CHAPTER XII. DIBTBISnTED CAPACITY, INDUCTANCE, BESISTANCE, AND liEAKAGE. 102. As far as capacity has been considered in the foregoing chapters, the assumption has been made that the condenser or other source of negative reactance is shunted across the circuit at a definite point. In many cases, how- ever, the capacity is distributed over the whole length of the conductor, so that the circuit can be considered as shunted by an infinite number of infinitely small condensers infi. nitely near together, as diagrammatically shown in Fig. 83. 8 3 S Fig, 83. Distributed Capacity. In this case the intensity as well as phase of the current,, and consequently of the counter E.M.F. of inductance and resistance, vary from point to point ; and it is no longer possible to treat the circuit in the usual manner by the vector diagram. This phenomenon is especially noticeable in long-distance lines, in underground cables, especially concentric cables, and to a certain degree in the high-potential coils of alternating- current transformers. It has the effect that not only the E.M.Fs., but also the currents, at the beginning, end, and different points of the conductor, are different in intensity and in phase. Where the capacity effect of the line is small, it may with sufficient approximation be represented by one con- §103] DISTRIBUTED CAPACITY. 151 denser of the same capacity as the line, shunted across the line. Frequently it makes no difference either, whether this condenser is considered as connected across the line at the generator end, or at the receiver end, or at the middle. The best approximation is to consider the line as shunted at the generator and at the motor end, by two condensers of J the line capacity each, and in the middle by a condenser of \ the line capacity. This approximation, based on Simpson's rule, assumes the variation of the elec- tric quantities in the line as parabolic. If, however, the capacity of the line is considerable, and the condenser current is of the same magnitude as the main current, such an approximation is not permissible, but each line element has to be considered as an infinitely small condenser, and the differential equations based thereon integrated. 103. It is thus desirable to first investigate the limits of applicability of the approximate representation of the line by one or by three condensers. Assuming, for instance, that the line conductors are of 1 cm diameter, and at a distance from each other of 50 cm, and that the length of transmission is 50 km, we get the capacity of the transmission line from the formula — c = microfarads, 4 log nat -^ where K = dielectric constant of the surrounding medium = 1 in air ;. / = length of conductor = 5 X 10* cm. ; d = distance of conductors from each other = 50 cm. ; 8 = diameter of conductor = 1 cm. Since C = .3 microfarads, the capacity reactance is 10« . 152 AL TERN A TING-CURRENT PHENOMENA, [$ 104 where N = frequency ; hence, at iV = 60 cycles, X = 8,900 ohms ; and the charging current of the line, at -£* = 20,000 volts, becomes, ^ to = — = 2.25 amperes. X The resistance of 100 km of line of 1 cm diameter is 22 ohms ; therefore, at 10 per cent = 2,000 volts loss in the line, the main current transmitted over the line is , 2,000 Q. I =r. — _ — = 91 amperes, representing about 1,800 kw. In this case, the condenser current thus amounts to less than 2\ per cent., and hence can still be represented by the approximation of one condenser shunted across the line. If, however, the length of transmission is 150 km and the voltage 30,000, capacity reactance at 60 cycles, x = 2,970 ohms ; charging current, /'o = 10.1 amperes ; line resistance, r = (S^ ohms ; main current at 10 per cent loss, / = 45.5 amperes. The condenser current is thus about 22 per cent, of the main current. At 300 km length of transmission it will, at 10 per cent, loss and with the same size of conductor, rise to nearly 90 per cent, of the main current, thus making a more explicit investigation of the phenomena in the line necessary. In most cases of practical engineering, however, the ca- pacity effect is small enough to be represented by the approx- imation of one ; viz., three condensers shunted across the line. 104. A.) Line capacity represented by one condetiser shunted across middle of line. Let — Y == g -{- j'b = admittance of receiving circuit; z =i r — j X = impedance of line ; be = condenser susceptance of line. §105] DISTRIBUTED CAPACITY. 15S Denoting, in Fig. 84, the E.M.F., viz., current in receiving circuit by E^ /, the E.M.F. at middle of line by E\ the E.M.F., viz., current at generator by EoyJo\ t r n Fig. 84. Capacity Shmrttd acroat MtMli of Uin, We have, E' = E+ ''-J^ I 2 = E^g^jb-j.\i^^I-:^n^^±m]X E, = E ^".^^I, o '" -e\i \ <^ ->^) ^'^ +>^) I (^ ->-^> <^ +>^) I ^ 2 ^ 2 jb ,{r-j.x) _ .. (r-jxYig+Jb) \ . 2 ^ ' 4 y or, expanding, = ^ 1 1 + (r -yx) ^^+ y* -i^-ih (r -jx)* 105. ^.) Z««^ capacity represented by three condensers, in the middle and at the ends of the line. Denoting, in Fig. 85, the E.M.F. and current in receiving circuit by E, I, the E.M.F. at middle of line by E', 154 ALTERNATING-CURRENT PHENOMENA, [§ 105 the current on receiving side of line by /', the current on generator side of line^by /", the RM.P'., viz., current at generator by -fo* />» D iTE ZUI ITi JJT 3t II! Pig. 85. Distributed Capacity. Otherwise retaining the same denotations as in A,), We have, G =^{'+=^i^+>'-f)}' J ft jf zJ-£s P' ^E\g^jb-^J^-i^{r-jx)[g^jh-lhYy E^^E' + 'L^^I"; ^, = ^ 1 1 + (r -jx) (g + Jb- ^') - i^ (r -jx)* 7 — /" — /^« ^ • As will be seen, the first terms in the expression of E^ and of 7^ are the same in A.) and in B.). § 106] DISTRIBUTED CAPACITY. 155 106. C) Complete investigation of distributed capacity, indnctanccy leakage, and resistattce. In some cases, especially in very long circuits, as in lines conveying alternating power currents at high potential over extremely long distances by overhead conductors or un- derground cables, or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance — which consumes E.M.Fs. in phase with the current — and of the line reactatice — which con- sumes E.M.Fs. in quadrature with the current — is not sufficient for the explanation of the phenomena taking place in the line, but several other factors have to be taken into account. In long lines, especially at high potentials, the electro- static capacity of the line is sufficient to consume noticeable currents. The charging current of the line condenser is proportional to the difference of potential, and is one-fourth period ahead of the E.M.F. Hence, it will either increase or decrease the main current, according to the relative phase of the main current and the E.M.F. As a consequence, the current will change in intensity as well as in phase, in the line from point to point ; and the E.M.Fs. consumed by the resistance and inductance will therefore also change in phase and intensity from point to point, being dependent upon the current. Since no insulator has an infinite resistance, and as at high potentials not only leakage, but even direct escape of electricity into the air, takes place by " silent discharge," we have to recognize the existence of a current approximately proportional and in phase with the E.M.F. of the line. This current represents consumption of energy, and is therefore analogous to the E.M.F. consumed by resistance, while the condenser current and the E.M.F. of inductance are wattless. Furthermore, the alternate current passing over the line induces in all neighboring conductors secondary currents, 166 AL TERNA TING-CURRENT PHENOMENA, [§ 1 06 which react upon the primary current, and thereby intro- duce E.M.Fs. of mutual inductance into the primary circuit. Mutual inductance is neither in phase nor in quadrature with the current, and can therefore be resolved into an energy component of mutual inductance in phase with the current, which acts as an increase of resistance, and into a wattless component in quadrature with the current, which decreases the self-inductance. This mutual inductance is by no means negligible, as,, for instance, its disturbing influence in telephone circuits shows. The alternating potential of the line induces, by electro- static influence^ electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence re- quires the expenditure of a current proportional to the E.M.F., and consisting of an energy component, in phase with the E.M.F., and a wattless component, in quadrature thereto. The alternating electromagnetic field of force set up by the line current produces in some materials a loss of energy by magnetic hysteresis, or an expenditure of E.M.F. in phase with the current, which acts as an increase of re- sistance. This electromagnetic hysteretic loss may take place in the conductor proper if iron wires are used, and will then be very serious at high frequencies, such as those of telephone currents. The effect of eddy currents has already been referred to under "mutual inductance," of which it is an energy component. The alternating electrostatic field of force expends energy in dielectrics by what is called dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is comparatively large, the dielectric hysteresis may at high potentials consume far greater amounts of energy than the resistance does. The dielectric hysteresis § 107] DISTRIBUTED CAPACITY. 157 appears in the circuit as consumption of a current, whose component in phase with the E.M.F. is the dielectric energy currents which may be considered as the power component of the capacity current- Besides this, there is the increase of ohmic resistance due to unequal distribution of current, which, however, is usually not large enough to be noticeable. 107. This gives, as the most general case, and per unit length of line : E.M.Fs. consumed in phase with the current /, and = r/, representing consumption of energy, and due to : Resistance, and its increase by unequal current distri- tribution ; to the energy component of mutual inductafue ; to induced currents ; to the energy component of self-inductatue ; or to electromag- netic hysteresis, E.M.Fs. consumed in quadrature ivith the current /, and" = ^ /, wattless, and due to : Self -inductance, and Mutual inductance. Currents consumed in p/uise with the E.M,F., E, and = gEy representing consumption of energy, and due to : Leakage through the insulating material, including silent discharge; energy component of electro- static influence ; energy component of capacity, or of dielectric hysteresis. Currents consumed in quadrature to the E.M.F., E, and = bE, being wattless, and due to : Capacity and Electrostatic influence. Hence we get four constants : — Effective resistance, r. Effective reactance, x. Effect iv'e conductance, g^ Effective susceptance, ^ = — b^y 158 ALTERJ^ATING-CURRENT PHENOMENA. [§§108,109 per unit length of line, which represent the coefficients, per unit length of line, of E.M.F. consumed in phase with current ; E.M.P\ consumed in quadrature with current ; Current consumed in ph*ase with E.M.F. ; Current consumed in quadrature with E.M.F. 108. This line we may assume now as feeding into a receiver circuit of any description^ and determine the current and E.M.F. at any point of the circuit. That is, an E.M.F. and current (differing in phase by any desired angle) may be given at the terminals of receiving cir- cuit. To be determined are the E.M.F. and current at any point of the line ; for instance, at the generator terminals. Or, Zi = ri — y xi ; the impedance of receiver circuit, or admittance, and E.M.F"., E^, at generator terminals iare given. Current and E.M.F\ at any point of circuit to be determined, etc. 109. Counting now the distance, x, from a point, 0, of the line which has the E.M.F., ^\ = ^1 +/^i'> and the current : /i = /'i + jii-, and counting x positive in the direction of rising energy, and negative in the direction of decreasing energy, we have at any point, x, in the line differential, dx : Leakage current : Ej^dx\ Capacity current : — J E bfdx\ hence, the total current consumed by the line element, ^x, ///= E(g—Jbc)d%, or, iL^R{g^jK). (1) EM.F. consumed by resistance, Irdx\ E.M.F. consumed by reactance, ^jlxdx\ is §110] DISTRIBUTED CAPACITY, 159 hence, the total E.M.F. consumed in the line element, rfx, is dE = I{r ^jx) //x, or, dE J. . , T\iQSQ fundamental differential equations : (2) d/ dx = ^U-J<^c), dE -dT =^(''-^^)' are symmetrical with respect to / and E. Differentiating these equations : d^/ dE , ... (1) (2) dx dx d 'E dl / . , V -7^ ^17^"-'''^' and substituting (3) in (1) and (2), we get : (3) dx' d^E \ -^ = E(g^jb,){r-jx), d^l (4) (5) the differential equations of E and /. llO. These differential equations are identical^ and con- sequently I and E are functions differing by their limiting conditions only. These equations, (4) and (5), are of the form : '^=wig-Jb,)(r-jx). (6) and are integrated by w •= at *"*, where c is the basis of natural logarithms ; for, differen- tiating this, we get, d^w dx- = v^at^ = i^w\ 160 ALTERNATING-CURRENT PHENOMENA. [§110 hence, v^ = {g -j'^c) ir - Jx) -, or, V = ± ^ {g-'jf^c){^-jx) ; hence, the general integral is : w = {(«// ~iS A) -(^.'I'-^.^IJ a and j3 being determined by equations (11). (16) (16) 112. li Z = R — j X is the impedance of the receiver circuit, E^ = c^ -\- J c^ is the E.M.F. at dynamo terminals (17), and / = length of line, we get at hence or At X = /, X = 0, j^ A + B E^d^'^-, ^ ^E ^ A — B a —jp /~A + B g-Jb/' A - B ^ j} S- J be A+B a-jP (18) — {(^c«'-^c-«0cos/8/-y(/^€«^ + Bc-^^ (19) sinp/}. 162 ALTERNATING-CURRENT PHENOMENA. [§ 113 Equations (18) and (10) determine the constants^ and B, which, substituted in (14), give the final integral equations. The length, x^ = 2ir/^ is a complete wave length (20), which means, that in the distance 2ir/^ the phases of cur- rent and E.M.F. repeat, and that in half this distance, they are just opposite. Hence the remarkable condition exists that, in a very long line, at different points the currents at the same time flow in opposite directions, and the E.M.Fs. are opposite. 113. The difference of phase, ^, between current, /, and E.M.F. , E^ at any point, x, of the line, is determined by the equation, D (cos u* + y sin u>) = -~ j where Z^ is a constant. Hence, ^ varies from point to point, oscillating around a medium position, wx, which it approaches at infinity. This difference of phase, wx, towards which current and E.M.F. tend at infinity, is determined by the expression, Z>(coswx +ysinwx) = /j2 / or, substituting for E and / their values, and since €~ ■* =0, and A c" (cos )9 x — y sin /3 x), a cancels, and D (cos w« + y sin wx) = -** — 4^- g - jiu _(a ^-f/g/v)~y(a^c-/g^) . hence, tan ix = Zi^JliAll. , (i>i ) ^g-V^K This angle, wx, = ; that is, current and E.M.F. come more and more in phase with each other, when «^c — fig = ^ J that is, oi-^ li=g-^ Kj or, §114] DISTRIBUTED CAPACITY, 16S substituting (10), gives, gx + b,r 2gbc hence, expanding, '^ -s- ^ = ^ -7- ^r ; (22) that is, the ratio of resistance to inductance equals the ratio of leakage to capacity. This angle, wx, = 45° ; that is, current and E.M.F. differ by ilh period, if - a ^^ + Pg = oig + pb,\ or, a ^K + g . ■ P K+g' which gives : rg + xly^ = 0. (23) That is, tivo of the four line constants must be zero ; eitlur g and -T, or g and b^ . 114. As an instance, in Fig. 86 a line diagram is shown, with the distances from the receiver end as abscissae. The diagram represents one and one-half complete waves, and gives total effective current, total E.M.F., and differ- ence of phase between both as function of the distance from receiver circuit ; under the conditions, E.M.F. at receiving end, 10,000 volts ; hence, ^^ = {." (27.4 cos /Jx - 47.3 sin ^x) + «- (32.6 cos ^x - 22.3 sin ^x)}; E^ {€"{W50cos/3x + 4410 sin ^x) + < (3530 cos /3x - 4410 sin ^x)} +y {." (4410 cos ;8x - 6450 sin ^x) - « (4410 cos /3x + 3530 sin ^x)}; t.n „-. = - -^' + ^^ «^+/ 1 = - .073, ». = - 4.2 ,_ L 1 St io... i \ ' !•/ "N 1/ i. >. L \ ^ ~' •- V 1 '•- r_ / 1,- -w ( ^ , / J "t; / ■" '~ / _j ... r k. / / !« "JH 7 1 /■' K 1 .. L .,. 1 I ^ _..-- ,.„,L r 1 / ^, 7 -■' / :r /. X / ■^^ --'' I"i J- |.B|0.00 •«_!.»! \ / l-,A -' = T r r * §115] DISTRIBUTED CAPACITY. 165 116. The following are some particular cases : A,) Open circuit at end of lines : X = : /i = 0. A = {ge^^- Ke() ^Jige^- b,e,) ^ - R\ hence, ^= ^ ^{(c«» + €-*«(cos/3x — y(€" -€-")sinj3x};| C = — L_^{(c«« - €-**)cos)9x -y(c" + €-«*)sin)3x}. a —jP J B.) Line grounded at ejid: X = : ^1 = 0. A = (a/\ + /8//) +y (a// ^pi,)^B E^ — ^^ — ^{(c*'^ - €-«*)cos/8x -y(««« + €-•*) sin/8x}; g - A J = — ?^ — ^{(c** + €-**)cos/8x ->(€«« — €-»«)sin^x}. C) Infinitely long conductor : Replacing x by — x, that is, counting the distance posi- tive in the direction of decreasing energy, we have, X = X : /= 0, ^ = 0; hence ^ = 0, and ^^ 1 ^c-««(cosi3x+ysin/3x); g-jf>c I = A €"•* (cos/3x +ysin)3x), involving decay of the electric wave. The total impedance of the infinitely long conductor is _ g ->/? g—JK _ (g -y;8) Cr+y^c) _ (a;f +j3^c) -jifig-ab,) g" + V -«•* + ^^ 166 AL TERN A TING-CURRENT PHENOAfENA, [§115 The infinitely long conductor acts like an impedance that is, like a resistance p _ g^ + pi>c g^ + ^c^ combined with a reactance We thus get the difference of phase between E.M.F. and current, tan cj = — = ^--^ , which is constant at all points of the line. If ^ = 0, ;r = 0, we have, hence, tan w = 1, or, (i> = 4o ; that is, current and E.M.F. differ by Jth period. D,) Generator feeding into elosed circuit : Let X = be the center of cable ; then, -fi'x = — ^_x ; hence : if = at x = ; which equations are the same as in B, where the line is grounded at x = 0. § 116, 117] AL TERN A TING-CURRENT TRANSFORMER, 167