CHAPTER IX. KBSISTANCi: AND KBACTANCE OF TRANSMISSION IINE8. 57. In alternating-current circuits, E.M.F. 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 E.M.F. consumed by the resistance is in phase, while the E.M.F. consumed by the reactance is in quadrature, with the current. Hence their influence upon the E.M.F. at the receiver circuit depends upon the difference of phase between the current and the E.M.F. 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 E.M.F. The change of potential due to line reactance is small if the current is in phase with the E.M.F., 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 potential due to a line of given re- sistance and inductance 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 + Jb, 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 reg- ulation. Its susceptance, by however, can be changed by shunting the circuit with a reactance, and will be increased by a shunted inductance, and decreased by a shunted con- densance. Hence, for the purpose of investigation, the 84 AL TERN A TIXG-CURRENT PHENOMENA, [§ 68 receiver circuit can be assumed to consist of two branches, a conductance, g^ — the non-inductive part of the circuit, — shunted by a susceptance, by which can be varied without expenditure of energy. The two components of current can thus be considered separately, the energy component as determined by the load on the circuit, and the wattless component, which can be varied for the purpose of regu- lation. Obviously, in the same way, the E.M.F. at the receiver circuit may be considered as consisting of two components, the energy component, in phase with the current, and the wattless component, 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 assump- tion as to whether the wattless part of the receiver circuit is in shunt, or in series, to the energy part. Let — ^u = ^o — j^o = impedance of the line ; y = ^ -^-j'b = admittance of receiver circuit ; jE^ = ^^ +j^/ = impressed E.M.F. at generator end of line ; E —, c '\-jt'' — K.M.F. at receiver end of line; E = V^-'' + e'^\ I,, == /p +yV = current in the line ; /^ = V/V^ + //*''. The simplest condition is that of a non-inductive receiver circuit, such as a lighting circuit. 1.) XoH-iudnctivc Receiver Circuit Supplied over an Indue til 'c L inc. 58. In this case, the admittance of the receiver circuit IS F = ^, since ^ = 0. ■§58] RESISTANCE OF TRANSMISSION LINES, 85 We have then — •current, /« = Eg\ impressed E.M.F., ^^ = ^ + Z^/„ = ^ (1 + Z^g). Hence — E.M.F. at receiver circuit, E = ^ = ^ ; 1 + Z,^ l+^r,-/^:r^ current, /. = . f^^ = ^_^^'^ - ' Hence, in absolute values — E.M.F. at receiver circuit, E ^ ** • ^(y+gr:)^+gw E z current, T. = ""^ . The ratio of E.M.Fs. at receiver circuit and at genera- tor, or supply circuit, is — ^^^^ 1 . E. ^i^\-^grY^^y and the power delivered in the non-inductive receiver cir- cuit, or output, p^ T^E= ^"^ ^ — (1 + groY + g^x. 2 As a function of g, and with a given E^, r^, and ;r^, this power is a maximum, if — ^ig that is — hence — conductance of receiver circuit for maximum output, _ 1 1 gm — Resistance of receiver circuit, '*/» = — = -o ; gm 8G A/. TERXA TIXG-CURRENT PHENOMENA. [§ 69 and, substituting this va P — E'^ E^ Maximum output, /*»» = ^ ~ ** and — ratio of E.M.F. at receiver and at generator end of line. ^1* rt v/- (' + 5) efficiency, m *'o That is, the output which can be transmitted over an inductive line of resistance, r^, and reactance, x^, — that is, of impedance, ^^ , — into a non-inductive receiver circuit, is a maximum, if the resistance of the receiver circuit equals the impedance of the line, r = s^, and is — E^ •* m — 2 (r, + z:) The output is transmitted at the efficiency of and with a ratio of E.M.Fs. of 1 fi... 59. We see from this, that the maximum output which can be delivered over an inductive line is less than the output delivered 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. 57 are shown, for the constants £, = 1000 volts, Z^ = 2.5 — 6/; that is, r, = 2.5 ohms, x^ = 6 ohms, Zo = ^-5 ohms, with the current A as abscissae, the values — 1 601 A'ES/STAA-C/i OF T/^A/iSM/SS/OX L/.\£S. 87 E.M.F, at Receiver Circuit, F., (Curve I.); Output of Transmission, /', (Curve II.) ; Efficiency of Transmission, (Curve III.)- The same quantities, E and P, for a non-inductive line of resistance, r„ = 2.5 ohms, .r, = 0, are shown in Curves IV„ v., and VI. •n:^;m™isTSi::, ' -..».^-^ .-.- 7 TI y ij. . / Zt i I : . ^ — ^ -/y^ '^i' IS™ » /^-'^'tti. V ™™ t ^^ 35-ffi". . J^ ^-^ ^ Ss- t N, r^™ . / \^\a» T ^^»» , / .„,.,,„uL.: ,....,.„ ^r'" n^ S7. Uan-lnAatliit Mtttlatr t It Suppliti Ootr /ndMt/M U 2.) Maximnm Pomer Snpplitd over an Inductive Line. 60. If the receiver circuit contains the susceptance, b, in addition to the conductance, g, its admittance can be written thus: — Then — current, /„ = EY\ Impressed E.M.F., E, = B ■\- I„Z, = £ (1 + l'-?.)- 88 AL TERN A TING-CURRENT PHENOMENA. [§61 Hence — E.M.F. at receiver terminals, 1 + KZ, ( 1 + r, ^ + ^, ^) ~ y (^, ^ - r, b) ' current, / _ E.y ^ E^^g^jb) . • 1 + rZ. (1 + r.g + xj) ^j(x^g^rj) ' or, in absolute values — E.M.F. at receiver circuit, p £ = ti± . V(l + r,g + xjf + (x,g - r^bf ' current. r.^E.^. L^E.J _^+^ {^ + r,g^.x,bY^(x.g-r,bY' ratio of RM.Fs. at receiver circuit and at generator circuit, ^^E^^ 1 . E. V(r+7;/qh^j^'+ {xlg-r,^^ ' and the output in the receiver circuit is, 61. a) Dependence of the output upon the susceptance of the receiver circuit. At a given conductance, g, of the receiver circuit, its output, P = E^(j?g, is a maximum, if a^ is a maximum; that is, when — /= 1 = (1 + r^g + xjy + {x^g - r^by a is a minimum. The condition necessary is — or, expanding. ^^ ^^ _^ ^^^ ^ ^. ^^ _ ^^ ^^^ _ ^^^^ ^ ^^ Hence — Susceptance of receiver circuit, r^ + x^ e„ or ^ + ^^ = 0, § 62] RESISTANCE OF TRANSMISSION LINES. 89 that is, if the sum of the susceptances of line and of receiver circuit equals zero. Substituting this value, we get — ratio of KM.Fs. at maximum output, E 1 «! = maximum output, _ ^.y current, ^o_U_^jJ^o) . and, expanding, phase difference in receiver circuit, . ^ b bo tan (!» = -= 2 ; phase difference in generator circuit, tan o)>» = '^ + ^O ^c;/ + gy'o^ 62. ^.) Dependence of the output upon the conductance of the receiver circuit. At a given susceptance, b^ of the receiver circuit, its output, Z' = ^^^a^^, is a maximum, if — dg 'dg\a^g) i/g\ g ) 90 ALTERNATING-CUR REXT PHENOMENA, [§ 63 that is, expanding, — or, expanding, — Substituting this value in the equation for a, page 88, we get — ratio of E.M.Fs., 1 Og ^ — '' V^ {f,* + (* + *.)* + ^.V^.» +"(* + *.)»} _ 1 _ y, . -_.. y power, -' (^ + ^.) 2 {^. + V;r„» + (^ + boY) v/'--' +(-•+- ^ J • As a function of the susceptance, b, this power becomes a maximum for dP^j db = 0, that is, according to § 61, if — Substituting this value, we get — ^ = —^oyg = gofy=yof hence: K= ^ + /^ = ^^ — y^^; x= —Xoyr=royZ = Zof ^=r—jx = ro + jx^; substituting this value, we get — ratio of E.M.Fs., yp __ ^0 . 2 go 2n <*m= t: — = 71 — > £9 power, Pm = T-^ ; that is, the same as with a continuous-current circuit ; or, in other words, the inductance of the line and of the receiver circuit can be perfectly balanced in its efifect upon the output. 63. As a summary, we thus have : The output delivered over an inductive line of impe- 163] RESISTANCE OF TRANSMISSION LINES, 91 dance, Z^ = r^ —J-^o » '"^^ ^ non-inductive receiver circuit, is a maximum for the resistance, r = s^, or conductance, ^ = j^^ , of the receiver circuit, or — at the ratio of potentials, 1 a = With a receiver circuit of constant susceptance, d, the out- put, as a function of the conductance, ^, is a maximum for the conductance, — and is 3 at the ratio of potentials, a = - With a receiver circuit of constant conductance, ^, the output, as a function of the reactance, 6, is a maximum for the reactance, ^ = — ^o> ^"d is at the ratio of potentials, a = The maximum output which can be delivered over an in- ductive line, as a function of the admittance or impedance of the receiver circuit, takes place when Z = r^ +J-^o> or y =^ go— j f>o'y ^^^ ^s» when the resistance or conductance of a receiver circuit and line are equal, the reactance or sus- ceptance of the receiver circuit and line are equal but of opposite sign, and is, P = E^ j \r^, or independent of the reactances, but equal to the output of a continuous-current 92 AL TEKNA TINU-CUKKEA-T PHENOMENA. [S64 circuit of equal line resistance. The ratio of potentials is, in this case, a = ::^j 2 r,,, while in a continuous-current circuit it is equal to J. The efficiency is equal to 50 per cent. _ UpLiS_iLEljI.E^ ^SS 5i3m '"Y"' -^ '""ja^lioL -^ --^BEss Den. .. i.-ilxjo sz it ^» /-^ ^..,ve.. ^^^^^ '' ^ \ „/^ z ^ ^^ ti L S. ^ L "s-^ 1 ^ f / T "^ -— ^ "" ir ll''==~- /„^ "V . / ]7? !5^ — Llfiy ' :. ^ -~^- ' Jf """^----L — " — '~^^^^^=*k— _ j^ 1 ~"-''-~-^m_i \ ~" — ' IV j^ J— ■^-'''■"■'^~~*~- / .-t-" i !-4— _1_ - ^ '-'"' IC ' L ^^^ J~ ^^^^ T fiff. M. r/sl/cw o/ (At PoUBtW la Unt at DIfftren 64. As an instance, in Fig. 58 are shown, for the constants — £, = 10(10 volts, and Z„ = 2.5 - &j; that is, for r, = 2.5 ohms, .r„ = Cohms, s„ = 6.B ohms, and with the variable conductances as abscissas, the values of the — output, in Curve I., Curve III., and Curve V. ; ratio of potentials, in Curve II., Curve IV., and Curve VI.; Curves I. and 11. refer to a non-inductive circuit ; 1 30] RESISTANCE OF TRANSMISSION LINES. Curves III. and IV. refer to a constant su seep tan ce Curves V. and VI. refer to a constant susceptance b = — .142 \ Curves VII. and VIII. refer to a non-inductive re- ceiver circuit of a non-inductive line. In Fig. 59, the output is shown as Curve I., and the . ratio of potentials as Curve II., for the same line constants, fora constant conductance, ^ = .0592 ohms, and for variable susceptances, b, of the receiver circuit. •TCa.lT*NIIIlPHCUEDLII.r. (f-lOOO / \ \ oJi'p'gT J !_ _; r 1 \ t tv^v W ^1 i ^L 3- t t ^^ -' J. SI y7 U ^^' v^ 7 S^., / ^^ --^ s^ 3.) Maximtnii Efficiency. 65. The output, for a given conductance, g, of a receiver circuit, is a maximum if ^ = — b^. This, however, is gen- erally not the condition of maximum efficiency. 94 AL TERN A TING-CURRENT PHENOMENA. [ § 66 The loss of energy in the line is constant if the cur- rent is constant; the output of the generator for a given current and given generator E.M.F. is a maximum if the cur- rent is in phase with the E.M.F. at the generator terminals. Hence the condition of maximum output at given loss, or of maximum efficiency, is — tan a)<, = 0. The current is — multiplying numerator and denominator by (1 + r^g + x^b) + j(x^g — ^'o^), to eliminate the imaginary quantity from the denominator, we have — ({giX + rog + Xob) - b(x,g - rob)} +\ J=£ \ J {^ iX + ^og + x ^b) +g {xpg - r^ b)} ) ^ (1 + rog + Xoby + (xog^r.by The current, /^, is in plvise with the E.M.F., E^, if its quadrature component — that is, the imaginary term — dis- appears, or ^ (1 + rog+xpb) +g{xog'-rob) = 0. This, therefore, is the condition of maximum efficiency. Expanding, we have, b g^ + b^ Hence, the condition of maximum efficiency is, that the reactance of the receiver circuit shall be equal, but of oppo- site sign, to the reactance of the line. Substituting x = — ;r^, we have, ratio of E.M.Fs., ^o (r+ro) (r+r,) ' power. p^Eo^ga'^-^-, se6] A'£SJSTAACE OF Th'AA'SAffSSJO.V LIA'FS. and depending upon the resistance only, and not upon the reactance. This power is a maximum if ^ s=^,, as shown before; hence, substituting ^ =£'a> '' = ''„. maximum power at maximum efficiency, J'n = '" , at a ratio of potentials, (T„ = — 2_ , or the same result as in § 62, s '; ^ - , "SS v>, P'^" ""-N ■•" ^ V \^ ... i.«..«. V ^^ i^ . 1. -»..«. a. I,^^ I J ^s, ^M f " i: 7 ^^ / ]3 the susceptance of receiver circuit, and g — —go + ~ > the conductance of receiver circuit ; a P^Eo'ga^ = a^ Eo^(^ - ^o^ , the output. 67. If tf = 1, that is, if the voltage at the receiver cir- cuit equals the generator potential — ^ g=yo—goi P=a^E,^{yo-goy If dr = 1 when ^ = 0, ^ = when ^ > 0, ^ < ; if a >1 when ^ = 0, or ^ > 0, ^ < 0, that is, condensance ; if a <1 when ^ = 0, ^ > 0, when g^^g, + J(^\- K\ ^ = ; v/(?)'- ' when ^ > - ^0 + J(^\- V, * < 0, \/(f or, in other words, if ^x < 1, the phase difference in the main line must change from lag to lead with increasing load. 68. The value of a giving the maximum possible output in a receiver circuit, is determined by dP I da = ; expanding : \ / hence, yo = 2agoy and a-/'- 1 = 2^» 2Vg7n, 2r.' 98 AL TERN A TING-CURRENT PHENOMENA. [ § 69 the maximum output is determined by — a and is, P=^^. 4 r From : ^ = Ji^ = Ji. , the line reactance, x^, can be found, which delivers a maximum output into the receiver circuit at the ratio of potentials, a, and Zo^= 2 r^ a^ for a = 1, Xo = ^o V4 a^ — I'y z — 2 r ' X. o = n V3. If, therefore, the line impedance equals 2 a times the line resistance, the maximum output, P = £^^/ 4t r^, is trans- mitted into the receiver circuit at the ratio of potentials, a. If ^^ = 2 r^, orjr^ = r^ V3, the maximum output, P = E^l^r^^ can be supplied to the receiver circuit, without change of potential 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 the load ; or, still more generally, — to cause any desired variation of the potential at the receiver circuit indepen- dently of any variation of the generator potential, as, for in- stance, to keep the potential of a receiver circuit constant, even if the generator potential fluctuates widely. 69. In Figs. 61, 62, and 63, are shown, with the output, P = E^ g cfi, as abscissae, and a constant impressed E.M.F., E^ = 1,000 volts, and a constant line impedance, Z^ = 2.5 — 6y, or, r^ = 2.5 ohms, jr^ = 6 ohms, z = 6.5 ohms, the following values : -j " ■yo Spf ?-• atitBEHWO -1. 1 *L .-itx» oSr£e°t?n NON-1«DUOTIVE RECEIVER OiBOUIT WITHOUT COMPENEiT on' >u "" ■*= =SS5 =^ u 1 n ?;;^ 4^ -£!. rvJ 1 ' *«S?K iW.n - . * BE EIV HO cine n ^ \ .,■ ~\ "" - -■-, -^ tA' N , fe ?-* — -n :;: iS- ^ 1 ' &< - '7>-' y l[ ^ ^ 1 1 . --^ r' -1- ,-' =g ^ fSS- p-; ?^ ,^- 5^ -- ^ ^ _, !=s: ::^ ^-^ ■ . u y & < — ^ 5L r iJL U ^ i_ — _ _ _ _ ) 1 REWmE Cj"eIt '=°''^^*"^ QE«EB*rO» POTENTl.L .= 1000 X OUHBENTIN NON-IBDUCtlVE OIRCUIT WITHOUT COHPEHflATtO lit ==^ OLTi L^ "■ k ... i" ■^ "■- -<-/ .™ ^ « k^ -. , \ 1 y ' ^ y i -^ ' ] ,^, ^ / " -^ i::^ / ~- -- .' 1 / ., ' - - -^ ^ -f ^ " _ _ L_£ f^ Ll ;„.t,i ■01 ^ .oU. L _ /7ff. « Hif/adon o/ CoHdb* r«n*m/M;wi Unu. ALTF.RNATING-CVRFENT PHENOMENA. MTIO OF RECIIVEK VOLTAGE TO BE DE'HvoLTVa£';a-ia 1 | 1 TQEKEiUrOS POTENTIAL E.-IOOO _ R cifrcuiT wiTHoin- coMPEBWTin r- 3„ "» ' ""^'^^s Z ^§^ I ^ „ - ""--^ J -: z ^J 1 IJM^i^lIU:: ' ---'^-'^ -" "= = --"==^- — ^^■""' ■:^j^4^:iijj nOl IT, Kll oJ^TTB FIj. 93. Yarlatlim of Vtltag* Energy component of current, gB, (Curve I.) ; Reactive, or wattless component of current, bE, (Curve II.) ; Total current, yE, (Curve III.) ; for the following conditions : a = 1.0 (Fig, r,8) i a= .7 (Fig. 59); a = 1.3 (Fig. 60). For the non-inductive receiver circuit (in dotted lines), the curve of E.M.F., /;, and of the current. I =gE, are added in the three diagrams for comparison, as Curves IV. and V. As shown, the output can be increased greatly, and the potential at the same time maintained constant, by the judi- cious use of shunted reactance, so that a much larger out- put can be transmitted over the line at no drop, or even at a rise, of potential. I 70] RESISTANCE OF TRANSMISSION LINES, ' 101 5.) Maximum Rise of Potential at Receiver Circuit, 70. Since, under certain circumstances, the potential at the receiver circuit may be higher than at the generator, it is of interest to determine what is the maximum value of potential, j5*, that can be produced at the receiver circuit with a given generator potential, E^ . The condition is that 1 a = maximum or — - = mmimum ; that is, dg db substituting, ^ = (1 + '•o-S- + x^by + {x,g - ub)\ a and expanding, we get, — a value which is impossible, since neither r^ nor g can be negative. The next possible value is ^ = 0, — a wattless circuit. Substituting this value, we get, and by substituting, in db ' zo^ ^ + ^0 = 0; that is, the sum of the susceptances = 0, or the condition of resonance is present. Substituting, we have 102 AL TERNA Tlh'G-CUKKENT PHENOMENA. \% 71 The current in this case is, or the same as if the Hne resistance were short-circuited without any inductance. This is the condition of perfect resonance, with current and E.M.F. in phase. s \ *(sy; \ \ \ \ s \, ks \ \ "^"^ 'Sn'Sn^Mt'iVa S"., ""ISE'.S'S^'.'t"-" ■kmVt™ / - "^ ^ - / J m ■^ s. m '•i K 1^' '%' 1 / ^ ^ / ' >// ^ A^ ,• '/ ,6- ' / « ^ c- r -V"' K.v. ~ Fig. 64. EffieltiKg ami Output of JraniKilitlim tint. 71. As summary to this chapter, in Fig. 64 are plotted, for a constant generator E.M.F., £"„ = 1000 volts, and a line impedance, ^„ = 2.5 — 6j, or, r, = 2.5 ohms, :r„ = 6 ohms, s, B 6.5 ohms ; and with the receiver output as §71] RESISTANCE OF TRANSMISSION LINES. 103 abscissae and the receiver voltages as ordinates, curves representing — the condition of maximum output, (Curve I.) ; the condition of maximum efficiency, (Curve II.) ; the condition ^ = 0, or a non-inductive receiver cir- cuit, (Curve III.) ; the condition ^ = 0, ^j, = 0, or a non-inductive line and non- inductive receiver circuit, or a non-inductive receiver circuit and a non-inductive line. In conclusion, it may be remarked here that of the sources of susceptance, or reactance, a choking coil or reactive coil corresponds to an inductance ; a condenser corresponds to a condensance ; a polarization cell corresponds to a condensance ; a synchronizing alternator (motor or generator) corresponds to an inductance or a condensance, at will ; an induction motor or generator corresponds to an inductance or condensance, at will. The choking coil and the. polarization cell are specially suited for series reactance, and the condenser and syn- chronizer for shunted susceptance. 104 ALTERNATING-CURRENT PHENOMENA. . [§ 72