Theory and Calculation of Transient Electric Phenomena and Oscillations — part 16 of 20

420 TRANSIENT PHENOMENA and in the same manner, . de de de pe et e+ gl) + Wa: (4) | These differential equations, of the second order, of current i and voltage e are identical; that is, in an electric circuit current | and e.m.f. are represented by the same equations, which differ | by the integration constants only, which are derived from the terminal conditions of the problem. Equation (3) is integrated by terms of the form ; t= AgU-, (5) . Substituting (5) in (3) gives the identity ! a@ = 1g — (rC + gL) b + LCP | = (OL — r) (0C — g). (6) | ; In the terms of the form (5) the relation (6) thus must exist | between the coefficients of J and t. Substituting (5) into (1) gives : S = (r — bL) Ae#-™, : (7) | and, integrated, . e = BHT 4 -al-w, (8) ‘ | The integration constant of (8) would be a function of t, and since it must fulfill equation (4), must also have the form (5) for the special value a = 0, hence b= ror b= 4 and_ therefore can be dropped.. In their most general form the equations of the electric circuit are 4 = Dnf Anema mt}, (9) _ n bab — 7 — dal — bat ? | e= > re Ane (’ (10) a,? — (bnL — 1) (b2C — g) = 0, (11) Lm

' GENERAL EQUATIONS 421 where A, and a, and 6, are integration constants, the last two being related to each other by the equation (11).

8. These pairs of integration constants, A, and (ap, b,), are determinated by the terminal conditions of the problem.

Some such terminal conditions, for instance, are:

Current 7 and voltage e given as a function of time at one point J, of the circuit — at the generating station feeding into the circuit or at the receiving end of the transmission line.

Current 1 given at one point, voltage e at another point — as voltage at the generator end, current at the receiving end of the line.

Voltage given at one point and the impedance, that is, the complex ratio oe at another point — voltage at the gen- erator end, load at the receiving end of the circuit.

Current and voltage given at one time ¢, as function of the distance / — distribution of voltage and current in the circuit at the starting moment of an oscillation, etc.

• Other frequent terminal conditions are:

Current zero at all times at one point /,— the open end of the circuit.

Voltage zero at all times at one point J,— the/grounded or the short-circuited end of the circuit. .

Current and voltage, at all times, at one point J, of the circuit, equal to current and voltage at one point of another circuit — connecting point of one circuit with another one. —

As illustration, some of these cases will be discussed below.

The quantities 7 and e must always be real; but since a, and b, appear in the exponent of the exponential function, a, and b, may be complex quantities, in which case the integration constants A, must be such complex quantities. that by com- bining the different exponential terms of the same index n, that is, corresponding to the different pairs of a and b derived from the same equation (15), the imaginary terms in A, and ny cancel. |

Qn |

In the exponential function

e7al ~ bt ~ \

422 TRANSIENT PHENOMENA writing . a@=h+jk and b=p+jq, (12) we have | grant gM m,-i +e) and the latter term resolves into trigonometric functions of the angle kl + qt; . kl + qt = constant (13) | therefore gives the relation between / and ¢ for constant phase | of the oscillation or alternation of the current or voltage. .

With change of time ¢ the phase thus changes in position / in the circuit, that is, moves along the circuit.

Differentiating (13) with respect to ¢ gives

dl . ke +q=0, or

d_ 4.

di = k , (14) that is, the phase of the oscillation or alternation moves along the circuit with the speed — d, or, in other words,

q =-—-t 1 is the speed of propagation of the electric phenomenon in the cir- curt.

(If no energy losses occur, r = 0, g = 0, in a straight con- ductor in a medium of unit magnetic and dielectric constant, that is, unit permeability and unit inductive capacity, S is the velocity of light.) .

4. Since (11) is a quadratic equation, several pairs or corre- sponding values of a and b exist, which, in the most general case, are complex imaginary. The terms with conjugate complex imaginary values of a and b then have to be combined for the elimination of their imaginary form, and thereby trigonometric functions appear; that is, several terms in the equations (9) and

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GENERAL EQUATIONS 428 (10), which correspond to the same equation (11), and thus can be said to form a group, can be combined with each other. Such a group of terms, of the same index n, is defined by the equation (11), a,” = (6,L — 1) (bnC — 9). For convenience the index n can be dropped in the investiga- tion of a group of terms of current and voltage, thus: a = (bL — r) (WC — 9), (16) and the following substitutions can be made: ; a=a, VIL, (17) a=h + jk, a, = h, + jk, (18) b=pt+ id from which h =h, VIC and k=k, VIC. (19) Substituting (18) in (16), oti =[o+io-F][@+i- 8] eo Carrying out and separating the real and the imaginary terms, equation (20) resolves into the two equations thus: r g Mt = (»-2)(2-2)-¢ and L C (21) . - a s) hk, =q (2 P-z-¢l Substituting l(t 49 u-3(7+9) (22) -5(; - 2) m= (7 - $), (23)

424 TRANSIENT PHENOMENA and p=st+u (24) into (21) gives hAvp-kPZ=8 -—¢ —m’, roto ¢ 5 or hk, = sq, and ¥-G =hP-ke tm, (26) or sq = h,k,. Adding four times the square of the second equation to the square of the first equation of (25) and (26) respectively, gives hetkeZ=V(e —-@ -— my +49¢ =Vilit+¢g—m)+4¢m (27) and Ft G =V(h? — kh? + my + 4h2k? = Vii +k? + my — Tk emt (3) = R,, and substituting (19), gives, by (25), (26) and (27), (28) be DV aye eh 1 29 k= VICVS {RY +g + mh, ) RR=Vieit+g—-myP+4¢m, or s= ty 1 2 2 2 1 A (30) q= Vig 5 {Re —h? + — LCm'}, RZ =V(v +h + LCmy — 4 LC. | | Lo

• | GENERAL EQUATIONS 425 | If, however (+ h + jk) and (w + s + jg) satisfy equation (16), | then any other one of the expressions ! (+h + jk) and (wuts + jg) also satisfies equation. (16), providing also the second equation of | (24) or (25) is satisfied, , hk = VLC sq; (31) that is, if s and g have the same sign, h and k must have the same sign, and inversely, if s and g have opposite signs, h and k must have opposite signs. This then gives the corresponding values of a and b: ; (1) a= +h-+ jk b=u-—s— jq +h — jk u—s+ jq (2) a= —h — jk b=u-—s-— jq —h+ jk u-SsStie q 14 (32) (3) @= —h+ jk b=uts— jg —h—jk utst iq | (4) a=+h-—jk b=ut+s— qq +h + jk ut+s+ jq | or eight pairs of corresponding values of a and b.

6. Substituting the values (1) of (32) into one group of terms of equations (9) and (10), i = Ac U-% and . 33 | e= bL = 1 4 .-at-m ¢ a gives a, = A,e~ +i I~ (usta) t + A,/e~ hi I- (u a tiat = eM (un adey 4 ig ikl +iat + A,’ tiH— iat} | | and substituting for the exponential functions with imaginary exponents their trigonometric expressions by the equation et? = cosx + jsinz

426 TRANSIENT PHENOMENA gives i, = e~M-M—##t 4 [cos (qt _ kl) + jsin (qt _ kD)

• A,’ [cos (gt — kl) — jsin (gt — K))}} =e M-™—911 (4 +A’) cos (gt—kl) +7 (A, —A,’) sin (gt—-H)}; hence, A, and A,’ must be conjugate complex imaginary quap- tities, and writing . C,=A,+ A, and (34) Cy’ = 7(A, - Ay) _ gives i, = e-™-@-9OUC, cos (qt — kl) + C,’ sin (gt — H)}. (35) Substituting in the same manner in the equation of e, in (33), ; gives _~M=8 -IDL—-T, asi —e-ioet “= ha jk 4 (u-stjgLl—r,, — (h+ik)l—(y—s +39)
• h — jk Aye = e7h-(u-s)t {i ~s 7 i) Zz r] (h = Ik) 4 ,ti@-io [us +i L—-rNG+ ib yy sqm): | ay 2s i } hence expanding, and substituting the trigonometric expressions | gen Natu-ant y(n) L-r)h—qkL _ ters) L-r] an , rt V+ +R . . [(u—s) L—r] h—qkL A, [cos (qt — kl) +7 sin (gt—k)}4+(“ | [(u—s) L—r] k+ gk - |
• RD ATE ED) 4 soos (gt HD) —jsin (tH, (88) +#? and introducing the denotations | o — eh thir - u-s)L)_gk+h(m+s), Pte Rte on | ora tir = (u—s)L}—ghL k(m+s) = qhy

• P+ - R+R , | hp.

GENERAL EQUATIONS 427 and substituting (37) in (36), gives | e, =e NOH (—6, + Jey’) A, [cos (gt + kl) + j sin (gt — Hy] | +(—¢, — je) A,’ [cos (gt — HM) — j sin (gt — ki))} | meT Numa eyy C, (A, + A,’)+ je, (A, ~ A, cos (qt 7 kl) | +[- jc,(A, — A,’)— ¢,/(A, + A,’)] sin(g@ — X)}. (88) Substituting the denotations (34) into (38) gives e, = e-M-@—-9' EC! — ¢,C,) cos (gt— kl) —(c,C,’ + ¢/C,) sin gt — k)}. (39) The second group of values of a and b in equation (32) differs from the first one merely by the reversal of the signs of h and k, and the values 7, and e, thus are derived from those of 1, and e, by reversing the signs of h and k. . Leaving then the same denotations c, and c,’ would reverse the sign of e,, or, by reversing the sign of the integration con- stants C, that is, substituting C, =—- (A, + A,’) and } (40) Cy’ =-7(A, - A,), the sign of 2, reverses; that is, i, = — etB- ONC, cos (gt + M) +C,/ sin (gt +kl)} (41) and ; eé, = e thot elcy _ c,C,) cos (qt + kl) — (¢,C,’ + ¢,/C,) sin (gt + k)}. (42) The third group in equation (32) differs from the first one by the reversal of the signs of h and s, and its values 7, and e, there- fore are derived from 7, and e, by reversing the signs of h and s. Introducing the denotations gk — h(m — s) I ¢= BR+te L, (43) ota Bam = 8) + gh ‘ 2 +h and C, =A,+ Ay, a =7 ; ‘ , (44) s =) (A,— Ay),

428 | TRANSIENT PHENOMENA gives 1, = eth- +911) cog (gt — kl) + C, sin (qt —k)} (45) and e, = etTh-@ tt CE ICY — c,C,) cos (gt — Kl) — (c,C,’ + ¢,/C,) sin (gt — kl)}. (46) The fourth group in (32) follows from the third group by the reversal of the signs h and k, and retaining the denotations c, and c,’, but introducing the integration constants, C, =~ (A, + A,) and (47) : CY =~ 7 (A, _ A,), gives ig = — eB 49! (Cy cos (gt + kl) + Cy sin (gt + Hd)} (48) *‘and 4 = eM tt EC! — ¢,C,) cos (gt + Al) ; — (¢,C/ + ¢,/C, sin gt + k)}. _ (49) 6. This then gives as the general expression of the equations of the electric circuit: i= DleM MOC, cos (gt — kl) + C,’ sin (gt — kl)} (@,) . — eth-U-9UC cos (gt + kl) + C,/ sin (qt + kl)} (i,) (50)

• et¥-@+914C, cos (gt — kl) + Cy sin (gt — kl)} (i) HOC, cos Gt MD) + CY sin (gt + DH) Gd) and e = dle! “44 (eC! — ¢,C,) cos (gt — Hd) — (¢/C, + ¢,C,’) sin (qt — K)} (e,)
• eth MOE tCY’ — ¢,C,) cos (gt + kl) — (¢/C, + ¢,C,’) sin (gt + kl)} (e,) (51)
• eth—UtOt eC /CY — ¢,C,) cos (qt — kl) — (c,/C, + ¢,C,’) sin (gt — kl)} (e,) +e MFO IC! — 6,C,) cos (gt + kD) — (¢/C, + ¢,C,’) sin qt + HK) } 1 (e), |

GENERAL EQUATIONS 429 where C,,, C,’, C,, C,/, Cy, C,’, Cy, C,/ and two of the four values 8, q, h, k are integration constants, depending on the terminal conditions, and , k+h(m+s) ; ¢= CF L, k (m +s) — gh = ed (52) c _ hk —h(m—s), 7 UWB , k(m—s) +qh ‘= ey a Bee” : and l(t" £) u=3(7 +6) lyr (53) 2 Cc; | and h, k and s, q are related by the equations h= VIC V3 [R?7 + 2 —¢ —m'}, k=VICVYRZ—-F +P + my, (54) and RR aV(e t+ ¢ — my +4¢m’; hence, +h = LCR, (55) | or , 1 8 = ag VITRE Fs Daey, 1 9= SVE RY PF Ley, (56) and RZ =V(W +h + Lm) — 4 LC’; | hence, += Ry (57) . LC

430 TRANSIENT PHENOMENA Writing D (qt + kl) = C cos (qt + Al) + C’ sin (gt + Kl) (58) and H (gt + kl) = (c'C’ — CC) cos (gt + Hl) — (CC + CC’) sin (gt + kl), (59) equations (50) and (51) can be written thus: = ~M-(u-at py _ _ +M—(u-s)t Z) + uA i=Dle (qt—Md) -e +H) CO

• etl (40! Dy (qt— kl) eM 4" D+ Hd) (2) and' e= Die“ -- H, (qt—kl) +et+¥-(@u-oe H,(qt +k) (e’) 61)
• et BOO H, (gt— kl) +e-M- 491 H (qt+kl)) (e")

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, g, h, k. |

Of the four terms of each group, 1,, 2,, 15, % OF €,, €3, Cs, &% | respectively (equations (50) and (51)), two contain the angles

(qt — kl), 7,, e, and 2,, e,; and two contain the angles (qt + xi), t,, €, and 14, &.

In the terms 7,, e, and 7,, e,, the speed of propagation of the phenomena follows from the equation qt — kl = constant, Co GF ng

Kk, thus: dl q di tk? | hence is positive, that is, the propagation is from lower to higher values of J, or towards increasing /.

In the terms #,, e, and %,, e,, the speed of propagation from |

gt + kl = constant

is . d_4g | | dk | hence is negative, that’ is, the propagation is from higher to | lower values of 1, or towards decreasing 1.

Considering therefore 7,, e, and 7,, e, as direct or main | waves, 2,, €, and 7,, e, are their return waves, or reflected waves, | and 7,, é, is the reflected wave of 1,, e,; 14, €, is the reflected wave of 2,, e,- !

. 481 | |

432 TRANSIENT PHENOMENA

Obviously, 7,, e, and z,, e, may be considered as main waves, and then 7,, e, and 7,, e, are reflected waves. Substituting (— 2) for (+ J) in equations (50) and (51), that is, looking at the circuit in the opposite direction, terms 7,, e, and 2,, e, and terms 1, e, and 7,, e, merely change places, but otherwise the equations remain the same, except that the sign of 7 is reversed, that is, the current is now considered in the opposite direction.

Each group thus consists of two waves and their reflected waves: 2, — 7, and e, + e, is the first wave and its reflected wave, and 7, — 1, and e, + e, 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: 1’ = 1, — 1, and & = e, + e, is the first wave, and 7” = 1, — 1, and e” =e, + & is the second wave.

In the first wave, 7’, e’, the amplitude decreases in the direction of propagation, «™ for rising, e+™ for decreasing J, and the wave dies out with increasing time ¢ by «~~! = e“ e+ #,

In the second wave, 7’, e’, the amplitude increases in the _ direction of propagation, «+™ for rising, «™ for decreasing J, but the wave dies out with the increasing time ¢ by «“**)! =e“ ¢~® 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 «~“.

It thus follows: In general, two waves, with their reflected waves, traverse the circuit, of which the one, 7”, 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, 7’, 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, 7’, 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 2”, e” duration in time is sacrificed to | duration in distance, and inversely in 7’, e’. |

DISCUSSION OF GENERAL EQUATIONS 433 It is interesting to note that ina 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=0,k =0 and toh =0,s =0. | 8. In the equations (50) and (51) qi = 2a gives the time of a complete cycle, that is, the period of the wave, 2a t= v7 ’ ; and the frequency of the wave is -/., a f-£ c kl =22 gives the distance of a complete cycle, that is, the wave length, 2x . — & > | l= ke? : L $ t rf | (u-s)t=1 and (ut+s)t=1 | give the time, | . r_ 1 We 1 ore and pe? , 1 | during which the wave decreases to 77 0.3679 of its value, and hl =1 gives the distance, 1 on over which the wave decreases to ; = 0.3679 of its value; that is, g is the frequency constant of the wave, 4 - 2%, f ~ 9x to q ’ . (62)

484 TRANSIENT PHENOMENA k is the wave length constant, 2x. ; ly = k , (63) (u — s) and (uw + s) are the time attenuation constants of the wave, if = —— uns’ 6 i (64) ti” = ——, utes and h is the distance attenuation constant of the wave, 1 haz. (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., g 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),

R2=V(Et+ C—m) +4 Gm =VGF+myp+2eq—m

29 (g — m?))} = 2 ——— @ +m) }1 + @ + my g-m

=> 2 eo

=Ptmis etm

=Gtmt+s |

-¢, | h=VICV¥{[RI+ 9% —¢ —m} =sVIC, 6s) | k=VIO V3 [R2 — 24+ +m} =VLC (G+ m) =qV IC, |

+B = (e+ ¢) LC = gle,

and ; kth), oh vfh ' R+h ik Y@

DISCUSSION OF GENERAL EQUATIONS 435 c, te ah (m= 8), _ gh / L | 2 V+ kb YC’ | ,k(m+s)-—qgh, qgVIC(m+3)—qsVIC m/E ¢/ = ———_ L= Le - w+ kh GLC q’C ,_ k(m—38)+ qh q VIL (m — s) + qs VLC m./L. | = ——a a eB ore a | + GLC qg'C | that is, , | yz | C, = Cc, = ral . m,. /L c/=c/= Ve: . Writing o=VIC, vi (68) . c= C’ where o is the reciprocal of the frequency of propagation (velocity . - 7 ' of light), we have h = @s, 69) k = a9, } | and C, =c, =¢ | c/=c/ =e (70) | r= =o and introducing the new independent variable, as distance, . A=al, (71) | we have Kl = qa and } (72) hl = 8A; |

; | 436 TRANSIENT PHENOMENA hence, the wave length is given by gi=2x as 2a ‘ A, q (73) and since the period is . . 22 t, = 7? it follows that by the introduction of the denotation (71) dis- tances are measured with the velocity of propagation as unit Jo. SH. length, and wave length J, and period ¢, thus have the same _* "numerical values. Substituting now in equations (50) and (51) gives b= eM D fet Dy fg CD] Dy lg C49) @)

• e729 Dy [g(t — a)]—e7* 4+® Dy [g @+))} @) (74) and e= en Di fete H, [q (t-A)]+e**¢™ A, [g(t+2)] (e) | + e@" Hy [q (t—a)) +679" Hy [g (t+ a}, (”) (75) where D{g(t+ Aj =Ccosg (t+ 44+ C’sing (t + A) and yi m H(g(t +a) = ala — C) cos g + ¥)

• (Fe + c’)sin g (+a) . (76)

10. As seen from equations (74) and (75), the waves are products of «~“ and a function of (t — 4) for the main wave, (t + 4) for the reflected wave, thus: a, +4, =e “f, t - 4) and (77) i, +%,=e°“f, (t+ a)3

| | DISCUSSION OF GENERAL EQUATIONS - 487 | hence, for constant (¢ — 4) on the main waves, and for constant (t + 4) on the reflected waves, we have , ti, = Be“ and (78) * 1,+%, = Be; that is, during its passage along the circuit the wave decreases by the decrement «~“, 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 l/r eo = ealetat and therefore is independent of the character of the wave, for instance its frequency, etc.

11. The physical meaning of the two waves 7’ and e’ can best be appreciated by observing the effect of the wave when travers- ing a fixed point A of the circuit.

Consider as example the main wave only, 2’ = 7, + 2,, and neglect the reflected waves, for which the same applies.

From equation (74),

t= eT —@ "9D [g(t — a) + e~ 49D, [g(t — a); (79) or the absolute value is ;

l= Dg78-%- 9 + Dygt- tot (80) | where D, and D, have to be combined vectorially.

Assuming then that at the time ¢ = 0, I’ = 0, for constant A

| we have | 1=D (e7 uae _ e7 (utayty (81) the amplitude of J at point 2.

Since (81) is the difference of two exponential functions of different decrement, it follows that as function of the time ¢, J rises-from 0 to a maximum and then decreases again to zero, as shown in Fig. 98, where

I, = De~%—-94 I, = De~ut# I=I1,-1, and the actual current 7 is the oscillatory wave with J as envelope.

438 TRANSIENT PHENOMENA

The combination of two waves thus represents the passage of & wave across a given point, the amplitude rising during the arrival and decreasing again after the passage of the wave.

SREP cen. See Pr Ty TTT TT PN ETT ET ee PT Vit Tt ttt tT PreAL ET Te PTA TTT ATT TT PT7T TT TT TT Te Peet PVT TT TT TT PET tT tt TTT TT ET TE Fig. 98. Amplitude of electric traveling wave.

12. If h and so also s equal zero, 7’, e and 2”, e” coincide in equations (74) and (75), and C, and C, thus can be combined into one constant B,, C, and C, into one constant B,, thus: |

: C, +C, =B,, C,+C, =B,,

• ‘ ” (82) C/+Cy = By, Cc, +C/= By, and (74), (75) then assume the form . i =e“ ){[B, cos g (t — A) + BY sing (t — a] . — = ([B,cosg (t+ 4) + B/sing@é + A]}, (83) e= vi mya, _ B,)cos 9 ((—2)— (7B, +B.) | | . m m . sing (t-2) ]+[(B,—B, )eosa(+4)-("B,+B,)sing +a} | (84)

These equations contain the distance 4 only in the trigono- metric but not in the exponential function; that is, i ande 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

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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 7 and e can be assumed as uniform . throughout the circuit, the equations (83) and (84) assume the . form | i =e-“{B, cos gt + B,’ sin gt} | and (85) . m m e= Vee { (Br — B) 000 - (7B +B)sin ibs a“ UG a- (7 q 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 g equals zero, the functions D and 7 in equations (74) and » ; (75) become constant, and these equations so assume the form | jae DillCer-» + Cyne] 3 — [Cyettt+ 4 Cente} : and (86) | | e= Vee {[Betet-* 4 Ben#t-a) |

• [Bye t? t+ + Be~*** }}, | where | B="@'-¢. (87) | q i} This gives expressions of current and e.m.f. which are no . longer oscillatory but exponential, thus representing a gradual change of 7 and e as functions of time and distance, corresponding to the gradual or logarithmic condenser discharge. For 4 = 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, l/r g m=3(7-¢)% 8) that is, r = g ’ . L7G (89) or r+g=L+C, 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 ___ of the electrostatic field of the circuit, 4, then u= : = 6 = time constant of the circuit, (90) and from equation (54) . Rk; =9+ 9, h =VICs = os, (91) k = VICq = «4, and from equation (52) L vz ¢°= - = C =C, c’ = 0, (92) o ale vz ; . 2=-=VrA-S and ° c . c,/ = 0; ; | | |

DISCUSSION OF GENERAL EQUATIONS 441 hence, substituting in equations (50) and (51), dae EB fete-9 Di [gt — D) - Dy ig b+ 4]

• 9 Dy lg (t — A] — 8 Dlg e+ DY} (98) and e=- Veet fet D [g(t -D] + Dy g t+ D]
• 6" Dlg @—2)] +e -°* Dy [gt + A}. (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 g and the other quantities ‘“distortionless circuit.”

These equations (93) and (94) therefore also apply for g = 0, and then assume the form inet! {[Cyet8¢-Y 4 Cy MJ —[Cye HON 4 Ce} (95)

L -¥! +8 (t—a) —a(t—a) e=- Vase {[Cy + Cg 7]

• [Cyetee*” + Cet}. (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.

CHAPTER III. STANDING WAVES. 14. If the propagation constant of the wave vanishes, h=0, . the wave becomes a stationary or standing wave, and the equa- tions of the standing wave are thus derived from the general . equations (50) to (61), by substituting therein h = 0, which gives | R2 = VE — Leny; (7) hence, if > LCm’, RZ = — LCn’; and if P< Cm’, RZ? = LCm? — P. Therefore, two different cases exist, depending upon the rela- tive values of k? and LC'm’, and in addition thereto the inter- mediary or critical case, in which ? = LCm’. These three cases require separate consideration. 1 [yr ar {3 ye vat 2 = _ —_—o = — — — — Lom wo} {(F Opn taVe-Vey is a circuit constant, while & is the wave length constant, that is, the higher k the shorter the wave length. A. Short waves, RB > LCm’, (99) hence, RZ =P - Wm (100) | and s=0, | q= Ven - (ot)

STANDING WAVES ; 443

or approximately, for very large k,

• • . 102

Herefrom then follows

a 1 k oT Gy mL “ me ° (103) Cc, = qh c 3 k — mL

and c,/= = c’.

Substituting now h = 0 and (101), (103) in equations (50), : (51), the two waves 1’, e’ and 2”, e” coincide, and all the expo- nential terms reduce to e«~“; hence, substituting

. B, = C, + Cs, B,=C,+C, 2 2 4 (104) BY = CY + Cy, and B/ =Cj/+C, . gives i = e—™ {[B, cos (gt — kl) + B,’ sin (gt — kl)] . — [B, cos (qt + kl) + B,’ sin (gt + kl)}} (105) and L .

é “5 et {[(mB,’ - qB,) cos (qt — kl) _ (mB, +9B,’) sin (qt —kl)]

• [(mB,’ —qB,) cos (qt + kl) — (mB, + qB,’) sin (gt+kl))}. (106) ° Equations (105) and (106) represent a stationary electrical oscil-

lation or standing wave on the circuit.

B. Long waves, .

kB? < LCm’; (107)

I 444 TRANSIENT PHENOMENA hence, R2 = LCm - B, (108) and $= Vn - # Le (109), q= 0, . ! or approximately, for very small values of k, l/r _ 9). | s=m= 3(F - ar (110) herefrom then follows c,=c, = 0, (m+ s)L a | and (111) (m — s) L cf = | Substituting now A = 0 and (109), (111) into (50) and (51), the | two waves 7’, e’ and 2”, e” remain separate, having different expo- | nential terms, «~ ““** and «“®, but in each of the two waves | the main wave and the reflected wave coincide, due to the vanishing of q. Substituting then B,=C,-C, B/ =C/+Cy, 1 1 2 (112) B,=(,-Cy and B/=C/+C, | gives | i =e“ {(B, e+” + Bye~") cos kl—(B,'e** + B,! e~) sin kl} (113) |

STANDING WAVES 445 | and ‘L —ul 4 tat B 7 Wat kl e=ze {[(m + s) B/ et" + (m — 8) B,/ e~"] cos

• [(m + 8) B,et" + (m — 8) B,e~*) sin Al} —lZe-m {m[(By et + Bye") cos Hd
• (B, et" + B, e~®) sin kl] (114)
• s[(B/et* —B, e~*) cos kl
• (B,e*" — Bye ~®) sin ki)}.

Equations (113) and (114) represent a gradual or exponential circuit discharge, and the distribution still is a trigonometric funetion of the distance, that is, a wave distribution, but dies out

gradually with the time, without oscillation.

C. Critical case,

= LCm’; (115) hence, R; = 0, s=0, (116) q= 0, and c¢,=c, = 0, mL \ /L (117) ¢! = ¢/ == ono and all the main waves and their reflected waves coincide when substituting h = 0, (116), (117) in (50) and (51). Hence, writing , | : B=C, -C, +€,-C 7} / and (118) | ‘B’ = CY + Cy + Cy + Cf | gives i= e~“ {Bos kM — B’ sin kl} (119)

446 TRANSIENT PHENOMENA and Liisa . e=VGe {B’ cos kl + Bsin kl}. (120) In the critical case, (119) and (120), the wave is distributed as a trigonometric function of the distance, but dies out as a simple exponential function of the time. 15. An electrical standing wave thus can have two different forms: it can be either oscillatory in time or exponential in time, , that is, gradually changing. It is interesting to investigate the conditions under which these two different cases occur. ‘ The transition from gradual to oscillatory takes place at Y k, = mLC; (121) for larger values of & the phenomenon is oscillatory; for smaller, exponential or gradual. .

Since & is the wave length constant, the wave length, at which the phenomenon ceases to be oscillatory in time and becomes a gradual dying out, is given by (63) as

ley = = ‘ 122 | bs (122) mVIC In an undamped wave, that is, in a circuit of zero r and zero g, in which no energy losses occur, the speed of propagation is q 1 S = 7 ST k VIC (123) and if the medium has unit permeability and unit inductivity, it is the speed of light, S, = 3 X 10”. (124) In an undamped circuit, this wave length J, would correspond to the frequency, _ fo Lm: qo Ny /TC ?

STANDING WAVES 447 | hence, from (62), a _™, fi oo” 2 x 2 n (125) The frequency at the wave length J, is zero, since at this wave length the phenomenon ceases to be oscillatory ; that is, due to the energy losses in the circuit, by the effective resistance r and effective conductance g, the frequency f of the wave is reduced below the value corresponding to the wave length l,,, the more, the greater the wave length, until at the wave length 1,, the frequency becomes zero and the phenomenon thereby non-oscillatory. This means that with increasing wave length the velocity of propagation of the phenomenon decreases, and becomes zero at wave length /,,. If mLC =0, k, = Oandl,, = ; that is, the standing wave is always oscillatory. If mLC = », .

: k, = ~ andl, = 0; . that is, the standing wave is always non-oscillatory, or gradually ; dying out.

In the former case, m?LC = 0, or oscillatory phenomenon, substituting for m?, we have | l/r g¥ 4 ( ~ a) = 0, C yz rVr-0Vg-0 and rb. | g C’ . or rC — gL = 0 (distortionless circuit). In the latter case, m?LC = o, or non-oscillatory or exponen- | tial standing wave, we have | re qVE- « L Ive" |

448 TRANSIENT PHENOMENA and since neither r, g, L, nor C can be equal infinity it fol- lows that either L = OorC = 0. |

Therefore, the standing wave in a circuit is always oscillatory, regardless of its wave length, if

rC — gL =0, (126) or r OL -=—'° 2 7G (127) that is, the ratio of the energy coefficients is equal to the ratio | of the reactive coefficients of the circuit.

The standing wave can never be oscillatory, but is always . exponential, or gradually dying out, if either the inductance L or the capacity @ vanishes; that is, the circuit contains no capacity l or contains no inductance. In all other cases the standing wave is oscillatory for waves shorter than the critical value 1, = =, |

, . where 1 ye L)’ 2 2 = _ — — ky = mLC= 5 L Vol’ (128) and is exponential or gradual for standing waves longer than the critical wave length l,,: or for k < k, the standing wave is exponential, for & > k, it is oscillatory. |

The term k, = m VLC thus takes a similar part in the theory of standing waves as the term r,> — 4L,C, in the condenser discharge through an inductive circuit; that is, it separates the exponential or gradual from trigonometric or oscillatory conditions.

The difference is that the condenser discharge through an inductive circuit is gradual, or oscillatory, depending on the circuit constants, while in a general circuit, with the same circuit constants, usually gradual as well as oscillatory standing waves exist, the former with greater wave length, or

nVIC > k, (129) the latter with shorter wave length, or mVIC <k. (130) |

STANDING WAVES 449

An idea of the quantity ,, and therewith the wave length 1, at which the frequency of the standing wave becomes zero, or the wave non-oscillatory, and of the frequency f,, which, in an -undamped circuit, will correspond to this critical wave length 1,,, can best be derived by considering some representative numerical examples.

As such may be considered:

(1) A high-power high-potential overhead transmission line.

(2) A high-potential underground power cable.

(3) A submarine telegraph cable.

(4) A long-distance overhead telephone circuit.

(1) High-power high-potential overhead transmission line.

16. Assume energy to be transmitted 120 miles, at 40,000 volts between line and ground, by a three-phase system with grounded neutral. The line consists of copper conductors, wire No. 00 B. and S. gage, with 5 feet between conductors.

Choosing the mile as unit length,

r = 0.41 ohm per mile. The inductance of a conductor is given by L= 1(2 log, # + ‘) 10~*, in henrys, (131) where J = the length of conductor, in cm.; J, = the radius of conductor; 7; = the distance from return conductor, and p» = the permeability of conductor material. For copper, # = 1.

As one mile equals 1.61 X 105 cm., substituting this, and reducing the natural logarithm to the common logarithm, by the factor 2.3026, gives

L = (0.7415 log # + 0.0805), in mh. per mile. (132) For l, = 0.1825 inch and 1, = 60 inches, L = 1.95 mh. per mile.

The capacity of a conductor is given by

, C=l ate 10°, in farads, (133) |

2S, log,

450 TRANSIENT PHENOMENA | where S, = 3 X 10° = the speed of light, and 3 = the allow- | ance for capacity of insulation, tie wires, supports, etc., assumed as 5 per cent. Substituting S,, and reducing to one mile and common loga- rithm, gives C= one in mf.; (134) ba logy . hence, in this instance, C = 0.0162 mf. ! Estimating the loss in the static field of the line as 400 watts per mile of conductor gives an effective conductance, . 400 - ! g = 70,000 ~ 2-5 x 10 mho, which gives the line constants per mile asr = 0.41 ohm; L = 1.95xX10~ henry; g = 0.25 X 10 mho, and C = 0.0162 x 10* farad. Herefrom then follows , lyr g\ 1 u= a(z+ $)- 5 (210 + 15.5) = 113, | m = 3(z - g) = 97 | 2 ch? o = VIC = V316 X 10* = 5.62 x 107, k, = mV IC = 545 X 10°; ‘hence, the critical wave length is Lae = 2 = 11,500 miles, > 0 k, and in an undamped circuit this wave length would correspond to the frequency of oscillation, . m to= ae 15.7 cycles per sec.

| STANDING WAVES 451 | Since the shortest wave at which the phenomenon ceases to be oscillatory is 11,500 miles in length, and the longest wave which | can originate in the circuit is four times the length of the circuit, or 480 miles, it follows that whatever waves may originate in this | circuit are by necessity oscillatory, and non-oscillatory currents or voltages can exist in this circuit only when impressed upon it by some outside source, and then are of such great wave length that the circuit is only an insignificant fraction of the wave, and great differences of voltage and current of non-oscillatory nature cannot exist. Since the difference in length between the shortest non- oscillatory wave and the longest wave which can originate in the . circuit is so very great, it follows that in high-potential long- | distance transmission circuits all phenomena which may result . in considerable potential differences and differences of current throughout the circuit are oscillatory in nature, and the solution case (A) is the one the study of which is of the greatest importance in long-distance transmissions. With a length of circuit of 120 miles, the longest standing wave which can originate in the circuit has the wave length 1, = 480 miles, and herefrom follows . k= ha = 0.0134 | and | RP 0.0134? T6736 x 108 = OX 10: hence, in the expression of g in equation (101), | _ / Ps, q= LC m = V5.7 x 10° — 0.00941 x 108, m? is negligible compared with ss ; that is, k 0.0134 = Tie ~ 562 x 19-«~ 795: O25)

452 TRANSIENT PHENOMENA or = .o = I an 380 cycles per sec.

Hence, even for the longest standing wave which may origi- nate in this transmission line, g = 2380 is such a large quantity compared with m = 97 that m can be neglected compared with q, and for shorter waves, the overtones of the fundamental wave, this is still more the case; that is,in equations (105) and (106) the terms with m may be dropped. In equation (106) A thus become common factors, and since from equation (135)

lq _ Vf, (136) k C by substituting m = 0 and (136) in (105) and (106) we get the general equations of standing waves in long-distance transmission lines, thus: i =e“ {[B, cos (qt — kl) + B,’ sin (qt — kl)] — [B, cos (qt + kl) + B,’ sin (gt + kl)}}, (137) e=-— Vem [B, cos (qt — kl) + B,’ sin (gt — kl))

• [B, cos (qt + kl) + B,’ sin (gt+kl)}}, (138) or e =e “{[A, cos (gt + kl) + A,’ sin (gt + K)]
• [A, cos (gt — kl) + A,’ sin (gt — kl)]}, (139) i VS eH 4, 008 (qt + kl) + Aj’ sin (qt + K)] — [A, cos (gt — kl) + A,’ sin (qt — kl)]}, (140) where A,=- Vez, A/ =- VaBy, ete.

(2) High-potential underground power cable.

17. Choose as example an underground power cable of 20 miles length, transmitting energy at 7000 volts between con-

e

.

STANDING WAVES 453 ductor and ground or cable armor, that is, a three-phase three- conductor 12,000-volt cable.

Assume the conductor as stranded and of a section equiva- lent to No. 00 B. and S. G.

Calculating the constants in the same manner, except that the expression for the capacity, equation (119), multiplies with the dielectric constant or specific inductive capacity of the cable insulation, and that is very small, about three or less; or taking the values of the circuit constants from tests of the cable,

we get values of the magnitude, per mile of single conductor r= 0.41 ohm; L = 0.4 X 10° henry; g = 10-* mho, corre- sponding to a power factor of the cable-charging current, at 25 cycles, of 1 per cent; C = .6 X 10-° farad.

Herefrom the following values are obtained: u = 513,m.= 512, ¢ = VIC = 15.5 X 107°, k, = mV LC = 7.95 X 107°, and the

. critical wave length is J, = 790 miles, and the frequency of an | undamped oscillation, corresponding to l,,, is f, = 81.5 cycles per second. As seen, in an underground high-potential cable the critical wave length is very much shorter than in the overhead long- distance transmission line. At the same time, however, the length of an underground cable circuit is very much shorter than that of a long-distance transmission line, so that the critical wave length still is very large compared with the greatest wave length of an oscillation originating in the cable, at least ten times as | great. Which means that the discussion of the possible phe- | nomena in any overhead line, under (1), applies also to the under-

ground high-potential cable circuit. | In the present example the longest standing wave which may

originate in the cable has the wave length . L, = 80 miles,

which gives

7 k = 0.0785 and k Vit 5070,

454 TRANSIENT PHENOMENA or about ten times as large as m, so that m can still be neglected in equation (87), and we have k = — = 507 1 ge ~ 0M i or J = 810 cycles per second, and the general equations of the phenomenon in long-distance transmission lines, (123) to (125), also apply as the general equa- tions of standing waves in high-potential underground cable circuits.

(3) Submarine telegraph cable.

18. Choosing the following values: length of cable, single stranded-conductor, ground return, = 4000 miles; constants per mile of conductor: r = 3 ohms, L = 107 henry, g = 10-* mho, and C = 0.1 < 10~* farad, we get u=1500;m = 1500;0 = VIC = 10 x 107, and k, = m VLC = 15 X 107, from which the critical wave length is /,, = 418 miles, and the corresponding frequency fo = 239 cycles per second.