CHAPTER VII. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 50. The free oscillation of a complex circuit differs from that of the uniform circuit in that the former contains exponential functions of the distance A which represent the shifting or transfer of power between the sections of the circuit. Thus the general expression of one term or frequency of current and voltage in a section of a complex circuit is given by equations (290); - £~SA [C cos q (A + 0 + D sin q (J + t)]} and /7 +SA [A cos q (A — t) -f B sin q (A — £)] where q = nq0, q0 = — , A = total length of circuit, expressed in the distance coordinate A = o-lt I being the distance coordinate of the circuit section in any measure, as miles, turns, etc., and r, L, g, C the circuit constants per unit length of I, a- = VIC, u = -(-=• + — ) = time constant of circuit section, 2 YL/ C ' UQ= u + s = resultant time decrement of complex circuit, s = u0 - u = energy transfer constant of circuit section. 613 514 TRANSIENT PHENOMENA The instantaneous value of power at any point X of the circuit at any time t is p = ei [A cos q(X-t) + B sin q (X - t)]2 [C cos q (X + 0 + D sin g (J + O]2} + [e+2sA(A2-£2) cos 2q (X-t) -e~2s* (C2-D2) cos 2 g (l + t)] + 2 [ABe+*s*sm 2 g (X-t) -CDe-2'* sin 2-g (A + *)]} ; (303) that is, the instantaneous value of power consists of a constant term and terms of double frequency in (X - t) and (A + t) or in distance A and time t. Integrating (303) over a complete period in time gives the effective or mean value of power at any point X as p = r*M {fi+2« (A2 + J52) - s-2s (C2 + £>2) } ; (304) .2 * C that is, the effective power at any point of the circuit is the difference between the effective power of the main wave and that of the reflected wave, and also, the instantaneous power at any time and any point of the circuit is the difference between the instantaneous power of the main wave and that of the reflected wave. The effective power at any point of the circuit gradually decreases in any section with the resultant time decrement of the total circuit, £-2uotf and varies gradually or exponentially with the distance A, the one wave increasing, the other decreasing, so that at one point of the circuit or circuit section the effective power is zero ; which point of the circuit is a power node, or point across which no energy flows. It is given by e+2sA (A2 + B2) = e~2s* (C2 + D2), or A2 + B2 C2 + D2 (305) POWER AND ENERGY OF THE COMPLEX CIRCUIT 515 The difference of power between two points of the circuit, ^ and A2, that is, the power which is supplied or received (depend- ing upon its sign) by a section X' = A2 — Xl of the circuit, is given by equation (304) as - (£~2S^ - £-28^) (C2 + D2)}. (306) If P0 is > 0, this represents the power which is supplied by the section X to the adjoining section of the circuit; if P0< 0, this is the power received by the section from the rest of the complex circuit. If sX2 and s^ are small quantities, the exponential function can be resolved into an infinite series, and all but the first term dropped, as of higher orders, or negligible, and this gives the approximate value = ± 2 s (;2 _ jg = ± 2 SA'; (307) hence, ^o - #y%'-** [A* + B2 + O* + D2} ; (308) that is, the power transferred from a section of length A' to the rest of the circuit, or received by the section from the rest of the circuit, is proportional to the length of the section, A', to its trans- fer constant, s, and to the sum of the power of main wave and reflected wave. 51. The energy stored by the inductance L of a circuit element dXj that is, in the magnetic field of the circuit, is 'LV dwl =-^~A where U = inductance per unit length of circuit expressed by the distance coordinate A. Since L = the inductance per unit length of circuit, of distance coordinate Z, and X = ^ (ouyj In general, the circuit constants r, L, 0, C, per unit length, I = 1 give, per unit length, X = 1, the circuit constants L. £. £. 5 j or Vic ' VLC c VLC Substituting (290) in equation (309) gives (310) + e~2sA [C cos q (X + 0 + D sin 0 (4 + Of - 2 [A cos q (A - 0 + B sin 0 (4 - 0] [C cos q (A + 0 + D sin 0 (J + 01} + [e+2«*(Aa -£2) cos 2 g (A -0 + fi-^C8- D2) cos 2 ^ (4 + 0] + 2 [A£e+2sA sin 2 0 (4 - 0 + CDe~2sX sin 2 g (4 + 0] - 2 [(AC - BD) cos 2 04 + (AD + BC) sin 2 g/l] - 2 [(AC + £D) cos 2qt+ (AD - BC) sin 2qt]}. (311) Integrating over a complete period in time gives the effective energy stored in the magnetic field at point A as a w j j. i*u/j 7 7T == ^ I — TT~ Ctfr (A - 2 [(AC - BD) cos 2qX+ (AD + BC) sin 2 ql]}, (312) POWER AND ENERGY OF THE COMPLEX CIRCUIT 517 and integrating over one complete period of distance A, or one complete wave length, this gives (313) The energy stored by the inductance L, or in the magnetic field of the conductor, thus consists of a constant part, dl a part which is a function of (X — t) and (X + t), (A2 - B2) cos 2 q (X - t) (C2 - D2) cos 2 q (X + 0] n2g(/l - 0 n2g(yl + 01} , (315) a part which is a function of the distance X only but not of time t, cos 2 ^ + (AD + BC) sin 2 ^ (316) and a part which is a function of time £ only but not of the distance ^, 1 2 (317) and the total energy of the electromagnetic field of circuit element dX at time t is Aw'rr 1 /7 "~ = V £~2""'{ (4(7+BI)) cos 2 9' + (^0-JSC) sin 2 qt\, dX d^ dl dX dX 52. The energy stored in the electrostatic field of the conductor or by the capacity C is given by CV dw2 = — dl\ 518 TRANSIENT PHENOMENA or, substituting (310), and substituting in (319) the value of e from equation (290) gives the same expression as (311) except that the sign of the last two terms is reversed ; that is, the total energy of the electro- static field of circuit element dX at time t is dw2 dwn dw' dw" dw'" ~df = ~df + ~dT + ~dA~ + ~dT' (32°^ and adding (318) and (320) gives the total stored energy of the electric field of the conductor, dw dw, dw2 cydw^ dw' and integrated over a complete period of time this gives « 2^ = dw" dw"' The last two terms, — and — , thus represent the energy which is transferred, or pulsates, between the electromagnetic and the electrostatic field of the circuit; and the term — repre- sents the alternating (or rather oscillating) component of stored energy. 53. The energy stored by the electric field in a circuit section ^, between A, and A2, is given by integrating - - between A2 and AI} U/A as - (€-2«J, _ ^-2.^ (Ca + £>2) I . (323) or, substituting herein the approximation (307), 1 ... II 2 W = 2 2 2 2 (324) POWER AND ENERGY OF THE COMPLEX CIRCUIT 519 Differentiating (324) with respect to t gives the power sup- plied by the electric field of the circuit as P = _ = uj> e-^ {A2 + B* + C* + D2}, (325) at or, more generally, p = 2. y£-2^ { (£+2S2 _ £-f2) (Aa + £2) _ (e-ad, _ e-a^) (C8 + D2)}. (326) 54. The power dissipated in the resistance r'dX = - r of a VLC conductor element dX is dp? = Mdl (327) 2r hence, substituting herein equation (318) gives the power con- sumed by resistance of the circuit element dX as ^ 2r (flfojo du/ _2) / , (337) POWER AND ENERGY OF THE COMPLEX CIRCUIT 521 the energy stored in the electric field of the circuit section of length X is ^; (338) 2 the power supply to the conductor by the decay of the electric field of the circuit is P = V'#2£~2w°'; (339) the power dissipated in the circuit section X' by its effective resist- ance and conductance is P» = ul'IPe-*1*, (340) and the power transferred from the circuit section A' to the rest of the circuit is P = sl'IPe-*"*', (341) u that is. — = ratio of power dissipated in the section to that u0 supplied to the section by its stored energy of the electric field. o — = fraction of power supplied to the section by its electric u0 field, which is transferred from the section to adjoining sections (or, if s < 0, received from them). o - = ratio of power transferred to other sections to power dissipated in the section. u0 -f- u -s- s thus is the ratio of the power supplied to the sec- tion by its electric field, dissipated in the section, and transferred from the section to adjoining sections. These relations obviously are approximate only, and applicable to the case where the wave length is short. 56. Equation (306), of the power transferred from a section to the adjoining section, can be arranged in the form (A2 + B2) - e-2 (C2 + D2)] (A2 + B2) - £-2sA' ((T2 + Z)2)]} ; (342) that is, it consists of two parts, thus : Po> - li/5f-** {e+2^ (A2 + £2) - e-2'*» (C2 + D2)}, (343) 2 " C 522 TRANSIENT PHENOMENA which is the power transferred from the section to the next fol- lowing section, and p* = + ye-** {e+a., (A2 + ff)_e-* (C2 + £>2)}? (344) which is the power received from the preceding section, and the difference between the two values, ^o - P,' ~ p»f> (345) therefore, is the excess of the power given out over that received, or the resultant power supplied by the section to the rest of the circuit. An approximate idea of the value of the power transfer con- stant can now be derived by assuming H2 as constant throughout the entire complex circuit, which is approximately the case. In this case, as the total power transferred between the sections must be zero, thus: hence, substituting (341), 2X-V = °> (346) and, since O fit , ^ /)/ *» — ^o »# w0A = 5)1^'; (347) that is, the resultant circuit decrement multiplied by the total length of the circuit equals the sum of the time constants of the sections multiplied with the respective length of the section, or, if ?!, ?2 • • • ?i = length of the circuit section, as fraction of the total circuit length A, KO = Dfct*t, (348) Whether this expression (348) is more general is still unknown. 57. As an example assume a transmission line having the following constants per wire :rl = 52;LX = 0.21 jg^ = 40 X 10~6, and Cl = 1.6 X 1Q-6. Further assume this line to be connected to step-up and step- down transformers having the following constants per trans- POWER AND ENERGY OF THE COMPLEX CIRCUIT 523 former high-potential circuit: r2 = 5, L2 = 3; g2 = 0.1 X 10 6, and C2 = 0.3 X 10~6; then A/ = a1 = vT/7i = 0.58 X 10~3, J2' = *2 = 0.95 X 10~3, u, = 136, u2 -= 1. The circuit consists of four sections of the lengths V = 0-58 X 10~3, A2'= 0.95 X 10~3, V =0.58 X 10~3, V = 0.95 X 10~3 ; hence a total length A - 3.06 X 10-3, and the resultant circuit decrement is M» = ^A 51.6 + 0.59 - 52.2; hence, sl = - 83.8 and s2 = + 51.2. If now the current in the circuit is i0 = 100 amperes, the e.m.f. e0 = 40,000 volts, the total stored energy is W = ;02 (L, + L2) + e* (C, + C2) = 32,000 + 3000 = 35,000 joules, and from equation (338) then follows, for t = 0, 2 - 35,000, = 22.8X10', which gives u0 = 52.2, IP = 22.8 X 106, W = 35,000. Line. Step-up Transformer. Line. Step-down Transformer. Length of section, X' = 0.58xlO~3 0.95X10~3 0.58X10~3 0.95X10-3 Time constant, u = 136 1 136 1 Transfer constant, s = -83.8 + 51.2 -83.8 + 51.2 Energy of electric field, W = 6.650 10.850 6.650 10.850kilojoules Power supplied by electric field, P= 690 1132 690 1132 kilowatts Power dissipated, P,° = 1800 22 1800 22 kilowatts Power transferred, P0 = - 1110 1110 - 1110 11 10 kilowatts 524 TRANSIENT PHENOMENA Thus, of the total power produced in the transformers by the decrease of their electric field, only 22 kw. are dissipated as heat in the transformer, and 1110 kw. transferred to the transmission line. While the power available by the decrease of the electric field of the transmission line is only 690 kw., the line dissipates energy at the rate of 1800 kw., receiving 1110 kw. from the transformers.