CHAPTER II. DISCUSSION OF GENERAL EQUATIONS. 7. In the preceding chapter the general equations of current and voltage were derived for a circuit or section of a circuit having uniformly distributed and constant values of r, L, g, C. These equations appear as a sum of groups of four terms each, characterized by the feature that the four terms of each group have the same values of s, q, h, k. Of the four terms of each group, iv iv i3, i4 or ev ev es, e4 respectively (equations (50) and (51)), two contain the angles (qt — kl): iv e1 and iz, e3; and two contain the angles (qt + kl): i2, e2 and i4, e4. In the terms iv e^ and iz, e3, the speed of propagation of the phenomena follows from the equation qt - kl = constant, thus: ti q dt r.*1 hence is positive, that is, the propagation is from lower to higher values of I, or towards increasing I. In the terms iv e2 and i4, e4, the speed of propagation from qt + kl = constant is dl_ _q Jt~ ~k hence is negative, that is, the propagation is from higher to lower values of I, or towards decreasing I. Considering therefore iv el and i3J e3 as direct or main waves, iv e2 and i4, e4 are their return waves, or reflected waves, and iv e2 is the reflected wave of iv e^ i4, e4 is the reflected wave of iv ey 431 432 TRANSIENT PHENOMENA Obviously, i2J e2 and i^ e± may be considered as main waves, and then iv et and i3, e3 are reflected waves. Substituting ( - I) for (+ I) in equations (50) and (51), that is, looking at the circuit in the opposite direction, terms i2, e2 and iv e1 and terms i4, 64 and iv e3 merely change places, but otherwise the equations remain the same, except that the sign of i is reversed, that is, the current is now considered in the opposite direction. Each group thus consists of two waves and their reflected waves: ^ - i2 and et + e2 is the first wave and its reflected wave, and i3 - i4 and e3 + e4 is the second wave and its reflected wave. In general, each wave and its reflected wave may be con- sidered as one unit, that is, we can say: if = il — i2 and e' = e^ + e2 is the first wave, and i" = i3 — i4 and e" = e3 + e4 is the second wave. In the first wave, i', e', the amplitude decreases in the direction of propagation, e~wfor rising, e+hl for decreasing I, and the wave dies out with increasing time t by e -<«-*>* = £~ut £+stf In the second wave, i", eft ', the amplitude increases in the direction of propagation, e+w for rising, s~hl for decreasing I, but the wave dies out with the increasing time t by £-(tt+s>< = s""* e~st, that is, faster than the first wave. If the amplitude of the wave remained constant throughout the circuit — as would be the case in a free oscillation of the circuit, in which the stored energy of the circuit is dissipated, but no power supplied one way or the other — that is, if h = 0, from equation (56) s = 0; that is, both waves coincide and form one, which dies out with the time by the decrement e~ut. It thus follows: In general, two waves, with their reflected waves, traverse the circuit, of which the one, i", e", increases in amplitude in the direction of propagation, but dies out corre- spondingly more rapidly in time, that is, faster than a wave of constant amplitude, while the other, i', e' , decreases in amplitude but lasts a longer time, that is, dies out slower than a wave of constant amplitude. In the one wave, i", e" ', a decrease of amplitude takes place at a sacrifice of duration in time, while in the other wave, i', e', a slower dying out of the wave with the time is produced at the expense of a decrease of amplitude during its propagation, or, in i", e" duration in time is sacrificed to duration in distance, and inversely in i', e'. DISCUSSION OF GENERAL EQUATIONS 433 It is interesting to note that in a circuit having resistance, inductance, and capacity, the mathematical expressions of the two cases of energy flow; that is, the gradual or exponential and the oscillatory or trigonometric, are both special cases of the equations (60) and (61), corresponding respectively to q = o, k = 0 and to h = 0, s = 0. 8. In the equations (50) and (51) qt = 2x gives the time of a complete cycle, that is, the period of the wave, and the frequency of the wave is / = -L 2 kl = 27T gives the distance of a complete cycle, that is, the wave length, W 7 7 k (u — s) t = 1 and (u + s) t = 1 give the time, */'- — and t"= -*—, during which the wave decreases to - = 0.3679 of its value, and hi = 1 gives the distance, over which the wave decreases to - = 0.3679 of its value; £ that is, q is the frequency constant of the wave, f - - I I: «'—'•• (62) > 2V °~' 434 TRANSIENT PHENOMENA k is the wave length constant, (63) (u - s) and (u -f s) are the time attenuation constants of the wave, 1 ) (64) U + S and h is the distance attenuation constant of the wave, L -I. (65) 9. If the frequency of the current and e.m.f. is very high, thousands of cycles and more, as with traveling waves, lightning disturbances, high-frequency oscillations, etc., q is a very large quantity compared with s, u, m, h, k, and k is a large quantity compared with h, then by dropping in equations (50) to (61) the terms of secondary order the equations can be simplified. From (54), ^ = V(s2 + q2 - m2)2 + 4 q2m2 = V(q2 + m2)2 + 2 s2 (q2 - m2) = q2 + m2 + s2 = q2 -f m2 + s2 -m (f - q -m = +rn2} =VLO(q2+ m2)=qVLO, h2 + tf = (s2 + q2) LC = q2LC, •j (66) and + h (m + s) _qL DISCUSSION OF GENERAL EQUATIONS 435 qk — h (m — s) qL IL = A' + yfc2 ~L="T \C' , _ k (m + s) - qh J _ q VLC (m + s) - qs VLC q2LC r _m /L ~V' f __ k (m - s) + qh q VTC (m - s) + qs VTC h2 + k2 that is, and Writing q2LC \/i Cl - C2 q l). (76) 10. As seen from equations (74) and (75), the waves are products of £~ut and a function of (t - X) for the main wave, (t + X) for the reflected wave, thus : *\ + *'. = *~utL (t - V } and (77) i2 + i4 = e~utf2 (t + A);J DISCUSSION OF GENERAL EQUATIONS 437 hence, for constant (t - X) on the main waves, and for constant (t + X) on the reflected waves, we have and (78) that is, during its passage along the circuit the wave decreases by the decrement e~ut, or at a constant rate, independent of frequency, wave length, etc., and depending merely on the circuit constants r, L, g, C. The decrement of the traveling wave in the direction of its motion is and therefore is independent of the character of the wave, for instance its frequency, etc. 11. The physical meaning of the two waves i' and e' can best be appreciated by observing the effect of the wave when travers- ing a fixed point X of the circuit. Consider as example the main wave only, i' = i^ + ia, and neglect the reflected waves, for which the same applies. From equation (74), i = e —*-("-*>« DJg (t - ;)] + e+s*-(u+s»D3[q(t - /I)]; (79) or the absolute value is (80) where Dt and D3 have to be combined vectorially. Assuming then that at the time t = 0, 7 = 0, for constant X we have / = D (£-<«-•>« - fi -<"+«>«), (81) the amplitude of 7 at point L Since (81) is the difference of two exponential functions of different decrement, it follows that as function of the time t, I rises from 0 to a maximum and then decreases again to zero, as shown in Fig. 98, where 73 - De - r- £ - 1* and the actual current i is the oscillatory wave with 7 as envelope. 438 TRANSIENT PHENOMENA The combination of two waves thus represents the passage of a wave across a given point, the amplitude rising during the arrival and decreasing again after the passage of the wave. Fig. 98. Amplitude of electric traveling wave. 12. If h and so also s equal zero, i' , er and i" ', e" coincide in equations (74) and (75), and Cl and C3 thus can be combined into one constant Bv C2 and C4 into one constant J52, thus : C3 = Bv Ct - Bv Cs' = Bt', (82) and (74), (75) then assume the form i = s-ut ^ j[JB1 cos q(t- X) + £/ sin q(t - X)] - [B2 cos q(t + *) + B2' sin q (t + A)] } , (83) (84) These equations contain the distance X only in the trigono- metric but not in the exponential function; that is, i and e vary in phase throughout the circuit, but not in amplitude; or, in other words, the oscillation is of uniform intensity throughout the circuit, dying out uniformly with the time from an initial maximum value; however, the wave does not travel along the DISCUSSION OF GENERAL EQUATIONS 439 circuit, but is a stationary or standing wave. It is an oscillatory discharge of a circuit containing a distributed r, L, g, C, and therefore is analogous to the oscillating condenser discharge through an inductive circuit, except that, due to the distributed capacity, the phase changes along the circuit. The free oscilla- tions of a circuit such as a transmission line are of this character. For A = 0, that is, assuming the wave length of the oscillation as so great, hence the circuit as such a small fraction of the wave length, that the phase of i and e can be assumed as uniform throughout the circuit, the equations (83) and (84) assume the form i = £-^{B0 cos qt + BQ' sin qt\ and (85) these are the usual equations of the condenser discharge through an inductive circuit, which here appear as a special case of a special case of the general circuit equations. If q equals zero, the functions D and H in equations (74) and (75) become constant, and these equations so assume the form and e = (86) where B=^0'-C. (87) This gives expressions of current and e.m.f. which are no longer oscillatory but exponential, thus representing a gradual change of i and e as functions of time and distance, corresponding to the gradual or logarithmic condenser discharge. For A = 0, these equations change to the equations of the logarithmic con- denser discharge. 440 TRANSIENT PHENOMENA These equations (86) are only approximate, however, since in them the quantities s, u, h have been neglected compared with q, assuming the latter as very large, while now it is assumed as zero. 13. If, however, that is, or L C r -H g = L H- C, (89) or, in words, the power coefficients of the circuit are proportional to the energy storage coefficients, or the time constant of the electromagnetic field of the circuit, — , equals the time constant L of the electrostatic field of the circuit, -^ , then u = — = — = time constant of the circuit, (90) L C and from equation (54) R* = s2 + (f, h = VWs = as, k = VWq = aq, and from equation (52) = L . /L c/ m 0, (91) and / = 0; (92) DISCUSSION OF GENERAL EQUATIONS 441 hence, substituting in equations (50) and (51), e L (_+s(£ — A) T~) r~ /* i\"i + s (t + A) T~) r~ /* i J\T I * i Li \ / J "^2 LI \ • / J + c-'<'-A> D.b (f - X)] - £-'«+*> DJg (t + i)]} (93) and + *~J A)z)3b(^->i)] + £-< '>D4fe(r-f (94) These equations are similar to (74) and (75), but are derived here for the case m = 0, without assumptions regarding the relative magnitude of #and the other quant it ies : " distortionless circuit." These equations (93) and (94) therefore also apply for q = 0, and then assume the form S^-A)]-[C2£+S('+A) + C4£-S(^A)]},(95) e=- + [C2e+8«+» + C4£-S(<+A)]}. (96) These equations (95) and (96) are the same as (86), but in the present case, where m = 0, apply irrespective of the relative values of the quantities s, etc. Therefore in a circuit in which m = 0 a transient term may appear which is not oscillatory in time nor in space, but changing gradually. If the constant h in equations (50) and (51) differs from zero, the oscillation (using the term oscillation here in the most general sense, that is, including also alternation, as an oscillation of zero attenuation) travels along the circuit, but it becomes stationary, as a standing wave, for h = 0; that is, the distance attenuation constant h may also be called the propagation constant of the wave. h = 0 thus represents a wave which does not propagate or move along the circuit, but stands still, that is, a stationary or standing wave.