CHAPTER IX CIRCUITS CONTAINING RESISTANCE, INDUCTIVE REACTANCE, AND CONDENSIVE REACTANCE 53. Having, in the foregoing, re-established Ohm's law and Kirchhoff 's laws as being also the fundamental laws of alternating- current circuits, when expressed in their complex form, E = ZI, or, 7 = YE, and "EE = 0 in a closed circuit, S/ = 0 at a distributing point, where E, I, Z, Y, are the expressions of e.m.f., current, impe- dance, 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 quanti- ties— calculate alternating-current circuits and networks of circuits containing resistance, inductive reactance, and conden- sive reactance in any combination, without meeting with greater difficulties than when dealing with continuous-current circuits. It is obviously not possible to discuss with any completeness all the infinite varieties of combinations of resistance, inductive reactance, and condensive reactance which can be imagined, and which may exist, in a system of network of circuits; there- fore only some of the more common or more interesting combina- tions will here be considered. 1. Resistance in Series with a Circuit 54. In a constant-potential system with impressed e.m.f., Eo = eo-\- je'o, Eo = V^TT^ let the receiving circuit of impedance, Z = r -\- jx, z = y/f^ -\- x^, be connected in series with a resistance, Tq. 60 CIRCUITS CONTAINING RESISTANCE 61 The total impedance of the circuit is then Z + ro = r -\- ro -{- jx; hence the current is _ -go _ Eo _ Eo(r + ro - jx) _ . ~ Z + To " r + To -\- jx " (r + ro)2 + x^ ' and the e.m.f. of the receiving circuit becomes Eo{r + jx) Eo { r(r + ro) -\- x^ -\- jrpx} E ^ IZ = r-\-rQ-\-jx (r + ro)2 + x2 EqIz"^ + rro-\-jrQx] 3^ ' or, in absohite values we have the following: Impressed e.m.f., Eo = ^^7T^; current, £"0 Eo I = V(> + ro) 2 + x^ Vz^ + 2 rro + ro^ ' e.m.f. at terminals of receiver circuit, , 7m^^^~ Eoz £/ = £/o (r + ro)- + a;2 Vz^- + 2n-o + ''o- ' x difference of phase in receiver circuit, tan 6 = - x difference of phase in supply circuit, tan ^0 = — r — since in general, imaginary component tan (phase) = , — ' 7 ^ real component (a) If X is negligible with respect to r, as in a non-inductive receiving circuit, / = — ; > E — Eo ro r + ro and the current and e.m.f. at receiver terminals decrease steadily with increasing ro. (6) If r is neghgible compared with x, as in a wattless receiver circuit, / = — . — , E = Eo — /-- — ; Vro' + x^ Vro^ + x^' or, for small values of ro, I = ^, E = Eo; X 62 ALTERNATING-CURRENT PHENOMENA that is, the current and e.m.f. at receiver terminals remain approximately constant for small values of ro, and then de- crease with increasing rapidity. In the general equations, x appears in the expressions for / and E only as x^, 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 lag- ging current, or an inductive circuit. For a given impedance, z, of the receiver circuit, the current, /, and e.m.f., E, are smaller the larger the value of r; that is, the less the difference of phase in the receiver circuit. IMPRESSED E. M.F. CONSTANT, ^0-- lOO 100 90 IMPEDANCE OF RECEIVER CIRCUIT CONSTANT, LINDRESISTAJUGECONSTAJ^T n =.2 V._^ n = .8 2 = (.0 y ■— n .2 ' ' g80 u |70 < Q 60 s. J s ^ ^ V _ Co -..8 ■ n DU( iTAN CE RE/ CT* NCE CON DEN SAN CE- •20 ao 0 3*= OH IS +1. .9 .8 .7 .6 ^ A .S .2 .1 -.1 -.2 - -.3 ■ -.4 - -.5- -.6 - -.8 • -.M Fig. 50. — Variation of voltage at constant series resistance with phase relation of receiver circuit. As an instance, in Fig. 50 is shown the e.m.f., E, at the re- ceiver circuit, for £"0 = const. = 100 volts, z = \ ohm; hence / = E, and (a) To = 0.2 ohm (Curve I) (6) ro = 0.8 ohm (Curve II) with values of reactance, x = \/z^ — r^, for abscissae, from X = -1- 1.0 to a; = — 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 resist- ance causes but a very small drop of potential. Hence the control of a circuit by series resistance depends upon the difference of phase in the circuit. CIRCUITS CONTAINING RESISTANCE 63 For ro = 0.8 and x = 0, x = -^ 0.8, x = - 0.8, the vector diagrams are shown in Figs. 51 to 53. In these Figs. OEq is the supply voltage, OE3 the voltage con- sumed by the line resistance, and OE the receiver voltage, with its two components, OEi in phase and OE2 in quadrature with the current. Eo E Fig. 51. M 0 E, Ej E2 El E, Eo Fig. 52. Fig. 53. 2. Reactance in Series with a Circuit 52. In a constant potential system of impressed e.m.f., Eo = eo-\- je'o, Eo = VeTT eo' let a reactance, Xo, be connected in series in a receiver circuit of impedance, Z ^ r -^jx, z = Vr2"+x2- Then, the total impedance of the circuit is Z -i- jxo = r -{- j {x + Xo), and the current is J _ Eq Eo Z -\- jxo ~ r -{- j (x + Xo) while the difference of potential at the receiver terminals is r 4- jx E = IZ = E^ Or, in absolute quantities, current, £^0 / = • r -\- j{x + Xo) Eo Vr^ + (a; + Xo)^ Vz"^ + 2 xxo + Xo^' e.m.f. at receiver terminals, E ^2 _j_ y.1 Eoz W^+{x-\- Xo)2 Vr^ -\-2xxo-\- Xo"^' 64 ALTERNATING-CURRENT PHENOMENA difference of phase in receiver circuit, X tan 6 — -; r difference of phase in supply circuit, tan do = - r (a) If X is small compared with r, that is, if the receiver circuit is non-inductive, / and E change very little for small values of Xo; but if X is large, that is, if the receiver circuit is of large re- actance, / and E change considerably with a change of Xq. (b) If X is negative, that is, if the receiver circuit contains condensers, synchronous motors, or other apparatus which produce leading currents, below a certain value of Xq the de- nominator in the expression of E becomes Eo', that is, the reactance, Xo, raises the voltage. (c) E = Eo, or the insertion of a series reactance, Xo, does not affect the potential difference at the receiver terminals, if V^' + 2x Xo + xo^ = z; or, Xo = — 2x. 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 = £"0, / = — If Xo < — 2x,it raises, ifxo > — 2x, it lowers, the voltage. We see, then, that a reactance inserted in series in an alter- nating-current circuit will always lower the voltage at the receiver terminals, when of the same sign as the reactance 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 E = ^0^ _ Eo Vr^ + xo^ L ' /Xo\ 2 p j. l/xo\2 , 3/a:o\^ , 1 = ^°ii-2(7) +8(7-) - + •••! ( /. . = = (1 -f- x) " ^- expanded by the binomial theorem (1 -f x)" = 1 -f- nx + ''^\~ ^^ X' -f- . CIRCUITS CONTAINING RESISTANCE 65 Therefore, if Xo is small compared with r, Eo- E _ 1 /^\ - Eo ~ 2\r/ ' That is, the percentage drop of potential by the insertion of reactance in series in a non-inductive circuit is, for small values of reactance, independent of the sign, but proportional to the square of the reactance, or the same whether it be induc- tive reactance or condensive reactance. 56. As an example, in Fig. 54 the changes of current, /, and of e.m.f. at receiver terminals, E, at constant impressed e.m.f., VOLTS E OR AMPERES I IMPRESSED E.M.F. CONSTANT, Eo 100 17 0 I r=1.0 x = 0 ■^ = n r = .6 a;=f.8 HI r = -6 x=-.8 1.0 r\ ^ 16 0 / ^, / \ 15 0 / V / \ 14 0 / \ 1 \ 13 0 / \ 1 \ 12 0 / \ / yi o/ 100 90 in 80 / \ / I "/ ^ ^ / IK^ \ ■s / / A / \ s 0 s. s, \ / / y y / 7 0 s s \ |60 O w40 / r ^ y A / 6 0 s N. N ^ ^ / ^ ^ '^ / 6 0 \ N, --' ^ ^ ^ 4 0 ^ . rl 3 0 ^30 >20 10 t— ■ 2 0 1 0 ajfl + 3.0 2,8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 .8 .6 .4 +.2 0 -.2 .4 .6 .8 1.0 1.2-1.4 ohms inductance-'— reactance-^condensance Fig. 54. E^, are shown for various conditions of a receiver circuit and amounts of reactance inserted in series. Fig. 54 gives for various values of reactance, a^o (if positive, inductive; if negative, condensive), the e.m.fs., E, at receiver terminals, for constant impressed e.rn.f., £'0 = 100 volts, and the following conditions of receiver circuit: z = 1.0, r = 1.0, a; = 0 (Curve I) z = 1.0, r = 0.6, X = 0.8 (Curve II) z = 1.0, r = 0.6, X = - 0.8 (Curve III). 66 ALTERNATING-CURRENT PHENOMENA As seen, curve I is symmetrical, and with increasing Xo the voltage E remains first almost constant, and then drops off with increasing rapiditj^ In the inductive circuit series inductive reactance, or in a condenser circuit series condensive reactance, causes the voltage to drop off very much faster than in a non-inductive circuit. Series inductive reactance in a condenser circuit, and series condensive reactance in an inductive circuit, cause a rise of potential. This rise is a maximum for a:o = ± 0.8, or Xq = — X (the condition of resonance), and the e.m.f, reaches the value £■ = 167 volts, or E = Eq— This rise of potential by series reactance continues up to Xo = + 1.6, or, Xo = — 2x, where E = 100 volts again; and for Xq > 1.6 the voltage drops again. At rro = ± 0.8, x = + 0.8, the total impedance of the circuit is r — j {x + Xo) = r = 0.6, x -{- Xo = 0, and tan do = 0; that 4 Eo Ex / E ^^ r Er 0 Fig. 55. Fig. 56. Fig. 57. is, the current and e.m.f. in the supply circuit are in phase with each other, or the circuit is in electrical resonance. Since a synchronous motor in the condition of efficient work- ing acts as a condensive reactance, we get the remarkable result that, in synchronous motor circuits, choking coils, or reactive coils, can be used for raising |he voltage. In Figs. 55 to 57, the vector diagrams are shown for the conditions Eo = 100, Xo = 0.6, X = 0 a; = + 0.8 a; = - 0.8 (Fig. 48) E = 85.7 (Fig. 49) E = 65.7 (Fig. 50) E = 158.1. CIRCUITS CONTAINING RESISTANCE 67 57. In Fig. 58 the dependence of the potential, E, upon the difference of phase, 9, in the receiver circuit is shown for the constant impressed e.m.f., Eq = 100; for the constant receiver impedance, z = 1.0 (but of various phase differences 6), and for various series reactances, as follows: xo = 0.2 (Curve I) Xo = 0.6 (Curve II) Xo = 0.8 (Curve III) a;o - 1.0 (Curve IV) Xo = 1.6 (Curve V) Xo = 3.2 (Curve VI). Since z = 1.0, the current, /, in all these diagrams has the same value as E. 180 170 160 150 140 130 H-120 ^110 ^100 1 90 tr O 80 UJ w TO § 60 " 50 40 30 20 10 ol \ ^ < 1 I I 1 ^ 1 — J > ' ' ' 1 • > > ' — ' ^w=+90 80 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 degrees lag-«- phase difference in consumer circuit-*- lead Fig. 58. In Figs. 59 and 60, the same curves are plotted as in Fig. 58, but in Fig. 59 with the reactance, x, of the receiver circuit as abscissas; and in Fig. 60 with the resistance, r, of the receiver circuit as abscissas. As shown, the receiver voltage, E, is always lowest when Xa and X are of the same sign, and highest when they are of opposite sign. IMPRESSED E.M.F. CONSTANT, Eo = 100 IMPEDANCE OF RECEIVER CIRCUIT CONSTANT, I, a;o='.2 IV, a:o=1.0 ^= ^•° ni,Xo=-8 VI, a;o=3.2 / ' 1 / / / , / // J^ ^ i ^ / ^ / / } /; :/ y / _l,3 ILJ — ' ^ 'A 40 V.'xo =r 1- i ^ y\ = 3" ^ ^^ 30 Vi;a;o 2 -^ 10 0 .1 .2 .3 .4 .5 .6 LAGGING CURRENT -«- .7 .8 .9 10 .9 .8 _ resistance of . consumer circuit Fig. 60. ,7 .6 .5 .4 .3 .2 .1 .0 LEADING CURRENT CIRCUITS CONTAINING RESISTANCE G9 The rise of voltage due to the balance of Xo and a; is a maxi- mum for a;o = + 1.0, a: = — 1.0, and r = 0, where £ = c» ; that is, absolute resonance takes place. Obviously, this condi- tion cannot be completely reached in practice. It is interesting to note, from Fig. 60, that the largest part of the drop of potential due to inductive reactance, and rise to condensive reactance — or conversely — takes place between r = 1.0 and r = 0.9; or, in other words, a circuit having a power-factor cos 6 = 0.9 gives a drop several times larger than a non-inductive circuit, and hence must be considered as an inductive circuit. 3. Impedance in Series with a Circuit 58. 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 hysteretic loss, which, as will be seen hereafter, can be represented as an effective resistance. Hence the impedance of a reactive coil (choking coil) may be written thus: Zo = ro -i- jxo, Zo = -y/ro^ + Xo^, where ro is in general small compared with .ro. From this, if the impressed e.ra.f. is Eo ^ eo + je'o, Eq = \/eo^ + eo'^ and the impedance of the consumer circuit is Z = r -\- jx, z = \/rM-~^> we get the current , _ Eq _ ^ Eo . Z + Zo ~ (r + ro) +j{x + x^) and the e.m.f. at receiver terminals, F - F ^ - F ''+-^'^ Z -\-Zo . " (r + ro) + j (x -\- Xo) Or, in absolute quantities, the current is, Eo Eo I = V(r + ro)2 + (x + xo)^ Vz^ + 2o' + 2 (rro + xxo) the e.m.f. at receiver terminals is Eoz Eoz E = V{r + ro)2 + {x + Xo)2 Vz^ + ^o' + 2 (rro + xxo) 70 ALTERNATING-CURRENT PHENOMENA the difference of phase in receiver circuit is tan 6 = —', r and the difference of phase in the supply circuit is X + Xq tan 0 r + ro 59. In this case, the maximum drop of potential will not take place for either a; = 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, -EJ is a minimum if the denominator of ^ is a maximum; or, since z, 2o, ^o, Xq are constant, if rro + xxo is a maximum, that is, since x = s/ z^ — r^, if rro + xiis/ f- — r^ is a maximum. A function, / =^ rro + Xq ■s/z^ — r^, is a maximum when its differential coefRcient equals zero. For, plotting / as curve with values of r as abscissas, at the point where / is a maximum or a minimum, this curve is for a short distance horizontal, hence the tangens-f unction of its tangent equals zero. The tangens-function of the tangent of a curve, however, is the ratio of the change of ordinates to the change of abscissas, or is the differential coefficient of the function represented by the curve. Thus we have / = rro + X(s-\/z^ — r^ is a maximum or minimum, if 2r) = 0; ro\/z^ — r^ — XqT = VoX — Xor = 0, or, r -7- X ^ ro -T- Xo. That is, the drop of potential is a maximum, if the reactance X Xn factor, -, of the receiver circuit equals the reactance factor, — , of the series impedance. 60. As an example. Fig. 61 shows the e.m.f., E, at the receiver terminals, at a constant impressed e.m.f., Eo = 100, a constant dr Differentiating, we get ,1 Xo ^ ^« + 2 V.^ - r^^ or, expanded. CIRCUITS CONTAINING RESISTANCE 71 impedance of the receiver circuit, z = 1.0, and constant series impedances, Zo = 0.3 + j 0.4 (Curve I) Zo = 1.2 +j 1.6 (Curve II) as functions of the reactance, x, of the receiver circuit. 150 140 130 120 110 100 90 80 70 / / 1 / / / / A / ,.-- ^ Zo __ c +.4 j^ / 60 50 40 / ^ / :+-! Ri . ^ -^ ^ lo — 30 20 10 1. .9 .8 .7 .6 .5 ,4 .3 .2 .1 0 -.1 -.2 -.3 -.1 -.5 -.6 -.7 -.8 -.9-L X — ^ Fig. 61. E // Uo / /> 'Uo // / f I 0 ] r tro Fig. 62. Fig. 63. 72 ALTERNATING-CURRENT PHENOMENA Figs. 62 to 64, give the vector diagram for ^o = 100, x = 0.95, X =. 0,x = - 0.95, and Zo = 0.3 + 0.4 j. 4. Compensation for Lagging Currents by Shunted Condensive Reactance 61. We have seen in the preceding 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 with 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 resistance of the receiver circuit, the current in the receiver circuit becomes more or less inde- pendent of the resistance — that is, of the power consumed in the receiver circuit, which in this case approaches the conditions of a constant alternating-current circuit, whose current is I = Eo ■\/r^ -}- xo^ , or, approximately, / Eo Xq 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 current in leading phase, a condenser shunted across such a circuit carrying cur- rent in lagging phase compensates for the lag, the leading and the lagging current combining to form a resultant current more CIRCUITS CONTAINING RESISTANCE 73 or less in phase with the e.m.f., and therefore proportional to the power expended. In a circuit shown diagrammatically in Fig. 65, let the non- inductive receiver circuit of resistance, r, be connected in series with the inductive reactance, Xo, and the whole shunted by a condenser C of condensive reactance, Xc, entailing but a negligible loss of power. Fig. 65. Then, if Eo = impressed e.m.f., the current in receiver circuit is T Eo I = . r— > r + jxo the current in condenser circuit is y Eo li = : — y JXc and the total current is /o = / + /, = £;« ' ^ = Ed I = Eo Vr2 + Xo"" /i ^0. . r + jxo J. jx Xo or, in absolute terms, /o = Eq 'ir^ + xo^ xj\' \ \r2 + xoV "^ V Xo while the e.m.f. at receiver terminals is r E = Ir = Eo E = xj ' Eor r + jxo Vr^ + xo^ 62. The main current, 7o, is in phase with the impressed e.m.f., Eo, or the lagging current is coznpletely balanced, or supplied by, the condensive reactance, if the imaginary term in the expression of 7o disappears; that is, if Xo 1 r^ + .ro^ = 0. 74 ALTERNATING-CURRENT PHENOMENA This gives, expanded, r^ + xo^ Xq Hence the capacity required to compensate for the lagging current produced by the insertion of inductive reactance in series with a non-inductive circuit depends upon the resistance and the inductive reactance of the circuit. Xo being constant, with increasing resistance, r, the condensive reactance has to be increased, or the capacity decreased, to keep the balance. Substituting r^ + Xq^ Xc = > Xq we get, as the equations of the inductive circuit balanced by condensive reactance, Eo ^ Eo(r - jxo) J ^ -^ r + jxo ~ r^ + .To" ' ' "" jEoXo , £"03^0 / = ^ r2 + V' ^' " r^ + Xo'' J _ Epr _ Eor , r' + xo" r' + Xi?' r + jxo A/r^ + Xo^' and for the power expended in the receiver circuit, 72 _ _E0, and the main current will be- come leading. IMPRESSED E.M.F. CONSTANT, E„ = i6oo'voiiTs. F VARIABLE RESISTANCE IN RECEIVER CIRCUIT. BALANCED BY VARYING THE SHUNTED CONDENSANC 1. CURRENT IN RECEIVER CIRCUIT. II. CURRENT IN CONDENSER CIRCUIT. III. CURRENT IN MAIN CIRCUIT. IV. E.M.F. AT RECEIVER CIRCUIT. 10 .100( !)00 9 F^ - = 8 800 \ "^ IV , . L— — - r 7 700 s s ■^ ^ <-^ ^ Sfi 600 \> ^ ^ .1 n ° 500 400 h /I N ^ III ^ J .^ s <4 > / ^ \ ^ — - — 1 — — ' -— . 3 300 ^ ry- Jl ^ ~~ 2 200 / "~" ■^ ~- =^ 1 100 / / resist'ance — •-''l OF receiv 1 er circuit 1 1 OHMS Fig. 66. — Compensation of lagging currents in receiving circuit by variable shunted condensance. We get in this case. Xc — Xq', r + ja^o I Eo ■y/r^ + xo^' 7. = i^\ Xo Ii J T 1 T Eor h Eor Xoixo - jr) XoVr^ -h xo^ E = Ir = r — > E Eor r + jxo V r^ + xo^ The difference of phase in the main circuit is tan do = r — > Xo 76 ALTERNATING-CURRENT PHENOMENA which is = 0, 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, /, is approximately con- stant for small values of r, and then gradually decreases. In Fig. 67 are shown the values of /, /i, 7o, E, in Curves I, II, III, IV, similarly as in Fig. 60, for Eq = 1000 volts, Xc — Xq = 100 ohms, and r as abscissas. MPRESSED E.M.F. CONSTAINT Eo =1000 VOLTS. SHUNTED CONDENSANCE CONSTANT, C= 100 OHMS, VARIABLE RESISTANCE. IN RECEIVER CIRCUIT. 1. CURRENT IN RECEIVER CIRCUIT. II. CURRENT IN CONDENSER CIRCUIT. III. CURRENT IN MAIN CIRCUIT. IV.E.M.F. AT RECEIVER CIRCUIT. AMPERES VOL 6 II. 10 -^ ^ _ 800 ^ — ' - 7 6 J^ r^ ■^ "^ £00 fiOO 400 -300 .200 -lOO, ^ ^ ^ 6 'V. ^ '^ -^ l'> / ■~~- .^ , / / / RESISTANCE r— OF RECEIVER CIRCUH r, OHMS. 1 1 1 M M 1 1 1 Fig. 67. 5. Constant Potential — Constant-current Transformation 64. In a constant potential circuit containing a large and constant reactance, x^s, and a varying resistance, 7', the current is approximately constant, and only gradually drops off with increasing resistance, r — that is, with increasing load — but the current lags greatly behind the voltage. This lagging current in the receiver circuit can be supplied by a shunted condensance. Leaving, however, the condensance constant, Xc = Xo, so as to balance the lagging current at no-load, that is, at r = 0, it will overbalance with increasing load, that is, with increasing r, and thus the main current will become leading, while the receiver current decreases if the impressed voltage, Eo, is kept constant. Hence, to keep the current in the receiver circuit entirely con- stant, the impressed voltage, Eo, has to be increased with in- CIRCUITS CONTAINING RESISTANCE 77 creasing resistance, r; that is, with increasing lead of the main current. Since, as explained before, in a circuit with leading current, a series inductive reactance raises the potential, to maintain the current in the receiver circuit constant under all loads, an inductive reactance, X2, inserted in the main circuit, as shown in the diagram, Fig. 68, can be used for raising the voltage, Eo, with increasing load, and by properly choosing the inductive and the condensive reactances, practically constant current at varying load can be produced from constant voltage supply, and inversely. Fig. 68 The generation of alternating-current electric power almost always takes place at constant potential. For some purposes, however, as for operating series arc circuits, and to a limited extent also for electric furnaces, a constant, or approximately constant, alternating current is required. Such constant alternating currents can be produced from constant potential circuits by means of inductive reactances, or combinations of inductive and condensive reactances; and the investigation of different methods of producing constant alternating current from constant alternating potential, or inversely, constitutes a good illustration of the application of the terms "impedance," "reactance," etc., and offers a large number of problems or examples for the application of the method of complex quantities. A numl)er of such are given in "Theory and Calculation of Electric Circuits."