Theory and Calculation of Transient Electric Phenomena and Oscillations — part 19 of 20

The effective power at any point of the circuit gradually decreases in any section with the resultant time decrement of the total circuit, «?““, 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*?0h (4? + BY) = 24? + DA), or asd C+ D ft" = ————__, A? + B 30 . i) 1 Cc? + DP ‘ (305) A = — log ——} 4s °A? + B

POWER AND ENERGY OF THE COMPLEX CIRCUIT 616 The difference of power between two points of the circuit, A, and 4,, that is, the power which is supplied or received (depend- ing upon its sign) by a section 4’ = A, — A, of the circuit, is given by equation (304) as P, = VE ew { (c+ etaeh) (A? + B?) — (e729 — e-2#h) (C7 + D*)}. (806)

If P,is > 0, this represents the power which is supplied by the section 2’ to the adjoining section of the circuit; if P,< 0, this is the power received by the section from the rest of the complex circuit.

If sd, and sd, 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 .

ett _ gttsh - 428(4,—-A) = +28; (807) hence, .

Pym at Het [a + B+ + DY: (308) that is, the power transferred from a section of length J’ 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, 2’, to its trans- fer constant, s, and to the sum of the power of main wave and reflected wave.

61. The energy stored by the inductance L of a circuit element dA, that is, in the magnetic field of the circuit, is

Li? dw, = 2 di, where L/ = inductance per unit length of circuit expressed by the distance coérdinate A. ;

Since L = the inductance per unit length of circuit, of distance coérdinate 1, and 4 = ¢l,

,L tL yi ; I! =—=—>=V5; ° V0 '@

; | 516 TRANSIENT PHENOMENA hence, 1, /L \ dw, = iV, ra Pda. (309) In general, the circuit constants r, LZ, g, C, per unit length, 1 = 1 give, per unit length, 4 = 1, the circuit constants r Lig C oc’ o’oa'a’ or _ (310) so, b fk a co fe VIC’ VIC C’ VIC’ VIC L Substituting (290) in equation (209) gives . Ws _ 1 /Lsust 2394 cos g (a — 0 + Bsing (A — t)F +e (Ccosg (A+ 2) + Dsing (A + OF — 2[A cosg (A — t) + Bsing (A — 8] (C cos q (A + t) + Dsing (4 + O)} = iV Ee {{et?#4 (4? + BY) + 6-724 (C? + D*)]

• [e+?* (4? — B*) cos2q (A — 2)
• 27%? — D) cos2q(4t 0)
• 2[ABe*?* sin 2q (A — t) + CDe~?" sin 2g (A + 0] — 2[((AC — BD) cos2 qd + (AD + BC) sin 2 qd] — 2[(AC + BD) cos 2 qt + (AD — BC) sin 2 qt]}. (311) Integrating over a complete period in time gives the effective energy stored in the magnetic field at point 4 as aw, _ 1 foe dA 2nd, aa -1yi eg 2 tot {[et?” (A? + BY) 4.724 (C? + D)) — 2[(AC — BD) cos 2 qi + (AD+ BC) sin 2 g4}}, (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

es fo ~1v —2ugl ff, +284 (42 hed, ae AV GE MEA + BY +e (C2 + D)}. (313) The energy stored by the inductance L, or in the magnetic field of the conductor, thus consists of a constant part, dw, 1,/L _ ’ _ a Vee aut fe+204 (4? 4 BY) + «24 (C7 + DA), (814) @ part which is a function of (A — ¢) and (A + 0), du’ iVvz —2uet +28 2 2 ma 74iVa! {[e (A? — B?) cos2q (A — £)

• e724 (C? — D*) cos2q (A + 2)]
• 2[ABet?" sin 2 (A — 2)
• CDe~** sin 2qg(a +d}, (315) a part which is a function of the distance A only but not of time ¢t,

• ~ = VE eo (AC — BD) cos2 qd + (AD + BC) sin2 qA, (316) and a part which is a function of time ¢ only but not of the distance 4, A . = = 5 VE e204 (AC+ BD) cos 2 gt + (AD—BC) sin 2 qt}, (317) and the total energy of the electromagnetic field of circuit element di at time t is dw, dw, dw dw’ dw” a ata a dk G18)

52. The energy stored in the electrostatic field of the conductor or by the capacity C is given by Ce du, = 2 da;

618 * TRANSIENT PHENOMENA or, substituting (310), dw, _ 1 ye | a 7aVi é, (319) 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 dA at time ¢ is . dw, dw, dw dw’ du”

a at ata a (820) and adding (318) and (320) gives the total stored energy of the electric field of the conductor,

. dw dw, dw, _ dw, du! a dat a a tae’ (21) and integrated over a complete period of time this gives . dW _ dw, 1/E —Dugtf +284 7 43 —ad a 2 a7 2VC! {e+?* (42+ B*) +<-™ (C?+ D’)}. (822) Ud mM :

The last two terms, a and o 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.

63. The energy stored by the electric field in a circuit section ’, between A, and ,, is given by integrating li between A, and 4, as

1 L -2 20, 3A,

W-=z, at wt {(e +80 _ @t80h) (4? + B?) — (e730 — 20h) (C2 + DA}; (323) or, substituting herein the approximation (307), Ww ah yee te {A?+ B+C74+D*}. (324)

POWER AND ENERGY OF THE COMPLEX CIRCUIT 519 Differentiating (324) with respect to ¢ gives the power sup- plied by the electric field of the circuit as P --% = ud’ Veer {A? + B?+C? + D*}, (325) or, more generally, P= Me [Ets { (gt? _ e t2eh) (A? + B’) — (e7 2 — e- 2h) (C2 + D*)}. (326) 64. The power dissipated in the resistance r’dd = = of a conductor element dA is dp’ = r’?da (327) = = dw,; hence, substituting herein equation (318) gives the power con- sumed by resistance of the circuit element dA as dp’ 2r(dw, du’ dw’ du” | a Etat t a a al ;, G23) g and the T ed by th ductance g/dA = ——da power consum y the conductance g’ Vit of a conductor element da is dp” = ged (329) = 79 dw, ; hence the power consumed by conductance of circuit element dA is dp _ 29 {diy , dit , du”, du} a Cla tatata Ss’ (330) and the total power dissipated in the circuit element dA is dp, _ dp’ | dp” dw, , dw’ dul’ du’” aM at da 4u (+ )-4m( +), (331)

520 TRANSIENT PHENOMENA where, as before, us s(z + g 2 a) 0-6-8 . 2b cr and integrating over a complete period . dP, _ dw, du’ mnt Ht @ (333) the power dissipated in the circuit thus contains a constant term, tu we, and a term which is a periodic function of the distance 4, 4 man, of double frequency. Averaged over a half-wave of the circuit, or a multiple thereof, the second term disappears, and dP? dw, . . a Aa) or, substituting (314), aPY weet fet? (4? + BY) + e-2%@ (C7 + D*)}, (334) da C , thus the power dissipated in a section 1’ = a, — A, of the circuit is, by integrating between limits 4, and A,, o_ u yz — uot +28A, +28, 2 ; P, r3VG {(e et?) (A? + B?)

• (c7?— eth) (@ + D)}, (335) or, approximately, PY= uit et {AP + BP + OF + D*}. (336)

65. Writing, therefore, Pat B +e DZ, - (337)

POWER AND ENERGY OF THE COMPLEX CIRCUIT 521

the energy stored in the electric field of the circuit section of length 1’ is

W = 5a He-t4; (338)

the power supply to the conductor by the decay of the electric jield of

the circuit is

P = u,l’H*e- 2; (339)

the power dissipated in the circuit section 1’ by its effective resist- ance and conductance is ,

Po = ul’ He? (340) and the power transferred from the circuit section 1’ to the rest of the circuit is

P, = sl He-?; (341) that is, = = ratio of power dissipated in the section to that

0 supplied to the section by its stored energy of the electric field.

= = fraction of power supplied to the section by its electric

‘0

field, which is transferred from the section to adjoining sections (or, if s < 0, received from them).

= = ratio of power transferred to other sections to power dissipated in the section. .

u, + u + 8 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.

66. Equation (306), of the power transferred from a section to the adjoining section, can be arranged in the form

P, _ bE ew {[et2ede (A? + B?) _ g 72th (C? + D’)) — (et? (4? + BY) — eC? + DI}; (342)

that is, it consists of two parts, thus: PJ = Vet {+s (A? + B’) _ e728, (c? + D*)}, (343)

522 , TRANSIENT PHENOMENA which is the power transferred from the section to the next fol- lowing section, and PJ" = + Ae {e+e (A? + BY) — 2% (C? + D’)}, (344) which is the power received from the preceding section, and the difference between the two values, P, = PY’ — P,”, (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 H? 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: LP. = 0; hence, substituting (341), Dad! = 0, (346) and, since 8; = Uy — Uy . UA = Dens (347) that is, the resultant circuit decrement multiplied by the total length of the circuit equals the sum of the time constants of the section multiplied with the respective length of the section, or, if . &,, & ...§, = length of the circuit section, as fraction of the total circuit length A, Uy = Dome (348) Whether this expression (331) is more general is still unknown. 67. As an example assume a transmission line having the following constants per wire:r, = 52;L, = 0.21;g, = 40 x 10%, and C, = 1.6 X 10. ; Further assume this line to be connected to step-up and step- down transformers having the following constants per trans- a,

!

POWER AND ENERGY OF THE COMPLEX CIRCUIT 528 former high-potential circuit: r, = 5, L, = 3; g, = 0.1 x 10~, and C, = 0.3 X 107*; then

1! =¢, = VEC, = 0.58 x 107, a! = 2, = 0.95 x 10°, u, = 136, u =1. The circuit consists of four sections of the lengths A,’ =0.58 X 10~°, A,’=0.95 x 107, A,’ =0.58 X 107, A,’=0.95 x 10; hence a total length A = 3.06 X 10°,

and the resultant circuit decrement is

, ,

u, = atu, + of u, = 51.6 + 0.59 = 52.2; hence, 8, = — 83.8 and s, = + 51.2. If now the current in the circuit is 7, = 100 amperes, the e.m.f. e, = 40,000 volts, the total stored energy is ‘ W =12(L,+ L,) +e27(C, +C,)

= 32,000 + 3000 = 35,000 joules,

and from equation (321) then follows, for ¢ = 0, 4 AH? = 35,000,

H = ne = 22.8 x 10$,

which gives u, = 52.2, H? = 22.8 x 108, W = 35,000.

Line. er Line. qorepdown Length of section, 1’ = [0.58 x 107 | 0.95 X 10770. 58 x 107? | 0.95 x 107 Time constant, u=| 136 1 136 1 Transfer constant, $= — 83.8 +51.2 —83.8 +51.2 Ene 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, P= |— 1110 1110 |- 1110 1110 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.

3 CHAPTER VIII. REFLECTION AND REFRACTION AT TRANSITION POINT. 68. The general equation of the current and voltage in a sec- tion of a complex circuit, from equations (290), is t =e“ fet474 cosg (A — t) + Bsing (A — 8) — «~"[Ccosq (A + t) + Dsing (A + 2)}} (290) e=ce—** {e+e [A cos g (A — t) + Bsing (A — 4]

• e~(C cos g (A + t) + Daing (A +2)]}, where A = ol = distance variable with velocity as unit; c= V IL; ’ c= VE; = Vai u,= u + s = resultant time decrement; u= s(; + z) = time constant, and ~ 20 ep > ame’ s = energy transfer constant of section. At a transition point A, between section 1 and section 2 the constants change by A,=efaet*A +b,e—%4 (C, cos 2 ga, + D, sin 2 ga,)}

• By=e~faethB +b,e-% (C, sin 2 gd, —D, cos 2 gA,)} (285) Cet fae*hC, +b,et#h (A, cos 2g’, +B, sin2qa,)} [~~ Dy=et™{ae-4D, +b,e+ (A, sin 2 gd, —B, cos 2gA,)}, where | ute =a, | | a= ~5 G, and 0b, De, (286) 525 . | | |

526 TRANSIENT PHENOMENA

Choosing now the transition point as zero point of A, so that A< 0 is section 1, A>0 is section 2, equations (285) assume the form A, = a,A, + b,C,,

B, = a,B, — 6,D,, C, = a0, + b,A,, (349) . D, = a,D, — 6,B,. From equations (286) it follows that c, (A — CY) = ¢, (AZ - C/’) and } (350) c, (BZ? — DZ) = c, (B? — D)).

If now a wave in section 1, A B, travels towards transition point A = 0, at this point a part is reflected, giving rise to the reflected wave C D in section 1, while a part is transmitted and appears as main wave A B in section 2. The wave C D in sec- tion 2 thus would not exist, as it would be a wave coming towards 4 = 0 from section 2, so not a part of the wave coming from section 1. In other words, we can consider the circuit as com- prising two waves moving in opposite direction:

' (1) A main wave A,B,, giving a transmitted wave A,B, and reflected wave C’,D,. (2) A main wave C,D,, giving a transmitted wave C,’D,’ and reflected wave A,’B,’.

The waves moving towards the transition point are single main waves, A,B, and C,D,, and the waves moving away from the transition point are combinations of waves reflected in the sec- tion and waves transmitted from the other section.

69. Considering first the main wave moving towards rising A: in this C, = 0 = D,, hence, from (349),

a,C, + b,A,=0 and (351) ; a,D, — 6,B,= 0, ; and herefrom , ee C,= a, Ay=t oe . and (852) —, or pair ap. D,=+ a, By=+ c, Fe, BY

REFLECTION AND REFRACTION 527 | which substituted in (349) gives by , a — 6? 2c, A,=4,A, a, A,= a, A,= c+ 6, and (353) _ by , af —b? 2c, B,= 4,B, a, B,= a B,= c+ 0,5 Then for the main wave in section 1, i, = eo etl (4 cosg (A — t) + B, sing (A — 0} and ‘ (354) e, =c,e et fA, cosg (A— f) + B, sing (A— #)}. When reaching a transition point 4 = 0, the wave resolves into the reflected wave, turned back on section 1, thus: i = — C2 St ut e841 A cosg (A+) ~B,sing (A+0)} ce, +c, and (355) e, = +c, 2—te-mte-"4f A cosg (A+t) —B, sing (A+4)}. c,t¢, : The ‘transmitted wave, which by passing over the transition point enters section 2, is given by i, = 241 pat +14 cos g (A — t)+B,sing (A—-t)} ¢,+¢, and (356) by = Co ee tlt A. cos q(A—t) +B, sing (A—0)}. . Cc, +e, . . B,. The reflection angle, tan (7,’) = — A ,is supplementary to the 1 . B ao . impact angle, tan (t,) = + q? and transmission angle, tan (7,) 1 iB A, Reversing the sign of 4 in the equation (355) of the reflected wave, that is, counting the distance for the reflected wave also in the direction of its propagation, and so in opposite direction as

528 TRANSIENT PHENOMENA in the main wave and the transmitted wave, equations (355) become i"=+ Bo cten wt tA cos q(x —1) + B, sin g (A—1)}, a (357) c,—¢ , . e,” = +0, te et 14! 4 cosg (W’—t) + B, sing (V’-2)}, c,+e, and then . \ 1, + a,” = i or

• ree (358) 0,2 + e,” =e,, e, +e,” =e,.

(1) In a single electric wave, current and e.m.f. are in phase with each other. Phase displacements between current and e.m.f. thus can occur only in resultant waves, that is, in the com- bination of the main and the reflected wave, and then, are a function of the distance 4, as the two waves travel in opposite direction.

(2) When reaching a transition point, a wave splits up into a reflected wave and a transmitted wave, the former returning in opposite direction over the same section, the latter entering the adjoining section of the circuit.

(3) Reflection and transmission occur without change of the phase angle; that is, the phase of the current and of the voltage in the reflected wave and in the transmitted wave, at the transi- tion point, is the same as the phase of the main wave or incoming wave. Reflection and transmission with a change of phase angle can occur only by the combination of two waves traveling in opposite direction over a circuit; that is, in a resultant wave, but not in a single wave.

(4) The sum of the transmitted and the reflected current equals the main current, when considering these currents in their respective direction of propagation. .

REFLECTION AND REFRACTION 629 —«

The sum of the voltage of the main wave and the reflected wave equals the voltage of the transmitted wave.

The sum of the voltage of the reflected wave and the voltage of the transmitted wave reduced to the first section by the ratio of voltage transformation , equals the voltage of the main wave.

1 (5) Therefore a voltage transformation by the factor o 1 iL, C, wa . . = Va 7, occurs at the transition point; that is, the trans- 2 1 mitted wave of voltage equals the difference between main wave and reflected wave multiplied by the transformation ratio=; 1 e, = <t (e, — e,”). As result thereof, in passing from one section 2 of a circuit to another section, the voltage of the wave may decrease or may increase. If 2 > 1, that is, when passing from 1 a section of low inductance and high capacity into a section of high inductance and low capacity, as from a transmission line . into a transformer or a reactive coil, the voltage of the wave is increased ; if 2 < 1, that is, when passing from a section of high

. 1 inductance and low capacity into a section of low inductance and high capacity, as from a transformer to a transmission line, the voltage of the wave is decreased.

This explains the frequent increase to destructive voltages, when entering a station from the transmission line or cable, of an impulse or a wave which in the transmission line is of relatively harmless voltage.

The ratio of the transmitted to the reflected wave is given by

ho 24 2b, POE 6 VEE VEG DC | L, C, and (359) &_ 2 2VvEe, a Gn8” VER, VEG 1 TG L,C,

580 TRANSIENT PHENOMENA 60. Example: Transmission line Transformer - L, = 1.95 x 107 ‘ Ly=1 ; CG, = 0.0162 x 10-* C, = 0.4 x 107° c, = 346 c = 1580 4 os jn ~ 0.56 gn 7 2-56 And in the opposite direction 4k fs a= — 2.56 gm ~ 0.56. . @, . L, Oo, . The ratio becomes a maximum, = ~, for = = =?, but in e,” C, ¢C, this case e,” = 0; that is, no reflection occurs, and the reflected wave equals zero, the transmitted wave equals the incoming wave. 1, _ 2e, . t, 6, +e,’ hence, becomes a maximum for c, = 0, or c, = © and (360) then = 2; in which case e, = 0. @_ 2% , & ¢,+e¢,’ hence, becomes a maximum for c, = 0, orc, = © and then = 2; in which case 7, = 0. From the above it is seen that the maxi- mum value to which the voltage can build up at a single transi- tion point is twice the voltage of the incoming wave, and this occurs at the open end of the circuit, or, approximately, at a point where the ratio of inductance to capacity very greatly increases. a” _ on 7, < vy C, + C , hence, becomes a maximum, and equal to 1, for c, = 0, (361) orc, = &. e,” = 7 ae & C, + C, has the same value as the current-ratio.

REFLECTION AND REFRACTION 531 61. Consider now a wave traversing the circuit in opposite direction; that is, C,D, is the main wave, A,B, the reflected wave, CD, the transmitted wave, and A,= 0 = B,. In equa- tion (349) this gives , A,= b¢,; ° . B, =—},D,; C, = aC,, and D,= 4,D,; hence, 1, 2¢, . G, arr ra , 1 2C, Dy a= og Pai «0 b, 6 362) A, a°? oe, CL and p op 3 Sp 2p. ya te te” that is, the same relations as expressed by equations (352) and (353) for the wave traveling in opposite direction. . The equations of the components of the wave then are: Main wave: . a, =— ee (C, cosg (A+ t) + D,sing (A+ 1)} (368) e,=+e,e~- «4 (C, cosg (A+ t) + D,sing (A+ dD}; Transmitted wave: : i= 21 ou e~*4IC, cos g (A+4)+D, sing (A+t)} GQ + Cc,

• 2c (364) €,= te, —2 ee 4IC cos g (A+4) + D, sin g (A+¢)}; . c,+c, Reflected wave: 1! = C1 Oa wot g 01d {C, cos g (A—t) —D, sing (A—t)} c¢, +e, (365) e/= cyt Be uat gtk {C, cos g (A—t) —D, sing (A—t) }; C,+¢, ;

582 TRANSIENT PHENOMENA or, in the direction of propagation, that is, reversing the sign of A: iY =— O17 Oa wot ga! {C, cos g (4° +t) +D, sin g (1 +4)} | c,te, e,”= ¢, M1769 5 — at aul {C, cos g (+t) +D, sing (+0}. | ec, te, (366) 62. The compound wave, that is, the resultant of waves pass- ing the transition point in both directions, then is a,° =i, + 1, a eo =e, te’ te 1 1 | - 1 (367) 1,° = ty + ty + 1,’ e, =e, +e, +e, In the neighborhood of the transition point, that is, for values A which are sufficiently small, so that e+" and e~“ can be dropped as being approximately equal to 1, by substituting equations (354) to (356) and (363) to (366) into (367) we have ° io =e [{A,cosg (A — t) + B, sing (A— 8} c,—C¢ . eye, {A,cosg (A+) — B,sing(A+d} 2c . _ att, {C, cos g (A + 2+ Dy sing (A+ }); eo =ce™ [{A, cosg (A — t) + B,sing(A— 4} ¢-¢ . '

• ¢, n c, {A,cosg (A+ t)— B,sing(A+ 0} 9
• lis {C, cos g (A+ + D, sing (A+0)}):} (368) 1 2 1,9 =—e " [{C,cosg (A+ t) + D, sing (A+ d} | ¢,- ¢, .

• cee, }C, cosg (A — t) — D, sing (A — 1} | 2c, . ; | _ ote, {A, cosg (A — t)+ B,sing (A—1)}}: |

REFLECTION AND REFRACTION 533 ef =ce~ [{C,cosqg (A+ 0) + D,sing(At+ 6} ¢,-—¢

• +—{C, cos q (A — t) — Dy sing (A — t) c+, 12 2sing 2c ;
• — {A, cosq (A — t) + B, sing (A — t)}]. C, + c, | 1 q 1 q }]

In these equations the first term is the main wave, the second term its reflected wave, and the third term the wave transmitted from the adjoining section over the transition point.

Expanding and rearranging equations (368) gives 5 Ben . i= GA, — ¢,C,) cos ga + c,(B,— D,) sin qa} cos gt

1 2 — {(c,B, + ¢,D,) cos ga — c, (A,+ C,) sin gd} sin gt]; eo = Lele, (A,+C,) cos q4+ (¢,B, + ¢,D,)sin gA} cos gt 1 2 — {c, (B, —D,) cos qa— (c,A,—¢,C,)sin gd}sin gt]; (369) . 2 eu . i? = [1 ed, — eC.) 008 94 + ¢, (By — Dy) sin ga} c08 of 1 2 — {(¢,B, + ¢,D,) cos gd — ¢, (A, + C,) sin gA} sin gé); eo = ee [fe tC) cos A+ (¢,B, +c¢,D,) sin gd} cos gt 1 2 — {c,(B, —D,) cos ga — (c,A,—¢,C,) sin gd} sin gt] 63. This gives the distance phase angle of the waves: tan 19 = —2 {B, — D,) cos gt + (A, + C,) sin gt} ' (¢,A, — ¢,C,) cos gt — (c,B, + ¢,D,) sin gt 3 _ ¢,{(B, — D,) cos gt + (A, + C,) sin gt} (370) tan a o— tt : 7 ©,4,— ¢,C,) cos gt — (¢,B, + ¢,D,) sin gt’ hence, tant» ¢, ,/L,C,, fants co, VLC,’ (371) tane® = (c,B, + ¢,D,) cos gt + (c,A, — ¢,C,) sin gt 1 e,{(A, + €,) cos gt — (B, — D,) sin gt} ’ tane.! = (c,B, + c,D,) cos gt + (c,A, — ¢,C,) sin gt. (372) 7 ~~ ¢,{(A, + C,) cos gt — ¢, (B, — D,) sin gt} ’

584 TRANSIENT PHENOMENA hence,

tane,? cy _ pie ;

tane, c, VLC,’ (873)

that is, at a transition point the distance phase angle of the wave

changes so that the ratio of the tangent functions of the phase angle is constant, and the ratio of the tangent functions of the phase angle of the voltages is proportional, of the currents inversely proportional to the circuit constants c = vi .

In other words, the transition of an electric wave or impulse from one section of a circuit to another takes place at a constant ratio of the tangent functions of the phase angle, which ratio is a

constant of the circuit sections between which the transition occurs.

This law is analogous to the law of refraction in optics, except that in the electric wave it is the ratio of the tangent functions, while in optics it is the ratio of the sine functions, which is con- stant and a characteristic of the media between which the tran- sition occurs,

Therefore this law may be called the law of refraction of a wave at the boundary between two circuits, or at a transition point.

The law of refraction of an electric wave at the boundary between two media, that is, at a transition point between two circuit sections, is given by the constancy of the ratio of the tangent functions of the incoming and refracted wave.

CHAPTER IX. INDUCTIVE DISCHARGES.

64. The discharge of an inductance into a transmission line may be considered as an illustration of the phenomena in a complex circuit comprising sections of very different constants; that is, a combination of a circuit section of high inductance and small resistance and negligible capacity and conductance, as a generating station, with a circuit of distributed capacity and inductance, as a transmission line. The extreme case of such a discharge would occur if a short circuit at the busbars of a gen- erating station opens while the transmission line is connected to the generating station.

Let r = the total resistance and L = the total inductance of the inductive section of the circuit; also let g = 0,C=0, and L, = inductance, C, = capacity, r, = resistance, g, = conduc- tance of the total transmission line connected to the inductive circuit.

In either of the two circuit sections the total length of the section is chosen as unit distance, and, translated to the velocity measure, the length of the transmission line is

4=e7=VLC,, ) and the length of the inductive circuit is

4, = 9, = VLC, = 0; (374) that is, the inductive section of zero capacity has zero length when denoted by the velocity measure 4, or is a “massed induc- tance.”’ ‘

It follows herefrom that throughout the entire inductive section A=0, and current 7, therefore is constant throughout this section.

Choosing now the transition point between the inductance and the transmission line as zero of distance, 4 = 0, the inductance

586

• 586 TRANSIENT PHENOMENA is massed at point A = 0, and the transmission line extends from A=0tod=A,.

Denoting the constants of the inductive section by index 1, those of the transmission line by index 2, the equations of the two circuit sections, from (290), are

i, =e7“* {(A, — C,) cos gt — (B, + D,) sin gt}, (375) e, =c,e~“"{(A, + C,) cos gt — (B,—D,) sin qt}; i, =e {et (A, cosg (A — t) + B,sing (A — 2)] — e*(C, cosqg (A+ ft) + Dysing (A+ é))}, (376) e, = c,67 {fet [A, cosg (A —¢) + B, sing (A—2)] +e" (C, cos g (A+ t) + D,sing (A + d))}}, and the constants of the second section. are related on those of the first section by the equations (285): A, =4,A,+0C,, C, = 4,6, + 0,A,, (349) B, = a,B, — 6,D,, D, = a,D, — 6,B,, where — C, + Cy . a,= a C, ’ ; : (286) = 1, O 2c, and _ L =(/x. 287 = Vi (287)

65. In the inductive section having the constants L and r, that is, at the point 4 = 0 of the circuit, current 7, and voltage | e, must be related by the equation of inductance,

. ai é =n, — L7 . (377) Substituting (375) in (377), and expanding, gives q {(A, + C,) cos gt — (B, — D,) sin gt} = (r + u,b) {(A, — C,) cos gt — (B, + D,) sin gt}

• gL {(A, — C,) sing + (B, + D,) cos gt},

INDUCTIVE DISCHARGES 537 , and herefrom the identities c, (A, + C,) = (r + u,b) (A, — C,) + gh (B,+ D,), (378) C, (B, —D,) =(r+ ul) (B, + D,) — qb (A,- C,). Writing A,-C,=M and (379) B,+D,=N gives . ce, (A, + C,) = (7 + u,b) M + gLN and (380) c, (B, — D,) = (r + u,b) N — qLM, which substituted in (349) gives 1 A, = {¢c +r + ub) M + qLN} =5(M + pN), 1 ; _1 B, = 5, {+r + ub) N - glM} = 5(N — pM), ; ; (381) C, “3 {qLN —(c —r — mL) M} = 5 PN — M), 1 Dy = 5 {gh + (¢ —r— mL) N} =5(OM + N), where in the second expression terms of secondary order have been dropped. .

• 2 _ ob c= GC,’ p= c . Then substituting in (375) gives the equations of massed inductance: i,=e-“*{ M cos gt — N sin gt} } e,=e“{[(r+u,L)M+qLN)cosgt—[(r+u,L) N —gLM)sin gt}. (382) If at t = 0, e, = 0, that is, if at the beginning of the transient discharge the voltage at the inductance is zero, as for instance the inductance had been short-circuited, then, substituting in

° 588 TRANSIENT PHENOMENA (382), and denoting by z, the current at the moment ¢ = 0, or at the moment of start, we have t = 0, 4,= 2,, e, = 0; hence, M =i, (383) nN a th tel iy qL . and . i, = te {eos qt + ie sin ats (384) (qlY + (r+u by. _.,. é = gers eater ae sin gt. In this case Lo A, = >> B, =— 5 {QL + (r+ ugh) + 0 + uh) }; = (385) a C,=- a ; _ 0 f(gp» 2 D, Sql {(qL) + (r + u,b)? — ¢(r + u,L)}. 66. In the case that the transmission line is open at its end, at point 4 = A,, 4=A, ; 1, = 0; hence, this substituted in (376), expanded and rearranged as function of cos gt and sin gt, gives the two identities et (A, cos gd,+ B, sin gA,) = ¢** (C, cos gd, + D, sin gA,) and (386) eto (A, singd, — B, cos g,) = —e**(C, sin gd, — D, sin qA,). Squared and added these two equations (386) give et2sd (A? + B?) = ‘¢—2 8 (C2 + D;). (387)

INDUCTIVE DISCHARGES 589 Divided by each other and expanded equations (386) give (A,C, — B,D,) sin 2 qa, = (A,D, + B,C,) cos 2 gA,. (388) Substituting (381) into equations (387) and (388) gives , e+e (GL)? + (ct+r+ugL)} =~? %{ GL) + (c—r—ugb)} (389) {(qL)? + (r + ub) — ¢} sin 2g, = 2 cg cos 2 gd, (390) Since 2 sd, is a small quantity, in equation (389) we can sub- stitute . et2t 4 2 sA,; hence, rearranging (389) and substituting s=U,—U ‘ gives c(r + Ugh) —(u — uy) A, { (QL)? +r + ub)? + ?}=0. (391) Since (r + u,L) is a small quantity compared with c’ (gLZ)’, it can be neglected, and equations (390) and (391) assume the form { (gL)? — ce} sin 2 gd, = 2 cgL cos 2 ga, (392) c(r + uh) — (wu — u) aA, {(qLl)? + 2} =0, (393) and, transformed, equation (392) assumes the form _ 2qh_. tan 2 qa, ~ (qL)? —~?¢’ or q=- Z tan gA,, (394) or c ° q=t ZL cot qA,, | hence tan 2 gd, is positive if gL > c, as is usually the case. Expanded for u,, equation (393) assumes the form a = to { (ql)? + 2} cr ° Ay { (QL)? + 2} + cL or ; (395) c (1 + “) | ye SN" u cL+A, {qb +¢} S= — (u— u,).

540 TRANSIENT PHENOMENA

From equations (394) g is calculated by approximation, and then from (395) u, and s. .

As seen, in all these expressions of g, u,, s, etc., the integration constants M and N eliminate; that is, the frequency, time atten- uation constant, power transfer, etc., depend on the circuit con- stants only, but not on the distribution of current and voltage in the circuit.

67. At any point J of the circuit, the voltage is given by equa- tion (376), which, transposed, gives

, e = ce“ fe+1(A, cos gd + B, sin g) cos gt ; + (A,sin gd — B, cos gA) sin qt]

• «~4([(C, cos gd + D, sin qd) cos gt — (C, sin gd — D, eqs gd) sin gt]}, or approximately, e = ce“ {[(A, + C,) cos gd + (B, + D,) sin gd) cos gt
• [(A, — C,) sin ga — (B, — D,) cos ga] sin gt}. Similarly to equation (381), . A, + C, = pN; A,-C,=M; 393 B,+D,=N; (38) B, -D 2 pM, where ’ en gL | p= ma ’ | then | e, = e~ (gL cos gi+csin gd) (N cos gt + M sin qt),
  . Li, 397

i, = e- (cos gd — c sin gd) (M cos qt — N sin gt); (97) \

e, = qLe~““ (N cos gt + M sin qt), a , (398) 1, =e“ (M cos gt — N sin qt). |

INDUCTIVE DISCHARGES 541 If e, = 0 for ¢ =0, N =0; hence, 1, = 1,¢~ cos gt, e, = qlie™ sin gt; 399 1, = te" (cos gd — sin qA) cos qt, 399) e, = t,¢-“* (gL cos gd + csin gd) sing!. ; Writing V4 (M? + N*) = 1, (400) the effective values of the quantities are I, =I", E, = qlle~““; ; 401) I, = Le (cos ga - © sin wi), E, = I~ (qL cos qa + ¢ sin ga). Herefrom it follows that I, = 0 for A = A, by the equation cos gd, — qsin ga, = 0, or . q= peot Go, (402) . while q=- = tan @, (403) ! . gives | gL cos gd, + csin gd, = 0; that is, | E, = Oats = A,.

| 542 TRANSIENT PHENOMENA

At the open end of the line 4 = A, the voltage E, by substi-

tuting (402) into (401) is 6 = gh — uot « | E, cos gh Ig (404) At the grounded end of the line 2 = A, the current /,, by sub- stituting (403) into (401), is ot Fie I =—-—.- (405) cos gf,

An inductance discharging into the transmission line thus gives an oscillatory distribution of voltage and current along the line.

68. As example may be considered the three-phase high- potential circuit, comprising a generating system of r = 2 ohms and L = 0.5 henry per phase and connected to a long-distance transmission line of r,= 0.4 ohm, L, = 0.002 henry, g,= 0.2 X 10-* mho, C,= 0.016 x 10-* farad per mile of conductor or phase, and of J, = 80 miles length.

c= V2! = 354, 2 = 125,300; . 0 o, = VLC, = 5.66 x 10-°; A, = 1,0, = 0.453 x 10°; 1/ft% ra — 1ne: u =3(R+¢ = 106; c Zo 708, and herefrom, substituting in equations (394) and (395), q =— 708 tan (0.0259 g)° (zero voltage) =+ 708 cot (0.0259 g)° (zero current), Uy 1 ; 1 —-—? = ——__ u «0.618 g 10-* + 1.28 qh = 100.35° | 185.64° | 273.83° |362.80° | 452.32° | 541.94° q= 3875 7168 10,572 |14,010 | 17,463 | 20,920
pi “2 = 0.0946} 0.0302} 0.0142} 0.00816] 0.0047] 0.0037 | Uy = 95.8 | 102.8 | 104.5 105.1 105.5 | 105.6 } 0 ag 10.2 3.2 1.5 0.87 0.5 0.4

INDUCTIVE DISCHARGES 548 | By equation (401) the effective values of the first six har- monics are given as (1) Quarter-wave: 100.35°. q, = 3875; Uy, = 95.8; I = i -™ (cos gd — 5.48 sin gd); ; E = 19397,¢.™ (cos gd + 0.182 sin gd). (2) Half-wave: 185.64°. q, = 7168; . u, = 102.8; Is te“ (cos gd — 10.14 sin gd); FE = 35851, ** (cos gd + 0.098 sin qa). (3) Three-quarter wave: 273.83°. q, = 10,572; | u, = 104.5; oo I = ie™ (cos gd — 14.90 sin qd); , | | E = 5287 t,¢* (cos gd + 0.067 sin g). ! (4) Full wave: 362.89°. q = 14,010; | uy = 105.1; I = ig* (cos gd — 19.8 sin gd); E = 7005 i,¢“* (cos ga + 0.050 sin qa). (5) Five-quarter wave: 452.32°. qs = 17,463; | u, = 105.5; I = i¢** (cos gd — 24.65 sin qd); FE = 8732 i,¢—** (cos gd + 0.040 sin g).

544 TRANSIENT PHENOMENA (6) Three-half wave: 541.94°. qs = 20,920; u, = 105.6; I = i~** (cos qd — 29.6 sin gd); E = 10,460 t,<~* (cos gd + 0.033 sin qa).

INDEX TT PAGE Acceleration constant of traveling wave. ............000ceeeeeeeeeee 466 Air blast, action in oscillating-current generator..................... 75 pressure required in oscillating-current generator................. 75 Alternating-current circuit and transient term of fundamental frequency 473 distribution in conductor.....................--0---+-. 369 transformer operating oscillating-current generator ....... 87 transmission, equations of traveling wave................ 477 wave as traveling wave without attenuation............. 472 Alternator control by periodic transient term of field excitation....... 223 Aluminum cell rectifier... 0.0000... c cece eee e eee e renee. 222 effective penetration of alternating current............... 378 . Amplitude of traveling wave............... 0.000000 cece e cece e eee 465 Of WAVE... eet et eee e eee eeee 488 Are, and spark .. 0.0... ec eee e cette ete re eee e ee 249 continuity at cathode ..... 0... kk ccc cece eee e eee e eens 249 lamp, control by inductive shunt to operating mechanism........ 131 machine............ 00. ec cece tte tence eeeteeeeeees 230 as rectifier...... 0.0.0... cece eee cece eee eee e teeter cess 221 current control........ 0.0. cee eee cee cece ee ee ees 220 properties... 0... eect cece en cece 249 rectification.......0.00.0 00.0 e tte eeeeeeeees 249 rectifiers 0.0.00 2. cece eee eee ee eteeeeees 222 resistivities... 02.060. kee eee eee eet t eens 9 starting... 0... cee ete cette eee e ee cees DAD Arcing ground on lines and gables, as periodic transient phenomenon .. 23 Armature reactance, reaction and short-circuit current of alternator..... 199 Attenuation of alternating magnetic flux in iron .................... 361 of traveling wave, and loading......................... 462 Booster, response to change of load..............0..0ceeeecesevesee 158 Brush are machine..................-02.-0-0++-+++2+- 221, 230, 242, 248 Building up of direct-current generator.......................-...-. 82 of overcompounded direct-current machine.............. 49 Cable, high-potential underground, standing waves.................. 452 opening under load...........................2.2....-. 112, 118 short-circuit oscillation................................. 118, 118 starting. ......... 0.00000. c cee ee cee eee eee VND, 117 transient terms and oscillations............................ 98, 102

546

546 INDEX PAGE Capacity, also see Condenser. and inductance, equations..................cccccceeeeees 48 and velocity of propagation.......................... 400, 401 distributed series... 6... eee eee e cece eee 348 energy of complex circuit................................ 617 in mutual inductive circuit.............................. 161 of electric circuit. ............0.. 0.0.0. cece eee eee eee 112 range in electric circuit...........0..............2..5.8.. 18 representing electrostatic component of elcctric field........ 5 shunting direct-current circuit........................... 183 specific, numerical values ....................2....0----. Il suppressing pulsations in direct-current circuit............. 184 Cast iron, effective penetration of alternating current................ 378 Cathode of arcs... 0... ccc cece t ene eeenceees 249 Charge of condenser................0. 0c cece cece et eeeeereeees Ol of magnetic field... 2.2.0.0... ccc cece cece ee eees 27 Circuit, complex, see Complex circuit. control by periodic transient phenomena................ 220, 223 electric, speed of propagation in............................ 422 Closed circuit transmission line. ..............0 0.00: cece e eee ee ees 806 Col al cece cee e cece cee cecneseeres 392, 394 Commutation and rectification............0..00.0.00 ccc ceceeeeeeeus. 222 as transient phenomenon............................ 40 Commutator, rectifying ©... 00.0... eee eee ee ne es 229 Complex circuit, of waves... 0.0.0... 6 6c cece ete eee eee e es 498 power and energy. ................--.0eeeeee eee 613 resultant time decrement.......................... 6504 traveling wave. .... 6.0.0.0... e eee eee eee eee eee = 468 Compound wave at transition point................. ccc eeee ee ees 582 Condenser, also see Capacity. charge, inductive...................... eee eee eee ees = 18 noninductive.............. 0.0 cece cece eee ee eeees = 18 Circuit of negligible inductance............-............. 55 equations. ............ 006 cece bec cece e eee eae = 48 oscillation, effective value of voltage, current and power.... 70 efficiency, decrement and output.............. 72 frequency... ............. 2. eee e eee eee eee = 62 general equations............................ 60 size and rating............. 0.000 chee eee eee eee eee §=©669 starting on alternating voltage.......................... 94 voltage in inductive circuit............................. 49 Conductance, shunted, effective.................0..0000-220eeeeee. = 12 Conductors at high frequency............00.0 0000 c cece cece eee eee 403 Constant-current mercury arc rectifier ............................ 250 rectification. ...........0.00020..2000220eee+. 221, 230 potential-constant-current transformation hy quarter-wave line 308 ; mercury arc rectifier... ...........0....0-.0022.. 251 rectification... .........00ccc cece cere recesses 221, 230