CHAPTER VIII. CIRCUITS CONTAINING RESISTANCE, INDUCTANCE, AND CAPACITY. 42. Having, in the foregoing, reestablished Ohm's law and Kirchhoff's laws as being also the fundamental laws of alternating-current circuits, when expressed in their com- plex form, E = ZS, or, / = YE, and *%E = 0 in a closed circuit, S/ = 0 at a distributing point, where E, I, Z, Y, are the expressions of E.M.F., current, impedance, and admittance in complex quantities, — these values representing not only the intensity, but also the phase, of the alternating wave, — we can now — by application of these laws, and in the same manner as with continuous- current circuits, keeping in mind, however, that E, I, Z, Y, are complex quantities — calculate alternating-current cir- cuits and networks of circuits containing resistance, induc- tance, and capacity in any combination, without meeting with greater difficulties than when dealing with continuous- current circuits. It is obviously not possible to discuss with any com- pleteness all the infinite varieties of combinations of resis- tance, inductance, and capacity which can be imagined, and which may exist, in a system or network of circuits ; there- fore only some of the more common or more . interesting combinations will here be considered. 1.) Resistance in series with a circuit. 43. In a constant-potential system with impressed E.M.F., o = e. +/V, E. = RESISTANCE, INDUCTANCE, CAPACITY. 59 let the receiving circuit of impedance Z = r —jx, z = Vr2 + x'2, be connected in series with a resistance, r0 . The total impedance of the circuit is then Z + r0 = r + r0—jx\ hence the current is ____ •" Z + r0 r+r0 -jx (r + r0)2 -f *2 ' and the E.M.F. of the receiving circuit, becomes E = IZ = ^° (r ~J^ = ^° or, in absolute values we have the following : — Impressed E.M.F., current, zr zr V(r + ;-0)2 + x2 -Vz2 + E.M.F. at terminals of receiver circuit, E = EnJ >* + *2 . Eo Vs2 + 2rr0 + r02 difference of phase in receiver circuit, tan w = - ; difference of phase in supply circuit, tan o>0 = since in general, tan (phase) = ^aginary component ^ real component a.} If x is negligible with respect to r, as in a non-induc- tive receiving circuit, 1= -=3_ r+ r. and the current and E.M.F. at receiver terminals decrease steadily with increasing r0 . 60 ALTERNATING-CURRENT PHENOMENA. b.} If r is negligible compared with x, as in a wattless receiver circuit, 7= E° , £ = £. X - or, for small values of r0 , /=— °, ^ = ^0; that is, the current and E.M.F. at receiver terminals remain approximately constant for small values of r0, and then de- crease with increasing rapidity. 44. In the general equations, x appears in the expres- sions for / and E only as xz, so that / and E assume the same value when x is negative, as when x is positive ; or, in other words, series resistance acts upon a circuit with leading current, or in a condenser circuit, in the same way as upon a circuit with lagging current, or an inductive circuit. For a given impedance, z, of the receiver circuit, the cur- rent /, and E.M.F:, E, are smaller, as r is larger; that is, the less the difference of phase in the receiver circuit. As an instance, in Fig. 37 is shown the E.M.F., E, at the receiver circuit, for E0 = const. = 100 volts, s = 1 ohm ; hence / = E, and — a.) r0 = .2 ohm (Curve I.) b.) r0 = .8 ohm (Curve II.) with values of reactance, x = V^2 — r2, for abscissae, from x = + 1.0 to x = — 1.0 ohm. As shown, / and E are smallest for x = 0, r = 1.0, or for the non-inductive receiver circuit, and largest for x = ± 1.0, r = 0, or for the wattless circuit, in which latter a series resistance causes but a very small drop of potential. Hence the control of a circuit by series resistance de- pends upon the difference of phase in the circuit. For r0 = .8, and x = 0, x = + .8, x = — .8, the polar diagrams are shown in Figs. 38 to 40. RESISTANCE, INDUCTANCE, CAPACITY. 61 2.) Reactance in series witJi a circuit. 45. In a constant potential system of impressed E.M.F., let a reactance, x0 , be connected in series in a receiver cir- cuit of impedance Z = r — jx, z = -\/r2 -|- x'2. IMPRESSED E.M.F. CONSTANT, E0=IOO IMPEDANCE OF RECEIVER CIRCUIT CONSTANT, Z - 1.0 LINE RESISTANCE CONSTANT n =.2 3 - -.4 T-5 ' '.6 T.7 r-8 Fig. 37. Variation of Voltage at Constant Series Resistance with Phase Relation of Receiver Circuit. Then, the total impedance of the circuit is Z -jx0 = r—j(x +#e). Er Er0 Fig. 38. and the current is, /= E Fig. 39. Z-jx0 r—j(x + x0}' /hile the difference of potential at the receiver terminals r—jx 62 ALTERNATING-CURRENT PHENOMENA. Or, in absolute quantities : — Current, /_ Eo EQ •* ~ Vr* -f- (x + x0)'2 V 'z'1 + 2xx0 -\- xa2 E.M.F. at receiver terminals, r / r' + *« = J^ ° V ra + (* + *„)* V** + 2*.r0 + *.a 5 difference of phase in receiver circuit, x tan E0 ; that is, the reactance, x0 , raises the potential. c.) E = E0 , or the insertion of a series inductance, x0 , does, not affect the potential difference at the receiver ter- minals, if ^z*-\-2xx0 + x02 = 2; or, x0 = — 2 x. That is, if the reactance which is connected in series in the circuit is of opposite sign, but twice as large as the reactance of the receiver circuit, the voltage is not affected, but E = E0,I= E0/z. If x0 < — 2 x, it raises, if x0 > — Zv, it lowers, the voltage. We see, then, that a reactance inserted in series in an alternating-current circuit will lower the voltage at the RESISTANCE, INDUCTANCE, CAPACITY. 63 receiver terminals only when of the same sign as the reac- tance of the receiver circuit ; when of opposite sign, it will lower the voltage if larger, raise the voltage if less, than twice the numerical value of the reactance of the receiver circuit. d.} If x = 0, that is, if the receiver circuit is non- inductive, the E.M.F. at receiver terminals is : = (!-}- *)•'* expanded by the binomial theorem = nx Therefore, if x0 is small compared with r : — That is, the percentage drop of potential by the insertion of reactance in series in a non-inductive circuit is, for small Fig. 40. values of reactance, independent of the sign, but propor- tional to the square of the reactance, or the same whether it be inductance or condensance reactance. 64 AL TERNA TING-CURRENT PHENOMENA. 46. As an instance, in Fig. 41 the changes of current, /, and of E.M.F. at receiver terminals, E, at constant im- pressed E.M.F., E0, are shown for various conditions of a receiver circuit and amounts of reactance inserted in series. Fig. 41 gives for various values of reactance, x0 (if posi- tive, inductance — if negative, condensance), the E.M.Fs., E, at receiver terminals, for constant impressed E.M.F., VOLTS E OR AMPERES I 100 IMPRESSED E.'M.F! CONSTANT, E IMPEDANCE OF RECEIVER CIRC.UI I. r=l.o x=o II. r=.6 X=H-,8 111. r=.e i=-.8 =160 r CONS ^ FAN ^ T.Z = l n 1" 0 if 0 r "V V \ U o J \ \ n 0 / \ ^ \ i? 0 / \ / \ 12 n / / \'l "/ / \ / / . X /n \" ^, ^ ., 'ill X / S n \ ^> \ £ ^ / |X . / - 0 \ ^ \ |?° ^x ' Lj / x / . D S \ \ ^ a. 60 O Y/ . X II X" | 0 \ so 10 Xo •*•» ^ ^ ^ ^ -^ . n *< _- — ' _~. — --- , — — - | o . 0 1 0 0 '0> x= 0 (Curve j) 2=1.0, r= .6,^= .8(CurveII.) 2= 1.0, r= .6, AT= — .8 (Curve III.) As seen, curve I is symmetrical, and with increasing x0 the voltage E remains first almost constant, and then drops off with increasing rapidity. In the inductive circuit series inductance, or, in a con- denser circuit series condensance, causes the voltage to drop off very much faster than in a non-inductive circuit. RESISTANCE, INDUCTANCE, CAPACITY. 65 Series inductance in a condenser circuit, and series con- densance in an inductive circuit, cause a rise of potential. This rise is a maximum for x0 = i .8, or, x0 = — x (the condition of resonance), and the E.M.F. reaches the value, E = 167 volts, or, E = E0z] r. This rise of potential by series reactance continues up to x0 = il.6, or, x0 = — %x, Fig. 42. where E = 100 volts again ; and for x0 > 1.6 the voltage drops again. At x0 = ± -8, x = =f .8, the total impedance of the circuit is r — j (x -f x0} = r = .6, x + x0 = 0, and tan S>0 = 0 ; that is, the current and E.M.F. in the supply circuit are in phase with each other, or the circuit is in electrical resonance. \ Fig. 43. Since a synchronous motor in the condition of efficient working acts as a condensance, we get the remarkable result that, in synchronous motor circuits, choking coils, or reactive coils, can be used for raising the voltage. In Figs. 42 to 44, the polar diagrams are shown for the conditions — E0 = 100, x0 = .6, x = 0 . (Fig. 42) E = 85.7 x = + .8 (Fig. 43) E = 65.7 (Fig. 44) E = 158.1 66 ALTERNA TING-CURRENT PHENOMENA. 47. In Fig. 45 the dependence of the potential, E, upon the difference of phase, oi, in the receiver circuit is shown for the constant impressed E.M.F., E0 = 100 ; for the con- stant receiver impedance, z = 1.0 (but of various phase differences to), and for various series reactances, as follows : x0 = .2 (Curve I.) x0 = .6 (Curve II.) x0 = .8 (Curve III.) xo = 1.0 (Curve IV.) Xo = 1.6 (Curve V.) x0 = 3.2 (Curve VI.) Fig. 44. Since z = 1.0, the current, /, in all these diagrams has the same value as E. In Figs. 46 and 47, the same curves are plotted as in Fig. 45, but in Fig. 46 with the reactance, .*•, of the receiver circuit as abscissas ; and in Fig. 47 with the resistance, r, of the receiver circuit as abscissae. As shown, the receiver voltage, E, is always lowest when x0 and x are of the same sign, and highest when they are of opposite sign. The rise of voltage due to the balance of x0 and x is a maximum for x0= +1.0, x = — 1.0, and r = 0, where RESISTANCE, INDUCTANCE, CAPACITY. L Q. 4— PHASE D FFERENCE IN CONSUMER SIR UIT l-90 80 70 bO 50 40 30 20 10 0 10 20 30 10 50 60 70 bO 90 OEUHE fig. 45. Variation of Voltage at Constant Series Reactance with Phase Angle of Receiver Circuit. Fig. 46. Variation of Voltage at Constant Series Reactance with Reactance of Receiver Circuit. 68 AL TERN A TING-CURRENT PHENOMENA. E = oo ; that is, absolute resonance takes place. Obvi- ously, this condition cannot be completely reached in practice. It is interesting to note, from Fig. 47, that the largest part of the drop of potential due to inductance, and rise to condensance — or conversely — takes place between r = 1.0 and r = .9 ; or, in other words, a circuit having a power Volts E or Amperes I. 160 150 140 130 120 110 100 90 80 70 sfl Fig. 47. Variation of Voltage at Constant Series Reactance with Resistance of Receiver Circuit. factor cos & = .9, gives a drop several times larger than a non-inductive circuit, and hence must be considered as an inductive circuit. 3.) Impedance in series witJi a circuit. 48. By the use of reactance for controlling electric circuits, a certain amount of resistance is also introduced, due to the ohmic resistance of the conductor and the hys- teretic loss, which, as will be seen hereafter, can be repre- sented as an effective resistance. RESISTANCE, INDUCTANCE, CAPACITY. 69 Hence the impedance of a reactive coil (choking coil) may be written thus : — &Q = ro JXoi ZQ = V f0 -j- Xo , where r0 is in general small compared with x0. From this, if the impressed E.M.F. is E0 = e0 +je0'> E0 = Ve02 + e0'2 and the impedance of the consumer circuit is we get the current, /= ^- = -. — and the E.M.F. at receiver terminals, . . ° 7 \ 7 "° (r \ *-\ //„_!_ „ \ ' •^I^o \r ~T ' o) J \*- ~T •*<>/ Or, in absolute quantities, the current is, ~\/(r -f- roy2 -|- (x -j- ^;0)2 V^2 + z02 + 2 (rr0 the E.M.F. at receiver terminals is, E0z E0z V(r + r0)'2 + (x + xoy V^2 + Z0* + 2 the difference of phase in receiver circuit is, x tan oi = - ; r and the difference of phase in the supply circuit is, 49. In this case, the maximum drop of potential will not take place for either x = 0, as for resistance in series, or for r = 0, as for reactance in series, but at an intermediate point. The drop of voltage is a maximum ; that is, E is a minimum if the denominator of E is a maximum ; or, since. zy z0, r0, x0 are constant, if rr0 + xx0 is a maximum, that is, since x = ~Vz2 — r2, if rr0 -f- x0 ~\/z2 — r2 is a maximum. 70 AL TERN A TING CURRENT-PHEXOMENA. A function, f = rr0 -+- x0 V^2 — r2 is a maximum when its differential coefficient equals zero. For, plotting f as curve with r as abscissae, at the point where f is a maxi- mum or a minimum, this curve is for a short distance horizontal, hence the tangens-function of its tangent equals zero. The tangens-function of the tangent of a curve, how- ever, is the ratio of the change of ordinates to the change of abscissae, or is the differential coefficient of the func- tion represented by the curve. / / / / ^ / / ^^«- , " •*^ '"^— ^^~ Z^ £L ,~-— — ' _---* / / ^__ • • •~~ ^ . • ^ — • ,--- J^- ~~ - SiL 9- <-* I. .9 .8 Tf .0 J .4 .3 .2 ., - -.1 - -.2 -.3 - -.4 - -•} ' -.fi -.? -.* 2J Off. 48. Thus we have : — f = rr0 + *0 Vs2 — r2 = maximum or minimum, if Differentiating, we get : — RESISTANCE, INDUCTANCE, CAPACITY. 71 That is, the drop of potential is a maximum, if the re- actance factor, x I r, of the receiver circuit equals the reac- tance factor, *0/r0, of the series impedance. Fig. 49. ''o Fig. 50. 50. As an example, Fig. 48 shows the E.M.F., E, at the receiver terminals, at a constant impressed E.M.F., E0 = 100, a constant impedance of the receiver circuit, s = 1.0, and constant series impedances, Z0= .S-/.4 (Curve I.) Z0 = 1.2 — / 1.6 (Curve II.) as functions of the reactance, x, of the receiver circuit. Fig. 51. Figs. 49 to 51 give the polar diagram for E0 = 100, x = .95, x = 0, x = - .95, and Z0 = .3 -/ .4. 72 ALTERNATING-CURRENT PHENOMENA. 4.) Compensation for Lagging Currents by Shunted Condensance. 51. We have seen in the latter paragraphs, that in a constant potential alternating-current system, the voltage at the terminals of a receiver circuit can be varied by the use of a variable reactance in series to the circuit, without loss of energy except the unavoidable loss due to the resistance and hysteresis of the reactance; and that, if the series reactance is very large compared with the resis- tance of the receiver circuit, the current in the receiver circuit becomes more or less independent of the resis- tance,— that is, of the power consumed in the receiver Fig. 52. circuit, which in this case approaches the conditions of a constant alternating-current circuit, whose current is. /= — " . or approximately, / = — ° . This potential control, however, causes the current taken from the mains to lag greatly behind the E.M.F., and thereby requires a much larger current than corresponds to the power consumed in the receiver circuit. Since a condenser draws from the mains a leading cur- rent, a condenser shunted across such a circuit with lagging current will compensate for the lag, the leading and the lagging current combining to form a resultant current more or less in phase with the E.M.F., and therefore propor- tional to the power expended. RESISTANCE, INDUCTANCE, CAPACITY. 73 In a circuit shown diagrammatically in Fig. 52, let the non-inductive receiver circuit of resistance, r, be connected in series with the inductance, x0 , and the whole shunted by a condenser of condensance, c, entailing but a negligible loss of energy. Then, if E0 = impressed E.M.F.,— the current in receiver circuit is, the current in condenser circuit is, and the total current is — Jxo Jc or, in absolute terms, I0 '•=VfeJ+fe-'/; while the E.M.F. at receiver terminals is, r 52. The main current, 70, is in phase with the impressed E.M.F., E0, or the lagging current is completely balanced, or supplied by, the condensance, if the imaginary term in the expression of I0 disappears ; that is, if This gives, expanded : Hence the capacity required to compensate for the lagging current produced by the insertion of inductance- in series to a non-inductive circuit depends upon the resis- tance and the inductance of the circuit. x0 being constant, 74 ALTERNATING-CURRENT PHENOMENA. with increasing resistance, r, the condensance has to be increased, or the capacity decreased, to keep the balance. r2 4- r2 Substituting c = ^/ " , we get, as the equations of the inductive circuit balanced by condensance : — 7 = r — Jxo and for the power expended in the receiver circuit : — that is, the main current is proportional to the expenditure of power. For r = 0 we have c = x0, or the condition of balance. Complete balance of the lagging component of current by shunted capacity thus requires that the condensance, <:, be varied with the resistance, r; that is, with the varying load on the receiver circuit. In Fig. 53 are shown, for a constant impressed E.M.F., E0 = 1000 volts, and a constant series reactance, x0 = 100 ohms, values for the balanced circuit of, current in receiver circuit (Curve I.), current in condenser circuit (Curve II.), current in main circuit (Curve III.), E.M.F. at receiver terminals (Curve IV.), with the resistance, r, of the receiver circuit as abscissae. RESISTANCE, INDUCTANCE, CAPACITY. 75 IMPRESSED E.M.F. CONSTANT, E0 = IOOO VOLTS. SERIES REACTANCE CONSTANT, X0= IOO OHMS. VARIABLE RESISTANCE IN RECEIVER CIRCUIT. BALANCED BY VARYING THE SHUNTED CONDENSANCE, I. CURRENT IN RECEIVER CIRCUIT. II. CURRENT IN CONDENSER CIRCUIT. III. CURRENT IN MAIN CIRCUIT. JV. E.M.F. AT RECEIVER CIRCUIT. 100 / r. OF RECEIVER CIRCUIT OHMS 10 20 30 40 50 60 70 80 90 100 110 120 130 HO 150 160 170 180 190 200 Fig. 53. Compensation of Lagging Currents in Receiving Circuit by Variable Shunted Condensance. 53. If, however, the condensance is left unchanged, c = x0 at the no-load value, the circuit is balanced for r = 0, but will be overbalanced for r > 0, and the main current will become leading. We get in this case : — r-jx The difference of phase in the main circuit is, — tan u>0 = , «0 which is = 0. 76 ALTERNA TING-CURRENT PHENOMENA. when r = 0 or at no load, and increases with increasing resistance, as the lead of the current. At the same time, the current in the receiver circuit, 7, is approximately con- stant for small values of r, and then gradually decreases. IMPRESSED E.M.F. CONSTANT, EO—IOOO VOLTS. SERIES REACTANCE CONSTANT, Xt, - E, • .V current in condenser circuit, main current, r ° *.(*.+./>) ' ( proportional to the load, T JZI0 — , ' «-— — VI "^ X - ~ ^ -^ - son X -- 581 ^ '' ^xi ? 700 / x^ X-1 / ^ 600/ ^ ' ^ \ •^ 4 ^ -'' ^ •^ L --'• ^ ^ 1 }joo ,x X ^ IMO ^x ^> ^ 100 ,. ^ ^ =iES ST* NCE — r c F R ECE VE H Cl RCL IT, OH AS X 1) . 1 .0 1 (1 1 I. V- 1 V 1) 2 » () HM8 F/3. 50. Constant-Potential — Constant-Current Transformation. Let — ri = 2 ohms = effective resistance of condensance ; r0 = 3 ohms = effective resistance of each of the inductances. We then have : — Power consumed in condensance, I* r± = 200 + .02 r2 ; power consumed by first inductance, 72 r0 = 300 ; power consumed by second inductance, /02r0 = .03 r*. Hence, the total loss of energy is 500 + -05 r2 ; output of system, /2 r = 100 r input, 500 + 100 r -\ effidenCy' 500 + 1W M It follows that the main current, f0, increases slightly by the amount necessary to supply the losses of energy in the apparatus. 82 ALTERNATING-CURRENT PHENOMENA. This curve of current, I0, including losses in transforma- tion, is shown in dotted lines as Curve V. in Fig. 56 ; and the efficiency is shown in broken line, as Curve VI. As shown, the efficiency is practically constant within a wide range. RESISTANCE OF TRANSMISSION LINES.