the power factor is corrected by bringing the stator current into phase with the stator impressed voltage by neutralizing the mag- netizing component of the stator current (see Figs. 238 and 239, pages 536 and 537, under "Single-phase Induction Motors"). The same method of power-factor adjustment could be applied to the series motor but it would require the use of an additional transformer. In the series motor the effect of the magnetizing current in the stator on power factor may be compensated by properly adjusting the two fields (ps and