CHAPTER XV DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE, AND LEAKAGE 127. In the foregoing, the phenomena causing loss of energy in an alternating-current circuit have been discussed; and it has been shown that the mutual relation between current and e.m.f. can be expressed by two of the four constants: power component of e.m.f., in phase with current, and = current X effective resistance, or r; reactive component of e.m.f., in quadrature with current, and = current X effective reactance, or x; power component of current, in phase with e.m.f., and = e.m.f. X effective conductance, or g; reactive component of current, in quadrature with e.m.f., and = e.m.f. X effective susceptance, or b. In many cases the exact calculation of the quantities, r, x, g, h, is not possible in the present state of the art. In general, r, x, g, b, are not constants of the circuit, but depend — besides upon the frequency — more or less upon e.m.f., current, etc. Thus, in each particular case it becomes necessary to dis- cuss the variation of r, x, g, b, or to determine whether, and through what range, they can be assumed as constant. In what follows, the quantities r, x, g, b, will always be consid- ered as the coefficients of the power and reactive components of current and e.m.f. — ^that is, as the effective quantities — so that the results are directly appHcable to the general electric circuit containing iron and dielectric losses. Introducing now, in Chapters VIII, to XI, instead of "ohmic resistance," the term "effective resistance," etc., as discussed in the preceding chapter, the results apply also — within the range discussed in the preceding chapter — to circuits containing iron and other materials producing energy losses outside of the electric conductor. 128. As far as capacity has been considered in the foregoing chapters, the assumption has been made that the condenser or 168 DISTRIBUTED CAPACITY 169 other source of negative reactance is shunted across the circuit at a definite point. In many cases, however, the condensive react- ance is distributed over the whole length of the conductor, so that the circuit can be considered as shunted by an infinite num- ber of infinitely small condensers infinitely near together, as diagrammatically shown in Fig, 100. liiilliiiiiiiiiiiiiiiiiii JTTTTTTTTTTTTTTTTTTTTTTT- Fig. 100. In this case the intensity as well as phase of the current, and consequently of the counter e.m.f. of inductive reactance 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, and to a certain degree in the high-poten- tial coils of alternating-current transformers for very high vol- tage and also in high frequency circuits. 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 condenser of the same capacity as the line, shunted across the line at its middle. 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. A better approximation is to consider the line as shunted at the generator and at the motor end, by two condensers of one- sixth the line capacity each, and in the middle by a condenser of two-thirds the line capacity. This approximation, based on Simpson's rule, assumes the variation of the electric 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. Or the phenomena occurring in the circuit can be investigated graphically by the method given in Chapter VI, §39, by dividing the circuit into a sufficiently large number of sections or fine 170 ALTERNATING-CURRENT PHENOMENA elements, and then passing from line element to line element, to construct the topographic circuit characteristics. 129. It is thus desirable to first investigate the limits of appli- cability 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 = 1.11 X 10-« kl H- 4 loge 2- microfarads, where k = dielectric constant of the surrounding medium = 1 in air; I = length of conductor = 5 X 10" cm.; ■ d = distance of conductors from each other = 50 cm.; 5 = diameter of conductor = 1 cm. Hence C = 0.3 microfarad, the condensive reactance is x = ^ — 7f< ohms, where/ = frequency; hence at/ = 60 cycles, X = 8,900 ohms; and the charging current of the line, at £' = 20,000 volts, be- comes, E to = — = 2.25 amp. X The resistance of 100 km. of wire 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 T 2,000 _. 1 = ~99~ = 91 amp. representing about 1,800 kw. In this case, the condenser current thus amounts to less than 2.5 per cent., and hence can still be represented by the approxi- mation of one condenser shunted across the line. If the length of transmission is 150 km., and the voltage, 30,000, condensive reactance at 60 cycles, x = 2,970 ohms; charging current, ?'o = 10.1 amp.; line resistance, r = 06 ohms; main current at 10 per cent, loss, / = 45.5 amp. DISTRIBUTED CAPACITY 171 The condenser current is thus about 22 per cent, of the main current, and the approximate calculation of the effect of line capacity still fairly accurate. 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 investiga- tion of the phenomena in the line necessary. In many cases of practical engineering, however, the capacity effect is small enough to be represented by the approximation of one; or, three condensers shunted across the line. 130. {A) Line capacity represented hy one condenser shunted across middle of line. Let Y = g — jh = admittance of receiving circuit; Z = r -\- jx = impedance of line; he = condenser susceptance of line. iEo Fig. 101. Denoting in Fig. 101. the e.m.f., and current in receiving circuit by E, 7, the e.m.f. at middle of line by E' , the e.m.f., and current at generator by Ea, h; we have, I = E(g-jh); E' = E + "^i^/ pJi , (r + jx) (g - jh) ] /o = 7 + jhcE' ■ =^j,-i5+A[i+^-+^"-y^-^-^>]}; Eq = E' -] 2 — -^0 = e\i-\- (^ + -^'^^ (^ ~ ^^^ 1 (^ + J^) (d - J^) * i ^ ^ , jhc(r + jx) (r + jxy- (g - jb) 1 + 2 + ^ ^^ 4 1 ' 172 ALTERNATING-CURRENT PHENOMENA or, expanding, /o = ^[ j^ + ^^{rb - xg)] - j[{b - h) - ^^{rg + xb)] } ; Eo = ^^ jl + (r+jx) (g - jb) +^^ (r+jx) = e{ 1 + (r + jx) [g - jb + ^-^) + ^-jir+jxY (g-jb) 131. Distributed condensive reactance, inductive reactance, leak- age, and resistance. 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 under- ground cables, or with very feeble currents at extremely high frequency, such as telephone currents, the consideration of the line resistance— vfhich. consumes e.m.fs. in phase with the current — and of the line reactance — which consumes 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 electrostatic capacitij 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 changes in intensity as well as in phase, in the line from point to point; and the e.m.f. con- sumed by the resistance and inductive reactance therefore also changes 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 electricitij into the air, takes place by corona, 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 power, and is, therefore, analogous to the e.m.f. consumed by resistance, while the condenser current and the e.m.f. of self- induction are wattless or reactive. DISTRIBUTED CAPACITY 173 Furthermore, the alternating current in the Hne produces in all neighboring conductors secondary currents, which react upon the primary current, and thereby introduce 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 a power coni'ponent of mutual induct- ance in phase with the current, which acts as an increase of resistance, and into a reactive component in quadrature with the current, which decreases the self-inductance. This mutual inductance is not always negligible, as, for in- stance, its distur))ing influence in telephone circuits shows. The alternating voltage of the line induces, by electrostatic influence, electric charges in neighboring conductors outside of the circuit, which retain corresponding opposite charges on the line wires. This electrostatic influence requires a current pro- portional to the e.m.f. and consisting of a power component, in phase with the e.m.f., and a reactive 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 resistance. This electromagnetic hysteretic loss may take place in the con- ductor 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 inductive reactance," of which it is a power component. The alternating electrostatic field of force expends energy in dielectrics by corona and dielectric hysteresis. In concentric cables, where the electrostatic gradient in the dielectric is com- paratively large, the dielectric losses may at high potentials consume appreciable amounts of energy. The dielectric loss appears in the circuit as consumption of a current, whose com- ponent in phase with the e m.f. is the dielectric power current, 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. Furthermore, the electric field of the conductor progresses with a finite velocity, the velocity of light, hence lags behind 174 ALTERNATING-CURRENT PHENOMENA the flow of power in the conductor, and so also introduces power components, depending on current as well as on potential difference. 132. This gives, as the most general case, and per unit length of line: e.m.fs. consumed in phase with the current, I, and = rl, repre- senting consumption of power, and due to: Resistance, and its increase by unequal current distri- bution; to the power component of mutual inductive reactance or to induced currents; to the power component of self-inductive reactance or to electromagnetic hysteresis, and to radiation. e.m.fs. consumed in quadrature with the current, I, and = xl, wattless, and due to: Self -inductance, and mutual inductance. Currents consumed in phase with the e.mf., E, and = g E, representing consumption of power, and due to: Leakage through the insulating material, including silent discharge and corona; power component of electrostatic influence; power component of capacity or dielectric hysteresis, and to radiation. 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, Effective conductance, g, Effective susceptance, — h, per unit length of line, which represents the coefficients, per unit lenght of line, of e.m.f. consumed in phase with current; e.m.f. consumed in quadrature with current; current consumed in phase with e.m.f.; current consumed in quadrature with e.m.f.; or, Z = r -\- jx, Y ^ g+ jb. and, absolute. = \/r^ + x^, = Vg- + b\ DISTRIBUTED CAPACITY 175 The complete investigation of a circuit or line contain- ing distributed capacity, inductive reactance, resistance, etc., leads to functions which are products of exponential and of trigonometric functions. That is, the current and potential difference along the line, I, are given by expressions of the form : e+«'(^ COS ^1 + B sin pi). Such functions of the distance, I, or position on the line, while alternating in time, differ from the true alternating waves in that the intensities of successive half-waves progressively increase or decrease with the distance. Such functions are called oscillating waves, and, as such, are beyond the scope of this book, but are more fully treated in "Theory and Calculation of Transient Electric Phenomena and Oscillations," Section III. There also will be found the discussion of the phenomena of distributed capacitj^ in high-potential transformer windings, the effect of the finite velocity of propagation of the electric field, etc. For most purposes, however, in calculating long-distance transmission lines and other circuits of distributed constants, the following approximate solutions of the general differential equation of the circuit offers sufficient exactness. 133. The impedance of an element, dl, of the line is: Zdl and the voltage, dE, consumed by the current, /, in this line ele- ment dl: JE7 VTJ7 dE = Zldl The admittance of the line element, dl, is: Ydl hence the current, dl, consumed by the voltage, dE, of this line element dl: ,^ ,^„ „ dl = YEdl This gives the two equations of the transmission line: Differentiating the first equation, and substituting therein the second, gives: J = ZYE (1) 176 ALTERNATING-CURRENT PHENOMENA and from the first equation follows: J _ I dE ^ ~ Z dl Equation (1) is integrated by: E ^ A Bl (2) (3) and, substituting (3) in (1), gives: B^ = ZY hence: B = -\- VZY and - VZF There exist thus two values of B, which make (3) a solution of (1), and the most general solution, therefore, is: E ^A^e + ^^'+A,e'-^^' (4) Substituting (4) in (2) gives: 7 = V r +^/zYi -VzYi] (5) where I is counted from some point of the line as starting point, for instance, from the step-down end as Z = 0. If then: £"0 = voltage at step-down end of the line, 7o = current at step-down end, it is, for: hence: I = 0; Eo ^ Ai -\- A2 h = Ai - A' A ^ \ F [^ T A2 - 2 .'~\y (6) and, substituting (6) into (5): + VZYl - y/ZYl € + e + VZYl - y/ZYl \ e — e ■VVZYl - VZYl rjy +VZYl -y/ZYl i-i" 2 '^Vz^' 2 (7) DISTRIBUTED CAPACITY 111 Substituting in (7) for the exponential function tlie infinite series: ^^ryyi , , ZYX" , ZYVZYI^ , Z-^YH" , ±VzH _ 1 + VZFZ + TT,- ± h, V -r-A V ■■■ e gives: 134. If then : I = k is the total length of line, and Zo = loZ — total line impedance, Fo = loY = total line admittance, the equations of voltage Ei and current Ii at the end k of the line are given by substituting I = lo into equations (8), as: (8) ^x = ^0 j 1 + ^ + . . . } + Zo7o { H- ^'p + . . . /x- /o { 1 + ^ + . . . } + Foi^o { 1 + ^-" + . . . (9) Since Zq is the line impedance, and thus ZqI the impedance Zol . . voltage, —— is the impedance voltage, as fraction of the total voltage. Since Fo is the line admittance, YqE is the charging Y W current, and — j— the charging current as fraction of the total current. The product of these two fractions is: ^ V 1^ - 7 V . p ■^ J — .^oi^ 0 ZqFo thus is the product of impedance voltage and charging current of the line, expressed as fraction of total voltage and total current, respectively, hence is a small quantity, and its higher powers can therefore almost always be neglected even in very long transmission lines, and the equation (9) approximated to: E, = Eo\l+^]+Zoh !l+^"^°^ 6 /i = Zo ] 1 + ^^ \ + YoEo I 1 + ^ ZqYo ] , -- _ f ^ , ZoYo 6 12 (10) 178 ALTERNATING-CURRENT PHENOMENA These equations are simpler than those often given by repre- senting the hne capacity by a condenser shunted across the middle of the line, and are far more exact. They give the generator voltage and current, Ex repectively /i, by the step-down voltage and current, -E'o and Zo respectively. Inversely, if E^ and 7o are chosen as the values at the generator end, the values at the step-down end are given by substituting Z = — Zo in equations (8), as: E, = E,[l + ^^]-Z,h {l+^ A = /o j 1 + ^ ) - Fo^o { 1 + ^" (11) Neglecting the line conductance: go = 0, gives: Fo - + jh and : Zo ^ n + jxo hence, substituted in equations (10) and (11), and expanded, gives El = £'o|l - -Y -^J^\ ± /o(^0+jXo)|l - -Q- = J-Q-\ where the upper sign holds, if Eq, Iq are at the step-down end. El, 1 1 at the generator end of the line, and the lower sign holds, if Eq, /q are at the generator end, Ei, Ii at the step-down end of the line. As seen, the equations (12) are just as simple as those of a circuit containing the resistance, inductance and capacity lo- calized, and are amply exact for practically all cases. Where a still closer approximation should be required, the next term of equations (8) and (9) may be included. Z Y In many cases, the — w— term in (10) and (11) may also be dropped, giving the still simpler equation: ^q/ 0 (13) h=U\l + ^\± WE,