CHAPTER X RESISTANCE AND REACTANCE OF TRANSMISSION LINES 65. In alternating-current circuits, voltage is consumed in the feeders of distributing networks, and in the lines of long- distance transmissions, not only by the resistance, but also by the reactance, of the line. The voltage consumed by the resistance is in phase, while the voltage consumed by the react- ance is in quadrature, with the current. Hence their in- fluence upon the voltage at the receiver circuit depends upon the difference of phase between the current and the voltage in that circuit. As discussed before, the drop of potential due to the resistance is a maximum when the receiver current is in phase, a minimum when it is in quadrature, with the voltage. The change of voltage due to line reactance is small if the current is in phase with the voltage, while a drop of potential is produced with a lagging, and a rise of potential with a leading, current in the receiver circuit. Thus the change of voltage due to a line of given resistance and reactance depends upon the phase difference in the receiver circuit, and can be varied and controlled by varying this phase difference; that is, by varying the admittance, Y = g — jh, of the receiver circuit. The conductance, g, of the receiver circuit depends upon the consumption of power — that is, upon the load on the circuit — and thus cannot be varied for the purpose of regu- lation. Its susceptance, b, however, can be changed bj' shunt- ing the circuit with a reactance, and will be increased by a shunted inductive reactance, and decreased by a shunted con- densive reactance. Hence, for the purpose of investigation, the receiver circuit can be assumed to consist of two branches, a conductance, g, — the non-inductive part of the circuit — shunted by a susceptance, h, which can be varied without expenditure of energy. The two components of current can thus be considered separately, the energy component as deter- 78 TRANSMISSION LINES 79 mined by the load on the circuit, and the wattless component, which can be varied for the purpose of regulation. Obviously, in the same way, the voltage at the receiver circuit may be considered as consisting of two components, the power component, in phase with the current, and the wattless com- ponent, in quadrature with the current. This will correspond to the case of a reactance connected in series to the non-inductive part of the circuit. Since the effect of either resolution into components is the same so far as the line is concerned, we need not make any assumption as to whether the wattless part of the . receiver circuit is in shunt, or in series, to the power part. Let Zo = To -{- jxo = impedance of the line; Zo = V ro^ + Xo^', Y = g — jb = admittance of receiver circuit; y = ^g' + b'; Eo = Co -{- je'o = impressed voltage at generator end of line ; Eo - Veo' + eo'2; E ^ e -\- je' = voltage at receiver end of line; E = Ve2 + e'2. h = H + ji'a = current in the line; /o = -nAo- + io'^. The simplest condition is the non-inductive circuit. 1. Non-inductive Receiver Circuit Supplied over an Inductive Line 66. In this case, the admittance of the receiver circuit is Y = g, since 6 = 0. We have then current, lo ^ Eg; impressed voltage: Eo ^ E + Zoh = E{1 -|- Zog). Hence — voltage at receiver circuit, ^ ^ Eo ^ Eo I -{-Zog I -\- gro -{- jgxo' current, .° 1 + Zog 1 + f/ro + jgxo 80 ALTERNATING-CURRENT PHENOMENA Hence, in absolute values — voltage at receiver circuit, h = ^ V(l + gror + g^xo^' current, J _ Eog " V(l + groP + g^xo' The ratio of e.m.fs. at receiver circuit and at generator, or supply circuit, is E 1 Eo V(l + gro)' + g^xo" and the power delivered in the non-inductive receiver circuit, or output, P - IE - ^"'^ As a function of g, and with a given E^, n, and Xo, this power is a maximum, if dg "• that is, — 1 + gV + g^xo'^ = 0; hence, conductance of receiver circuit for maximum output, 1 1 gm = Resistance of receiver circuit, r^ = — = 2o; gm and, substituting this in P, Eo^ Eo Maxnnum output, P, 2 (ro + zo) 2 {to + Vro^ + xo^}' and ratio of e.m.f. at receiver and at generator end of line, _ E^ ^m 771 IJO #+i-:)' efficiency, : — = — -. — ■' That is: The output which can he transmitted over an inductive line of resistance, ro, and reactance, Xo — that is, of impedance, Zo — into a TRANSMISSION LINES 81 non-inductive receiver circuit, is a maximum if the resistance of the receiver circuit equals the impedance of the line, r — Zo, and is P -— ^?1_. ^'^ 2 (ro + Zo) The output is transmitted at the efficiency of Zo ro + Zo and with a ratio of e.m.fs. of #+:-:) NON-INDUCTI SUPPLIED OVER IN VE RECEIVER CIRCUIT DUCTIVE LINE O." IMPEDANCE ■OD' K AND OVER NON-INDUCtTvE LH^/e OF RESISTANCE K = 2.5 CURVE 1. E. M. F. AT RECEIVER CIRCUIT, INDUCTIVE LINE II IV. II 1' |i II NON-INDUCTIVE " ■^ " II. OUTPUT IN II II INDUCTIVE' 11 V. II II II I. NON-INDUCTIVE I ^ ^ 'z o K II VI. II .1 II NON-INDUCTIVE / y CO O / A to 7 o / a. > / O o / >- o H / u co / u. U- O ->- 1000 ^ y^ ^ \ 100? "^^ ==* /, / /> \ 90^ 900 ^•~^ ^ '^ ■^ \ 80% soo / ^ ^ 70?; 700 / ■\ N ^ m% GOO / f > N. s \, 50^ 500 ^ / \ \ iOt 400 / \ \ 30^ 300 / \ 20? 200 / tio.^ 100 / cu ?RE ^T n l INE ; \o AMP ERE S \ 0 10 20 30 40 50 60 70 fiO SO 100 110 120 130 140 150 ICO 170 180 Fig. 69. — Non-inductive receiver-circuit supplied over inductive line. 67. We see from this that the maximum output which can be delivered over an inductive Hne is less than the output de- livered over a non-inductive line of the same resistance — that is, which can be delivered by continuous currents with the same generator potential. In Fig. 69 are shown, for the constants, 82 ALTERNATING-CURRENT PHENOMENA Eo = 1000 volts, Zo = 2.5 + 6j; that is, ro = 2.5 ohms, Xo = 6 ohms, Zo = 6.5 ohms, with the current 7o as abscissas, the values. e.m.f . at receiver circuit, E, (Curve I) ; output of transmission, P, (Curve II) ; efficiency of transmission, (Curve III). The same quantities for a non-inductive line of resistance, To = 2.5 ohms, Xo = 0, are shown in Curves IV, V, and VI. 2. Maximum Power Supplied over an Inductive Line 68. If the receiver circuit contains the susceptance, b, in addition to the conductance, g, its admittance can be written thus: Y = g - jh, y = Vg^ + b^. Then, current, /o = EY; Impressed voltage, E^ =^ E -\- IqZq = £"(1 + YZq). Hence, voltage at receiver terminals, p, _ __j^0 Eq . ^ ~ 1 + FZo " (1 + roir + xob) -{-jixog - rob)' current, EoY ^ Eo{g-Jb) . . " 1 + YZo (1 + Tog + Xob) + j{xog - nb) ' or, in absolute values, voltage at receiver circuit, E= ^' V(l + rog-{-xob)^-\- (xog - nb)'' current. T = F I ^' + ^' ^0 ^«^ (1 + ;.„^ + a;o6)2 + (xog - rob)'' ratio of e.m.fs. at receiver circuit and at generator circuit, ^E_^ 1 " ^0 V(l + rog-i-x,b)'+{xog- rob)" and the output in the receiver circuit is P = E'g = Eo'a'g. 69. (a) Dependence of the output upon the susceptance of the receiver circuit. At a given conductance, g, of the receiver circuit, its output, P = Eo'a'g, is a maximum if a' is a maximum; that is, when / = ~ = (1 + re? + Xob)' + (xog - ro6)2 a is a minimum. TRANSMISSION LINES 83 The condition necessary is db ^' or, expanding, a;o(l + rog + Xoh) - ro(xog - rob) = 0. Hence Susceptance of receiver circuit, ^ - "ro^ + xo^ ~ ~ Zo'~ ~^'' or b -\- bo = 0, that is, if the sum of the susceptances of hne and of receiver circuit equals zero. Substituting this value, we get ratio of e.m.fs. at maximum output, _ E^ 1 ""'- Eo~ zoig + goV maximum output, p, = ^"'^ current. h = zoKg + go)" EoY Eo{g + jbo) ' l + ZoK I -^ in + jxo) ig -{- jbo) Eojg + jbo) . {1 -\-rog - xobo) - j (robo + xog) ' -^4 g' + W and, since. (1 + rofif — xobo)^ + (robo + xog) , Xo ro ^0 = ^2.^2' ^0 2' ro' + xo^ ^" ro^ + xo^ it is, (1 + rog - xobo^ + (ro6o + XogY = (vog + 1 - ^^2 ^ 52) + (,-r^^p^2 + ^og) = ro- (g + go)' + Xo'- (g + go)' = 2o' (g + go)', Thus, it is, current, ^ ^ ^oVg' + bo'. ° 2o(g + go) ' 84 ALTERNATING-CURRENT PHENOMENA phase difference in receiver circuit, b bo tan 9 = - = : g g phase difference in generator circuit, x + xo _ bojy^ - ijo^) tan c7o — ; — 9 1 9 * 70. (b) Dependence oj the output upon the conductance of the receiver circuit. At a given susceptance, b, of the receiver circuit, its output, P = Eo^ a^g, is a maximum if = 0, or ^(p) = 0. c^g ' dg or / 1 \ ^ d_ /(I + ro{7 + xoby + (^oflr - ^06)^ ^ that is, expanding, (1 + rog + xoby + (a:oSr - ro6)' - 2 ^(ro + ro'g + x^'^g) = 0; or, expanding, {b + 6o)2 = g^ - go'; g = Vg^' + (6 + bo)'. ' Substituting this value in the equation for a, §G8, we get- ratio of e.m.fs., 1 zo^[2{go- + (6 + 60)^ + goVgJ^ + 6o)T = 1 _ yo power, • EoV EoW _ ' 2(^ + ^0) 2Msfo + \/sfo^+(6 + 6o)'} As a function of the susceptance, 6, this power becomes a db dPo maximum for -777 = 0, that is, according to §69 if 6 = - 60. Substituting this value, we get b = - bo, g = go, y = yo, hence: Y = g - jb = go + jbo] x = — Xo, r = To, z = Zo, Z ^ r -\- jx = ro — jxo] TRANSMISSION LINES 85 substituting this value, we get — ratio oi e.m.ls., «,„ = ^ — = ^ — • Z Qq Z /'o power, P,„ = ^^; that is, the same as with a continuous-current circuit; or, in other words, the inductive reactance of the hne and of the receiver circuit can be perfectly balanced in its effect upon the output. 71. As a summary, we thus have: The output delivered over an inductive line of impedance, ^0 = ?'o + i^^'o, into a non-inductive receiver circuit, is a maxi- mum for the resistance, r = z^, or conductance, g = ijq, of the receiver circuit, and this maximum is 2 (ro + 2o) at the ratio of voltages. With a receiver circuit of constant susceptance, h, the out- put, as a function of the conductance, g, is a maximum for the conductance, and is 2 Cg + (J.)' at the ratio of voltages, V2g{g + g^) With a receiver circuit of constant conductance, g, the output, as a function of the susceptance, 6, is a maximum for the susceptance 6 = — 6o, and is 2;o^ ((7 + g^y at the ratio of voltages, _ 1 " ~ Zo (fif + f/u) 86 ALTERNATING-CURRENT PHENOMENA The maximum output which can be delivered over an induc- tive line, as a function of the admittance or impedance of the receiver circuit, takes place when Z = r^ — jxo, or Y = go -\- jho; that is, when the resistance or conductance of receiver circuit and line are equal, the reactance or susceptance of the receiver Eo^ circuit and line are equal but of opposite sign, and is P = ^TT' or independent of the reactances, but equal to the output of a RA7 10 OF P )TENTI OU iL f TPU AT XPlANb 1 heceivInq'anc se (DING E .ND OF .IN : OF IMF ED/l NCE 242.5+6/ AT 1 CONST^ NT ^ ^ ^mPRESSED E MJ ■> E 0=1 300 \ /' "^ r<^ \ ' 11 III IV NOl^-INDUCTI RECEIVER Clf /E RECEIVER ClRCUIT^J CUITi CF 6USCEPTA^^CE ?>s U2 a 1.5 lA 1.3 1.2 1.1 T n / Vll VIII NON-INDUCTIVE REf I" 1 '{> = EiVER circuit; i45j ^ „^ / \ ' ■ sh NC N-IN Due IVE Line y ^ ■«s >/ \ ^o ^ \ ^> ^ \ / \ ^ \ P / N ><■ "" ^ **= =^ / A ^ \J ■ n / ^ . ^ '\fM n x \^ ,o / / '^ ■^ — ^^ ia. — / ^/ "~" -«. -~~. Ji_ .6 4 ._ IV ,^ — - -- / J z^ ~ — ~ ? V / ^ ^ «> 1 / \^ < ,1 1 / ^ ^ 0 ^ c ONC >uc "AN ;e c L5 •CE vef Clf cut T L .01 .03 .03 .01 .05 .06 ,07 .OS .09 .10 .11 .13 .13 .11 .15 16 ,1T Fig. 70. — Variation of the potential in line at different loads. continuous-current circuit of equal line resistance. The ratio of voltages is, in this case, a = ^ — , while in a continuous- ^ To current circuit it is equal to 0.5. The efficiency is equal to 50 per cent. 72. As an example, in Fig. 70 are shown for the constants Eo = 1000 volts, and Zo = 2.5 + Gj; that is, for To = 2.5 ohms, Xo = Q ohms, Zq = 6.5 ohms, TRANSMISSION LINES 87 and with the variable conductances as abscissas, the values of the output, in Curve I, Curve III, and Curve V; ratio of voltages, in Curve II, Curve IV, and Curve VI; Curves I and II refer to a non-inductive receiver circuit ; Curves III and IV refer to a receiver circuit of constant susceptance 6 = 0.142 output" p Xnd RATid OF poteJjtiaI a AT receiv^mgjInd SFNniNG FND OF LINE OF IMPEDANCa 7„=5. E+ V' AT CONSTANT IMPRESSED E.M.F. E- — ^ 900 40 SO / / ^ ■^ ^■i- 800 70/ / / -^ N 700 80 / ^ ^ 600 '50 > y III 500 SO i 40 / ' -<-' -^ 400 30 / / ^ ^ 300 ^ 20 / ^ 200 lu / ^ ^ 100 u ^ C ;oNC 1 1 5UC 1 1 ■AN :e c F B :cE VER ^ CUJ ■■£7 0 Fig. 72. .03 ,03 M. .05 .00 .07 .OS -Load characteristics of transmission lines. 4. Control of Receiver Voltage by Shunted Susceptance 74. By varying the susceptance of the receiver circuit, the voltage at the receiver terminals is varied greatly. Therefore, since the susceptance of the receiver circuit can be varied at will, it is possible, at a constant generator voltage, to adjust the receiver susceptance so as to keep the voltage constant at the receiver end of the line, or to vary it in any desired manner, and independently of the generator voltage, within certain limits. 90 ALTERNATING-CURRENT PHENOMENA The ratio of voltages is E 1 \2 Eo V(l + rog + a;o& y + (xog - nhy If at constant generator voltage Eo the receiver voltage E shall be constant, a = constant; hence, (1 + Tog + a^ob)^ + (xog - roh)^ = ^; or, expanding, = -^«+v(^) -(^+^°)^ which is the value of the susceptance, 6, as a function of the receiver conductance — ^that is, of the load — which is required to yield constant voltage, aEo, at the receiver circuit. For increasing g, that is, for increasing load, a point is reached where, in the expression 2 the term under the root becomes negative, and b thus imaginary, and it thus becomes impossible to maintain a constant voltage, aEo. Therefore the maximum output which can be transmitted at voltage, aEo, is given by the expression a yoV hence the susceptance of receiver circuit is 6 = — &o, and the yo — ) a -)'-(? + go)— 0; ■ receiver circuit is conductance of receiver circuit is g = — go + P = Eo-ga"- = a-Eo^ I— - go) , the output. 75. If a = 1, that is, if the voltage at the receiver circuit equals the generator voltage, g = 2/0 - go; P = Eo'^iijo - go). If a = 1, when g = 0, 6 = 0 when g > 0, 6 < 0; if a > 1, when g = 0, or g > 0, 6 < 0, that is, condensive reactance; if a < I, when g = 0, 6 > 0, TRANSMISSION LINES 91 when g = — go + \j {-^j — &o", "+m' when g > — go + -yl {~j — ho", b < 0, or, in other words, if a < I, the phase difference in the main line must change from lag to lead with increasing load. 76. The value of a giving the maximum possible output in dP a receiver circuit is determined by -7— = 0: •^ da ' expanding /I/O ^\ ahf 2a( go) 0 = 0 \ a 1 a- hence 2/0 = 2 ago. and yo 1 2o . 2 go 2\/goro ^^0' the maximum output is determined by , yo g = - go + — = go; and is, p _ Eo^ 4ro' From 1/0 2o 2^0 ~ 2ro' the line reactance, Xo, can be found, which delivers a maximum output into the receiver circuit at the ratio of voltages, a, as 20 = 2 roa, Xo = roVl a^ — 1; for a = 1, Zo = 2ro; Xo = roVS. If, therefore, the line impedance equals 2 a times the line ^0^ . resistance, the maximum output, P = j — , is transmitted into the receiver circuit at the ratio of voltages, a. Eo^ If ^'o = 2 To, or Xo = ro\/3j the maximum output, P = -. — , 4 To can be supplied to the receiver circuit, without change of voltage at the receiver terminals. Obviously, in an analogous manner, the law of variation of the susceptance of the receiver circuit can be found which is required to increase the receiver voltage proportionally to 92 ALTERNATING-CURRENT PHENOMENA the load; or, still more generally, to cause any desired varia- tion of the voltage at the receiver circuit independently of any variation of the generator voltage, as, for instance, to keep the voltage of a receiver circuit constant, even if the generator volt- age fluctuates widely, 77. In Figs. 73, 74, and 75 are shown, with the output, p = Eo-ga", as abscissas, and a constant impressed voltage. — RATIO OF RECEIVER VOLTAGE TO SENDER VOLTAGE: a = 1.0 LINE IMPEDANCE^ Zo=2.6-j-ej 1 ENERGY CURRENT CONSTANT GENERATOR POTENTIAL E o= II REACTIVE CURRENT III TOTAL CURRENT IV CURRENT IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION V POTENTIAL II 11 II 1' M .1 000 ■z 'SO ■"" "= ■= , ^ "fiO ", fie^ CTll ^ r- ?10 § ^^^t ■^ if 1000 900 E ^ .F. REC :lvl^ GCI ?CU T N N fiyn — — '- ;^ s4^l ii N 180 "-^^^ 25K °]^ III ino 700 — — f\ ,7-%^li^Jfc. j * II i?rt '-HO ^^'^ / ^ -^ ino , f ^ ■\^ 0> ^ 1 80 ,^' y ?r_. "^ ;pg _co 60 800 ■:^ =^ ^ >^ -^ ^ 40 iisS ^ -^ -^ W) ^ eS. »S !^ 0 10 20 80 40 50 60 70 80 90 OUTPUT IN RECEIVER CIRCUIT, KILOWATTS Fig. 73. — Variation of voltage of transmission lines. Eo = 1000 volts, and a constant line impedance, Zq = 2.5 -f 6 j, or ro = 2.5 ohms, a^o = 6 ohms, z = 6.5 ohms, the following values: power component of current, gE, (Curve I) ; reactive, or wattless component of current, hE^ (Curve II) ; total current, yE^ (Curve III), and power factor at generator for the following conditions: a = 1.0 (Fig. 73); « = 0.7 (Fig. 74); a = 1.3 (Fig. 75). For the non-inductive receiver circuit (in dotted lines), the curve of e.m.f., E, and of the current, / = gE, are added in the three diagrams for comparison, as Curves IV and V. As shown, the output can be increased greatly, and the voltage at the same time maintained constant, by the judicious TRANSMISSION LINES 93 — RATIO OF RECEIVER VOLTAGE To'sEn'oERVOLTAGE: a = 7 ■ ~ ~ t- 1 ENERGY CURRENT CONSTANT GENERATOR POTENTIAL E II REACTIVE CURRENT III TOTAL CURRENT IV POTENTIAL IN NON-INDUCTIVE CIRCUIT WITHOUT COMPENSATION 0 = 1000 § =5 kk 1^ ^^ ^^ 900 800 700 600 WO — -■ ~k y^ ^N s > '^^ r p* ^ 11 N s / ^> N \ / \ N ,^ / > / / €r 't t / ,> ''^ ^ ^ ^ 800 / ^ ^ ^ ^ f-^ / 200 100 — -» ■^ ■5 ^ >• d. -- ^ ^ ^-^ "-. ^ y L^ L^ "^ "^ OUTPUT IN RECEIVER CIRCUIT, KILOWATTS Fig. 74. — Variation of voltage of transmission lines. RATIO OF RECEIVER VOLTAGE TO SENDER VOLTAGE ■ a I1. 3 LINE IMPEDANCE: Zo=2.6 -f- 6/ I ENERGY CURRENT CONSTANT GENERATOR POTENTIAL E„= II REACTIVE CURRENT III TOTAL CURRENT IV POTENTIAL IN NON-INDUCTIVE RECEIVER CIRCUIT WITHOUT COMPENSATION lopj ^ »- ■ "*= =a < ^== == 5-; ;j^ "flO ^^i::-. ■>»^ "^v ^ "^ uoo 1000 900 T^ ^ 220 "s\ ~~- -- — - -*.. IV ) 180 160 110 ■^> ^.^ // 700 ^^ "'s III ^ y ^ ^ V _^ \ .> ^ 120 100 80 60 500 400 SOO 200 100 1 r c. ^ --' u ^ ^ ^ j== ■ ■^ — ^ - _--: =2 £=: _^ Z^ ^ .^^ "* ^ ■ ' ^ ^ /- ^ ^^ -'■' ,^ . — ' --f^ — 20 0 /^ r- 1 10 20 30 40 50 00 70 SO UO OUTPUT IN RECEIVER CIRCUIT, KILOWATTS Fig. 75. — Variation of voltage of transmission lines. 94 ALTERNATING-CURRENT PHENOMENA use of shunted reactance, so that a much larger output can be transmitted over the Hne with no drop, or even with a rise, of voltage. Shunted susceptance, therefore, is extensively used for voltage control of transmission lines, by means of synchronous condensers, or by synchronous converters with compound field winding. 5. Maximum Rise of Voltage at Receiver Circuit 78. Since, under certain circumstances, the voltage at the receiver circuit may be higher than at the generator, it is of interest to determine what is the maximum value of voltage, E, that can be produced at the receiver circuit with a given generator voltage, £'o. The condition is that 1 . . a = maxnnum or ^, = minimum; a" that is, substituting,