■im^ 
 
CORNELL UNIVERSITY 
 
 LffiRARIES 
 
 ITHACA, N. Y. 14853 
 
 Engineering Library 
 Carpenter Hall 
 
Cornell University Library 
 TK1141.B41 1917 
 
 Alternating currents :an analytical and 
 
 3 1924 004 884 866 
 
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/cu31924004884866 
 
ALTERNATING CURRENTS: 
 
 AN ANALYTICAL AND GRAPHICAL TREATMENT 
 FOR STUDENTS AND ENGINEERS. 
 
 BY 
 
 FEEDEEICK BEDELL, Ph.D., 
 
 AND 
 
 ALBEET GUSHING CEEHOEE, Ph.D. 
 
 FIFTH EDITION. 
 
 MoGKAW PUBLISHIIvrG COMPANY, 
 
 New York. 
 
 CJOKNBLL Co-OPEBATIVE SoCIETy, 
 
 Ithaca, N. Y. 
 1917 
 
Copyright, 1901, 
 
 BY 
 
 P BEDELL AND A. C. CBEHOKE. 
 
 All rights reserved. 
 
PEEFACE. 
 
 The recent advances made in the utilization of alter- 
 Dating currents and alternating current apparatus of all 
 descriptions have been of such importance that there 
 are now many interested in this field of work who desire 
 to become conversant with the underlying principles of 
 the subject in order that they may become better 
 equipped to undertake the vast engineering problems 
 which are constantly arising. In its newness, the theory 
 of alternating currents has been developed this way and 
 that, added to here and there, so that it is necessary 
 for one to stop and consider the basis from which cer- 
 tain conclusions are reached, and the logical sequence 
 by which the results are attained. Although many of 
 the problems which arise have been fully treated by 
 various writers, the solutions have as a rule been limited 
 in their application to certain special cases, and have 
 for the most part been presented in a fragmentary man- 
 ner. This lack of a clear and succinct treatment, suffi- 
 ciently broad to be general in its application, has been 
 strongly felt, and it is in order to meet this demand for 
 definite information in regard to the fundamental prin- 
 ciples governing the flow of variable or alternating cur- 
 rents that this work is now presented to the public. 
 
 The purpose has been to use such mathematical terms 
 and analytical methods as make it possible for the dem- 
 
2 PREFACE. 
 
 onstrations to be exact and rigorous, and at the same 
 time to express the results in such a way as to be per- 
 fectly intelligible to those who do not desire to follow 
 the methods of proof, but are only interested in the con- 
 clusions reached. 
 
 There are some to whom graphical methods appeal 
 more strongly than analytical processes, and the cases 
 of simple circuits have accordingly been fully treated in 
 both ways. The problems of divided circuits and net- 
 works of conductors yield the more readily to graphical 
 treatment, inasmuch as analytical methods necessarily 
 become cumbersome and involved and do not appeal 
 directly to the senses. The subject is therefore capable 
 of two natural divisions, the analytical, which constitutes 
 Part I., and the graphical, which constitutes Part II. 
 
 In Part I. the discussion of circuits containing re- 
 sistance and self-induction only is first taken up, and the 
 first chapter contains the elementary principles neces- 
 sary for the establishment of the equation of energy for 
 such circuits. This equation is logically developed 
 from the experiments of Coulomb, Faraday, Joule, and 
 Ohm, upon which depends all the modern science of 
 electricity. The treatment is based upon simple ele- 
 mentary ideas and is complete in itself, so that no pre- 
 vious knowledge of the theory of electricity and mag- 
 netism is requisite. Taking the equation of energy as a 
 basis, in the following chapters the general solution for 
 the current is obtained, after which the various particu- 
 lar cases are taken up, in which the electromotive force 
 is assumed to vary as some definite function of the time. 
 The solution for each particular case is derived inde- 
 pendently from the differential equations and also from 
 the general integral equation. 
 
 Inasmuch as the assumption of an harmonic electro- 
 motive force often approximates to the truth, a chapter 
 
PREFACE. 3 
 
 has been devoted to the discussion of harmonic func 
 tions in order that the solutions obtained under such an 
 assumption may be the more clearly understood. 
 
 As is explained in the introductory chapter, the coeffi- 
 cient of self-induction L is considered constant, whereas 
 this is only strictly true if the permeability of the sur- 
 rounding medium is also constant. That this assump- 
 tion is nearly correct is readily seen by noting the curves 
 of magnetization for various commercial irons given by 
 Prof. Ewing, and by Mr. Steinmetz and Mr. M. E. 
 Thompson in this country, for it is not until a higher 
 degree of magnetization is reached than is ordinarily 
 met with in actual practice, that these curves deviate 
 materially from a straight line. 
 
 After the completion of the treatment of circuits con- 
 taining resistance and self-induction, the discussion of cir- 
 cuits containing resistance and capacity is taken up and 
 developed in a similar manner from elementary princi- 
 ples. From the simple ideas of static charge, the mean- 
 ing of potential and work is shown, leading up to the 
 derivation of the equation of energy and electromotive 
 forces for a circuit containing a condenser. Following 
 the same plan as in the treatment of circuits containing 
 self-induction, the general solution is first obtained, and 
 we thus have the expression for the current and charge 
 at any time for any impressed electromotive force what- 
 soever. Particular electromotive forces are then as- 
 sumed, and the solutions for these cases are obtained 
 from the general integral equation, and also independ- 
 ently by particular solutions. 
 
 The general case of circuits containing resistance, self- 
 induction, and capacity is next taken up, and the same 
 order of treatment is followed as iu the discussion of 
 circuits containing resistance and self-induction only, 
 and resistance and capacity only. Now that the con- 
 
4 PREFACE. 
 
 denser, as well as its older brother, the transformer, is 
 being applied to practical uses, the question of the action 
 of a condenser in a circuit with self-induction becomes 
 an important one, and the discussion of this case is given 
 at length, the same method of giving particular cases 
 after the general solution being followed as before. The 
 case of oscillatory and non-oscillatory charge is treated 
 at length as well as the corresponding case of discharge. 
 In order that the effects caused by the variation of the 
 constants of a circuit may be clearly understood,, curves 
 are drawn showing these effects for certain particular 
 cases. The nature of the flow of current immediately 
 after making a circuit is then investigated, and the re- 
 sults shown by plotting the instantaneous values for a 
 particular case. The neutralizing effects of self-induc- 
 tion and capacity are next discussed, and the necessary 
 conditions ascertained under which not only the instan- 
 taneous values of the current will be the same as though 
 the self-induction and capacity were absent, but likewise 
 the thermic and dynamometric effects. 
 
 The first part closes with an investigation of the 
 nature of wave propagation in a conductor possessing 
 self-induction and distributed capacity, a subject which 
 assumes importance in submarine cables and in ex- 
 tended telephone circuits. 
 
 The results obtained by analytical processes too often 
 fail to carry their full significance while in symbolic 
 form, and for this reason it has been found advisable to 
 give applications to concrete cases, and to draw curves 
 illustrating the points involved. In order that the full 
 significance of the results may be grasped, the values of 
 the quantities used in these numerical examples have in 
 all cases been given, so that the curves plotted show not 
 only the general nature of the relations between the 
 various quantities, but also the value of these quantities 
 
PREFACE. 5 
 
 in the particular cases assumed. The advantage of this 
 is especially shown in the discussion of the effects of the 
 variation of the constants in a circuit containing resist- 
 ance, self-induction and capacity, for it is the illustra- 
 tions which here bring out the true significance of the 
 effects. 
 
 In Part II. the same order is followed as in Part I. 
 The graphical method of treating problems of simple 
 circuits containing resistance and self-induction is first 
 fully established from the analytical results obtained in 
 Part I., and is then extended to problems arising in the 
 case of combination circuits. Problems arising in the 
 case of simple and combination circuits containing re- 
 sistance and capacity but no self-induction are then 
 solved, and finally the general case of circuits containing 
 resistance, self-induction and capacity is taken^ up, and 
 the graphical solutions given for series, parallel and 
 combined circuits. 
 
 The graphical methods are rigorously proved by the 
 analytical solutions obtained in the earlier part of the 
 book, but the development is such that those who do 
 not follow through the analytical proof may readily 
 apply these graphical methods to the solution of practi- 
 cal problems. 
 
 In order to avoid ambiguity, the same symbols are 
 used throughout with the same signification, and a list 
 of symbols used, together with their meanings, is given 
 in an appendix. 
 
 There have been many valuable papers on subjects 
 relating to alternating currents, among others those by 
 Dr. Duncan and Prof. Eyan in this country, and by Prof. 
 Ayrton, Dr. Sumpner, Dr. Fleming, and Mr. Blakesley in 
 England, and the electrical public has gained much in- 
 formation from the excellent works of the last two 
 writers. The subject has not, however, been hitherto 
 
6 PREFACE. 
 
 developed in the way followed in the succeeding pages, 
 and it is in order to meet the demand for a logical treat- 
 ment of the theory of alternating currents that this book 
 has been prepared. 
 
 Much of the matter here contained has already been 
 given by the writers in various papers, some of which 
 originally appeared as a series of articles in the Electrical 
 World, and others in the London Electrician, the Philo- 
 sophical Magazine, the American Journal of Science, and 
 the Transactions of the American Institute of Electrical En- 
 gineers. We have been permitted to use some of the 
 cuts from the latter, for which courtesy we desire to 
 extend our thanks. 
 
 The matter contained in the second part now appears 
 for the first time, with the exception of the method for 
 obtaining the equivalent resistance, self-induction and 
 capacity of parallel circuits, which was first given in the 
 Philosophical Magazine. 
 
 In all cases these papers have been carefully revised 
 and rewritten, and in many cases amplified to suit the 
 requirements of the book. 
 
 COBNELL UNrVBKSITY, ItHACA, N, Y., 
 
 PREFACE TO FIFTH EDITION. 
 
 In this edition, some minor changes have been made and 
 a few notes added, with no material alteration in either tlie 
 subject matter or the arrangement. 
 
 Ithaca, n: Y., 1909. 
 
CONTENTS. 
 
 PAET I. ANALYTICAL TREATMENT. 
 
 CHAPTER I. 
 
 Introductory to Circuits containing Resistance and Self- 
 induction. 
 
 Magnet. Lines of force. Field of force. Pole. North pole. Like poles 
 repel, unlike attract. Unit pole. Law of attraction. Intensity of 
 a field of force. Uniform iield. Unit line of force. Unit pole 
 has 4« lines of force. Induction. Current develops a field. Unit 
 current. Number of lines proportional to current. Self-induction. 
 E. M. P. Ohm's law. Quantity. Quantity definite for definite change 
 in lines. Joule's law. Energy dissipated in, heat. Total energy im- 
 parted to a circuit. Energy expended in field. Equation of energy. 
 Equation of E. M. P. 's. Pages 17-31. 
 
 CHAPTER II. 
 
 On Harmonic Functions. 
 
 Harmonic E. M. P. often assumed. Simple harmonic motion. Ampli- 
 tude. Period. Angular velocity. Frequency. Epoch. Phase. Lag. 
 Advance. Graphical representation of simple harmonic functions. 
 Average value of ordinates of sine-curve. Value of mean square of 
 ordinates of sine-curve. Periodic functions composed of several simple 
 sine-functions of like period, — of unlike period. Fourier's theorem. 
 
 Pages 33-41. 
 CHAPTER III. 
 
 Circuits containing Resistance and Self-induction. 
 
 Equations of energy and E. M. P. 's. Criterion of integrability. General 
 solution when e —f(t). 
 
 Case I. E. M. F. suddenly Removed, Solution from differential equa- 
 tion, — from general solution. Geometric construction of 
 logarithmic curve. 
 Case II. E. M. P. suddenly Introduced. Solution from differential 
 equation, — from general solution. 
 
 7 
 
8 CONTENTS. 
 
 Case III. Simple Harmonic E. M. F. Solution from general equation. 
 
 Impedance. Lag. Effect of exponential term at "make."' 
 Case IV. Any Periodic E. M. F. Sum of two sine-functions. Sum of 
 
 any number of sine-functions. Pages 43-59. 
 
 CHAPTER IV. 
 
 Introductory to Circuits containing Resistance and Capacity. 
 
 Plan to be followed. Charge. Law of force. Unit charge. Work in 
 moving a charge. Potential. Capacity. Energy of charge. Con- 
 denser, — energy of and capacity of. Capacity of parallel plates ; of 
 continuous conductor. Equation of energy, in terms of i; in terms 
 of q. Equation of E. M. P.'s. Pages 60-69. 
 
 CHAPTER V. 
 Circuits containing Resistance and Capacity. 
 
 Equation of E. M. F.'s; Differential equation in linear form. Criterion of 
 integrability. General solution when e =f{t). 
 
 Case I. Discbarge. Quantity and current from general solution, — from 
 differential equations. 
 
 Case II. Charge. Quantity and current from general solution, — from dif- 
 ferential equations. 
 
 Case III. Simple harmonic E. M. F. Quantity and current from general 
 solution. Discussion. 
 
 Case IV. Any periodic E. M. F. Pages 70-80. 
 
 CHAPTER VI. 
 
 Circuits containing Resistance, Self-induction, and Capacity. 
 General Solution. 
 
 Equation of energy in terms of e, i, and t ; in terms of e, q, and t. Equa- 
 tion of B. M. F.'s in terms of e, i, and t; in terms of e, q, and t. 
 Equations transformed for solving in terms of i and t ; in terms of 
 q and t. Complete solution for i in terms of t ; complete solution for q 
 in terms of t. Four cases will be considered: I. e=f{t) = 0; II. 
 e =f(t) = B; III. 6 =f{l) = .^sin not; IV. e =f{t) = 2 E ain (b co v + 6) 
 
 Pages 81-89. 
 
 CHAPTER VII. 
 
 Circuits containing Resistance, Self-induction, and Capacity. 
 Case I. Discharge. 
 
 Integral and differential equations when e =f(i) = 0. Sir Wm. Thomson's 
 solution, i equation with value of T replaced. Three forms of i and q 
 equations. To transform the j-equation to a real form when B^G is 
 less than 4-C. To derive the solutions from the differential equations 
 when H'C^z 4Z. 
 
CONTENTS. ■ 9 
 
 Non-oscillatory Discharge. 
 
 Determination of constants. Complete solution. Value of T re- 
 placed. Current and charge curves for a particular circuit. Time of 
 maximum current. EcLuation (125) applied to a circuit containing resist- 
 ance and self-induction only, and to a circuit containing resistance 
 and capacity only. 
 
 Oscillatory Discharge. 
 
 Determination of constants. Complete solution for i and q. Current 
 and charge curves for a particular circuit. 
 
 Discharge of Condenser when B^G = 4£. 
 
 Determination of constants. Complete solutions for i and q. Figure 
 showing method of constructing the current and charge curves. Curves 
 for i and g in a particular circuit. Pages 90-111. 
 
 CHAPTER VIII. 
 
 Circuits containing Bbsistance, Sblf-induction, and Capacity. 
 Case II. Chakgb. 
 
 Differential equations with e = /«) = E. Solution of these equations. 
 Solution from the general integral equation. Three forms of i and q 
 equations. 
 
 Non- oscillatory Gliarging. 
 Determination of constants. Complete solutions for i and q with 
 constants determined. Curves for i and ? In a particular circuit. Equa- 
 tion (101) applied to a circuit containing resistance and self-induction 
 only ; also to a circuit containing resistance and capacity only. 
 
 Oscillatory Charging. 
 Determination of constants. Complete solutions for i and q with 
 constants determined. Curves for i and g in a particular circuit. 
 
 Charge, of the Condenser when R'C=4:L. 
 Detei-mination of constants. Complete solutions for i and q with 
 constants determined. Curves for i and g in a particular circuit. 
 
 Pages 113-123. 
 
 CHAPTER IX. 
 
 Circuits containing Resistance, Self-induction, and Capacity. 
 Case III. Solution and Discussion for Harmonic E. M. F. 
 
 To find from the general solutions the particular equations in the case of an 
 harmonic E. M. F. Complete solutions for i and q. These same solu- 
 tions obtained directly from the differential equations. 
 
 Discussion of Case III. Harmonic E. M. F. 
 The impediment. Case A. Circuits containing resistance and self- 
 induction only. Case B. Circuits containing resistance and capacity 
 
10 CONTENTS. 
 
 only. Case C. Circuits containing resistance only. Case D. Circuits 
 containing capacity only. 
 
 ElffecU of Varying the Constants of a Circuit. 
 
 First. Electromotive force varied. Second. Resistance varied. 
 Tliird. Coefficient of self-induction varied. Fourth. Capacity .varied. 
 Fifth. The frequency varied. 
 
 Tlie energy expended per second upon a circuit in which an har- 
 monic circuit is flowing. Pages 124^143. 
 
 CHAPTER X. 
 
 Circuits containing Resistance, Self-induction, and Capacity. 
 
 Case III. (continued.) Current at the "Make" fob an 
 
 Harmonic E. M. F. 
 
 Complete equations for i and q with the complementary function in 
 the oscillatory form. To determine the constants A' and #'. To 
 determine the constants A and $. Complete equation for i with 
 constants determined. Examples to explain the general equation 
 in case of particular circuits. Curves showing the current at the make 
 for a particular circuit. The phase at which the E. M. P. should be 
 introduced to make the oscillation a maximum. Pages 144^155. 
 
 CHAPTER XI. 
 
 Circuits containino Resistance, Sblf-induction, and Capacity. 
 Case IV. Any Periodic E. M. F. 
 
 Fourier's theorem. General equations for i and q with any periodic 
 E. M. F. If the self-induction and capacity neutralize each other 
 at every point of time and the current is therefore the same as if 
 both self-iuduction and capacity were absent, the impressed E. M. F. 
 must be a simple harmonic E. M. F. If the heating effect, or any 
 effect which depends upon Jhdt, in a circuit, is the same when the self- 
 induction and capacity are present as it is when they are absent, the 
 impressed E. M. F. must be a simple harmonic E. M. F. Various 
 types of current curves. When curves are not symmetrical, although 
 the quantity flowing in the positive direction is equal to the quantity 
 in the negative direction, yet Xhefi^dt effect will generally be different 
 in these two directions. Illustration from a particular curve. Alter- 
 nating-current arc-light carbons. Pages 156-175. 
 
 CHAPTER XII. 
 
 Circuits containing Distributed Capacity and Self-induction. 
 General Solution. 
 
 Derivation of the differential equation for circuits containing distributed 
 capacity only. This equation extended so as to represent a particu- 
 lar case of distributed capacity and self-induction. Differential 
 
CONTENTS. 11 
 
 equation for E. M. P. is of the same form as that for current. The 
 general solutions of the differential equations. Particular assumption 
 of harmonic E. M. P. Constants of the general equation determined 
 under this assumption; first, from the exponential solution; second, 
 from the sine solution. Current determined from the E. M. P. 
 equation. Pages 176-193. 
 
 CHAPTER XIII. 
 
 CmCUITS CONTAINING DISTRIBUTED CAPACITY AND SbLP-INDUCTION. 
 
 Discussion. 
 
 Cireuita with no self-induction. Particular form of e and i equations. 
 Nature of waves. Rate of propagation. Wave-length. Decreasing 
 amplitude. Rate of decay with distance; with time. 
 
 Circuits with self-induction. Phase difference. Rate of propagation. 
 Diminishing amplitude. Rate of decay. Limitations of the tele- 
 phone. 
 
 Wave propagation in Closed Circuits. 
 Positive and negative waves travel around the circuit until they 
 vanish. Resultant effect. Potential zero at middle point of the cable. 
 Expression for potential simplified if the length of the cable is 
 a m\iltiple of a wave-length. Same results may be applied to the 
 current equation. Pages 193-207. 
 
 PART II.— GRAPHICAL TREATMENT. 
 
 CHAPTER XIV. 
 
 Intkoductoey to Part II. and to Circuits containing Resistance 
 AND Self-induction. 
 
 Introductory. Analytical solutions of Part I. for simple circuits extended 
 to compound circuits by graphical method. Arrangement of Part II. 
 Graphical representation of simple harmonic E. M. P. 's. Graphical rep- 
 resentation of the sum of simple harmonic E. M. F.'s of same period. 
 Triangle of E. M. F.'s for a single circuit containing resistance ana 
 self-induction. Impressed E. M. F. Effective E. M. P. Counter 
 E. M. P. of self-induction. Direction shown from differential equation. 
 Graphical representation. Methods to be used and .symbols adopted 
 in the graphical treatment of problems. First method (the one used 
 throughout this book), employing E. M. F. necessary to overcome self- 
 induction. Second method, employing E. M. F. of self-induction. 
 System of lettering and conventions adopted in graphical construction. 
 ' Pages 311-221. 
 
12 CONTENTS. 
 
 CHAPTER XV. 
 
 Pboblems with Circuits containing Resistance and Self-induction. 
 Sebies Circuits and Divided Circuits. 
 
 Prob. I. Efiects of the Variation of the Constants R and £ in a Series 
 Circuit. jB varied. L varied. 
 
 Prob. II. Series Circuit. Current given. 
 
 Prob. III. Series Circuit. Impressed E. M. F. given. 
 
 Prob. IIIo. Measurements on a Series Circuit. 
 
 Prob. IV. Divided Circuit. Two Branches. Impressed E. M. P. given. 
 Equivalent Resistance and Self-induction defined. 
 
 Prob. V. Divided Circuit. Any Number of Branches. Impressed 
 E.M. F. given. Equivalent Resistance and Self-induction 
 obtained for Parallel Circuits. 
 
 Prob. VI. Divided Circuit. Current given. First Method: Entirely- 
 Graphical. Second Method: Solution by Equivalent R 
 and L. 
 
 Prob. VII. Effects of the Variation of the Constants R and Z in a Divided 
 Circuit of Two Branches. R varied. L varied. Limiting 
 Cases. Constant Potential Example. Constant Current 
 Example. Pages 323-247. 
 
 CHAPTER XVI. 
 
 Problems with Circuits containing Resistance and Self-induction. 
 Combination Circuits. 
 
 Prob. VIII. Series and Parallel Circuits. Impressed E. M. F. given. 
 
 Solution by Equivalent R and L. 
 Prob. IX. Series and Parallel Circuits. Current given. Solution by 
 
 Equivalent R and L. 
 Prob. X. Extension of Problems VIII. and IX. 
 Prob. XI. Series and Parallel Circuits. Entirely Graphical Solution. 
 Prob. XII. Multiple-arc Arrangement. Pages 248-259. 
 
 CHAPTER XVII. 
 
 Problems with Circuits containing Resistance and Self-induction. 
 Moke than One Source of Electromotive Force. 
 
 Prob. XIII. Electromotive Forces in Series. 
 
 Prob. XIV. Direction of Rotation of E. M. F. Vectors. 
 
 Prob. XV. Electromotive Forces in Parallel. 
 
 Prob. XVI. Electromotive Forces having Different Periods. 
 
 Pages 260-266. 
 
CONTENTS. 13 
 
 CHAPTER XVIII. 
 
 Introductory to Circuits containing Resistance and Capacity. 
 
 Problems with B and G analytically and graphically analogous to prob- 
 lems with B and L. Triangle of E. M. F.'s for a single circuit 
 containing resistance and capacity. Impressed E. M. F. Effec- 
 tive E. M. F. Condenser E. M. I". Direction shown from differ- 
 ential equations. Graphical representation. Two methods used. 
 First method (the one used throughout this book), employing E. M. P. 
 necessary to overcome that of condenser. Second method, employing 
 E. M. F. of condenser. Further identification of analytical and 
 graphical relations. Mechanical analogue. Pages 267-273. 
 
 CHAPTER XIX. 
 Problems with Circuits containing Resistance and Capacity. 
 
 Prob. XVII. Effects of the Variation of the Constants B and C in a 
 
 Series Circuit, i? varied. varied. 
 Prob. XVIII. Series Circuit. Current given. Equivalent B and C in 
 
 Series. 
 Prob. XIX. Series Circuit. Impressed E. M. F. given. 
 Prob. XX. Divided Circuit. Two Branches. Impressed E. M. F. 
 
 given. Equivalent B and G for Parallel Circuit. 
 Prob. XXI. Divided Circuit. Any Number of Branches. Impressed 
 
 E. M. F. given. Equivalent B and G obtained for 
 
 Parallel Circuits. 
 Prob. XXII. Divided Circuit. Current given. First Method: Entirely 
 
 Graphical. Second Method: Solution by Equivalent B 
 
 and G. 
 Prob. XXIII. Effects of the Variation of the Constants B and C in a 
 
 Divided Circuit of Two Branches. 
 Prob. XXIV. Series and Parallel Circuits. Impressed E. M. P. given. 
 
 Solution by Equivalent B and G. 
 Prob. XXV. Series and Parallel Circuits. Current given. Solution by 
 
 Equivalent B and C. 
 Prob. XXVI. Series and Parallel Circuits. Entirely Graphical Solution. 
 Prob. XXVII. Multiple-arc Arrangement. Pages 274-291. 
 
 CHAPTER XX. 
 
 Circuits containing Resistance, Self-induction, and Capacity. 
 
 Introductory. Graphical methods for circuits with B, L, and G based 
 upc-i 'eraphical methods for circuits with B and L, and B and G. 
 Diagram of four E. M. F.'s. Triangle of E. M. F.'s. Method con- 
 sistent with analytical results obtained for circuits with B, L, and C. 
 
14 
 
 CONTENTS. 
 
 Capacity or self-induction which is equivalent to a combination of 
 capacity and self-induction. 
 Prob. XXVIII. Effects of the "Variation of the Constants in Series Circuit. 
 
 B, L, C, and eo varied. 
 Prob. XXIX. Series Circuit. Current given. Equivalent B, L, and G 
 
 of Series Circuit. 
 Prob. XXX. Series Circuit. Impressed E. M. F. given. 
 Prob. XXXI. Divided Circuit. Impressed E. M. F. given. Equivalent 
 
 B, L, and of Parallel Circuits. 
 Prob. XXXII. Example of a Divided Circuit. Impressed E. M. F. 
 
 given. 
 Prob. XXXIII. Divided Circuit. Current given. 
 Prob. XXXIV. Series and Parallel Combinations of Circuits. 
 
 Pages 392-311, 
 
 APPENDIX A, 
 Relation between Practical and C. Or. S. Units. 
 
 Page 312. 
 
 APPENDIX B. 
 Some Mechanical and Electrical Analogies. 
 
 Pages 313-315. 
 
 APPENDIX C. 
 
 System of Notation adopted. 
 
 INDEX. 
 
 Pages 316-318. 
 Pages 319-325. 
 
PAET I. 
 ANALYTICAL TREATMENT. 
 
CHAPTER I. 
 
 INTRODUCTORY TO CIRCUITS CONTAINING RESISTANCE 
 AND SELF INDUCTION. 
 
 CoNTEKTS: — Magnet. Lines of force. Field of force. Pole. North pole. 
 Like poles repel, unlike attract. Unit pole. Law of attraction. In- 
 tensity of a field of force. Uniform field. Unit line of force. Unit 
 pole has 4« lines of force. Induction. Current develops a field. Unit 
 current. Number of lines proportional to current. Self-induction. 
 E. M. F. Ohm's law. Quantity. Quantity definite for definite change 
 in lines. Joule's law. Energy dissipated in heat. Total energy im- 
 parted to a circuit. Energy expended in field. Equation of energy. 
 Equation of E. M. F.'s. 
 
 In order that circuits containing resistance and self- 
 induction may be properly discussed, a brief review will 
 first be given of the elementary theory of magnetism, the 
 nature of the magnetic field, and the relation between a 
 current of electricity and magnetism. Those well-known 
 elements of the subject will be presented which enable us 
 to obtain expressions for the energy imparted to a circuit, 
 the energy dissipated in heat, and the energy expended in 
 the magnetic field, and finally to establish the equation of 
 energy and the equation of electromotive forces for circuits 
 with resistance and self-induction. 
 
 If a needle is magnetized uniformly in the direction of 
 its length and placed in iron filings, the filings are at- 
 tracted to the ends of the needle and become attached 
 thereto in clusters. The attractive power of the magnet- 
 
 17 
 
18 INTRODUCTOBT TO CIBCUIT8 
 
 ized needle is apparently concentrated at the ends, whioli 
 are called poles. The filings in the space around the 
 magnet tend to gather in lines, called lines of force, extend- 
 ing from one pole of the needle to the other. Thus the 
 magnet is seen to be surrounded by &fiM of force, in which 
 the lines indicate the direction of the force at any point of 
 the field. When a compass-needle is placed in the field, it 
 always assumes a definite position, tangent to the line of 
 force passing through that point. The earth acts like a 
 huge magnet, producing a magnetic field in which the 
 lines of force have a direction nearly north and south. A 
 magnetized needle freely suspended in the earth's magnetic 
 field assumes a definite position tangent to the earth's lines 
 of force. This position is usually nearly in the geographical 
 meridian, the magnet having one pole toward the north 
 and the other toward the south. The pole that is toward 
 the north is called the positive pole, marked -\- ; and the 
 opposite pole the negative, marked — . When magnetic 
 poles are brought near one another there is found to be 
 either an attraction or a repulsion between them, and two 
 poles which have the same sign tend to repel one another, 
 while two poles of opposite sign tend to attract one another. 
 The definition of a unit magnetic pole would therefore 
 naturally be : a magnetic pole which exerts a force of one 
 dyne* upon another equal pole at a distance of one centi- 
 
 * Our knowledge of the physical universe is obtained from our per- 
 ception of matter in its relations to time and space; and physical measure- 
 ments are, accordingly, measurements of mass, length, and time. Any 
 quantity can be expressed in terms of these three, and the units in which 
 the quantity is measured can be expressed in the three fundamental units 
 of length, mass, and time. The fundamental units commonly used to 
 measure length, mass, and time are the centimetre, gramme, and second; 
 these are arbitrarily selected, and give rise to the C. G-. S. system of units. 
 All other units are readily obtained from these and are called derived units. 
 The wlocity of a body moving uniformly is the space passed over in a unit 
 time. For a body having a variable motion, the velocity is equal to an 
 
CONTAINma BB8ISTANCE AND SELF INDUCTION. 19 
 
 metre. Such a magnetic pole as this just defined forms the 
 foundation upon which is based the whole system of electro- 
 magnetic units, those of current, electromotive force, etc.; 
 and it therefore deserves attention. 
 
 This definition depends upon the exact measurement of 
 the distance between two poles. But in reality magnetic 
 poles have finite dimensions and it is necessary to deter- 
 mine, the mean distance between them. The distance 
 taken is that between two points so situated that the action 
 between the two poles would be the same if they were 
 concentrated at these two points. "We therefore think of 
 a magnetic pole as concentrated at a point. This concep- 
 tion is no more strained than the conception of' centre of 
 gravitative attraction of a body, where we consider the 
 whole mass of the body as concentrated at a point. 
 
 The length of a compound pendulum is measured in a 
 similar way, by considering that the mass of the pendulum 
 is concentrated at such a point that the time of oscillation 
 is not changed. 
 
 element of the distance ds, divided by the time dt, in which the distance 
 
 ds is traversed; that is, velocity equals the rate of change of length with 
 
 ds 
 respect to time, v — -z-. In the C. G. S. system velocity is measured in 
 
 centimetres per second. The acceleration of the body is the rate at which 
 
 dv d^s 
 the velocity is changing; that is, a = —- = -—, In the C. G. S. system 
 
 acceleration is measured in centimetres per second. 
 
 By Newton's first law, every body continues in a state of rest, or of uni- 
 form motion in a straight line, except in so far as it may be compelled by 
 impressed forces to change that state. Foi'ce may be defined as that which 
 causes or tends to cause a change in the velocity of a body. The unit of 
 force is that force which causes a unit change in velocity of a unit mass 
 in unit time, that is, produces unit acceleration. In the C. G. S. system 
 the unit of force is the dyne, and is the force which, when acting for one 
 second, will give a mass of one gramme a velocity of one centimetre per 
 second. 
 
20 INTRODUCTOBT TO CIRCUITS 
 
 LAW OK ATTEACTION. 
 
 The law of the action between magnetic poles, as ex- 
 perimentally determined by Coulomb, is that the attraction 
 or repulsion between two poles is inversely as the square 
 of the distance between them, and directly as the product 
 of their strengths ; that is, 
 
 7-, mm' 
 ^« -;^> 
 
 where m and m' are the strengths of the poles, that is, the 
 number of unit poles to which each is equivalent, and r 
 the distance between them. 
 
 A unit pole being as previously defined, the sign of 
 variation may be changed for one of equality if the dis- 
 tance r is measured in centimetres and the force F is 
 measured in dynes. The force between two magnetic poles 
 is then 
 
 „ mm' 
 F = • 
 
 When the poles considered have the same sign, and are 
 both north poles or both south poles, the product m m' is 
 positive, and a force of repulsion has the positive sign. 
 Similarly, a force of attraction has a negative sign, 
 
 INTENSITY OF A FIELD OF FOECE. 
 
 The strength of a magnetic field of force at any point is 
 measured by its action on a unit positive magnetic pole at 
 that point. 
 
 If we could place a free magnetic pole in a magnetic 
 field, it would always be urged in a certain direction ; and, 
 if free to move, would actually move in this direction. 
 The direction in which a positive pole would be urged is 
 called the positive direction of the line of force which 
 
CONTAINma RESISTANCE AND SELF INDUCTION. 21 
 
 passes through the pole. The force with which a unit 
 positive pole would be urged at any point of a magnetic 
 field is the strength of the field at that point, and is usually- 
 denoted by H. 
 
 Usually it is found that H varies at different points in 
 the field ; but if H has the same value at every point, both 
 in magnitude and in direction, the field is said to be uni- 
 form. 
 
 If the uniform field be one of unit intensity, then H = 1, 
 and there is said to be owe Zime of force per square centi- 
 metre ; and when the intensity is H, there are H lines of 
 force per square centimetre. Thus the intensity of a mag- 
 netic field is thought of as being determined by the number 
 of lines which pass through one square centimetre of a 
 surface normal to the direction of the lines of force. 
 
 As an example, by the definition of a unit pole the in- 
 tensity of the field H is unity at a distance of one centime- 
 tre from the pole. If a sphere be described about the unit 
 pole as a centre, having a radius of one centimetre, there is 
 consequently one line of force passing through the surface 
 of the sphere for every square centimetre. As the surface 
 of the sphere contains 4 n square centimetres, there are in 
 all 4 n lines of force that emanate from a unit pole, and 
 4 TT m lines from a pole whose strength is to. 
 
 INDUCTION. 
 
 The number of lines of force in air is the same as the 
 number of lines of magnetizing force. In a magnetic sub- 
 stance, such as iron, the number of lines of force is greatly 
 increased, and they are then called lines of magnetization, 
 or lines of induction. 
 
 The number of lines of induction N, passing through 
 any area, is called the total magnetic induction through 
 this area. The total number of lines of force per square 
 
22 INTRODUCTORY TO CIRCUITS 
 
 centimetre of area normal to these lines is called the in- 
 duction per square centimetre, or simply the induction, B. 
 In a non-magnetic medium the induction B is equal to the 
 magnetizing force, H. In a magnetic medium, such as iron, 
 the magnetizing force produces an induction B greater 
 than H. The ratio of the induction to the magnetizing 
 
 force is called the permeability, /x ; that is, /t = .e.. 
 
 H 
 
 A current of electricity flowing in a circuit always pro- 
 duces a magnetic field in the surrounding region. The 
 lines of force which constitute this field are always closed 
 lines which encircle the conductor. The total number of 
 lines passing through the area bounded by a closed electric 
 circuit is the total magnetic induction of the circuit. As 
 the current is increased in strength, the intensity of the 
 magnetic field at every point is increased, and if there is 
 no magnetic substance in the region, the intensity of the 
 field is increased in direct proportion to the strength of 
 current. 
 
 A unit current is defined in terms of the intensity of the 
 magnetic field which it generates. A unit current is that 
 which, flowing in a circuit of one centimetre radius, acts 
 on a unit magnetic pole, placed at the centre, with a force 
 of one dyne per centimetre length of the circumference. 
 This is a unit of current in the C. G. S. system. Each unit 
 length of conductor is acted upon by the unit pole, placed 
 at the centre, with a force of one dyne, which is the same 
 force as that by which a unit pole would be acted upon 
 when substituted for a unit length of the conductor at the 
 same distance. The practical unit of current, the ampere, 
 is one-tenth of the 0. G. S. unit. 
 
 NUMBEE OF LINES PKOPOETIONAL TO CUKEENT. 
 
 We have seen that when a current flows through a 
 closed circuit a field is set up consisting of a definite num- 
 
OONTAININQ BESI8TANCE AND SELF INDUCTION. 33 
 
 ber of lines threading tlie circuit. If there is no magnetic 
 substance in the vicinity, that is if the permeaMlity of the 
 surrounding region be constant, the number of lines pro- 
 duced by a current in a circuit is directly proportional to 
 the current, and any change in the current produces a pro- 
 portional change in the number of lines threading the cir- 
 cuit. 
 
 This may be expressed JV oc i, and -^7- a -j,- Since N 
 
 varies as i, we may say 
 
 N=Zi, 
 
 (1) and consequently —57- — -^ -jx- 
 
 The coefficient Z is called the coefficient of self-induction, 
 and is defined by the equation as the ratio of the total in- 
 duction threading a circuit to the current producing it. 
 When the current is unity, the coefficient of self-induction 
 is equal to the number of lines produced by the current. 
 If the permeability of the medium surrounding the con- 
 ductor is constant, this will be the value of L for all values 
 of the current, and L will be constant. Unless a high degree 
 of magnetization is reached, Z is approximately a constant 
 for a given circuit, and will hereafter be so considered. 
 
 faeaday's law of electeomotive foece. 
 
 When a conductor is moved in a magnetic field so as 
 to cut lines of force an electromotive force is produced in 
 the conductor. Faraday showed as the result of his re- 
 searches that this E. M. F. produced is directly propor- 
 tional to the rate of cutting the lines of force, and is in a 
 direction at right angles to the direction of motion and also 
 to the direction of the lines of force. He further showed 
 that, if the magnetic induction through any closed circuit 
 be varied by any means, an E. M. F. is developed in the 
 
24 INTROBUCTOBT TO GIBCUITS 
 
 circuit proportional at any instant to the rate of change 
 (decrease) of the magnetic induction at that instant. In 
 the 0. G. S. system of units this experimental law may be 
 expressed by the equation 
 
 where e denotes the E. M. F. developed and iTthe magnetic 
 induction of the circuit. This means that a 0. G. S. unit 
 E. M. F. is developed when there is a change in the induc- 
 tion of the circuit at the rate of one line per second. The 
 negative sign indicates that the E. M. F. is induced in such 
 a direction as to oppose the change in the number of lines 
 threading the circuit. The practical unit of E. M. F., the 
 volt, is 10" times the C. G. S. unit just defined. 
 
 ohm's law. 
 
 An electromotive force impressed upon a closed circuit 
 causes a current to flow which depends upon the resistance 
 of the circuit. Ohm first showed, and others have since 
 verified to a high degree of accuracy, that with a constant 
 E. M. F., the current is directly proportional to the E. M. F. 
 and inversely proportional to the resistance of the circuit. 
 
 Ohm's law may be expressed 
 
 where / denotes current ; E, electromotive force ; and B, 
 resistance. Since E. M. F. and current are already inde- 
 pendently defined, the unit of resistance is naturally taken 
 to be that resistance which allows a unit current to flow 
 in a circuit having a unit impressed E. M. F. Ohm's law 
 may then be expressed 
 
CONTAININQ BE8I8TANCE AND 8BLF INDUCTION. 25 
 
 From tLe relation of the practical units of E. M. F. and 
 current, the volt and the ampere, to the corresponding 
 C. G. S. units, it follows that the practical unit of resist- 
 ance, the ohm, is 10° times the 0. G. S.'unit. 
 
 QUANTITY. 
 
 A unit quantity of electricity is said to flow in a circuit 
 when unit current flows for one second. When a current 2 
 flows in a circuit for t seconds, a quantity It wil 1 flow. 
 And in a short interval of time dt, a quantity i dt will flow, 
 which is represented by dq, thus : 
 
 dq 
 
 q representing quantity. 
 
 We have seen that for a constant electromotive force, by 
 Ohm's law the current equals the E. M. F. divided by the 
 resistance. During a short interval of time dt, any E. M. F. 
 may be considered constant, and we may write 
 
 i = -p, during the time dt. 
 
 The capital letters E, I, and Q will be used to denote a 
 constant electromotive force, current, or charge. When 
 these are variable the small letters e, i, and q will be used. 
 
 When a closed conductor is moved from one position to 
 another in a magnetic field, so as to cause the number of 
 lines of force included by the circuit to change from one 
 value iVi to another value N^ , it will be found that the 
 quantity of electricity which flows in the circuit is always 
 a definite amount, being equal to the change in the number 
 of lines N, — N„ divided by the resistance of the circuit, 
 and is entirely independent of the manner of the change, 
 and of the time occupied in making the change. 
 
26 INTRODUCTORY TO CIRCUITS 
 
 This will be evident when we remember Faraday's law, 
 e = — -TT, and consider that the only E. M. F. acting in 
 
 Civ 
 
 the circuit during the motion of the conductor is this 
 • Hence the following relations are true : 
 
 dt 
 
 dq . e 1 dN 
 
 dt ~'^^ E^~BW 
 
 N - N 
 (3) Whence Q= '^ ' ■ 
 
 Here Q denotes the sum of all the small quantities, or 
 the total quantity of electricity flowing through the circuit 
 during the motion of the conductor, and is seen to be equal 
 to the change of the induction through the circuit diyided 
 by the resistance of the circuit, as stated above. 
 
 The earth inductor is a good example of an instrument 
 which depends for its use upon the principle just stated. 
 When a ballistic galvanometer is connected with an earth 
 inductor, the throw of the galvanometer is proportional to 
 the total change in the number of lines of force included 
 by the earth inductor coil as it turns from one position to 
 another, provided the needle does not start to swing until 
 the whole quantity of electricity has flowed through the 
 galvanometer. 
 
 joule's law. 
 
 The fourth and last great experimental law to be men- 
 tioned is the discovery by Joule that the heat liberated by 
 a conductor carrying a current of electricity is strictly pro- 
 portional to the product of the square of the current-strength 
 and the resistance of the conductor. 
 
 Now for the flrst time we have a means of telling how 
 much power is required to send a current of any desired 
 strength through a conductor, and we always expect to find 
 
CONTAINING BEBISTANOE AND SELF INDUCTION. 27 
 
 some source for the supply of energy when we see a current 
 flowing through a conductor. 
 
 The elementary principles, already given, upon which 
 the system of electromagnetic units is based, are deduced 
 from the experimental researches of Coulomb, Faraday, and 
 Ohm. When a current flows through a conductor there is 
 always heat liberated in the conductor and accordingly a 
 dissipation of energy. It therefore requires a certain 
 amount of power to send a current through a conductor. 
 The exact amount of this heating effect was first determined 
 by Joule. The results of his experiments show that the 
 energy liberated per second in the form of heat in a con- 
 ductor carrying a current of electricity is strictly propor- 
 tional to the product of the square of the current-strength 
 and the resistance of the conductor. Joule's law may be 
 expressed 
 
 WcxPB, 
 
 where W represents the energy expended per second. 
 
 The energy expended in the time t, during which the 
 current is constant, is P B t. If the current be a variable 
 i, it may be considered constant for the time dt, and so in 
 the time dt 
 
 (4) Energy dissipated in heat i& dW = i^ B dt. 
 
 When all the energy given to the circuit is expended in 
 heat, that is when there is no counter E. M. F. of any kind 
 and the current is constant, 7^ may be replaced by E, 
 according to Ohm's law, and the energy expended in the 
 time t may be written 
 
 Wo:EIt. 
 
 This becomes more definite in the units already dis- 
 cussed. If a conductor carrying a current I be placed at 
 right angles to the lines of force in a uniform field of 
 
28 INTRODVCTOBT TO CIRCUITS. 
 
 intensity H, each unit of length will be acted upon with a 
 force HZ. HI be the length of the conductor, the force 
 will be Z H /. When moved with a velocity v against this 
 force, work will be performed at the rate oilHIv ergs per 
 second, or in the time t the work done is 
 
 W=lWIvt. 
 
 This must be equal to the rate at which work is done in 
 generating a current /, by moving the conductor through 
 the field. The conductor, when moving with a velocity v, 
 cuts I H V lines per second, and so produces an E. M. F. 
 
 E=l\\v. 
 
 Substituting above, we see that the amount of energy 
 expended in a circuit is equal to the product of the cur- 
 rent, electromotive force, and time, 
 
 W=EIt. 
 
 This is seen to be equivalent to Joule's law above and is 
 equally true for C. G. S. and for practical units. In the 
 C. G. S. system, energy is measured in ergs and the equa- 
 tion expresses the fact that energy in ergs is equal to the 
 product of current, E. M. F., and time in C. G. S. units. 
 The practical unit of energy is the jorde and is so defined 
 that the equation W = E It, true in ergs and other C. G. S. 
 units, shall be also true in practical units — the joule, the 
 volt, and the ampere. The equation is then interpreted as 
 meaning that energy in joules is equal to the product of 
 current, E. M. F., and time in amperes, volts, and seconds. 
 From the relation already given between the ampere and 
 volt and their corresponding C. G. S. units, the joule equals 
 10' times the C. G. S. unit the erg. 
 
 The rate of work is in electrical terms expressed in 
 watts : one watt equals one joule per second. The common 
 
CONTAININO BESISTANCE AND SELF INDUCTION. 29 
 
 English unit of rate of work is the horse-power : one horse- 
 power equals 745.9 watts. 
 
 The rate at which energy is imparted to a circuit multi- 
 plied by the time is the total energy imparted during that 
 time. If there is a variable E. M. F., e, from any source 
 whatever given to a circuit, and a current i flows, the energy 
 imparted to the circuit in the time dt from the source of 
 this E. M. E. is the product e i dt. Thus : 
 
 (5) Energy imparted to a circuit = ei dt. 
 
 This enables us to ascertain the energy possessed by a 
 magnetic field. By Faraday's law, when the magnetic 
 induction through any closed circuit changes from any 
 cause whatsoever, there is always an electromotive force 
 given to the circuit which is equal to 
 
 dN di 
 
 • ~ dt ~ dt 
 
 This E. M. F. is due to the existence of the magnetic 
 
 field. In creating the field, an equal and opposite E. M. F., 
 
 di 
 L ri,is necessary to drive the current. The work which 
 
 this force does is equal to the product of the force, the 
 current which flows in the circuit, and the time dt ; as ex- 
 plained above. The change in the energy possessed by a 
 magnetic field in the time dt is, therefore, 
 
 .dN .di 
 
 i^iT-dt = Li-ndt. 
 dt dt 
 
 di 
 {6) Energy expended in the magnetic field = Li-jj dt. 
 
 The change in the induction through any circuit may 
 be due to any external cause, as moving magnets, or it may 
 be due to a change in the current itself. "When the change 
 is due to a change in the current, an increase in the 
 
30 INTRODUCTORY TO CIRCUITS 
 
 strength of the current increases the energy of the 
 magnetic field ; and positive work is done by the current 
 in creating the field. When the current decreases, the 
 energy of the field decreases, and negative work is done by 
 the current on the field ; for, when the current decreases 
 
 with the time, 37 is negative. To say that the current is 
 
 doing negative work is equivalent to saying that the 
 magnetic field in decreasing is imparting energy to the 
 circuit. Thus we see that the energy may be stored up in 
 a magnetic field, and that this is not dissipated, but is 
 returned to the circuit when the field is diminished in 
 strength. 
 
 To find the expression for the total energy of a mag- 
 netic field which is due to a current i flowing in a circuit, 
 we need merely find the sum of all the small quantities of 
 energy imparted to the field as the current is increased 
 from zero to its final value /: this is found to be 
 
 (7) £Lidi = iLr. 
 
 THE EQUATION OF ENEKGT. 
 
 If e represents the impressed E. M. F. given to a circuit 
 which has a resistance ^ and a coefficient of self-induction 
 L, we have seen [equation (5)] that the total energy given 
 to the circuit from the source is e i dt. 
 
 A part of this energy supplied is dissipated in heating 
 the conductor, and in the time dt is equal to B i' dt [equa- 
 tion (4)]. A second part is stored up in the magnetic field, 
 
 di 
 and in the time dt is equal io Li -r, dt [equation (6)]. These 
 
 two ways are the only ones in which the energy of the 
 source is used, under the hypothesis made that the circuit 
 contains no statical capacity or counter electromotive force 
 
CONTAINING RESiaTANCE AND SELF INDUCTION. 31 
 
 of any kind other than that due to the field, but only 
 contains a resistance R and a self-induction L. 
 
 By the principle of the conservation of energy we may, 
 therefore, say that the energy supplied to the circuit is 
 the sum of the energy dissipated in heat and the energy 
 expended on the field. 
 
 We have, therefore, the equation of energy : 
 
 (8) eidt = Bi'dt-\-Li%dt. 
 
 When each member of the equation of energy is divided 
 by i dt, we obtain 
 
 (9) e = i?. + z| 
 
 This is an equation of electromotive forces : e is the E. M. F. 
 
 of the source impressed upon the circuit, B i the E. M. F. 
 
 di 
 necessary to overcome the ohmic resistance, and L -n the 
 
 (XZ 
 
 E, M. F. equal to the E. M, F. of self-induction. 
 
 KoTB. — The number of lines or the total Induction threading a circuit, 
 when the circuit consists of a coil of several turns, is equal to the number 
 of lines which pass through the coil as a whole, multiplied by the number 
 of turns. One line passing through a coil of s turns actually threads the 
 circuit s times; thus, if 3,000 lines pass through a coil of 50 turns, the total 
 induction of the circuit is iV= 3,000 X 50 = 150,000. The explanations on 
 p. 38 et seg. and equations (1), (3), and (3) are to be thus understood. 
 
 The coefficient of self-induction is defined in terms of the counter-electro- 
 motive force of self-induction as follows : e = L— ; hence, the coefficient 
 
 at 
 
 of self-induction (or inductance) is the ratio of the counter-electromotive 
 force of self-induction to the time-rate of change of the current producing 
 it. The unit of self-induction, the henry, as defined by the International 
 Electrical Congress, Chicago, 1893, is the self-induction of a circuit when 
 the electromotive force induced in the circuit is one volt, while the inducing 
 current varies at the rate of one ampere per second. 
 
CHAPTER II. 
 
 ON HARMONIC FUNCTIONS. 
 
 Contents : — Harmonic E. M. F. often assumed. Simple harmonic motion. 
 Amplitude. Period. Angular velocity. Frequency. Epoch. Phase. 
 Lag. Advance. Graphical representation of simple harmonic functions. 
 Average value of ordinates of sine-curve. Value of mean square of 
 ordinates of sine-curve. Periodic functions composed of several simple 
 sine-functions of like period, — of unlike period. Fourier's theorem. 
 
 If a conductor revolves with uniform velocity about 
 some fixed axis in a uniform field, the rate at which it cuts 
 the lines of force is different at different parts of the revo- 
 lution and varies directly as the sine of the angle of rota- 
 tion. The electromotive force set up in the conductor at 
 any instant is numerically equal to the rate of cutting lines 
 at that instant and is accordingly a sine-function of the 
 angle of rotation and, since the rotation is uniform, a sine- 
 function of the time. Inasmuch as the assumption of 
 such an electromotive force often closely approximates to 
 the truth, and since, as will be shown later, any electro- 
 motive force whatever may be expressed as a sum of terms 
 each of which is a sine-function of the time, it has been 
 found convenient to express electromotive forces in terms 
 of sines. 
 
 In order that sine-functions may be clearly understood 
 when used in the following chapters, it is considered 
 advisable to digress and devote the present chapter to the 
 discussion of harmonic or sine-functions. 
 
 33 
 
ON HARMONIC FUNCTIONS. 
 
 33 
 
 HARMONIC MOTION. 
 
 If a point moves uniforinly around the circumference of 
 a circle, the motion of the projection of that point upon 
 any fixed diameter is said to be harmonic. The radius of 
 the circle is called the amplitvde of the motion, and is 
 designated by a. The time T of making one complete 
 revolution is called the period. Positive rotation will be 
 considered as counter-clockwise. 
 
 If a uniformly revolving radius of a circle is projected 
 upon any fixed diameter, its projection is said to vary 
 harmonically. The maximum value of this projection is 
 the amplitude, or radius of the circle. This is represented 
 ill Fig. 1. P is a point moving uniformly about the centre 
 
 Fig. 1.— Harmonic Motion. 
 
 0, and OP' is the projection of the radius OP upon the 
 fixed diameter BD. When OP is in the position OA at 
 right angles to BD, the projection OP' is zero ; and when 
 OP is in the position OP, the projection OP' has its 
 maximum value and is equal to the radius OP. The 
 projection is again zero at OG, and a negative maximum 
 at OD. 
 
34 ON HARMONIC FUNCTIONS. 
 
 The angular velocity of the point P is denoted by w, 
 and at any point is c» = -rr. Since the motion of the point 
 
 is uniform, ca = — , or os^ = 0, where is the angle described 
 
 in the time t. As the tiine occupied in describing a circum- 
 
 ference is T, the uniform velocity gj = -=7 ; hence = -^t. 
 
 The second is taken as the unit of time. The number of 
 revolutions made by the moving point P, in one second is 
 
 -^ and is called the periodicity or frequency, often denoted 
 
 by n. The frequency is the reciprocal of the period, i.e., 
 
 n = -~. It is evident that the angular velocity may be 
 
 expressed in terms of the frequency; thus, co = ^nn, and 
 therefore, = 'innt. 
 
 If we begin to count the time from the point A (Fig. 1), 
 where the projection of OP' is zero, denoting OP' by y, we 
 have at any time 
 
 y ^ a sin (p = a sin oo t, 
 
 where a denotes the amplitude and the angle described 
 in the time t; ?/ is an harmonic or sine-function of the angle 
 or the time t. 
 
 Suppose that the time is counted from some point Q, 
 Fig. 2, other than the point A at which the projection of OP 
 is zero. There is an angle 0, called the angle of epoch, be- 
 tween the point from which time is reckoned and the point 
 at which the projection of the radius is zero. The time in 
 which this angle is described is called simply the epoch. 
 As before, the angle is that described in the time t. The 
 angle (0 + 6), through which the point P has revolved 
 from the point A where the projection of OP is zero, is 
 called the angle of phase or briefly the phase. More strictly 
 
Olf EABMQNIG FUNCTIONS. 
 
 35 
 
 defined, the phase is the ratio of the arc PA to the circum- 
 ference of the circle. 
 
 If we denote by y the projection of OF upon BD, and 
 count time from Q, 
 
 (10) 2/ = a sin (0 + (9) = a sin {oot-\-(f). 
 
 When 6 is positive, — that is when it is in the positive or 
 counter-clockwise direction from A, as in Fig. 2, — it is often 
 called the angle of advance. When 6 is negative, — and the 
 
 
 ^ 
 
 p' 
 
 - ^ 
 
 ^p 
 
 
 - -V 
 
 ""K 
 
 
 
 
 ; 
 
 / 
 
 A 
 
 3 
 
 V- - 
 
 -/ 
 
 V 
 
 +a\ 
 
 j/ = OBin(ut+e) 
 
 / 
 
 a 
 
 
 /^ 
 
 V 
 
 \ 
 
 7 
 
 ^-. 
 
 
 \ 
 
 / 
 
 c 
 
 V 
 
 
 
 
 y 
 
 A 
 
 A 
 
 a' 
 
 o 
 
 II 
 
 a' p' 
 
 y. 
 
 S^' Time 
 
 d' / 
 -a / 
 
 O' 
 
 
 c 
 
 ) 
 
 
 
 
 
 TT 
 
 ^ d' 
 
 ^' Phase 
 
 !jr 
 
 Fig. 2.— Simple Sine-curve. 
 
 time is counted from some point Q' at a distance in the 
 negative or clockwise direction from A, — it is often called 
 the angle of lag. When the angle of phase is zero, OP co- 
 incides with OA and the projection y = 0. When the 
 phase is 90°, the projection is a maximum, and y =. -\-a. 
 At 180°, again, j/ = ; and at 270°, y = — a, a maximum in 
 the negative direction. This cycle is repeated every revo- 
 lution. 
 
 In the equation y = a sin {Got-\- 0), the amplitude, a, 
 angular velocity, oo, and angle of epoch, 6, are constants, 
 and the variable y is said to be expressed as a simple sine- 
 function of the variable t. The time t is directly propor- 
 tional to the angular distance passed through. A variable 
 whose value at any time can be expressed as a constant 
 multiplied by the sine of an angle changing uniformly with 
 
36 ON HARMONIC FUNCTIONS. 
 
 the time, is called a simple sine-function, or simple har- 
 monic function of the time. 
 
 In Fig. 2, y is plotted as a sine-function of t. At any- 
 time, t, when ^e revolving point has the position P, y has 
 a -value OP'. 
 
 Angle of epoch = AOQ = d. 
 
 Time of epoch = a'q'. 
 
 Angle described in time t = QOP = (f) ^ oat. 
 
 Angle of phase — (l>-\- (angle of epoch) — cp -\- 8. 
 
 Time of phase =t-\- (time of epoch) =-t-\- a'q'. 
 
 Amplitude = OA = OB — a. 
 
 When the term " harmonic function" or " sine-function" 
 or " sine-curve" is used, such a function or curve as shown 
 in Fig. 2 is meant. 
 
 TO FIND THE AVERAGE VALUE OF THE OEDINATE OF A 
 
 SINE-OUEVE. 
 
 A sine-curve repeats itself symmetrically and the aver- 
 age ordinate for the whole period is, therefore, algebraically 
 zero, as it is negative and positive alternately for equal 
 intervals of time. We can, however, obtain the average 
 value for one half a period and, since the second half is a 
 repetition of the first half with sign reversed, this will give 
 the arithmetical average value for the whole period. 
 
 The average ordinate is equal to the sum of all the 
 vertical elements of area divided by their number, or, what 
 is the same thing, it is equal to the area included between 
 the curve and the axis of abscissae, divided by the base. 
 For half a period the limits are and tt, so the 
 
 J ydx 1/1, 
 
 Average Ordinate = -^ = - / vdx. 
 
 fjdx '''^" 
 
ON HARMONIC FUNCTIONS. 37 
 
 But for a sine-curve, y = a sin x. ; therefore 
 
 Average Ordinate = - / sin xdx = — 
 
 I-" 2a 
 — cos X = • 
 
 -0 '^ 
 
 2 
 But a is the maximum ordinate, and — = .6369; so we 
 
 It 
 
 may write 
 
 /■nx Average Ordinate „.„ 
 
 (11) ^TF — -■ — ~ — 7^-^- — r- = .6od9, 
 
 Maximum Ordinate 
 
 which determines the value of the average ordinate. 
 
 TO FIND THE VALUE OF THE MEAN SQUABE OF THE OEDINATES 
 OF A SINE-CUEVE. 
 
 Although it is often useful to know the value of the 
 average ordinate of a sine-curve, it is more often desirable 
 to know the value of the mean square of the ordinates, or 
 the square root of the mean square. Since the square of 
 an ordinate is positive irrespective of the sign of the 
 ordinate, we can find the mean square of the ordinates by 
 integrating for the whole and not for half a period as was 
 necessary in finding the average ordinate. 
 
 "•air 
 
 Mean Square 
 
 of 2^ = -—n^ir- = 2^7 sm» X dx. 
 / dx *^' 
 
 •^ . . , 1 COS 2a; „!. c 
 But sm' 33 = 2 — — 2 — Therefore 
 
 /2ir r^" Sa 
 
 sin'a;c^a; = |_^ 2~ 
 
 ^"flj sin 2a3 
 
 = n. 
 
 4 
 
 Substituting n for the integral above, we have 
 
 2;^^'^=2- 
 
 (12) Mean Square oiy = ■^y.n = 
 
38 ON HARMONIC FUNCTIONS. 
 
 The square root of the mean square of the ordinates is, 
 therefore, 
 
 ^ = .707a. 
 
 V2 
 
 This means that the square root of the mean square of 
 the instantaneous values of y, which varies harmonically 
 with the time, is equal to .707 of the maximum ordinate. 
 
 The square root of the mean square of the instantaneous 
 values of a variable current or electromotive force is called 
 the virtual current or electromotive force and is equal to .707 
 times the maximum value of the current or electromotive 
 force. [ Virtual value is also called effective value.] 
 
 Inasmuch as the heating and dynamometer effects of 
 any current depend directly upon its mean square value, 
 this virtual value is of much more importance than the 
 average value in the measurement of an alternating current. 
 
 PERIODIC FUNCTIONS COMPOSED OP SEVEEAL SIMPLE SINE- 
 
 FUNCTIONS. 
 
 A single-valued function is one which has but one value 
 at any one point of time, as represented in Fig. 3. A mul- 
 tiple-valued function is one which 
 may have more than one value at 
 one point of time, as represented 
 in Fig. 4. A periodic function is 
 one which repeats itself after a 
 Fig. 3.-SINQLE-VALUBD defijiite. time or period. If any 
 Function. t p 
 
 number of simple sme-functions of 
 
 the same period be added, the resultant sum will be a 
 
 X-Axis 
 
 Fig. 4. — MuLTiPLE-TALtrED Function. 
 simple sine-function of the same period. This is rigorously 
 
ON HARMONIC FUNCTIONS. 
 
 39 
 
 shown in Chap. XIV., Part II., for the addition of two 
 simple sine -functions, as illustrated in Fig. 47 ; and it is evi- 
 dent that, if true for the addition of two, it is true for the ad- 
 
 Fis. 5. — Addition of Simple Hakmonic Curves of Same Period. 
 dition of any number of simple sine-functions. An example 
 of the addition of two simple sine-functions of the same 
 
 Fig, 6. — Addition of Simple Harmonic Curves of Different Periods. 
 
40 ON HABMONIC FUNCTIONS. 
 
 period is shown in Fig. 5. The resultant curve, represented 
 by the heavy line, is likewise a sine-curve. 
 
 If a number of simple sine-functions of periods which 
 are different but commensurable, are added together, the 
 resultant sum is a function which is periodic, but not har- 
 monic, with a period equal to the least-common-multiple of 
 the periods of the several component sine-functions. The 
 two heavy curves in Fig. 6 are obtained by adding two 
 simple sine-curves of the same amplitude and with periods 
 in the ratio 1 : 2. The equation for the lower heavy curve is 
 
 y = a sin aot -{-a sin 2aot, 
 
 the two component curves, shown by dotted lines, being 
 zero at the start. The upper curve has the equation 
 
 y ■= a sm oo 
 
 t -\-a sin(2a3i-|--^j, 
 
 the component dotted curves never being zero at the same 
 time. 
 
 The addition of two sine-curves with different amplitudes 
 
 Fig 7.— Addition of Simple Habmonic Cubveb of Different Periods. 
 
 and with periods in the ratio 1 : 3 is illustrated in Figs. 7 
 and 8. The component curves in Fig. 7 have no phase 
 
ON HARMONIG FUNCTIONS. 
 
 41 
 
 difference at the start and the resultant curve reptesents 
 the equation 
 
 y = asin oot — hsm^cot. 
 
 The curve in Fig. 8 represents the equation 
 
 2/ = a sin £» ^ -(- 6 sin (3 03 « -j- ^). 
 
 By adding a number of component simple' sine-curves 
 with different periods and amplitudes, resultant periodic 
 
 Fig. 8. — Addition of Simple Harmokic Curves of Different Periods, 
 
 curves of all manner of forms are obtained. Fourier has 
 shown that any single-valued periodic curve may be built 
 up by combining a number of simple sine-curves. Analyt- 
 ically this means that any single-valued periodic function 
 may be expressed as the sum of a series of sine-terms ; thus,. 
 
 y =/(a;) = A sin ax-\- B sin 2ax -\- C sin Sax -\- . . . etc., 
 -\- P cos ax-]- Q cos 2 apj -|- J? cos 3ax-{- . . . etc., 
 
 where / is a single-valued function. This is true for any 
 single-valued periodic function, even one represented by 
 an irregular series of straight lines. Each coefficient A, 
 B, C, etc., is independent of x and has only one value 
 which Fourier has shown how to find. 
 
CHAPTER III. 
 
 CIRCUITS CONTAINING RESISTANCE AND SELF INDUCTION. 
 
 Contents; — Equations of energy and E. M. F.'s. Criterion of integra- 
 bility. General solution when e =f(t). 
 
 Case I. E. M. P. suddenly Removed. Solution from differential equa- 
 tion, — from general solution. Geometric construction of 
 logarithmic curve. 
 
 Case II. B. M. P. suddenly Introduced. Solution from differential 
 equation, — from general solution. 
 
 Case III. Simple Harmonic E. M. F. Solution from general equation. 
 Impedance. Lag. Effect of exponential term at " make." 
 
 Case IV. Any Periodic E. M. F. Sum of two sine-functions. Sum of 
 any number of sine-functions. 
 
 In the first chapter the equation of energy for a circuit 
 containing self-induction and resistance was derived, and 
 from it the equation of electromotive forces 
 
 (9) e = i?i+i|; 
 
 that is, the electromotive force applied to the circuit is 
 equal to the sum of the electromotive force necessary to 
 overcome resistance and the electromotive force necessary 
 to overcome the counter electromotive force of self-induc- 
 tion. 
 
 This equation of electromotive forces, being regarded 
 as a differential equation containing three variables e, i, 
 and t (of which the general type of the first order is 
 
 (13) Pdx-\-Qdy-\-Sdz^ 0, 
 
 43 
 
HESISTANCE AND SELF INDUCTION. 43 
 
 where P, Q, and xS* are any functions of x, y, and z) does 
 not satisfy the condition of integrability. That condition, 
 which must hold true when there exists a single integral 
 equation of which (13), or a multiple of (13), is the exact 
 differential,* is 
 
 d% dyJ'^^Kdx d^) + ^\d^~d^)^^' 
 
 If we put (9) in the form of (13), we have 
 
 Ode — Ldi + (e — Bi) dt = 0. 
 
 Here e, i, and t correspond to x, y, and z respectively, 
 and P = 0, Q = - i/, iS = e - ^i. 
 
 The criterion of integrability reduces to 
 
 - i (1 - 0) = 0, or -L = 0, 
 
 and is not satisfied. 
 
 The meaning of this is that, unless some relation exists 
 between two or more of the variables, there is no single 
 equation of which (9) is the exact differential. 
 
 We know that the impressed E. M. F., e, has one single 
 value at any particular point of time, and may therefore be 
 expressed as a function of the time thus, 
 
 (14) e=f{t), 
 
 where/ is any arbitrary single-valued function. 
 
 By equating (14) to (9) the equation (9) of E. M. F.'s is 
 reduced to a linear equation, having constant coefficients 
 
 with the second member equal to —j^- Thus, 
 
 /-lev di , B . 1 
 
 (15) dt^L'^-Lf^^- 
 
 * See Johnson's Differential Equations, p. 370. 
 
44 CIRCUITS CONTAINING 
 
 The general type of this equation is 
 
 where P and Q may be functions of x only. The solution 
 of equation (16), which is a linear differential equation of 
 the first order,* is 
 
 (16a) 2/ = e "A*" Cj^^ Qdx-\-ce "A*". 
 
 e denotes the base of the Naperian system of logarithms 
 and is equal to 2.718. c is the arbitrary constant of in- 
 tegration. Both of these letters will be thus used when- 
 ever they occur. 
 
 With the particular values of the coeflScients in (15) its 
 solution is, therefore, 
 
 (17) i=j.e~''fe''f{t)dt + ce''. 
 
 This is the general solution for the current flowing in a 
 circuit containing resistance and self-induction and any 
 impressed E. M. F. 
 
 The integration indicated in (17) can only be performed 
 when we assume e to be some particular function of t. 
 We proceed then to assume several ways in which the 
 E. M. F. varies with the time. 
 
 Case I. Dying Away op Cubbent on Eemoval op E. M. F. 
 
 PKOM A ClECUIT CONTAINING KeSISTANOE AND SeLF-INDUC- 
 TION. 
 
 Suppose that a current has been flowing in a circuit 
 until it has reached its steady state, and that the source of 
 E. M. F. is then suddenly removed while the resistance 
 * See Johnson's Differential Equations, p. 31. 
 
BE8ISTANGE AND SELF INDUCTION. 45 
 
 and self-induction remain the same. The equation of 
 electromotive forces (9) becomes, under this hypothesis, 
 
 The solution of this equation is readily found since the 
 variables admit of separation. Thus, 
 
 di _ ^ , 
 i ~ L ' 
 
 i Bt 
 
 or 
 
 i = c e 
 
 Rt 
 
 The constant of integration c is determined by the par- 
 ticular supposition introduced that when we .begin to count 
 the time, the current has its steady value /. This gives 
 c = I. Hence we have 
 
 (18) i = Ie 
 
 Rt 
 
 Eef erring to the general solution (17), we might have 
 written (18) at once. For as/(i) = 0, [see (14),] the integral 
 vanishes, and we have (18) as an immediate result. 
 
 This equation (18) is graphically represented in Fig. 9, 
 
 where the ordinates represent the values of the current at 
 
 any time after the E. M. F. is removed. The self-induction 
 
 of the circuit prevents the current from falling immediately 
 
 to zero. It is evident that it woiild do so if there were no 
 
 self-induction from equation (18) ; for, if we make X = 0, 
 
 i becomes zero. The current which flows after the removal 
 
 of the E. M. F. is called the .extra current of self-induction. 
 
 The energy required to cause such a current to flow is that 
 
 energy which was previously stored up in the field and is 
 
 * Naperian logarithm (base e) is used here and in corresponding cases 
 which follow. 
 
46 
 
 CIRCUITS GONTAININO 
 
 now returned to the circuit. When t has the value ^ the 
 exponent of e becomes minus unity, and we have the rela- 
 
 PlG. 9. — CunVB SHOWING THE DYING AWAY OF CURKBNT AT ANY TiME 
 AFTER THE REMOVAL OF THE IMPRESSED E. M. F. FROM A CIRCUIT 
 
 ■WHOSE Resistance i? is .1 Ohm and Coefficient of Self-induc- 
 tion i IS .01 Henry. 
 
 tion - = e = 2.71828. -p represents, therefore, the time 
 that it takes for the current to fall to one eth part, that is to 
 9~7TS98 °^ its original value. This is sometimes called the 
 time-constant of the circuit, and denoted by T, that is 
 ^ — T. The curve represents an exponential function of 
 
 the time and approaches the cc-axis as an asymptote. This 
 means that the current becomes smaller and smaller, but 
 is never zero until an infinite time has elapsed. 
 
 GEOMETEICAL METHOD OP CONSTRUCTING THE LOGAEITHMIC 
 
 CUEVE. 
 
 The following method shown in Figs. 10 and 11 will be 
 found to be a convenient way to construct a curve graphi- 
 cally whose equation is of the form 
 
 (19) y = c6-'^, 
 
 where o and or have any real values whatever. 
 
RESISTANCE AND SELF INDUCTION. 
 
 47 
 
 Lay off OA equal to c. Then OA is the value of y 
 when a; = and may be called 
 y, ; that is, ?/„ = c = OA. 
 
 Lay off OB equal to c e""*'* 
 Then OB is the value of y 
 when 33 = a;,, and may be 
 called 2/, ; that is, y^ = ce~'^^ 
 = OB. 
 
 Vr OB 
 
 Hence Ml = ^ 
 
 Fig. 10.— Graphical Methc^''of 
 
 Constructing a Logarithmic 
 
 Curve. 
 
 If arcs AA' and BB' are described from the centre 0, and 
 a line BG drawn parallel with A'B\ thence another line 
 CD drawn parallel with AB, and so on, lines parallel 
 with A'B' and with AB being alternately drawn, as in the 
 figure, then the distances UA, UB, DC, OD, etc., will rep- 
 resent the values of y respectively as x takes the values 
 0, x„ 2a3„ 3a7,, etc. For if y„ y„ y„ y„ etc., denote the 
 values of y when x takes the values 0, x^, 2a3,, dx^, re- 
 spectively, we have 
 
 y, = Ci 
 
 y. 
 
 - = e"*'. 
 
 — SeuEt 
 
 "i oa;, 
 
 2/, = c e 
 
 ' SuXi 
 
 ^2 gaiC] 
 
 Hence 
 
 ^ = ^I = ^i = -^-5 = etc. = e"^ 
 2/. 2/, 2/3 ^4 
 
48 
 
 CIRGUIT8 CONTAINING 
 
 From the construction of the figure, and remembering 
 that OA — y, and 0B = y„ we see 
 
 OD . 
 
 = etc. = 6 
 
 OE 
 
 Hence 
 
 y, = OD, etc. 
 
 X-Axls 
 
 -Xr-f 
 
 FiG. 11.— Logarithmic Curyb. 
 
 Therefore to construct the curve y = ce~'^, Fig. 11, 
 we may proceed as follows : Upon two intersecting lines, 
 as in Fig. 10, lay off the distances y^ = c, and ^, = c e "" "*', 
 which latter must be calculated, and obtain the values of 
 OC, OD, etc., as described. Then y,, y„ y„ etc., or OA, 
 OB, C, etc., will be the successive ordinates of the loga- 
 rithmic curve. Fig. 11, at distances 0, a;,, 2x^, 3^,, etc., and 
 the curve may be drawn. 
 
 Case II. Establishment of a Cureent on Inteodxjction 
 OF A Constant Electeomotive Foece into a Ciecuit 
 
 CONTAINING EeSISTANCE AND SeLF-INDUCTION. 
 
 .Suppose a source of constant E. M. F. is suddenly 
 introduced into a circuit of resistance H and self-induction 
 L. The differential equation in this case is 
 
 (20) 
 
 E=Ri-\rL 
 
 di 
 dt 
 
KESISTANGE AND SELF INDUCTION. 49 
 
 where ^ is a constant. The variables may be separated 
 here as in the previous case, thus : 
 
 di R 
 
 and iogl(i_--) = _^^. 
 
 E -~ 
 
 Therefore i — ^+ c e ^ . 
 
 The constant of integration, c, is determined by , the 
 condition that, when t — 0, i = 0, and therefore c — — =• 
 We have then as a result, 
 
 (21) i = f(l-.--) = /(l-,-^). 
 
 Referring to the general solution (17) we might have 
 substituted /(^) =■ E, a. constant, and written at once equa- 
 tion (21). For in this case we easily find the required 
 integral : 
 
 Rt 7- Bt 
 
 jEe^dt = E^e';. 
 
 1 ■?* 
 Multiplying this by the coefficient t^^ (17) becomes 
 
 E 
 
 i = ^ + ce 
 Eeplacing c by — -q, we have 
 
 L 
 
 a result identical with (21) 
 
 = f(-^-') 
 
50 
 
 CIRCUITS CONTAINING 
 
 Here we notice that, if the self-induction is zero, the 
 equation becomes simply Ohm's law ; that is, it is the 
 self-induction of the circuit which prevents the current 
 from reaching its full value immediately after the intro- 
 duction of the E. M. F. 
 
 •o' 
 
 10 
 
 
 
 
 
 
 A 
 
 n 
 
 
 
 
 
 
 
 "™™" 
 
 a 
 
 
 
 
 ,^-^^,---^ 
 
 
 
 
 -5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 
 
 y 
 
 j|. 
 
 
 
 
 g 
 
 
 
 / 
 
 II 
 
 
 
 
 " 
 
 / 
 
 / 
 
 
 +3 
 
 
 
 
 
 
 
 .02 
 
 .04 
 
 — , 1 
 
 .06 .08 .10 
 
 m 
 
 Seconds 
 
 
 Fig. 13. — Curve showing the Establishment op Cubrent at ant 
 Time after the Introduction of an B. M. F. into a Circuit 
 WHOSE Resistance ij is .1 Ohm and Coefficient of Self-induc- 
 tion L IS .01 Henry. 
 
 The increase of the current with the time is shown by 
 the curve Fig. 12. This is a logarithmic curve similar to 
 that in Fig. 9, with the ordinates measured downward from 
 the horizontal line O'A, at a distance above the axis equal 
 to the maximum value /, of the current. 
 
 Case III. Haemonic Impeessed E. M. F. in a Ciecuit 
 
 CONTAINING A EeSISTANCE AND SeLF-INDUCTION. 
 
 Let us now suppose that in a circuit containing re- 
 sistance and self-induction there is a simple harmonic 
 impressed E. M. F., that is that the E. M. F. is a sine-func- 
 tion of the time, thus : 
 
 (22) 
 
 e—f{t) = £:sm cot. 
 
 Here ^is the amplitude or maximum value of the im- 
 pressed E. M. F., and co is the angular velocity, equivalent 
 
BESISTANGE AND SELF INDUCTION. 51 
 
 to ^Litn, or -=f-, where n denotes the ntimber of complete 
 
 periods per second, and T the time of one complete period. 
 The general solution for the current, equation (17), is 
 
 1 fit H£ Rt_ 
 
 (17) i = ^e'^Je''f{t)dt-\-ce'''. 
 
 Substituting in (17) the value for f(f) in (22), the general 
 expression becomes, according to the particular hypothesis 
 of a sine E. M. R, 
 
 TT _ R t ^^ Bt _ R t 
 
 (23) i = ~e' "■ fe^smwtdt-\-ce 
 
 L 
 
 Before integrating this equation we will first obtain the 
 general integrals 
 
 f e"' sm {^x+ff)dx and J' e"' cos {/3x + 6) dx. 
 
 Applying the formula for integrating by parts, 
 
 I udv =^ uv — I V du, 
 
 these integrals become 
 
 f sin {^x -{-&). e-^dx 
 
 ea an 
 
 = sin (/?a; + (9). -J ^^ cos (ySas + 6) dx, 
 
 fcos{0x-{-e).e'^dx 
 
 = ooB{/3x + ff). — -\--J 6"'sm{/3x-\-e)dx. 
 
 Eliminating / e"* cos {/3x -{- 0)dx between these two equa- 
 tions, we obtain as one of the integrals sought 
 
 <24) ye" sin (/?a; + S) dx 
 
 = -^^,{c^ sini^x + ff) - /3 cosi^x + e)\ . 
 
53 CIIiCVITS CONTAININQ 
 
 Eliminating / e^ sin {fix -{- ff)dx between the same two 
 equations, we obtain in the same way the integral 
 
 (25) y^e" cos {fix -\-ff)dx 
 
 Replacing a by -^, /3 by oo, B by 0, and x by t, in equa- 
 tion (24), we have the integration indicated in (23), and 
 equation (23) then becomes 
 
 
 /««N . E IB . \ , 
 
 (26) I = — ^-d5 r {^Y sm cot — aocos ootj -j-ce 
 
 This may be written in simpler form by the use of the 
 trigonometric formula 
 
 (27) ^ sin ^ + jB cos ^ = VA' + B' sin ((9 + tan " * ^)- 
 
 This formula is established as follows : 
 ^ sin ^ + -^ cos 6 
 
 If tan = -r, then sin = ,_ =, and cos = 
 
 ^ A' ^ VA'-^B' ^ VA'-\-B'' 
 
 Making these substitutions, we have 
 
 ^ sin 9 + 5 cos 6/ = VA" + B'' (cos sin 6^ -j- sin cos ff) 
 =: VA^'+B' Bin {d ^ -((>), 
 which establishes the truth of (27). 
 
BESI8TAN0E AND SELF INDVGTION. 53 
 
 Eeducing equation (26) to its simplest form by means 
 of formula (27), we have from (26) the value of the current 
 at any instant of time. 
 
 (28) i=-T====sin(a3«-tan-i^)+ce"-^ 
 
 DISCUSSION OF THE CUBBENT EQUATION. 
 
 After a very short time the exponential term in this 
 equation, containing the arbitrary constant of integration, 
 becomes inappreciably small, and may be neglected. Just 
 what effect the exponential term has during this short time 
 will be considered later. The equation shows that, where 
 there is an impressed sine electromotive force in a circuit, 
 the current is likewise a sine-function of the time, and that 
 the current lags behind the electromotive force by an angle 
 
 whose tangent is —5-. If there is no self-induction and 
 Z = 0, equation (28) becomes 
 
 E 
 
 i = -5 sin GO t, 
 
 which is a direct result of Ohm's law. Thus the self- 
 induction not only causes the current to lag behind the 
 impressed E. M. F., but also diminishes the maximum 
 value of the current. 
 
 "When sin (ca f — tan"'-^) becomes unity, the current 
 
 has its maximum value I, and 
 
 ^^^) ~ VE' + Z" CO'' 
 
 The term "impedance " has been applied to the expression 
 ^B' + L' £»", the apparent resistance of a circuit contain- 
 ing ohmic resistance and self-induction, and an impressed 
 sine electromotive force. 
 
54 
 
 CIRCUITS CONTAINING 
 
 (30) 
 
 The equation (29) may be written 
 
 Maximum E. M. F. 
 
 Maximum current 
 
 Impedance 
 
 Since virtual current = -7^ maximum current, and vir- 
 
 i/2 
 
 tual E. M. F. = — =^ maximum E. M. P., see equation (12), 
 
 T 2 
 
 we may write 
 
 (31) 
 
 Virtual current = 
 
 Virtual E.M.F. 
 Impedance 
 
 Fig. 13. — Value of Impedance. 
 
 The value of impedance is graphically represented in 
 Fig. 13. i ft? is sometimes called the inductive resistance in 
 contradistinction to the ohmic resistance JR. (See note p. 59.) 
 
 It has been shown above that the tangent of the angle 
 
 of lag is —p-- The angle of lag is therefore represented 
 by in Fig. 13. 
 
 Fig. 14.— Value aw Impsessbd E. M. F. 
 The triangle may be drawn so that the three sides rep- 
 resent E. M. F. as in Fig. 14. Here B I represents the 
 
BE8I8TANCE AND SELF INDUOTION. 55 
 
 E. M. F. necessary to overcome tlie olimic resistance, and 
 is in the same direction as the current. L oo lis at right 
 angles to this and represents the counter E. M. F. of self- 
 induction. I VB' + -^^ ^^ is the impressed electromotive 
 force E. 6 is the angle by which the current lags behind 
 the impressed E. M. F. 
 
 Full discussion of the triangles of current and E. M. F. 
 is given in the graphical treatment of circuits with re- 
 sistance and self-induction, Chap. XV. 
 
 It is convenient to consider the impedance as a resist- 
 ance, and the propriety of doing so is shown by its dimen- 
 sions, which are the same as those of resistance, that is a 
 velocity in the electromagnetic system of units. 
 
 length 
 The dimensions of resistance, B, are —. = velocity. 
 
 The dimension of the coefl&cient of self-induction, L, is 
 
 length. The dimension of an angular velocity m is —. — . 
 
 . _ length , . . 
 
 Therefore the dimensions oi L co are -j- = velocity, and 
 
 thus the impedance has the same dimensions as a resistance. 
 
 EXPLANATION OF THE EXPONENTIAL TEBM. 
 
 Let us return to the solution for current, equation (28), 
 
 and consider the effect of the exponential term, c e ^ , 
 during the short time after " make," that is, after the intro- 
 duction into the circuit of a simple harmonic impressed 
 electromotive force. The equation (28) for current may be 
 written 
 (32) i = 7sin^ + ce i; 
 
 and tp = Got — tan -g- ; 
 
56 
 
 CIRCUITS CONTAINING 
 
 that is, /represents the maximum value and tf> the phase of 
 the current. The E. M. F. is introduced at a time f,. At 
 that time the current is zero, for the circuit is just made. 
 If we call ip^ the value of tp when t = t,,ht the introduction 
 of the E. M. F. equation (32) becomes 
 
 Rt, 
 
 = 7 sin ip^-^c e ^, 
 
 . Rt, 
 
 (33) 
 
 and c = — Je 
 
 sin ^i- 
 
 Substituting this value of c in (82), the equation for cur- 
 rent becomes 
 
 (34) 
 
 t = I sm ip — le ^ sm ^,. 
 
 Fig. 15. — Cukyb showing the Effect of the Exponential Tbkm 
 _ ?? 
 e e ^ UPON THE Current at the Make, in a Circuit where 
 Z = 1 Henry, i? = 50 Ohms, ca = 1000, ^, = 30°. 
 
 This equation may best be explained by referring to 
 Fig. 15, which represents the plot of the equation. The par- 
 ticular values assumed in this case are L=l henry, i? = 50 
 ohms, oa = 1000, and ^, = 30°. The resultant current curve 
 III. is made up of two component parts, / sin ^, and 
 
 — -(,t -t) . 
 — le ^ ' sin ^, , which are represented by the curves 
 
RESISTANCE AND SELF INDUCTION. 57 
 
 I. and II. respectively. Curve I. is a sine-curve and curve 
 
 II. a logarithmic curve, the effect of which upon the result- 
 ant current becomes inappreciable after a very short space 
 of time, in this particular case after five or ten periods. 
 The initial value of this logarithmic curve is equal and op- 
 posite to the value of the ordinate of the component sine- 
 curve I. at the time t, when the E. M. F. is introduced. 
 This is evident from the equation, since the initial value of 
 the logarithmic curve is — 7 sin tp^ , and the value of the 
 sine- curve, when i = f , , is -{- 7 sin ijj^ . 
 
 If another curve IV. is constructed so that its ordinates 
 represent the initial values of the logarithmic curve, when 
 the E. M. F. is introduced at different points in the period, 
 it is seen to be simply a sine-curve, corresponding with the 
 component curve I. but reversed, or, what is the same 
 thing, differing from it by 180° in phase. 
 
 To conclude, we see that the effect of the exponential 
 term in the equation is a maximum if the E. M. F. is intro- 
 duced at that point of its phase at which the current has 
 its maximum, value when everything has reached its per- 
 manent state ; this term has no effect if the E. M. F. is 
 introduced at that point of its phase at which the current 
 has its zero value when everything has reached its per- 
 manent state. 
 
 Case IV. — Peeiodic E. M. F. which is not Harmonic, in 
 CiKCUiTS containing Eesistance and Self-induction. 
 
 In Case III. the solution was given for a circuit contain- 
 ing an impressed E. M. F. which was a simple sine-function 
 of the time. Now let us suppose that the E. M. F. does 
 not follow a simple sine law, but that it is the sum of two 
 components each following a sine law, that is, 
 
 (35) e = E,Bia.Got-\-E, sin {hcot-\- 6). 
 
58 CIRCUITS CONTAININQ 
 
 Substituting in the general expression for current (17) 
 this Talue for/ (t), we have 
 
 Bt Bt 
 
 (36) i=^e~ '^fe'' sincotdt 
 
 _B^t_ Rt _Rt 
 
 + ^e ^ f€''sm(boot-{-ff)dt+ce ^. 
 
 Performing the indicated integrations by use of the 
 formula of integration (24), we have 
 
 /QFTN * = 7^3 ; { ^ sin a? i — CO cos oat y 
 
 (37) Zf5, + c.^) l^ i 
 
 "i 'Tpsi — ' ^ \ 7sin(6oi5i4" ^) — ^<«'C0s(6(a<+ ^) >■ 
 
 _Rt 
 -}-C6 i. 
 
 Simplifying by formula (27) this becomes 
 
 (38) i = ' sin [go t - tan-'-^- 
 
 ' VE' + L-'oa' ^ R ' 
 
 + -=i= s.m(poot + e- tan-i^) 
 
 Vi?'' + Z' b' oo' 
 
 c e i. 
 
 By this equation it is seen that each simple sine impressed 
 E. M. F. gives rise to a simple sine term in the resulting 
 current equation. The result may therefore easily be 
 extended, and we may say that, if there are n simple sine 
 impressed E. M, F.'s of the form ^ sin {boot-^- 6), where 
 
RESISTANCE AND SELF INDUCTION. 59 
 
 E, b, and 6 have different values in each component term, 
 the current equation will be the sum of n terms of the form 
 
 E 
 
 sin 
 
 VB' + Z" If oo' 
 
 in •< b Got -\- 6 ~ tan ' — „— l 
 
 Rt 
 
 plus the term c e l containing the arbitrary constant. 
 
 Here £^, h, and 6 have the same values in each term as 
 they do in the corresponding term of the impressed E. M. F- 
 
 Expressing the current by a summation, we have 
 
 (39)i = ^> — ^==z sin -^ icjt-i-ff — tan"^ — ^ [ 
 
 „ VW+Z' b' CO 
 
 _Bt 
 
 when the impressed , .„. ^^ „ . p, ^ , m 
 
 E. M. F. 13 ^ ^ ^— "- 
 
 E, b, 9. 
 
 In these sums E, b, and 6 may have n values, but they 
 must be the same values in each sum, giving rise to the 
 same number of terms in each. 
 
 It was first shown by Fourier that such a sum of simple 
 sine terms as that represented in equation (39) may express 
 any single-valued function whatever, and thus we see that 
 the equation expresses the most general case of a current 
 flowing in a circuit with resistance and self-induction, and 
 may represent the current caused by any E. M. F. what- 
 soever. 
 
 The consideration of this most general expression for 
 the current will be deferred until the case has been taken 
 up where the circuit not only contains resistance and self- 
 induction, but also a condenser. 
 
 Note.— Since the first publication of this volume the quantity Loo has 
 been termed reactance, and Lao I the reactive electromotive force. The com- 
 ponent electromotive force in phase with the current may be termed the 
 power electromotive force. The ohmic electromotive force iJ/is a power 
 electromotive force. [Effective now usually means virtual, see p. 38.] 
 
CHAPTER IV. 
 
 INTRODUCTORY TO CIRCUITS CONTAINING RESISTANCE 
 AND CAPACITY. 
 
 Contents:— Plan to be followed. Charge. Law of force. Unit charge. 
 Work in moving a charge. Potential. Capacity. Energy of charge. 
 Condenser, — energy of and capacity of. Capacity of parallel plates; 
 of continuous conductor. Equation of energy, in terms of i ; in terms 
 of q. Equation of E. M. F.'s. 
 
 In the first chapter the fundamental principles neces- 
 sary to lead up to the derivation of the equation of energy 
 for circuits containing resistance and self-induction only 
 were given ; then followed, in the third chapter, the solu- 
 tion of this differential equation, which enabled us to ascer- 
 tain the current flowing in the circuit at any time. Follow- 
 ing a similar plan, there will be given in this chapter the 
 necessary fundamental principles which lead up to the 
 derivation of the differential equation of energy for circuits 
 containing resistance and capacity, and in the following 
 chapter the general solution of this differential equation 
 and its application to various particular cases. 
 
 LAW OF FORCE. 
 
 Every one is familiar with the fact that bodies may be 
 charged with electricity, and that two like charges repel 
 and two unlike charges attract one another. It was found 
 from experiment by Coulomb that if we have two charges, 
 each concentrated at a point, the force of attraction or 
 
 60 
 
0IBCUIT8 CONTAINING RESISTANCE AND CAPACITY. 61 
 
 repulsion between them varies directly witli the product of 
 the two charges and inversely as the square of the distance 
 between the two points, that is, 
 
 where q and q' represent the quantities of the charges, r the 
 distance, and i^ the force between them. When the quantities 
 considered have the same sign, the product q q' is positive, 
 and therefore a force of repulsion has a positive sign. 
 Similarly a force of attraction has a negative sign. 
 
 If the distance between these points is unity, the 
 charges being equal, and if the force between them is a 
 unit force, each charge is called a unit charge. The defini- 
 tion of the electrostatic unit of quantity of electricity, in 
 the 0. G. S. system, is then : that quantity which, when 
 placed at a distance of one centimeter from an equal quan- 
 tity (in a medium whose specific inductive capacity is unity 
 — that is, in air or vacuo), repels it with the force of one 
 dyne. "Where these units are used, and the medium is 
 a vacuum, the law of force may be written 
 
 TT- ^^' 
 
 !> = — ^• 
 
 Where the medium is not a vacuum, the force is found to 
 be less and equal to 
 
 KT 
 
 where x- is a constant quantity called the spedfic inductive 
 capacity of the medium. 
 
 POTENTIAIv. 
 
 Since there exists a force between two charges of elea 
 tricity, mechanical work is done if either is moved so as to 
 change the distance between them. The work done in 
 
62 INTRODUCTORY TO CIRCUITS 
 
 moving any body against a uniform force is equal to the 
 
 product of the force and the distance through which the 
 
 body is moved against that force. The force between the 
 
 Q q' 
 electrical charges q and q' is ^-f-. If they be moved in any 
 
 direction whatsoever, so that the distance r between them 
 
 is changed to r + dr, the work done in moving them is the 
 
 q q' 
 product of the force — — and the change in the distance dr, 
 
 since the force may be considered constant throughout the 
 small distance dr. Therefore the work is 
 
 dW= ^4- dr. 
 
 'p, 
 Fio. 16. — Work done in moving a Chakgbd Body. 
 
 Suppose a charge q is situated at the point A (Fig. 16), 
 and a charge q' is moved from the point Pj to P,. The 
 work done by the electric force in moving the charge is 
 
 or, the work done against the electric force \s,qq'[ J. 
 
 It is seen that the work done in moving a charge from 
 one point to another is independent of the path by which it 
 is moved, and simply depends on the initial and final dis- 
 tances between the charge q and q'. If the distance r^ is 
 infinite (meaning that the charge q' is carried from an in- 
 
CONTAINING RESISTANCE AND CAPACITY. &i 
 
 finite distance to a point at a distance r,), the work done 
 against the electric force becomes simply 
 
 If q' is unity and a unit charge is moved, the work becomes 
 
 It is seen that each point in the region surrounding an 
 electric charge possesses a certain characteristic which 
 determines the amount of work done in bringing a charge 
 from infinity to that point. This characteristic of the 
 point has been called its potential. The potential F" at a 
 point is therefore defined as the work done in moving a 
 unit positive charge from an infinite distance to that point ; . 
 
 thus, V=: —. This potential is positive when the work 
 
 is positive, that is, when work is done, in moving the 
 charge, by some agent external to the system. 
 
 The potential at a point due to a number of charges, 
 each concentrated at a point, is the sum of the potentials 
 at that point due to each charge independently ; thus, 
 
 If there is a charge distributed upon any surface and 
 dq is the charge upon an element of that surface, the poten- 
 tial at any point due to this charged surface is equal to the 
 sum of the potentials due to each elemental charge ; that is. 
 
 The potential at every point of a good conductor is the 
 same, since the electricity will so distribute itself on 
 the body that no work would be done by transferring a 
 
64 INTRODUCTORY TO CIRCUITS 
 
 charge from one point of the conductor to anothei point of 
 it. This potential Fis called the potential at the conductor, 
 and the conductor is said to be at potential V. The poten- 
 tial at a conductor may be due partly or wholly to the 
 charge on the conductor itself. 
 
 CAPACITY OF A CONDUCTOB. 
 
 The potential of a charged body is directly proportional 
 to its charge, that is, V cc q, or q = C V, where C is some 
 constant; for, suppose the body possesses a unit charge 
 and its potential is V; a second unit charge brought from 
 infinity to the body doubles its original charge. The po- 
 tential is then 2 V, for the potential is the work done in 
 bringing a unit charge from infinity to the point, and the 
 work in bringing a unit charge to a body with a quantity 
 2q is twice the work in bringing a unit charge to a body 
 with a quantity q. We thus see that q is proportional to 
 V, and is consequently equal to V multiplied by some con- 
 stant, thait is, 
 
 (41) q= CV. 
 
 If a body is charged to a unit potential and the quan- 
 tity is q, 
 
 q = a 
 
 C is therefore defined as the quantity of electricity upon 
 a body when at a unit potential. This is called the capacity 
 of the conductor. The capacity depends upon the size and 
 geometrical form of the conductor and the specific inductive 
 capacity of the surrounding medium. 
 
 ENEEGT OF A CHAEGED CONDUCTOB. 
 
 Suppose a body is charged with a quantity of electricity 
 o, and is at a potential V. The work done in bringing a 
 unit quantity of electricity from an infinite distance up to 
 the body is V by definition. (This is provided q is so large 
 
CONTAININa BESISTANCE AND CAPACITT. 65 
 
 in comparison witli a unit quantity that its potential is not 
 appreciably altered by the addition of the unit quantity.) 
 If, under the same conditions, we bring up, not a unit 
 quantity, but a quantity dq, the work done is Vdq, and this 
 represents the increment of the energy of the charge g- 
 That is, 
 
 (42) dW^ Vdq. 
 
 Eeferring to equation (41), we may always replace V by 
 its equal -^, or dq by its equal CdV, and obtain the equa- 
 tions 
 
 and 
 
 dW= CVdV. 
 
 The integrals of these equations, taken between the 
 limits zero and q, and zero and V, respectively, are ^^ 
 
 ^ - 2 C 
 W=kCV\ 
 Siuce q = CV, each of these equations may be written 
 
 (43) W=iqV. 
 
 Here ^is the potential energy possessed by the charged 
 body, as the limits of integration were taken from zero 
 charge to charge q, and from zero potential to potential V. 
 
 CAPACITY AND ENEEGY OP A CONDENSEE. 
 
 A condenser is a device for increasing the capacity of a 
 conductor by bringing it near another similar conductor, 
 which is separated from it by any non-conducting medium 
 or dielectric. This dielectric will be considered to be a 
 
66 INTRODUCTORY TO CIBOUITS 
 
 perfect non-conductor ; that is, the condenser is not leaky. 
 A condenser usually consists of two sets of parallel plates 
 alternately connected, and separated by a distance very 
 small as compared witli the dimensions of the plates. The 
 two sets of plates are usually called simply the two plates 
 of the condenser. When the condenser is charged, the two 
 plates have equal quantities of electricity upon them, but 
 of the opposite sign. 
 
 The total energy of a charged condenser may readily be 
 found by taking the algebraic sum of the energies of the 
 charge on each plate, as given by the equation (43). 
 
 If the plates of a condenser have charges -j- q and — q 
 at potentials F, and V, , respectively, the total energy is 
 
 (44) 'W=\qV-iqV^ = ^q{V,-V:); 
 
 that is, the energy of a charged condenser is equal to one- 
 half the product of the charge of one of the plates and the 
 difference of potential between the plates. If this differ- 
 ence of potential between the plates is simply V, the ex- 
 pression for the energy of a charged condenser is 
 
 (45) W^iqV. 
 
 The capacity C of a condenser is the quantity of elec- 
 tricity on one plate when there is a unit difference of poten- 
 tial between the plates ; and when there is a difference of 
 potential Fthe charge is 
 
 (46) q=OV. 
 
 It can be shown that the capacity of a condenser, com- 
 posed of parallel plates of equal area, whose distance apart 
 is small as compared with the dimensions of the plates, is 
 directly proportional to the area of the plates, and inversely 
 
CONTAINING BESI8TANCE AND CAPACITY. 67 
 
 proportional to the distance between them, and that the 
 capacity is 
 
 (47) C = j^^ , [See note, p. 69.] 
 
 where A is the area of each plate and d the distance be- 
 tween the plates. 
 
 As the plates of a condenser approach nearer and nearer 
 together, the capacity C becomes larger and larger. In the 
 limit, when the plates come into contact, the capacity be- 
 comes infinite, which means that, no matter how much one 
 plate is charged, there can exist no difference of potential 
 between them. If, then, a circuit is a continuous con- 
 ductor and has no condenser in it, it may be said to have 
 a condenser of infinite capacity in series with it. 
 
 By combining equations (45) and (46) the energy of the 
 charge of the condenser may be expressed in terms of the 
 capacity and the potential V, or in terms of the capacity 
 and the charge g. Thus, 
 
 (48) W=iCV'=:l^^. 
 
 The increment of the energy d W, as the potential and 
 charge vary simultaneously, is 
 
 (49) dW=CVdV=^' 
 
 THE EQUATION OF ENEBGY. 
 
 We can now write the equation of energy for an electric 
 circuit having a resistance B, and having in series with 
 that resistance a condenser of capacity G. 
 
 The total energy given to the circuit by the source of 
 E. M. F. is eidt; and that part of the energy used in heat- 
 ing the conductor in the time dt is Bi'dt, as shown in 
 equations (5) and (4). The amount of energy required ir 
 
 , . dW _,^ 
 
 the time dt to change the charge of the condenser is -^dt. 
 
68 INTRODUCTOBT TO CIRCUITS 
 
 Since, under the conditions supposed, these two are the 
 only ways in which the energy imparted by the source is 
 used, we have the equation of energy, 
 
 dW 
 
 (50) eidt = Bi^dt-\--^dt. 
 
 We have seen that dW = -^r (equation 49) ; therefore, 
 
 (51) eidt = Bi^dt^^^dt. 
 
 When a current i flows into a condenser for a time dt, 
 the quantity which flows during this time is % dt, but this 
 is the increment dq of the charge of the condenser, that is, 
 
 dg — idt; 
 hence 
 
 (52) q = fidt. 
 
 Substituting these values of q and i in equation (51) we 
 may write the equation of energy in two forms, in terms of 
 i or in terms of q, thus : 
 
 idt I idt 
 
 (53) eidt = B i'dt -\ ^, ; 
 
 Dividing (53) through by idt and (54) by its equal dg, 
 we have 
 
 fidt 
 
 (55) e = iZt + ^^-; 
 
 (56) ' = ^W + -C- 
 
CONTAINING RESISTANCE AND CAPACITY. 69 
 
 These are equations of electromotive forces, where e is 
 
 the impressed E. M. F. of the source, Ri or B^ the 
 
 E. M. F. necessary to overcome the ohmic resistance, and 
 
 / 
 
 idt Q 
 
 jy— = -p = V, the E. M. F. necessary to oppose the 
 
 E. M. F. of the condenser. 
 
 When C is infinite, that is, as explained above, when the 
 plates of the condenser come into contact, we have a circuit 
 with resistance only, in which case equation (55) gives 
 
 e = Hi, 
 which is Ohm's law. 
 
 Note. — ^This expression for capacity, equation (47) (page 67), is true for 
 C. G. S. electrostatic units. To find the value in electromagnetic or prac- 
 tical units, consult Appendix A, page 312. 
 
CHAPTER V. 
 
 CIRCUITS CONTAINING RESISTANCE AND CAPACITY. 
 
 Contents. — Equation of E. M. P.'s. Differential equation in linear form. 
 Criterion of integrability. General solution when e = 'f(t). 
 
 Case I. Discharge. Quantity and curreul from general solution, — from 
 differential equations. 
 
 Case II. Charge. Quantity and current from general solution, — from dif- 
 ferential equations. 
 
 Case III. Simple harmonic E. M. F. Quantity and current from general 
 solution. Discussion. 
 
 Case IV. Any periodic E. M. P. 
 
 In the previous chapter the equation of energy for 
 a circuit containing ohmic resistance and capacity was 
 derived, and, by dividing the equation of energy through 
 by i dt or dq, it was found that the equation of electro- 
 motive forces thus obtained may be expressed in terms of 
 current, i, or charge, q, thus : 
 
 (55) e = Bi+f-^*. 
 
 Differentiating (55), to free it from the integral sign, 
 and transposing, the two equations may be written : 
 
 (57) Cde ^BCdi — idt = 0. 
 
 (58) Ode-B Cdq -^ {e C - q)dt = 0. 
 
 70 
 
RESISTANCE AND CAPACITY. 71 
 
 Each of these equations is a differential equation of the 
 first order with three variables, e, i, and t, and e, q, and t, 
 respectively, of the form 
 
 Pdx-\- Qdy-{-Sdz = 0. 
 
 If there exists a single integral equation of which this 
 is the exact differential, the condition of integrability * 
 
 (dQ dS\ fdS dP\ (dP dQ\_ 
 
 must be satisfied. 
 
 Applying this criterion of integrability to the equations 
 
 Fig. 17.— Cibcuit having Ohmic Resistance and Capacity. 
 
 (57) and (58), it is found that the coridition is not satisfied 
 by either equation. No single equation exists, therefore, 
 of which (57) or (58) is an exact differential. 
 
 But, as was previously stated, we know that the electro- 
 motive force e may always be expressed as a single-valued 
 function of the time, since it must have some one value at 
 each point of time, and we have 
 
 (59) e=/(0. 
 
 where / is an arbitrary single-valued function. By differ- 
 entiation (59) becomes 
 
 (60) S =/'('> 
 
 * See Johnson's Differential Equations, p. 270. 
 
72 CIRCUITS CONTAINING 
 
 Equations (57) and (58) may now be written in the linear 
 form thus : 
 
 The solutions of these linear equations * are 
 
 J ^ 
 
 (63) I = ^5-y e / (0 di + c. € 
 
 6 I ^ BC y.,,^ 1^ , BO 
 
 (64) g=^J ^ ''y{t)dt+c,e'' 
 
 The integrals here expressed cannot be found unless we 
 know in what particular way the electromotive force varies 
 with the time. When we know this, these equations will 
 give the values of the current and charge at any time, 
 provided the integral sought can be obtained. We will 
 now assume several ways in which the E, M. F. varies 
 with the time, which will allow the integration to be easily 
 performed. 
 
 Case I. Dischaege op a Condenseb. 
 
 Suppose that a constant source of E. M. F., E, has been 
 acting upon a circuit containing in series a resistance, and 
 a condenser with capacity C, until everything has reached 
 its steady state. No current will be flowing, and the con- 
 denser will be cnarged with a quantity Q, and have a dif- 
 ference of potential E at its terminals. Now suddenly 
 remove the source of E. M. F. from the circuit and suppose 
 its resistance then is B. The condenser will immediately 
 
 * See Johnston's Differential Equations, p. 31. 
 
RESISTANCE AND CAPACITT. 73 
 
 begin to discharge through the conductor, and we wish to 
 find the value of the charge q and current i at any time 
 after the discharge begins. 
 
 When the E. M. F. was removed from the circuit the 
 impressed E. M. F., e =/(<), became equal to zero at every 
 point of time after the removal ; hence, substituting 
 
 (65) e =f{t) = 
 
 in the general equations (63) and (64), we find that the in- 
 tegral vanishes, and we*have the immediate results, 
 
 % = c, e 
 
 q = c,e 
 
 t 
 
 ' RC 
 
 t 
 
 The arbitrary constants c, and c, are determined by the 
 initial conditions. If the charge is Q when the time is 
 zero, the charge equation becomes 
 
 t 
 (66) g=Qe~'"'; 
 
 and since dq = i dt, the current equation becomes 
 
 Q - ' 
 
 (67) i=-m' 
 
 BC 
 
 If, instead of substituting e =f{t) = in the general 
 solutions (63) and (64), we had substituted in the differ- 
 ential equations (61) and (62), it is seen that the second 
 member of each becomes zero, and that the solutions are 
 merely the " complementary functions," namely, the terms 
 in the general solutions containing the arbitrary constants, 
 as pointed out. 
 
 It may be of interest to derive the solution directly from 
 the differential equations, since the variables easily admit 
 of separation. 
 
74 
 
 CIRCUITS CONTAINING 
 
 Equation (62), vfh%nf{t) = 0, is 
 
 dq 
 di' 
 
 = 0. 
 
 dq 
 1 
 
 BG 
 dt 
 
 t_ 
 
 BG 
 
 Hence — = — 
 
 and 
 
 or 
 
 q = ce 
 
 which is identical with (67). 
 
 In Fig. 18 is shown a curve of discharge of a condenser 
 for a particular case. The rapidity of discharge is shown 
 by the value of the time-constant T, which gives the time 
 in which the charge of the condenser is reduced to one eth 
 of its initial value. 
 
 T' = ^ C = 100 X 10' X 4 X 10-" = .0004 seconds. 
 
 Fig. 18. — Cukve showing Discharge op a Condenser whose Ca- 
 pacity C = 4 Microfarads, through a Resistance B = 100 Ohms. 
 
 Case II. Charge op a Condenser. 
 
 Suppose that a constant source of E. M. F., E, is sud- 
 denly introduced into a circuit, and that the resistance 
 when it is introduced is B, the capacity of the condenser 
 in series with the resistance being O. The values of the 
 
RESISTANCE AND CAPACITY. 75 
 
 current i and charge q at any time after the introduction 
 of the E. M. F. will be given by equations (63) and (64) if 
 we suppose 
 
 (68) e =/(«) = ^, a constant, 
 
 de 
 and consequently -j- =f'{t) = Q. 
 
 Substituting these values, (63) and (64) become 
 
 t_ 
 
 (69) i=c,e ^^. 
 
 (70) q=GE+c,e~^. 
 
 Determining the constants of integration c, and c, by the 
 condition that there was no charge in the condenser when 
 i = 0, we have 
 
 c, = - CE. 
 
 But since CE= Q, the final charge of the condenser when 
 everything has reached its steady state, (70) becomes 
 
 (71) q=Q[l-e~^), 
 
 and by the relation dq = idt equation (69) becomes 
 
 (72) ^ = Sc^~^- 
 
 It is noticeable that the equations for the current (67) 
 and (72) are identical in the case of charge and discharge 
 of a condenser, except that the sign of i, i.e., the direction 
 of the current, is reversed. 
 
 Equation (71) may easily be derived from the differ- 
 ential equation (62) directly, upon substituting /(i) = ^, 
 as the variables easily admit of separation ; thus, 
 
 dt '^'BC~ B 
 
76 
 
 may be written 
 
 CIRCUITS CONTAINING 
 
 dq 
 
 dt 
 
 q-GE~ BO' 
 
 ajid log 
 
 {q-GE) 
 
 BO' 
 
 Hence q = C E -\- c, e 
 
 RC 
 
 which is identical with (70). 
 
 The curve representing the charge of a condenser is 
 shown in Fig. 19. The time-constant B0= .0004. The 
 final charge is 
 
 = ^ r= 4 X 10-" X 200 X 10' X 10 = .0008 coulombs. 
 
 Fig. 19. — Cubve showing the Charge of a Condenser whose Ca- 
 pacity C = 4 MiCROPAKADS when SUBJECTED TO A DIFFERENCE OP 
 
 Potential op 200 Volts through a Resistance op 100 Ohms. 
 
 The curve of discharge for the same condenser under 
 the same conditions was given in Fig. 18. 
 
 Case III. Electeomotive Force a Simple Haemonio 
 Function of the Time. 
 
 Let us now suppose the impressed E. M. F. to be a 
 simple harmonic function of the time, as in Case III., Chap, 
 
RESIST ANGE AND CAPACITY. 11 
 
 III., in the discussion of circuits containing resistance and 
 self-induction ; that is, 
 
 (73) e=f{t) = Esmcot, 
 
 where E is the amplitude, or maximum value of the 
 E. M. F., and go the angular velocity. By differentiation, 
 
 de 
 
 jj =/' {t) = E GO cos, cot. 
 
 Substituting these valuos in the general equations (63) and 
 (64), we obtain 
 
 (74) i^E^e' ^/e^ '^ cos co t dt + c, e" ^. 
 
 (75) ? = J e" ^y*e"^^ sin ootdt-{-c,€~ ^. 
 
 These integrals may be found by the formulae of reduc- 
 tion, obtained by integrating by parts, given in equations 
 (25) and (24). 
 
 Applying these formulae of reduction to equations (74) 
 and (75), they become 
 
 (76) i = j^^^,|c»sinc»« + -^cos««f[+c.e~^. 
 
 (77) q = ^^ ^, ^. ^. \^^smo^t- oocosGot\+c,e-^o ^ 
 
 These equations (76) and (77) may be simplified by the 
 trigonometric formula 
 
 (27) 
 
 ^ sin ^ + i? cos ^ = VA' + B' sinj 6/ + tan"' j i. 
 
E --^ 
 
 q = — sm\oot—i&-D.-'CBGo\+c,e ^° . 
 
 78 CIRCUITS CONTAINING 
 
 By the application of this formula to (76) and (77) we 
 have the complete solutions of the differential equations, 
 namely, 
 
 (78) i= _^_^_ sin|a,^ + tan-^[+c.e"^"^, 
 
 V^' + o^' 
 
 and 
 
 V^' + ct" 
 
 The last equation is equivalent to 
 
 (79) q = -^A__cos]a,^ + tan-;^[ + c,e"'^. 
 
 These equations (78) and (79) are the complete solutions, 
 expressed in their simplest forms. It will be noticed that 
 the differential of (79) is (78), according to the relation dq 
 — i dt. It was not necessary to carry both equations 
 through together, as one may be directly derived from the 
 other by integration or differentiation. It is thought it 
 may add interest to the case if we have the two to compare, 
 so that any differences that exist become more apparent. 
 
 After a very short time the last term of each of these 
 equations, containing the arbitrary constant of integration, 
 becomes inappreciably small and may be neglected. Then 
 it is seen that the current and charge are both harmonic 
 functions of the time ; but the current, instead of lagging 
 behind the impressed E. M. F., as it did in the case where 
 there was self-induction in the circuit, advances ahead of it 
 
 by an angle whose tangent is ytb — • When the capacity 
 C is infinite (and there is no condenser in the circuit, as ex- 
 
RESISTANCE AND OAPACITT. 79 
 
 plained on page 67) the tangent 7=-^- is zero, and the cur- 
 
 Kj IC GO 
 
 rent is in phase with the E. M. F. "When the condenser 
 alone is in circuit, so that the resistance is negligible, 
 
 ^-D — becomes very large and the angle of advance is 
 
 nearly 90°. -pj— has been termed the reactance- 
 The equation of the current then becomes 
 
 (80) i= CEgo sin (co t + 90°), 
 and of charge 
 
 (81) 9- = - Qcos((»f + 90°); 
 
 and the charge will always be a maximum when the current 
 is zero and vice versa, as the cosine is a maximum when the 
 sine is zero. 
 
 When the sin ] t»i + tan~' „ „ \ becomes unity the 
 
 current has its maximum value /, and 
 
 M 
 (82) 
 
 ^R 
 
 C'co' 
 
 The radical \/ E'A is the apparent resistance of 
 
 V C of' 
 
 the circuit ; and, upon comparing with equation (29), we 
 
 see that it corresponds to the radical ■\/B' + £' oo\ which 
 
 has been called the " impedance" of the circuit, in the case 
 
 where there is self-induction and resistance only. 
 
 Case IV. Any Peeiodic Electeomotive Foece which is 
 NOT Habmonio. 
 
 If the impressed electromotive force is any periodic 
 function whatsoever of the time, then— as was mentioned 
 
80 CIllOVITS CONTAINING BESISTANCE AND CAPAOITT. 
 
 in the discussion of circuits containing self-induction — this 
 E. M. F. may be expressed, according to a theorem due to 
 Fourier, as the sum of terms of the form 
 
 Thus, 
 
 (83) e = ^> E sin {b cot +0) 
 
 E, b, e. 
 
 may represent any electromotive force whatsoever, where 
 E, b, and ff have n different values corresponding to n terms 
 of the sum. As was previously shown in the case of self- 
 induction, each term of the E. M. F. impressed gives rise 
 to a corresponding term in the resultant current equation 
 of the form 
 
 == Sin J 6 c^ « + .9 + tan-'-^^j-^ I , 
 
 V^ 
 
 B'-{- 
 
 C" b' oa" 
 
 where E, b, and 6 have values equal to their values in the 
 corresponding term of the E. M. F. equation. 
 
 The expression for current, then, when (83) is the im- 
 pressed E. M. F., is 
 
 (84) i = ^>—j- — - =-sin]&ojf + g 
 
 E,b, 
 
 This gives the general solution for the current in a 
 simple circuit containing resistance and capacity, and any 
 impressed E. M. F. The discussion of this general solution 
 will be deferred until circuits containing resistance, self- 
 induction, and capacity have been considered. 
 
CHAPTER VI. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF INDUCTION, 
 AND CAPACITY. GENERAL SOLUTION. 
 
 Contents. — Equation of energy in terms of e, i, and i ; in terms of e, q, and 
 t. Equation of E. M. F.'s in terms of e, i, and t ; in terms of e, q, and t. 
 Equations transformed for solving in terms of i and t ; in terms of 
 q and t. Complete solution for i in terms of t ; complete solution for q 
 in terms of t. Four cases will be considered: I. e =/(<) = 0; II. 
 e =f{t) = E; III. 6 =f{t) = Eain cot; IV. e =f(t) = 2Esin{ba>t + 6). 
 
 In the preceding chapters the formation of the differ- 
 ential equations for circuits containing resistance and self- 
 induction alone, and resistance and capacity alone, has been 
 discussed, and the solution of these differential equations 
 obtained and discussed for these two particular cases. It 
 is now proposed to consider a circuit containing all three, 
 resistance, self-induction, and capacity, in series, and in the 
 present chapter to derive from the differential equations 
 two general solutions which express, respectively, the cur- 
 rent flowing in the circuit and the charge of electricity in 
 the condenser, at any moment, when the circuit is sub- 
 jected to any impressed electromotive force whatsoever. 
 The succeeding five chapters of Part I will then be devoted 
 to a discussion of these general equations, now to be ob- 
 tained, and their application to various particular cases of 
 impressed electromotive forces. 
 
 The differential equation of energy for a circuit contain- 
 ing all three, resistance, self-induction, and capacity, may 
 
 81 
 
82 CIRCUITS CONTAININO 
 
 be written at once, since we have already derived expres- 
 sions which represent the energy used in heating the 
 conductor [see equation (4)], in creating the magnetic field 
 around the conductor [see equation (6)], and in charging 
 the condenser [see equation (53)]. 
 The equation of energy is 
 
 , c idt I idt 
 
 (85) eidt-Bi'dt-\-Li^dt-\ 'L 
 
 d t G 
 
 The first member of this differential equation eidt 
 
 represents the total energy supplied to the circuit in the 
 
 time dt. A part of this energy represented by ^i'*^^ is 
 
 di 
 used in heating the conductor. A second part Li-ridt is 
 
 expended in creating a magnetic field in the space sur- 
 rounding the conductor, A third part, represented by 
 
 id tj id t 
 
 -^ , is expended in charging the condenser. Equa- 
 
 tion (85) is the general differential equation of energy, in 
 terms of the current which flows in the circuit, the E. M. F. 
 which drives the current, and the time, for a circuit con- 
 taining resistance, self-induction, and capacity in series. 
 
 This equation of energy may be expressed as a differen- 
 tial equation in terms of the quantity of electricity in the con- 
 denser, that is, the charge of the condenser, the E. M. F., and 
 
 the time, by means of the relation dq^idt, ovq= fidt. 
 On substituting in (85) i = -tt, we have 
 
 Each term in this equation is equal to the correspond- 
 ing term in equation (85), since it is obtained by direct 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 83 
 
 substitution. The first member, e-sjdt, is the total energy 
 
 supplied to the circuit, and the three terms of the second 
 member represent the three ways in which this energy is 
 expended, viz., in heat, creating the field, and charging the 
 condenser. 
 
 If equation (85) is divided through by id t, it becomes 
 an equation of E. M. F.'s, thus : 
 
 (87) e=Bi-\-L 
 
 dt'^ C ' 
 
 If equation (86) is divided through by -jj d t, it likewise 
 becomes an equation of E. M. F, 's, thus : 
 
 ^dq , ^ d' g , a 
 
 These are equations of E. M. F.'s: equation (87) in 
 terms of current, E. M. F., and time ; and equation (88) in 
 terms of the charge of the condenser, E. M. F., and time. 
 Each term in (88) is equal to the corresponding term in 
 (87). The first member, e, is the E. M. F. impressed upon 
 the circuit. That part of e necessary to overcome the 
 
 resistance is Hi, or Ji -ri. That part of e necessary to 
 
 di 
 overcome the counter E. M. F. of self-induction is L -ji, or 
 
 d'a 
 £ — i. The third part of e, necessary to overcome the 
 
 r idt 
 counter E. M. F. of the condenser, is — —- , or i-. 
 
 These differential equations may be written in forms 
 more convenient for solving. Differentiating equation (87) 
 
84 CIRCUITS CONTAINING 
 
 with regard to t, to free it from the integral sign, we 
 obtain 
 
 d'i H di i _ Ide 
 (89) dt'^L'dt'^rO^Xdt' 
 
 By transposition (88) becomes 
 
 *-^"-' df^Ldt^ZC~r 
 
 We know that the impressed E. M. F. has one value at 
 one particular time and is therefore a single-valued func- 
 tion of the time, that is, e =f(t). When we introduce this 
 relation into (89) and (90), the general solution of each of 
 these equations may be readily obtained. The solution of 
 equation (89) will give the value of the current at any time, 
 and the solution of equation (90) will give the value of the 
 charge of the condenser at any time. 
 
 de 
 If e =f{t), and jr —f (t), upon substitution in (89) and 
 
 (90), we have 
 
 GENEBAX SOLUTION FOE CUEKENT AT ANY TIME. 
 
 In solving equation (91) to obtain the value of the cur- 
 rent at any time, it is convenient to make use of the sym- 
 bolic method for linear equations. (See page 101, Johnson's 
 Differential Equations.) 
 
 dt ' df 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 85 
 "Writing (91) in symbolic form, we have 
 
 Eesolving the inverse operator, „ =— , into 
 
 partial fractions, we have the identical equation 
 1 LC 
 
 (94) 
 
 
 ^ , EG- V E'C'-4.LG 
 
 ^ , BG+VR'C'-4:LG 
 ^+ 2Z(7 
 
 2ZC 
 (95) Let2'.=-— 
 
 and T„ = 
 
 RG-VR'G'-4:LG 
 
 2LG 
 
 BC-\-VE'C'-4:LC' 
 
 Placing these values in (94), and substituting (94) in (93), 
 we obtain 
 
 Each term of equation (96), equated separately to i, 
 forms a linear equation of the first order. This will be 
 evident when we consider the linear equation of the first 
 
86 CIRCUITS CONTAINING 
 
 d n ' 
 order between tlie variables x and y, viz., --. 1- ay =/(»). 
 
 When written in the symbolic form this becomes 
 
 {D + a)y=f{x), 
 
 (97) or y = ^.J(x). 
 
 The solution of this linear equation of the first order is 
 known to be (see Johnson's Diff. Equations, page 31) 
 
 (98) 2/ = 6-«=» /'6»*/(a;)da; + ce-°*. 
 
 Here c is the arbitrary constant of integration, and none 
 other must be added when the integration is performed. 
 By equating (97) and (98), we have 
 
 jf^fi^) = e" «*ye«^ f(x)d x-i-ce'"". 
 
 If we replace a, in this general formula, by the constant 
 Tjr, and/(a;) by/' (t), we have 
 
 ■'- 1 
 
 i> + ^ 
 
 But this is the value of the first term ia the parenthesis 
 of equation (96). The value of the second term in that 
 parenthesis may be found in a similar manner, and (96) 
 may finally be written 
 
RESISTANCE, SELF INDUCTION, AND GAPACITT. 87 
 
 This is the general solution of equation (91) and gives 
 the current which flows at any time in a circuit having 
 resistance, self-induction, and capacity. 
 
 Since the differential equation (92) for the charge be- 
 comes identical with the differential equation (91) for the 
 current when we write /' (t) instead of / (t), and since / 
 denotes any arbitrary single-valued function whatever, we 
 may in the general solution (99) suppress the accents on 
 the arbitrary functions and write the solution for q. Thus, 
 
 PABTICULAE ELECTEOMOTIVE FOKCES. 
 
 These equations, (99) and (100), express the values of 
 the current and charge at any time, when the impressed 
 E. M. r. is anything whatever, since / is any arbitrary 
 single-valued function whatever. 
 
 There are four cases, covering all possible ones, which 
 arise according to the nature of the impressed E. M. F. 
 These are : 
 
 Case I. e=f{t) = 0. 
 
 Case II. e =/(^) = j?= constant. 
 
 Case III. e =f{t) = -fi'sin oo t. 
 
 Case IV. e =f{t) = ^> Esm.{hoot + 8). 
 
 The meaning of the first assumption is that the im- 
 pressed E. M. E. is to be zero at every point of time. This 
 condition is fulfilled if we charge a condenser with some 
 
88 CIRCUITS CONTAINING 
 
 quantity Q, and then suddenly remove the impressed 
 E. M. E., that is, if we connect the two plates of the con- 
 denser by a conductor so as to discharge it. The im- 
 pressed E. M. F. remains zero at every point of time after 
 the removal of the source of E. M. F., and consequently 
 satisfies the condition e =f{t) = 0. The solutions of _ the 
 differential equations under this assumption give the cur- 
 rent at any time flowing in the circuit, and' the charge 
 a»any time remaining in the condenser, when an im- 
 pressed E. M. F. is suddenly removed from the circuit. 
 Xt may be any circuit whatever containing any combination 
 of resistance, self-induction, and capacity, that is, a circuit 
 containing M and Z alone, Ji and G alone, or H, L, and C 
 together. In case the circuit has R and G, or R, L, and C, 
 the solutions will give the current i and quantity q at any 
 time during the discharge of the condenser. If the circuit 
 contains R and L alone, the solution will give the current 
 at any time as it dies away after the removal of the 
 E. M. F. 
 
 "When we assume e =f{t) — E = a, constant, we mean 
 that the E. M. F. is to be equal to ^ at every point of 
 time. This condition will be fulfilled if the source of 
 E. M. F. in any circuit is suddenly changed from one con- 
 stant value to another constant value, either of which may 
 be zero. If the circuit contains R and G, or R, L, and G, 
 the solutions give the current fiowing in the conductor 
 and the charge of the condenser at any time after the 
 change in the E. M. F. If the circuit contains i? and L 
 only, the solution gives the value of the current at any time 
 as it changes to its final steady value. 
 
 The third assumption, e^ E sin co t, means that the 
 circuit contains an impressed E. M. F. varying harmoni- 
 cally with the time. The solutions of the general equa- 
 tions for g and i show that when the impressed E. M. F. is 
 harmonic, both the current and the charge are likewise 
 
RBSI8TANCE, SELF INDUCTION, AND OAPACITY. 89 
 
 simple sine-functions of the time, having the same period 
 as the E. M. F. 
 
 The fourth assumption, e = ^> ^ sin (b a)t-{- 6),— 
 
 E, b, fl. 
 
 where h takes in succession any integer values, — means that 
 the circuit contains an impressed E. M. F. which is any 
 periodic function of the time whatsoever. 
 
 The solution and discussion of these four cases will be 
 considered in the following chapters. 
 
CHAPTER VII. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF INDUCTION, 
 AND CAPACITY. 
 
 Case I. Discharge. 
 
 Contents : — Integral and differential equations -when e =f{i) = 0. Sir 
 Wm. Thomson's solution, i equation with value of T replaced. Three 
 forms of i and q equations. To transform the i-equation to a real form 
 when -ff' G is less than 4 L. To derive the solutions from the differen- 
 tial equations when R' C = i L. 
 
 Non-oscillatory Discharge. 
 Determiuation of constants. Complete solution. Value of T re- 
 placed. Current and charge curves for a particular circuit. Time of 
 maximum current. Equation (125) applied to a circuit containing resist- 
 ance and self-induction only, and to a circuit containing resistance 
 and capacity only. 
 
 Oscillatory Discharge. 
 Determination of constants. Complete solution for 8 and g. Current 
 and charge curves for a particular circuit. 
 
 Discharge of Condenser when JS'' C— iL. 
 Determination of constants. Complete solutions for i and g. Figure 
 showing method of constructing the current and charge curves. Curves 
 for i and g in a particular circuit. 
 
 In this chapter the case will be discussed in which the 
 impressed electromotive force is suddenly removed from 
 the circuit or reduced to zero ; that is, e =f{t) = 0. When 
 a current has been flowing in a circuit and the source of 
 electromotive force has been suddenly removed, the cur- 
 rent continues to flow for an appreciable time before 
 
 90 
 
BESI8TANCE, SELF INDUCTION. AND CAPACITY. 91 
 
 dying entirely away. The value of the current at any 
 time may be ascertained by applying the general equation 
 (99) to this particular case. 
 
 As another example, we may have a condenser or 
 Leyden jar charged to a certain difference of potential, and 
 the source of potential then removed. If we now connect 
 the two plates of the condenser or coatings of the jar with 
 a conducting wire, a current flows through the wire and 
 the condenser is discharged. The source of potential was 
 previously removed, and so e =f(t) = 0. The general 
 equations (99) and (100) can be applied to this particular 
 case, enabling us to ascertain the current which flows at 
 any time in the circuit and the charge remaining in the 
 condenser. 
 
 Since /(t) = 0, the first derivative is f'(t) = ; and if 
 the value/' {t) = is substituted in the general equation 
 (99) for current, and the value /{t) = in equation (100) 
 for charge, we have 
 
 (101) i = c.6 ^' + c,e \ 
 
 -1 -1 
 
 (102) q=c,e ^' + c, e ^". 
 
 Had the value e = =/{t) been substituted in .the 
 differential equation (92), and f'{t) = in equation (91), 
 we should have had 
 
 d'i , H di , i n 
 
 (103) d¥ + Idt+Ta=^'^- 
 
 (104) di+zdi+rc=^- 
 
 It is to be noted that the form of the differential 
 equation for i is identical with that for q. Hence their 
 
92 CIRCUITS GONTAimm 
 
 integrals (101) and (102) have tlie same form, although with 
 different arbitrary constants of integration. The solutions 
 of the differential equations (103) and (104) — which are 
 identical with (91) and (92) when their second members 
 are zero — give what is called the " complementary function " 
 (see Johnson's Differential Equations, Art. 94). The 
 complementary function contains all the arbitrary con- 
 stants of integration. The sum of the particular integral 
 — found to satisfy equations (91) and (92) when the second 
 member is not zero — and the complementary function 
 gives the complete integral of the general differential 
 equations (91) or (92). 
 
 The particular case of the discharge of a condenser 
 through a circuit possessing resistance and self-induction 
 has been fully discussed by Sir Wm. Thomson and was 
 published as early as 1853 in the Philosophical Magazine. 
 He obtained equation (102) as his result, which he showed 
 could be expressed in two different forms, according as T^ 
 and T, are real or imaginary. 
 
 Writing equation (101) in full, by replacing the values 
 of y, and T, given in (95), we have 
 
 (lUo) I = c, e + Cj c 
 
 If the value of B' C is greater than 4 L, the value of i is 
 real ; but if IS' C is less than 4 Z, i apparently assumes an 
 imaginary form. It will be shown, however, that i can 
 by a trigonometric transformation be expressed in a real 
 form when R' O is less than 4Z. 
 
 When R' O is equal to 4:1 and we have the critical case, 
 it is evident that the two terms of equation (105) may be 
 written as one, and thus the two arbitrary constants com- 
 bine into one. The complete solution, which must contain 
 two arbitrary constants, inasmuch as it is derived from a 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 93 
 
 differential equation of the second order, cannot be readily 
 obtained in this case from (105) ; but it will be directly 
 obtained from the differential equations (103) and (104). 
 
 TO TRANSFORM EQUATION (105) TO A REAL FORM WHEN 
 B'C IS LESS THAN 4Z. 
 
 After factoring out the common factor e ^^, we may 
 write (105) in another forin, thus : 
 
 (106) i=e %,e '^^ +c,i '^^ 'j 
 
 Here y is used to represent V— 1. If we write 
 
 (107) e = 2Z77 *■■ 
 
 then (106) becomes 
 (108) 
 
 Rt 
 
 2L ( J6 , -j6 I 
 
 The sine and cosine may be written in exponential form* 
 
 thus: 
 
 j6 -j6 je -jB 
 e — e e -f- e 
 (109) sin = 5-: , and cos ff = s 
 
 * By Maclaiiriu's theorem for the expansion of a function into a series, 
 the sine and cosine may be developed into the following series : 
 
 (1) sin 9 = 9 - -«;- +^-^^ - i.g.3;;.5.6., + etc. 
 
 9' B* 96 
 
 <2) cos9 = l-j^ + j,2:3^_j,^^-— g + etc. 
 
 Also the development of e into series gives 
 
 iQ e» 93 6* 9' 
 
 {3) e =1+^9-1:3-^1:373 + 1:3:31+^1.3.3.4.5 
 
 9' 9^ 9^ , 
 
 1.3.3.4-5-6 ■'1-3.3.4-5-6-7 +l-2-3-4-5-6-7-8"^' ' " 
 
94 CIBGUITS CONTAINING 
 
 Therefore cos 6 -{-j sin 6 = e , 
 
 and cos — j sin ^ = e 
 
 Multiplying throngli by c, and c,, respectively, and add- 
 ing, we have 
 
 (110) c,e -)- c^e = (c, + c,) cos ^ + (c, — c,)^ sin ft 
 
 If c, and Cj are conjugate imaginary quantities, they 
 may be written 
 
 A + Bj 
 
 c. = 
 
 2 ' 
 
 A-Bj 
 
 Multiplying (1) by j and adding to (3), we find that the resulting series is 
 identical with (3). Hence we obtain 
 
 (4) cos e +^'sin 6 = e . 
 The expansion or e into series gives 
 
 (5) e ^l__;9- — +j— -+— -— -^ 
 
 1-3 ' •'l-3-3^1-3-3.4 •'1-3-3-4-5 
 
 1.3.3-4-5-6^ 
 
 Multiplying (1) hyj and subtracting from (3), the resulting series is iden- 
 tical with (5). Hence we have 
 
 (6) cos 6 — ^' sin 6 = 6 
 
 Adding equations (4) and (6) and dividing by 3, we get 
 
 ja -j9 
 
 (7) cos e = ^- + ^ 
 
 3 
 Subtracting (6) from (4) and dividing by 2j, we have 
 
 (8; SID 9 = 
 
 ^J 
 
RESIST ANOE, SELF INDUCTION, AND CAPACITY. 95 
 
 ■where A and B are both real quantities. Taking the sum 
 and difference, 
 
 c,-]-c,^ A, 
 
 c, — c, = Bj; 
 
 and substituting these values in (110), we have 
 
 (111) c, e -\-c^e = A cos ^ + -^ sin (9, 
 
 where Cj and c, are imaginary, while A and JB are real 
 quantities. Substituting (111) in (108), we obtain 
 
 (112) i = e ^ ^(^ cos (? + ^ sin 6). 
 
 By the trigonometric formula [see Chapter III, equa- 
 tion (27)], 
 
 ^cos6'+JSsin(9= VA" + £' sin (^-f tan"^-g j, 
 
 we may finally write equation (112), after restoring the 
 value of B from (107), in the form 
 
 (113) i = ^e'^sinj 2X^ < + ^}. 
 
 where A and ^ are the arbitrary constants of integration. 
 Here A is not the same as in equation (112), but stands for 
 
 V A' -{-£', and # stands for tan -g . This equation is 
 
 the equivalent of (105). It is real when (105) is imaginary 
 and imaginary when (105) is real. 
 
96 CIRCUITS. OONTAININQ 
 
 TO DEBIVE THE SOLUTIONS FEOM THE DIFFEBENTIAL EQUA- 
 TIONS WHEN R'C—LL. 
 
 If WC is equal to 4Z, then the differential equations 
 (103) and (104) become 
 
 d'i Rdi , B' . ^ 
 
 (11^) dF + Ldi + -^^'-^- 
 
 Upon substituting i = e"'*, we have 
 
 (116) ra^^~m + ~=0, 
 
 which is seen to be a perfect square as it stands, and con- 
 sequently the two values of m become equal, and 
 
 TO 
 
 m = — 5"^. When there are equal roots, the solution is 
 
 of the form 
 
 i = c, e^' + c^fe"*' 
 
 (see Johnson's Diff. Equations, page 95); or, replacing m by 
 
 R 
 
 its value, — ^-y , we have as the complete solutions 
 
 (117) i = c.e"'^ + c,«e"'^, 
 
 _Rt_ _Bt 
 
 (118) ■ y = c'e ^^J^d'te ^^. 
 
 Eeturning to equation (103), we may write its solution 
 (101), the complementary function, in three different real 
 forms, according as the value of .S' C is greater than, less 
 than, or equal to 4 L. These forms are : 
 
BE8ISTANCE, SELF INDUCTION, AND OAPACITT. 97 
 WhenB'0> 4:Z, 
 
 (119) i = C,6 2^Cr *_j_^^g zLC '. 
 
 When B'O < 4:L, 
 
 (120) ^ = ^e-^'^sinjJ^i?^lZ,+ ^|. 
 WhenE''C=4:L, 
 
 _Rt Bt 
 
 (121) i = c,e ^'' + cje '^^ 
 
 The value of the charge q given by equation (102), being 
 of the same form as (101), may take three different forms 
 according as ^'Cis greater than, less than, or equal to 4Z ; 
 and these forms only differ from the above in the arbitrary 
 constants, thus : 
 
 WhenR'C>4:L, 
 
 Rc- vWci-iLc ^ BCj-yn^c -Ilc 
 
 (122) g = c'e ^^^ +c"e ^^^ \ 
 
 WhenE'0<4:L, 
 
 (123) g = A'e-^- sin { ^^^^^^t+^' J . 
 
 WkenB'G = 4:L, 
 
 -El -El 
 
 (124) q = c'e '^ + c"«e 'f 
 
 The constants of integration in these equations are de- 
 termined by the initial conditions imposed by the problem. 
 For instance, if a condenser charged with a quantity Q is 
 suddenly discharged through a circuit with resistance and 
 self-induction, we may count the time from the moment of 
 discharge, and thus have q = Q and i= when f = 0, and 
 q = and i = when t = ao . 
 
ye CIRCUITS CONTAINma 
 
 NON-OSCILLATOBY DISCHAEGE. 
 
 Determination of Constants. — The equations (119) and 
 (122) may be written as in (101) and (102), in terms of the 
 time- constants J', and T^ [see (95)], thus : 
 
 _ i. -L 
 
 (125) i = c, 6~ ^' + c, e ^». 
 
 _1 _ L 
 
 (126) q = c'e ^•+c"e ^'. 
 
 The arbitrary constants c, , c^ , c', c" of these equations 
 "will be determined according to the conditions mentioned 
 above, viz., when ^ = 0, 4 = and q = Q; when t = ao , 
 i = and q = 0. Substituting in (125) i = when ^ = 0, 
 and in (126) q = Q when i = 0, we have 
 
 = c, + c„ or c, = — c,. 
 
 (127) <3 = C + c". 
 
 Since we have the relation dq = idt, -we may differen- 
 tiate (126) and write 
 
 *--^e - e 
 
 Equating this and (125), we find 
 
 c 
 
 c 
 c, = - ^, or c" = - c, 2;. 
 
 Remembering that Cj = — c^ , we may write 
 
 c" =^c,T,. 
 
 Adding c' and c", 
 
 c' + c" = c. ( r, - rj = Q. [See (127)]. 
 
REaiSTANGB, SELF mDUGTlON, AND CAPACITY. 99 
 Hence c, = 
 
 o, — 
 
 T, - t: 
 
 
 Q 
 
 Cj — 
 
 T,-T^ 
 
 -,' 
 
 QT, 
 
 
 T,-T,' 
 
 „" — 
 
 QT, 
 
 Substituting in (125) and (126) the constants c^,c^, c', c", 
 as finally determined, we have 
 
 (128) i=T^yi-e-k\, 
 
 (129) q^—9-^^T,e^^-T,e~^^\. 
 
 Discussion of Non-oscillatory Discharge. — These equa- 
 tions give the complete solution and express the current 
 or the charge at any time after discharge (see Fleming's 
 " Alternate Current Transformer," Vol. I. page 376). They 
 show that if we have the relation IfG > 4:L, the dis- 
 charge is a gradual dying away without oscillation. Since 
 T^ and T^ are each of them positive when B^G > ^L [see 
 (95)], i or q may be represented geometrically as the 
 difference of two decreasing logarithmic curves. To see 
 this more clearly, the values of the time-constants T^ and 
 T^ may be substituted in the coefficients of equations (128) 
 and (129). The result is 
 
 (130) ^ je"^-^"^4 
 
100 CIRCUITS OONTAININR 
 
 (131) g- gj ^^ r^"^- 
 
 _ ^ i ^^ - 1 1 e"^' 
 
 These equations may be more easily explained by refer- 
 ring to Figs. 20 and 21, which represent the plot of these 
 equations for particular assumed values of R, L, and G. 
 The values assumed for the constants of the circuit are 
 
 R = 100 ohms, L = .0016 henrys, C = 1 microfarad. 
 
 By calculating the values of T, and T, [equation (95)], 
 r. = 8 X 10-^ and T; = 2 X 10 -5, the equations (130) and 
 (131), with these particular values, become 
 
 - < _ < 
 
 m ^•=:63no^«l^"""^-^''"""1- 
 
 (133) y = |(l| + l) e'^'^-Klf - l) 
 
 2x10-6 
 
 If the condenser was charged to a potential of 2000 
 volts, the capacity being .000001 farads, the charge is .002 
 coulombs. Substituting this value for Q, we have 
 
 i = 33.33 (e ^"^^"-^ -e '^ '<"'), 
 
 - 2X10-6 
 
 where i is in amperes and q in coulombs. 
 
 In Fig. 20, curves I. and II. represent the two compo- 
 nent logarithmic curves, corresponding to the first and 
 
RESISTANCE, SELF INDUOTION, AND 0APACIT7. 101 
 
 second terms, respectively, of equation (130),whose difference 
 gives the resultant current curve III. Curve II., cor- 
 responding to the second term, has the larger time-constant, 
 and is therefore the more important curve. The area m- 
 
 Itf^S Seconds SOxlo"' 
 
 Fig. 20.— Curve showing Current during Non-oscillatory Dis- 
 charge OF Condenser with Capacity 0=1 Microfarad. 
 THROUGH A Circuit with Resistance R - 100 Ohms, and Selp- 
 induction L = .0016 Hbnrts, when origin ally charged to a 
 Potential of 2000 Volts. 
 
 eluded between curve III. and the axis of abscissae is equal 
 to fidt= Q, and is therefore independent of the constants 
 of the circuit through whict the condenser is discharged. 
 
 The current is a maximum at a point which may be de- 
 termined by differentiating equation (130) and equating the 
 first derivative to zero in the usual manner for a maximum. 
 
102 
 
 CIRCUITS CONTAINING 
 
 The time t^ at whicli the current is a maximum is thus 
 found to be 
 
 Pig. 21. — Curve showing non-oscillatory Discharge op a Con- 
 denser, WITH Capacity 0=1 Microfarad, through a Circuit 
 WITH Besistancb R = 100 Ohms, and Self-induction L = .0016 
 Henrys, when originally charged to a Potential of 3000 
 Volts. 
 
 Substituting in (134) the particular values used in plot- 
 ting Fig. 20, we find the time when the current is a maxi- 
 mum to be 
 
 f,„= 3.78X10-=. 
 
 In Fig; 21 curves I. and II. are the two component loga- 
 rithmic curves, corresponding to the first and second terms, 
 respectively, of equation (131) for charge. Curve III. is 
 plotted by subtracting II. from I., and represents the 
 
RESISTANCE. SELF INDUCTION, AND CAPACITY. 103 
 
 charge of the condenser at any time. It is noticeable that 
 the upper curve, I., has the larger initial value, and as T^ 
 is larger than T^, decreases the slower. It is therefore 
 this curve which is the more important in determining the 
 discharge of the condenser. 
 
 EQUATION (125) APPLIES TO A CIKCUIT CONTAINING BESISTANCE 
 AND SELF-INDUCTION ONLY. 
 
 If there is no condenser in the circuit, as explained in 
 Chapter IV., it is equivalent to saying that there is a con- 
 denser of infinite capacity in the circuit. Substituting 
 C = 00 in the equation (95) for the time-constants, we have 
 
 2Z6' 
 
 T, = = c» . 
 
 According to equation (101), we have the value of the 
 
 current at any time 
 
 _t_ __* 
 
 i = c^e ^' -f- c, e '. 
 
 Substituting in this equation the values of 2', and T, 
 above, we have 
 
 i = C, + C, 6 
 
 Bt 
 
 ' L 
 
 When t = 0,i = I, that is, the current flowing previous 
 to the removal of the E. M. F. This gives 
 
 But when f = oo , * = c, = 0. Substituting these values 
 for the constants, we have 
 
 _Bt 
 
 i = Ie ^ 
 a result which is well known [see equation (18)]. 
 
104 CIRCUITS CONTAINING 
 
 EQUATION (125) APPLIES TO A CIECUIT CONTAINING EESISTANOE 
 AND CAPACITY ONLY. 
 
 Upon substituting Z = in the values of the time-con- 
 stants T^ and T^ (95), the expressions become indeterminate, 
 but can readily be evaluated by differentiating numerator 
 and denominator, and then substituting Z = as in ordi- 
 nary vanishing fractions. 
 
 Differentiating numerator and denominator with respect 
 to L, we have 
 
 ^iRC-VE^C^-ALC) 2y^'^'-4Zg 
 
 Now letting Z = 0, we have 
 
 T, = RO. Similarly, T^ = -BC. 
 
 Substituting these values in equations (101) and (102), 
 we have 
 
 (135) i = c.e"^ + c,e"^^. 
 
 (136) ^=,03e"^+c,e"^^^. 
 
 c, and c, must each be zero, or else when i = oo we would 
 have i = 00 and ^ = 00. When f = 0, q =: Q = c^. By 
 differentiating (136) and equating to (135), we have 
 
 ^ dt- BO -'^' ' 
 
 and, therefore, c, = — -^77 = — "wji' 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 105 
 
 Substituting in (135) and (136) the values found for the 
 constants c, , c, , c, , c, , we have 
 
 (137) i=_^e"^=/e~^. 
 
 (138) q=Qe 
 
 t 
 
 These are the well-known results for the case of discharge 
 through a circuit with no self-induction [see equations (67) 
 and (68)]. 
 
 OSCILLATOEY DISCHABGE. 
 
 Determination of Constants. — In the case of oscillatory 
 discharge, the equations for current and charge at any 
 time are 
 
 Rt 
 
 <120) ^ = Ae ^ sin | ^^"^ * + * } 
 
 Rt 
 
 <123) q = A'e '^sin j ^X"^ <+* }• 
 
 The arbitrary constants A, A', 0, and *' will be deter- 
 mined according to the same conditions as those mentioned 
 above, viz., when i = 0, i = and q =: Q; also when < = oo , 
 i = and q = 0. Substituting in (120) i = when t = 0, 
 and in (123) q = Q when < = 0, we have 
 
 = -(4 sin 0i 
 
 (139) and Q = A' sin <P'. 
 
 Since A and sin are constants, and their product is 
 zero, one of them must be zero. But if A is zero, i is zero 
 for every point of time, which is impossible. Therefore 
 
 (140) = 0. 
 
106 CIRCUITS CONTAININO 
 
 Differentiating (123) and remembering that i z= -^ . we 
 have 
 
 _Rt 
 
 (1^1) ' = dt = 2Z— ^^" 1 ^LG ^+* I 
 
 fit 
 
 + 2z^ °°M — 2X^ — '+ r 
 
 Substituting i = when ^ — 0, 
 
 = — JS sm <?' -|- p9 cos *'. 
 
 i/4Z ^— B'G' 
 (142) Hence jg' = tan'' ^c ' 
 
 By (139), ^'=il^- And. by (142), 
 
 sin tan 
 
 ■ 1 
 
 ,G-Ii'G'' r WW 
 EG \I^~4.LG 
 
 To determine the constant A, transform (141) by the 
 formula (27) so as to write it in terms of a sine only. The 
 coefficient of the sine in the equation as transformed will be 
 
 '\Itg' 
 
 Since equations (141) and (120) are each equations for i, 
 we may equate the coefficients of the sine, and have 
 
 vlg 
 
 And, by (143), 
 
 (144) A= ^^ 
 
 V4.LG-R'G' 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 107 
 
 Substituting A and <5, as determined in (144) and (140), 
 in equation (120), and substituting A' and <?', as determined 
 in (143) and (142), in equation (123), we have 
 
 (14:5) I = _ e sm < — ' — n r /-, i 
 
 Bt 
 
 (146) q 
 
 'iQVTC -E 
 
 sm 
 
 V4:ZG-Ii'G\ , , _i V4:ZG-B'C^ 
 
 -^LG * + *^'' -RG ]• 
 
 Discussion of Oscillatory Discharge. — These equations 
 may be more readily understood by referring to Fig. 22, 
 in which curves showing the current and charge, according 
 to these equations, are drawn for the discharge of a con- 
 denser for particular values of R, L, and G, assumed. The 
 particular constants assumed are 
 
 R = 100 ohms, L = .0125 henrys, C = 1 microfarad. 
 If the condenser be originally charged to 2000 volts, Q = 
 .002 coulombs. On substituting these values in (145) and 
 (146), the equations for current and charge become 
 
 - 4000 1 . „„„„ , T 
 
 i = 20 e sm 8000 1, and 
 
 q = .00224 e ~ '"^* sin (8000 1 + tan" ^ 2), 
 where i is in amperes and q in coulombs. Curve I., repre- 
 sentine the current, is a sine-curve with an amplitude de- 
 
 - 4000 t 
 
 creasing according to the logarithmic curve 20 6 
 The period is -^=.000785 seconds; that is, there are 
 1275 complete oscillations per second. A very few oscil- 
 lations are sufficient for a complete discharge. 
 
 The charge at any time is shown in curve II., which is 
 likewise a sine-curve with an amplitude decreasing accord- 
 
108 
 
 CIRCUITS CONTAINING 
 
 -4000* 
 
 ing to the logaritlimic curve, in this case .00224 e 
 The scale in Fig. 22 is such that the same logarithmic curve 
 is an envelope lor the current and the charge curve. The 
 periods of the two are the same, but the curves differ in 
 
 Fig. 22. — Oscillatory Discharge op a Condenser with Capacity 
 (7=1 Microfarad, through a Circuit with Resistance R -= 
 100 Ohms, and Self-induction = L .0125 Henrys, when origi- 
 nally CHARGED to A POTENTIAL OF 2000 VOLTS. 
 
 phase by- an angle = tan"* 2, that is, the charge is ahead of 
 the current by an angle of advance of 63° 27'. 
 
 DISCHAKGE OF THE CONDENSEE WHEN R'C—^L. 
 
 Determination of Constants. — This is the critical case when 
 the discharge is just non-oscillatory. The equations for 
 the current and charge, as previously determined, are 
 
 (121) 
 (124) 
 
 = c, e 
 
 C e 
 
 Rt 
 
 + c,te' 
 
 Bt 
 
 'sL 
 
 Bt 
 
 -{-c"te 
 
 Bt 
 '2L 
 
RESISTANGB, SELF INDUCTION, AND CAPA0IT7. 109 
 
 The arbitrary constants of integration, c„ c„ c', c", of these 
 equations will be determined by the same conditions as in 
 the previous cases, namely, -yvhen t = 0,i = Q and q= Q. 
 Equations (121) and (124) then become 
 
 (147) = c, 
 
 Q = c'. 
 
 Differentiating equation (124) and substituting Q for c', we 
 obtain 
 
 (-) ••=if=(-ii+«"-m"^- 
 
 But when < = 0, i = ; therefore 
 
 (149) ^" = |t' 
 
 Equating equations (121) and (148) and replacing the values 
 for the constants given in (147) and (149), we have 
 
 (150) c, = - 
 
 4i= 
 
 Substituting for Q its value E C, and for E" its equivalent, 
 
 4.L 
 in this particular case -^ • (150) becomes 
 
 E 
 
 Having thus determined the values of the arbitrary con- 
 stants, the equations for current and charge may be written 
 
 (151) ' = -T*^ ' 
 
 (152) ^=(l + g)(2, 
 
 Bt 
 ' 2L 
 
 Discussion of Discharge lohen E' — ^ L. — These equa- 
 tions give the value of the current and charge at any 
 time during the discharge of the condenser in the case 
 
110 
 
 CIRCUITS CONTAINING 
 
 where the discharge is just non-oscillatory. This case 
 is often called the case of quickest discharge, as was pointed 
 out by Dr. "W. E. Sumpner in the Philosophical Magazine, 
 and afterwards discussed by Dr. Oliver Lodge in the Elec- 
 trician for May 18, 1888. 
 
 The curve representing the plot of the equation (151) 
 for the current may be drawn as indicated in Fig. 23. A 
 
 Fig. 23.— Showing Method of constructing the Cubrbnt Cubtb 
 IN THE Case where iZ" (7 = 4 Z. 
 
 logarithmic curve I., having its initial value y- and time- 
 
 2Z 
 
 constant —^ , is drawn to represent the equation as it would 
 
 be with t omitted from the coefficient ; and each ordinate 
 is then multiplied by the ordinate of a straight line II., 
 which passes through the origin and represents the uniform 
 increase of the time t. The product of the ordinates of 
 curves I. and II. at each point gives the ordinate of the 
 current curve III. at that point. When actual values of Ji, 
 L, and G are assumed, it is found to be difficult to repre- 
 sent these curves to scale, so that Fig. 23 is shown simply 
 as an illustration of the method of constructing the current 
 curve. Curve I., Fig. 25, represents the cxirrent in an 
 actual case where R = 100 ohms, Z = 2.5 henrys, and C = 
 1000 microfarads, the condenser being originally charged 
 to a potential of 2000 volts. 
 
 The method of constructing the curve, showing the 
 
BESISTANOB, SELF INDUCTION, AND CAPACITY. Ill 
 
 charge left in the condenser at any time, is given in Fig. 24, 
 and is similar to the method just shown for constructing 
 
 Time 
 
 Pig. 24. — Showing the Method of consteucting the Cckve bep- 
 resbnting the charge left in the condenseb at any timb 
 AFTER Discharge. 
 
 the current curve. The difference is that the straight line 
 passes through a point one unit above the origin, on the 
 vertical axis, instead of through the origin as before. 
 
 Fig. 25.— Just Non-oscillatory Discharge of a Condenser with 
 Capacity C = 1000 Microfarads, throttgh a Circuit with 
 Resistance It — 100 Ohms, and Self-induction L = 2.5 Henrys. 
 
 The logarithmic curve has the initial value Q and a time- 
 
 constant -^- . The curve showing the charge for the actual 
 
 SI 
 
 case where H = 100 ohms, Z = 2.5 henrys, and C= 1000 
 microfarads, the condenser being originally charged to a 
 potential of 2000 volts, is represented by curve II., 
 Fig. 25. 
 
CHAPTEE VIII. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF INDUCTION, AND 
 
 CAPACITY. 
 
 Case II. Charge. 
 
 Contents:— Differential equations with e = f(t) = S. Solution of these 
 equations. Solution from tlie general integral equation.. Three forms 
 of i and q equations. 
 
 Nbn- oscillatory Olmrging. 
 
 Determination of constants. Complete solutions for i and ? with 
 constants determined. Curves for i and g in a particular circuit. Equa- 
 tion (101) applied to a circuit containing resistance and self-induction 
 only ; also to a circuit containing resistance and capacity only. 
 
 Oscillatory Charging. 
 
 Determination of constants. Complete solutions for i and g with 
 constants determined. Curves for i and g in a particular circuit. 
 
 Charge of tlie Condenser when IPC= 4.L. 
 
 Determination of constants. Complete solutions for i and g vpith 
 constants determined. Curves for i and 5 in a particular circuit. 
 
 The E. M. F., instead of being zero, as in Case I., is 
 assumed to be a constant B, and e =f(t) = E. This is the 
 case when an E. M. E. is suddenly changed from one con- 
 stant value to another constant value in a circuit, and it 
 includes Case I. as a particular case, since U raaj be zero. 
 Since e is a constant, the first derivative of e is zero, and, 
 
 113 
 
BE8ISTANGE, SELF INDUCTION, AND CAPACITY. 113 
 
 therefore,/' («) = 0. Substituting these values of/(i) and • 
 /' (f) in the differential equations (89) and (90), they become 
 
 a54^ and ^,Rdq q E 
 
 (154) and — +__+^^ =_ 
 
 It is seen that the equation for current (153) is identical 
 with that of the previous case (103), while the equation for 
 
 charge (154) has its second member equal to -j, a constant, 
 
 instead of being zero as in equation (104). By substituting 
 a new variable, q' = q — E O, this equation may be trans- 
 formed into one having its second member zero, thus : 
 
 nKK^ d'q' Rdq' q' _ ' 
 
 (155) -dF + LU^Z-O-^- 
 
 The solutions of (153) and (155)' are, as in the previous case, 
 
 _ t_ _± 
 
 (101) i = c,e ^' + c,e =^'. 
 
 _ 1 _1 
 
 q'=c,e~''^-\-c,e ^ 
 
 Eeplacing the value of g'', and remembering Q = EC = 
 the final charge, we may write 
 
 1 - L 
 
 (156) q=Q+c,e~'^^ + c,e ^'. 
 
 These equations, (101) and (156), for current and charge 
 might have been obtained directly from the general solu- 
 tions (99) and (100) by substituting/(0 = E, and/' (t) = 0. 
 Upon substituting/' (0 = in (99), we obtain (101) directly, 
 and upon substituting/ («) = ^in (100), we have 
 
114 .CIBOUITS CONTAINING , 
 
 But, by the values of T, and T, in (95), we find that 
 
 T,— T, = V H'C -4:LC; &nA hence this equation is 
 identical with (156), as Q ^ EC. 
 
 As in Case I, where/(i) = 0, the equations just obtained 
 for current (101) and charge (156), when / (t) = E, assume 
 three forms. 
 
 WhenB''C>4:L, 
 
 _ — - L 
 
 (157) i = c, e ^' + c,6 ^». 
 
 _ i. _ ' 
 
 (158) j= ^ + c'e ^'-\-c"e ^». 
 
 WJienR' C<4:L, 
 
 (159) % = Ae -^sm I ^^-j^ ^ + «^ | 
 
 Rt 
 
 (160) ^=g+^'e"«^sin|ili^^^^?l^^+^'| 
 
 Bt Bt 
 
 (161) i = c,6 ^^ + cje ^^. 
 
 Bt _ Bt 
 
 (162) ^=Q + c'e '^+c"te 
 
 The constants of integration must be determined by the 
 conditions of the problem as to the previous state of the 
 circuit, the changes made, and the final state, 
 
 NON-OSCILLATOEY CHAEGING. 
 
 Determination of Constants. — The constants c^, c^, c', and 
 c" of equations (157) and (158) will be determined by the 
 following conditions : 
 
 When ^ = 0, i = and q=Q,. 
 
 When t = CO , i = and q= Q. 
 
BESISTANCE, SELF INDUCTION, AND CAPACITY, 115 
 
 This means that the condenser is suddenly charged or dis- 
 charged from the initial charge Q„ to the final charge Q. 
 Determining the constants by the same method as in Case 
 I., we find that ; 
 
 c, = 
 
 c, = 
 
 c = 
 
 c" = 
 
 Q.-Q 
 
 T,- 
 
 t: 
 
 Qo- 
 
 Q 
 
 T.- 
 
 t: 
 
 («o- 
 
 -Q)T, 
 
 T. 
 
 -T, ' 
 
 («c- 
 
 -Q)T, 
 
 T.-T, 
 
 Substituting in (157) and (158) the values of the constants 
 just determined, we hare 
 
 (163) '^•=1^1 -"^-^'^1- 
 
 (164) q = QJ^^^^[Ty'^^-Tj^. 
 
 For Q„ , the original charge, we may write C E, , and for Q 
 the final charge, we may write C E. 
 
 These equations give the value of the current and 
 charge at any time after the change of E. M. F. from E^ to 
 E in & circuit with Ji'C> 4:L. As the equations now 
 stand in their general form, they hold true for either total 
 or partial charge or discharge according to the values of 
 E„ and E, and consequently Q, and Q, assumed. If the 
 final charge is Q = 0, we have the case of complete dis- 
 charge and the equations take the form of (128) and (129). 
 If the original charge Q, = 0, we have the case of charge 
 from zero to Q. 
 
116 
 
 CIRCUITS CONTAINING 
 
 Discussion of Non-oscUlatory Charge. — These equations 
 •will perhaps be better understood by referring to Fig. 
 26, which represents the equations with particular values 
 
 Fig. 26.— Non-oscillatory Dischakgb of a. Condenser with Ca- 
 pacity C — 1 MiCBOFAKAD, THROUGH A CIRCUIT WITH RESISTANCE 
 
 a = 100 Ohms and Self-induction L — .0016 Henrys when sub- 
 jected TO A Potential of 3000 Volts. 
 
 assumed. These values are the same as in the pre- 
 ceding case, namely, H = 100 ohms, (7=1 microfarad, 
 L = .0016 henrys. The condenser originally had no charge, 
 and when charged to a potential of 2000 volts, has a 
 charge of .002 coulombs. The current curve I., Fig. 26, 
 is identical with curve III., Fig. 20, which represents the 
 current during discharge. Curve II. representing the 
 charge is the same as curve III., Fig. 21, inverted and 
 plotted downwards from the horizontal line Q = .002. It 
 is noticeable that the ordinates of curve I., expressing the 
 current, are proportional to the tangents of the angle of 
 inclination of curve II. at every point, since the current 
 
 i = -^ , and -^ is the tangent of the angle of inclination of 
 
 the curve of charge II. It is seen that the point of inflec- 
 tion on curve II. comes at the maximum value of the current 
 curve I., as the tangent is a maximum at this point. In- 
 deed, curve I. might be constructed geometrically simply 
 from the foregoing consideration. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. Ill 
 
 EQUATION (101) APPLIES TO A CIECUIT CONTAINING EESISTANCE 
 AND SELF INDUCTION ONLY, IN THE CASE OF THE ESTABLISH- 
 MENT OF A CUEBENT UPON INSEETING AN E. M. F. 
 
 In this case there is no condenser in the circuit, that is, 
 the capacity is infinite. Substituting C = oo in the values 
 
 of the time-constants (95), we have T, = aD, T, = ^,aaiii 
 
 Case I., where the current dies away after the removal of , 
 the E. M. F, Substituting these values in (101), we have 
 
 _ Et 
 
 i = c. -f c, e ^ . 
 
 When t — 0, i = = c, + c,. 
 
 When f = 00 , i = /= c^. .•. c,:= — T, 
 
 Substituting these values for the constants c, and c,, we 
 have 
 
 <--^) 
 
 / is the final steady value of the current, and is equal to 
 -5; hence 
 
 (21) 
 
 = l(-^-) 
 
 which is the well-known expression for the establishment 
 of a current in a circuit with self-induction [see equation 
 (21), Chap. III.]. 
 
 EQUATION (156) APPLIES TO A CIECUIT CONTAINING EESISTANCE 
 AND CAPACITY ONLY IN THE CASE OF CHAEGING A CONDENSEE. 
 
 Upon substituting L = in the values of the time-con- 
 stants T^ and T,, the expressions become indeterminate, but 
 
118 CIBGUIT8 CONTAINING 
 
 can be evaluated as before by differentiation of numerator 
 and denominator before substituting L = 0. We thus find 
 the values, when Z = 0, 
 
 Substituting these values in the equations of current (101) 
 and charge (156), we have 
 
 (165) i = c,e"^ + c,e'^^^. 
 
 (166) q=Q + o,e~^^ + o/^^. 
 
 Q^ is the previous charge of the condenser, and Q the final 
 charge. The constants c,, c^ must be zero, or else when 
 f = Qo we would have i = oo , g" = oo . When t = 0, equa- 
 tion (166) becomes 
 
 By differentiating (166) and equating to (165), we have 
 
 Ǥ;_ _C3_ BC _ BC 
 
 dt~ EC^ ~ "^ ■ 
 
 Therefore c. = -^=-%^. 
 
 Substituting in (165) and (166) the values for the constants 
 c,, Cj, Cj, c,, as determined, 
 
 (167) z=- ^°~^ e"^. 
 
 (168) q^Qj^(_Q^^Q)e 
 
 t 
 
 ' RC 
 
 These equations are true for the charge or discharge 
 from Q^ to Q, through a resistance with no self-induction. 
 When the final charge Q is zero, we have the case of com- 
 plete discharge, and the equations become the same as (137) 
 
RESiaTANCE, SELF INDUCTION, AND CAPACITY. 119 
 
 and (138). When the original charge Q^ is zero, we have 
 the case of charging from zero to Q, and equations (167) 
 and (168) become 
 
 i — -^ F~ ^^ 
 
 g=^(l-e"^). 
 
 These equations are identical with (72) and (71), already 
 obtained in Chap. V. It is noticeable that the current 
 equation is the same as that for discharge equation (137), 
 and that the charge equation is analogous to that in the 
 case of the establishment of the current in a circuit with 
 resistance and self-induction, equation (21). 
 
 OSCILLATOEY CHARGING. 
 
 Determination of Constants. — The constants A, A', ^, 
 and 0' in equations (159) and (160) will be determined by 
 the same conditions as before, namely, 
 
 When ^ = 0, i = and q = Q^. 
 
 When t =^ CD , i:=0 and q ^ Q. 
 
 The meaning of this supposition is the same as in the pre- 
 ceding case, namely, that the condenser is suddenly charged 
 or discharged from the initial charge Q^ to the final charge 
 Q. The constants, determined by the same method as in 
 Case I., are 
 
 ^ HQ.-Q) . 
 
 V4.LG -R'C 
 
 V4:LG-E'C' 
 # = 0. 
 
 A' 
 
 g, _tan ^-^ 
 
120 CIRCUITS CONTAINING 
 
 With the constants thus determined, equations (159) 
 and (160) become 
 
 Rt 
 
 (169) i^ /(^.-^) e-^^sin ^^f/;-/'^:^. 
 
 Rt 
 
 (170) q = Q -\-'^^^^^^^£ e ^^ 
 
 sm -^ - ^ ' ^ — ' 
 
 2ZC 
 
 .^+tan-^ ^^ } 
 
 We may write OE^ for the original charge Q,, and (7^ for 
 the final charge Q. 
 
 Discussion of Oscillatory Charge. — These equations give 
 the value of the current and charge at any time after 
 the change of the electromotive force from E„ to E in 
 a circuit with E" G <.4:L. As the equations now stand 
 in their general form, they are true for either total or 
 partial charge or discharge, according to the values 
 assigned to Q^ and Q. If the final charge Q is zero, we 
 have the case of complete discharge, and the equations take 
 the form of (145) and (146). When Q is less than Q„ we 
 have partial discharge ; if ^ is greater than Q^, we have 
 partial charging. If the original charge Q„ = 0, we have 
 the case of charge from zero to Q. 
 
 Fig. 27 illustrates the case of oscillatory charge through 
 a circuit having the same constants as those of Fig. 22. 
 The current curve I. is the same as that in Fig. 22, and the 
 charge, represented by curve II., is the same as in that 
 figure, but inverted and plotted from the horizontal line 
 Q = .002. It is seen that in charging the condenser, the 
 charge rises at first higher than its final value, and then 
 oscillates about that final value until it has become steady. 
 
RESISTANCE, SELF INBUGTION, AND OAPAGITT. 121 
 CHABGE OP THE CONDENSEK WHEN B' G = 4 L. 
 
 Determination of Constants.— This is the critical case, 
 where the charging is just non-oscillatory. The equations 
 for current and charge are 
 
 (161) 
 
 (162) 
 
 * = c, e 
 
 Bt 
 ' 2L 
 
 + cje 
 
 Bt 
 '2L 
 
 y= Q-\-c'e ^^-^c"te 
 
 Bt _Bt 
 
 2L I .//^ . 2£ 
 
 The initial charge is Q„, and the final charge Q. To 
 determine the arbitrary constants of integration, let ^ = 0. 
 
 10x10"* Seconds 
 
 Fig. 27. — Oscillatoky Charge of a Condenseb with Capacity G = 
 1 Microfarad, through a Circuit with Resibtakob B = 100 
 Ohms and Self-induction L = .0135 Henrys when subjected 
 TO A Potential of 2000 Volts. 
 
 Then i — 0, and ^ = Q,. Equations (161) and (162) then 
 become 
 
 c, = 0. 
 
 c'=Q,- Q. 
 
122 CIRCUITS OONTAININQ 
 
 Differentiating equation (162) and substituting the value of 
 c', we have 
 
 When t = 0,i — 0; therefore 
 
 c" =■ 
 
 2L 
 
 Equating equations (161) and (171), and replacing the values 
 for Cj, c', and c", we have 
 
 c,= - {Qo- Q)-^^- 
 
 If E^ and E are the initial and final potentials, respectively, 
 we may write E^C for Q^, and EC for Q. Making this sub- 
 stitution and remembering that in this particular case 
 
 B^ = -p- , we have 
 
 E-E, 
 
 Replacing the values of the arbitrary constants, the equa- 
 tions (161) and (162) for current and charge may be written 
 
 _ Rt 
 
 (172) i = - -' t e 
 
 E — E^ . - 2£ 
 
 (173) ?=e + (^.-0(l + |4)e"^. 
 
 Discussion of Charge when B' C = 4, Z. — The current 
 curve in the case of charging a condenser, represented 
 by equation (172), is the same as in the case of discharge, 
 equation (151). It is represented in Fig. 28, curve I., 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 123 
 
 and may be constructed by tlie method shown in Fig. 23. 
 Curve II., Fig. 28, showing the charge is constructed in 
 
 .'20 Seconds .25 
 
 Pig. 28.— Just Non-oscillatory Chabgb ot' a Condenser with 
 Capacity C = 1000 Microfarads throttgh a Circuit with 
 Ebsistancb B = 100 Ohms, and Self-induction i = 2.5 Henrys. 
 
 a similar manner to curve II., Fig, 25, and, indeed, curve 
 II. of Fig. 28 is identical with curve II. of Fig. 25, it 
 being inverted and plotted downwards from the horizontal 
 line. 
 
CHAPTER IX. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF-INDUCTION, 
 AND CAPACITY. 
 
 Case III. SoLttnoN and Discussion for Haemonio E. M. F. 
 
 Contents : — To find from the general solutions the particular equations in 
 the case of an harmonic E. M. F. Complete solutions for i and q. 
 These same solutions obtained directly from the differential equations. 
 
 Discussion of Case III. Harmonic B. M. F. 
 
 The impediment. Case A. Circuits containing resistance and self- 
 induction only. Case B. Circuits containing resistance and capacity 
 only. Case C. Circuits containing resistance only. Case D. Circuits 
 containing capacity only. 
 
 Effects of Varying the Constanta of a Circuit. 
 
 First. Electromotive force varied. Second. Resistance varied. 
 Third. Coefficient of self-induction varied. Fourth. Capacity varied. 
 Fifth. The frequency varied. 
 
 The energy expended per second upon a circuit in which an har- 
 monic current is flowing. 
 
 The Equations fob an Harmonic E. M. E. Obtained from 
 THE Geneeal Solution. 
 
 In tlie preceding cases considered, those of discharge and 
 charge, the solutions for the value of the current and charge 
 at any time were obtained in two ways, first from the general 
 solution, and then directly from the differential equations, by 
 substituting e —/(t) = 0, and e =f(t) = E, respectively. 
 
 124 
 
BESISTANCE, SELF INDUCTION, AND CAPACITY. 125 
 
 The case of a circuit containing resistance, self-induc- 
 tion, and capacity, in which there is an impressed E. M. F. 
 varying harmonically, will now be considered, and the solu- 
 tion derived first from the general equations (99) aTnd (100), 
 and then directly from the differential equations (89) and 
 (90). In this case, 
 
 (174) e=/(«) =E&mQ!)t, 
 
 de 
 
 (175) and ^ =/' (t) = Eco cos oat. 
 
 Substituting these values in (99) and (100), we have 
 
 (176) i = ^■^'^ \ e" ^- A"^^' cos cotdt 
 
 -e ^' fe ^^ cos cotdt\-{-c,e ^' + c,e ^\ 
 
 and 
 
 (177) g =—==££== \e ^' fe ^'sincatdt 
 
 — € ^'/"e ^'sin cofdf t-l-c.e ^'-Hc.e ^'. 
 
 The solution for q being similar to that for i, we will 
 give the integration and reduction of (176) alone, and simply 
 give the resulting expression for q^. The integrals may be 
 found by the formulae of reduction [see equations (24) and 
 '(25), Chapter III.]i obtained by integrating by parts. The 
 integration of each term in (176) is 
 
 (178) 
 e ^ le ^ cos ootdt = —^ I yp cos cot -{-oosmmty. 
 
126 CIRCUITB CGNTAINIJfG 
 
 For convenience in transformation and reduction, put 
 T^ =-^ and r^z= —. After making these substitutions in 
 equation 1(176), we have 
 
 (179) i = ^^'^ I (-^_ -^) CO, cot 
 
 We may simplify (179) by substituting the values of r^ and 
 T, [see (95)]. 
 
 _ 1^ _ BC- VB'G' -4:LC 
 '^'~T~ 'iLG 
 
 1 IiG-\-VJ^G'-4:ZG 
 ^'~T,~ 2LG 
 
 Then, afi^er a few simple algebraic transformations in the 
 coefficients of the sine and cosine, (179) becomes 
 
 EBoo' 
 (180) I = ^-T y sm cot 
 
 S'a)'^[-^-ZGo') 
 
 Eoo[~ -Lco^ _L _ 1 
 
 cos Oat-j-C, € +C, 6 
 
 i2't»' + 
 
 [h-^-1 
 
 This may be transformed into a more convenient form by 
 means of the trigonometrical formula [see equation (27), 
 Chapter III.] 
 
 ^ sin cc + 5 cos a? = VA^ + B'' sin [x -f- tan"^ -^j, 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY 127 
 and when transformed is written 
 
 (181) i= ^ 
 
 sm 
 
 This is the complete solution for the current in a circuit 
 Tffith resistance, self-induction, and capacity when the 
 ^. M. F. is harmonic and equal to E sin oa t. The discussion 
 of this equation is deferred to the latter part of the chapter. 
 
 To Find the Equation foe Chaege. 
 The corresponding equation for charge, being the inte- 
 gral of the current according to the relation q = I idt, may 
 be written 
 (182) q= -^ 
 
 cos 
 
 
 This equation is the complete solution for the charge in a 
 circuit with resistance, self-induction, and capacity, when 
 there is an harmonic impressed E. M. F. 
 
 To Obtain the Solution Dieectly feom the Dippeeen- 
 TiAL Equation. 
 
 Let us now proceed to obtain this same solution, equa- 
 tion (181), by solving the original differential equation, 
 with the assumption that the E. M. F. varies harmonically. 
 
128 OIRCUITS CONTAINING 
 
 de 
 that is, e = ^ sin co t. Substituting jj = E oa ao& oot in the 
 
 differential equation (89), we have 
 
 cfi , Edi , i Eoo 
 
 This is a linear equation of the second order with constant 
 coefficients. [See Johnson's Differential Equations, page 
 91]. The complete integral of such an equation consists of 
 the sum of two parts, namely, the particular integral and 
 the complementary function. The complementary function 
 is the integral obtained by equating the first member to 
 zero, and contains two arbitrary constants. The particular 
 integral contains no arbitrary constants. The comple- 
 mentary function, obtained by equating the first member to 
 zero and solving, is 
 
 (101) i = c,e ^"+0,6" 
 
 
 To find the particular integral, it is convenient to use the 
 symbolic notation 
 
 ^ - df ^ - de- 
 With this notation (183) is written 
 
 Ego 
 
 (184) or i = 
 
 J cos cot 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 129 
 Next, to find the value of Z>', we have 
 d cos w t 
 
 dt 
 d' cos Got 
 
 = D cos cot = — GO sin 00 1, 
 
 = D' cos cot = — 00^ cos 03 1. 
 
 dt 
 
 Therefore D^ — — go'. 
 
 Substituting in (184) D' = — oo*, we have 
 Ego 
 
 (185) i = -j^ J c 
 
 cos cot 
 
 n.-f\ 
 
 LC 
 
 Ego 
 
 cos 0> t. 
 
 RI)-\--^-Lcd 
 
 Multiplying numerator and denominator of the coefficient 
 of cos coth-^ BD —\-T^ — L co'j, we obtain 
 
 Ego\bD-{^~ -Zoj'] J 
 
 i = ■■ TT Ta COS cot. 
 
 B'l)'- [^-Zco') 
 Substituting — oo' for D', and separating into two terms, 
 — EaoED cos Got-\- Ego f-^ — LGa'\ cos cot 
 
 I = 
 
 B'go' -^{^ -Z go') 
 
 But D cos cof = — GO sin co t. Hence 
 
 Eoa'B sin 03 i 
 
 Ego f p- ~ Lgo] cos oot 
 
 + (-i - i go)' Ra>' + [l--L GO')' 
 
130 CIRCUITS CONTAINING 
 
 This is the particular integral, to which must be added the 
 complementary function (101) in order to obtain the com- 
 plete integral. The complete integral is thus found to be 
 
 , JEJao^B sin aot 
 
 (186) i = - — Yj 
 
 1 
 
 Fgo 
 
 (^ 
 
 - L ca' j cos 
 
 , cot 
 
 1 
 
 B 
 
 '(»" + 
 
 (1 
 
 — Loo 
 
 ')■ 
 
 + 
 
 c.e 
 
 t 
 
 %e 
 
 i 
 
 
 B'co'Ji-[^--Zoo'' 
 
 This solution for the current obtained from the differential 
 equation (89) is seen to be identical with (180), the result 
 obtained from the general solution (99). The solution for 
 charge could be obtained in a similar manner from the dif- 
 ferential equation (90). 
 
 Discussion op Case III. — Haemonic E. M. F. 
 
 These solutions, (181) and (182), show that, after a very 
 short time has elapsed, so that the exponential terms con- 
 taining the arbitrary constants of integration become in- 
 appreciably small and can be neglected, both the current 
 and the charge are simple harmonic functions and may 
 either lag behind or advance ahead of the impressed 
 E. M. F. The current lags behind the impressed E. M. F., 
 
 when Z 07 > -7= — , and advances ahead of it when Z go <.-p7-~' 
 O CO (J 00 
 
 When Loa^ —■ — that is, when ca = — ; ■ , there is no 
 
 Coo' i/ZC 
 
 lag or advance, and the current is exactly in phase with 
 the impressed E. M. F. In this case the current equation 
 becomes 
 
 . E . 
 
 I = -p- sm 00 1, 
 
 which is identical with the current equation obtained from 
 Ohm's law, without considering either self-induction or 
 
BE8ISTANCE, SELF INDUCTION, AND OAPACITT. 131 
 
 capacity. When the sine is unity in (181), the maximum 
 value of the current, represented by /, is 
 
 (187) 1= ^ 
 
 , 00 
 
 From the analogy of this equation to Ohm's law, we see 
 
 that the expression V E' -{- y^j Loa) is of the nature 
 
 of a resistance, and is the apparent resistance of a circuit 
 containing resistance, self-induction, and capacity. This 
 expression would quite properly be called "impedance," 
 but the term impedance has for several years been used as 
 a name for the expression Vlf + U oo', which is the appa- 
 rent resistance of a circuit containing resistance and self- 
 ■ induction only [see equation (29), Chapter III.]. We would 
 suggest, therefore, that the word " impediment " be adopted 
 
 as a name for the expression V B^ -{- (7; L(^] , which 
 
 is the apparent resistance of a circuit containing resistance, 
 self-induction, and capacity, and that the term impedance 
 be retained in the more limited meaning it has come to 
 have, that is, VB'' -f- L' 00^, the apparent resistance of a cir- 
 cuit containing resistance and self-induction only. Equa- 
 tion (187) may be written 
 
 „__, __ . , Maximum E. M. F. 
 
 (loo) Maximum current = — ^ :r. r — • 
 
 impediment 
 
 Since the virtual current (the square root of the mean 
 square of the instantaneous values of the current) is equal 
 
 to — -pr times the maximum value of the current, and since 
 
 |/2 
 
 the virtual E. M. F. = --= times the maximum E. M. F., 
 
 ,,^^v ,r. , , Virtual E. M. F. 
 
 (189) Virtual current = — =f -r-. 7-^— " 
 
 '' ^ impediment 
 
132 CIRCUITS CONTAININQ 
 
 It is convenient to consider tlie impediment as a resistance, 
 and we are justified in so doing inasmucli as it has the 
 same dimensions as a resistance, that is, a velocity in the 
 electromagnetic system of units. 
 
 27r 
 
 "' - Time 
 
 L — Length. 
 
 _, , ^ Length , ., 
 Therefore, Loo= -j^. = velocity. 
 
 ^_ (Time)' 
 Length 
 
 1 Length , ., 
 
 This gives the dimensions of a velocity to the whole expres- 
 sion for the impediment, which may therefore be considered 
 as a resistance. 
 
 The several particular cases of circuits containing vari- 
 ous combinations of resistance, self-induction, and capacity 
 may readily be found by means of the general solution, 
 equation (181). 
 
 Case A. Cibcdits containing Eesistance and Self- 
 induction ONLY. 
 
 In this case the circuit has resistance R and self-induc- 
 tion L, and an harmonic E. M. F., ^sin cot. There being 
 no condenser in the circuit, the ca,pacity O is infinite [see 
 page 67, Chapter IV.]. After the lapse of a very small time 
 the terms containing the constants of integration in the 
 general solution may be neglected as explained above. 
 Substituting in (181) C= oo , we have 
 
 E . ( . , ,Loo 
 
 I = 
 
 */B' -f D 
 
 sin loot — tan"^— o~f. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 133 
 
 Ttis equation has been independently obtained from the 
 
 ' differential equation [see equation (28), Chap. III.]. The 
 
 current must always lag behind the impressed E. M. F. by 
 
 L GO 
 
 an angle whose tangent is — p- . In this case the impedi- 
 
 ment takes the particular value VH'' -\- Z" ao', which is 
 known as the impedance of the circuit. 
 
 Case B. Ciecdits containing Besistance and Capacity 
 
 . ONLY. 
 
 In this case the circuit has resistance JR and capacity 
 C, with an harmonic E. M. F., e = E sin ao t. Substituting 
 X = in the general equation (181), we have 
 
 E 
 
 V 
 
 B' + 
 
 = sin|a,« + tan-^^[. 
 
 Coo' 
 
 This equation has been independently obtained from the 
 differential equation [see equation (78), Chapter V.]. The 
 current must always advance ahead of the impressed 
 E. M. F., when there is resistance and capacity only in the 
 
 circuit, by an angle whose tangent is -^Tp" 
 
 ',00 
 
 Case C. Cikcuits containing Besistance Only. 
 
 In this case the self-induction Z = 0, and the capacity 
 C = (X3 . Substituting these values in the general solution 
 (181), we have 
 
 iz^ -= sin cot. 
 
134 CIRCUITS CONTAINING 
 
 This result is immediately derivable from Ohm's law. Thus, 
 Siace e = ^ sin wt, 
 
 or * = ^5- sm GO t. 
 
 SI 
 
 Case D. Cikcuits containing Capacity only. 
 
 In this case S = 0, and Z = 0. Substituting in the 
 general equation (181), we have 
 
 i = CEoo sin | (»i -)- ^ i . 
 This is identical with equation (80), Chapter V. 
 
 Effects of Varying the Constants op a Circuit. 
 
 The general equation (181) enables us to ascertain the 
 current which will flow in a circuit when we know its re- 
 sistance, self-induction, and capacity, the value of the im- 
 pressed E. M. F. and its frequency. It is important to know 
 two things about the current ; first, its maximum value I, 
 and, second, the angle B by which it lags behind or advances 
 ahead of the impressed E. M. F. The mean square value 
 is readily obtained from the maximum value. We are given 
 R, C, L, E, and w. The angle of lag or advance is 
 
 (190) ^ = *-"(w^-^)' 
 
 1 Lgd 
 
 or tan 6 — -^-^ -^-. 
 
 This is an angle of advance or lag, according as -7^^^ — is 
 ° 00 ^j 2i GO 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 135 
 
 greater or less than -^^- • The maximum value for the 
 current is 
 
 (191) J= ^ = ^ = f cos 9. 
 
 It is interesting to note how any change in R, L, C, go, or 
 E affects the value of 6 and the current. 
 
 First. If the impressed E. M. F. E is varied, and S, L, 
 C, and 00 are maintained constant, is not affected, and 
 the angle of lag or advance remains unchanged. The value 
 of the current is varied in direct proportion to E. 
 
 I<xE. 
 
 Second. If the resistance H of the circuit is varied, and 
 Z, C, GO, and E are maintained constant, as H is increased, 
 the angle of lag or advance is diminished. 
 
 tan 9 <x -p- 
 
 The sign of tan 6 is positive or negative, and the angle 
 therefore one of advance or lag, according to the values of 
 L, G, and oo, and is independent of any variations in the 
 resistance. The current is in all cases diminished by an 
 increase of resistance, but the amount of this decrease 
 depends not only upon B, but upon the relation between 
 
 -p^ and L go. 
 Coo 
 
 In Fig. 29 are shown two particular cases of the varia- 
 tion in the current produced by change in the resistance. 
 Curve I. is for a circuit in which 
 
 Self-induction Z = 2 henrys = 2 X 10^ C. G. S. units. 
 Capacity G = .55 microfarads = .55 X 10"'^ C. G. S. units. 
 Impressed E. M. F. -B" = 200 volts = 200 X 10' 0. G. S. units. 
 2 TTTO = 00= 955. 
 
136 
 
 CIRCUITS CONTAINING 
 
 The abscissae represent resistance in ohms (1 ohm = 10' 
 C. G. S. units of resistance). The ordinates represent cur- 
 rent in amperes (1 ampere = 10"^ C. G. S, units). The 
 
 7. 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 a 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 <?. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8r 
 
 
 
 
 
 
 
 S' 
 
 ■Y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 <^' 
 
 L 
 
 
 
 
 
 
 
 
 
 
 
 
 
 £. 
 
 
 
 
 
 
 
 
 
 \ 
 
 vC 
 
 „ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 M- 
 
 
 
 
 
 
 
 
 
 
 1. 
 
 
 
 
 G 
 
 -101 
 
 lO.o 
 
 r91 
 
 
 Ci 
 
 PV( 
 
 M 
 
 
 
 
 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ohms 
 
 
 
 
 2 
 
 ». 
 
 k 
 
 u. 
 
 6 
 
 0. 
 
 8 
 
 0. 
 
 1 
 
 w. 
 
 1 
 
 0. 
 
 1^ 
 
 Vi. 
 
 1 
 
 0. 
 
 It 
 
 10. 
 
 ■m. 
 
 Fig. 39.^Vakiation op Curkent with CHANaE in Resistance in a 
 CiKCurr IN WHICH E = 200, (7 = .55, L = 2. 
 
 relation between L, C, and co here taken is such that 
 
 : , which is the relation that gives 
 
 79- = Z GO, or oa= —- 
 
 no angle of lag or advance. The relation between current 
 and resistance is the same as in Ohm's law, and when 
 plotted gives the hyberbola curve I. In the same figure, 
 curve II. represents the value of the current with different 
 resistances in a circuit in which 
 
 Z = 2 henrys, 
 
 G = .55 microfarads. 
 
 F = 200 volts, 
 
 GO = either 1000 or 912. 
 
 The constants here are the same as in the previous case, 
 with the exception of go, which has been changed from 
 
 955, that is, , to either 1000 or 912. Any change in 
 
 V L G 
 
BESISTANOB, SELF INDUCTION, AND CAPACITY. 137 
 
 CO from the value , whether it be an increase or de- 
 
 VL G 
 
 crease, causes the curve to depart from the hyperbola, curve 
 
 I. It is to be noticed that a change in frequency of only 
 
 seven alternations per second will change the curve from I. 
 
 toll. 
 
 Third. If the coefficient of self-induction L is varied 
 
 while H, C, 03, and E are maintained constant. 
 
 When L < „ ^ , tan 6 is positive and 6 is an angle of ad- 
 
 vance. 
 
 6 becomes less 
 I becomes greater 
 
 ^ as Z increases. 
 
 When L > p; — j , tan 6 is negative and 6 is an angle of lag, 
 
 6 becomes greater ) j- . 
 
 f > a.s L mcreases. 
 
 /becomes less ; 
 
 These changes in the angle of lag or advance and the cur- 
 rent, due to change in the self-induction, are better seen 
 from the consideration of a particular case. In Fig. 30 the 
 values of 6 and / are plotted for various values of i in a 
 circuit in which 
 
 B = 50 ohms, oa = 1000, 
 
 G = .55 microfarads, E = 200 volts. 
 
 When Z = 79 — 5 = 1.82, the current has its maximum 
 
 E 
 value equal to -^ , and 5 = 0. This is a critical point, and 
 
 a slight change of L in either direction will cause B to reach 
 a considerable value and the current to fall to a small part 
 of the maximum value. If Z be increased from 1.82 to 
 1.92, B changes from zero to — 68°, an angle of lag, and the 
 current falls from 4 to 1.8 amperes. If L be made 1.72, 
 
138 
 
 OmOUITS GONTAININO 
 
 6 becomes an angle of advance of 63° and the current will 
 be 1.8 amperes. It is thus seen that an exact balance of 
 self-induction and capacity would be exceedingly hard to 
 maintain in this case, for a slight change in the self-induc- 
 tion would cause a large angle of lag or advance and a large 
 diminution in the current. Just how critical the curves 
 will be in the vicinity of the point of equilibrium depends 
 
 90 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 An 
 
 w 
 
 of Adva 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 g 
 
 £. 
 
 
 
 I 
 
 
 
 
 
 
 
 
 f\ 
 
 
 
 
 
 
 
 
 
 
 
 a 
 
 
 30° 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 p. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 ^ 
 
 P"'- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -— 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ] 
 
 
 
 
 
 
 
 
 
 
 
 i. 
 
 i. 
 
 enr, 
 
 S 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30° 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 «0° 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 „ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 Am 
 
 rie 
 
 ?^ 
 
 ag- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Fig 30.— Value of Cukrbnt, and Angle op Advance ok Lag for 
 Different Amounts of Self-indtjction in a Circuit in which 
 R = 50, 0— .55, E - 200, co - 1000. 
 
 upon the constants of the circuit. The curves will always 
 be of a form similar to those in Fig. 30, but will often be 
 decidedly modified by the particular values of R, C, and co. 
 The critical parts of the curves may be more or less marked 
 according to these particular values. 
 
 Fourth. If the capacity C is varied while B, L, oo, and 
 E are maintained constant. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 139 
 
 When C < j-^^ , tan d is positive, and 6 is an angle of ad- 
 
 vance. 
 
 becomes less 
 /becomes greater 
 
 I 
 
 as C increases. 
 
 When G > -j-^ , tan 8 is negative and is an angle of lag. 
 
 6 becomes greater ) „ 
 I becomes less ) 
 
 mcreases. 
 
 These changes of current and lag, with the variation in 
 capacity, are shown in Fig. 31 for a particular case in which 
 
 JS = 50 ohms, 
 Z> = 2 henrys, 
 
 GO = 1000, 
 E= 200 volts. 
 
 to' 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 ngl 
 
 J of 
 
 Ad 
 
 an 
 
 :s 
 
 
 S 
 
 
 
 
 
 
 
 
 
 
 
 
 1. 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 eS 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 3. 
 
 
 » 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B 
 
 S 
 
 a 
 
 a 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ■) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 h 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 ^ 
 
 * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 — 
 
 
 
 
 
 
 
 
 f 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 5 
 
 . 
 
 
 
 Ml 
 
 .8 
 ro-fara< 
 
 » 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 „ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 N 
 
 4n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 <li 
 
 
 ?lk 
 
 r^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pig. 31. — Valtjb of Cdekent, and Angle of Advance or Lag for 
 Different Capacities in a Circuit in which B =50, L = 2, 
 E = 300, 00 = 1000. 
 
140 CIRCUITS CONTAimNO 
 
 1 
 
 The maximum value for the current occurs when G =--r , — 
 
 .5 microfarads. This is a critical point in the curve similar 
 to that in the curves where the self-induction was varied. 
 Here B is zero, and so the current, being in phase with the 
 impressed E. M, P., has a value of 4 amperes in accord- 
 ance with Ohm's law. The critical nature of the curves 
 here is seen by the fact that when G = .55 there is an angle 
 of lag of 75° and I = 1.07 ; when G = .458, there is an angle 
 of advance of 75°. When G is changed from .5 to .488, the 
 current falls from 4 to 2.83 amperes and is put 45° out of 
 phase in advance of the E. M. F. 
 
 Fifth. If the frequency is varied while S, G, L, and E 
 are maintained constant, still more marked changes occur 
 in the values of / and B. 
 
 When 03 < .^^ , tan 8 is positive and 9 is an angle of 
 y LG 
 
 advance. 
 
 6 becomes less 
 I becomes greater 
 
 When CO > — — ==, tan ti is negative and 6 is an angle of lag. 
 
 |- as oj increases. 
 
 6 becomes greater ) 
 
 f f as ffi» increases, 
 
 /becomes less ) 
 
 In Fig. 32 the values of the current and angle of lag are 
 shown for different values of a? in a circuit in which 
 
 5 = 50 ohms, G = .55 microfarads, 
 
 Z = 2 henrys, E = 200 volts. 
 
 1 
 
 When CO = , = 955, the current has its maximum 
 
 VZ G 
 
 value of 4 amperes, in accordance with Ohm's law. Here 
 
 ^ = 0. A change of five per cent, one way or the other in 
 
 this critical value for co causes an angle of lag or advance 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 141 
 
 of 75°, and the current falls to one-fourth of the maximum. 
 Just how critical the curves are, in the vicinity of this point 
 
 «> 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ng 
 
 a-c 
 
 fA 
 
 5^ 
 
 ice 
 
 
 N 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 tS 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 Sf 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 6 
 
 i 
 
 af 
 
 a 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 / 
 
 s r 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 — ■ 
 
 ^ 
 
 
 
 Sni 
 
 
 
 
 
 
 
 
 
 
 
 
 — 
 
 
 
 
 b 
 
 K) 
 
 
 
 
 1000 
 
 
 
 
 >,7t> 
 
 Fr< 
 
 Hue 
 
 loy 
 
 
 W 
 
 M) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 30° 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 eo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 S 
 
 4 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Zijgvs^ 
 
 tL 
 
 h^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pia. 33.— Valtje op Cttbbent and Angle of Advance on Lag for 
 Different Frequencies in a Circuit in which iJ = 50, L — %, 
 = .55, E = 300. 
 
 of equilibrium depends upon the particular values of B, C, 
 and L. 
 
 In Fig. 33 is shovirn the E. M. F. necessary to cause a 
 constant current to flow in a circuit in which Ji, C, and oo 
 are constant. In the particular case plotted, 
 
 ^ = 50 ohms, 
 
 C = .55 microfarads. 
 
 / = 1 ampere, 
 aj = 1000. 
 
 As the self-induction is increased up to the value 
 Z = Tp-, = 1.82, the E. M. F. needed to drive the current 
 
 G GO 
 
 becomes less and less, and when L = 1.82 the E. M. F. 
 needed is only 50 volts. As Z increases past this critical 
 value, the value of the E. M. F. needed increases. Except 
 
142 
 
 CIRCUITS CONTAINING 
 
 very near tlie critical point, the change in the necessary 
 E. M. F. is almost directly proportional to the change in 
 the self-induction, that is, the curve is formed of two straight 
 lines with a rounded point. This curve is the reciprocal of 
 
 SOOOi, 
 
 "^~ 
 
 
 
 
 
 
 _ 
 
 ~"~ 
 
 
 
 
 
 
 
 "~ 
 
 
 
 
 
 7\ 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 / 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 \, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 / 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 tn 
 
 
 
 
 
 S 
 
 s 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 2. 
 
 
 
 
 
 
 H 
 
 enrj 
 
 s 
 
 _, 
 
 Pig. 33.— Relation between Impressed E. M. F. and Self-induction 
 WHEN 1 Ampere flows in a Circuit in which if = 50, = .55, 
 o = 1000. 
 
 the corresponding curve for current, with E constant and 
 L variable, as shown in Fig. 30. 
 
 The Enebgy Expended pek Second upon a Ciecuit in which 
 AN Habmonic Cueeent is Flowing. 
 
 The energy expended in any circuit in the time dt is 
 the product of the E. M. F. and current at that instant by 
 the time ; that is, 
 
 dW— eidt. [See equation (5), Chap. I.] 
 
 When the E. M. F. is harmonic the instantaneous value of 
 it is e = -2* sin cot. The current at the same instant is 
 i = Ism\oot — 6]. Therefore the differential equation of 
 energy is 
 
 a92) dW =i: I sm Got sin [cot — e\dt. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 143 
 
 Integrating between the limits zero and T, the time of one 
 complete period, we obtain 
 
 (193) W^EI f smoDts,ia.\Got-6}dt. 
 
 Expanding sin \oot — 6\,y}e obtain 
 
 pT pT 
 
 PF= jB'Zcos 6 I sin' cotdt— ^7 sin S I sin go t cos co t. 
 
 ■^ ■, . . , , 1 -. . , , sin 2 a?^ 
 Keplacing sin cot cos oathj its eqmvalent ^ , 
 
 />^ I! I sin 6 pT 
 W— El cos B I sin' cotdt — „ J sni2 ootdt. 
 
 Between the limits zero and T the second integral vanishes 
 
 T 
 and the first integral is equal to -^. [See page 37.] The 
 
 value of the energy expended per period is therefore 
 
 (194) w=^-^T. 
 
 The energy expended per second^s therefore 
 
 (195) ' W=^^; 
 
 that is, the energy per second is half the product of the 
 maximum E. M. F. by the maximum current by the cosine 
 of the angle of difference between the E. M. F. and cur- 
 rent. Since the effective E. M. F. or current is equal to 
 
 — — times the maximum value, we have 
 V2 
 
 (196) W=^Icose, 
 
 meaning by ^ and /the square root of the mean square 
 values of E. M. F. and current. 
 
CHAPTER X 
 
 CIRCUITS CONTAINING RESISTANCE, SELF INDUCTION, AND 
 
 CAPACITY. 
 
 Case III. (Continued.) Cueeents at the "make" foe 
 AN Harmonic E. M. -F, 
 
 Contents: — Complete equations for i and q with the complementary 
 function in the oscillatory form. To determine the constants A' and 
 $'. To determine the constants A and $. Complete equation for ^ 
 with constants determined. Examples to explain the general equation 
 in cases of particular circuits. Curves showing the current at the make 
 lor a particular circuit. The phase at which the E. M. F. should be 
 introduced to make the oscillation a maximum. 
 
 In the discussion of the current equation in Chapter 
 IX. for an harmonic E. M. F., it was stated that after the 
 lapse of a very short time the exponential terms, equation 
 (181), become inappreciably small and can be neglected, 
 and the discussion of the equation there given only applies 
 after the current has been flowing for a short time. It is 
 proposed in this chapter to investigate the effect of these 
 exponential terms in modifying the current during the very 
 short time after the " make," or, in other words, after the 
 harmonic E. M. F. is suddenly introduced into the circuit. 
 The E. M. F. may be introduced at any point of its phase, 
 that is, it may be zero or may have its maximum or any 
 intermediate value, but, in any case, the complete equations 
 (181) and (182) show just what happens, provided we de- 
 termine the constants c, and c, of the complementary func- 
 
 144 
 
HEaiSTANOE, SELF INDUCTION, AND CAPACITY. 145 
 
 tion, so that they correspond to the particular hypothesis 
 
 made. 
 
 It has been noted (120) that the complementary func- 
 
 _i- _i- 
 
 tion c^e ^' + c, e ^' may be written in another form, viz. : 
 
 Ae-^.in[^±lLC^S^t+$]. 
 
 This latter form must be used when we have the relation 
 4Z > H'C, for, under this hypothesis, the time-constants 
 T^ and T, of the first form become imaginary. To make 
 this supposition is equivalent to saying that the character 
 of the discharge from the circuit is oscillatory [see Chap- 
 ter YII.]. Inasmuch as this relation 4^L> IfCis true for 
 most ordinary circuits in which L has an appreciable value, 
 and since the results obtained are rather more interesting 
 under this supposition than under the supposition that 
 4Z < jffi'C, which would give "dead beat" discharge, we 
 will confine our attention to the oscillatory case only. The 
 plan to be followed in the discussion of this subject will be 
 to determine the constants A and <P of the general equa- 
 tion, and write the general result. The application of this 
 result to a particular circuit will then be made, and curves 
 drawn showing the current as it starts in this circuit before 
 it has reached its final harmonic form. 
 
 The general equation for current, under the assumption 
 made that 4Z > R'C, may be written 
 
 (197) i = —j====siu \oot + tan- (^^g- 
 
 _^)}+^.-t.„|i2?^Z5'<+.|, 
 
146 CIRCUITS CONTAINING 
 
 where A and <5 are tlie constants of integration to be deter- 
 mined and are each of them real. Likewise, the equation 
 expressing the quantity of charge on the condenser at any 
 moment may be written [see (182) and (123)] 
 
 — E { 
 
 (198) q = — „ cos ] fij^-|-tan"* 
 
 CO , 
 
 I 1 LooW , ,-Wl. { V4:LC-B'C\ , ^,) 
 \-0B^-^)\-^^' ^^H 2X^ '+"^1. 
 
 To determine the constants A' and ^'; — Eemembering the 
 relation dq = idt, we may differentiate (198) and write 
 
 da E 
 
 ^^''^ ' = dt= / /I V 
 
 sin {a,^ + tan->(-^--^)| + -^e 
 
 Rt 
 ' 2L 
 
 sm i oTTy t-\- Q' — tan"' 
 
 BC 
 
 Equating (199) with (197), we obtain the relations 
 
 A' 
 
 (200) A = 
 
 VlU' 
 
 (201) #=*'-tan-iM^IL^?!E 
 
 H 
 
 Tor simplification make the following substitutions : 
 E_ 
 
 (202) /= / f, =^,. [See (191).J 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 147 
 (203) i, = oot + tan- (-^ - ^) = a>t + 6. 
 
 (204) a = ^^^^^- ^'^' 
 
 2L0 
 
 In 
 
 a 
 
 The frequency of oscillation is -^ — , and the period 
 
 Then we may write, after substituting in (197) and (198) 
 the values of A' and <?' as determined, 
 
 Bt 
 
 (205) i = /sin^ + ^e ^^ sm{at-{- $\. 
 
 Bt 
 
 (206) q = --cosrp-\-AVZG6 '^ 
 
 00 
 
 sin i at -^ -\- tan-' ^77 c 
 
 To determine the constants A and ^: — ^In these equations 
 time is counted from the point when the impressed E. M. F. 
 is zero. Let i, be the time when the E. M. E. is introduced. 
 We know then that the current and the charge of the con- 
 denser are each zero at the time f, , the condenser having 
 no initial charge. These conditions alone, namely, that 
 i = and q = when t = t,, are suflScient to determine the 
 constants. In equations (205) and (206) make i = 0, g = 
 when t =: t^, and call ^, the value of ip when t = t^, and we 
 have 
 
 _Bti 
 
 (207) = /sin^, + Je ^^ sin {at, + 0\. 
 
 (208) = cos ^, + J vTGe ^^ 
 
 sin 
 
 {««.+ ^ + tan- ^ [ 
 
148 CIRCUITS CONTAINING 
 
 Eliminating A between these equations, we obtain 
 ( 2cot?/', + ^Caj 1 
 
 Substituting this value of <? in (207), 
 
 Bt 
 
 (210) A = -Ie 
 
 , Jill . 
 
 , 2eot^ + ^C<» 
 sm cot"^ 
 
 --{ 
 
 ooV4.LO-B'0' 
 
 This expression for A may be reduced by simple trigo- 
 nometrical operations to the form 
 
 (211) ^ = - 
 
 2/6^-^ 
 
 ooV4:LG-E'C' 
 
 \f\L Coo' - 1) sin= f, + ^ROoa sin 2 ^, + 1. 
 
 Substituting these values of A and ^ in equation (197), we 
 may write the complete solution with constants determined. 
 
 ,„,„,. ^. , 2/|/(Z(7(»=-l)sin=(^,+ii?(7<»sin2^, + l 
 (212) 1=1 sm tb i i-h = ^-i— ! — 
 
 sin \a{t-t,) + cot-' - r 
 
 CDV4:ZC~Ii'C'J 
 
 There are several general conclusions which can be made 
 
 in interpreting the meaning of this equation. It is evident 
 
 that there will be an oscillation of the current when the 
 
 E. M. F. is first introduced, which gradually dies away, the 
 
 rate of dying away depending upon the exponent of e in the 
 
 equation or, in other words, upon the time-constant of the 
 
 2Z 
 circuit, namely, -p- ' The initial value of this logarithmic 
 
 decrement curve, that is at the make when i = f, ,.is ex- 
 pressed by the coefficient of e in the equation. It is evident 
 
BE8I8TANCE, SELF INDUCTION; AND CAPACITY. 149 
 
 that this initial value depends upon the value of ^, for its 
 value, or is a function of ^,. The initial value of the log- 
 arithmic curve has, then, a different value for every value 
 of ^,, i.e., for every point of the phase of what the current 
 vrould have been if it had started out at the make as an 
 harmonic current having the same phase difference with 
 the E. M. F. as it finally assumes. Again, at the time t^t^ 
 the value of the last term of the equation becomes —/sin ^,. 
 This will be evident upon replacing the coefficient of e by 
 its value given in (210). The first term becomes /sin ^, , 
 when f = ^, , and the two terms together show that the 
 equation makes the value of the current zero at the time t^, 
 that is, at the time the E. M. E. is introduced. 
 
 In order to show the meaning of this equation more 
 clearly, a particular example will be assumed. Suppose 
 we have a circuit with a resistance of 50 ohms, a self-induc- 
 tion of 2 henrys, and a capacity of ,55 microfarads, all in 
 series. Such a circuit would correspond nearly to the fine 
 wire coil of a small 10-light Westinghouse transformer 
 connected in series with a condenser of .55 microfarads 
 capacity. Let an E. M. F. of 100 volts (maximum value), 
 having a periodicity of 159, be impressed upon the circuit ; 
 that is, the angular velocity ca = 2 ;r X 159 = 1000, approx- 
 imately. We have, then, with these values assumed, 
 
 E = 100 volts (max.) = 100 X 10' C. G. S. units. 
 
 R=m ohms = 50 X 10' 0. G. S. units. 
 
 Z = 2 henrys = 2 X 10' C. G. S. units. 
 
 C = .55 microfarads = .55 X lO"'' 0. G. S. units. 
 
 ca = 1000. 
 
 ^ 2Z 4X10' „„ , 
 
 ^=-W = 50^W =-08 seconds. 
 
 / = .53 amperes (max.). [See equation (202). J 
 e = - 74° 30'. 
 
 a = 955 = 2 TT X frequency of oscillation = 2 tt X 151. 
 [See (204).] 
 
150 CIRCUITS CONTAINING 
 
 The equation for the current in this particular case is 
 
 (213) i = .53 sin tp — All V.l sin' ip, + .0137 sin 2 ^. + 1 
 
 e ■'^ sin|955(i!-0 + ;i:}. 
 
 Curve III., Fig. 34, represents the plot of this equation 
 when the particular value of ^, is 30°. This means that 
 the E. M. F. is introduced into the circuit at that particular 
 time at which the normal current curve is 30° from its zero 
 value. The value of the coefficient of e when ^, is 30° is 
 .495, and the equation reads 
 
 t-tj 
 
 (214) i = .53sin^-.495e ■"* sm{955{t -t,) + x]- 
 
 Here x stands for angle of lag expressed in equation (212), 
 and is not expressed in figures inasmuch as it is not neces- 
 sary to know it in order to draw the curves, because the 
 phase is determined by the fact that we know the distance 
 0' A', Fig. 34, it being equal but of opposite sign to the 
 distance A. It will be noticed that the initial value of 
 the logarithmic decrement is nearly the same for any value 
 of ^, in this particular case. Moreover, as it happens, the 
 initial value of the logarithmic decrement is nearly the 
 same as the maximum value of the current I. Curve I. is a 
 sine curve representing the first term in equation (214), and 
 curve II. a sine curve with logarithmic decrement repre- 
 senting the second term in the equation. The current 
 curve, III., is the sum of curves I. and II. After about 
 one-tenth of a second, curve II. becomes inappreciable and 
 the current follows a simple sine curve. 
 
 As a second example, let us consider the same circuit 
 as before. But now suppose the frequency is just half 
 what it was in the first example, namely, 79.5, or that 
 a? = 500. Furthermore, suppose the E. M. F. is such that 
 it will send a maximum current of .5 of an ampere through 
 
RE8iaTANCB. SELF INDUCTION; AND CAPACITY. 151 
 
 ". < 
 1^ ^ 
 
 IS 
 
 
 
 5 
 
 » 
 
 t 
 
 B 
 
 u 
 
 P O 
 
 P 
 
 tf 
 
 H 
 
 
 M 
 
 
 
 g 
 
 *^ 3 
 
 H ^ 
 
 H 
 
 
 
 
 Hi 
 
 tif 
 
 
 ■< 
 
 ■< 
 
 t" 
 
 n 
 
 S 
 
 1 
 
 
 
 i-l 
 
 
 ^ 
 
 
 
 i 
 
 tJ* 
 
 p 
 
 
 ^ 
 
 !/J 
 
 <! 
 
 B 
 
 S 
 
 «! 
 
 1 
 
 s 
 
 " 
 
 O 
 
 M 
 
 p 
 
 g 
 
 rS 
 
 1^ 
 
 H 
 
 
 U 
 
 1 
 
 
 <1 
 
 t— ( 
 
 a? 
 
 fr. 
 
 
 w 
 
 < 
 
 1 
 
 
 
 p 
 ^ 
 
 
 is 
 
 
 i;9 
 
 > 
 
 
 B 
 
 « 
 
 CO 
 
 
 ni 
 
 
 J 
 
 o 
 
152 CIRCUITS CONTAINING 
 
 the circuit. It will be found, upon calculation, that the 
 E. M. F. must be 1320 volts maximum. With these values, 
 then, 
 
 E = 1320, 
 
 Z = 2, 
 
 C = .55, 
 
 CD = 500, 
 
 T = .08 seconds, 
 
 / = .5 amperes, 
 
 e = 88° 55', tan (9 = 52.8, 
 
 a = 955, 
 
 the equation for the current becomes 
 
 (215) i = .5 sin f — .955 V- .725 sin" tp, + .0069 sin 2 ^. + 1 
 
 t-t, 
 , e ■•"* sin|955(i5-«,)+;i:}. 
 
 The plot of this equation, when f^ is taken equal to 180° 
 (that is, the E. M. E. is introduced when the normal current 
 curve is zero), is shown in Fig. 35. It will be noticed that 
 the initial value of the logarithmic curve has considerable 
 variation according to the particular point of time at which 
 the E. M. F. is introduced. This variation is represented 
 in the curve IV., Fig. 35. The initial value of the logarith- 
 mic decrement at 0° or 180° is almost twice as much as the 
 
 955 
 maximum value of the current /, their ratio being ■ ' _ . 
 
 The equation, when ^, is 180°, reduces to 
 
 Ct-ti) 
 
 (216) i = .5sin^-.955e ""^ sm{955{t -t,) + x}^ 
 ln each of the above examples the current follows the 
 
 sine law in about one-quarter of a second after the periodic 
 E. M. F. is introduced, during which time somewhere in the 
 neighborhood of forty oscillations have been made. 
 
BE8I8TANGE, SELF INDUCTION, AND CAPACITY. 153 
 
 The phase at which the E. M. F. should he introduced to 
 make the oscillation a maximum : — It may be interesting to 
 inquire at what point the E. M. F. should be introduced 
 
 goiadniy ^ "S^"' 
 
 into the circuit to render the effect of the oscillation a 
 maximum. This point may readily be found by referring 
 to equation (212). The coefficient of e becomes a maximum 
 
154 
 
 CIRCUITS CONTAINING 
 
 (for a variation in t^, when the quantity under the radical 
 sign is a maximum. Differentiating the quantity under the 
 
 Fig. 36.-'-Showing how to tind geometrically the Angle ^i 
 
 WBaCH MAKES THE EFFECT OF THE EXPONENTIAL TeBM A MAXIMUM. 
 
 radical, then, with respect to t^, and equating to zero, we 
 obtain 
 
 (217) (Z Coa' - 1) sin 2 ^, + 5 Ceo cos 2 ^, = 0. 
 Whence tan2^, = :j t n i ' 
 
 1 — JL Li CO 
 
 But it will be remembered that [see equation (190)] 
 
 l-LCoo' 
 
 tan 6 = 
 
 BCoo 
 
BESISTANCE, SELF INDUCTION, AND CAPACITY. 155 
 Hence tan 2^, = cot 6 = tan ( ^ — ^)> 
 
 (218) or ^. = J-|. 
 
 And since f^= cot^-\- 6 [see (138)], we find 
 
 (219) -^. = f-¥- 
 
 Suppose ^ is an angle of lag of — 75°, as in the first example 
 
 n 75° 
 cited, then its sign is negative and ^, = -j- + —n~ — 82° 30' 
 
 for a maximum. If 6* is -|- 88° 55', as in the second exam- 
 ple, ip, = 45° — 44° 27'.5 = 32'.5 for a maximum. 
 
 The curve IV., Fig. 35, shows that the maximum point 
 is nearly at the position where ^ = 0, and thus agrees with 
 this result. The exact form which the current curve 
 assumes at the introduction of an harmonic E. M. F. depends 
 upon the time of its introduction and the constants of the 
 circuit. The curves shown in Figs. 34 and 35 give an idea 
 of what may be expected Jn other cases. In all cases, after 
 a very few periods, the current reaches the simple sine 
 form. 
 
 The current which flows upon mating a circuit which 
 contains resistance and self-induction, but no capacity, is 
 shown in Fig. 15, Chapter III., to which the reader is 
 referred. 
 
CHAPTER XI. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF-INDUCTION, 
 AND CAPACITY. 
 
 Case IV. Any Pebiodio E. M. F. 
 
 Contents: — Fourier's theorem. General equations for i and g with any 
 periodic E. M. F. If the self-induction and capacity neutralize each 
 other at every point of time and the current is therefore the same as if 
 both self-induction and capacity were absent, the impressed E. M. F. 
 must be a simple harmonic E. M. F. If the heating effect, or any 
 effect which depends upon Ji'dt, in a circuit, is the ?ame when the self- 
 induction and capacity are present as it is when they are absent, the 
 iftipressed E. M. F. must be a simple harmonic E. M. F. Various 
 types of current curves. When curves are not symmetrical, although 
 the quantity flowing in the positive direction is equal to the quantity 
 in the negative direction, yet theji'dt effect will generally be different 
 in these two directions. Illustration from a particular curve. Alter- 
 nating-current arc-light carbons. 
 
 If we suppose that the impressed E. M. F. is made up 
 of a number of simple harmonic E. M. F.'s added together, 
 the impressed E. M. F. may be written 
 
 (220) e = E, sin (&.(»< + ^,) + E, sin (6, <» f + 6,) 
 
 + E, sin (5, 05 < -f (9,) + etc. 
 and, therefore, 
 
 de 
 
 -rj = E,b,a} cos (5, cot -\- 6,) -{- E^b,oo cos (6, cat -(- 6,) + etc. 
 
 156 
 
BBSISTANCE, 8ELF-INBUCTI0N, AND CAPACITY. 157 
 Expressed as a summation, we have 
 
 (221) e = I^ ^ sin (6 ca « + l9) =f{i). 
 
 E,b,B 
 
 (222) T^ = <» ^^J cos {bcot+0)=f' (t). 
 
 ^* E,b,8 
 
 In this summation it is to be understood that U and 6 take 
 in succession any values, fractional or integral, but that b 
 may only have positive integral values as the E. M. F. is 
 supposed to be periodic, and consequently the periods of 
 the component sine-curves must be commensurable. It 
 was shown by Fourier, in his treatise on the Analytical 
 Theory of Heat, published in 1822, that such an expression 
 as (220) or (221) represents any single-valued periodic func- 
 tion whatever, and is therefore an expression which repre- 
 sents any possible E. M. F. whatever. If (222) is substituted 
 in the general equation for current (99), and (221) in the 
 general equation for charge (100), it will be found, upon 
 integrating, that each component term in the E. M. F. gives 
 a term in the current or charge similar to that given in 
 equations (181) and (182) in Case III., and consequently the 
 resultant current may be expressed as a summation thus : 
 
 (223) i=^^ ^ 
 
 E,b,e 
 
 ;- sin ibaot-\-6 
 
 t t 
 
 and the charge 
 
 (224) g= ^ / fi ,oosiboot + 
 
 
158 CIUGUITS CONTAININO 
 
 In these sums for i aud q there must be as many terms in 
 each as there are in the expression for the E. M. F., and 
 the values of E, h, and 6 must be the same in corresponding 
 terms. These equations express the current and charge in 
 a circuit whose E. M. F. is any periodic E. M. F., as in equa- 
 tion (221). 
 
 If the self-induction and capacity neutralize each other at 
 every point of time, and the current is therefore the same as if 
 both self-induction and capacity tvere absent, the impressed 
 E. M. F. must be a simple harmonic E. M. F. — In the discus- 
 sion of Case III., where the E. M. F. was harmonic and the 
 resulting current was shown to be harmonic also, it was 
 
 pointed out that if the relation oa = —-^i=. existed, the 
 
 current was the same as if there was no self-induction and 
 no condenser in the circuit, and the same as if it simply 
 followed Ohm's law. This was shown by substituting the 
 
 relation a> = — , , or -77 i a» = 0, in the current or 
 
 VLC <^«> 
 
 charge equations (181) and (182) and neglecting the com- 
 plementary function. Those equations, with these substi- 
 tutions, become 
 
 . E . ^ 
 * = ^ sin 00 1. 
 
 cos 00 t. 
 
 ^- R 
 
 It is seen that the current and charge are the same at 
 every point of time as if the self-induction and capacity 
 were absent. Now, since the current is the same at every 
 point of time, the effects of this current will be the same ; 
 namely, the quantity which flows in a half period, being 
 
 T 
 
 I idt = Q, is the same as when there is no self-induction 
 
BE8I8TANCE, SELF-INDUCTION, AND CAPACITY. 159 
 
 and capacity, and the energy expended in the circuit in 
 performing work, or in heating effects, is likewise the same, 
 
 being proportional to ji^d t. 
 
 In order to ascertain whether some similar relation be- 
 tween self-induction and capacity would cause them to 
 neutralize each other when the impressed E. M. F, is 
 not a simple harmonic function of the time, consider the 
 case where the E. M. F. is composed of two parts, each a 
 sine-function of the time. Suppose 
 
 (225) e — E^smaoot-^-E^ sin b cat, 
 
 where a and h are integers. In the circuit there is re- 
 sistance, self-induction, and capacity. Then at any time 
 the value of the current is [see (223)] 
 
 (226) i= ^' 
 
 sin \ a<jot-{- tan"' ^7 l-p: Zaw] t 
 
 I B \Caa> I ) 
 
 sin{6a,«-ftan-i(^-Z&a,)|. 
 
 Suppose the self-induction and capacity have the relation 
 
 a li) = , • Then they will neutralize each other in 
 
 VLG ^ 
 
 the first term of the above expression for the instantaneous 
 
 value of the current. But in the second term the relation 
 
 6 <» = — ^=:^ is necessary to cause the self-induction and 
 
160 0IR0UIT8 CONTAINING 
 
 capacity to neutralize each, other. Now, if one of the above 
 terms is changed by the introduction of self-induction and 
 capacity, while the other term is unaffected, the value of 
 the current which is equal to the sum of the two terms must 
 be changed. It therefore follows that neither the relation 
 
 a £» = , ■ ■ nor 60 = — , will cause the self-induc- 
 
 VLG VLO 
 
 tion and capacity to neutralize each other when introduced 
 into a circuit containing an impressed E. M. F. composed of 
 two simple harmonic E. M. F.'s with angular velocities a oa 
 and b 00, respectively. If a = 6, the two terms in the expres- 
 sion for the instantaneous value of the current may be writ- 
 ten as one, and we have a simple harmonic function of the 
 
 time. The relation « oj = hco= — , will then cause the 
 
 VLO 
 
 self-induction and capacity to neutralize each other. 
 
 If E^ = 0, or if E, = 0, then we have a simple sine-func- 
 tion, and the relation J aa = — , , or a a? = — , , re- 
 
 VL G VLG 
 
 spectively, will cause the balancing of the self-induction 
 and capacity. 
 
 In order to ascertain the conditions under which there 
 may be self-induction and capacity in a circuit, just neutral- 
 izing each other, so that the instantaneous values of the 
 current will be the same as though there were no self-induc- 
 tion and capacity in the circuit, we will consider the general 
 differential equation of E. M. F.'s 
 
 idt 
 
 ' = ^' + ^dt+—G-' 
 
 [See equation (87).] We wish to ascertain the conditions 
 by which the current will be the same as when there ip. 
 neither self-induction nor capacity, that is, the conditions 
 
BESISTANCE, SELF-INBUGTION, AND CAPACITY. 161 
 
 by -whicli * = -s ^^^ e = Bi, according to Ohm's law. 
 Substituting in the above equation, we have 
 
 , . I idt 
 
 This is the same as saying that the E. M. F.'s of self-induc- 
 tion and capacity are equal and opposite. By differentia- 
 tion, 
 
 d' i idt 
 
 ~di~~ lAj' 
 
 Multiplying by 
 
 di 
 dt' 
 
 
 
 (di\ (di\ 
 
 idi 
 -~ LC 
 
 By integrating 
 
 we have 
 
 
 
 d:T=- 
 
 di 
 
 dt ~ ^' 
 
 
 
 y'- LC 
 
 The variables may be readily separated, thus ; 
 
 di 
 
 (228) 
 
 y/c 
 
 — = dt. 
 
 I' 
 
 To 
 
 The integral of (228) is obtained by the formula of integra- 
 tion. 
 
 / 
 
 dx . 1 35 
 = sm"^ — - 
 
 Va' -x' » 
 
162 CIRCUITS CONTAININQ 
 
 Upon integration it becomes 
 
 . . i t 
 
 VcLC VLG 
 
 + c. 
 
 Taking the sine of each member and writing c' for VcLC, 
 
 (229) t = c'sin(-^ + c.). 
 
 The only two variables in this equation are i and t, and 
 the current is seen to be a sine-function of the time. When 
 the current is a maximum, the sine is unity and we have 
 
 I=c'. 
 
 If the time is reckoned from the point where the current is 
 zero, ^ = when i = 0, and we have 
 
 c. = 0. 
 
 Substituting these values for the constants c' and c„ we 
 have 
 
 (230) i = Ism-^=t. 
 
 In an harmonic function, as this, the coeflScient of the vari- 
 able t is the angular velocity which we designate by oa. 
 Equation (230) then becomes 
 
 (231) i — 7sin oat. 
 
 We have, then, the necessary conditions by which the self- 
 induction and capacity will just neutralize each other at 
 every point of time. The current must be a simple sine- 
 function of the time, and the self-induction and capacity 
 
 must have such values that go — . By no other con- 
 
 r X 
 
 ditions, with self-induction and capacity in a circuit, can 
 
/ 
 
 RESISTANCE, SELF-INDUCTION, AND CAPACITY. 163 
 
 the instantaneous values oi. the current be the same as 
 though the capacity and self-induction were absent. 
 
 If the heating effect, or any effect which depends upon 
 
 i' d t, in a circuit is the same when the sdf-indvxiion and 
 
 capacity are present as it is when they are absent, the impressed 
 E. M. F. must be a simple harmonic E. M. F. — Since we have 
 found that there is no possible relation between L and C, 
 so that the instantaneous values of the current are unchanged 
 by their introduction into a circuit with an impressed 
 E. M. F. which is not an harmonic functi6n, it is interesting 
 to inquire whether any relation can be given L and C so 
 that the energy spent in the conductor in a given time is the 
 same before as after the introduction of L and G. 
 
 Before attempting to investigate such a relation, it will 
 be well to first consider some different classes of current 
 
 curves, then ascertain the I Vdt effect for some particular 
 
 current curves, and afterwards consider the energy of any 
 periodic curve whatever. 
 
 Fig. 37 represents a curve which has an equal area above 
 and below the axis every period. This means that the in- 
 
 FiG. 37. 
 
 tegral fidt for one period is zero, that is, the quantity of 
 
 electricity which flows each period in the positive direction 
 is equal to that which flows in the negative direction. 
 Moreover, if the lower half of the current curve is inverted 
 
164 CIRCUITS CONTAININO 
 
 and represented by the dotted line, it is an exact repetition 
 of the first half of the curve. This curve may represent 
 the type of current curves given by alternating generators 
 in circuits with resistance, self-induction, and capacity ; 
 for, it is evident that, as the armature revolves, the number 
 of lines introduced into the circuit every period equals 
 those taken from the circuit. Now, the quantity of current 
 which flows is strictly proportional to the change in the 
 number of lines threading the circuit. This is equivalent 
 to saying that the quantity which flows in the positive direc- 
 tion is exactly equal to the quantity flowing in the negative 
 direction, or the total algebraic quantity per period is zero. 
 Now, if the generator is exactly symmetrical, the current 
 curve in the second half of the period is, if inverted, an 
 exact repetition of the curve in the first half. Any irregu- 
 larities in the symmetry of the machine might cause slight 
 differences in the two parts of the curve, but hardly enough 
 to prevent this curve from representing the type of curves 
 given by alternating machines. During every complete 
 revolution of the armature, the total algebraic quantity of 
 current flowing must be rigorously equal to zero, no matter 
 how many irregularities there may be in the machine ; for, 
 the number of lines introduced into the circuit exactly 
 equals those subtracted from the circuit, because after a 
 complete revolution the number of lines is the same as at 
 the start. It is possible that adjacent positive and negative 
 areas may be unequal in a multipolar machine, due to some 
 irregularity in the machine, but after a complete revolution 
 of the armature the sum of the positive areas equals the 
 sum of the negative. 
 
 Fig. 38 represents a current curve which has equal areas 
 above and below the axis every period, but the negative 
 area, when inverted, is not necessarily a repetition of the 
 positive area. This represents the type of current curve 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 165 
 
 when there is a non-leaky condenser in the circuit, since 
 the total algebraic flow here is necessarily equal to zero. 
 
 Fig. 38. 
 
 Fig. 39 represents a current curve in which the negative 
 area is neither equal to the positive area nor symmetrical 
 with it when inverted. 
 
 Fig. 39. 
 
 It is interesting to inquire whether the I i^ dt effect is 
 
 the same in a circuit while the current flows in the positive 
 direction as it is while flowing in the negative direction. 
 We can see that it is the same for a current of the 
 type represented in Fig. 37, for, squaring the ordinate at 
 
 each point and drawing a new curve, h, Fig. 40, the f i' dt 
 
 effect is proportional to the areas of this new curve. Since 
 the current curves a, a are exact repetitions, these areas, 6, h, 
 
 are identical, and the I i'dt effect is the same when the 
 
 current is positive as it is when negative. 
 
 Let us inquire how this is for a current of the type of 
 
166 
 
 CIRCUITS CONTAINING 
 
 Fig. 38, where the areas are equal, that is, the / id t is the 
 same for positive as for negative current, but the negative 
 
 Fig. 40. 
 
 part, when inverted, is not an exact repetition of the positive 
 part. In Fig. 41 the areas between the axis and the cur- 
 
 — V.6 
 
 Fig. 41. 
 
 rent curve a, a are equal for each half period. The curve 
 h, h is drawn by squaring each ordinate of the curve a. The 
 
 areas 6, b represent the I i'dt effect, and we wish to find 
 
 whether they are equal. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 167 
 
 Illustration from a Farticvlar Case. — To show that this 
 
 I i'dt effect is not necessarily the same when the current 
 
 is positive as when it is negative, it will suffice to take one 
 
 Fig. 42. 
 
 particular case of a current curve. Suppose the positive 
 curve is a parabola (Fig. 42) whose equation, referred to 
 as an origin, is 
 
 (232) 
 
 t' = -3i+3. 
 
 Suppose that the negative curve is a sine-curve Vfh^e equa- 
 tion, referred to 0" as origin, is 
 
 (233) i = f4/3sin<. 
 
 It is easily shown that the areas of these curves are equal. 
 
 Area parabola = | [base X height]. 
 
 One-half of the base of the parabola is found by ^akinp 
 i = in equation (232) and finding the value of t. 
 
 Therefore, Jbase = 4/3, 
 
 Base = 2 VS. 
 
168 CIRCUITS CONTAINING 
 
 The height is found by making t = and finding the value 
 of i. 
 
 Therefore, Height = unity. 
 
 (234) Hence Area parabola = f [base X height] = f -/S. 
 
 The area of the sine-curve is equal to the mean ordinate 
 multiplied by the base ; therefore 
 
 Area sine-curve = mean ordinate X n. 
 
 The mean ordinate of a sine-curve equals twice the maxi- 
 mum ordinate divided by n. [See p. 37.] By equation (233), 
 the maximum ordinate equals | V^ and, therefore, mean 
 ordinate = f tt -/S, and 
 
 (235) Area sine-curve = ^ Vd, 
 
 which is the same as the area of the parabola given in (234) 
 above. Moreover, the tangents of the angles which these 
 two curves make at the point with the axis are equal, 
 and the curves consequently blend into one another with- 
 out any abrupt change in continuity. This is easily shown 
 as follows : Differentiating (232) and (233) respectively, we 
 have 
 
 (236) ^= -If = tan 6^. 
 
 di - 
 
 (237) _ = I ^3 cos t = tan 6'. 
 
 Making t — V3 m (236), we have the tangent of the inclina- 
 tion of the parabola at the point 0. Making ^ = — ;r in 
 (237), we have the tangent of the inclination of the sine- 
 curve at the point 0. These values, it is noticed, reduce 
 (236) and (237), respectively, to tan ^ = tan (9' = - -f VS, 
 which is the value of the tangent of inclination of either 
 curve at the point 0. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 169 
 
 It remains to find the fi'dt for each of these curves. 
 By transposition, the equation of the parabola (232) is 
 
 • 1 *' 
 
 i' = l-lf^*^- 
 
 By squaring, 
 
 Integrating between the limits — V3 and VS, we have 
 
 This is the fi'dt effect for the parabola. 
 
 For the sine-curve the equation is 
 4 = 14/3 sin t. 
 
 fi' dt = ^ y sin' t. 
 
 Integrating between the limits and tt, 
 
 J. *^^ = 3><.L2-2 ''''^°°'^J = 3^2=3''- 
 
 This gives the 1 i'dt effect for the sine-curve. Hence we 
 
 find that, although the area of the current curve is the same 
 for the positive and the negative current — that is, the total 
 
 algebraic quantity of flow is zero — yet the I i'dt effect is 
 
170 CIRCUITS CONTAININO 
 
 different in the positive and negative directions. In the 
 case supposed, the ratio of the two effects is 
 
 ^""^ = 1.135. 
 
 if 4/3 
 
 This may afford an explanation for the fact that in many 
 cases one carbon of an alternating-current arc lamp is con- 
 sumed more rapidly than the other, depending upon the 
 way it is connected up. 
 
 General Proof. — Let us now return to the consideration 
 of the energy in a conductor when any periodic E. M. F. is 
 applied, and ascertain whether there is any condition by 
 which self-induction and capacity may be introduced into 
 
 the circuit without changing the energy or I i'dt effect. 
 
 The energy expended in a conductor is proportional to 
 
 Ji^dt. When the E. M. F. in the circuit is 
 
 e = ^^sin (bcot-\- 0), [see (221),] 
 £, 6, e 
 
 which represents any periodic E. M. F., it has been shown 
 that the current is 
 
 W 
 
 (238) i = ^ 
 
 sin|ia,i+6/ + tan-(^;^^ --^ 
 
 neglecting the complementary function [see (223)]. And, 
 when there is neither self-induction nor capacity, the cur- 
 rent is 
 
 (239) % = ^^^sax(bwt-\-e). 
 
 E,b,Q 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 171 
 If we put 
 (240) 1= ^ 
 
 \/^"+(^-^^-y' 
 
 (241) I, = §, 
 
 we may abbreviate (238) and (239) as follows : 
 
 (242) i = ^ 7 sin (& t» f + a). 
 
 (243) i„ = ^ J. sin (6 oj f + 60. 
 
 The subscript „ indicates the absence of self-induction 
 and capacity. Eemembering that the energy is proportional 
 to / i'dt, we have 
 
 (244) W= fi'dt =y[2 Ism{hcat-\-a)\dt, 
 and 
 
 (245) W, = fi: d t =f I" ^i„ sm{ho!>t-{-(f)Jd t, 
 
 where W is proportional to the energy expended in the 
 circuit with L and C, and W„ bears the same relation to 
 the energy when they are absent. In order to find wha^ 
 relation must exist between L and C to cause the energy 
 expended during a certain time to be the same in both 
 cases, we must integrate (244) and (245) between the same 
 limits of time, and equate them. In .order to simplify (244) 
 and (245), express as follows : 
 
 (246) W = f\rsm(b,(at+a^)-\-r8mQ),ojt-\-a^-\.r' etc^dt. 
 (247)F;=y[//sin(6,(a«+^,)+^."sm(6,(»H^,)+^/"etc.]df. 
 
172 CIRCUITS CONTAINING 
 
 Since the square of any polynomial is equal to the sum of 
 the squares of each term separately plus twice the product 
 of each term by every other term, we have as a result to 
 find the integrals of only two forms, thus : 
 
 (248) y ^^'i' Q)oot^a)dt, and 
 
 (249) / sin (b^wt -\- «,) sin {b^wt -\- a,). 
 
 If the limits are taken from t = to t = T, a. complete 
 
 period, — the E. M. F. being periodic with a period T = — 
 
 — it can be shown that all the integrals of the form of (249) 
 vanisn ; for, expressing the sine of the sum of two angles 
 in terms of the sines of the angles themselves, 
 
 (250) sin(6, 03^ + a,) = sin b^oat cos a, + sin a^ cos boat, 
 and 
 
 (251) sin(6,aj t -\- a^) = sin b,a}t cos a^ -{- sin a, cos b, go t. 
 
 Multiplying (250) and (251), we obtain terms of the follow- 
 ing forms : 
 
 (252) I sin b^oat cos b^cotdt, 
 
 (253) J cos b^cot cos b^wtdt 
 
 (254) / sin b^aat sin b^ootdt, 
 
 which are to be integrated between the limits and T, or 
 
 2 7C 
 
 — . Substituting for b oot, ax, and for \wt, bx, we have 
 
 00 
 
 made the integral in (249) depend upon the three forms, 
 
 /in 
 sin a X cos bxdx, 
 
BE8I8TANCE, SELF INDUCTION, AND CAPAGITT. 173 
 
 (256) / cos a a; cos bxdx, 
 
 (257) / sin ax sin bxdx. 
 
 To show that each of these three forms vanishes between 
 the limits zero and 2 7t, we can reduce as follows : 
 
 /^^ 1 />*" 1 air 
 
 sin ax cos bocdx = « / sin(a + b)xdx-{- ^ P 
 
 • / TV J 1 ^^ fcosfa + 6)03 , cos(a — b)x~\ 
 sin(a — b)xdx = —^ ^^ — !-_^ t v U^ _ n 
 
 COS 005 cos bxdx = o / cos(a -\-b)xdx-\--^ C 
 
 cos(»-»)...=i';[55^^+"j^j=„. 
 
 /S" 1 ^2t 1 8ir 
 
 sin aaj sin bxdx = o / cos(o — b)xdx — ^ f 
 
 , , ,. , 1 ^"■rsinfa — 6)03 sin(a + 6)03-1 
 -os{a + b)xdx=^ X—aZrr- ^6 J = «• 
 
 Since, therefore, the integral in (249) is zero in every case, 
 we have only to find the integral expressed in (248), This is 
 
 2ir 2ir 
 
 (261) J^\in'(boot^a)dt = ''[^-^^ 
 
 -4^^^'^2(6^^ + «)]=| = |, 
 
 which is obtained by the formula 
 
 /x 1 . „ 
 sin' xdx = ^ —-TSin2x, 
 
174 CIRCUITS CONTAINING 
 
 upon replacing xhj hcot -{- a, and dx by hoodt. Returning 
 to equations (246) and (247), and replacing the value of the 
 integral in (261), as determined, we have now found the 
 values of W and W^ in equations (246) and (247) to be 
 
 and 
 
 w= 
 
 . /' 
 
 ' + ■ 
 
 Til'' 1 Jill'' 
 
 + etc. J 2". 
 
 or 
 
 = /. 
 
 W = 
 
 2 «^ ° 
 
 -1 m 
 
 '" + etc.J|. 
 
 Equating TTand W^, as before explained, to determine the 
 condition necessary to mate the energy the same, we obtain 
 
 (262) ^i^=^/„S 
 
 which, written in full, is 
 
 (263) 
 
 I \ ,, \" -^i?» 
 
 '^+\-Uh^-L^^) 
 
 [See (240) and (241).] This equation expresses the relation 
 which must be true if the I i''dt effect is the same when 
 the self-induction and capacity are present as it is when 
 they are absent. This equation expressed without the sign 
 of summation is 
 
 (264) ' ' ^" 
 
 ^"+(-^-^^--y ^'+(^-^^«-y 
 
 + etc. = ^ + -]^ + etc. 
 
RESISTANCE, SELF INDUCTION, AND CAPACITY. 175 
 
 It is evident that the parenthesis in the denominator of each 
 term of the first member, being squared, is always positive 
 no matter what values L and G may have. Each term, 
 then, of the first member is less than the corresponding 
 term in the second member, unless the expression in the 
 parenthesis is zero. And in order that the first member 
 shall be as large as the second member, each parenthesis 
 must be separately eaual to zero ; that is, we must have 
 
 
 VLC- 
 
 VLO' 
 
 and h^co = — -=^^ , etc. 
 
 VL U 
 
 Therefore b, = b,=:b, = etc. But this condition is equiv- 
 alent to saying that the impressed E. M. F. can only be a 
 simple harmonic E. M. F., and that we must have the rela 
 
 tion CO = — in order to have the A'" d t effect the same 
 
 VLG J 
 
 in a circuit when the self-induction and capacity are pres- 
 ent as when they are absent. There is, then, no relation 
 between the self-induction and capacity which can be given 
 
 that will make they^'" A t effect the same in a circuit when 
 they are present as when they are absent, if the impressed 
 E. M. F. is not an harmonic E. M. F. 
 
CHAPTER Xn. 
 
 CIRCUITS CONTAINING DISTRIBUTED CAPACITY AND SELF 
 INDUCTION. GENERAL SOLUTION.* 
 
 Contents: — Derivation of the diflfereutial equations for circuits containing 
 distributed capacity only. Ttiis equation extended so as to represent 
 a particular case of distributed capacity and self-induction. Differ- 
 ential equation for E. M. F. is of the same form as that for current. 
 The general solutions of the differential equations. Particular 
 assumption of harmonic E. M. F. Constants of the general equation 
 determined under this assumption; first, from the exponential solu- 
 tion; second, from the sine solution. Current determined from the 
 E. M. F. equation. 
 
 In former chapters the only capacity considered has 
 been that due to a condenser placed at some particular 
 point of the circuit, thus introducing an actual break in the 
 continuity of the conducting metal. It is possible to have 
 the effects of capacity without thus introducing a condenser 
 into the circuit. The problem of the propagation of the 
 electric current in a cable containing distributed static 
 capacity was first discussed by Sir William Thomson, and 
 
 * The purpose in writing this book has been to give concisely such 
 principles as are necessary for a clear understanding of alternate-current 
 phenomena, and to make the work one connected unit, dealing with the 
 various pi-oblems in turn, so that no portion could be omitted without in- 
 terfering with the logical sequence. This and the following chapter con- 
 stitute, however, a separate discussion which may be read alone, and 
 without which the rest of the book is logically complete. 
 
 Note. — The authors' thanks are due to Prof. Merritt for calliDg attention to certain 
 discrepancies in the signs of some of the equations from 273 to 317 in the first edition. 
 These discrepancies did not affect the results and have been rectified in the present 
 edition. 
 
 176 
 
DISTRIBUTED OAPACITT AND SELF INDUCTION. 177 
 
 afterwards by Mascart and Joubert,* Blakesley, f and 
 others. The solution for the variation in the current and 
 potential at different points of a conductor containing self- 
 induction as well as distributed capacity was given by the 
 authors in the American Journal of Science | and some of 
 the effects of the self-induction noted, and a fuller dis- 
 cussion was given in the London Electrician.^ 
 
 When a current of electricity flows in a wire, the po- 
 tential of the wire at any point is generally different from 
 the potential of the surrounding medium, and in order 
 that this potential may be different it is necessary that the 
 exterior surface of the wire should become charged with a 
 certain amount of electricity. A portion of the current, 
 then, as it flows along the wire, is used to charge the sur- 
 face of the wire. Indeed, the wire must be charged with 
 its proper amount before the current can flow on to more 
 distant parts of the circuit. It is evident, then, that the 
 larger the capacity of the wire to hold a charge, the greater 
 will be its effect in modifying the flow of current. The 
 capacity per unit length of the wire (the wire being re- 
 garded as one plate of the condenser) depends upon its 
 superficial area and upon the thickness of the dielectric 
 (usually between it and the conducting earth near it), as 
 well as upon its nature. In Fig. 43 is represented the 
 longitudinal section of a cable, A being the conducting wire 
 and £-B the insulating sheath around it. Suppose it to 
 be submersed in water ; the other conductor is the water, 
 which, with the wire, forms the condenser. 
 
 Let the capacity of a unit length of the wire be denoted 
 by C, and the capacity of an element FQ, whose length is 
 
 * Mascart and Joubert, Le9ons sur I'electricite et le magnetisme. Vol. I., 
 §233. 
 
 f T. H. Blakesley, Alternating Currents of Electricity, Chap. VIII. 
 
 t Vol. XLIV., page 389. 
 
 § Vol. XXIX., pages 619 and 634. 
 
178 CIRCUITS CONTAINmC 
 
 d X, by Cd x. Let i? denote the resistance of a unit length 
 of the -wire. The resistance of the element PQ is Rdx. 
 Suppose a current i is flowing across the section of the 
 wire at P in the positive direction indicated by the arrow. 
 Let the potential of P at that instant be denoted by e. 
 
 Fig. 43.— Longitudinal Section of Cable. 
 
 Since the current always flows from the higher to the lower 
 
 potential, the potential at Q, the other end of the element, 
 
 must be less than that at P, and the potential therefore 
 
 diminishes in the positive direction. This fall of potential 
 
 de 
 from P to is denoted hj — -r-dx. By Ohm's law the 
 
 current i, at any moment through the element PQ, equals 
 the difference of potential divided by the resistance, and 
 is, therefore, 
 
 de 
 
 (265) i = - 5r^ = _ 1 ^. 
 
 ^ ' Rdx Rdx 
 
 If the current remained constant, having this value i all the 
 time, the potential of the element and its charge would 
 continually remain the same, and the flow of electricity 
 across the section Q would be the same as that at P, since 
 as much must flow out from as into the element, unless the 
 charge of the element be changed. Now, considering that 
 the current does not remain constant but changes every 
 moment of time, the potential e of the element, and conse- 
 quently its charge, must change with the time. When the 
 charge changes, it means that more electricity is flowing 
 
DISTRIBUTED CAPACITY AND SELF INDUCTION. 179 
 
 into tlian out from the element, or vice versa, and conse- 
 quently the flow of current across P is different from that 
 across Q by just such an amount as the element gains or 
 
 di 
 loses. The current at Q is then denoted hj i-\--j- dx. 
 
 Let the quantity flowing across the section F, in the 
 time d t, be denoted by d Q, and that across the section Q 
 hj dQ — dq, where dq is the change in the charge of the 
 element in the time d t. The quantity of electricity flowing 
 across the section P is equal to the current flowing at P 
 multiplied by the time ; that is, 
 
 (266) dQ = idt, or jy = *• 
 
 Similarly the flow across Q is the current flowing at Q 
 multiplied by the time ; that is, 
 
 (267) 
 
 dQ — dq = \i -\- -j—d xjd t. 
 
 Subtracting (267) from (266), we obtain 
 
 dq di , 
 
 (268) T^=-J^^^- 
 
 This equation may be interpreted to mean that the rate of 
 change of the charge on the element is equal to the differ- 
 ence of the currents flowing into the element and out from 
 it. "We might at once have written this equation from this 
 consideration. 
 
 The charge of the element, as of any condenser, is equal 
 to its capacity multiplied by its potential. The charge 
 being denoted by q, the potential by e, and the capacity, as 
 stated above, by Gd x, we have 
 
 (269) g= Gedx. 
 
180 CIRCUITS CONTAINING 
 
 The rate of change of the charge with the time is, by dif- 
 ferentiation,. 
 
 Equating this result to equation (268), we have 
 
 Equations (265) and (271) are the differential equations 
 which are sufficient to determine the problem of the propa- 
 gation of the current along a cable containing distributed 
 capacity such as that described, when the impressed 
 E. M. F. of the source is known. The solution of these 
 equations may be obtained for the most general case, al- 
 though the arbitrary constants of integration can only be 
 determined in certain particular cases where the impressed 
 E. M. F. is known. 
 
 When the impressed E. M. F. is harmonic and equal to 
 e=z E sin oo t, the arbitrary constant may be found. 
 
 These two differential equations may be expressed as a 
 single equation by differentiating (265) with respect to x 
 and equating to (271), thus : 
 
 di _ 1 d' e 
 dx R dx'' 
 
 d'e de 
 
 (272) and, therefore, -r-^=CR-T-.- 
 
 In the foregoing discussion no account has been taken 
 of the self-induction of the circuit, but it necessarily has a 
 certain effect upon the flow of the current which it would 
 be well, if possible, to consider. The effect of the self- 
 induction must be felt as a back E. M. F. opposing the 
 current and depending upon its rate of change. We shall 
 assume that the back E. M. F. per unit length of the con- 
 
BISTBIBUTEB CAPACITY AND SELF INDUCTION. 181 
 
 ductor is equal to the rate of change of the current, multi- 
 
 di 
 plied by a constant ; that is, it is equal to L -r-- In some 
 
 cases this assumption may approximately represent the 
 true effect of self-induction, and it is thought that this par- 
 ticular assumption may show the nature of the effect of 
 self-induction even in cases where the assumption is not 
 justifiable. 
 
 Instead of leaving equation (265) as it stands, therefore, 
 without taking into account the effect of the back E. M. F. 
 of self-induction, we may introduce this effect into the 
 equation by subtracting from the difference of potential 
 
 de 
 between P and Q, viz., — -j-dx, the internal E. M. F. of 
 
 di 
 self-induction, L-^^dx, and so may write, still in accord- 
 
 ance with Ohm's law, 
 
 {^^^)^- Bdx ~ Bdx B df 
 
 The relation in equation (271) is not changed by the con- 
 sideration of the self-induction, and these two equations, 
 (271) and (273), are sufficient to determine the problem of 
 the flow of current, taking into account both the capacity 
 and self-induction. These equations, now containing four 
 variables, may be expressed as two differential equations 
 containing three variables by eliminating first i and then e. 
 After transposing and arranging, we may write (271) and 
 (273) 
 
 ^de di ^ 
 
 (274) f^d-J + d-x = ^- 
 
 de , ^ di ^ . . 
 
 (275) d^ + ^rf-< + ^^ = 0- 
 
182 CIRCUITS CONTAINING 
 
 Operating upon (274) by -r-:, that is, differentiating with 
 respect to t, we obtain 
 
 Operating upon (275) ^J Yd~' ^® ^^^ 
 
 .277^ l^«4.fki). :«li-0 
 
 From (276) subtracting (277), we have 
 (27«) ^dt'~Ldx'-Zdx-^- 
 
 Substituting here the value of t— , namely, — C j-r , in 
 
 (274), we eliminate i and finally have for the differential 
 equation of potential 
 
 To eliminate e from (274) and (275), operate upon (274) 
 by T-, and upon (275) by C'-ji, and we have 
 
 'de\ 
 (280) 6^41^ + 5-: = 0. 
 
 '^IfiJ . (Z^' 
 
 
 (281) and C -p- + L 0^^, + BG~ = 0. 
 
 ^^~' ,d'i . ^^di 
 
BI8TBIBUTEB CAPACITY AND SELF INDUCTION. 183 
 
 Subtracting (281) from (280), we have the differential 
 equation for current 
 
 It is evident from the similarity of equations (279) and 
 (282) that the integral current equation will be the same 
 as the integral potential equation, except for the arbitrary 
 constants that enter in integration. 
 
 To Find the Solutions of the Dipfebentiaij Equations. 
 
 Assume that the solutions of the pair of differential 
 equations (274) and (275) are 
 
 mx-\-nt 
 
 (283) e=he 
 
 mx + nt 
 
 (284) and i=e 
 
 where m, n, and k are constants which must be determined, 
 and X is the distance from the source of E. M. F. These 
 constants may be determined by differentiation so that the 
 equations satisfy the differential equations (274) and (275), 
 and are, therefore, correct solutions. Differentiating (283) 
 and (284) with regard to x and t, we obtain 
 
 de , mx+nt 
 
 dx 
 
 
 .de , yn, mx+nt 
 
 It 
 
 C -y-2 = n kV e , 
 
 di mx + nt 
 
 T- = me ; 
 
 dx 
 
 .,. di T mx+nt 
 
 L — = L%e 
 dt 
 
184 CIBCUITS CONTAINING 
 
 Substituting these values in (274) and (275), we obtain the 
 simultaneous equations 
 
 (285) nTeC+m = 0, 
 
 (286) and mh -\- Ln-\- B = 0. 
 
 If these equations are satisfied, the differential equations 
 are likewise satisfied. Solving for m and n, we find 
 
 (287) m = - -^-^^-j. 
 
 (288) n=+ -^ 
 
 Substituting these constants in (283) and (284), we have 
 (289) e = ke ""'-^ 
 
 * : (( - Ckx) 
 
 (290) and i = e 
 
 ^ r(t-Ckx) 
 
 Ck^—L 
 
 These equations are solutions of equations (274) and 
 (275), and they may be easily verified by differentiation. 
 But a more general solution might be obtained by assum- 
 ing the E. M. F., e, to be a sum of several terms such as 
 that already assumed, thus : 
 
 (291) e = A, i, e +Kk,e -\- ... = ^hke 
 
 h,k. 
 
 (292) i = \e +li,e + . . . = >A e 
 
DISTRIBUTED CAPACITY AND SELF INDUCTION. 185 
 
 Determining m and n as before, (291) and (292) may be ex- 
 pressed 
 
 (293) 
 
 ft, fc.. 
 
 (294) 
 
 h,k. 
 
 These equations may also be verified by differentiation 
 and found to satisfy the differential equations (274) and (275), 
 and they are the complete integrals of those differential 
 equations. If we know how the current or the potential 
 varies with the time at any one point of the wire, the arbi- 
 trary constants h and k can be determined, and we have 
 the complete solution of the problem, and are enabled to 
 tell the potential or current at every point of the wire at 
 any time. 
 
 Haemonic E. M. F. 
 
 The general solutions, (293) and (294), hold true in case 
 the constants to be determined are real or imaginary ; if 
 they are imaginary, the equations may be transformed into 
 a real form consisting of some function of the sine. 
 
 Suppose the cable before described is indefinitely long, 
 and that at the point P (arbitrarily selected as the zero 
 point of the wire, the positive direction being indicated by 
 the arrow) the potential is caused to vary harmonically 
 with the time and is always equal to 
 
 (295) e = Esma)t. 
 
 Since equation (293) expresses the E. M. F. at every 
 point of the wire and at every moment of time, we may, by 
 making ce = in that equation, find an expression giving 
 
186 CIRCUITS CONTAINING 
 
 tlie potential at the origin at every moment of time. This 
 expression is 
 
 (296) e= ^hhe'''^-'' 
 
 h,k. 
 
 But, since we have supposed this potential to be harmonic, 
 we may equate equation (295) to (296) and determine the 
 constants Ji and k so as to make the expressions identical. 
 Equating the equations thus, we have 
 
 ^ t 
 
 (297) E sin o3t= ^hke""''^ 
 
 h,k. 
 
 In order to determine the constants, we write the sine in 
 its exponential form, thus : 
 
 +ju>t —jot 
 
 , e — e 
 
 Sin <» I = — 
 
 y 
 
 where j stands for V — 1. (See equation (109), Chapter VII., 
 with footnote.) We may, therefore, substitute in (297) the 
 exponential value of the sine and write only two terms of the 
 summation, thus : 
 
 (298) e = Esmcot = 
 
 This becomes an identity if 
 
 (299) KK^jj' ^"^ '^.^. = -27" 
 
 n n 
 
 (300) Also, if y oj = -^jrzTX' ^°^ - i <» = gT'"^^ ' 
 
DI8TBIBUTED CAPACITY AND SELF INBUCTION. 187 
 Solving equations (300) for h^ and \, we have 
 
 (301) u,= ±^VLoojJ^B. 
 
 VG 
 
 w 
 
 y^ 
 
 (302) \=± -—^ VL ooj^=-B. 
 
 V U GO 
 
 Since all the constants are found to be imaginary, this 
 imaginary exponential expression for the E. M. F. should be 
 transformed into a real expression involving some function 
 of the sine. This sine-function may be found by continuing 
 the method already indicated. The next step necessary is 
 to transform the complex imaginary values of A by a 
 rather laborious process until the imaginary j is removed 
 from under the radical sign. 
 
 It will be evident that the following equations are 
 identically true, either by squaring each member and seeing 
 that they are identical, or by supposing either B ox Lio 
 be zero, when they reduce to an identity. 
 
 (808) vT^5T^ = ^<^+.^tbB 
 
 +j^(«i+£^l^. 
 
 
188 CIRCUITS CONTAINING 
 
 Substituting these expressions in (301) and (302), and writ- 
 ing Im for the impedance {R^ -\- Z" co")*, we have 
 
 « "'= ± 1 -1^J'^^^^+^, ♦'^•^^^^ I ■ 
 
 Since we know that 
 
 1 4- / 1 — 1 
 
 ■1/ 4- J = — =^ > and i/TTv = -, 
 
 we may substitute these values in (305) and (306) and 
 write 
 
 (307) *. = ± { ~-j==-l Vlm-B-{- Vim + R\ 
 ^ A V \j GO 
 
 
 2/0 
 
 00 
 
 TB]^ 
 
 f 1 
 
 (308) A,= ±]— — =[i//m + J2 + 4/J^5rr 
 \ Av (J oa 
 
 B] 
 
 J 
 
 ^[Vlm + B-Vlm-B'l \- 
 
 GO ' 
 
 2/(7: 
 
 These values of \ and h, may be simplified, for we have 
 the identities 
 
 (309) Vim -B-\- Vlm-\-B = /2 Vim + Loo, 
 and 
 
 (310) Vlm-B-Vlm-\-B = V2Vlm — L 
 
 00. 
 
BI8TBIBUTED CAPACITY AND SELF INDUCTION. 189 
 
 These may be verified by squaring both members. Upon 
 the substitution of these values, the expressions for \ and 
 A, become 
 
 (311) K = ± y^-^ l Vlm^Lco +j Vlm-LoD\- 
 
 1 , 
 
 (312) K=± ^2"^' ^-^™ -\-Loo-j Vim — Loo\. 
 
 Returning to equation (293) of E. M. F.'s and writing two 
 terms of the summation, we have 
 
 Bt Ck^Sx Rt _ Ck,Rx 
 
 (313) e = h,k,e'""'-'' '"'''-'' + h,k,e'""'-^ ^"«'-^. 
 Substituting in (313) the values of h,, k,, h^, k, , 
 
 Ck^—r 
 
 and -frri~ — T gi^^i^ i^^ (299) and (300), we have 
 G », — Ij 
 
 (314) e = ^.\e -e }, 
 
 Substituting in (314) the values of the constants i, and 
 yfc, , already given in equations (311) and (312), and factoring 
 
 out the common laotor e * , we have 
 
 ± ■\/9^(Im-L<o)*x 
 
 (315) e=:Ee ^ 
 
 /Cat 1 /C<^ t 
 
 jiotTjy -^ilm + Luyx —iatt±jy-— (Im+Lay : 
 
 E — e ^ 
 
 2y 
 
290 
 
 CIRCUITS CONTAINING 
 
 Eemembering tlie exponential value of tlie sine, equation 
 (109), we may express (315) as a sine-function, thus : 
 
 (316) e=Ec 
 
 :[-/^4/7SrrLi;J« 
 
 sin|a3i!± \/ "2- V/m + ZcGJ'» [■ 
 
 This equation gives the value of the potential at any point 
 of the conductor at any time. Its interpretation and dis- 
 cussion will be taken up in the following chapter. 
 
 Second Method of Obtaining the Solution. 
 
 We might have assumed the solution to be some func- 
 tion of the sine, since the potential at the origin is supposed 
 to vary harmonically ; and it is much easier to determine 
 the constants if we do make such an assumption, inasmuch 
 as we need not deal with imaginaries. Let us assume that 
 the solution is of the form 
 
 e = « e sm {pt -\- a x), 
 
 and determine the arbitrary constants a, /?, h, r, and p so 
 as to satisfy the differential equation (279). We see that 
 the constant r must be zero, for when x is zero the E. M. F. 
 is E sin oo t. Therefore h = E, and /? = <». The constants 
 p and a remain to be determined, and the E. M. F. is 
 
 (317) e = Ee^''sin{oot-^ax). 
 
 By differentiation we obtain (with E omitted) 
 d'e 
 
 dx' 
 
 .d'e 
 
 ■ p — a' 
 
 
 px 
 
 sm{Got-\-ax)-{-2pa 
 
 px 
 
 e sin{oot-{-ax) 
 
 -OB 
 
 00 
 
 e cos {oot-\- ax) 
 
 px 
 e co^{o3t-\-ax). 
 
DISTRIBUTED GAPACITT AND SELB' INDUCTION. 191 
 
 Equating the coefficients of the sine and cosine, sepa- 
 rately, to zero, we obtain the simultaneous equations 
 
 p» - a= + CL go' = 0, 
 and 2pa — C B oo =0. 
 Solving these equations for p and a, we find that 
 
 (318) p = ± \J~ V{B' + Boo-')i-LoD, 
 
 I'C^ 
 
 (319) and a = ± W -_ ^(^= _|_ jjoay + Loo. 
 
 Substituting in (317) these values of the constants p and a, 
 we find that the result is identical with (316) already ob- 
 tained by a different method. 
 
 To Obtain the Oukeent. 
 
 The current may be obtained from the potential equa- 
 
 tion by means of the relation ^ = — t- [see (271)]. 
 
 Differentiating (317) with respect to t and multiplying by 
 G, we obtain 
 
 C-j—z= C E 00 e'^" cos {go t -\- a x) = — j— - 
 at "' 1^ 
 
 Integrating this result with respect to x, we get 
 
 i= — ■ \ pcos{oat-\-ax)-\- asvsi{pot-\-ax)\. 
 
 Transforming this into an equation containing the sine 
 only, by means of formula (27), Chap. III., we obtain 
 
192 DISTRIBUTED CAPACITY AND SELF INDUCTION. 
 
 V C 00 px { 
 
 (320^ i= — E ^ — — e smJ wt + ax 
 
 -\- tan 
 
 / Im — L Qo) 
 V Im-\-LooY 
 
 Here p and a represent the expressions (318) and (319). 
 Equation (820) may be written 
 
 i = -Ej^e^^" sin L«± «!r+tan-i0 
 
 This equation gives the value of the current at any time 
 at any point of a conductor containing distributed capacity 
 and self-induction vi^hen subjected to an harmonic source 
 of electromotive force. The discussion of this equation 
 and the potential equation will be taken up in the follow- 
 ing chapter. 
 
CHAPTER Xni. 
 
 CrRCTJITS CONTAINING DISTRIBUTED CAPACITY AND 
 SELF INDUCTION.— DISCUSSION. 
 
 Contents : — Circuits with no self-induction. Particular form of e and i 
 equations. Nature of waves. Rate of propagation. Wave-length. 
 Decreasing amplitude. Rate of decay with distance, — with time. 
 
 Circuits with self-induction. Phase difference. Rate of propagation. 
 Diminishing amplitude. Rate of decay. Limitations of the tele- 
 phone. 
 
 Wave-propagation in Closed Circuits. 
 
 Positive and negative waves travel around the circuit until they 
 vanish. Resultant efEect. Potential zero at middle point of the cable. 
 Expression for potential simplified if the length of the cable is 
 a multiple of a wave-length. Same results may be applied to the 
 curreat equation. 
 
 In order to ascertain the physical effects of distributed 
 self-induction and capacity in a circuit, we will first discuss 
 the analytical results obtained in the preceding chapter, as 
 applied to a circuit in which the self-induction is neglected, 
 and then, after investigating the nature of the wave-propaga- 
 tion, the wave-length, rate of propagation, and rate of decay, 
 consider circuits containing distributed capacity and self- 
 induction, noting the changes caused by the introduction of 
 the self-induction. 
 
 193 
 
194 CIRCUITS OONTAININO 
 
 ClBCUITS WITH DiSTBIBUTED CAPACITY BUT NO SeLF INDUCTION. 
 
 When the effect of self-induction is not considered in 
 the cable, we may put Z = in the current and potential 
 equations (316) and (320) and reduce them to more simple 
 forms, thus : 
 
 (321) e^Ee '' sm\oDt± |/^^'^ x | - 
 
 (322) i=±E\/^r^ ' 
 
 sm 
 
 [.t±^9^. + A^^]. 
 
 These results agree with those given by Mr. T. H. Blakesley 
 in his book on " Alternating Currents of Electricity," page 
 60, second edition. They may be directly obtained from 
 the differential equation (272), and upon differentiation will 
 be found to satisfy it. 
 
 Nature of the Wave-propagation. — Equation (321) shows 
 that at any point of the conductor the potential varies har- 
 monically with the time . At the origin where x — 0, the 
 potential is always equal to e = ^sin oo t, its maximum value 
 being E; but as we proceed from the origin the potential 
 
 ~ V — o — » 
 becomes less, being equal to ^e , an expression 
 
 which decreases as x increases. The double sign is retained 
 in the exponent in the equation, since it represents two 
 waves, one going in the positive and the other in the 
 negative direction from the origin, which is the source 
 of alternating potential. The maximum value of the po- 
 tential at any point of the cable may be represented 
 as in Fig. 44 by an ordinate to the logarithmic curve ; 
 
DISTRIBUTED CAPACITY AND SELF INDUCTION. 195 
 
 thus, OA represents the maximum value of the harmon- 
 ically varying potential at the origin, and 5 Cits maximum 
 value at a distance x from the origin. The distance OB is 
 taken as that in which the logarithmic curve has decreased 
 
 to — of its original value, OA. The distance OD is a 
 
 quarter wave-length, and OE a half wave-length. It will 
 
 Fig. 44.— Ctteve I. — Instantaneous Wave nsr Infinite Cable. 
 
 CUKVE II. — LOGAEITHMIC DbCBEASE IN AmPLITITDE. 
 
 presently be shown that the amplitude decreases to almost 
 
 JTTTr 
 
 of its original value in one complete wave-length. 
 
 One of the most striking results shown by these formulae 
 is that the current always precedes the potential by one 
 eighth of a period, and this difference in phase is not altered 
 by any change in the resistance or capacity of the cable. 
 
 Bate of Propagation. — Curve I. (Fig. 44) represents an 
 instantaneous position of the potential wave travelling along 
 the wire in each direction from the origin. The distance 
 
196 CIRCUITS CONTAINING 
 
 along the cable from one maximum potential to the next, or 
 in other words the length of one complete wave, may be 
 found by equating to 2 ;r the angle containing x in the 
 equation. This gives 
 
 OR CO 
 2 — X =171:, 
 
 or «' = ^ = 2;r^^|- = 2y/I^-, 
 
 where \ denotes the length of one complete wave, and 
 
 CO 
 
 n = ^ — the frequency of alternation. 
 
 The time of one complete period is represented by 
 
 T = — , and in that time the wave advances a distance \, 
 
 equal to one wave-length. The rate at which the wave 
 advances is found by dividing the wave-length by the time 
 taken in advancing that distance ; thus, 
 
 Bate of propagation = ^ = 2 A/-^ = a/"^ • 
 
 It is seen that the rate of propagation not only depends 
 upon the character of the cable, but likewise varies as the 
 square root of the frequency of alternation. A wave with a 
 frequency of 400 will travel twice as fast as one with a 
 frequency of 100 alternations per second. 
 
 Decreasing Amplitude. — The frequency affects not only 
 the rate of propagation, but also has a marked influence 
 upon the rate at which the amplitude of the waves decreases 
 with the distance. The distance, which we will call x', at 
 
 which the amplitude of the wave will have - of its original 
 
DISTRIBUTED CAPACITY AND SELF INDUCTION. 197 
 
 value, is the reciprocal of the coefficient of x in the 
 exponent, thus : 
 
 ^'=\/c 
 
 It w 
 
 This distance, at which the wave decays to - of its value 
 
 at the origin, varies inversely as the square root of the 
 frequency ; this means that a wave with a frequency of 100 
 alternations per second can go twice as far as one with a 
 frequency of 400 and experience the same decay in ampli- 
 tude. Comparing this value of x' with that of a wave-length 
 X, we may write 
 
 A. 
 x' = s— • 
 
 The amplitude of a wave, therefore, decreases to -^^ = .00187 
 
 of its value in advancing a distance equal to one wave- 
 length. 
 
 In order to find the time, f, in which the amplitude de- 
 cays to - of its value, we must divide this distance, x', by 
 the rate of propagation, thus : 
 
 We see that the time of decay varies inversely as the 
 frequency ; thus a wave with a frequency of 400 alternations 
 per second will decrease a certain amount in one fourth 
 
198 CIRCUITS CONTAINING 
 
 the time a wave with a frequency of 100 is decreasing the 
 same amount. 
 
 ClECUITS WITH DiSTKIBUTED CAPACITY AND SeLF INDUCTION. 
 
 It will be noticed that the introduction of self-induction 
 into the cable, in the manner before described, does not 
 materially alter the character of the wave propagation. 
 This is evident from the equations (316) and (320). At 
 every point of the conductor the potential, or current, varies 
 harmonically with the time, and, as before, the amplitude of 
 the wave decreases with a logarithmic decrement as it pro- 
 ceeds from the origin. 
 
 Phase Differefice. — One effect of the self-induction is to 
 change the angle of advance of the current ahead of the 
 potential. This angular difference is no longer a constant 
 angle of 45°, as formerly, but is a function depending upon 
 B, L, and w, viz., 
 
 (323) tan B = a/j 
 
 Im — Lea p 
 
 m,-\- IjOo a 
 
 When Z = 0, tan 6 = 1, and 6 = 45°. As L increases from 
 zero to infinity the expression changes from unity to zero, 
 and the angle &, consequently, from 45° to 0°. The effect 
 of the self-induction is, therefore, to decrease the phase 
 difference between the current and potential. This difference 
 of phase also becomes less with an increase in the frequency. 
 Bate of Propagation. — The distance along the cable 
 between two maxima is found by equating the angle con- 
 taining X in equation (316) to 2 tt and solving for x, thus : 
 
 2 71 2 7tV2 
 
 a; = A, = — 
 
 « VGcoVlm-\-Loo' 
 
BISTBIBUTEB CAPACITY AND SELF INDUCTION. 199 
 
 where \ denotes the wave-length. Subscripts are here 
 used to denote circuits with self-induction. The time oc- 
 
 cupied in travelling this distance is 2" = ; hence, 
 
 Col 
 
 Bate of propagation = -^ = — = a / 
 
 2(» 
 
 It is to be noticed that the wave-length and the rate of 
 propagation are each less than that found for circuits con- 
 taining no self-induction. "When Z = 0, these expressions 
 just found reduce to those previously given. 
 
 Another point to be noticed is that a change in frequency 
 will have a greater effect in altering the wave-length but 
 not so great an effect in changing the rate of propagation 
 as in the case of a circuit with no self-induction. Two 
 waves of different periods will, therefore, go more slowly 
 but with less difference in their rates of propagation than 
 with no self-induction. As before, the wave of higher fre- 
 quency will have the shorter wave-length and advance the 
 faster. 
 
 Decreasing Amplitude. — As before, the amplitude of the 
 harmonic wave has a logarithmic decrement, decreasing 
 with the distance from the origin. The distance at which 
 
 the amplitude has — of its original value is the reciprocal 
 of the coefficient of the exponent, thus : 
 
 . V'-^ 
 
 Coo 
 
 P Vim — Lw 
 
 This is larger on account of the self-induction. The 
 substitution of Z = reduces it to the value of x' found 
 before. An increase in frequency will cause this value to 
 
200 CIRCUITS CONTAINING 
 
 decrease, and the decay in a certain distance to increase ; 
 but the frequency has not so great an influence upon this 
 decay as it has in a circuit with no self-induction. 
 
 Comparing with the values found for X^ and d, we see 
 that 
 
 x: = 
 
 2 n tan d ' 
 
 In order to find the time, </, in which the amplitude 
 decays to — of its original value, we divide the distance 
 »/ by the rate of propagation, thus : 
 
 _ 2 Tt tan e T 1 
 
 \ 2 ar tan Q w tan d' 
 
 'T 
 
 The self-induction causes tan 6 to have a value less than 
 unity, thus increasing the time for a certain decay, — that is, 
 decreasing the rate. An increase in the frequency causes 
 6 to become smaller. The exact effect upon the rate of 
 decay caused by a variation in frequency in a circuit with 
 self-induction depends upon the constants of the circuit. 
 The wave of higher frequency will always decay the more 
 rapidly, but with self-induction in the circuit there is less 
 difference in the rates of decay of waves of different periods 
 than there is in circuits without it. 
 
 Limitations of the Telephone. — The effects of distributed 
 capacity in a conductor upon the wave-propagation have 
 been given, and the way in which these effects are altered 
 by the introduction of self-induction or by a change in 
 frequency. A consideration of the results with reference 
 to telephone circuits is valuable inasmuch as it is just such 
 effects that cause the limitations to telephony. In all cases 
 
DISTRIBUTED OAPACITT AND SELF INDUCTION. 201 
 
 the waves of higher frequency travel the faster, and so the 
 several harmonic components of a complex tone are con- 
 stantly changing in their relative phases. The waves of 
 higher frequency are likewise subject to the more rapid 
 decay, and so when the several components are recombiued 
 the resultant tone may be materially altered from the orig- 
 inal complex tone. These effects may be modified by the 
 presence of self-induction, but in all cases they will be 
 present to a certain extent, thus defining the limits of the 
 use of the telephone. 
 
 ■Wave-pkopagation in Closed Cikcuits. 
 
 Let us consider that a dynamo giving an harmonic alter- 
 nating E. M. F. is inserted at some point of a cable, such 
 as that described, which forms a continuous closed circuit. 
 There will be a forward wave of positive potential starting 
 from one pole of the machine which will travel around and 
 around the circuit, continually diminishing in amplitude, 
 until it finally vanishes. At the same time a backward 
 wave of negative potential will start from the other pole of 
 the machine and travel around in the opposite direction 
 until it too vanishes. The potential at any particular point 
 of the circuit is thus the sum of the potentials due to all 
 the positive and negative waves. Let I denote the length 
 of the cable. When the first positive wave reaches a point 
 at a distance x from the pole of the dynamo, the potential 
 due to this forward wave is 
 
 e p = Ee'^" wa.{oot — arcc), 
 
 where j5 and a have the values given in (318) and (319). 
 When the wave has travelled completely around the circuit 
 and comes to this point a second time, its value is 
 
 e^ -z^Ee-^^'^^siXiioat- oLX-ocl), 
 
202 CIRCUITS CONTAINING 
 
 which may be obtained by substituting x -{-I for x in the 
 above. After going n times around, it has become 
 
 e^^ = Ee-P"-'^^'' s,m{Got - ax - aln). 
 
 The resultant of all these forward waves at any one point 
 may be written, therefore, 
 
 (324) e^=e^^ + e^,+ ...+e^^ 
 
 ^Ee-P" |^e-^'"sin(fi3i5 — ace- aln). 
 
 n=0 
 
 When we consider all the backward waves, they may be 
 represented by a similar expression in which E has the 
 negative sign. When the first backward wave has travelled 
 around the circuit so as to be a positive distance x from the 
 origin, it has travelled a distance I — x; this distance from 
 the origin in a negative direction we will call x'. We may 
 therefore write for the first backward wave 
 
 e^ = —Ee~^'' sa\{wt — ax'), 
 
 and for the sum of all the backward waves 
 
 (325) e^=-^e-P'"'^e-^'" &m{a>t - ax' - aln). 
 
 n = Q 
 
 These expressions (324) and (825) may be siipplified 
 since we may put 
 
 (326) sia(cof — ax — aln) ■= sva.{oat — a x)cos aln 
 
 — cos {cot — ax) sin a I n. 
 
DISTRIBUTED CAPACITY AND SELF INDUCTION. 203 
 Substituting (326) in (324), we have 
 
 n=otj 
 
 (327) e^= Ee-P' &m{Gat - ax) ^e-^'^cosaZw 
 
 m = 
 
 m=oo 
 
 -Ee-^" cos,{o3t-ax) ^e"^^" smaln. 
 
 B = 
 
 Similarly we may reduce (325) to 
 
 n = Qo 
 
 (328) e^ = - ^6-^=^' s,m{Got - ax') ^e"^^" cos aln 
 
 7»=0 
 
 n = ao 
 
 4-^e-^''' cos(apf-aa;')^6-^'"sina?w. 
 
 n = 
 
 The resulting potential at any point due to the forward 
 and backward waves is the sum of E-^ and E^. Writing 
 Z — 33 for x' in (328) and adding to (327), we obtain 
 
 n=ao 
 
 (329) e = ^^e-^'"cos«U \ e-^" s,m{Got - ax) 
 
 + ^^e-^'" sinaZw 
 
 n = 
 
 I £+P»'-P'cos(<»« + aa;-aZ)— e~^'"cos(a3« -a£c)j-- 
 
 By means of the exponential values of the sine and cosine, 
 the values of the two summations expressed in (329) are 
 found * to be 
 
 n-x 
 
 € 
 
 ^' sin al 
 
 (330) ^e "'"sin aln = i _ ge^' cos «Z+ e**^" 
 
 91 = > 
 
 * Equations (330) and (381) may be verified thus : For brevity put 
 pl = h, and al = ik. Writing the exponential value of the sine [see equa- 
 tion (109), Chap. VII.]. ■we have 
 
204 CIBCUITS CONTAINING 
 
 and 
 
 .QQ1X ^^ -P'- 7 e"~P'-eP' COS al 
 
 (331) ^e cos«Z« = ^_2^,,^^^^^_j_^,,,- 
 
 "^%- *» sin *» ="^%-''» . ffl^Lz^lfllL 
 
 n =0 n = ^J 
 
 n =00 n = 00 
 
 •^ » = •' n = 
 
 Thus we have the given series equivalent to the difference of two infinite 
 decreasing geometrical series. The sum of such a series is known to be 
 equal to the first term divided by unity minus the common ratio, i.e., 
 
 s — z , where s denotes the sum, a the first term, and r the common 
 
 1 — r ' 
 
 ratio. Applying this formula, the sum of the first series is — • tie-h ' 
 
 2J l-e^» 
 
 and of the second - . • -n-r-—r^ . Hence 
 
 n = CD 
 
 """2/|i_e.>*-h i_e-<''* + ''>r 
 
 Multiplying both numerator and denominator by e and reduclngthe terms 
 in the brackets to a common denominator after factoring out the factor e , 
 we have 
 
 n = CO 
 
 
 n = ■ ' 
 
 Replacing the exponential values of the sine and cosine, we have 
 
 pi . , 
 e aiaal 
 
 n = 
 
 h ft • 7 ®^" 
 
 ^' '"" *" = .^"-ae^cosA + l ~ e^'^'-Ze^'coaal+i 
 
 In a similar manner we may verify equation (331), thus : 
 
 n = 00 
 
 ^ e " cos A« = - ^ e + - 
 
 n s n = 
 
 2 ji-e^^-^+l- e- <•'* + '"[ 
 
DISTRIBUTED GAPACITT AND SELF INDUCTION. 205 
 
 Let POP (Fig. 45) represent the cable which is sup- 
 posed to form a closed circuit, the ends at P being joined 
 
 Pig. 45. — Forward ahd Backward Waves, and KEStrLTANT Poten- 
 tial, IN A Closed Condtictor. 
 
 together. The maximum value of the potential at the posi- 
 tive pole of the dynamo is represented by O A. As we go 
 from A, this decreases along the logarithmic curve 
 A B CD E, until it finally vanishes altogether. Similarly, a 
 
 3 \e"~^ -%e^ coik^l \' 
 
 Therefore 
 
 n = DO 
 
 ;^6--P'" cosa«» = 
 
 .ipi 
 
 — e''' cos al 
 
 e^^'-Se^'cosai+l 
 
 Q.B.D. 
 
206 OIBCUITS CONTAINING 
 
 backward wave coming from the negative pole decreases 
 along the curve A' B' G' D' E' . At the point P, half-way 
 between the poles of the dynamo, the middle point of the 
 cable, it is evident that the potential remains continually 
 
 zero, for, at the point P, the distance a; is ^ , which reduces 
 
 equation (329) to zero. 
 
 If the length of the cable happens to be some multiple 
 of a wave-length, the expression for the potential takes a 
 simpler form. In this case each successive forward wave 
 travels around the circuit in the same phase as the first, 
 and all these forward waves may, therefore, be added 
 together algebraically. The maximum resultant potential 
 at any point will be the sum of the maxima of the separate 
 waves. 
 
 In Fig. 45, e , e^, represent successive forward waves, 
 and e^ and e^ the corresponding backward waves. In the 
 case where the length of the cable is a multiple of the wave- 
 length, the sum of the maxima of all the forward and of all 
 the backward waves is represented by the dotted lines e^, 
 and e^, respectively. The solid line e, the sum of e^, and 
 e^, represents the resultant maximum potential along the 
 
 conductor. 
 
 2 n 
 We have seen that the wave-length is A = — . The 
 
 length of the cable is a multiple of the wave-length 
 
 2 TT /f 
 
 Z = /f A, = , and a Z = 2 TT /C-, where /<• is a positive in- 
 
 a 
 
 teger. This value reduces (330) to zero, since sin 2 /c ;r = 0, 
 
 and reduces (331) to 
 
DI8TBIBUTED CAPACITY AND 8BLF INDUCTION. 207 
 
 These values cause the second term in (329) to vanish, and 
 the whole becomes 
 
 gP« ^pl-px 
 
 (332) e = E —^ — sin (cat — a x), 
 
 which expresses the resultant potential, represented by the 
 solid line e in Fig. 45, at any point of the cable, provided its 
 length is some multiple of a wave-length. When a; = 0, this 
 reduces to e = ^sin oot, the expression for the potential 
 
 at the terminals of the dynamo. When x ■=-^ , the ex- 
 
 pression vanishes, showing that the potential is constantly 
 zero at the middle point of the conductor. 
 
 This last simplification was made possible by consider- 
 ing the length of the cable to be a multiple of the wave- 
 length ; otherwise the algebraic addition of the maxima of 
 the several waves would not be possible, since they would 
 diifer in phase. The construction of the resultant curves 
 would not be so simple, but the nature of the results would 
 not be materially modified. 
 
 The phenomena in connection with the flow of current 
 are similar to those just discussed relating to the propaga- 
 tion of potential, and are obtained in the same manner from 
 the current equation. 
 
PART II. 
 GRAPHICAL TREATMENT. 
 
CHAPTEE XIV. 
 
 INTRODUCTORY TO PART II. AND TO CIRCUITS CON- 
 TAINING RESISTANCE AND SELF INDUCTION. 
 
 Contents: — Introductory. Analytical solutions. of Part I. for simple 
 circuits extended to compound circuits by graphical method. Arrange- 
 ment of Part II. Graphical representation of simple harmonic 
 E. M. P.'s. Graphical representation of the sum of simple harmonic 
 E. M. F.'s of same period. Triangle of E. M. F.'s for a single circuit 
 containing resistance and self-induction. Impressed E. M. F. Ef- 
 fective E. M. F. Counter E. M. F. of self-induction. Direction shown 
 from diflEerential equation. Graphical representation. Methods to be 
 used and symbols adopted in the graphical treatment of problems. 
 First method (the one used throughout this book), employing E. M. P. 
 necessary to overcome self-induction. Second method, employing 
 E. M. F. of self-induction. System of lettering and conventions 
 adopted in graphical construction. 
 
 The analytical solutions derived in Part I. apply merely 
 to a single circuit having resistance, self-induction, and ca- 
 pacity in series. The problems which arise in case there 
 is not simply a single circuit but a complicated network of 
 conductors might be treated analytically, though the pro- 
 cess would be exceedingly laborious and the results too 
 cumbersome to handle. Fortunately, however, by making 
 use of the analytical solutions already given in Part I., and 
 extending them by graphical methods, we are enabled to 
 solve problems with compound circuits which offer con- 
 
 311 
 
212 CIRCUITS CONTAINING 
 
 siderable difficulty to analytical investigation. These 
 graphical methods are most easily and advantageously 
 adapted to problems in which we deal with an harmonic 
 impressed E. M. R 
 
 The object of this Part is to show how to solve by 
 graphical methods any problems arising with any combina- 
 tion of series and parallel circuits, in any branch of which 
 there may be an harmonic impressed E. M. F. 
 
 The plan to be followed is similar to that adopted in 
 the first Part. First are considered various compound cir- 
 cuits which contain resistance and self-induction only, and 
 then circuits containing resistance and capacity only, and 
 finally circuits containing all three, resistance, self-induc- 
 tion, and capacity. The problems to be considered in each 
 case are similar, first a series circuit, then a divided circuit 
 with two branches and with any number of branches, then 
 any combination of series and parallel circuits. 
 
 Before giving the solutions of these problems, the way 
 in which this graphical method corresponds to and is a 
 substitute for the analytical method, and the manner in 
 which it is to be used, will be explained. 
 
 Geaphical Eepeesentation of a Simple Habmonio 
 Electkomotive Fokce. 
 
 An harmonic impressed electromotive force is repre- 
 sented by the equation 
 
 e = E sin co t, 
 
 as was explained in Chap. II. on harmonic functions. The 
 plot of this equation, in which t is the independent and e 
 the dependent variable, gives the sine-curve represented in 
 Fig. 46. A diagrammatic method of representing this har- 
 monic E. M. F. is seen in the same figure. The line OA 
 is supposed to revolve in the counter-clockwise direction 
 
BE8I8TANCE AND SELF INDUCTION. 
 
 213 
 
 about the point with uniform angular velocity. Its pro- 
 ]'ec;tion P at any moment corresponds to the ordinate 
 O'P' of the sine-curve. If the circle be moved horizontally 
 with a constant velocity, the projection P would trace a 
 sine-curve the ordinates of which represent the value of the 
 impressed E. M. F. at any instant. Diagrammatically we 
 may represent the impressed E. M. F. by the line OA 
 
 Fig. 46. — Graphical Representation op a Simple Harmonic 
 Eleotromotivb Force. 
 
 alone, which is equal in length to its maximum value, K 
 In this sense, then, we may represent harmonic E. M. F.'s 
 by lines in the graphical constructions which follow. 
 
 Gbaphical Eepbesentation op the Sum op Simple Hab- 
 MONic Electeomotive Fobces having the Same Pebiod. 
 
 If an E. M. F. is the sum of two simple harmonic E. M. F.'s 
 of the same period, it may be represented by the equation 
 
 (333) 
 
 e = ^, sin Ci7 f -}- ^, sin {cot-j- 6). 
 
 It can easily be shown analytically that this sum is a 
 simple harmonic E. M. F., differing in phase and amplitude 
 from each of the two components, and having the same 
 
214 
 
 CIRCUITS CONTAININO 
 
 period ; for, upon expanding sin {oot-\- 0), the equation 
 becomes 
 
 e = (,£j 4" ^1 cos ff) sin cot -\- E^ sin d cos oa t. 
 
 This may be transformed by means of the trigonometric for- 
 mula (27), Part I., to 
 
 (334) e = VE,' + ^," + 2 E, E, cos ti 
 
 sm 
 
 This equation represents a simple harmonic E. M. F., since 
 it is of the form 
 
 &■=■ E sin {a)t-\- 0), 
 
 in which E and 4> are constant quantities. Moreover, this 
 equation shows that the diagonal of the parallelogram 
 formed by the two component lines which represent the 
 two component terms of equation (333) is the line which 
 graphically represents equation (334), and is therefore the 
 sum of the two components. 
 
 Fig. 47.— Rbsultaht of Two Harmonic Electromotive Forces. 
 
 In Fig. 47, curve I., generated by the line A, represents 
 the first term of equation (333). Curve II., generated by 
 B, represents the second term. Curve III. is the sum of 
 
RESISTANCE AND SELF INDUCTION. 315 
 
 curves I. and II., and is generated by the diagonal C ol tlie 
 parallelogram formed upon the two components A and 
 OB. That curve III., the geometrical sum, represents 
 equation (334), the analytical sum, is seen by the fact that 
 the analytical relations, as shown by the equation, agree 
 with those readily obtained from the geometry of the 
 figure. Thus, from the equation, the amplitude of the 
 resultant harmonic function must be 
 
 E = •v/^/ + ^/ + 2^,^,cosft 
 
 But from the geometry of the figure this same relation is 
 evident, for 'OA = E, , OB = E,, a.nd A OB = 6. 
 
 Again, from the equation the resultant E. M. F. differs 
 in phase from ^, by an angle 
 
 ^ ^ , ^„sin^ 
 <p = tan" 
 
 ^, +^, cosfii- 
 From the figure we see 
 
 ^ nn ^ + - ^^ ^ . ^.sin^ 
 = G A — tan ' ^=- = tan ■ 
 
 DO E.-^E^cosO' 
 
 This agreement of the analytical and graphical relations 
 establishes the correctness of the construction, and we can, 
 therefore, conclude that the sum of any two sine-curves of 
 the same period represented by two lines revolving about 
 a common centre is also a sine-curve of the same period 
 represented by the diagonal of the parfl,llelogram formed 
 on the two component lines. 
 
 When the component E. M. F.'s are more than two in 
 number the sum is represented by the vector, which Is the 
 geometrical resultant of all the component vectors. This 
 evidently follows from the preceding, since any two compo 
 
216 
 
 CIRGUITS GONTAININQ 
 
 nents are equivalent to a single E. M. F., and this combined 
 ■with a third and fourth component gives the geometrical 
 resultant as the sum of all the components. Thus, in Fig. 
 
 Figs. 48 and 49.— Addition op Harmonic Electromotive Forces. 
 
 48, we have a number of vectors A, B, G, D, each repre- 
 senting one component E. M. F. and drawn from the same 
 origin 0. The sum is found in the usual manner by con- 
 structing a parallelogram on any two and then combining 
 the resultant with a third, and so on until all the compo- 
 nents are reduced to a single resultant vector E. This 
 process is equivalent to that indicated in Fig. 49, where the 
 vector A is first drawn from the origin 0, then B from the 
 extremity of A, C from the extremity of B, and so on until 
 all the lines are drawn. The resultant or geometrical sum 
 is then the vector, R, drawn from the origin to the last 
 point found, thus completing a closed polygon. 
 
resistance and self induction. 217 
 
 Teiangle of Electeomotive Foeoes fob a Single Ciecuit 
 
 CONTAINING EeSISTANCE AND SeLF-INDUCTION. 
 
 In Chapter III., in whicli circuits containing resistance 
 and self-induction were analytically treated, it was shown 
 that if a circuit contains an harmonic impressed E. M. F., 
 
 e = ^sin 00 1, 
 
 the value of the current is also harmonic and is 
 
 (335) i = sin [cat — tan^' -„-). 
 
 VB'+Z'oa' ^ Bl 
 
 This current equation was derived from the differential 
 equation of electromotive forces 
 
 in which e is the instantane ous value of_the^mpressed 
 
 E. M. F. of the source, and B i that part, usually called the 
 
 effective E. M. F., necessary to overcome^ the_Qlimic_xesist- 
 
 di 
 ance, and Z -r- that part necessaryjto_overcQmg_thejCQ]jnter- 
 
 ' — Cv z 
 
 E. MJf\jjj[sel|JadJic±ian. 
 
 In Fig. 50 let the vector A represent the harmonic 
 
 impressed E. M. F. Then, by equation (335), we know that 
 
 the current is represented by a vector O B lagging behind 
 
 . L CO 
 
 Ahj au angle 6 whose tangent is -^. 
 
 The effective E. M. F. must be represented by a vector 
 G, equal to BI, in the same direction as the current, and 
 equal to the current vector, OB, multiplied by E. The 
 
 counter E. M. F. of self-induction, Z ji, is at right angles 
 
218 
 
 CIRCUITS CONTAINING 
 
 to the current and must therefore be represented by the 
 vector C A perpendicular to B. It can be shown to be 
 
 Fig. 50.— Triangle of Electromotive Forces. First Method— the 
 one used throtjghout this book— employing e. m. f. to over- 
 COME Self-induction. 
 
 at right angles to the current, as follows. Equation (335) 
 may be written thus : 
 
 i = I sin {pat — ff). 
 
 By differentiation, 
 
 (336) 
 
 di ^ 
 
 -Tj = 00 1 cos {cot — 6). 
 
 Multiplying this equation by Z and writing in terms of the 
 sine, we have 
 
 (337) z|*=:Za3 7sin(a3«- (9 + 90°). 
 
 di 
 By this equation it is seen that the E. M. F., L -r-, necessary 
 
 to overcome that of self-induction is represented by a vector 
 (J A, whose length i&LooI, ninety degrees in advance of the 
 
BE8ISTAN0E AND SELF INDUCTION. 219 
 
 current. The E. M. !P. of self-induction' is equal and oppo- 
 site to that which is necessary to overcome it, and is conse- 
 quently ninety degrees behind the current, represented by 
 the vector A G. 
 
 That the foregoing construction represents the case and 
 fulfils the analytical conditions expressed by the current 
 equation (335) may be shown again by a further comparison 
 of the geometrical with the analytical relations. Thus in 
 Fig. 50 or 51, 
 
 A G Lool Lo3 
 tan A G = -= = „ j = -75- = tan 0. 
 
 This is seen to correspond to the angle of lag in equation 
 (335). Also the impressed E. M. F. A, being the hypote- 
 nuse of the triangle OAG,is equal to the square root of 
 the sum of the squares of the two sides, and therefore 
 
 OA = '^OG'+CA'; 
 
 that is. E= VB' r + r go' r = iVb' + z^ ^'. 
 
 This is seen to correspond to the maximum value of the 
 current given in equation (835). 
 
 Method to be Used and Symbols Adopted in the 
 Geaphical Teeatment of Peoblems. 
 
 In the graphical treatment of circuits with resistance 
 and self-induction there are two methods, each equally 
 correct, for obtaining the same results depending upon 
 whether we consider the E. M. F. of self-induction or the 
 equal and opposite E. M. F. necessary to overcome it. 
 
220 
 
 CIRCUITS CONTAINING 
 
 The method in which the E. M. F. necessary to over- 
 come the self-induction is used is shown in Fig. 50 and has 
 been fully discussed. In this method of construction, the 
 impressed E. M. F. is regarded as made up of the sum of 
 two components, one the effective E. M. F. in the direction 
 of the current, and the other that necessary to overcome self- 
 induction ninety degrees ahead of the current. 
 
 Fig. 51.— Triangle op Electromotive Forces. Second Method, 
 Emploting E. M. p. of Self-induction. 
 
 The method in which the E. M. F. of self-induction is 
 used is shown in Fig. 51. The point of difference is that 
 the line A C represents the E. M. F. of self-induction in- 
 stead of the E. M. F. necessary to overcome it, and is ninety 
 degrees behind the current instead of ahead of it. 
 
 In this method of drawing, the effective E. M, F. which 
 drives the current is regarded as the resultant of the two 
 other E. M. F.'s in the circuit, viz., the impressed E. M. F. 
 and that of self-induction. 
 
 Either of these methods, if carried throughout the whole 
 drawing, is correct and finally brings the same results ; but 
 unless one method is adopted and carried throughout, there 
 is apt to be confusion. In all cases the method used in 
 this book is the first one, namely, that which considers 
 
RESISTANCE AND SELF INDUCTION 231 
 
 the E. M. F. as that necessary to overcome self-induction, as 
 illustrated in Fig. 50. 
 
 In order that the diagrams may be readily understood, 
 the arrangement and system of lettering adopted will be 
 explained. In all cases the positive direction is counter- 
 clockwise, and the diagram is supposed to revolve counter- 
 clockwise around the centre 0. Lines will be designated 
 by letters in small capitals placed at their extremities. 
 The letters therefore designate points and will be used 
 alphabetically, beginning with A, in the order in which the 
 points are determined. Thus in Fig. 54 the line 0-^ is 
 first drawn, then the points B, 0, D, etc., are determined, 
 and the lines OB, G, B 6' CD, etc., drawn in order. 
 All revolve counter-clockwise about 0. 
 
 The direction of lines representing E. M. F. or current 
 will be indicated by arrows, and, where possible, these 
 arrows will he placed so as to show where the lines ter- 
 minate. In order that lines representing current and 
 E. M. F. may be distinguished, the arrows for current will 
 have a closed head, as in the case of the line A, Fig. 54, 
 and the arrows for E. M. F. will have an open head, as on 
 the line B C. Dotted lines are needed only for the con- 
 struction of the figure or to make clear some point that 
 would otherwise be ambiguous or doubtful. When a 
 number of lines terminate at one point and are each di- 
 rected toward the point, it has been found convenient to 
 avoid the confusion of the many arrows coming thus to- 
 gether by omitting the arrows and drawing a small circle 
 at the point, as a,t G, Fig. 54. 
 
CHAPTER XV. 
 
 PROBLEMS WITH CIRCUITS CONTAINING RESISTANCE AND 
 
 SELF INDUCTION. SERIES CIRCUITS AND DIVIDED 
 
 CIRCUITS. 
 
 Prob. I. Effects of the Variation of the Constants R and i in a Series 
 Circuit. B varied. L varied. 
 
 Prob. II. Series Circuit. Current given. 
 
 Prob. III. Series Circuit. Impressed E. M. P. given. 
 
 Prob. Illa. Measurements on a Series Circuit. 
 
 Prob. IV. Divided Circuit. Two Branches. Impressed. E. M. P. given. 
 Equivalent Resistance and Self-induction defined. 
 
 Prob. V. Divided Circuit. Any Number of Branches. Impressed 
 E. M. P. given. Equivalent Resistance and Self-induction 
 obtained for Parallel Circuits. 
 
 Prob. VI. Divided Circuit. Current given. First Method: Entirely- 
 Graphical. Second Method: Solution by Equivalent B 
 and L. 
 
 Prob. VII. Effects of the Variation of the Constants R and i in a Divided 
 Circuit of Two Branches. B varied. L varied. Limiting 
 Cases. Constant Potential Example. Constant Current 
 Example. 
 
 Problem I. Effects of the Variation of the Constants 
 -B and £ in a Series Circuit. 
 
 Befoee taking up the problems proper whicli arise in 
 connection with the investigation of circuits containing 
 resistance and self-induction, it will be well to first con- 
 sider the changes which occur when the resistance is varied 
 and the coefficient of self-induction kept constant, and those 
 
 332 
 
RESISTANCE AND SELF INDUCTION. 
 
 233 
 
 ■which occur when the self-induction is varied and the 
 resistance kept constant. The limiting cases, when the 
 resistance or the self-induction approaches zero or infinity, 
 will be shown, so that the following problems may be 
 applied to such limiting cases without the confusion which 
 might otherwise arise. 
 
 Eesistance Vaeied. 
 
 Let us suppose that the ohmic resistance is varied in a 
 circuit in which the self-induction is kept constant. 
 
 Let OAC, Fig. 52, represent the triangle of E. M. F.'s 
 
 Fig. 52. — Variation op Resistance and Self-induction in a Series 
 Circuit. Problem I. 
 
 for the circuit when the resistanc e is B. Divide C hjH 
 to obtain the current I equal to B. Draw the line O D 
 of indefinite length perpendicular to the E. M. F. A in 
 the direction of lag. The angle BOG, being the comple- 
 
 ment oiAOC, is therefore tan-^-= — Draw B E perpendicu- 
 
 lar to OB and let it meet the line UD in E. Then in the 
 
 rieht triangle B E 'Hh.Q side BE equals -y— ; for OB 
 
 a <=> ±,00 
 
 equals /, and iwa.EOB equals 
 
 R 
 
224 CIRCUITS CONTAINING 
 
 It follows that the hypotenuse, E, of this triangle is 
 
 E 
 equal to -j — , and is, therefore, a constant entirely inde- 
 pendent of any variation in the current I, or resistance It. 
 Taking the square root of the sum of the squares of the 
 sides B and B E, we obtain 
 
 OE=^fOB' + BE' = I^l + -^,. 
 
 From equation (29) we have 
 
 
 E 
 
 
 VE'+L'od" 
 
 
 Therefore 
 
 
 rr / pa 
 
 E 
 -La 
 
 Now, since the side B oi the right triangle OBE 
 always represents the current /, and the hypotenuse Ehs, 
 independent of the current or the resistance, it follows that 
 the current is always represented by a vector B inscribed 
 in the semi-circle OBE, for any possible variation in the 
 resistance. The arrow shows the direction of change as B, 
 increases. 
 
 In the particular cases when R is infinite or zero we 
 see clearly by this figure the limiting values of the current. 
 When B, is infinite the current is evidently zero. When 
 R approaches zero (or, what is approximately the same 
 thing, becomes very small compared with the self-induc- 
 tion) B approaches E, and in the limit the current 
 becomes 
 
 Li CO 
 
RESISTANCE AND SELF INDUCTION. 225 
 
 When the circuit contains no ohmic resistance we see, 
 first, that CA= A, that is, the impressed E. M. F. is 
 equal to Zoo I, the E. M. F. of self-induction ; and, second, 
 that the current lags 90° behind the impressed E. M. F. 
 These relations, here geometrically shown, are analytically 
 expressed in equation (337). 
 
 Self Induction Vabied. 
 
 Suppose the coefficient of self-induction is yaried in a 
 circuit in which the resistance is constant ; we wish to find 
 how the current changes. 
 
 In the same figure, 52, prolong the line ^B to F until 
 it meets the impressed E. M. F. A prolonged. Then 
 
 the line ^i^must equal ~-b^, since tan BOF equ.&\s -^. 
 
 The hypotenuse OJ*' is, therefore, 
 
 0F=V0£"-{-£F' = 7a/i + ^- 
 But from (29) we find 
 
 V'+ 
 
 Bod" _E 
 E 
 
 Therefore, O F = -^. 
 
 Since the hypotenuse OF is independent of the current 
 I or the self-induction L, and is a constant for any variation 
 in L, it follows that the current is always represented by a 
 vector, OB, inscribed in the semi-circle B F, for any 
 possible variation in the self-induction Z. In the figure 
 the arrow shows the direction of change as L increases. 
 
 We easily see what the value of the current is in the 
 
226 CmCUITS CONTAININa 
 
 limiting cases where L is infinite and zero. When L ap- 
 proaches infinity, the current approaches zero. When L 
 approaches zero, the vector B approaches F, the 
 E. M. F. necessary to overcome self-induction is zero, 
 
 and the current follows Ohm's law, being equal to -p. 
 
 That the construction of Fig. 52 is consistent with the 
 equations is further shown by the following relations. 
 
 (338) EF'={EB+BF\'={^+^y 
 
 (339) OE'+Or = ^+§. = jr^,(B' + Z'<.^). 
 Equating (338) and (339), we find 
 
 I'(R'A-Z'gd') = I;\ or 1= ^ _ , 
 
 a result which is identical with that analytically expressed 
 in equation (335). 
 
 It is seen that in the limiting cases, where the resist- 
 ance or the self-induction approaches zero or infinity, the 
 triangle of electromotive forces becomes two superimposed 
 straight lines, that is, one side becomes zero. In most of 
 the following problems only the general cases are discussed 
 in which the circuit contains a finite resistance and self- 
 induction. The constructions may be modified, however, 
 according to the principles just set forth, so that the solu- 
 tions given may be applied to the limiting cases referred 
 to. Although in some cases it may require a little thought 
 and care to make this modification, it has been deemed 
 unnecessary to show its application to each particular 
 problem. 
 
BE8I8TANCE AND SELF INDUCTION. 
 
 227 
 
 Problem II. Series Circuit. Current Given. 
 
 Let there be a circuit, Fig. 53, having n different coils 
 
 € 
 
 ■M Bfl Tn i T i -i ; — s; 
 
 Fig. 53.— Problem II. and Pkoblem III. 
 
 in series, with resistances Bi, B,, etc., and self-inductions 
 Zi , Zj , etc. It is required to find the impressed E. M. F. 
 necessary to cause a current / to flow through the coils. 
 In Fig. 54 make OA equal to the current flowing. 
 
 Fig. 54.— Problem II. and Problem III. 
 
 Multiply this by B, and lay oS B equal B,I, which is then 
 the effective E. M. F. in the flrst coil. Draw JB perpen- 
 dicular to OA in the positive direction, or direction of ad- 
 vance, and make the angle BOO equal to 0^ = tan'" --^. 
 Then B O O is. the triangle of E. M. F.'s for coil one, and 
 Ea is its impressed E. M. F. Similarly lay off VB parallel 
 to O^ and equal to B^I, and then make the angle DOE 
 
228 CIRCUITS CONTAINING 
 
 equal to 6^ = tan"' -^. This triangle CDE then repre- 
 -"2 
 
 sents the triangle of E. M. F.'s for the second coil, and E^, 
 its impressed E. M. F. In a similar way we may go on 
 constructing triangles of E. M. F.'s for each of the n coils 
 until we finally reach a point G, which is the end of the line 
 representing the impressed E. M. F. in the last coil. If we 
 draw the line Q, it must be the impressed E. M. F. of 
 the source, which we wished to find, as it is the sum of all 
 the n different falls in potential for each coil. Indeed, this 
 will be evident from the following. If we lay o& B H 
 = GD, and HK= ~EF, we find- that WK = BJ-\- R, I 
 -\- etc. = 12 H. And, similarly, K G ^ L^ca I-\- L,oo I 
 -\- etc. = 00 12 L. If we replace all the n different coils by 
 a single coil whose resistance is the sum of all the n resist- 
 ances, viz., SB, and whose coefficient of self-induction is 
 the sum of all the n coefficients, viz., 2 L, we find that G 
 is the impressed E. M. F. necessary to cause the given 
 current I to flow, and K G is the triangle of E. M. F.'s 
 for the equivalent coil. 
 
 Problem III. Series Circuit. Impressed E. M. F. Given. 
 
 First Method. — The circuit being the same as in Peob- 
 LEM II., Fig. 53, it is required to find the current, I, which 
 a given impressed E. M. F. will cause to flow. 
 
 We may solve this problem by constructing upon the 
 given E. M. F. 0~G, Fig. 54, the triangle OKGso that the 
 
 2 L GO 
 
 angle at is tan"' -^^- The side OK is then equal to 
 
 12 R. The current I, = OA, is then found by dividing 
 'OKhj 2R. 
 
 The impressed E. M. F.'s, E^, E^,, etc., of the several 
 parts of the circuit are found as in the previous problem. 
 
BESI8TANGE AND SELF INDUCTION. 229 
 
 The total effective E. M. F. represented by OK is divided 
 in proportion to the resistances B,, B,, etc., into the parts 
 OB, BH, etc., representing the effective E. M. F. in the 
 several parts of the circuit. The impressed E. M. F.'s are 
 obtained by erecting upon OB, B H, etc., the E. M. F. 
 triangles B 0, G D E; etc. 
 
 Second Method. — It is sometimes more convenient to 
 solve this problem in the following way. Assume that a 
 certain current is flowing in the circuit ; then find the 
 E. M. F. required, by the method of Problem II. Now if 
 the whole figure be magnified or diminished in proportion 
 until the E. M. F. thus found is made equal to the given 
 E. M. F., then the current will be that due to this given 
 E. M. F., which is the required current ; for, it is evident 
 that if we change either the E. M. F. or the current, the 
 other is changed in proportion, and indeed the whole dia- 
 gram is changed in proportion. 
 
 Problem Ilia.— Measurements on a Series Circuit. 
 
 One of the simplest and also one of the most important 
 cases of series circuits which is met with is that of a non- 
 inductive resistance in series with an inductive resistance 
 as illustrated in Fig. 54a. The corresponding diagram of 
 E. M. F.'s is given in Fig. 546, in which OB and B A rep- 
 resent E^ and E^, the E. M. F.'s impressed upon the non- 
 inductive and inductive resistances, respectively, and A 
 represents E, the total impressed E. M. F. The inductive 
 circuit R, Z, may be the primary of a transformer or any 
 inductive circuit whatsoever. From the values of E, E^ , 
 and E^ , which are readily obtained from three voltmeter 
 readings, and the value of the non-inductive resistance R^ , 
 we can ascertain the following quantities : the angle d by 
 which the current lags behind the impressed E. M. F., E; 
 the angle Q^ by which the current in the inductive part of 
 
230 
 
 CIRCUITS CONTAINING 
 
 the circuit lags behind the E. M. F., E^ , impressed upon that 
 part ; the impedance, resistance, and self-induction of the 
 inductive circuit (in the case of a transformer it is the 
 apparent resistance and self-induction which is found) ; and 
 
 ^ i > 
 
 •^ r *-^ e:^ ^' 
 
 B'l Rs La, 
 
 Figs. 54a, 54J. 
 
 R.i C 
 
 the power expended in each part of the circuit and in the 
 whole circuit. 
 
 From the values E, E, , and E^ , the triangle A £ is 
 drawn, and upon OA the right triangle O A is erected by 
 producing B io O. 
 
 The resistance R, is obtained thus. B = B^I, B G 
 = B,I. Therefore, R,: R,:: OB : BC. For R, we may 
 
 write J. The resistance i?, is then 
 
 R, 
 
 BO ^ = ^^ 
 ~0B '~ OB I' 
 
 The angle • 6 by which the current lags behind the 
 E. M. F. impressed upon the whole circuit is found from 
 the geometry of the figure. By trigonometry, 
 
 E,' = ^» + ^; - 2 EE, cos e. 
 
RESIST ANOE AND SELF INDUCTION. 231 
 
 Whence cos d = ' \^ ^ ' ■ 
 
 The angle 6^ , by which the current lags behind the 
 E. M. F. impressed upon the inductive part of the circuit is 
 similarly found from the trigonometrical expression 
 
 ^" = i:; -\-E^-1 E, E, cos OB A. 
 
 From this it follows that 
 
 E' - E^ - E^ 
 
 cos Q,-= — cos B A 
 
 •iE^E^ 
 
 In the non-inductive portion of the circuit, the current 
 is in phase with the E. M. F. and i9, = 0. 
 
 The value of the self-induction of the inductive circuit 
 is obtained from the values for R^ and d^ given above, 
 
 and from the relation tan 6^ = p . The value of the 
 
 expression L^ go, called the inductive resistance in contra- 
 distinction to ohmic resistance, may be given in ohms. 
 To find Z,, 0^ is first found from the expression given 
 above for cos 6, by means of trigonometry tables, and 
 the tangent of the angle is then found, also from the tables, 
 
 L w 
 and equated to ^— , from w'hich Z, may be readily cal- 
 
 culated. 
 
 An explicit expression for Z, in terms of E^, E,, and E 
 may be found as follows. From the figure it is seen that 
 
 E. . „ z; 
 
 Z, = f^ sin e,= jj:Vl- cos" ^,. 
 
 When the expression given above for ^, is substituted in 
 this expression, it becomes 
 
 A = ^looE, v^e,^e:+ 2Z"z;'-i- 2 e'e:-e'-e:-e:- 
 
232 CIROUITS CONTAINING 
 
 Inasmuch as tlie expression involves the differences 
 of the fourth powers, it does not afford as accurate a 
 method for determining self-induction as that given in the 
 preceding paragraph. 
 
 In these expressions, E, E^, E^, and I represent maxi- 
 mum values, but in the above cases the expressions would 
 be the same if the virtual values were used, that is, the 
 square root of the mean square of the instantaneous values 
 [see page 38], which are represented by 7, E, etc. This is 
 because the values of the above expressions all depend 
 upon the ratio of the quantities in such a way that if each 
 quantity were multiplied by the same constant, the values 
 of the expressions themselves would remain unchanged. 
 It is therefore immaterial whether maximum or virtual 
 values are used. 
 
 In obtaining the expressions for the power expended in. 
 each portion of the circuit the virtual values will be used, 
 inasmuch as these are the values usually obtained from 
 alternating-current measuring instruments. The general 
 expression for the power imparted to a circuit is [see (195) 
 and (196)] 
 
 Tr= i Er cos e = j'Jcos e, 
 
 where 6 is the angle of lag between the E. M. F. and the 
 current. 
 
 In the non-inductive resistance the angle of lag is zero 
 and the power is, therefore, 
 
 W^ = EJ. 
 
 The power expended in the inductive part of the circuit is 
 
 w,=ejcos0^=^{e'-e:-'^:) 
 
 =. ^cr -e: -e:). 
 
 ^B 
 
RESISTANCE AND SELF INDUCTION. 233 
 
 The power expended in the whole circuit is 
 
 W^'E /cos e = ~(S-\-E" - £') 
 2E ^ ~ ' "' 
 
 =^(w+E:-E:y 
 
 25 
 
 It is evident that the power expended in the whole circuit 
 is the sum of the power in each part, or 
 
 W= TF,+ W,. 
 
 This method of measuring power is known as the three- 
 Toltmeter method and was apparently suggested by Mr. 
 Swinburne and by Prof. Ayrton and Dr. Sumpner at about 
 the same time. The method is applicable to any circuit 
 whether the E. M. F. is harmonic or not.* For maximum 
 accuracy E^=- E^. 
 
 Problem IV. Divided Circuit. Two Branches. 
 Impressed E. M. F. Given. 
 
 Let us consider the problem of a divided circuit having 
 two branches in parallel as indicated in Fig. 55. Each 
 
 Fig. 55.— Peoblem IV. 
 
 branch contains self-induction and resistance, and there is 
 an impressed E. M. F., E, between the terminals Jf and N\ 
 
 * See " The Measurement of the Power given by any Electric Current 
 to any Circuit :" Prof. Ayrton and Dr. Sumpner ; Proa. Boy. Soc, Vol. 
 XLIX., 1891, p. 434. 
 
234 
 
 CIRCUITS CONTAININO 
 
 it is required to find the main current, I, and the currents 
 /, and Z, in the branches. 
 
 Fig. 56 shows how to find graphically the main and 
 branch currents when the impressed E. M. F. and the resis- 
 tance and self-induction of each branch are given. 
 
 Fig. 56.— Pkoblem IV. 
 
 Since the impressed E. M. F. at the terminals of each 
 branch is known, each may be separately treated as a 
 simple circuit containing resistance and self-induction by 
 the method previously given in Fig. 50. 
 
 Draw A equal to the impressed E. M. F., E. Make the 
 
 angle A B =6^= tan"' -^- in the negative direction such 
 
 that it is an angle of lag. Then the right triangle OB A 
 is the triangle of E. M. F.'s for the first branch, B is the 
 E. M. F. necessary to overcome resistance, and B A that 
 necessary to overcome the self-induction. In a similar way 
 
 we may lay off the angle A O ^ 6^=. tan"' -^— to repre- 
 
 sent the angle of lag in the second branch, and then 
 construct the triangle OCA, which will represent the 
 E. M. F.'s in the second branch. Since these are right 
 
BE8I8TANCE AND SELF INDUCTION. 235 
 
 triangles, the points B and G lie on the circumference of a 
 circle whose diameter is A. Since the effective E. M. P., 
 B^ /, , in the first branch is B, the current is Z>, equal 
 to OjB divided by R^. Similarly the current /, is 0~E, 
 equal to C divided by B^. Now the current in the main 
 circuit at any instant is equal to the sum of the currents in 
 the branch circuits at that instant. Construct, therefore, 
 the parallelogram upon the sides D and E. The diag- 
 onal F represents the main current, /, for its projection 
 at any moment equals the sum of the projections of the 
 two sides D and E, which projections represent the 
 instantaneous values of the current in the two branches. 
 From the geometry of the figure it follows that 
 
 E=i, vb: + A' oo" = /, vb:+ l: cd = i^r = +z"(»'. 
 
 It is seen that the current in each branch is inversely pro- 
 portional to the impedance. 
 
 This diagram gives the complete solution of the problem 
 of the divided circuit. The currents I^ and 7, in the 
 branches lag behind the impressed E. M. E., E, by angles 
 6^ and ^,. The main current, I, lies between these, making 
 an angle (9 with E. It is evident that the maximum value 
 of the main current, I, being the longest diagonal of the 
 parallelogram whose sides represent the currents in the 
 branches, is greater than the current in either branch. 
 Since the currents differ in phase, at certain parts of a 
 period it happens that the current in a branch is greater 
 than the main current, for when the main current is zero 
 the branch current may have a considerable value. 
 
 Equivalent Eesistanoe and Self Induction. 
 
 Suppose that instead of the two parallel branches which 
 have been considered, a single circuit be substituted for them 
 whose resistance, B\ and self-induction, L' , are such that 
 
236 CIRCUITS CONTAINING 
 
 the same current as before will flow in the main line. Then 
 G A must represent the triangle of E. M. F.'s for this cir- 
 cuit, since the impressed E. M. F. is A, and the effective 
 E. M. r. is OG, in the direction of the current, and the E. M. E. 
 G Aio overcome self-induction is at right angles to the 
 current. The resistance, H', and self-induction, L', of the 
 equivalent simple circuit — that is, a circuit which allows the 
 same current to flow in the main line — are called the 
 equivalent resistance and equivalent self-induction of the 
 divided circuit. 
 
 The values of this equivalent resistance, R', and self- 
 induction, Z', may easily be found in terms of the resist- 
 ances and self-inductions of the branches. This will be 
 deferred until after the discussion of the following problem, 
 in which the solution is given for any number of circuits 
 connected in parallel. 
 
 Problem V. Divided Circuit. Any Number of Branches. 
 Impressed E. M. F. Given. 
 
 Let the divided circuit MN, Fig. 57, have n branches 
 
 L, R, 
 
 innr^ 
 
 L, 
 
 000 '• ^ 
 
 ^~1S1S6 '» \ 
 
 s^ ^~^ 
 
 Fig. 57. — ^Problem V. and Problem VI. 
 
 in parallel, each containing resistance and self-induction, 
 with an impressed E. M. F., E, between the terminals M 
 and N. The currents 7, , Z, ,.../„ in each branch may be 
 constructed as in Problem IV., where there were only two 
 
BESISTANCE AND SELF INDUCTION. 
 
 237 
 
 branches, and the resultant current, /, in the main line 
 found, since it is the geometrical resultant of the n branch 
 currents. Fig. 58 is constructed as follows. Draw a semi- 
 
 FiG. 58. — Problem V. and Pboblem VI. 
 
 circle upon the impressed E. M. F., OA, and lay off the n 
 different angles 6*, , ^, , . . . ^„ in the negative or lag direc- 
 tion, which represent the lag of the current in each branch 
 behind the impressed E. M. F., E. This will give n differ- 
 ent right triangles B A, OCA, D A, etc., which repre- 
 sent the E. M. F.'s in each branch, the sides of which 
 represent the effective E. M. F., the E. M. F. to overcome 
 self-induction, and the impressed E. M. F. Now the cur- 
 rents /,,/,, /s , etc., ov F, O G, IT, etc., are found by 
 dividing the effective E. M. F.'s B, 7, , B^I,, B, I, , etc., or 
 OB, C, D, etc., by the resistances R^, R^, R,, etc. 
 The resultant current, /, or L, is found by taking the 
 geometrical resultant of all the branch currents, F, O, 
 OH, etc. This construction is shown by the closed poly- 
 gon O FJKL, each side of which is equal to a branch 
 current. By this construction it is evident that, since the 
 angles F J, FJK, etc., must be each greater than a right 
 angle, the maximum resultant current, I, or L, is greater 
 than any of the branch currents. During a certain portion 
 of each period, as before explained, the instantaneous value 
 
238 CIRCUITS OONTAININa 
 
 of the resultant current is less than the instantaneous value 
 of the current in any one branch. 
 
 Equivalent Eesistance and Self-induction of Parallel 
 
 ClEOUITS. 
 
 In this case, as in the previous one, suppose that a 
 single equivalent circuit is substituted for the n parallel 
 branches having such a resistance, R', and self-induction, 
 L', that the current in the main line is not changed either 
 in magnitude or phase. The values of this equivalent 
 resistance, JR', and equivalent self-induction, Z', may 
 easily be found in terms of the resistances and self-induc- 
 tions of each branch. . In Fig. 58 the triangle MA must 
 represent the triangle of E. M. F.'s for the single equivalent 
 circuit substituted for the system of parallel branches, if 
 the resultant current (9 Z is to be the same ; for, the effect- 
 ive E. M. F., R'l, is in the direction of the current L, and 
 is therefore equal to M, since the E. M. F. to overcome 
 the self-induction, L'ool or MA, is perpendicular to the 
 current. 
 
 To find R' and L', as well as the tangent of the angle d 
 which the main current makes with the impressed E. M. F., 
 we may proceed as follows. 
 
 If we take the projections of the currents /, /,,/,, etc., 
 upon the line A, we obtain the equation 
 
 (340) / cos 6* = /, cos (9, + 7, cos ^, -f . . . = 2 Jcos 6. 
 
 If we consider the projections of the currents upon a 
 line perpendicular to A, we obtain 
 
 (341) 7 sin (9 = J, sin (9, + /, sin (9, + . . . = ^ /sin 6. 
 Since all the triangles B A, OCA, etc., are right 
 
RESISTANCE AND SELF INDUCTION. 239 
 
 triangles, the following values for I, 7, , Z, , etc., and for 
 cos e, cos (9, , etc., sin 6, sin 6', , etc., will be evident. 
 
 (342) 
 
 (343) 
 
 (344) 
 
 I— 
 
 E 
 
 
 VIi" + L"oiJ'' 
 
 I — 
 
 E 
 
 ■'■I 
 
 VB,' + Z.= CO"' 
 
 T — 
 
 E 
 
 -'-t 
 
 VB.'-ir.L.'oa'' 
 
 Cos — 
 
 B' 
 
 
 VB"-{-L" co''' 
 
 POS — 
 
 B, 
 
 
 Vi?/ + Z/Gj'' 
 
 
 B. 
 
 
 vb:-\-l:cJ' 
 
 sin 9 = 
 
 L'w 
 
 VB^ + L'^of' 
 
 sin 6^ = 
 
 Z, CO 
 
 VB,' + L,'oo'- 
 
 
 L^co 
 
 etc. 
 
 etc. 
 
 '^b: + l: co"' 
 
 Substituting these values in (340), we have 
 J cos B' 
 
 etc. 
 
 (345) 
 
 E - B" + L" a>' 
 
 _B, , B^ , _ "^__^ . 
 
 'B^+ Z," 00-' + B^' + A" ca'^ "f" ^1B' +'Z" o' 
 
240 CIRCUITS CONTAINING 
 
 Making a similar substitution in (341), we have 
 
 / sin L' GO 
 
 (346) 
 
 Z, CO Z," '^ i^ _ ^T" Z 00 
 
 - ^.^ +z>" ^e:^l,'oo'^ ^:^B' + n oo' 
 
 For brevity, let 
 
 (347) ^B'^Z'oo' ^ ^' 
 
 (348) and ^^^=^-. 
 Dividing (346) by (345), we have 
 
 (349) tan = ^. 
 
 From equations (345) and (346), we have 
 
 B' 
 
 B'^-\-L"w- 
 
 = A, 
 
 IJw 
 
 ^^•^ B"^L"co'~^'^- 
 
 Comparing these with the values of cos and sin d in (343) 
 and (344), we obtain 
 
 (350) ^ = ^-5^, or i2' = ^4^, 
 
 (d51) and B oo— r, ■ or L' oo = -^5— • 
 
 ^ ^ X GO ^fij 
 
BE8I8TANCE AND SELF INDUCTION. 341 
 
 For cos' & and sin' 6 we may substitute the values 
 1 1 A' 
 
 cos' 6 = 
 
 1 + tan' d~ B'ao' ~ A' -[- B' w'' 
 
 1 + 
 sin' e = 
 
 A' 
 1 B'co' 
 
 1 + cot'^ A' -A'+B~^'' 
 
 ^ B'oo' 
 
 Making these substitutions, equations (350) and (351) be- 
 come 
 
 A 
 
 (352) B' = 
 
 (353) Z'co = 
 
 A' + £'(»'• 
 
 Bcj 
 A' -{-B'co" 
 
 These expressions, (352) and (353), enable us to calculate 
 the equivalent resistance and self-induction of any number 
 of parallel circuits when we know the resistance and self- 
 induction of each. The angle of lag of the main current is 
 found from (849). These same analytical results were 
 otherwise obtained by Lord Eayleigh, and given by him in 
 a paper on " Forced Harmonic Oscillations of , Various 
 Periods" in the Philosophical Magazine, May 1886. The 
 present demonstration was first given by the authors in the 
 Philosophical Magazine for September 1892. 
 
 Problem VI. Divided Circuit. Current Given. 
 
 Suppose we have a number of circuits, each containing 
 resistance and self-induction, connected in parallel as in 
 Fig. 57, and we know the value of the current, /, in the 
 main line. It is required to find the current in each of the 
 several branches. The value of the impressed E. M. F. is 
 not known, and so the construction cannot be made in the 
 same manner as in the problem just discussed. 
 
242 CIRCUITS CONTAINING 
 
 FiEST Method. Entirely Graphical. 
 
 We can, however, assume any value for the impressed 
 E. M. F., E, and make the construction accordingly, as in 
 the previous problem. We would thus obtain a value for 
 the main current, /, different from the one given. The 
 diagram will be correct in all respects except the scale, and 
 this must be changed in the ratio of the given value of / 
 to the value of / obtained from the assumed impressed 
 E. M. F. The true value of the impressed E. M. F. and 
 the current in each branch may thus be obtained and the 
 solution is complete. 
 
 Second Method. Solution by Use of Equivalent 
 Eesistance and Self Induction. 
 
 Another solution for this same problem is obtained by 
 the use of equivalent resistance and equivalent self-induc- 
 tion of parallel circuits. These values for R' and L' are 
 calculated according to the expressions (352) and (353). 
 Draw M, Fig. 58, equal to R'l, and draw MA perpen- 
 dicular to M andi equal to L'aol. The hypotenuse A 
 of the right triangle MA gives us the value of the im- 
 pressed E. M. F., E. The further construction is the same 
 as before. The angles of lag ^, , 0^, 6*3, etc., are laid off, 
 and the E. M. F. triangle for each branch circuit is drawn. 
 The effective E. M. F. and the current in each branch are 
 thus readily found. 
 
 Problem VII. Effects of the Variation of the Constants 
 JB and X in a Divided Circuit of Two Branches. 
 
 Eesistance Alone Varied in Either Branch. 
 
 Suppose the resistance of one branch of a divided cir- 
 cuit to be varied and the other constants to remain un- 
 changed ; it is required to find the changes in the currents 
 due to this variation in resistance when there is a constant 
 
BBaiSTANCE AND SELS' INDUCTION. 243 
 
 impressed E. M. F. Let the diagram for the divided circuit 
 shown in Fig. 55 be represented in Fig. 59, where the same 
 letters represent the same points as in the diagram, Fig. 5&, 
 already given for the divided circuit. 
 
 If the resistance B^ is varied, it is evident that the effec- 
 tive E. M.F. ^ always lies on the feemi-circle B A, and, 
 as this branch may be regarded as a single circuit having a 
 constant E. M. F. and a resistance which is varied, the cur- 
 
 I. — Vabiation of Resistance and Sblf-indtjction in a 
 Divided Circuit. Problem: VII. 
 
 rent /, always lies on the semi-circle D H, whose diameter 
 
 E 
 
 is ZT or -J — (see Problem I.). If B^ is the only quantity 
 
 varied, it is evident that the resultant main current must 
 lie on the semi-circle ^J^eT", whose diameter ^j/ is equal 
 to OH. 
 
 Similarly, if B^ is varied alone, the current Z, must lie 
 
244 CIRCUITS CONTAININQ 
 
 E 
 
 on the semi-circle E K, whose diameter is 0^ or -^ — . 
 
 I/^OO 
 
 The resultant main current will then lie on the semi-circle 
 D F L. When both resistances are varied at the same 
 time the currents /, and I^ lie on their semi-circles D H 
 and EK; but the resultant or main current has no par- 
 ticular locus. 
 
 The arrows on the curves, showing the effects of a varia- 
 tion of the resistance, indicate the direction of the change 
 as the resistance increases. 
 
 Self Induction Alone Varied in Either Branch. 
 
 Eegarding each branch of the divided circuit, ha\'ing a 
 
 constant difference of potential at its terminals, as a single 
 
 circuit, it is evident that any variation of L^ alone will cause 
 
 the current vector /, to lie upon the semi-circle DM, 
 
 W 
 whose diameter is -5- (see Problem I.). Any variation of 
 
 i, alone will cause the resultant main current vector, I, to 
 lie upon the semi-circle EFN. 
 
 Similarly, when L^ alone is varied, the current /, lies 
 upon the semi-circle OEP, and the resultant current 1 
 upon the semi-circle D F Q. If both L^ and L^ are simul- 
 taneously changed, the currents /, and /, still lie on their 
 circles D M and OEP, respectively, but the resultant 
 current / has no particular locus. 
 
 The arrows on the curves, showing the effects of a varia- 
 tion of the self-induction, indicate the direction of the 
 change as the self-induction increases. 
 
 Limiting Cases. 
 
 This diagram enables us to see what the currents will 
 be in the divided circuit in the limiting cases when the 
 resistances or self-inductions approach infinite or zero 
 values. As a particular instance, suppose it happens that 
 
RESISTANCE AND SELF INDUCTION. 
 
 245 
 
 Zj is zero, and B^ is very small compared with Zj. This 
 means that there is self-induction alone in one branch and 
 resistance alone in the other. The current 7, would then 
 be represented by H, and /^ by P, and the main cur- 
 rent, /, by the resultant of these. 
 
 Constant Potential Example. — ^As an example, suppose 
 -there is an incandescent lamp. Fig. 60, of 50 ohms re- 
 sistance B^, and a coil whose self-induction Z, is .5 
 henrys shunted around the lamp, the terminals of which 
 are subjected to a constant difference of potential of 50 
 volts. What are the currents through the lamp, coil, and 
 
 o 
 
 SiSiSL-h- 
 
 L, 
 
 Fig. 60.— Problem VII. 
 
 the main line ? Suppose that oo = 1000. We may calcu- 
 late 
 
 and 
 
 Z;_50 
 E 50 
 
 Lgo~.5x 1000 
 
 E 
 
 = 1=1., 
 
 = 1 = 1,. 
 
 Make O P, Fig. 61, equal to -^ = 1, and ninety degrees 
 
 E 
 
 behind it make H = j^— - 
 
 OS is easily calculated, thus ; 
 
 7^ 
 behind it make OH = f — = .1. The resultant current 
 
 OS = VOP' -f ir' = a/1 + -01 = 1-005. approx. 
 
246 
 
 CIRCUITS CONTAININO 
 
 If the incandescent lamp should break, the current I, 
 through it would be stopped and the main current reduced 
 to OH, equal to .1 amperes. 
 
 -r-l Ampere- 
 
 FiG. 61.— Constant Potential Example, Problem VII. 
 
 Constant Current Example. — Suppose that, instead of 
 being subjected to a constant potential, a divided circuit, as 
 Fig. 60, is supplied with a constant current. Let the main 
 current be maintained constantly at ten amperes. It is re- 
 quired to find the branch currents and the difference of 
 potential at the terminals. Let it have a resistance S^ of 
 two ohms, and let the self-induction of the choke-coil be. 02 
 
 \ 
 
 10 Amperes 
 
 Fig. 62.— Constant Current Example, Problem VII. 
 
 henrys. Using the first method of solving the problem of 
 the divided circuit when the current is given, Problem VI., 
 we may assume an impressed E. M. F. A of ten volts. 
 Following the same construction in Fig. 62 as in Fig. 61 for 
 
RESISTANCE AND SELF INDUCTION. 247 
 
 the solution of the constant potential example, we may 
 
 calculate 
 
 and 
 
 ^ = -2-=5 = 7, = 0P, 
 
 E _ 1 r _ r _ 
 
 L,oo~ .02 X 1000 -'^ -1^-OH. 
 
 The resultant 8'w. calculated thus : 
 
 WB = VWW+TW = /25 + .25 = 5.025 amperes. 
 Since the main current should be ten amperes, it is neces- 
 sary to magnify the whole diagram in the ratio -^-tz^, in 
 
 order to find the true difference of potential at the ter- 
 minals, and the true branch currents. This makes the 
 
 impressed E. M. F. 10 X j^^ equal to 19.9 yolts ; the cur- 
 rent 7, equal to 9.95 amperes ; and /, equal to .995 amperes. 
 Since the current and the E. M. F. are in phase, the energy 
 consumed by the lamp is equal to 19.9 X 9.95 = 198 watts. 
 The energy consumed by the choke-coil is almost nothing, 
 since the current Z, is almost at right angles to the im- 
 pressed E. M. E. 
 
 If the lamp filament should break, the current /, would 
 be suddenly stopped and the whole current OB of ten 
 amperes would flow through the coil. The potential G at 
 the terminals would suddenly become much greater, large 
 enough to overcome the E. M. F. of self-induction Z, go I, 
 that is, .02 X 1000 X 10 = 200 volts. 
 
 Thus the choke-coil shunted around the lamp consumes 
 but little energy and prevents the current from being inter- 
 rupted when the lamp breaks. In case the lamp does 
 break, however, there is the sudden rise in potential as 
 shown above. 
 
CHAPTER XVI. 
 
 PROBLEMS WITH CIRCUITS CONTAINING RESISTANCE 
 AND SELF-INDUCTION. 
 
 COMBINATION CIRCUITS. 
 
 Prob. VIII. Series and Parallel Circuits. Impressed E. M. F. given. 
 
 Solution by Equivalent B and L. 
 Prob. IX. Series and Parallel Circuits. Current given. Solution by 
 
 Equivalent S and L. 
 Prob. X. Extension of Problems VIII and IX. 
 Prob. XI. Series and Parallel Circuits. Entirely Graphical Solution. 
 Prob. XII. Multiple Arc Arrangement. 
 
 Problem VIII. Series and Parallel Circuits. Impressed 
 E. M. F. Given. Solution by Use of Equivalent Kesist- 
 ance and Self Induction. 
 
 Peoblems arising from combinations of series and paral- 
 lel circuits are readily solved by the repeated application 
 of the foregoing methods. Let us consider the case where 
 two systems of parallel circuits are joined in series, as in 
 Fig. 63. The resistance and self-induction of each branch 
 is given and the total impressed E. M. F. It is required to 
 find the current in the main line and in the branches. 
 
 The equivalent resistance and self-induction i?„' and Z„' 
 between J/" and N, and i?^' and L^' between N and 0, are 
 readily found according to the formulae (352) and (353). 
 We can now treat the problem as that of a series circuit, as 
 
 848 
 
RESISTANCE AND SELF INDUCTION. 
 
 249 
 
 in Peoblem III., and ascertain the impressed E. M. F. 
 between M and N and between N and 0. 
 
 -X- 
 
 Ei 
 
 Fig. 63.— Pkoblbm VIII. and Problem IX. 
 
 Upon the impressed E. M. F., A, Fig. 64, draw the 
 right triangle OB A such that tan AO B = „ ' -i- R ' ' 
 
 =^i 
 
 Tig. 64.— Problem VIII. and Problem IX. 
 
 Then OB is the E. M. F. effective in overcoming the resist- 
 ance Ba + Rh and may be divided at C so as to show the 
 E. M. F. effective in overcoming each. 
 
 0G\ CB :'.RJ:B^'. 
 
250 OIROUITB CONTAINING 
 
 Cis the E. M. F. effective in overcoming the resistance 
 Ed . Draw C -D perpendiciilar to 0, and complete the 
 
 right triangle OGD,so that tan Z> 6* C = -~wt- OD is the 
 
 impressed E. M. F. between the points Jf and N, and DA 
 the impressed E. M. F. between N and 0. Knowing the 
 impressed E. M. F., Ea , between M and iT, we can obtain 
 the currents /, and /^ in the branches by the method fully- 
 explained in Pkoblem IV. and Peoblem Y. On the diameter 
 '01), the E. M.F. triangles OFD and GD are drawn 
 with angles of lag according to the constants of each 
 branch. The currents I^, I,, and / are found by dividing 
 the effective E. M. F.'s by the resistances H^, H^, and RJ , 
 respectively. In the same way, the E. M. F. triangles D LA 
 and DP A are erected on the line D A, which represents 
 E^, , the effective E. M. F, between N and 0. The currents 
 Jj and I^ are then found and we have the complete solution 
 of the problem. 
 
 Problem IX. Series and Parallel Circuits. Current 
 Given. Solution by Use of !Eciuivalent Kesistance 
 and Self-induction. 
 
 Suppose that we have the same arrangement of circuits 
 as that just described and shown in Fig. 63, and that the 
 main current, I, is given. It is required to find the current 
 in each branch. The solution for the part between M and 
 N and for the part between N and can be obtained inde- 
 pendently according to the second method given in Peob- 
 lem VI. 
 
 In Fig. 64, O Gis drawn equal to -Ba'/, and GD equal to 
 LJ GO I. On OD the E. M. F. triangles OFD, OGD are 
 drawn and the solution for branches one and two obtained. 
 D E is then drawn parallel io O and equal to R^ I, and 
 'EA equal to L^ oo I. The E. M. F. triangles, DLA and 
 
RESISTANCE AND SELF INDUCTION. 
 
 251 
 
 D P A, are then erected on IDA, and the solution for 
 branches three and four obtained in the regular way. The 
 line connecting and A gives the total impressed 
 E. M. F., E. 
 
 Problem X. Extension of Problems VHI. and IX. 
 
 The solution given in Peoblem VIII. may be applied to 
 any combination of circuits in series and parallel. Let us 
 consider a combination of circuits such as that shown in 
 
 Fig. 65. — Problem X. 
 
 Fig. 65, having a given impressed JS" IT i^ between the points 
 M and P. The circuits may be divided into three parts, 
 MN, JV O, and P, and the equivalent resistance and self- 
 induction of each obtained [see (352) and (353)]. The im- 
 pressed E. M. F's., Ea, E^, Eo, of each portion can now be 
 laid off, Fig. 66, according to the method given for series 
 
 Fig. 66.— Peoblem X. 
 
 circuits, Problem III., and a semi-circle erected upon each, 
 as was done upon Ea and E,, in Fig. 64. Each portion of the 
 
252 CIRCUITS CONTAINING 
 
 circuit is now treated separately according to tlie method 
 for parallel circuits, Problem V. In each semi-circle the 
 various E. M. F. triangles are drawn and the currents in the 
 several branches found. 
 
 If the main current is given in an extended system of 
 conductors, as in Fig. 65, the solution is obtained, as in 
 Peoblem VII., by dividing the system into its several sets 
 of parallel circuits and treating the separate sets, MN, 
 NO, OP, independently. 
 
 Proljlem XI. Series and Parallel Circuits. Entirely 
 Graphical Solution. 
 
 In the previous treatment of the problems arising from 
 combinations of circuits in series and parallel it was neces- 
 sary to find analytically the values of the equivalent resist- 
 ance and self-induction of each set of parallel circuits, and 
 the solutions were, therefore, partly analytical and partly 
 graphical. They may be obtained, however, by entirely 
 graphical methods, if we assume some value for the current 
 in a particular branch or assume its impressed E. M. F., 
 solve a portion of the system of conductors accordingly, and 
 then correct the scale as required by the given conditions of 
 the problem. Various ways of doing this suggest them- 
 selves as preferable according to the nature of the problem. 
 
 By Assuming Various Impressed E. M. F.'s. 
 
 Given the main current, /, in a system as shown in Fig. 
 67. Let us assume any value we please for the impressed 
 -^ — ^2-^« — h ^ E=_»,p ^i s. ^' „ 
 
 Jl. 
 
 ^^ I. _ 
 rSBBTT 
 
 UfiAfi.' 
 
 Is 
 
 I 
 
 Fig 67,— Problem XI. 
 
 E. M. F., El , then erect, on a line representing ^j, , the E. 
 M. F. triangles for branches one and two, and thus find 
 
RESIST ANGS AND SELF INDUCTION. 
 
 253 
 
 the currents 1^,1^, and I, due to this assumed E. M. F. 
 The value thus obtained for the main current, /, will be 
 different from the given value, and the assumed E. M. F., E^, , 
 must be changed in the ratio of the given value of I to the 
 value obtained from the assumed value of Ui,. This amounts 
 to changing the scale of the drawing. The solution is thus 
 obtained for the part between JS and G. The several 
 other portions G D, D E of the system are separately 
 treated in the same manner and thus a complete solution 
 obtained. 
 
 If we were given the total impressed E. M. F., E, and 
 not the main current, /, a convenient graphical solution 
 would be obtained by assuming some value for /, solving 
 as in the previous paragraph, and then changing the scale 
 according to the ratio of the given value of the impressed 
 E. M. F. to the value thus obtained. 
 
 By Assuming the Cueeent in Ceetain Beanches. 
 
 Instead of assuming the E. M. F. impressed at the ter- 
 minals of each part of the system, we may assume the cur- 
 rent flowing in any one branch of each parallel set of con- 
 ductors. The complete graphical solution by this method 
 of a combination circuit representing in Fig. 68 is given, 
 
 R,L, 
 
 LjiqjlqjlJ 
 
 H O } R3L, 
 
 R4L4 
 
 Pig. 68.— Problem XI. 
 
 to illustrate the principles already given, 
 pressed E. M. F., E, is given. 
 
 The total im- 
 
254 
 
 CIECUIT8 CONTAINING 
 
 In Fig. 69 any assumed line A h^ drawn to represent 
 the current in the branch whose resistance and self-induc- 
 
 Fm. 69.— Problem XI. 
 The Solution pok Cikcuits 
 Bbtwebit a and B, Pi&. 6 
 
 Fig. 70.— Problem XI. 
 
 The Soltjtion for Circtjits 
 
 Between and D, Fig. 68. 
 
 tion are B^ and Z, . Ais then multiplied by B^ and pro- 
 duced to B. Then OTS = B, I, is the effective E. M. F. in 
 branch one. Draw B O perpendicular to B in the direc- 
 tion of advance and make it equal to Z, caZ,. Then Cia 
 the impressed E. M. F. necessary to drive the assumed 
 current through branch one. Now having this impressed 
 E. M. F., we can draw the E. M. F. triangle, D G, for 
 branch two, and obtain the current E flowing in the second 
 branch by dividing Dhj B,. The total current 1= F 
 is then the vector sum of Z, and 7^ , or of OA and OE. 
 
 For the parallel system between D and (Fig. 68), the 
 same process is followed, and in Fig. 70 O ^ is first assumed 
 as the current branch in three, and F finally found to 
 be the total current flowing between O and D (Fig. 68). 
 Since these two parallel systems are in series, the total 
 current, WF (Fig. 69), flowing between A and B (Fig. 68), 
 must equal the total current, F (Fig. 70), flowing between 
 C and D (Fig. 68). Hence Fig. 70 is magnified until WF 
 becomes as large as OF, Fig. 69, and is represented in 
 Fig. 71. Next the two figures 69 and 71 are combined, as 
 
BEaiSTANCE AND SELF INDUCTION. 
 
 235 
 
 shown in Fig. 72, so that 0' F' is parallel with OF, since 
 each represents the same current I. O C ', the vector sum of 
 
 Fig. 71.— Pboblkm XI. Fig. 70 Enlarged. 
 
 Eis, and E^ , is the total impressed E. M. F. at the termi- 
 nals A and D necessary to send the current /, If this 
 
 Fig. 72.— Pkoblem XI. 
 
 figure is now magnified until C is equal to the given 
 impressed E. M. F., the solution of the problem is complete 
 
256 C1BCUIT8 CONTAININO 
 
 and we have found the currents in each branch for the 
 given impressed E. M. F. 
 
 Problem XII. Multiple-arc Arrangement. 
 
 Graphical solutions for circuits in series and for circuits 
 in parallel have been separately explained at length and it 
 has been shown how the solution of any combination of 
 circuits in series and parallel may be obtained by dividing 
 the system into its separate parts of series and parallel ar- 
 rangements and successively applying the foregoing meth- 
 ods. There are countless combinations which might arise, 
 but the solutions of all depend upon the principles already 
 given, and it will suffice to further illustrate them by their 
 application to one more problem of combined circuits. 
 
 Let us consider a system of parallel circuits, each with 
 resistance and self-induction, extending between two mains 
 containing resistance and self-induction. Such a system is 
 shown in Kg. 73. The circuits 1, 2, 3, etc., contain resist- 
 
 -R t- 
 
 -R t- 
 
 -R t- 
 
 FiG. 73.— Problem XII. 
 
 ance and self-induction B^ Z, , B^L^, R, L, , etc., respec- 
 tively. The resistance and self-induction of the mains are 
 
 R L 
 
 igic I i I i 
 
 Fig. 74.— Problem XII. 
 
 lla and i„ for the portion a, B,, and Z,, for the portion h, 
 between circuits 1 and 2, -S„ and Z„ for the portion c, etc. 
 
RE8ISTAN0E AND SELF INDUOTIOJSf. 257 
 
 The equiyalent resistance and self-induction of circuit one 
 and that portion of the system beyond circuit one — namely, 
 b, c, d, etc., and 2, 3, 4, etc. — is R' L' ; for circuit two and 
 the portion of the system beyond, the equivalent resistance 
 
 Fig. 75.— Problem XII. 
 
 and self-induction are R" and L" ; for circuits three and 
 beyond they are R'" and L'" ; etc. These values of the 
 equivalent resistance and self-induction are computed by 
 successive applications of the formulae (352) and (353), be- 
 ginning at the most distant end of the system. The equiva- 
 lent resistance R"" and self-induction L"" are found by 
 adding R^ and L^ to R^ and L^ , respectively, and finding the 
 equivalent resistance and self-induction when combined 
 in parallel with circuit four. B'" and L'" are found by 
 adding R"" and L"" to Rg, and L^ and finding the equiv- 
 alent resistance and self-induction of this when combined 
 in parallel with circuit three. R" and L", and B' and L', 
 are similarly found. 
 
 Let us now replace by a simple circuit with resistance 
 and self-induction R' and L' that part of the system to 
 which it is equivalent. The system then reduces to a series 
 circuit (Fig. 74), and its solution is obtained by the method 
 for series circuits, Peoblem III. The complete solution for 
 this problem is given in Fig. 76. 
 
 Draw A = K On A erect the right triangle B A 
 
 L -\- L' 
 so that tan AOB ^= „° . — sr oo. Find the point G such 
 Ra-r -ti 
 
 that 
 
 0: CB::R^:R\ 
 
258 
 
 CIRCUITS CONTAINING 
 
 Draw D C perpendicular to O B, and complete the triangle 
 CD s,o that tan D G = -jr <^- Then Ea represents the 
 
 Fig. 76.— Problem XII. 
 
 impressed E. M. F. of the portion a of the circuit, and E, the 
 impressed E. M. F. of the remaining portion. 
 
 tan AD E =^,w. 
 
 Now let us take the system as originally shown in Fig. 
 73, and replace by a simple circuit with resistance R" and 
 self-induction L" that part of the system to which it is 
 equivalent. The system then reduces to the form showw 
 in Fig. 75. The construction of Fig. 76 is continued as be- 
 fore. 
 
 On DA, which represents E^ , the E. M. F. impressed at 
 the terminals of the two parallel circuits, draw the right 
 triangle D F A so that 
 
 Divide Z> J' at (? so that 
 
 DG: GF::B.'.R". 
 
RESISTANCE AND SELF INDUCTION. 259 
 
 Construct the right triangle HD O so that 
 
 tan HD G = ^co. 
 
 -Clb 
 
 Then J5|, is the E. M. F. impressed in the portion 6 of the 
 circuit, and U^ that impressed on the part of the circuit 
 beyond b. 
 
 Repeated applications of this method of construction 
 finally give the complete solution of the problem, and we 
 have ^j, ^j, ^5*3, etc., as the E. M. F.'s impressed on the 
 circuits 1, 2, 3, etc. ; and Ea, E^,, E^, etc., as the E. M. F.'s 
 impressed on the portions a, b, c, etc. 
 
 Knowing the impressed E. M. F. on any simple portion 
 of the circuit, a triangle of E. M. F.'s can be drawn and the 
 current obtained. The E. M. F. triangles on^, ^6, and Eg 
 are already drawn and the effective E. M. F.'s, Ha la . -^s-^s i 
 Rclcj found. The current is found by dividing the effec- 
 tive E. M. F. by the resistance. In a similar way the cur- 
 rent in each of the branch circuits 1, 2, 3, etc., may be 
 found. For instance, on E^ the E. M. F. triangle LN is 
 drawn. The effective E. M. F, LN, divided by the resist- 
 ance gives the current, I,. 
 
 The solution of this problem by entirely graphical 
 methods could be gone through with, as in some of the 
 previous problems, and likewise the problem of the same 
 arrangement of circuits with the current in some portion of 
 the circuit given. 
 
CHAPTER XVn. 
 
 PROBLEMS WITH CIRCUITS CONTAINING RESISTANCE AND 
 
 SELF INDUCTION. MORE THAN ONE SOURCE OF 
 
 ELECTROMOTIVE FORCE. 
 
 Prob. XIII. Electromotive Forces in Series. 
 
 Prob. XIV. Direction of Rotation of B. M. F. Vectors. 
 
 Prob. XV. Electromotive Forces in Parallel. 
 
 Prob. XVI. Electromotive Forces having Different Periods. 
 
 Problem XIII. Electromotive Forces in Series. 
 
 Suppose that in diiferent parts of a single circuit there 
 are two sources of harmonic E. M. F. It is required to 
 find the current which flows and the various falls of poten- 
 tial in the different parts of the circuit. 
 
 ~T6ii — rrr 
 
 E,( O ) ( O )E, 
 
 RflLa 
 
 __JL2JL__ 
 
 E D 
 
 Fig. 77.— Problem XIII. 
 
 Let the circuit be that represented in Fig. 77, where JE^ 
 and £^, are two different sources of harmonic E. M. F of the 
 same period. Draw the lines OA and CTE, Fig. 78, to 
 represent the E. M. F.'s ^, and JS, respectively. 
 
 The total E. M. F. acting in the circuit is the geometric 
 sum of 6* ^ and B, that is, the diagonal C (see page 213). 
 
 260 
 
RESISTANCE AND SELF INDUCTION. 
 
 261 
 
 Kegarding O a.s the impressed E. M. F. in a single circuit, 
 whose resistance is 2 B, and self-induction 2 L, we may 
 
 Fig. 78.— Pkoblbm XIII. 
 
 construct the triangle of E. M. F.'s and thus find the current. 
 
 2 L 00 
 
 Make the angle GOD equal to tan"' -^~o~' 
 
 Then OD 
 
 equals 12 B, and I) O equals Ioo2 L. Dividing D by 
 2 B, we obtain the current I,ox E. To obtain the various 
 falls of potential between the points AB, B C, and ED 
 (Fig. 77), divide D at F and O into parts proportional to 
 H^, B,, and B, , and D Ca,t H and /into parts proportional 
 to Z, , Z, , and L,. This determines the points J and ^and 
 thus gives the falls of potential J, J K, and K C for eacn 
 part of the circuit. 
 
 Problem XIV. Direction of Kotation of E. M. F. Vectors. 
 
 When two harmonic E. M. F.'s of the same period are 
 connected in series, the question may arise whether it 
 may not happen that the vectors representing the two 
 E. M. F.'s revolve in opposite directions. It is evident that, 
 if they should revolve in opposite directions, the resultant 
 
262 
 
 CIRCUITS CONTAINING 
 
 at any instant, instead of lying on a circle, lies upon an 
 ellipse (Fig. 79). Here 0£ is an E. M. F. vector reTolving 
 
 Tig. 79.— Problem XIV. 
 
 counter-clockwise, and A one revolving with the same 
 angular velocity in the opposite direction. The resultant 
 O must always lie upon the ellipse. The major axis has 
 a fixed direction OD which bisects the angle between OA 
 and B. The magnitudes of the semi-major and the semi- 
 minor axes are equal, respectively, to the arithmetical sum 
 and the arithmetical difference of the vectors A and B. 
 If, instead of drawing A in the direction indicated, 
 we had drawn it in the position O (making the angle 
 O C^jH" equal to A OH), and caused it to revolve counter- 
 clockwise in the same direction as B, the projections, 
 H, oi A ox O O would be the same at every moment. 
 Consequently the vector G revolving counter-clockwise 
 represents the same E. M. F. at every moment as the vector 
 A revolving clockwise, and may therefore be substituted 
 
BB8I8TANCS AND SELF INBUCTION. 263 
 
 for it. But the resultant of (FG and WB gives Ol, whose 
 locus is a circle. Thus the projection of ^is the same 
 as the projection of 00, and the ellipse may therefore be 
 replaced by the circle. 
 
 It is never necessary, therefore, to consider vectors re- 
 volving in opposite directions, for a vector revolving in one 
 direction can always be replaced by a vector revolving in 
 the opposite direction. 
 
 Problem XV. Electromotive Forces in Parallel. 
 
 Suppose that in each branch of a divided circuit, such 
 as that represented in Fig. 80, there is a source of har- 
 monic E. M. F., and that all these E. M. F.'s have the same 
 period ; it is required to find the currents in the branches. 
 
 The currents may be found by making use of the gen- 
 eral principle* that, if the currents due to each E. M. F. 
 acting separately can be found, the current which flows 
 when all the E. M. F.'s are acting together is the geometri- 
 cal sum of all these partial currents. 
 
 nmfr^-'-0n 
 
 R« L, 
 
 Fig. 80.— Problem XV. 
 
 To find the currents due to all the E. M. F.'s acting to- 
 gether we may then proceed by regarding each branch, 1, 
 2, and 3, in turn, as the main line in which there is the im- 
 pressed E. M. F., and the other branches as a divided circuit. 
 
 * See Mascart and Joubert's Electricity and Magnetism, Vol. 1, Art. 202. 
 
264 CIRCUITS CONTAINING 
 
 Then, considering E^ to be the only E. M. F. acting, the 
 problem of finding the partial currents //, 7,', and /,' is 
 readily solved by the methods already given. Next, con- 
 sidering E, as acting alone, we may find the partial currents 
 7/', /,", and I,", and finally we find //", //", and /,'", due 
 to E, acting alone. 
 
 The actual currents in the branches T^,I,, and I, when 
 all the E. M. F.'s act together, by the principle just stated, 
 must be equal to the geometrical sum of the partial cur- 
 rents ; that is, 
 
 I^ = geometrical sum of I/, //, and /,' ; 
 
 I,= " " " //','/;', and//'; 
 
 I,= " " « J/", //", and //". 
 
 Problem XVI. Electromotive Forces Having Different 
 
 Periods. 
 
 Let there be two impressed harmonic E. M. F.'s in 
 series having periods which bear a ratio of three to one. 
 It is required to find the resultant impressed E. M. E. and 
 the current that flows in the circuit. 
 
 In Fig. 81 let A represent maximum value of E^ , and 
 B that of E^ , they being in the ratio of one to two. As 
 O A revolves around its circle three times as fast as B, 
 A arrives B,i C when B arrives at D, and the re- 
 sultant F traverses the curve EF G U. If the projection 
 of the resultant vector O Fi^ taken upon the axis Y ?Li 
 equal intervals of time, we may thus plot the curve of re- 
 sultant E. M. F., Fig. 82. This E. M. F. curve is the plot 
 of the equation 
 
 e-= E^ sin ^ oot -{- E, sin go t. 
 
 The curve is composed of two simple harmonic components. 
 
 To find the current which this resultant E. M. F. causes 
 
 to flow, it is only necessary to find the currents which each 
 
RESISTANCE AND SELF INDUCTION. 
 
 265 
 
 component E. M. P. acting separately would cause, and then 
 add these together geometrically. If there is self-induction 
 in the circuit, the tangent of the angle of lag of the com- 
 
 _Resulfa, 
 
 Fig. 81.— Problem XVI. 
 
 ponent currents behind their respective E. M. F.'s is —p-. 
 
 Let OPBhe the E. M. F. triangle upon F, , and 'OJ the 
 current Z, . J must lie upon the semi-circle JM, whose 
 
 diameter is ^^-^ (see Problem I.). The angle of lag, A Q, 
 
 of the component current due to the E. M. P., E^ , is now 
 determined, since its tangent is three times the tangent of 
 
 BOP, thus. 
 
 L 00, % LCfO, 
 
 R 
 
 R 
 
 -". Also the current K,ov I,, 
 
 due to the E. M. P. A, or JE^ , is now determined, since 
 ^must lie upon a semi-circle K N whose diameter ON 
 
 equals \ oi M. 'For O N ■- 
 
 h 0,1, 3 ^ 
 
 and 0M = 
 
 <», 
 
266 
 E. 
 
 RBHISTANCE AlfD SELF INDUCTIONS. 
 
 2K 
 
 T^- = V=^, and thus OM ^ 6 ON. The resultant of OK 
 1, 00, -L 00, 
 
 and O J gives O L. and this vector always follows the curve 
 
 marked " Eesultant Current." If the projection oiOL upon 
 
 the axis OY is taken at regular intervals as JL moves 
 
 around its curve, we may obtain the current curve Fig. 82. 
 
 Fig. 82.— Problem XVI. 
 
 This current curve is composed of two simple harmonic 
 curves each due to a simple harmonic E. M. F., but the two 
 component current curves lag behind their respective com- 
 ponent E. M. F. curves by different angles. For this reason 
 the resultant current curve is not symmetrical with the re- 
 sultant E. M. F. curve. 
 
CHAPTER XYin. 
 
 INTRODUCTORY TO CIRCUITS CONTAINING RESISTANCE 
 AND CAPACITY. 
 
 Contents:— Problems with B and G analytically and graphically analo- 
 gous to problems with S and L. Triangle of E. M. F.'s for a single 
 circuit containing resistance and capacity. Impressed E. M. F. 
 Effective E. M. P. Condenser E. M. P. Direction shown from differ- 
 ential equations. Graphical representation. Two methods used. 
 First method (the one used throughout this book), employing E. M. P. 
 necessary to overcome that of condenser. Second method, employing 
 E. M. P. of condenser. Further identification of analytical and 
 graphical relations. Mechanical analogue. 
 
 When Chapter III., giving the analytical treatment of 
 circuits containing resistance and self-induction, is com; 
 pared with Chapter V., which gives the corresponding 
 analytical treatment of circuits containing resistance and 
 capacity, the similarity leads us to infer tluit the graphical 
 solutions of problems will be v^rjjEinaLogous in the two 
 cases. Although the analogy is very close, which fact 
 makes it much easier to follow the solutions for resistance 
 and capacity and is a great help, yet, in many respects, the 
 contrast is so marked that it is considered advisable, in 
 discussing problems with circuits containing resistance 
 and capacity, to give the solutions for the same arrange- 
 ment of circuits as those which have been given for circuits 
 containing resistance and self-induction in the previous 
 pages, in order that the points of similarity and difference 
 may be clearly understood. 
 
 267 
 
268 CIRCUITS CONTAININO 
 
 Teiangle of Electbomotive Foeces for a Single Circuit 
 Containing Eesistance and Capacity. 
 
 In Chapter V., in whicli circuits containing resistance 
 and capacity were analytically treated, it was shown [equa- 
 tion (78)] that when the impressed E. M. F. is harmonic, 
 that is, 
 
 e = ^sin oat, 
 
 the resulting current which flows is also harmonic and is 
 
 (78) 
 
 E r 1 "1 
 
 The charge of the condenser is likewise harmonic and 
 is [equation (79)] 
 
 E 
 
 9 = 
 
 V 
 
 R'-\- 
 
 =^ sin <» i 4- tan~* -/m 90° . 
 
 1 L. (J ±i GO J 
 
 Coa' 
 
 These equations for the current and charge were de- 
 rived from the differential equation of electromotive forces 
 which may be written in any of the forms [see (55)] 
 
 fidt 
 
 Bi-\- 
 
 de ^di 
 
 G 
 
 q 
 
 °^ di = ^dt + G- 
 
 Here e is the instantaneous value of the impressed 
 E. M. F. of the source ; Ri is that part necessary to over- 
 
 1 idt 
 
 come the ohmic resistance; and ^^— ^ — or -^ the E. M. F. 
 necessary to overcome the E. M. F. of the condenser. 
 
RESISTANCE AND CAPACITY. 
 
 269 
 
 Let the vector, A, Fig. 83, represent the harmonic 
 impressed E. M. F. of the source. Then, by equation (78), 
 
 Fig. 83. — Triangle of Electromotive Forces. 
 
 First. Method — the One used thkoughout this Book — employing 
 
 E. M. F. to overcome that of the Condenser. 
 
 we see that the current must be represented by a vector, 
 
 0£, in advance of OAhj an angle 6, or tan"' „ „ . 
 
 The effective E. M. E., being equal to BI, has the same di- 
 rection as the current and must be represented by a vector 
 C equal to the current vector, OB, multiplied by E. 
 The E. M. E. to overcome that of the condenser, having the 
 
 instantaneous value -^ , is at right angles to the current, 
 
 and must therefore be represented by the vector C A per- 
 
 <1 
 
 pendicular to O B. 
 
 It may be shown that this E. M. E. jf is at right angles 
 to the current by the preceding equations, thus : 
 
 /*• 
 
 dt 
 
 O 
 
 O 
 
 E_ 
 
 i?' + 
 
 Cw' 
 
 sm 
 
 [c.^ + tan-^^-90'']- 
 
270 CIRCUITS CONTAININO 
 
 To simplify this expression substitute 
 
 y/^« 
 
 and 6 = tan"^ ^ „ • 
 
 Vjli 00 
 
 The equation then becomes 
 
 (354) |. = -^sin[c»f+^-90°]. 
 
 This equation shows that the E. M. F., -~, to overcome 
 
 that of the condenser is ninety degrees behind the current, 
 
 and that the maximum value of this E. M. F. is -p^- , 
 
 Coo 
 
 The vector (Fig. 83), whose length is 77- , ninety degrees 
 
 G CO 
 
 behind the current, 0£, therefore represents the E. M. F. 
 
 to overcome that of the condenser. 
 
 The E. M. F. 0/ the condenser is equal and opposite to 
 
 that which is necessary to overcome it, and is consequently 
 
 ninety degrees in advance of the current represented by the 
 
 vector, AC, Fig. 84. 
 
 The Method to be Used in the Gbaphioal SoLtrTiONS op 
 / Peoblems foe Cieoxjits Containing Eesistanoe and 
 Capacity. 
 
 In the graphical treatment of problems with circuits 
 containing resistance and capacity, just as was the case with 
 circuits containing resistance and self-induction, there are 
 two methods of drawing, each equally correct, which will, 
 if followed throughout, give identically the same results. 
 These two methods arise according to whether the E. M. F. 
 necessary to overcome the E. M. F. of the condenser is con- 
 
BESIBTANGE AND GAPACITT. 
 
 271 
 
 sidered, or the E. M. F. of the condenser. The first method 
 is illustrated by Fig. 83 ; the second by Fig. 84. 
 
 In order that uniformity may exist throughout all the 
 diagrams which represent cases where both self-induction 
 and capacity are considered in circuit, since the method of 
 
 Fig. 84. — Triakgle of Electromotive Forces. 
 Second Method, emploting E. M. P. of Condenser. 
 
 drawing was adopted which considered the E. M. F. neces- 
 sary to overcome the self-induction, here we are obliged to 
 adopt that method which employs the E. M. F. necessary 
 to overcome the E. M. F. of the condenser, as represented in 
 Fig. 83. 
 
 That the construction of the figures fulfils the condi- 
 tions expressed by the current equation (78) may be shown 
 again by a further comparison of the relations. Thus in 
 Fig. 83 or 84 it is evident that 
 
 I 
 
 AC Ceo 1 , ,, 
 
 = tan u. 
 
 ia,nAOC = 
 
 OC RI ~ CBoo 
 
 and this corresponds to the angle of advance. Again, the 
 impressed E. M. F., O A, being the hypotenuse of the 
 triangle A C, is equal to the square root of the sum of 
 the squares of the two sides, and, therefore, 
 
 0A = j/oc'-{- ca': 
 
272 
 
 CIRCUITS CONTAININa 
 
 that is, E 
 and /: 
 
 =\/^'^^+^=V^'+^' 
 
 E 
 
 y^^+ir^^- 
 
 This is seen to correspond to the maximum value of the 
 current given in equation (78). 
 
 Mechanical Analogue. 
 
 That the E. M. F. of the condenser is at right angles tc 
 the current may, perhaps, be best understood by the phy- 
 sical conception of the part played by the condenser in a 
 circuit. A good mechanical analogue of the condenser is 
 an air-chamber, as represented in Fig. 85, in which the air 
 is first compressed and then expanded. 
 The piston P moves back and forth, with 
 an harmonic motion, we will say, first 
 compressing and then expanding the 
 air in the chamber. When at its central 
 position, the air is at the atmospheric 
 pressure. The current may be repre- 
 sented by the motion of the piston, or 
 of the water in the tube which trans- 
 mits the pressure to the air-chamber. 
 The charge of the condenser may be 
 represented by the volume of water 
 which enters or leaves the air-chamber, 
 the charge being taken as zero when 
 the piston is at its central position, that 
 is, when the air is at the atmospheric 
 pressure. Considering the moment 
 when the piston is in the central posi- 
 tion P, moving upward, the charge is 
 zero, and the current is a maximum, as 
 here the piston moves with its maximum velocity. The cor- 
 
 Fra. 85.— Mechanical 
 Abtalogue of a Con- 
 
 DENSEB. 
 
RESISTANCE AND CAPACITY. 273 
 
 responding points on the curves, Fig. 84, are Hand K; that 
 is, the positive current is represented by the upward motion 
 of the piston. When the piston arrives at Q, the upper end 
 of the stroke, the current is zero and is represented by the 
 point N on the curve. The charge is here a positive max- 
 imum, and during the previous quarter of the stroke the 
 compressed air has exerted an outward pressure, cor- 
 responding to 'the E. M, F. of the condenser, opposed to 
 the . current. This pressure reaches a negative maximum, 
 together with the charge, when the current is zero. This 
 corresponds to the point M on the curve. During the re- 
 turn of the piston to the central position, both the current 
 and the pressure are in the same negative direction until 
 the current becomes a negative maximum, at the central 
 position, where the pressure becomes zero and then changes 
 sign. This example shows how the pressure exerted by 
 the air, corresponding to the E. M. F. of the condenser, is 
 just ninety degrees in advance of the current. The pres- 
 sure which must be exerted upon the piston to overcome the 
 pressure of the air chamber, corresponding to the E. M. F. 
 necessary to overcome that of the condenser, is evidently 
 equal and opposite to the pressure of the air-chamber, and 
 lags, therefore, ninety degrees behind the current. As be- 
 fore explained. Fig. 83 represents the manner of drawing 
 when the E. M. F. necessary to overcome that of the con- 
 denser is considered, and Fig. 84 when the E. M. F. of the 
 condenser is considered. 
 
CHAPTER XIX. 
 
 PROBLEMS WITH CIRCUITS CONTAINING RESISTANCE AND 
 
 CAPACITY. 
 
 Prob. XVII. Effects of the Variation of the Constants JB and C in a 
 
 Series Circuit. iJ varied. C7 varied. 
 Prob. XVIII. Series Circuit. Current given. Equivalent B and in 
 
 Series. 
 Prob. XIX. Series Circuit. Impressed E. M. F. given. 
 Prob. XX. Divided Circuit. Two Branches. Impressed E. M. P. 
 
 given. Equivalent B and C for Parallel Circuit. 
 Prob. XXI. Divided Circuit. Any Number of Branches. Impressed 
 
 E. M. P. given. Equivalent B and G obtained for 
 
 Parallel Circuits. 
 Prob. XXII. Divided Circuit. Current given. First Method: Entirely 
 
 Graphical. Second Method: Solution by Equivalent .B 
 
 and C. 
 Prob. XXIII. Effects of the Variation of the Constants B and C in a 
 
 Divided Circuit of Two Branches. 
 Prob. XXIV. Series and Parallel Circuits. Impressed E. M. P. given. 
 
 Solution by Equivalent B and C 
 Prob. XXV. Series and Parallel Circuits. Current given. Solution by 
 
 Equivalent B and C. 
 Prob. XXVI. Series and Parallel Circuits. Entirely Graphical Solution. 
 Prob. XXVII. Multiple-arc Arrangement. 
 
 Problem XVII. Eflfects of the Variation of the Constants 
 JS and C In a Series Circuit. 
 
 The Eesistance Varied. 
 
 When the olimic resistance is varied in a circuit con- 
 taining only resistance and capacity, the current is changed 
 
 274 
 
BESISTANGE AND OAPAOITT. 
 
 275 
 
 and it is of interest to investigate just how it changes 
 both in magnitude and in direction. The triangle AG, 
 Pig. 86, represents the triangle of E. M. F.'s for the circuit 
 
 CEU 
 
 Fig. 86.— Variation of Resistance and Capacity in a Sbkies 
 Circuit. Problem XVII. 
 
 when the resistance is R. The current B is equal to G 
 divided by H. Draw a line D, of indefinite length, perpen- 
 dicular to the E. M. F. A in the direction of advance. 
 The angle D O Gis the complement oi AOG, and is, there- 
 fore, tan" C B CO. Draw j& ^E" perpendicular to OB and let 
 it meet D at JE. The line ^^then equals CB co I; for, 
 OB equals I, and tan E B equals GBoa. 
 
 It can be shown that the hypotenuse OE oi this triangle 
 is equal to C Eoa, and is therefore a constant entirely inde- 
 pendent of any variation in the current /, or resistance R. 
 Taking the square root of the sum of the squares of the 
 sides OB and B E, we obtain 
 
 'OE=^/OB'-\-BE' = Iyf\-\-G'B 
 
 E 
 
 Substituting for / its value 
 
 /. 
 
 , we obtain 
 
 i?' + 
 
 (?"(»' 
 
 /4/l+^'i?"c»' = CEoo, 
 
 and, therefore, 
 
 0E= GEoa. 
 
276 CIRCUITS CONTAINING 
 
 Now since the side OB oi the right triangle B E always 
 represents the current I, and the hypotenuse E is inde- 
 pendent of current or resistance, it follows that the current 
 is always represented by a vector B inscribed in a semi- 
 circle OB E for any possible variation in the resistance. 
 In the figure the arrow indicates the direction of variation 
 as the resistance increases. 
 
 In the limiting cases when R is infinite or zero, we see 
 by this figure the limiting values of the current. When R 
 is infinite, the current is evidently zero. When li ap- 
 proaches zero, OB approaches E, and in the limit the 
 current becomes 
 
 1= CEoo. 
 
 When the circuit contains no ohmic resistance, we see, first, 
 
 that the impressed E. M. F. is equal to -^— , the E. M. F. 
 
 of the condenser; and, second, that the current is 90° in 
 advance of the impressed E. M. F. These relations, here 
 geometrically shown, are analytically expressed in equa- 
 tion (354). 
 
 The Capacity of the Condenseb Vaeied. 
 
 Suppose that the capacity of the condenser in the cir- 
 cuit is varied while the resistance remains the same ; we 
 wish to find how the current changes. 
 
 In the same figure, 86, prolong the line E B until it meets 
 the impressed B. M. F. 'OA prolonged at F. Then BF 
 
 equals -/^Ti — ■, since tan B F equals y^n — . 
 The hypotenuse E is, therefore, 
 
BE8I8TANGE AND CAPAOITT. 277 
 
 From the value for / in (82) it follows that 
 
 V'+ 
 
 1 ^E_ 
 
 — W 
 Hence, F = ^. 
 
 It 
 
 Since the hypotenuse OF'x^ independent of the current 
 / or the capacity C, and is a constant for any variation in 
 C, it follows that the current is always represented by a 
 vector B inscribed in the semi-circle B F iox any pos- 
 sible value of the capacity. In the figure the arrow indi- 
 cates the direction of variation as the capacity increases. 
 
 In the limiting cases when G is zero or infinite, we see 
 from the figure the value of the current. When O ap- 
 proaches zero, the current evidently approaches zero. 
 When G approaches infinity (which is equivalent to having 
 no condenser in the circuit), the current vector OB ap- 
 
 . . . E 
 
 proaches OF, and, in the limit, Z= -^, and the current 
 
 follows Ohm's law. 
 
 That the construction of Fig. 86 is consistent with the 
 equations is further shown from the following relations. 
 
 (355) 
 
 EF'={EB-^rBFy = [GB<^I^-^^\ 
 
 -^^'^1+^'^''^'J- 
 
 (356) 'OE'-^ Wr = O' F' 00'+ ~ = JV -f G' E' c»'). 
 Equating (355) and (356), we find 
 
 ^-,{l + C'E'c^^) = F\ or /= 
 
 ^'gj' 
 
278 
 
 CIRGUIT8 CONTAINING 
 
 a result which is identical with that analytically expressed 
 in equation (82). 
 
 Problem XVIII. Series Circuit. Current Given. 
 
 Let there be a circuit, Fig. 87, having in series n differ- 
 ent resistances R^, B,, etc., and n condensers with ca- 
 pacities C, , C'j , etc. It is required to find the impressed 
 E. M. F. necessary to cause a current / to flow. In Fig. 
 88, make A equal to the current flowing. Multiply this 
 
 Figs. 87 and 88. — Problem XVIII. and Problem XIX. 
 
 by R^ , and lay oS OJS equal to ^, /, which is, then, the ef- 
 fective E. M. F. to overcome the resistance S^. Draw Ji C 
 perpendicular to -^ in the negative direction, and make 
 
 the angle B G=e, = tan" ^ _g ^ - Then B C is the 
 
 triangle of E. M. F.'s for that part of the circuit between A 
 and B, Fig. 87. 
 
 The construction of the figure is similar to that of Fig. 
 
RESISTANCE AND CAPACITY. 279 
 
 54, in Pboblem II., but differs from Fig. 54 in the fact that 
 the E. M. F. triangles in the present construction are so 
 drawn that the various currents are in advance of their re- 
 spective electromotive forces. The triangles C D E, etc., 
 are drawn and the construction completed similar to the 
 corresponding case, Problem II., of a series circuit with 
 self-induction. We thus find the impressed E. M. F, to 
 be OQ. 
 
 Equivalent Eesistance and Equivalent Capacity in Series. 
 
 Suppose that we replace all the resistances in Fig. 87 
 by a single resistance, and all the condensers by a single 
 one ; it is required to find that resistance and capacity which 
 will allow the same current to flow. 
 
 It is evident that if 'OK, Fig. 88, is B'l, and KG is 
 
 -, where B' and C represent, respectively, the equiva- 
 
 lent resistance and equivalent capacity, the same current 
 
 / 
 
 OA will flow. But OK = I 2 B, and KQ= -2-^. 
 
 CO V 
 
 It therefore follows that B' = 2 R, and ^, = 2-^. We 
 may write it C = — r and have the equivalent capacity 
 
 equal to the reciprocal of the sum of the reciprocals of each 
 separate capacity. 
 
 Problem XIX. Series Circuit. Impressed B. M. F. 
 
 Given. 
 
 The circuit being the same as in Fig. 87 in the previous 
 problem, it is required to find the current and the fall of po' 
 -tential through each of the various parts of the circuit when 
 the impressed E. M. F. is given. From the remarks on equiv- 
 alent resistance and capacity immediately preceding, it is 
 
280 CmCUITS CONTAINma 
 
 evident that the same current will flow if these equivalents 
 are substituted for the separate resistances and capacities. 
 The triangle K O may now be drawn and the current 
 found. From this point we may proceed as in the preced- 
 ing problem to find the various falls of potential C, Q E, 
 and J^. 
 
 Problem XX. Divided Circuit. Two Branches. 
 Impressed E. M. F. Given. 
 
 Let us consider the problem of a divided circuit having 
 two branches in parallel, as indicated in Fig. 89. Each 
 
 I. 
 Fig. 89.— Pkohlbm XX. 
 
 branch contains resistance and capacity, and there is an 
 impressed E. M. F., E, between the terminals Jf and N ; it 
 is required to find the main current, I, and the currents 7", 
 and /j in the branches. 
 
 This problem corresponds very closely to Pboblem IV., 
 in which case the branches contain self-induction instead 
 of capacity. Fig. 90 represents the solution of the present 
 problem, and corresponds to Fig. 56, Peoblem IV. The 
 difference is that the E. M. F. triangles OB A and G A, 
 Fig. 90, lie on the positive or advance side of the impressed 
 E. M. F. A, instead of on the negative, side as in Fig. 56. 
 Otherwise the construction by which we obtain the two 
 currents D and E, and the resultant main current, 
 OF, is identical with that in Peoblem IV. 
 
besistance and capacity. 
 Equivalent Eesistance and Capacity. 
 
 281 
 
 Suppose that, instead of the two parallel branches Just 
 considered, a single circuit be substituted for them whose 
 resistance, B', and capacity, C, is such that the same cur- 
 
 FiG. 90.— Problem XX. 
 
 rent as before will flow in the main line. The triangle of 
 E. M. F.'s for this equivalent circuit must be G A, Fig. 
 90, since the impressed E. M. F. is OA, and the effective 
 E. M. "F. is O G in the direction of the current, and the 
 E. M. F. G A, to overcome that of the condenser, is at right 
 angles to the current. We may write, therefore, B'l for 
 
 O G, and 777— for GA. This equivalent resistance and 
 
 capacity may be expressed in terms of the resistances and 
 capacities of the branches, but the determination of these 
 values will be deferred until after the discussion of the fol- 
 lowing problem. 
 
282 
 
 CmCUITS CONTAINING 
 
 Problem XXI. Divided Circuit. Any Number of 
 Branches. Impressed E. M. F. Given. 
 
 Let the divided circuit, M N, Fig. 91, have n branches 
 in parallel, each containing resistance and capacity, with an 
 impressed E. M. F., E, between the terminals. It is re- 
 quired to find the main current /. 
 
 AAM/— ^ 
 
 Fig. 91.— Pkoblem XXI. and Pkoblem XXII. 
 
 The construction of Fig. 92 is similar to that of Fig. 58, 
 in Problem V., except that the E. M. F. triangles and all 
 the branch currents are laid off in the direction of advance 
 and not of lag. The main current Lis the geometrical 
 resultant of all the branch currents OF, G, OH, and 
 01, as before. 
 
 Fig. 92. — Problem XXI. and Problem XXII. 
 This diagram gives the complete solution of the problem 
 of the divided circuit containing resistance and capacity. 
 Here, too, as was the case with the divided circuit contain- 
 
RESISTANCE AND GAPACITT. 283 
 
 ing resistance and self-induction, it is evident that the 
 maximum main current, /, is greater than any of the branch 
 circuits. 
 
 Equivalent Besistance and Capacity op Paballel Circuits. 
 
 In this case, as in the previous one, we may suppose a 
 single circuit substituted for the parallel branches, having 
 such a resistance, R' , and capacity, ', that the current in the 
 main line is not altered in magnitude or phase. The values 
 of this equivalent resistance and capacity in terms of the 
 resistances and capacities of the branches may be found by 
 proceeding in the same way as was done to obtain the 
 values of the equivalent resistance and self-induction of 
 parallel circuits, Pkoblem Y. Equations are formed by 
 taking the projections of the currents first upon the line 
 O A, Fig. 92, and then upon a line perpendicular to A. 
 In these equations, values for /, I,, I,, etc. ; cos 6, cos ^, , 
 cos B^ , etc. ; sin 6, sin 0, , sin 0^ , etc., obtained from the 
 geometry of the figure, are substituted, and, after opera- 
 tions similar to those used in obtaining equivalent resist- 
 ance and self-induction, the following expressions are ob- 
 tained for the equivalent resistance and capacity of parallel 
 circuits. 
 
 (356 a) B' = j^» -\. B" QS" 
 
 1 Boo 
 
 (356 h) and -^j^ = j^j^r^^ , 
 
 ^ B 
 
 where A = ^^> j^j 
 
 and ^'»=^ f- =^C"i2't»" + l" 
 
284 CIRCUITS CONTAINING 
 
 The main current is in advance of the impressed E. M. F. 
 by an angle d such that 
 
 B CO 
 tan 6 = — T-. 
 
 The complete proof of this was first given by the authors 
 in the Philosophical Magazine for September, 1892. These 
 results may be obtained from the general expressions 
 for equivalent resistance, self-induction, and capacity which 
 are discussed in Problem XXXI. 
 
 Problem XXII. Divided Circuit. Current Given. 
 
 Suppose that we have a number of circuits, each with 
 resistance and capacity, connected in parallel as in Fig. 91, 
 and we know the value of the current I in the main line. 
 We wish to find the current in the several branches. There 
 are two solutions similar to the two given for the corre- 
 sponding case of circuits with self-induction. 
 
 FiEST Method. Entirely Graphical. 
 
 By assuming any value we please for the impressed 
 E. M. F., ^, we may. solve as in the foregoing problem. 
 The scale of the drawing must then be changed in the 
 ratio of the given value of the main current, /, to the value 
 of 1 thus obtained according to the assumed impressed 
 E. M. F. 
 
 Second Method. By Use op Equivalent Eesistance ane 
 
 Capacity. 
 
 The problem may be otherwise solved by use of equiva- 
 lent resistance and capacity of parallel circuits as given in 
 (356 a) and (356 6). OM, Fig. 92, is laid off equal to H'l. 
 
RESISTANCE AND CAPACITY. 
 
 285 
 
 The line ilf ^ is drawn perpendicular to M and equal to 
 
 7J7-. The hypotenuse OA is the impressed E. M. F. 
 
 The further construction is the same as in the foregoing 
 problem. Upon OA the E. M. E. triangle for each branch 
 is drawn and the current and angle of advance found. 
 
 Problem XXIII. Effects of the Variation of the Constants 
 M and C in a Divided Circuit of Two Branches. 
 
 If we compare Peobuems I. and XVII., in which the dis- 
 cussion is given of the effects of the variation of the con- 
 stants Ji and L, and H and G in series circuits, we see that 
 
 Fig. 93. — Variation of Resistance and Capacity in a Divided 
 Circuit. Problem XXIII. 
 
 the two problems are similar, and that the constructions in 
 Figs. 52 and 86 are the same except for direction, the 
 
286 
 
 CIRCUITS CONTAINING 
 
 former being in the direction of lag and the latter in the 
 direction of advance. The present problem is similar to 
 Pkoblem YII., in which the effect of the variation of R and 
 Z in a divided circuit is considered. The construction is 
 given in Fig. 93, which explains itself, and is exactly- 
 similar to that given in Fig. 59, which was fully described 
 in Pkoblem VII. The arrows in the figure indicate the 
 direction of the change as the resistance or capacity in- 
 creases. 
 
 Problem XXIV. Series and Parallel Circuits. Impressed 
 E. M. F. Given. Solution by Use of Equivalent Kesist- 
 ance and Capacity. 
 
 Problems arising from combinations of series and 
 parallel circuits with resistance and capacity are solved by 
 the repeated application of the methods used for the fore- 
 going simple problems in the same way as were solved the 
 problems involving combinations of circuits with resistance 
 and self-induction. Let us consider the case in which two 
 systems of parallel circuits are joined in series, as in Fig. 
 94. The resistance and capacity of each branch and the 
 
 Eb 
 
 RoCa RjCj 
 
 Fig. 94.— Problem XXIV. 
 
 total impressed E. M. F. are given. It is required to find the 
 current in the main line and branches. The problem is 
 similar to Pkoblem VIII., and the solution given in Fig. 95 
 
BE8I8TANCE AND CAPACITY. 
 
 287 
 
 is obtained by a construction similar to Fig. 63. The 
 equivalent resistance and capacity Ma and Od between M 
 and N, and Bb and 0^' between N and 0, are calculated ac- 
 cording to (356 a) and (356 &). The impressed E. M. F.'s Ea 
 and Eb are now found according to the method for series 
 circuits, Peoblem XIX. The part between M and N and 
 
 Fig. 95.— Problem XXIV. 
 
 the part between N and are now separately treated by 
 the method of parallel circuits, Peoblem XXI. The con- 
 struction is shown clearly by the figure. 
 
 A more extended system of circuits in series and parallel 
 is solved by the same methods. 
 
 Problem XXV. Series and Parallel Circuits. Current 
 Given. Solution by Use of ^Equivalent Resistance 
 and Capacity. 
 
 Let us suppose the same arrangement of circuits as that, 
 shown in Fig. 94, and that the main current, I, is given. 
 It is required to find the current in each branch. The 
 parts between M and N and between N and may be 
 separately solved according to the second method given in 
 Peoblem XXII. The solution of any number of circuits in 
 series and parallel could be readily obtained by the same 
 method. 
 
288 
 
 CIRCUITS CONTAINING 
 
 Problem XXVI. 
 
 Series and Parallel Circuits. Entirely 
 Graphical Solution. 
 
 In tlie foregoing treatment of problems involving series 
 and parallel combinations of circuits containing resistance 
 and capacity it was necessary to find analytically the 
 values of the equivalent resistance and capacity of each set 
 of parallel circuits, and the solutions were, therefore, partly 
 analytical and partly graphical. They may be obtained, as 
 in the corresponding cases of combinations of circuits with 
 resistance and self-induction (see Peoblem XI.), by entirely 
 graphical methods by assuming the value of the current in 
 a particular branch or assuming its impressed E. M. F. 
 After solving in this way, the values assumed and the scale 
 of the diagrams must be altered to agree with the given con- 
 ditions of the problem. Figs. 96, 97, 98, and 99 give the 
 
 Figs. 96 and 97.— Problem XXVI. 
 
 construction for the entirely graphical solution of two par- 
 allel sets of circuits connected in series, as in Fig. 94. The 
 method is to solve separately each parallel set of circuits 
 by assuming some value for the impressed E. M. F. or for 
 one of the branch currents. Figs. 96 and 97 give the con- 
 struction for the solutions of the parts M N and NO, re- 
 spectively, starting with assumed values for the branch 
 currents ij and I, . Fig. 97 is then magnified, as shown in 
 
ME8I8TANGE AND OAPAOITT. 
 
 289 
 
 Fig. 98, until the main current / is the same size as in Fig. 
 96. The two figures, 96 and 98, are now combined in Fig. 99 
 so that i^ is parallel to 0' F', since each represents the 
 
 Fig. 98 —Problem XXVI. 
 
 current L We thus find E, the impressed E. M. F. which 
 will cause the current / to flow. If the value of the im- 
 pressed E. M. F. had been given, the scale of the diagram 
 
 Fig. 99.— PKOBiiBM XXVI. 
 
 could be altered until E equaled the given value of the 
 E. M. F. The figure would then give the value of the main 
 and branch currents which flow when there is this given 
 E. M. F, 
 
 Problem XXVII. Multiple-arc Arrangement. 
 
 Of the many arrangements in which circuits with resist- 
 ance and capacity may be combined, let us consider, as a 
 
290 CIBCUITS OONTAINING 
 
 further example, the arrangement in multiple arc, as shown 
 in Fig. 100. The solution is obtained by dividing the 
 
 -R-O- 
 
 -R-C- 
 
 FiG. 100.— Pboblem XXVII. 
 
 system into different parts and successively applying the 
 foregoing solutions for series and for parallel circuits. This 
 
 % 
 
 
 Co 
 Fig. 101.— Pboblem XXVII, 
 
 -MfffH^f-M/A4ri^3-» 
 
 problem and its solution are similar to Pboblem XII., and 
 it will, therefore, suffice to merely outline the method to be 
 
 Ra 0, Rj Cj 
 
 Fi&. 103.— Problkm XXVII. 
 
 followed. The circuits 1, 2, 3, etc., have resistances and 
 capacities R,,B,, B,, etc., and 0„ C,,0,, etc. The resist- 
 ance and capacities of the mains are Ba and Oa for the por- 
 tion a ; Bb and Gb for the portion h between circuits 1 and 
 2 ; Be and Co for the portion 0; etc. B' and G' are the 
 equivalent resistance and capacity for circuit 1 and the part 
 of the system beyond, as indicated in Fig. 101. B" and 
 G" are the equivalent resistance and capacity for circuit 
 
RESISTANCE AND GAPAOITT. 
 
 291 
 
 2 and the part of the system beyond, as indicated in Fig. 
 102. Similarly, B'", B"", C", G"" have values as indi- 
 cated. The values for the equivalent resistances and ca- 
 pacities are found by the successive applications of the 
 formulae (356 a) and (356 6). The complete solution is given 
 in Fig. 103, and its construction is similar throughout to 
 
 Fig. 103.— Pboblbm XXVII. 
 
 that of Fig. 76, Peoblbm XII. E,, E^, E,, etc., give the 
 impressed E. M. F.'s of the several parallel branches. By 
 erecting an E. M. F. triangle on each, the effective E. M. F. 
 and so the current in each branch may be found in the 
 usual way. Thus in branch B,LNis the effective E. M. F., 
 and I, the current. Ea, Eb, Ec, etc., give the imipressed 
 E. M, F.'s in the portions a, h, c, etc., respectively, and the 
 currents are easily found from the effective E. M. F.'s 
 Ba la , Rbibi He Ic , etc. The full construction can best 
 be followed by comparing Peoblem XII., the similar case 
 of circuits with resistance and self-induction. 
 
CHAPTER XX. 
 
 CIRCUITS CONTAINING RESISTANCE, SELF-INDUCTION, AND 
 
 CAPACITY. 
 
 Contents : — Introductory. Graphical methods for circuits with S, L, and 
 G based upon graphical methods for circuits with B and L, and B 
 and 0. Diagram of four E. M. P.'s. Triangle of E. M. P.'s. Method 
 consistent with analytical results obtained for circuits with B, L, and 
 G. Capacity or self-induction which is equivalent to a combination of 
 capacity and self-induction. 
 Prob. XXVIII. EfEects of the Variation of the Constants in Series Circuit. 
 
 B, L, C, and ca varied. 
 Prob. XXIX. Series Circuit. Current given. Equivalent B, L, and G 
 
 of Series Circuit. 
 Prob. XXX. Series Circuit. Impressed E. M. P. given. 
 Prob. XXXI. Divided Circuit. Impressed E. M. P. given. Equivalent 
 
 jB, L, and G of Parallel Circuits. 
 Prob. XXXII. Example of a Divided Circuit. Impressed E. M. P. 
 
 given. 
 Prob. XXXIII. Divided Circuit. Current given. 
 Prob. XXXIV. Series and Parallel Combinations of Circuits. 
 
 In the foregoing chapters the complete graphical solu- 
 tions have been given for any combination of circuits in 
 series and parallel when the circuits contain resistance and 
 self-induction or when they contain resistance and capacity. 
 In the first, the impressed E. M. F. of the source is equal 
 to the E. M. E. necessary to overcome resistance plus the 
 E. M. E. necessary to overcome the counter E. M. F. of 
 self-induction ; in the second, the impressed E. M. E. is 
 
 292 
 
BE8I8TANGE, SELF-INDUCTION, AND CAPACITY. 293 
 
 equal to tjie E. M. F. necessary to overcome the resistance 
 plus the E. M. F. necessary to overcome that of the con- 
 denser. In each of these cases the three E. M. F.'s were 
 represented by the three sides of a triangle. 
 
 Where a circuit contains resistance, self-induction, and 
 capacity there are four E. M. F.'s to be considered. The 
 impressed E. M. F. is equal to the sum of the E. M. F.'s 
 necessary to overcome the resistance, the self-induction, 
 and the condenser E. M. F., respectively. 
 
 The E. M. F. to overcome resistance is B I; that to 
 overcome the self-induction is I, col and. is 90° ahead of 
 the current ; and that to overcome the E. M. F. of the con- 
 denser is -y^ — and is 90° behind the current. These may 
 
 O GO 
 
 be drawn as the lines A, AB, and B 0, respectively, in 
 Fig. 104, and the geometrical or vector sum C accord- 
 ingly represents the impressed E. M. F. 
 
 Fig. 104.— Diagkam of Elbctromotivb Forces ik a Circuit with 
 Resistance, Self-induction, and Capacity. 
 
 Now the E. M. F, to overcome that of self-induction and 
 that of the condenser are always in exactly opposite direc- 
 tions, and when combined give one E. M. F. at right angles 
 to the current. Thus, in Fig. 104, A C represents the com- 
 bined effect of the E. M. F.'s L oo Zand ^i — > represented by 
 
294 CIRCUITS CONTAINma 
 
 A B and B C, respectively, and is equal to ( - — — L oo\ J. 
 
 We may, therefore, represent the E. M. F.'sin a circuit con- 
 taining resistance, self-induction, and capacity by a triangle 
 whose sides represent, respectively, the impressed E. M. F., 
 that necessary to overcome resistance, and that necessary 
 to overcome the E. M. F. of self-induction and capacity 
 combined. Fig. 104 may then be drawn as Fig. 105, When 
 
 7^ is greater than L go, the current is ahead of the im- 
 
 (j GO 
 
 pressed E. M. F.; and when -~~ is less than Z co, the cur- 
 rent lags behind. 
 
 Fig. 105.— Triangle of Electkomotitb Fohces in a CiKctnT with 
 Resistance, Selp-induction, and Capacity. 
 
 The tangent of this angle of lag or advance is 
 
 \Ggo~ I _ 1 Lgo 
 
 tan - ^^ -^^-^ - -^. 
 
 When positive, the angle is one of advance ; when negative, 
 one of lag. Tan d = reactance -j- resistance. 
 
 The impressed E. M. F. O, being equal to the square 
 root of the sum of the squares of the two sides of the tri- 
 angle, is 
 
 E=I^E' + [1--Loo). 
 
BESI8TANCM, 8ELF-INBUGTI0N, AND CAPACITY. 295 
 
 But this radical is tte expression called the impediment 
 (see page 131), and we may therefore write 
 
 „ ^ E. M. F. 
 
 Current = 
 
 Impediment' 
 
 which corresponds to Ohm's law. 
 
 We have now established the graphical method of repre- 
 senting the E. M. F.'s in a simple circuit containing resist- 
 ance, self-induction, and capacity, basing it upon the 
 graphical solutions already given for circuits containing 
 resistance and self-induction, and circuits containing re- 
 sistance and capacity alone. These were separately ob- 
 tained from the analytical equations previously given. 
 
 Let us now compare these graphical methods with the 
 analytical results obtained in the discussion of circuits 
 containing resistance, self-induction, and capacity. The 
 general solution for current in a circuit with an harmonic 
 impressed E. M. F. is [see (181)] 
 
 This shows that the current has an angle of lag or advance 
 whose tangent is 
 
 Z 
 
 a> 
 
 GBoo B ' 
 
 the angle being advance when positive and lag when nega- 
 tive, which corresponds to the graphical construction just 
 given. The maximum value of the current is 
 
 E^ E 
 
 ~~ , / '. 7~~i V ~ Impediment' 
 
296 CIBCUITS CONTAINING 
 
 These equations, being identical with those just obtained 
 graphically, show that the analytical results are correctly 
 represented by this graphical method. 
 
 Capacity or Self-induction which is Equivalent to a Com- 
 bination OF Self-induction and Capacity. 
 
 Let C ox L' denote the capacity or self-induction which 
 is equivalent to a given combination of the two, that is, 
 which allows the same current to flow in the circuit when it 
 is substituted for the combination. Beferring to Fig. 105, we 
 
 see that the E. M. F. of the combination is -^ L oo I. 
 
 G GO 
 
 Eegarding this as a positive quantity, i.e., supposing 
 -p — > Zf t», we may put 
 
 (357) Hence ^'= l-=^_-^, 
 
 which is positive since 1 > Z C oo''. 
 
 If we suppose 7=— < L 00, then Loo — 7= — is positive. 
 
 V 00 U GO 
 
 We may then put 
 
 L' oal = Lool— -^ — ; or L'a) = Loo— -pq—. 
 O 00' Ggo 
 
 (358) Hence L' = L — ^-^, a positive quantity. 
 
 Problem XXVIII.— Effects of the Variation of the Con- 
 stants It, L, C, and 00. 
 
 Eesistance Varied. 
 If the resistance alone be varied in a circuit containing 
 self-induction and capacity, it is interesting to inquire how 
 
BESI8TANCE. SELF-INDUCTION, AND CAPACITY. 297 
 
 the current changes. Since the combination of the self- 
 induction and capacity is equivalent to a self-induction or 
 a capacity, we may substitute this equivalent for the com- 
 bination. The change in the current caused by any varia- 
 tion in the resistance must therefore be the same as that 
 before explained (Peoblems I. and XVII.) in the case of 
 self-induction or capacity alone. 
 
 Fig. 106.— Variation of Constants. Pkoblem XXVIII. 
 
 In Fig. 106, 00 represents the impressed E. M. F., E, 
 divided by the resistance R. The current 1 va&j either 
 advance ahead of or lag behind according to whether 
 
 Yi — is greater or less than Leo. For certain values of 
 
 resistance, self-induction, and capacity let OA represent 
 the current in advance of the impressed E. M. F., which 
 
 signifies that y, — > Lao. Make D equal to C Eoo, and 
 
 draw the semi-circle AD upon D as diameter. This 
 is then the locus of the current vector ^ as the resistance 
 alone changes, as explained in Problem XVII. 
 
 Similarly, if the quantity Ty— — Z go had been negative 
 
 and of an equal magnitude to its former positive value, the 
 current would have been represented hy OB lagging behind 
 O O, and any variation in the resistance alone would cause 
 the current vector to move upon the semi-circle B E, 
 being equal to ^ when the resistance is zero, as explained 
 
298 C1E0UIT8 OONTAININQ 
 
 in Problem I. It is to be noticed that Fig. 106 is the same 
 as Figs. 52 and 86 combined. 
 
 The arrows R, M, show the direction of change as the 
 resistance increases. 
 
 Self-induction ob Capacity Vaeied. 
 When either the self-induction or capacity alone is varied, 
 it is evident that the value of the quantity j^ -^ "^ ^'^d, 
 
 therefore, the value of the equivalent self-induction, Z ', 
 or equivalent capacity C, is changed. Now any varia- 
 tion in the equivalent self-induction will cause the cur- 
 rent vector to move on the semi-circle B C,2& explained 
 in Peoblem I., and any variation in the equivalent 
 capacity will cause the current vector to move on the 
 semi-circle A C, as explained in Peoblem XVII. Any 
 change, then, in self-induction or capacity will cause the 
 current to move through some part of the circle OA B, 
 
 Z* 
 
 whose diameter is O C equal to ^. 
 
 The arrow Z, C shows the direction of change as the 
 capacity or the self-induction increases. 
 
 Frequency Vaeied. 
 
 When the frequency of alternation is varied, it is equiv- 
 alent to a variation of go, the angular velocity, which is 
 equal to 2 tt times the frequency. Any increase in the fre- 
 quency increases the effect of the self-induction or the 
 capacity. If the self-induction is the more important ele- 
 ment and the circuit has an equivalent self-induction [see 
 equation (358)], 
 
RESISTANCE, SELF-INDUCTION, AND CAPACITY. 299 
 
 any variation in the frequency will cause a variation in the 
 equivalent self-induction according to this equation. If 
 the capacity is the more important element, the equivalent 
 capacity varies with oo according to the equation (357), 
 
 0'= ° 
 
 It has just been shown that any variation in the equivalent 
 self-induction or capacity causes the current vector to move, 
 between limits, on the circle A C B. This, then, is the 
 effect of a change in frequency. The direction of this 
 change, as the frequency increases, is shown by the arrow 
 Z,ainrig. 106. 
 
 Problem XXIX.— Series Circuit. Current Given. 
 
 Let there be a circuit having n different coils and con- 
 densers in series as represented in Eig. 107. It is required 
 to find the impressed E. M. F. necessary to cause the cur- 
 rent / to flow, and the difference of potential at the termi- 
 nals of each coil and condenser. 
 
 Q: 
 
 c, R.L. 
 
 
 Fig. 107.— Pkoblbm XXIX. and Problbm XXX. 
 
 In Fig. 108 make <TA equal to the given current /. 
 Lay off WB equal to J?, /, , and make^ C equal to Z, <» 7 
 perpendicular to 6~B in the positive direction. Then in 
 
 the negative direction make BD equal to -^-^ The 
 
 algebraic sum of WC and BD is BE. Then (5[Frepre- 
 sents the potential difference between M and 0, Eig. 107 ; 
 
300 
 
 CIRCUITS CONTAINING 
 
 O C, the potential difference between M and N ; and £ D, 
 or GE, the potential difference between N and O, the 
 terminals of the condenser. In a similar manner the lines 
 E G, EI, I K, and 1 M are drawn representing the poten- 
 
 FiG. 108.— Problem XXIX and Problem XXX. 
 
 tial difference between P,0 Q, QR, and Q S, respectively, 
 Fig. 107, until we finally reach the point M, Fig. JOS. OM 
 is then the required impressed E. M. F. necessary to cause 
 the current I to flow. 
 
 Equivalent Kesistance, Self-induction, or Capacity of 
 Seeies Ciecuits. 
 
 It is evident from Fig. 108 that if we had one coil only, 
 whose self-induction L' is such that the line JV Mis equal 
 to L' GO I, and whose resistance H' is such that the line 
 O JVis equal to R' I, the same current / would flow if this 
 coil be substituted for the combination of condensers and 
 coils. The resistance of this equivalent coil must evi- 
 dently be 
 
 (359) B' = Ii,-\-It,+R, + etc. = 2 B. 
 
 The self-induction of the coil, being represented by M~N 
 divided by a>I, is evidently found thus : 
 
BESiaTANCE, SELF-INDUCTION, AND CAPACITY.' 301 
 
 MN=BC -BD-\-Fa-FH-\-JK-JL, or 
 
 III 
 L' 00 1 =: L^oo I — p^ \- L^oo I — Yi — -\- L^OO I ■ 
 
 (360) L'=1^[l.--^). 
 
 If it happens that this sum is a negative quantity, the 
 self-induction cannot replace the combination, but a con- 
 denser can. It will be seen that the capacity of this con- 
 denser C may be found as follows : 
 
 (861) Hence C = — ^yi \- 
 
 These equations, (359), (360), and (361), give the means for 
 computing the equiTalent resistance, self-induction, and 
 capacity of series circuits. 
 
 Problem XXX.— Series Circuit. Impressed E. M. F. 
 
 Given. 
 
 Suppose the impressed E. M. F., represented by the 
 line OM, Tig. 108, is given, and the circuit is that shown 
 in Fig. 107. It is required to find what current will flow 
 and what is the E. M. F. at the terminals of each coil and 
 condenser. 
 
 If the equivalent self-induction L' given by equation (360) 
 above, or the equivalent capacity 0' given by equation (361), 
 is calculated, we may construct the triangle of E. M. F.'s 
 
303 CIRCUITS CONTAINING 
 
 OMJUfyin. which O N equals 1 2 B, and NM equals 
 
 12 (Za> — -77—]. The current O A is found by dividing 
 
 OJV by 2 R. After we have obtained the value of the 
 current, we may proceed, as in the preceding problem, to 
 find the E. M. F. in each part of the circuit. 
 
 Problem XXXI.— Divided Circuit. Impressed E. M. F. 
 
 Given. 
 
 Let us consider the problem of a divided circuit having 
 resistance, self-induction, and capacity in each branch, as 
 shown in Fig. 109. The impressed E. M. F., E, is given ; 
 
 I 
 
 .<-k r-^ , Jttk^-'JAA. 
 
 Fig. 109.— Problem XXXI. 
 
 it is required to find the main and branch currents. The 
 construction in Fig. 110 gives the complete solution. Since 
 the impressed E. M. F. at the terminals of each branch is 
 known, each may be separately treated as a simple circuit 
 containing resistance, self-induction, and capacity, as in 
 Problem XXX. Upon O A, which represents the impressed 
 E. M. F., E, a circle is drawn, and upon UA the several 
 E. M. F. triangles O B A, OCA, D A, axe erected with 
 
 angles ^, , 6,, 6,, of advance or lag according as -pv— 
 
 U oa 
 
 is greater or less than Z 00. The currents ^ , 7, , T, are 
 found by dividing the corresponding effective E. M. F.'s 
 by the resistance i?, , B^, R^, respectively, and the main 
 current, /, is found by taking the vector sum of the branch 
 currents. The problem is in every way the same as the 
 
RESIST ANOE, SELF-IJVDUCTION, AND CAPACITY. 303 
 
 problem of the parallel circuits with the resistance and 
 self-induction, or with resistance and capacity, except that 
 the current in any one branch may be either in advance or 
 behind the impressed E. M. F., according to the particular 
 values of the resistance, self-induction, and capacity of that 
 branch. 
 
 Equivalent Eesistance, Self-induction, and Capacity of 
 Parallel Oiecuits. 
 
 Let us suppose that for the parallel system there is sub- 
 stituted a simple circuit containing resistance and self- 
 induction, or resistance and capacity, such that the same 
 main current will flow. The investigation of the values of 
 equivalent resistance, self-induction, and capacity is similar 
 
 Pig. 110.— Problem XXXI. 
 
 to the determination of equivalent resistance and self- 
 induction, Pboblem v., and first appeared in a paper by 
 
304 CIRCUITS CONTAINING 
 
 the authors in the Philosophical Magazine for September, 
 1892. 
 
 If we take the projections of the currents I, 1^,1^, etc., 
 upon the line A, we obtain the equation 
 
 (362) 7cos B = I^ cos S, + i, cos ^, + . . . = -2 Zcos 6. 
 
 If we consider the projections of the currents upon a line 
 perpendicular to OA, we obtain 
 
 (363) /sin d = I, sin ^, + 7, sin (9,+ . . . = 2 /sin 6. 
 
 Since aU the triangles B A, O A, etc., are right tri- 
 angles, we get the following relations : 
 
 (364) /= ^ ^ 
 
 \/^'"+(^-^'-y 
 
 /.= 
 
 
 (365) cos d 
 
 cos 6^ = — ' 
 
 cos 0, = — , " — , etc. 
 
BE8I8TAN0B, BELF-INBUCTION, AND OAPAOITT. 305 
 
 (366) sin 6 = 
 
 w^-^''^ 
 
 ^^'"+(^-^'-y 
 
 sin e, = ^' 
 
 1 ^ 
 
 '^^^•'+(^V-A-y 
 
 1 ^ 
 
 — X, CO 
 
 sm 5, = — ■ = , etc. 
 
 Substituting these Talues in (362), we have 
 
 (367) 
 
 
 Making a similar substitution in (363), we have 
 
 = J. 
 
 /sin ^ C 07 ^' '^ 
 
 (368) -^ 
 
 1 
 
 Goo 
 
 L 
 
 00 
 
 O- G'Zco' 
 
 = t^C'li'oo' + {l-GLooy 
 
 GO = B 00. 
 
306 GIBCUITS CONTAINING 
 
 Here the letters A and B are introduced to simplify the 
 resulting expressions. 
 
 Dividing (368) by (367), we have 
 
 (369) tan 6 = -^^ 
 
 Comparing (365) and (367), we obtain 
 
 , cos' (9 „, cos'^ 
 
 (370) A = -^r> or B' = —^. 
 
 Comparing (366) and (368), we obtain 
 
 sin' (9 1 ^, sin'^ 
 
 (371) Boa = -^ , or -^r^- L oo = -^. 
 
 For cos' 6 and sin' d we may substitute the values 
 
 1 1 ^' _ 
 
 <^°^'^ = i + tan' </".,, B'oo'-J' + B'oo" 
 
 sin' e =. 
 
 A' 
 1 1 
 
 1 + cot' ^ ~ , , _^ - ^' + ^' (»' • 
 
 ■•• + ^' £»' 
 
 With these substitutions equations (370) and (371) be- 
 come 
 
 A 
 
 (372) B -^«^^iiQ,«> 
 
 1 ^, 5co 
 
 (373) W^-^'^-^ WT^B^- 
 
 Here ^ and B oo each stand for a summation, as ex- 
 pressed in (367) and (368), and are calculated from the 
 particular values of the resistance, self-induction, and 
 capacity of each branch. This gives a definite value to the 
 equivalent resistance, B', according to (372), and a de- 
 
RESISTANCE, SELF-INDUCTION, AND CAPACITY. 307 
 
 finite value to -pr~ — L' go, according to (373). There may 
 
 be an indefinite number of values assigned to Z' or C 
 according to values assigned to the other, that is, we may 
 assume any value for L' and by (373) determine the value 
 for C, or vice versa. 
 
 If the right-hand member of (373) is positive, we may 
 consider that the equivalent circuit has no self-induction, 
 i.e., Z' = 0, and calculate the equivalent capacity. If this 
 member is negative, we may consider that the equivalent 
 circuit has no condenser, i.e., C = oo, and calculate ac- 
 cordingly the equivalent self-induction. In any case, 
 therefore, we may speak of the equivalent resistance and 
 self-induction, or the equivalent resistance and capacity of 
 a combination of circuits, according as the equivalent 
 simple circuit would have self-induction or capacity. 
 
 The angle of lag or advance of the main current is ob- 
 tained from equation (369). 
 
 Beanch Oiecxtits with Eesistance and Self-induction only. 
 
 There is no condenser in any branch and the capacity 
 of each is, therefore, infinite. We can, accordingly, obtain 
 the expressions for A and B oo for this case by substituting 
 C = 00 in the summations in (367) and (368). This gives 
 
 J3. - 
 
 -^B'-^-L^co'' 
 
 -Boo = 
 
 ^^ Loo 
 
 -^ B' -\-L'od'' 
 
 From (372) and (373), we have 
 
 B' 
 
 A 
 
 ~ A'-^B'co''' 
 
 L'oo 
 
 — Boo 
 
 - A'+B'oo^' 
 
308 CIRCUITS CONTAININa 
 
 These results are seen to be identical with those obtained 
 in Pboblem V. and given in equations (352) and (353). 
 
 Bbahch Ciecuits with Eesistance and Capacity only. 
 
 In this case there is no self-induction in any branch, and 
 the expressions for A and B oa are found by substituting 
 Z = in the summations in (367) and (368). This gives 
 
 ^=5—^ 
 
 E- '' ' 
 
 £03 = 
 
 G CD' 
 
 1 
 
 Cod ^- C( 
 
 ^^ + C OP' 
 
 The expression for S' is the same as that in (372), and 
 from (373) we get an expression for the equivalent capacity, 
 thus: 
 
 £ 
 
 oa 
 
 C 00 ~ A' -\- £' go" 
 
 These results are identical with those previously given 
 in Peoblem XXI. 
 
 Problem XXXII.— Example of a Divided Circuit, 
 Impressed E. M. F. Given. 
 
 Suppose a divided circuit has a condenser with a 
 capacity G of one micro-farad in one branch, and a coil 
 whose self-induction Z is one henry and resistance H one 
 
RESISTANCE, SELF-INDUCTION, AND CAPACITY. 309 
 
 hundred ohms in the other branch, as in Fig. 111. Let 
 the impressed E. M. F. be one thousand volts, and 2 n times 
 
 C=1 Micro-Farad 
 
 R=10OOhms 
 L=l Heniy 
 
 -10O0 Vdts- 
 
 FiG. 111.— Pkoblbm XXXII. 
 
 the frequency be one thousand. What are the currents in 
 the main line and branches ? 
 
 Since there is no resistance in the condenser branch, 
 the current, B, Fig. 112, is ninety degrees in advance of 
 
 CEli) = 1 Ampere ° -J|jiz=.i Ampere 
 
 Fig. 112.— Problem XXXII. 
 
 the impressed E. M. F., O A, and is equal to G Em — 10"° 
 
 X 1000 X 1000 = 1 ampere. , The tangent of the angle of 
 
 .. . Lw , , 1X1000 ,^ 
 
 lag of the current m the coil is -^, equal to — ^^ — = 10, 
 
 and therefore the current, TTD, in the coil is almost ninety 
 degrees behind the impressed E. M.F. 
 
310 CIRCUITS CONTAINING 
 
 The current Bis almost equal to one ampere, for D 
 is almost equal to C, and 
 
 E 1000 
 
 ^^=r^=nnooo=^^"^p^^"' 
 
 We have, then, the condenser current OB and the coil 
 current D, each equal approximately to one ampere, 
 one in advance of the E. M. F. and the other lagging 
 behind. The resultant of these two branch currents is 
 OE and is equal to one tenth of an ampere, approx- 
 imately ; that is, each branch current is about ten times as 
 large as the main current. In this case the main current 
 is almost in phase with the impressed E. M. F., being in 
 advance of it by a small angle. 
 
 Problem XXXIII. Divided Circuit. Current Given. 
 
 If we have a number of parallel circuits, containing 
 resistance, self-induction, and capacity, and know the value 
 of the main current, /, the solution is similar to that given 
 in Problem YI. The first method of solution consists in 
 assuming an impressed E. M. F., solving as in the previous 
 problem, and then correcting the scale ,to agree with the 
 given value of the current. The second method consists in 
 computing the equivalent resistance and equivalent self-in- 
 duction or capacity of the parallel system, according to the 
 formulae (372) and (373), finding graphically the impressed 
 E. M. F., and then solving according to the last problem. 
 
 Problem XXXIV.— Series and Parallel Combinations of 
 
 Circuits. 
 
 In the graphical treatment of circuits with resistance 
 and self-induction, and of circuits with resistance and 
 capacity, the discussion was given first of series circuits 
 
BESI8TAN0E, SELF-INDUGTIOIT, AND CAPACITY. 311 
 
 and then of circuits connected in parallel. It was then 
 shown how problems arising from any combination of 
 circuits in series and parallel could be readily solved by 
 repeated applications of the methods given for the solution 
 of series and parallel circuits. In the problems given for 
 circuits containing resistance, self-induction, and capacity 
 the full solution has been given for series and for parallel 
 circuits. These principles may be applied in solving any 
 combination of series and parallel circuits, and to go 
 through particular examples of these would be needless. 
 The same problems as those given for a circuit with re- 
 sistance and self-induction or capacity can be solved in the 
 same way if the circuits contain all three. The problems 
 given have been selected as examples and not as exhaus- 
 tively representing all the problems which these graphical 
 methods are adapted to solve. The various combinations 
 which arise are endless and may often be solved in more 
 ways than one. The choice of method depends upon the 
 particular requirements of the problem. A clear idea of 
 the principles involved in the simple cases will enable one 
 to extend them with ease to whatever problems arise. 
 
APPENDIX A. 
 
 Belation between Practical and C. G. S. Units, 
 electeical units. 
 
 
 Practical 
 System. 
 
 C. G. S. System. 
 
 
 Electro- 
 magnetic. 
 
 Electrostatic. 
 
 Ouantitv 
 
 1 coulomb 
 1 ampere.. 
 
 1 volt 
 
 1 ohm 
 
 1 farad . . . 
 1 henry. . . 
 1 henry. . . 
 
 10-1 
 
 10-1 
 
 10^ 
 
 10' 
 
 10-9 
 
 10" 
 
 109 
 
 V X 10-1 — 3 X 109 
 
 Current 
 
 V X 10-1 — 3 X 109 
 
 Potential 
 
 10« -^ V = i X 10-'' 
 
 Kesistance 
 
 Capacity 
 
 «' X 10-9— 9 X 10" 
 
 Self-induction 
 
 
 Mutual induction . 
 
 
 
 
 {v = velocity of light = 3 X lOi".) 
 
 MECHANICAL UNITS. 
 
 Practical System. C. G. S. System. 
 
 Unit length = 109 ^^ 
 
 Unit mass = 10"" grms. 
 
 Unit time =1 sec. 
 
 One joule = lO'^ ergs. 
 
 One watt = t^ h. p =10' ergs per sec. 
 
 312 
 
APPENDIX B. 
 
 Some Mechanical and Electeical Analogies. 
 
 table i. — linear motion. 
 
 Notation. 
 
 1. Time = t. 
 
 2. Distance = s. 
 
 3. Linear velocity = v = — ; or, ds = vdt. 
 
 . T ■ 1 , . dv d''s 
 
 4. Linear acceleration = a = ^- = t—- . 
 
 dt dt" 
 
 5. Mass = M. 
 
 6. Momentum = Mv. 
 
 Frictional Besistance. 
 
 7. Frictional resistance = B, 
 
 8. Force to overcome resistance ^= Fg = Bv. 
 
 9. Energy expended in the time dt in overcoming resist- 
 
 ance = d Wb = Fads = Bv'dt. 
 
 Inertia, 
 
 d "D ' 
 
 10. Force to overcome inertia = F' = Ma = M-rr . 
 
 dt 
 
 11. Kinetic energy acquired in the time d t 
 
 = dW' = F'ds = Mv^dt. 
 
 dt 
 
 12. Kinetic energy = W = P Mvj^dt = ^Mv\ 
 
 Resistance plus Inertia. 
 
 13. Total force applied - F = Fs-\- F' = Bv -\- M-^. 
 
 14. Total energy supplied in the time dt 
 
 = dW=dWB-{-dW'; or, Fds = F^ds -{- F'ds; 
 
 d V 
 or, Fvdt = Bv'dt -{-Mv^dt. 
 
 318 
 
314 APPBNDIX B. 
 
 TABLE II. BOTABT MOTION. 
 
 Notation, 
 
 1. Time = t. 
 
 2. Angle = 0. 
 
 3. Angular velocity = a? = -^ ; or, d(p^ oadt. 
 
 4. Angular acceleration = a = -nrr = -ttj- - 
 
 5. Moment of inertia = I. 
 
 6. Angular momentum = loo. 
 
 Frictional Sesistance. 
 
 7. Frictional resistance = B. 
 
 8. Torque to overcome resistance = Tn = Rco. 
 
 9. Energy expended in the time dt in overcoming resist- 
 
 ance z=dWn=z Tjtdfp = Bco'dt. 
 
 Inertia. 
 
 10. Torque to overcome inertia — T' = la = I-rr - 
 
 11. Kinetic energy acquired in the time d t 
 
 = dW'= T'd<p=zIoo^^dt. 
 
 12. Kinetic energy = W = T loo-jjdt = J /a>°. 
 
 Resistance plus Inertia. 
 
 13. Total torque applied = T = Ts-\- T' = Boo + 1-^. 
 
 14. Total energy supplied in the time d t 
 
 = dW=dWR^dW'; or, T d(p= Ti,d4>+ T' d(f>; 
 
 or, TQodt=BGa'dt-\-Ioo-^dt. 
 
APPENDIX B. 315 
 
 TABLE m. — ^ELECTBIO CUREENT. 
 
 Notation. 
 
 1. Time = t. 
 
 2. Quantity = q. 
 
 3. Current =i = ~ ; or, dq = idt. 
 
 'di 
 
 4. Current acceleration = /J = -rr • 
 
 '^ dt 
 
 5. Coefficient of self-induction = L. 
 
 6. Electro-magnetic momentum = Li. 
 
 Ohmic Resistance. 
 
 7. Ohmic resistance =: 5. 
 
 8. Electromotive force to overcome resistance = e^ =Bi. 
 
 9. Energy expended in the time dt in overcoming resist- 
 
 ance = d TFr = e^dq == Bi' dt. 
 
 Sdf-indvction. 
 
 10. Electromotive force to overcome self-induction 
 
 ^ ^ ^di 
 = e'=L^ = L^. 
 
 11. Energy acquired by the magnetic field in the time dt 
 
 di 
 = dW' = e'dq = Li-jrdt. 
 
 ^ dt 
 
 12. Energy of magnetic field = W = C Lijrdt = ^LP. 
 
 Resistance plus Sdf-induction. 
 
 13. Total electromotive force applied 
 
 ^di 
 = e = ejt-\-e'=zRt-\-Zj^. 
 
 14. Total energy supplied in the time d t 
 
 = dW=dWs + dW'; or, edq = eitdq -{-e'dq; 
 
 di 
 or, eidt = Ri*dt-\- Li^dt. 
 
APPENDIX C. 
 
 Notation Used thboughout this Book. 
 
 (Numbers refer to page where ilirst used.) 
 
 A. Area, 67 ; or, constant, 41. 
 £. Constant, 41. 
 
 B. Induction per square centimeter, 22. 
 G. Capacity, 64 ; or, constant, 41. 
 
 C Equivalent capacity, 279. 
 
 D. Symbolic operator, 84. 
 
 E. Constant E. M. F., 25 ; or, maximum value of bar- 
 
 monic E. M. R, 50. 
 
 E. Virtual E. M. F., i.e., square root of mean square 
 
 value, 38 and 143. 
 
 F. Force, 20. 
 
 H. Magnetizing force, 21. 
 
 /. Constant current, 25 ; or, maximum value of har- 
 monic current, 53. 
 
 /. Virtual current, i.e., square root of mean square 
 value, 38 and 143. 
 Im. Impedance, 188. 
 
 Z. Coefficient of self-induction, 23. 
 
 Z'. Equivalent self-induction, 235. 
 
 N. Total induction, i.e., total number of lines, 21. 
 
 0. Origin. Center of revolution, 33. 
 
 Q. Constant quantity ; or, charge of electricity, 25. 
 
 Ji. Resistance, 24. 
 
 B'. Equivalent resistance,|235. 
 
 316 
 
APPENDIX C. 317 
 
 T. Period, 33 ; or, time constant, 46. 
 
 V. Potential, 63. 
 
 W. Work or energy, 28. 
 
 a. Amplitude, 33 ; or, constant, 86. 
 
 i. Constant, 57. 
 
 c. Arbitrary constant of integration, 44. 
 
 d. Distance, 67. 
 
 e. Instantaneous value of electromotive force, 25, 
 /. Arbitrary function, 43. 
 
 /'. First differential coefficient of/, 71. 
 
 h. Constant, 184. 
 
 i. Instantaneous value of current, 25. 
 
 j. V^^, 93. 
 
 k. Constant, 183. 
 
 I. Constant length, 201. 
 
 m. Strength of pole, 20 ; or, constant, 96. 
 
 n. Frequency, 34 ; or, constant, 58. 
 
 p. An abbreviation, 191. 
 
 q. Instantaneous value of charge, 25. 
 
 r. Distance, 20; or, constant, 190. 
 
 t. Time, 34. 
 
 X. Independent variable, 41 ; also length or distance, 
 178. 
 
 y. Dependent variable, 34. 
 
 z. Dependent variable, 42. 
 
 a. An abbreviation, 191 ; or, a constant, 41. 
 
 /3. Constant, 41. 
 
 y. Constant, 41. 
 
 e. Naperian base, (2.71828), 44. 
 
 --(- 0. Angle, usually of advance, 35. 
 
 — 0. Angle, usually of lag, 35. 
 
318 APPENDIX C. 
 
 K. Specific inductive capacity, 61 ; or, Constant, 206. 
 
 \. Wave-length, 196. 
 
 fx. Permeability, 22. 
 
 It. Eatio of circumference to diameter, (3.14159), 21. 
 
 2. Summation, 59. 
 
 r. 1 -T- time -constant, yp , 126. 
 
 #. Arbitrary constant, 95. 
 
 0. Angle, 34. 
 
 X' Angle, 150. 
 
 ip. Current angle, 55. 
 
 GO. Angular velocity, 2 ;r w, 34. 
 
 For graphical conventions, see 219. 
 
INDEX. 
 
 Acceleration, unit of, 18 
 
 Addition of harmonic electromotive 
 
 forces, 313 
 Addition of harmonic functions, 38 
 Advance, angle of, 35, 78, 134 
 Air-chamber analogue of condenser, 
 
 373 
 Ampere, 33 
 Amplitude, 33 
 Analogies, mechanical, 313 
 Analytical treatment {see Contents), 
 
 7, 17 
 Angle of advance, 35, 78, 134 
 Angle of lag, 35, 54, 134 
 Angle of phase, 34 
 Angle of epoch, 34 
 Angular velocity, 34, 50 
 Apparent resistance, 53, 79, 131 
 Arrows, meaning of, 331 
 Attraction, law of, 
 
 — for charged bodies, 60 
 
 — for magnetic poles, 30 
 Average value of sine-curve, 36 
 
 B 
 
 B, Induction, 23 
 Backward waves, 303, 305 
 Ballistic galvanometer, 36 
 
 C 
 
 Cable, distributed capacity of, 176 
 Capacity, distributed, 176 
 Capacity, eflfects of variation of, 
 
 — in parallel circuits with resist- 
 ance, 285 
 
 — in series circuit with resistance, 
 376 
 
 — in series circuit with resistance 
 and self-induction, 138, 398 
 
 Capacity, equivalent, 
 
 — for parallel circuits, 281, 283 
 
 — for parallel circuits with self- 
 induction, 303 
 
 — for series circuits, 379 
 
 — for series circuit with self-in- 
 duction, 396, 300 
 
 Capacity of a condenser, 65 
 Capacity of a conductor, 64 
 Capacity of continuous wire, 67 
 Capacity of parallel plates, 67 
 Capacity varied 
 
 — in parallel circuits with resist- 
 ance, 385 
 
 — in series circuit with resistance, 
 376 
 
 — in series circuit with resistance 
 and self-induction, 138, 398 
 
 C. G. S. units, 18, 313 
 Charge, energy of, 64 
 Charge equation 
 
 — for any periodic E. M. F., 157 
 
 — for charging a condenser, 118 
 
 — for circuits with resistance and 
 capacity, 73 
 
 — for circuits with resistance and 
 capacity and harmonic E. M.F., 
 78 
 
 — for discharge through circuits 
 ■with resistance, self-induction, 
 and capacity : 
 
 general forms, 97 
 
 non-oscillatory, 99 
 
 oscillatory, 107 
 
 when R^ = 4 L, 109 
 
 — for discharging condenser, 73 
 
 — for non-oscillatory charging, 
 115 
 
 — for oscillatory charging, 120 
 
 — when 2^0 = 4 £, 122 
 
 319 
 
320 
 
 INDEX. 
 
 Charge, for circuits with resistance, 
 self-induction, and capacity, 112 
 Charge, general solution for, 
 
 — in circuits with resistance and 
 capacity, 73 
 
 — in circuits with resistance, self- 
 induction, and capacity, 87 
 
 Charge of a condenser, 74 
 Charge, unit of, 25, 61 
 Charging a condenser, 117 
 Charging equations, three forms of, 
 
 114 
 Charging, non-oscillatory, 114 
 
 — determination of constants, 114 
 
 — discussion, 116 
 Charging, oscillatory, 119 
 
 — determination of constants, 119 
 
 — discussion, 130 
 Charging when R^O=iL, 121 
 
 — determination of constants, 131 
 
 — discussion, 122 
 Circles, meaning of, 231 
 Closed arrows, meaning of, 231 
 Closed circuits, wave-propagation in, 
 
 201 
 Coefficient of self-induction, 23 
 Combination circuits 
 
 — with resistance and capacity, 
 386, 287, 388 
 
 — with resistance and self-induc- 
 tion, 348 
 
 — with resistance, Self-induction, 
 and capacity, 310 
 
 Complementary function, 73, 93, 
 96 
 
 Composition of harmonic electromo- 
 tive forces, 218 
 
 Composition of harmonic function^ 
 88 
 
 Condenser, 65 
 
 — capacity of, 65 
 
 — discharge of, 73 
 
 — electromotive force of, 69 
 
 — energy of, 65 
 
 — mechanical analogue of, 272 
 Conductor, energy of, when 
 
 charged, 64 
 Conjugate imaginaries, 94 
 Constant current example, 246 
 Constant potential example, 245 
 Constants, variation of, 
 
 — in parallel circuits with resist- 
 ance and capacity, 385 
 
 — in parallel circuits with resist- 
 ance and self-induction, 343 
 
 — in series circuit with resistance 
 and capacity, 374 
 
 — in series circuit with resistance 
 and self-induction, 323 
 
 Constants, variation of, in series cir- 
 cuit with resistance, self-induc- 
 tion, and capacity, 134, 396 
 Construction of logarithmic cui've, 
 
 46 
 Continuous conductor, capacity of, 
 
 67 
 Conventions adopted 219, 316 
 Cosine expanded, 93 
 Cosine, exponential form of, 93 
 Coulomb's law 
 
 — for attraction between poles, 30 
 
 — for attraction between charged 
 bodies, 60 
 
 Counter-electromotive force of self- 
 induction, 39 
 Critical case of discharge, 108 
 
 — of charge, 131 
 Criterion of integi-ability, 43, 71 
 Current at the " malie," 55, 144 
 Current equation, 
 
 — any periodic E. M. P., 157 
 
 — for charging a condenser, 118 
 
 — for circuits containing resist- 
 ance only, 133 
 
 — for circuits with capacity only, 
 
 — for circuits with distributed 
 self-induction and capacity, 192 
 
 — for circuit with resistance and 
 capacity, 72 
 
 — for circuits with resistance and 
 capacity, and an harmonic 
 E.M.F., 78, 133 
 
 — for circuit with resistance and 
 self-induction, and harmonic 
 E. M. F., 53, 133 
 
 — for discharging condenser, 73 
 Current equation for discharge 
 
 through circuit with resistance, 
 self-induction, and capacity, 92, 97 
 
 — non-oscillatory, 99 
 
 — oscillatory, 107 
 
 — when i?' C = 4 £, 109 
 Current equation for establishment 
 
 of current in circuit with R and 
 £, 117 
 Current equation, 
 
 — for non-oscillatory charging, 
 115 
 
 — for oscillatory charging, 120 
 
 — when IP 0=4: L, 123 
 Current, general solution for, 
 
 — in circuits with resistance ana 
 self-induction, 44 
 
 — in circuits with resistance and 
 capacity, 73 
 
 — in circuits with resistance, self- 
 induction, and capacity, 84, 86 
 
INDEX. 
 
 321 
 
 Current graphically shown by closed 
 
 arrows, 321 
 Current, unit of, 23 
 Curves, types of, 163 
 
 D 
 
 D, symbolic operator, 84 
 Decay of waves 
 
 — in circuit with distributed ca- 
 pacity, 197 
 
 — in circuit with distributed ca- 
 pacity and self-induction, 200 
 
 Decreasing amplitude of waves 
 —: in circuits with distributed ca- 
 pacity, 197 
 
 — in circuits with distributed ca- 
 pacity and self-induction, 199 
 
 Direction of rotation, 221 
 Direction of rotation of E. M. F. 
 
 vectors, 261 
 Differential equations 
 
 — for charging, 113 
 
 — for circuits with resistance and 
 capacity, 73 
 
 — for circuits with resistance and 
 self-induction, 43 
 
 — for circuits with resistance, self- 
 induction, and capacity, 84 
 
 — for discharge, 91 
 Dimensions of impediment, 183 
 Dimensions of L oo, 55 
 Discharge through circuit with re- 
 sistance and capacity, 72, 104 
 
 Discharge through circuit with re- 
 sistance, self-induction, and ca- 
 pacity, 90 
 
 — non-oscillatory, 98 
 
 — oscillatory, 105 
 
 -- when B°- 0=4:L, 108 
 Distributed capacity, 176 
 
 — with no self-induction, 194 
 
 — with self-induction, 198 
 Divided circuit 
 
 — with resistance and capacity, 
 280, 282, 284 
 
 — with resistance and self-induc- 
 tion, 233, 236, 241 
 
 — with resistance, self-induction, 
 and capacity, 302, 308, 310 
 
 Dying away of current in circuit 
 with resistance and self-induc- 
 tion. 44, 103 
 
 Dyne, 18 
 
 E 
 
 S!, e. Electromotive force, 25 
 
 Earth inductor, 36 
 
 Effective electromotive force, 38, 55 
 
 Effects of varying constants 
 
 — in parallel circuits, 343, 285 
 
 — in series circuits, 134, 223, 374, 
 396 
 
 Electrical analogies, 313 
 Electrical horse-power, 39 
 Electromagnetic induction, 31 
 Electromotive force 
 
 — diagram for circuits with re- 
 sistance, self-induction, and ca- 
 pacity, 393 
 
 — equation for circuit with resist- 
 ance and capacity, 69, 70 
 
 — equation for circuit with resist- 
 ance and self-induction, 31, 43 
 
 — equation for circuit with resist- 
 ance, self-induction, and ca- 
 pacity, 83 
 
 — equation for circuit with dis- 
 tributed self-induction and ca- 
 pacity, 190 
 
 — graphically shown by open 
 arrows, 231 
 
 — law of, 23 
 
 — maximum value of, 50 
 
 — of condenser, 69 
 
 — of condenser graphically rep- 
 resented, 269, 371 
 
 — of self-induction, 29 
 
 — of self-induction graphically 
 represented, 220 
 
 — triangle of, for circuits con- 
 taining resistance and capacity, 
 368 
 
 — triangle of, for circuits with 
 resistance and self-induction, 
 317 
 
 — imit of, 34 
 Electromotive forces 
 
 — in parallel, 363 
 
 — in series, 260 
 
 — with different periods, 264 
 E. M. P. vectors, rotation of, 261 
 Energy dissipated in beat, 27 
 Energy, equation of, for circuit with 
 
 resistance and capacity, 67 
 
 — equation of, for circuits with 
 resistance and self-induction, 30 
 
 — equation of, for circuits with 
 resistance, self-induction, and 
 capacity, 82 
 
 — imparted to a circuit, 29, 142 
 
 — of a charged conductor, 64 
 
 — of a condenser, 66 
 
 — of magnetic field, 39 
 
 — unit of, 28 
 Epoch, 34 
 
 Equation of energy for circuits with 
 resistance and capacity, 67 
 
322 
 
 INDEX. 
 
 Equation of energy 
 — for circuits with resistance and 
 self-induction, 30 
 
 — for circuits with resistance, 
 self-induction, and capacity, 83 
 
 Equation of E. M. F.'s 
 
 — for circuits with resistance and 
 capacity, 69, 70 
 
 — for circuits with resistance and 
 self-induction, 31 
 
 — for circuits with resistance, 
 self-induction, and capacity, 83 
 
 Equivalent capacity 
 
 — of parallel circuits, 281, 288 
 
 — of parallel circuits with self- 
 induction, 303 
 
 — of series circuits, 279 
 
 — of series circuits with self-in- 
 duction, 396, 800 
 
 Equivalent resistance 
 
 — of parallel circuits, 235, 288, 
 281, 283, 303 
 
 — of series circuits, 379, 300 
 Equivalent self-induction 
 
 — of parallel circuits, 285, 338 
 
 — of parallel circuits with ca- 
 pacity, 803 
 
 — of series circuits, 296, 300 
 Erg, unit of energy, 28 
 Establishment of current 
 
 — in circuit with resistance and 
 capacity, 117 
 
 — in circuit with resistance and 
 self-induction, 48 
 
 — in circuit with resistance, self- 
 induction, and capacity, 113 
 
 Example of a divided circuit with 
 resistance, self-induction, and ca- 
 pacity, 309 
 Expansion of sine and cosine, 98 
 Expenditure of energy in a circuit, 
 
 83 
 Explanation of exponential term, 55 
 Exponential form of sine and co- 
 sine, 93 
 Exponential term, 
 
 — efEect of, at "malce," 56 
 
 — explanation of, 55 
 
 Faraday's law, 23 
 Field of force, 18 
 
 — intensity of, 20 
 
 — unit field, 21 
 Force, 
 
 — law of, for charged bodies, 60 
 
 — law of, for magnetic poles, 20 
 
 — unit of, 19 
 
 Formulae of reduction, 51, 53 
 Forward waves, 302, 305 
 Fourier's Theorem, 41 
 Frequency, 34 
 
 — variation of, 140, 298 
 Fundamental units, 18 
 
 G 
 
 Galvanometer, ballistic, 26 
 General solution for charge 
 
 — in circuits with resistance and 
 capacity, 73 
 
 — in circuits with resistance, self- 
 induction, and capacity, 87 
 
 General solution for current 
 
 — in circuits with resistance and 
 capacity, 73 
 
 — in circuits with resistance and 
 self-induction, 44 
 
 — in circuits with resistance, self- 
 induction, and capacity, 84, 86 
 
 Graphical representation 
 
 — of a simple harmonic E.M.F., 
 313 
 
 — of the sum of several harmonic 
 E. M. F.'s, 313 
 
 Graphical treatment, 11, 209 
 
 — symbols adopted, 319 
 
 H 
 
 H, magnetizing force, 31 
 Harmonic electromotive force 
 
 — discussion, 180 
 
 — general solution, 134 
 
 — graphical representation of,213 
 
 — in circuit with resistance and 
 capacity, 76 
 
 — in circuit with resistance and 
 seif-induction, 50 
 
 — solution from differential equa- 
 tions, 127 
 
 Harmonic functions, 32 
 
 — addition of, 88 
 Harmonic motion, 33 
 Heating efEect, 37 
 
 — same with as without self-in- 
 duction and capacity, 163 
 
 Horse power, electrical, 39 
 
 /, i, current, 35 
 
 I, maximum value of current, 53, 79, 
 
 131 
 Impedance, 53, 79, 181 
 — measurement of, 230 
 
INBEX. 
 
 323 
 
 Impediment, 131, 295 
 — dimension of, 132 
 Impressed electromotive force, 55 
 Induction, 21 
 Inductive resistance, 54 
 Infinite capacity, 67 
 Integrability, criterion of, 43, 71 
 Integration by parts, 51 
 Intensity of a field of force, 20 
 
 j = ^n^, 93 
 
 Joule, unit of energy, 28 
 
 Joule's law, 26 
 
 Just non-oscillatory 
 
 — charge, 121 
 
 — discharge, 110 
 
 K 
 
 K, specific inductive capacity, 61 
 
 L, coefficient of self-induction, 23 
 Lag, 
 
 — angle of, 35, 54, 138, 184 
 
 — measurement of, 280 
 Law of attraction 
 
 — for charged bodies, 60 
 
 — for magnetic poles, 20 
 Law of Coulomb, 20 
 
 — of Faraday, 23 
 
 — of Joule, 24 
 
 — of Ohm, 26 
 
 Linear equation, 43, 44, 86 
 Line of force, 18, 21 
 Lines of induction, 21 
 Limitations of the telephone, 200 
 Logarithmic curve, construction of, 
 
 46 
 LoD, dimensions of, 55 
 
 M 
 
 Magnetic field, energy of, 29 
 
 — intensity of, 20 
 
 Magnetic pole, 18 
 
 Magnetizing force, 21 
 
 Make, current at, 56, 144 
 
 Maximum oscillation, 153 
 
 Maximum value of harmonic cur- 
 rent, 53, 79, 131 
 
 Mean square value of a sine-curve, 
 
 37 
 Measurement by three-voltmeter 
 
 method, 230 
 Mechanical analogue of condenser, 
 
 272 
 Mechanical analogies, 813 
 
 Method used in graphical treatment, 
 
 219 
 Multiple-arc arrangement 
 
 — of circuits with resistance and 
 capacity, 289 
 
 — of circuits containing resistance 
 and self-induction, 256 
 
 Multiple-valued function, 38 
 jii, permeability, 22 
 
 N 
 
 JT, total- induction, 21 
 n, frequency, 34 
 
 Negative direction of rotation, 221 
 Neutralizing of self-induction and 
 capacity 
 
 — at every point of time, 158 
 
 — necessary conditions for, 162 
 Neutralizing of self-induction and 
 
 capacity impossible except for 
 sine-curve, 175 
 Non-oscillatory charging, 114 
 
 — determination of constants, 114 
 — discussion, 116 
 
 Non-oscillatory discharge, 98 
 
 — determination of constants, 98 
 
 — discussion, 99 
 
 Notation, 219 (see also Appendix) 316 
 
 O 
 
 Ohm's law, 24, 158 
 Ohm, unit of resistance, 24 
 Open arrows, meaning of, 221 
 Oscillation a maximum, 153 
 Oscillatory charging, 119 
 
 — determination of constants, 119 
 
 — discussion, 120 
 Oscillatory discharge, 105 
 
 — determination of constants, 105 
 
 — discussion of, 107 
 
 Parabola and sine-curve example, 
 
 167 
 Parallel circuits 
 
 — with resistance and capacity, 
 280, 282, 284 
 
 — with resistance and self-induo 
 tion, 238, 236, 241 
 
 — with resistance, self-induction, 
 and capacity, 802, 808, 810 
 
 Parallel plates, capacity of, 67 
 Particular E. M. F.'s, 87 
 Period, 83 
 Periodic functions, 88 
 
324 
 
 INDEX. 
 
 Periodic E. M. P. in circuit with 
 resistance and capacity, 79 
 
 — in circuit with resistance and 
 self-induction, 57 
 
 — in circuit with resistance, self- 
 induction and capacity, 124 
 
 Periodicity, 34 
 
 Permeability, 22 
 
 Phase, 84 
 
 Pole, unit magnetic, 18 
 
 Positive and negative flow of alter- 
 nating current equal, 164 
 
 Positive direction of rotation, 33, 221 
 
 Potential, 61, 68 
 
 Potential of a conductor with dis- 
 tributed self-induction and ca- 
 pacity, 190 
 
 Power, measurement of, by three 
 voltmeters, 282 
 
 Practical units, 812 
 
 Problems, see Contents, 11 
 
 Propagation of waves, rate of, 
 
 — in circuits with distributed 
 capacity, 195 
 
 — in circuits with distributed 
 capacity and self-induction, 198 
 
 Q 
 
 Quantity, Q, 
 
 — definition of, 25 
 
 — for half period, 164 
 
 — unit of, 25, 61 
 Quickest charge, 121 
 Quickest discharge, 110 
 
 R 
 
 Rate of decay of waves 
 
 — in circuit with distributed ca- 
 pacity, 197 
 
 — in circuit with distributed capa- 
 city and self-induction, 200 
 
 Rate of propagation of waves 
 
 — in circuit with distributed ca- 
 pacity, 195 
 
 — in circuit with distributed ca- 
 pacity and self-induction, 198 
 
 Rate of work, 28 
 Reactance, 54, 59, 79, 294 
 Resistance, effect of variation of, 
 
 — in parallel circuit with ca- 
 pacity, 285 
 
 — in parallel circuit with self-in- 
 duction, 242. 
 
 — in seriescircuit with capacity, 274 
 
 — in series circuit with self-in- 
 duction, 223 
 
 — in series circuit with resistance, 
 self-induction, and capacity, 185 
 
 Resistance equivalent, 
 
 — of parallel, circuits, 235, 238, 
 281, 288, 308 , 
 
 — of series circuits, 279, 200 
 Resistance, B, unit of, 24 
 Resultant of several harmonic 
 
 E. M. P.'s of the same period, 
 
 213 
 Rotation, direction of, 221 
 Rotation of E. M. F. vectors, 261 
 
 S 
 
 Self-induction, coefficient of, 23 
 — electromotive force of, 220 
 Self-induction , effect of variation of, 
 
 — in parallel circuits, 244 
 
 — in series circuit, 225 
 
 — in series circuit with resistance, 
 self-induction, and capacity, 
 187, 298 
 
 Self-induction, equivalent, 
 
 — of parallel circuits, 235, 288 
 
 — of parallel circuits with ca- 
 pacity, 303 
 
 — of series circuit with capacity, 
 296, 300 
 
 Self-induction, measurement of, 230 
 Series and parallel circuits 
 
 — with resistance and capacity, 
 286, 287, 288 
 
 — with resistance and self-induc- 
 tion, 248, 250, 251, 252 
 
 — with resistance, self-induction, 
 and capacity, 310 
 
 Series circuit 
 
 — with resistance and capacity, 
 278, 279 
 
 — with resistance and self-induc- 
 tion, 227, 228, 229 
 
 — with resistance, self-induction, 
 and capacity, 299, 301 
 
 Several sources of E. M. F., 260 
 Sine-curve, 35 
 
 — average value of, 36 
 
 — -mean square value of, 37 
 Sine-curve and parabola example, 
 
 167 
 Sine expanded, 93 
 Sine, exponential form of, 93, 186 
 Sine-functions, 32 
 Single-value function, 38 
 Specific inductive capacity, 61 
 Strength of a magnetic field, 20 
 Sum of harmonic E. M. F.'s of the 
 
 same period, 218 
 Symbols adopted in graphical treat- 
 ment, 219 
 Symbolic operator, 84, 128 
 
INDEX. 
 
 325 
 
 Telephone, limitations of the, SOO 
 
 T, period, 33 
 
 T, see Time-constant. 
 
 Three- voltmeter method, 230 
 
 Time-constant 
 
 — in circuit with resistance and 
 capacity, 74 
 
 — in circuit with resistance and 
 self-induction, 46 
 
 — in circuits with resistance, self- 
 induction, and capacity, 85 
 
 Transformation to real form, 93 
 Triangle of E. M. F.'s 
 
 — for circuits with resistance and 
 capacity, 268 
 
 — for circuits with resistance and 
 self-induction, 217 
 
 — for circuits with resistance, self- 
 induction, and capacity, 393 
 
 Two E. M. F.'s in series, 260, 264 
 Types of curves, 163 
 
 U 
 
 Unit charge, 61 
 Unit current, 33 
 Unit magnetic pole, 18 
 Unit of energy, 28 
 Unit quantity, 25 
 
 Variation of capacity 
 
 — in parallel circuits, 285 
 
 — in series circuits, 138, 276, 296 
 Variation of constants in parallel 
 
 — circuits, 385 
 
 Variation of constants in series cir- 
 
 — cuits, 134, 274, 396 
 Variation of frequency 
 
 — in series circuits, 140, 396 
 Variation of resistance 
 
 — in parallel circuits, 343, 285 
 
 — in series circuits, 135, 323, 374, 
 296 
 
 Variation of self-induction 
 
 — in parallel circuits, 344 
 
 — in series circuits, 137, 235, 396 
 Velocity, unit of, 18 
 
 Virtual values of E. M. F. and cur- 
 rent, 38, 54, 131, 143 
 Volt, 34 
 
 W 
 
 W, work or energy, 38 
 Watt, unit of work, 28 
 Wave-length, 306 
 
 Wave-propagation in circuits with 
 distributed capacity, 
 
 — decreasing amplitude of, 197 
 
 — nature of, 194 
 
 — rate of, 195 
 Wave-propagation in circuits with 
 
 distributed capacity and self-in- 
 duction, 
 
 — decreasing amplitude of, 199 
 
 — nature of, 198 
 
 — rate of, 198 
 
 Wave- propagation in closed circuits, 
 
 .301 
 Work done by harmonic current, 
 
 142 
 Work in moving charge, 62 
 GO, angular velocity, 34