Theory and Calculation of Transient Electric Phenomena and Oscillations — part 17 of 20

From the above it is seen that in a submarine cable the eritical wave length J, is relatively short, so that in long submarine cables standing waves may appear which are not oscillatory in time but die out gradually, that is, are shown by the equation of case B. In such cables, due to their relatively high resist- ance, the damping effect is very great; u = 1500, and standing waves, therefore, rapidly die out.

In the investigation of the submarine cable, the complete equations must therefore be used, and g cannot always be assumed as large compared with m and wu, except when dealing with local oscillations.

. (4) Long-distance overhead telephone circuit.

19. Consider a telephone circuit of 1000 miles length, metallic return, consisting of two wires No. 4 B. and S. G., 24 inches distant from each other.

Calculating in the same way as discussed under (1), the follow- ing constants per mile of conductor are obtained: r = 1.31 ohms,

L = 1.84 X 107° henry, and C = .0172 x 10™* farad.

As conductance, g, we may assume

(a) g = 0; that is, very perfect insulation, as in dry weather. |

(6) g = 2.5 X 107°; that is, slightly leaky line.

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STANDING WAVES 455

(c) g = 12 X 10°°; that is, poor insulation, or a leaky line.

(d) g = 40 X 10°*; that is, extremely poor insulation, as during heavy rain.

The condition may also be investigated where the line is loaded with inductance coils spaced so close together that in their effect we can consider this additional inductance as uni- formly distributed. Let the total inductance per unit length be increased by the loading coils to

| L, =9X 107A, or about five times the normal value.

Denoting then the constants of the loaded line by the index 1, we have:

| Quantity (a) (b) (c) (a) w= 356 429 706 1,518 . uo 73 146 423 1,236 m= 356 283 6 — 806 m= 3 1) 27 . — 1,080 o=VIC = 5.63 x10-¢ o= VIC = 12.45 x 10-8 ky = mVILC = 210-8 =) 1.6Xx10-* | 33.7 x10-8| 4.56 x10-8 k= m,V1,C= | 910x10-8 0 3.45 X10-3 | 13.5 x 10-3 | 7 : = 3,140 3,920 187,000 1,380 hey, iE - 6,900 « 1,820 464 h-- 55.6 45 0.96 128 Ju= 5 = 11.6 0 “4 173 | In a long-distance telephone line, distributed leakage up to a certain amount increases the critical wave length and thus makes even the long wave oscillatory. Beyond this amount . leakage again decreases the wave length. Distributed induc- tance, as by loading the line, increases the critical wave length if the leakage is small, but in a very leaky line it decreases the critical wave length, and the amount of leakage up to which an increase of the critical wave length occurs is less in a loaded line, that is, in a line of higher inductance.

456 TRANSIENT PHENOMENA

In other words, a moderate amount of distributed leakage improves a long-distance telephone line, an excessive amount of leakage spoils it. An increase of inductance, by loading the line, improves the line if the leakage is small, but may spoil the line if the leakage is considerable. The amount of leakage up to which improvement in the telephone line occurs is less in a loaded than in an unloaded line; that is, a loaded telephone line requires a far better insulation than an unloaded line.

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CHAPTER IV. TRAVELING WAVES. 20. As seen in Chapter III, especially in electric power cir- cuits, overhead or underground, the longest existing standing wave has a wave length which is so small compared with the critical wave length — where the frequency becomes zero — that the effect of the damping constant on the-frequency and the wave length is negligible. The same obviously applies also to traveling waves, generally to a still greater extent, since the . lengths of traveling waves are commonly only a small part of the | length of the circuit. Usually, therefore, in the discussion of | traveling waves, the effect of the damping constants on the fre- quency constant q and the wave length constant k can be | neglected, that is, frequency and wave length assumed as inde- pendent of the energy loss in the circuit. Usually, therefore, the equations (74) and (75) can be applied in dealing with the traveling wave. - In these equations the distance traveled by the wave per | | second is used as unit length by the substitution A= al, | where «= VIC, . as this brings ¢ and A into direct comparison and eliminates h and k from the equations by the equation (72). With this unit length the critical value of k, k, = mV LC, by substituting (69) and (68), gives g, = m, and the condition of the applicability of equations (74) and (75), therefore, is that ’ q be a large quantity compared with g,= m. In this case 7 is a small quantity, and thus can usually be neglected in equations (76) and (75), except when C and C’ are very different in magnitude. 457

458 TRANSIENT PHENOMENA This gives, under the limiting conditions discussed above, the general equations of the traveling wave, thus: t= e-™ fet?“ (C, cosg (t — A+ C, sing ¢ — d)] — et#¢+% 10) cos q (t + 2+ C,/sing (t + a]

• e~*¢-% [C, cos g (t — A+ C, sing (t — — 2" (Ccosg (t+ A+Ci/ sing +A]}} (141) and L e=- vi e~“ fet#"-% IC cosg (t — A+ C,' sing (¢ — A]
• et#@+*% IC) cos g (t + A+ C,/sing (t + d))
• e7et-A) [c, COS q ¢t _ A+ Cy sin g (t a 4)]
• "+9 1C, cos (¢ + A+ C,’sing (t + d)]}, (142) or e=eM fetl+2TA cogg (t+ A+ A, sing (t + a]
• et¢-%[A, cos q(t — 4) + A,’sing (¢.— 2)] 4 en et +) [A, cos q (t+ A+ A, sin g (ae 4)}
• e~"-% TA, cosg (t — a)+ A/sing (t — AJ} (143) and i= ve e7™ fet#+%7A cos g (t + A+ A,’ sing (t + d)] — et¢- (4, cosg (t — a+ A,’ sing (t — d)]
• e494 cosg (t + a+ A,’ sing (t + a] — «7° TA, cosg (t — 4)+ A,’ sing (t — A}, (144) where l/r sg «=3(5 +2) j=l, (145) and ¢ = VIL. In these equations (141) to (144) the sign of 4 may be reversed, which merely means counting the distance in opposite direction.

| TRAVELING WAVES 459 . . This gives the following equations: : t= e“f{et?-% (B, cosg (t — A)+ B,/ sing (t — 4)] — e**€+% (B cosg (t + 4)+ B, sing (t + A]

• e-**-) (B cosg ¢ — A)+ B, sing (t — 4] — e*@*% [By cos g (¢ + 4)+ B, sing (t+ A)]}, (146) and e= Veermfetee-s [B, cos q (t — 4)+ B,’ sing (¢ — 4)]
• et*@+% (B cos g (¢ + 4)+ B,/ sing ¢ + 4]
• e~**- (B, cos g (t — A)+ By,’ sing (t — A] ;
• e~*4" (B, cos q (t + A+ B, sing (t + dj}, | (147) | or e=e“fet#(-% 74 cosg (t — 4)+ A,’ sing (t — A]
• et#@*% (4, cos g (¢ + A)+ A,’ sing (¢ + A)]
• e**-[A, cosg (t — A+ A,’ sing (t — )]
• e~"4% TA, cosg (t + A)+ A,’sing (t+ A)]} (148) and . C —utf .+8(t—A) Sos t=V7e {e [A, cos gq (t — 4)+ A,’ sing (t — d)] — et#@*% TA cosg (t + A+ A,’ sing (t+ 4)]
• e--% TA, cosg (t — 4) + A,’ sing (t — a] — e¢4% TA, cosg (t + A) + A,’ sing (t + d)]}. (149)

In these equations (141) to (149) the values A, B, C, etc., are integration constants, which are determined by the terminal conditions of the problem.

The terms with (¢ — 2) may be considered as the main wave, the terms with (¢ + 4) as the reflected wave, or inversely, depend- ing on the direction of propagation of the wave. :

21. As the traveling wave, equations (141) to (149), consists

, of a main wave with variable (¢ — 4) and a reflected wave of the same character but moving in opposite direction, thus with the variable (¢ + A), these waves may be studied separately, and afterwards the effect of their combination investigated.

460 TRANSIENT PHENOMENA Thus, considering at first one of the waves only, that with the variable (t — 4), from equations 148 and 149 we have e =e“fet#¢- A, cosg (t — A+ A,’ sing (t — d)]

• e~**- (A, cos q (t — 4) + A,’ sing (t — 4)}} = e~™ {(Aet#¢-% + Aye*¢-%) cos (t — A)
• (Aset#O™ + Afe-*¢-®) sing (¢ — 2D} | , . (150) and i- v6 e: qs) that is, in a single traveling wave current and voltage are in | phase with each other, and proportional to each other with an . effective impedance e L =-=/-. 152) 2 = {= This proportionality between e and 7 and coincidence of phase obviously no longer exist in the combination of main waves and reflected waves, since in reflection the current reverses with the reversal of the direction of propagation, while the voltage remains in the same direction, as seen by (148) and (149). In equation (150) the time ¢ appears only in the term (¢ — 4) except in the factor e~“, while the distance 4 appears only in the term (¢ — 4). Substituting therefore b =t{— 4, hence t=4+a; that is, counting the time differently at any point 4, and counting it at every point of the circuit from the same point in the phase of the wave from which the time ¢ is counted at the starting point of the wave, A = 0, or, in other words, shifting the starting point of the counting of time with the distance 4, and substituting in (150), we have

. TRAVELING WAVES * 461

e =e“ {et (A, cos gt, + A,’ sin gt,)

• e~™ (A, cos gt, + A,’ sin gt))}

= eM eM ete (4 cos gt; + A,’ sin

{ ( 1 qh 1 qty) (1 53)

• e~* (A, cos gt, + A, sin qt,)}

=e Me “L(A, eth + Ay e—%) cos qt,

• (AJ et + A,’ e~™) sin gt}.

The latter form of the equation is best suited to represent the variation of the wave, at a fixed point A in space, as function of the local time ¢,.

Thus the wave is the product of a term «~“ which decreases with increasing distance 4, and a term

e, =e “# {e+ (A, cos qt, + A,’ sin ¢t,)

• e~ (A, cos qt, + A,’ sin gt,)} =e “i { (Att + A, e~™) cos gt,

• (A, are + A, e~*) sin qt} , (154) which latter term is independent of the distance, but merely a ‘ function of the time ¢, when counting the time at any point of the line from the moment of the passage of the same phase of the

wave.

Since the coefficient in the exponent of the distance decrement

e~“ contains only the circuit constant, _1(" $) ' “<3 (, top but does not contain s and q or the other integration constants, resubstituting from equations (71) to (68), h=ol=l1VIG, we have ud =uVICl _! Cc vz “3{rVE+ovi}e where / is measured in any desired length.

462 TRANSIENT PHENOMENA Therefore the attenuation constant of a traveling wave is — 1 ye / L
= al _ _ 1 . 3 Uy u LC 2 { r L + g C , ( 55) and henoe the distance decrement of the wave, eT = eu Vic! | depends upon the circuit constants r, L, g, C only, but does not depend upon the wave length, frequency, voltage, or current; | hence, all traveling waves in the same circuit die out at the same . rate, regardless of their frequency and therefore of their wave shape, or, in other words, a complex traveling wave retains its | wave shape when traversing a circuit, and merely decreases in amplitude by the distance decrement «~“. The wave attenua- tion thus is a constant of the circuit. The above statement obviously applies only for waves of.con- | stant velocity, that is, such waves in which q is large compared . with s, u, and m, and therefore does not strictly apply to ex- tremely long waves, as discussed in 13. 22. By changing the line constants, as by inserting inductance L in such a manner as to give the effect of uniform distribution (loading the line), the attenuation of the wave can be reduced, that is, the wave caused to travel a greater distance / with the same decrease of amplitude. | As function of the inductance L, the attenuation constant (155) is a minimum for | du, _.. . . a= 0; hence, ua — gL =0, or Lor (156) Cog’ and if the conductance g = 0 we have L = »; hence, in a per- a fectly insulated circuit, or rather a circuit having no energy losses depending on the voltage, the attenuation decreases with increase of the inductance, that is, by “loading the line,”’ and the more inductance is inserted the better the telephonic transmission.

TRAVELING WAVES 463

In a leaky telephone line increase of inductance decreases the attenuation, and thus improves the telephonic transmission, up to the value of inductance,

rC L=—, 157 9 (157) and beyond this value inductance is harmful by again increasing the attenuation.

For instance, if a long-distance telephone circuit has the following constants per mile: r=1.31 ohms, L = 1.84 x 107 henry, g = 1.0 x 10-* mho, and C = 0.0172 x 10-° farad, the attenuation of a traveling wave or impulse is

u, = 0.00217; hence, for a distance or length of line of J, = 2000 miles, gol = et — 0.0129; that is, the wave is reduced to 1.29 per cent of its original value. The best value of inductance, according to (157), is L= 5c = 0.0225 henry, and in this case the attenuation constant becomes : uy = 0.00114, and thus : e—telo = eM — 02,1055, or 10.55 per cent of the original value of the wave; which means that in this telephone circuit, by adding an additional inductance of 22.5 — 1.84 = 20.7 mh. per mile, the intensity of the arriving wave is increased from 1.29 per cent to 10.55 per cent, or more than eight times.

If, however, in wet weather the leakage increases to the value

g = 5 X 10-°, we have in the unloaded line Uy, = 0.00282 and e~*" = 0.0035, while in the loaded line we have u, = 0.00341 and e~“ = 0.0011, .

464 TRANSIENT PHENOMENA and while with the unloaded line the arriving wave is still 0.35 per cent of the outgoing wave, in the loaded line it is only 0.11 per cent; that is, in this case, loading the line with inductance has badly spoiled telephonic communication, increasing the decay of the wave more than threefold. A loaded telephone line, therefore, is much more sensitive to changes of leakage g, that is, to meteorological conditions, than an unloaded line. 23. The equation of the traveling wave (153), @ = eA gh {er* (A, cos qh + A, sin gti)

• @~% (A, cos gt, + A,’ sin qt;)}, can be reduced to the form e= a {Be (et — e—") sin gt,
• Eig 7 4 (e+ — 6—s) cos qt}, (158) where ‘ : 4=h—-y=t-A-y, and (159) h=h-y,=t-A-y, ; By substituting (159) in (158), expanding, and equating (158) with (153), we get the identities . E,e~™ cos gy, — Ey, ™ sin gy, = A,, E,e~™ sin gy, + E,s~™ cos gy, = A,’, . . (160) E,et™ cos gy, — Egt™ singy, =—As,| - Eyet™ sin gy, + E,e+™ cos gy, = —Ay, and these four equations determine the four constants E,, £,, Vu Yor Any traveling wave can be resolved into, and considered as consisting of, a combination of two waves: the traveling sine wave, €, = Eye~™ e~Mn(e th — @—%h) sin gt, (161) and the traveling cosine wave, ; . €, = Eye e— Mts (et oe — @—%h) cos gt. (162)

TRAVELING WAVES 465 | Since q is a large quantity compared with u and s, the two - component traveling waves, (161) and (162), differ appreciably from each other in appearance only for very small values of 4,

that is, near ¢,= 0 and¢,=0. The traveling sine wave rises

in the first half cycle very slightly, while the traveling cosine wave

rises rapidly; that is, the tangent of the angle which the wave . . de . .

makes with the horizontal, ora equals 0 with the sine wave and

has a definite value with the cosine wave.

All traveling waves in an electric circuit can be resolved into constituent elements, traveling sine waves and traveling cosine waves, and the general traveling wave consists of four component waves, a sine wave, its reflected wave, a cosine wave and its reflected wave.

The elements of the traveling wave, the traveling sine wave é,, and the traveling cosine wave e, contain four constants: the intensity constant, HZ; the attenuation constant, u, and u, respectively ; the frequency constant, g, and the constant, s.

The wave starts from zero, builds up to a maximum, and then gradually dies out to zero at infinite time. |

The absolute term of the wave, that is, the term which repre- sents the values between which the wave oscillates, is

| e, = Bee Mi (ete — gt), (163) The term e, may be called the amplitude of the wave. Itisa maximum for the value of ¢,, given by de, which gives ; _ (u _ 8) eB) hy + (u + s) ete ty 0; hence, et tety = uts u-s and 1 ut+s 4, = 7 log uns’ . (164)

466 TRANSIENT PHENOMENA and substituting this value into the equation of the absolute term of the wave, (163), gives 28 ut s_ er em = Ee-*% ——— (=) 165, Ve —s\u-s . (ibe) The rate of building up of the wave, or the steepness of the wave front, is given by de, ; a [F,« as G, = Ee“ [— (wu — 8s) e7 “O88 + (u + 8) OTOH), = 2sHe-™,; (166) that is, the constant s, which above had no interpretation, represents the rapidity of the rise of the wave.

Referring, however, the rise of the wave to the maximum

value e» of the wave, and combining (165) with (166), we have uts G, = em (ut 9) . - (167) (wu — 8) 2*

The rapidity of the rise of the wave is a maximum, that is,

t; @ minimum, for the value of s, in equation (164), given by dt,, a which gives lo ut+ts 2us . Sas we’ hence, s = 0, or the standing wave, which rises infinitely fast, that is, appears instantly.

The smaller therefore s is, the more rapidly is the rise of the traveling wave, and therefore s may be called the acceleration constant of the traveling wave.

24. In the components of the traveling wave, equations (161) and (162), the traveling sine wave,

e, = Hee4# (e+ — ¢-) sin gt, (161), | |

TRAVELING WAVES 467. and the traveling cosine wave, (162), e, = Be~ eM (e+ — 6%) cos gt with the amplitude, €, = Beet (eth — gH), (168) we have e, = e@, sin gl, and (169) . €, = @, Cos qt). If t,; = 0, e, = 0; that is, ¢, is the time counted from the beginning of the wave. ; It is t; = t _ a —Ts or, if we change the zero point of distance, that is, count the distance A from that point of the line at which the wave starts at time ¢ = 0, or, in other words, count time ¢ and distance A from the origin of the wave, i =t—- A, and the traveling wave thus may be represented by the amplitude, e, = He~™ (ett — e—%); the sine wave, e, = Ee~™ (e+ — ¢™) sin gt, = e, sin qui; (170) the cosine wave, e, = He™* (e+# — ¢~*) cos gt; = e€, cos gt,; and ¢, = t— A can be considered as the distance, counting backwards from the wave front, or the temporary distance; that is, distance counted with the point 4, which the wave has just | reached, as zero point, and in opposite direction to A. Equation (170) represents the distribution of the wave along the line at the moment ¢. As seen, the wave maintains its shape, but progresses along the line, and at the same time dies out, by the time decrement e™,

| | 468 TRANSIENT PHENOMENA Resubstituting, ty =t- A,
the equation of the amplitude of the wave is e, = Ee~™ (e+#@-4) _ galt ay, (171) As function of the distance A, the amplitude of the traveling | wave, (171), is a maximum for de, . a” which gives A=0; that is, the amplitude of the traveling wave is a maximum at all times at its origin, and from there decreases with the distance. This obviously applies only to the single wave, but not to a combination of several waves, as a complex traveling wave. For A=0, e, = Ee~™ (et — e-), , and as function of the time ¢ this amplitude is a maximum, according to equations (163) to (165), at 1 ut+s t, = 5, 06 uae ru and is iu (172) E, = p—2s_(2") - Vu — #\u—s At any other point A of the circuit, the amplitude therefore is a maximum, according to equation (164), at the time tm = ty + a, and is “ E 2se“* /u + s\ 2 (173) om = Ve =(; =)

TRAVELING WAVES 469

25. As an example may be considered a traveling wave having . the constants u = 115, s = 45, g = 2620, and E = 100, hence,

e, = 100 eT USE (e445 _ 454) = 100 718A (e— 704 _ 71004) : where t,=t—A.

In Fig. 99 is shown the amplitude e, as function of the dis- tance A, for the different values of time,

t = 2,4, 8, 12, 16, 20, 24, and 32 x 107°, Cassdentantastessesteniay NOT TT TT Totlé als. 8 a2, 06, 20, 2, 2 i a vb BANE CECT thas TTT Fh NANGEE | mapa c aaawe REARS EEE ee Sih ShARSER AREER e wee BLS 0S QE WA Q\S SS AREEERESSSEEEEHTH nt Be

Ls N |

PNT To heedinde XT Srarasitestass st cetars CACACHEREEARE REPRESS EEE W%2 «4 6 8 10 R M4 16 18 2 2 a Fig. 99. Spread of amplitude of electric traveling wave.

with the maximum amplitude em, in dotted line, as envelope of the curve of e,.

As seen, the amplitude of the wave gradually rises, and at the same time spreads over the line, reaching the maximum at the starting point A = 0 at the time ¢, = 9.2 X 10-* sec., and then decreases again while continuing to spread over the line, until it gradually dies out.

It is interesting to note that the distribution curves of the amplitude are nearly straight lines, but also that in the present

. instance even in the longest power transmission line the wave has reached the end of the line, and reflection occurs before the maximum of the curve is reached. The unit of length A is the distance traveled by the wave per second, or 188,000 miles, and during the rise of the wave, at the origin, from its start to the . maximum, or 9.2 X 10-* sec., the wave thus has traveled 1760 miles, and the reflected wave would have returned to the origin before the maximum of the wave is reached, providing the cir- cuit is shorter than 880 miles.

470 TRANSIENT PHENOMENA « iam aa5 SSSR RRRRR | | x|107" < [4 tt | | +20 ‘a 7 1 iva: SLR TCA A SCREEN Vaneia

CEPI VT TMs aaaen foOC uneuscu sce cbeneguazae = on TAN ARAN Nie Are wea -whf VEN WA i PCE Er PEE TET ett tt shots 2 2 8 2 4 3 8 4 @ 4 6 48

Fig. 100. Passage of traveling wave at a given point of a transmission line. P feesy te sy Tae pee] soxio Bec) Pt TT tT TT fet as TTT PTA TT pepe pnpet tt team | OLE tN PL Pee TP TATA TN BE aegaNER eee eee Pt ttt? N TT TAT TT PT tT TTT TE NET OAT TT peor ren AA PEt iT TPT tT tT YP tT iN To Etat TE TE TAT TT AT SERENE AeA PCCCCENCCV EC Pt ttt tT Prt Ar NET TIN | lesser toopel | | | | YT tT | KT TT pet TT TT TYE TET TN FT PT PIN ETA TT TT ATT Pi TT PNET TT pti ttt | TdT PAPEETE pemren tYA TT IN LN _t pit iTAT TT TAT TT TS PT AAT Tr TT IN TT TT A NL . Pt Tt tT tT et ET TT TY TT PT TTT tet ey eT TINY fii tit it et ete TTT

Fig. 101. Beginning of electric traveling waves. | | Ain.

TRAVELING WAVES 471 With s = 1 it would be ¢, = 8.7 x 107° sec., or nearly the same, and with s = 0.01 it would be ¢, = 3.75 x 107? sec., or, in other wofds, the rapidity of the rise of the wave increases very little with a very great decrease of s. . Fig. 100 shows the passage of the traveling wave, e, =e, sin gt, across a point A of the line, with the local time ¢, as abscissas and the instantaneous values of e, as ordinates. The values are given for A = 0, where ¢, = ¢; for any other point of the line 4 the wave shape is the same, but all the ordinates reduced by the factor e—"®4 in the proportion as shown in the dotted curve in Fig. 99. Fig. 101 shows the beginning of the passage of the traveling wave across a point 4 = 0 of the line, that is, the starting of a wave, ar its first one and one-half cycles, for the trigonometric functions differing successively by 45 degrees, that is, . e, = e€, sin gh, é, te . a a = ¢, sin ( +7) €, = €, COs gl; = e, Sin (at + s) “1 1g cos (gt, + Z)=e sin (qu'+ 32) V2 0 qh 4 0 qh 4r The first curve of Fig. 101 therefore is the beginning of Fig. 100. In waves traveling over a water surface shapes like Fig. 101 can be observed. | For the purpose of illustration, however, in Figs. 100 and 101 | the oscillations are shown far longer than they usually occur; the value q = 2620 corresponds to a frequency f = 418 cycles, while traveling waves of frequencies of 100 to 10,000 times as high are more common. . Fig. 102 shows the beginning of a wave having ten times the attenuation of that of Fig. 101, that is, a wave of such rapid decay that only a few half waves are appreciable, for values of the phase differing by 30 degrees. 26. A specially interesting traveling wave is the wave in which s8=4u, (174)

472 TRANSIENT PHENOMENA since in this wave the time decrement of the first main wave and its reflected wave vanishes, eno = 1; (175) that is, the first main wave and its reflected wave are not tran- sient but permanent or alternating waves, and the equations of ee de 7 COARKECCC CRS TE PTA TINT TTT TT Teen AT | TIN fe tinge | PTT | AY Pt Ty TT Netorkettayy TT TT TT LY SNe ZOCPN aes ec A, PTET TING TT yt ttt yt Tl PINT TNT AEN TT ; AGENE? At) Zn SNE Pi tT TT KET Ae ZaNEEb<Z2 67 4nRNeREe VN Teo sin (i ry | COONS ZEEE Ps BERNE Zee ZN Ae AeeeeNeeee _— CL Neeaaheyy TTT Seco pT TK TTA EN ETE TE TT Pt | INE TA TING TTT A | | Neoptera yt TT TT | | eT PT IN TAT eee TT letse Baan ae Fig. 102. Passage of a traveling wave at a given point of a line. the first main wave give the equations of the alternating-current circuit with distributed r, Z, g, C, which thus appear as a special case of a traveling wave. Since in this case the frequency, and therewith the value of q, are low and comparable with u and s, the approximations made

TRAVELING WAVES 473 in the previous discussion of the traveling wave are not per- missible, but the general equations (50) and (51) have to be used. Substituting therefore in (50) and (51), s=4, gives t = [e-™ {C, cos (gt — kl) + C/’ sin (qt — l)} soe — et {C, cos (gt + Al) + C,’ sin (gt + kl)}) — 4 [e-™ {C, cos (gt + kl) + C,/ sin (gt + kl)} — et {C, cos (gt — Al) + C,’ sin (gt — Kl)}] (176) and e =[e-" {(c,/C,’— ¢,C,) cos (gt — kl) ; — (c/C, + ¢,C,’) sin (gt — kl)}

• et {(c/C/— ¢,C,) cos (gt + kl) . — (c/C, + ¢,C,’) sin (gt + kl)}]
• ete { (6 /C, — c,C,) cos (gt + kl) — (c,/C, + ¢,C,) sin (gt + kl)}
• et fe/Cy — c,C,) cos (gt — kl) — (¢/C, + ¢,Cy) sin (gt — kl)}). (177) In these equations of current 7 and e.m.f. e the first term represents the usual equations of. the distribution of alternating | current and voltage in a long-distance transmission line, and can by the substitution of complex quantities be reduced to a form given in Section III. = The second term is a transient term of the same frequency; that is, in a long-distance transmission line or. other circuit of | distributed r, L, g, C, when carrying alternating current under an | alternating impressed e.m.f., at a change of circuit conditions, a transient term of fundamental frequency may appear which has . | the time decrement, that is, dies out at the rate , | pu 2 GTO In this decrement the factor o, =e “zt

474 . TRANSIENT PHENOMENA , is the usual decrement of a circuit of resistance r and inductance L, while the other factor, 2 dé,=e © ‘ ; may be attributed to the conductance and capacity of the circuit, and the total decrement is the product, d= 36,0,.

A further discussion of the equations (176) and (177) and the meaning of their transient term requires the consideration of the terminal conditions of the circuit.

27. The alternating components of (176) and (177),

. ti, = «-"{C, cos (gt — kl) + C,/ sin (gt — My} — e*"{C.cos (gt + kl) + C,/ sin (qt + A} (178) and . eo= eM (ele! ~ cC,) cos (qt— kl) - (eC, +¢,C,’) sin (qt—kD}

• t™LelCy = ¢,C,) cos (qt+k) —(c/C,+¢,C,/) sin (gt+k)}, (179) are reduced to their usual form in complex quantities by resolv- ing the trigonometric function into functions of single angles, qt and kl, then dropping cos gt, and replacing sin gt by the imag- nary unit j7. This gives t, =e" {(C, cos kl — C{ sin kl) cos gt |
• (C/ cos kl + C, sin kl) sin qt} —« "{(C, cos kl + C,/ sin kl) cos gt
• (C, cos kl — C, sin kd) sin qt} ; hence, in complex expression, . T=e™ {(C, + jC/) cds kl — (C/ — 5C,) sin kl} ; —et™ 1(C, + jC) cos kl + (C,’ — jC,) sin kl}, (180) and in the same manner, E=e™ fe! (C/— jC.) — ¢, (C, + iC) cos kl +(e’ (C, + 7C/) +¢,(C/- iC] sink} +et™ {[c,’ (C,/— jC,) — ¢, (C, + 3C,/)] cos kl : — [e’ C, + 7C/) + ¢, C,’— jC,)] sin Ka}. - (181) | yO ;

TRAVELING WAVES 475 - However, from equation (52), _gk +h(m+s) , k&(m+s8) — gh = yp b and Bae » since q = 27f, -yal(t £) s=[=Su= ( + C , _lyr sg (182) m= a(z ~ o) and stm= Z =7 we have = 2nfLk+rh — xk +rh 1 r+k _ P+ and (183) , th —27flh rk — xh ef =— OS ' W+Kh P+ where x = 2 2fL = reactance per unit length. (184) From equation (5+), Ro = Vie + @ — mil 4 Agim; ' * hence, substituting (182) and (184) and also : b = 2afC, (185) |. we have Ro = VOT GTA) | | _¥W = 50 where z= Vr + 2 = impedance per unit length and (186) y = Vg + & = admittance per unit length. |

«476 TRANSIENT PHENOMENA

• From the above it follows that h=VIC V3 {[R2t+ ?—¢G-my = Vk (zy + rg — xb) (187) and k =V 4zy — 1g + 2b). If we now substitute ‘ C,+49C/ =B,VY and (188) C, + icy =~ BYVY, or C/ — jC, = — jB,VY and (189) Cy — jC, = + jB, VY, where Z=7r-jx and (190) Y =g — jb; in (180) and (181) we have 1=VY {Be*™ (cos kl—j sin kl) +B,e~™ (cos kl +] sin kl)} (191) and E = (c, + je’) VY {Bye*™ (cos kl — j sin kl) — Byz~™ (cos kl + jsink)}, (192) and substituting (183) gives . h (r—jx)+k(x+jr) (r—jx) (h+jk) r—jz ree I NS NE C+ 7e, h? + ke h2 + ke h—jk (193) However, h—jk=V Waki = Var BY = 2 jk = V (rg — xb) — jV (zy) — (rg — 2bY = V (rg—2b) —j VP +2) GF +P) — (rg —2b)? = V (rg — xb) — 7 Vr? + 2g + 2 rgb = V(rg — xb) — 7 (rb + xy) = V(r — jx) (g — 9b); or h—-jk =VZY (194)

TRAVELING WAVES 477 substituted in (193) gives | . [Z c, + je,’ = y’ (195) | and (195) substituted in (192) gives : E= VZ {Bet (cos kl—j sin kl) —By~™ (cos kl +j sin kl)}, (196) i where B, and B, are the complex imaginary integration constants. | Writing h=aandk = 8, . B, = D, and B, = —D, 7 the equations (191) and (196) become identical with the equa- tions of the long-distance transmission line derived in Section III, equations (22) of paragraph 8. | It is interesting to note that here the general equations of : alternating-current long-distance transmission appear as a special case of the equations of the traveling wave, and indeed can be | considered as a section of a traveling wave, in which the accelera- ! tion constant s equals the exponential decrement wu.

CHAPTER V. FREE OSCILLATIONS.

28. The general equations of the electric circuit, (50) and (51), . contain eight terms: four waves, two main waves and their reflected waves, and each wave consists of a sine term and a cosine term.

; The equations contain five constants, namely: the frequency constant, g; the wave length constant, /; the time attenuation constant, u; the distance attenuation constant, h, and the time acceleration constant, s; among these, the time attenuation, u, is

. a constant of the circuit,independent of the character of the wave.

By the value of the acceleration constant, s, waves may be sub- divided into three classes, namely: s = 0, standing waves, as discussed in Chapter III; u > s > 0, traveling waves, as dis- cussed in Chapter IV; s = u, alternating-current and emf. waves, as discussed in Section III.

The general equations contain eight integration constants C and C’, which have to be determined by the terminal condi- tions of the problem.

Upon the values of these integration constants C and ("’ largely depends the difference between the phenomena occurring in electric circuits, as those due to direct currents or pulsating currents, alternating currents, oscillating currents, inductive dis- charges, etc., and the study of the terminal conditions thus is of

the foremost importance. :

29. By /ree oscillations are understood the transient phe- nomena occurring in an electric circuit or part of the circuit to neither which electric energy is supplied by some outside source ner from which electric energy is abstracted.

Free oscillations thus are the transient phenomena resulting

‘ from the dissipation of the energy stored in the electric field of the circuit, or inversely, the accumulation of the energy of the electric field; and their appearance therefore presupposes the possibility of energy storage in more than one form so as to allow

478 ; "

_FREE OSCILLATIONS 479° an interchange or surge of energy between its different forms, electromagnetic and electrostatic energy. Free oscillations occur only in circuits containing both capacity C and inductance, L. The absence of energy supply or abstraction defines the free oscillations by the condition that the power p = e at the two ends of the circuit or section of the circuit must be zero at all times, or the circuit must be closed upon itself. The latter condition, of a circuit closed upon itself, leads to a full-wave oscillation, that is, an oscillation in which the length of the circuit is a complete wave or a multiple thereof. With a cir- cuit of uniform constants as discussed here such a full-wave oscillation is hardly of any industrial inrportance. While the most important and serious case of an oscillation is that of a closed ‘circuit, such a closed circuit never consists of a uniform conductor, but comprises sections of different constants; generat- ing system, transmission line and load, thus is a complex circuit comprising transition points between the sections, at which par- -tial reflection occurs. The full-wave oscillation thus is that of a complex circuit, which will be discussed in the following chapters. Considering then the free oscillations of a circuit having two ends at which the power is zero, and representing the two ends of the electric circuit by / = 0 and I = 1,, that is, counting the distance from one end of the circuit, the conditions of a free oscillation are ; 1=0, p = 0. L=l,, p = 0. Since, p = et, this means that at / = 0 and / = lJ, either e or 7 must be zero, which gives four sets of terminal conditions: (1) e =Oatl =0; t=Oatl =1,. V2i= = 0: = = (2) 71 =Oatl =0; e=Oatl =1,. (197) | (3) e = Oatl =0; e = Oatl =1,. (4)7 =Oatl =0; 7=Oat/l =1.,. Case (2) represents the same conditions as (1), merely with the distance | counting from the other end of the circuit — a line open at one end and grounded at the other end. Case (3) repre- | | | |

480 TRANSIENT PHENOMENA sents a circuit open at both ends, and case (4) a circuit grounded at both ends.

30. In either of the different cases, at the end of the circuit l = 0, either e = 0, or i = 0. Substituting / = 0 into the equations (50) and (51) gives eg =e @-O(e’ (C,! + Cy) -— €, (C, + C,] cos gt — [ce (C, + C,) + ¢, (C+ C,)] sin gt}

• eT te (CL + CY) — ¢, (C, + C,)] c08 gt — [e, (Cr+ Cy) + & (Cy + C,)]sin gt} (198) and tp =e @-"' 4 (C,- C,) cosgt+ (C,’ — C,’) sin gt} .
• et (Cl — C,) cosgtt+ (C,’ — C,’) singt}. (199) If neither g nor s equals zero, for e, = 0, c/ C/ +C/) -—¢,C, +C¢,) =0 and c/ (C, + C,) + ¢, (CY + C,) = 0; hence, Cc, =-C C, = -C;, re and for 7, = 0, - = =C,. woe, een fm Substituting in (50) and (51), i =e "OF 1C, [e~™ cos (gt — kl) + e*™ cos (gt + K)) +C/ [e~™ sin (gt — Al) + e*™ sin (gt + &l))}
• ett (C [e+ cos (gt — kl) + ¢~™ cos (gt + kl))
• Cy [et™ sin (qt — kl) + ¢ "sin (gt + kl)]} (202)

FREE OSCILLATIONS 481 and ; e =e 4-9! 1 (¢/C,/— ,C,) [e~™ cos (gtk)

• e+” cos (gt + kl)) — (¢,/C, + ¢,C,’) [e~™ sin (qt — kl) F et" sin (gt + kl)}}

• e~ M49 CICS — 6,C,) [e*™ cos (gt —k) Fe" cos (gt+kl)] -— CC, + ¢,C,’) [e+ sin (gt — Kl)

• e™ sin (gt + kl))}, (208) where the upper sign refers to e = 0, the lower sign to7 = 0 for 1=0.

31. In a free oscillation, either e or 7 must be zero at the other end of the oscillating circuit, or at / = 1,. Substituting, therefore, / = J, in equations (202) and (203),

. and resolving and arranging the terms by functions of ¢t, the | respective coefficients of e~—®)Feos gt, «~~ f sin gt, «“*** cos gt, and e “+ sin gt must equal zero, either in equation (202), if7 = 0 at / = 1,, orin . equation (203), if e = 0 at / =1,, provided that, as assumed above, neither s nor q vanishes.

This gives, for i = O atl = 1,, from equation (202), | C, (e~™ + e+) cos kl, — C,(e7™ F e+") sin kl, = 0, (204) | C, (e~™ F e+) sin kl, + C,/(e-™ + e~™*) cos kl, = 0, | and analogously for C, and C,’. In equatipns (204), either C,, C,’, C,, C,’ vanish, and then the whole oscillation vanishes, or, by eliminating C, and C,’ from equations (204), we get (e~ Me + ¢ +)? cos? kl, + (7 F eth)? sin? kl, = 0; (205) hence, (eM + 6 +) eos kl, = 0 and (eM F +h) gin kl, = 0:

482 TRANSIENT PHENOMENA hence, for the upper sign, or if e = 0 for / = 0, . h = 0 and cos kl, = 0. thus: kl _ (2n + I) ‘ (206) rn ae and for the lower sign, or if 1 = 0 forl.= 0, h=0 and sinkl, = 0, 207) . 7 thus: kl, = new. In the same manner it follows, for e = 0 at / = l,, from equa- tion (208), if e = O for? = 0, h=0, sinkl, =0 (208) thus: kl = nz, . andif 7 = 0 forl = 0, . h=0 and cosH, = 0, . thus: ay 2D, (209) , o 2 From equations (206) to (209) it thus follows that kh = 0, that . is, the free oscillation of a uniform circuit 1s a standing wave. Also kl, = Gaete ae (210) if e = 0 at one, 7 = 0 at the other end of the circuit, and kl, = nz (211) | if either e = 0 at both ends of the circuit ori = 0 at both ends of the circuit. 32. From (210) it follows that : ; kl, = 2’ or an odd multiple thereof; that is, the longest wave which can exist in the circuit is that which makes the circuit a quarter-

FREE OSCILLATIONS 488 wave length. Besides this fundamental wave, all its odd multi- ples can exist. Such an oscillation may be called a quarter-wave oscillation.

The oscillation of a circuit which is open at one end, grounded at the other end, is a quarter-wave oscillation, which can contain only the odd harmonics of the fundamental wave of oscillation.

From (211) it follows that

| kl, = 2, or a multiple thereof; that is, the longest wave which can exist in such a circuit is that wave which makes the circuit a half- wave length. Besides this fundamental wave, all its multiples, odd as well as even, can exist. Such an oscillation may be called a half-wave oscillation.

The oscillation of a circuit which is open at both ends, or grounded at both ends, is a half-wave oscillation, and a half-wave oscillation can also contain the even harmonics of the funda- mental wave of oscillation, and therefore also a constant term forn = Qin (211). *

It is interesting to note that in the half-wave oscillation of a circuit we have a case of a circuit in which higher even harmonics exist, and the e.m.f. and current wave, therefore, are not sym- metrical.

From h = 0 follows, by equation (56),

s=0, if > LCm’,

and (212) qg=0, if M< LCm’. | The smallest value of & which can exist from equation (210) is

nu

| B=5 l,’

and, as discussed in paragraph (15), this value in high-potential |

high-power circuits usually is very much larger than LC'm’, so that the case g = 0 is realized only in extremely long circuits, ~

as long-distance telephone or submarine cable, but not in trans- | mission lines, and the first case, s = 0, therefore, is of most | importance. | x | |

484 TRANSIENT PHENOMENA Substituting, therefore, h = 0 and s = 0 into the equation (52) gives | ¢, =a, = iL =c and (213) m c/ =c, = ra =C¢, and substituting into equations (202) and (203) of the free oscilla- tion gives i = e—“{A, [cos (gt — kl) + cos (gt + kd)]

• A, [sin (qt — kl) + sin (gt + kl))} (214) and L e= rae {(mA,— qA,) [cos (gt—kl) ¥ cos (gt + H)] — (mA,+ qA,) (sin (qt—kl) ¥ sin (gt + kl)}}. (215) Since & and therefore q are large quantities, m can be neglected compared with g, and k = VICq; hence . Iq _/k k Cc and the equation (215) assumes, with sufficient approximation, . the form L e= ~ Vee {A, [cos (qt — kl) ¥ cos (gt + k)] —A, [sin (qt — kl) ¥ sin (gt+ kl)}}, — (216) __ where the upper sign in (214) and (216) corresponds to e = 0 at ~ "| = 0, the lower sign to 7 = 0 at 1 = 0, as is obvious from the equations. From A, =C,+C, and A, =C + Cy,

. FREE OSCILLATIONS 485 - substituting .

_A,=Acosy and A, = Asin; (217) into (214) and (216) gives the equations of the free oscillation, thus:

i = Ae~“ {cos (qt—kl—y) ¥ cos (qt + kl — y)} . and

L (218) e=-—A Vere feos (qt—kl—y) ¥ cos (gt + kl — y)}.

With the upper sign, or for e = 0 at / = 0, this gives

4 = 2 Ae~™ cos kl cos (gt — 7) and L (219) e=- 2ay/Eew sin kl sin (gt — 7).

• With the lower sign, or for + = 0 at / = 0, this gives

t= 2Ac~™“ sin kl sin (gt — y) ;

and . IL (220)

e=-2A Vee cos kl cos (qt — 7).

33. While the free oscillation of a circuit is a standing wave, the general standing wave, as represented by equations (139) and (140), with four integration constants A,, A,’, A,, A,’, is not necessarily a free oscillation.

To be a free oscillation, the power ez, that is, either e or 7, must be zero at two points of the circuit, the ends of the circuit or sec- tion of circuit which oscillates.

At a point J, of the circuit at which e = 0, the coefficients of cos gt and sin gt in equation (139) must vanish. This gives