Theory and Calculation of Transient Electric Phenomena and Oscillations — part 18 of 20

(A, + A,) cos kl, + (A, — A’) sinkl, =0 and . (221) — (A, — A,) sin kl, + (A, + A,’) cos kl, = 0. Eliminating sin kl, and cos kl, from these two equations gives (A? — AZ) + (A,? — A,”) = 0, or (222) A? + A,? = A? + A,’,

• 486 TRANSIENT PHENOMENA as the condition which must be fulfilled between the integration constants. . The value /, then follows from (221) as . A/+A/_ A, +4, | tan ly = a ASSAY (223) At a point J, of the circuit at which 7 = 0 the coefficients of . cos gt and sin gt in equation (140) must vanish. This gives, in the same manner as above, . (A? — Aj) + (A,? — A,”) = 0, that is, the same conditions as (221), and gives for /, the value _A/- Ay | A,—-A, tan Hl, = TA, AS HA, (224) From (223) and (224) it follows that . 1 tan kl, = tan ki, > (225) That is, the angles Mi, and kd, differ by one quarter-wave length or an odd multiple thereof. . Herefrom it then follows that if the integration constants of a standing wave fulfill the’ conditions A? + A,? =A? + A,? = B, (226) the circuit of this wave contains points /,, distant from each other by a half-wave length, at which e = 0, and points /,, distant from each other by a half-wave length, at which 7 = 0, and the points I, are intermediate between the points /,, that is, distant there- from by one quarter-wave length. Any section of the circuit, from a point J, or J, to any other point J, or J,, then is a freely oscillating circuit. In the free oscillation of the circuit the circuit is bounded by one point J, and one point /,; that is, the e.m.f. is zero at one end and the current zero at the other end of the circuit, case (1) or (2) of equation (197), and the circuit is then a quarter-wave or an odd multiple thereof, or the circuit is bounded by two points 1, or by two points J,, and then the voltage is zero at both ends of the circuit in the former case, number (3) in equation (197), or

FREE OSCILLATIONS 487 the current is zero at both ends of the circuit in the latter case, number (4) in equation (197), and in either case the circuit is one half-wave or a multiple thereof. Choosing one of the points /, or J, as starting point of the dis- tance, that is, substituting / — 1, or / — 1, respectively, instead of 7, in the equations (139) and (140), with some transformation these equations convert into the equations (219) or (220). In other words, the equation (226), as relation between the integra- tion constants of a standing wave, is the necessary and sufficient ' condition that this standing wave be a free oscillation. 34. A single term of a free oscillation of a circuit, with the dis- tance counted from one end of the circuit, that is, one point of zero power, thus is represented by equations (219) or (220), respectively. Reversing the sign of J, that is, counting the distance in the opposite direction, and substituting B = + 2 Ay. a these equations assume a more convenient form, thus: for : e=Oatl =0, e = Be“ sin kl sin (qt — y) and 227 i = BY e-+ cos cos (at - v) (227) and for t = Oatl =0, e = Be~“ cos kl cos (gt — y) nd np Costin tr (28) 7=B ze sin kl sin (gt — y). Introducing again the velocity of propagation as unit distance, A=al, — 229 from equation (66) and (229) we get: | kl = AVE Eo | m? |

• ayi+(*); q q | |

488 TRANSIENT PHENOMENA hence, if m is small compared with q, kl = gA, (230) and substituting (229) in (230) gives | k= og =qvlt, (231) and from (210) and (211), for a quarter-wave oscillation, we have _@nt+1)z k= "3 l, 930 and Qnt)e (232) 1 o1Vie” for a half-wave oscillation, . k = + ° (233)

• 7. . 1" | VIC Denoting the length of the circuit in a quarter-wave oscillation by 4, = al,, (234) and the length of the circuit in a half-wave oscillation by 4, = al,, (235) , the wave length of the fundamental or lowest frequency of oscillation is 4, = 44, =24: (236) or the length of the fundamental wave, with the velocity of prop- agation as distance unit, in a quarter-wave oscillation is 4,=41, VIL, and in a half-wave oscillation is (237) 4, = 21, VIC.

FREE OSCILLATIONS 489 Substituting (237) into (232) and (233) for a quarter-wave oscillation gives k = (2n +1) 2nVie and — (238) qg = (2n+4+ 1) —*, 4 and for a half-wave oscillation gives 2xVIC k= 1—— and ° (239) q A ° Writing now . 6= an t, | 4o 27x 2x7VLC (240) t=—A = ——— ], Ay Ay that is, representing a complete cycle of the fundamental fre- . quency, or complete wave in time, by 0 = 27, and a complete wave in space by t = 272, from (239) and (240) we have kl = nt (241) and gt = n8, | where n may be any integer number with a half-wave oscillation, | but only an odd number with a quarter-wave oscillation. : _ 35. Substituting (241) into (227) and (228) gives as the complete expression of a free oscillation the following equation A. Quarter-wave oscillation. (a) e=Oatl =O (ort = 0) e =e“ SinB, sin (2n + 1) rsin[(2n + 1) 6 — 7] , 0 and C . (242) t= Vo. Zr Bn cos (2n+ 1) rt cos[(2n+1)6 —7,);

490 TRANSIENT PHENOMENA (b) t = Oatl = O (ort = 0) e =e“ Sn B, cos (2n + 1) reos[(2n + 1) 6— 74] and _ ° (243) t= Vo.-u SB sin (2+ 1)rsin((27+ 1)0—z,]. L 0 ” | B. Half-wave oscillation. ! (a) e = Oatl = 0 (ort = 0) | e=e > Bn sin nz sin (nO — 7,) | and - (244) t= Vee 3." Bn cos nt cos (nO — 7a); . (6) 7 = Oatl = 0 (ort = 0) e=e“ > 3B: cos nz cos (nO — 7p) and oe (245) t= yoo Br sin nz sin (n6é — 7,), where 9 = 2%, Xs 240 22VIC (240) t= —— 1, oa A = 41, VLC ina quarter-wave (237) = 21, VIC in a half-wave oscillation, and eM =e 7 ae. (246) A, is the wave length, and thus + the frequency, of the funda- mental wave, with the velocity of propagation as distance unit. It is interesting to note that the time decrement of the free oscillation, «~“, is the same for all frequencies and wave lengths,

. FREE OSCILLATIONS 491 and that the relative intensity of the different harmonic compo- nents of the oscillation, and thereby the wave shape of the oscillation, remains unchanged during the decay of the oscillation. This result, analogous to that found in the chapter on traveling waves, obviously is based on the assumption that the constants of the circuit do not change with the frequency. This, however, is not perfectly true. At very high frequencies r increases, due to unequal current distribution in the conductor, as discussed in Section III, LZ slightly decreases hereby, g increases by the energy losses resulting from brush discharges and from electro- static radiation, etc., so that, in general, at very high frequency an increase of zand a and therewith of u, may be expected; that is, very high harmonics would die out with greater rapidity, which would result in smoothing out the wave shape with increas- ing decay, making it more nearly approach the fundamental and its lower harmonics. 36. The equations of a free oscillation of a circuit, as quarter- wave or half-wave, (242) to (245), still contain the pairs of inte- gration constants B, and 7p, representing, respectively, the intensity and the phase of the nth harmonic. These pairs of integration constants are determined by the ter- minal conditions of time; that is, they depend upon the amount and the distribution of the stored energy of the circuit at the starting moment of the oscillation, or, in other words, on the distribution of current and e.m.f. at t = 0. The e.m.f., e,, and the current, 7,, at time ¢ = 0, can be ex- pressed as an infinite series of trigonometric functions of the distance /; that is, the distance angle t, or a Fourier series of such character as also to fulfill the terminal conditions in space, as dis- cussed above, that is, e = 0, and 7 = 0, respectively, at the ends of the circuit. The voltage and current distribution in the circuit, at the starting moment of the oscillation, t = 0, or, 6 = 0, can be represented by the Fourier series, thus: . € = Sin (an cos nt + dy’ sin nz)

and ° (247) 1 = Sin (by cos nt + b,/ sin nt), . Vy py (b, cos )

492 FRANSIENT PHENOMENA . where 1 f’" oF a= f ert = avg [e,],’", | a, = * f e€, cos nt dt=2 avg [e, cos nt], ", $ (248) a,/ = * f sein nede=2avelessinnee| e and analogously for b. The expression avg [F—: denotes the average value of the function F between the limits a, and a,. Since these integrals extend over the complete wave 2 z, the wave thus has to be extended by utilizing the terminal conditions regarding t, but the wave is symmetrical with regard to / = 0 and with regard tol = J,, and this feature in the case of a quarter- wave oscillation excludes the existence of odd values of n in equations (247) and (248). 37. Substituting in equations (242) to (245), t=0, 9=0, and then equating with (247), gives, from (242), é) = Dn Basin (2n + 1) rsiny, = S'a[a, 00s (2 + 1)t 0 0 |

• a,’ sin (2n + 1) t} | and . | ye C) B ( ) i) | % =Vz >- B, cos (2n + 1) rcosy, = > n[b, cos (2n + 1)t “Vid To = Zr bbq cos ( )
• 0,’ sin (2 + 1) 7); . . hence, a, = 0, b,,” = 0, | . , C B,siny, = 4,’ and 7 Bn COS 7, = Dn.

FREE OSCILLATIONS 493 Equation (242) gives the constants an = 0; bn’ = 0, L . = R —f.2 Bu=V an" + Gon, (249) tan = Gn’ Vz. mo, VO’. in the same manner equation (243) gives the constants an’ =0; bn = 0, L . _ 2,“/pn . Ba = Vo, + Gon , . (250) b,’ ye tan fn = a, L Equation (244) gives the same values as (242), and (245) the | same values as (243). . | Examples. | . $8. ‘As first example may be considered the discharge of a transmission line: A circuit of length J, is charged to a uniform voltage E, while there is no current in the circuit. This circuit then is grounded at one end, while the other end remains ] insulated. i Let the distance be counted from the grounded end, and the time from the moment of grounding, and introducing the deno- tations (235). The terminal conditions then are: . (a) -=0 e=0, | t= 5 +=0. (b) at@ =0 e =O for « = 0; e = E for t# 0, 4 =0 for ct #0; 7 = indefinite for cr = 0.

494 TRANSIENT PHENOMENA The distribution of e.m.f.,e,, and current, z,,in the circuit, at the starting moment @ = 0, can be expressed by the Fourier series (247), and from (248), ° ' 4E ‘— i = ——__ a,’ = 2avg[E sin (2+ 1)r] @ntbs (251) and ba = 0, and from (249), 4E ; Ba = Gay Ty tnd tan re = @; hence, n- (252) : fa = 3 , and substituting (252) into (242), , _4E yws sin (2n + 1) rc0s (2n + 1)0 era >a — ontl and (253) ; 18 /C «5, cos (2m + 1) rsin (2n + 1)8 oar YD 7 2n+1 From (240) it follows that . = 24,VIC 6. | us 6 = 27 gives the period, = 4 l VIL; : and the frequency, : f- 1

• 41,VIC and t = 2 gives the wave length, ; lL, =41,, of the fundamental wave, or oscillation of lowest frequency and greatest wave length. | |

FREE OSCILLATIONS 495 Choosing the same line constants as in paragraph 16, namely: 1, = 120 miles; r = 0.41 ohm per mile; L = 1.95x10~ henry per mile; g = .25 x 10~* mho per mile, and C = 0.0162 x 10* farad per mile, we have u = 113, ul = 2 VIE yp = 0.0485 8, and the fundamental frequency of oscillation is J, = 371 cycles per second. If now the e.m.f. to which the line is charged is E = 40,000 volts, substituting these values in equations (253) gives : e = 51,000 e-°™** {sin rt cos 0 + $sin 3 + cos 30

• $sin5 7c0s50+...}, in volts and (254) t = 147 <-°™*? {cos rsin 6 + $cos3 rsin3 0
• $cos5rsin56+...}, in amp. : The maximum value of e is e = E = 40,000 volts, and the maximum current of 7 is a= ] = 115.5 amp. Since Yea eee Oa Gaobs oop Pan’ bo, ifb—F <axb, or b+ 5 <a<b+z, and (255) =+5) if b<a<b+> =—5,ifbe<a<b-5, | |

496 TRANSIENT PHENOMENA applying (255) to (254) we have at any point r of the line, at the time @ given by | 0<O<t: e=EKe“; i1=0. | r<O<rts: e=0; i= ke-*, r+ 5<O0<tte: e=-Ee“; 1=0. rte<OcrtS: e=0; t=— Ie“, 3x _ . tH SO <rt 2x; e=Ee“; i4=0, ete. At any moment of time @ one part of the line has voltage e = Ke~“ and zero current, and the other part of the line has current? = Je~“ and zero voltage, and the dividing line between the two sections of the line is at + = 0 +5 hence moves along the line at the rate z = 6. 89. As second example may be considered the discharge of a live line into a dead line: A circuit of length /,, charged to a uniform voltage /, but carrying no current, is connected to a circuit of the same constants, but of length /,, and having neither voltage nor current, otherwise both circuits are insulated. Let the total length of the circuit be denoted by | 2l=1, +1, | and let the time be counted from the moment where the circuits . l, and J, are connected together, the distance from the beginning of the live circuit 7,, whose other end is connected to the dead circuit J,. Introduce again the denotations (240), and represent the total length of the line 27 = 1, + l, by t = z, then write ~ 4, | TTT +h" As the voltage is E from t = 0 to t = ¢,, and 0 from + = 1, to t = 7, the mean value of voltage, or the voltage which will be left on the line after the transient phenomenon has passed, is Tt e, = aE,

| . - ee FREE OSCILLATIONS 497 | and the terminal conditions of voltage and current are | 6=0 ; e=E-e, for0<t< 14, e=-e fort,<t<z, 17 =0. | Proceeding then in the same manner as in paragraph 34, in the present case the equations (245) and (248) apply, and 1 oe Gn = 2 {fre — @,) cos no d0— J c, cos n6 do _ 2 Esin nz, -——, a,’ = b, = b,’ = 0; hence, - tn = 0 and | 2E§t, _y wa inne, | en hare Yr tee ne cos nO, | 2E /C i (258) | 2k a sin nt . i=— Vs. > — sin ne sin 06. | Choosing the same line constants as in paragraph 35, and | assuming | 1, = 120 miles and 1,= 80 miles, |

• we have 1 = 100 miles and 7, = 0.6 z. Let E = 40,000 volts, ut = 0.0404 0, and the fundamental frequency of oscillation, f,,= 445 cycles per second; then e = 24,000 + 25,500 «~°™™#{ sin 108° cos z cos 0+ 4 sin 216° cos 2 t cos 2 +4 sin 348° cos 3 t cos 3 0+: --} volts and (257) 1=73.5 e— {sin 108° sin r sin 9 + 4 sin 216° sin 2 t sin 20+ } sin 348° sin 3 rsin 3 0 +---} amp.

‘

CHAPTER VI.

TRANSITION POINTS AND THE COMPLEX CIRCUIT. |

40. The discussions of standing waves and free oscillations in . Chapters III and V, and traveling waves in Chapter IV, apply directly only to simple circuits, that is, circuits comprising a con- ductor of uniformly distributed constants 7, L,g,andC. Indus- trial electric circuits, however, never are simple circuits, but are always complex circuits comprising sections of different con- stants, — generator, transformer, transmission lines, and load, — and a simple circuit is realized only by a section of a circuit, as a transmission hne or a high-potential transformer coil, which is cut off at both ends from the rest of the circuit, either by open- circuiting, 1 = 0, or by short-circuiting, e = 0. Approximately, the simple circuit is realized by a section of a complex circuit, connecting to other sections of-very different constants, so that the ends of the circuit can, approximately, be considered as reflection points. For instance, an underground cable of low L and high C, when connected to a large reactive coil of high L and low C, may, approximately, at its ends be considered as having reflection points 1 = 0. A high-potential transformer coil of high Z and low C, when connected to a cable of low L and high C, may at its ends be considered as having reflection points e = 0. In other words, in the first case the reactive coil may be considered as stopping the current, in the latter case the cable considered as short-circuiting the transformer. This approximation, however, while frequently relied upon in engi- neering practice, and often permissible for the circuit section in which the transient phenomenon originates, is not permissible in considering the effect of the phenomenon on the adjacent sections of the circuit. For instance, in the first case above mentioned, a transient phenomenon in an underground cable connected to a high reactance, the current and e.m.f. in the cable may approx- imately be represented by considering the reactive coil as a reflection point, that is, an open circuit, since only a small current

498

TRANSITION POINTS AND THE COMPLEX CIRCUIT 499 exists in the reactive coil. Such a small current in the reactive coil may, however, give a very high and destructive voltage in the reactive coil, due to its high L, and thus in the circuit beyond the reactive coil. In the investigation of the effect of a transient phenomenon originating in one section of a complex circuit, as an oscillating arc on an underground cable, on other sections of the circuit, as the generating station, even a very great change of circuit constants cannot be considered as a reflection point. Since this is the most important case met in industrial practice, as disturbances originating in one section of acomplex circuit usually develop their destructive effects in other sections of the circuit, the investigation of the general problem of a com- plex circuit comprising sections of different constants thus becomes’ necessary. This requires the investigation of the changes occurring in an electric wave, and its equations, when passing over a transition point from one circuit or section of a circuit into another section of different constants.

41. The equations (50) to (57), while most general, are less convenient for studying the transition of a wave from one circuit to another circuit of different constants, and since in industrial high-voltage circuits, at least for waves originating in the circuits, q and k are very large compared with s and h, as discussed in . paragraph 16, s and h may be neglected compared with q and k. This gives, as discussed in paragraph 9,

h = as, k = «9, | a= - ene, (258) m, /L c, = c,/ ~ yi. 0, where _ c= VIC, (259) and substituting A=oal, (260) that is, kl = qA, } (261) hl = 8A,

500 TRANSIENT PHENOMENA gives toe“ {e~24-O.1C, cosg (A — t) + C,/sing (A — 4)

• et#44+0(C, cosg (A +t) + Cy sing (A+ t)] :

• et#4-9(C, cosg (A — 1) + C,’ sing (A — 8) — 84+ cong (A+ O+C,' sing (A + O]} (262) and L —-wf.,-8& qt ‘s o = Verma tt (0, c08 9 (4-0 + Cy’ sing (A — 0)
• et#449(C cosg (Att) + Cy sing (A + 0] . 4 et#4-91C, cosg (A — 2) + Cy sing (A — 8)
• e-A+9 (C, cos g (A + t) + C,’ sing (A + 8))}- (263) Substituting now C24+C,2 =A’, . C24+C,? = B, C24+C2% =C, 4) C2407 =D, | Cr = tan a | C, ) Gn tan, . . c! (265) C, = tany, Gh = tan, gives i= en {Ae 84-0 cos [g (A—2) —a]—Bet 4+ eos [gq (A+t)-8]
• Cet#4- eos [g (A—t) —7]—De74° cos [g (A+#) -2]] (266) |

I

. TRANSITION POINTS AND THE COMPLEX CIRCUIT 501 and e= Veerst{deA- cos (A—t) —a]+ Bet? @+%eos[q(d+t) — 8] +Ce#9-9 cos [q(A—t) —7]+ De-* 4+” cos [q(A+t)—2]}, (267) 42. In these equations (266) and (267) A is the distance coérdinate, using the velocity of propagation as unit distance, and at a transition point from one circuit to another, where the circuit constants change, the velocity of propagation also changes, and thus, for the same time constants s and q, h and k also change, and therewith kl, but transformed to the distance variable 4, gd remains the same; that is, by introducing the distance variable , the distance can be measured throughout the entire circuit, and across transition points, at which the circuit constants change, and the same equations (266) and (267) apply throughout the entire circuit. In this case, however, in any section of the circuit, Aaed 268) = l LL; (

where L,; and C;; are the inductance and the capacity, respect- ively, of the section 7 of the circuit, per unit length, for instance,

per mile. . In a complex circuit the time variable ¢ is the same throughout the entire circuit, or, in other words, the frequency of oscillation, as represented by qg, and the rate of decay of the oscillation, as represented by the exponential function of time, must be the same throughout the entire circuit. Not so, however, with the distance variable /; the wave length of the oscillation and its rate of building up or down along the circuit need not be the same, and usually are not, but in some sections of the circuit the wave length may be far shorter, as in coiled circuits as transformers, due to the higher Z, or in cables, due to the higher C. To extend the same equations over the entire complex circuit, it therefore becomes necessary to substitute for the distance variable / another distance variable A of such character that the wave length has the same value in all sections of the complex circuit. As the wave length of the section 7 is » this is done by changing the unit distance by the factor a; = VLC, The distance unit of

502 TRANSIENT PHENOMENA

the new distance variable 4 then is the distance traversed by the wave in unit time, hence different in linear measure for the different sections of the circuit, but offers the advantage of carrying the distance measurements across the entire circuit and over transition points by the same distance variable 2.

This means that the length /; of any section 7 of the complex circuit is expressed by the length 4, = ¢,J,.

The introduction of the distance variable A also has the advan- tage that in the determination of the constants r, L, g, C of the different sections of the circuit different linear distance measure- ments / may be used. For instance, in the transmission line,

. the constants may be given per mile, that is, the mile used as unit length, while in the high-potential coil of a transformer the turn, or the coil, or the total transformer may be used as unit of length J, so that the actual linear length of conductor may be unknown. For instance, choosing the total length of conductor in the high-potential transformer as unit length, then the length of the transformer winding in the velocity measure A is 4, = VLC. where L, = total inductance, C, = total capacity of transformer.

The introduction of the distance variable 4 thus permits the representation in the circuit of apparatus as reactive coils, etc., in which one of the constants is very small compared with the other and therefore is usually neglected and the apparatus . considered as “massed inductance,” etc., and allows the investi- gation of the effect of the distributed capacity of reactive coils

: and similgr matters, by representing the reactive coil as a finite (frequently quite long) section A, of the circuit. 43. Let 4,, 4,, 4,,--- An be a number of transition points at |

• which the circuit constants change and the quantities may be denoted by index 1 in the section from A, to 4,, by index 2 in the section from 4, to 4,, etc.

At A = A, it then must be 7, = 7,, e, = e,; thus substituting . A = A, into equations (246) and (247) gives

eH Ott A eh coslg, (A, —t) — a,]—B,e ***cos[g, (a, +t) —8,)}

te +H et eos(q,(A,—t)— nl —Dye~™ cos [9,(A, +t)—6,}} | ment A e-scos[q,(A,— t)—a,] — Byte cos [q,(2, +t)—8,)}

+e THC et *heos(q, (A,-4) at | —D ~™ cos CAE +4) —8,}}.

(269) |

|

TRANSITION POINTS AND THE COMPLEX CIRCUIT 508 Herefrom it follows that 92 = Us (270) that is, the frequency must be the same throughout the entire circuit as is obvious, and U,t8,=U, + 8. (271) Since u, ¥ u,, only one of the two waves can exist, the A B, or the C D, and since these two waves differ from each other only by the sign of s, by assuming now that s may be either positive or negative we can select one of the two waves, for instance, the second wave, but use A, B, «, 8 as denotations of the integration constants. 44. The equations (266) and (267) now assume the form 4 = e7{ Aet*A-9 oo [g (A — t) — a] — Be~*A+9 eog [q (A + t) — AI} and 7 (272) e= Vee {Aet#A-9 eos [g (A — t) — a]

• Be-4+9 cos [q (A + t)— Bl}, or t= e404 Ae +24 cos ig (A — t) — a] — Be~ cos [q (A + t)— B]} and T, (273) e= Ve eter { Ae**4 cos [g (A — t)— a]
• Be-*4 cos [g (A + t)—8}; or, using equations (262) and (263) instead of (266) and (267), ° the corresponding equations are of the form i = e~ fe-*44+974 cos g (A — t) + Bsing (A — 8] — e~*A+9 IC cos g (A + t) + Dsing (A+ 8))} and (274) e= Vee fe-2A+9(4 cosg (A — t)+Baing (A—2)]
• 6449 (C cos g (A + t)+Dsing (A+d)]},

504 TRANSIENT PHENOMENA or i= 7% + fe +474 cos g (A — t) + Bain g (A—2)] —e~(Ccosg (A + t) + Dsing (A + 8))} and ° (275) e = VE ewer {e"[A cosg (A — t)+Bsing (A—d)]

• e~"4IC cos g (A + t)+Dsing (A+d)}, where s may be positive or negative. From equation (269) it then follows that U, + S, = Uy + S, = Uy + 8, =... = Un + Sn = Uy (276) where u,, U,, Us, etc., Un are the time constants of the individual sections of the complex circuit, ; ( + ‘), and u, may be called the resultant time decrement of the complex circuit.

45. Equation (269), by canceling equal terms on both sides, then assumes the form Aethhs cos [9 (a, — t) _ a,] ~ Byeoths cos [9 (A, + t) - B,) = A,e*** cos [q (4 — 1) — a] — Bye~** cos [q (4, + ) — Bh and, resolved for cos gt and sin qt, this gives the identities A,e*"* cos (gi, — a,) — Bye~** cos (ga, — B,) = A,e*** cos (ga, — a) — B,e~** cos (ga, — 8), | A,et® sin gi, — a,) + Bye7"* sin (ga, — B,) = At sin (ga, — @,) + Bye7'™ sin (ga, — 8,). (277) | These identities resulted by equating 7, = 7, from equation | (272). Inthe same manner, by equating e, and e, from equation | (272) there result the two further identities | | L . vi {A,e+4 cos (ga, — a,) + Bye~* cos (ga, — ,)} = 1 VE: +d —agd | at {At cos (G2, — a3) + By cos (94, ~ 8,)}, i |

TRANSITION POINTS AND THE COMPLEX CIRCUIT 505 vi {Ata sin (ga, — a,) — Bye7"* sin (ga, — 8,) = 1 L . . V2 {A,e4 sin (qd, — a) — Bye~ sin (gd, — 8,)}. (278) : 2 Equations (277) and (278) determine the constants of any section of the circuit, A,, B,, a,, 8,, from the constants of the next section of the circuit, A,, B,, a,, B,. Let Ae ** cos (ga, — a) = A’; Ae** sin (qa, — a) = A’; ah (279) Be~*" cos (qa, — 8) = B’, Be~** sin (qi, — 8) = B’; L, ¢°= vai (280) 2 C; Then ; . 2c,A,’ = (c, + ¢,) A,’ + (c, —¢,) BY, 2c,B, = (c, + Cy) BY + (c, _ c,) A,’ (281) 2c,A,” = (c, + ¢,) A,” — (c, — ¢,) B,”, 2c,B,” = (c, + c,) B,” — (ce, — ¢,) Ay’, and since A? + A? = Atet?h ete,, (282) substituting herein (281), 402A, tam —(c, +¢,)? Ate teh +(c,- ¢,)? Bre73h, (283)

• 2(A/B,’ — A,’B,”) (¢,'—c,?)

506 TRANSIENT PHENOMENA and 1- ¢,-& B, —2,A, sin (94, — 8,) c,t+¢, A,‘ sin (qd, —a@,) (qa ) tan (ga, —a,) = —-2-+-____=1 © itan (g,-@ q 1 a,) 1+ C,—¢, By and cos (qa, - 8.) q4, 1 c,te, A, cos (qA,—@,) 1 (7% A, et 2nd sin (gA, —4,) ¢ +c, B, sin (qA,— £,) (a )

• tan (ga, — 8,) =——- 2 +41 © lan (gi, —8,). Ci 1 8.) 14 6103 As, +2mt, 008 (41-44) qJ 1 B, c,+c, B, cos (94, — ,) (284) In the same manner, equating, for 4 = A,, in equations (275) the current 7,, corresponding to the section from 4, to 4,, with the current 7,, corresponding to the section from A, to 4,, and also the e.m.fs., e, = e,, gives the constants in equations (275) and (274), of one section, 4, to 4,, expressed by those of the next adjoining section, A, to 4,, as Aaa fae th A +b," (C, cos 2 g4,+D, sin 2 94,)} B,=e~{a,e+% BL +b (C, sin 2 ga, —D, cos 2 g,)} (85) C,=etfa en th C,+b,e% (A, cos 2 g4,—B, sin 2 qi,)} D,=e**{a,e-% D,+b,e7% (A, sin 2 gd, —B, cos 2 gA,)} . where c, +c, a. = 1 2 Cs ? a (286) b, = 1 Cy , 2c, L = c ' 1 — (287) ve c, =>’. C, | |

TRANSITION POINTS AND THE COMPLEX CIRCUIT 507

46. The general equation of current and e.m.f. in a complex circuit thus also consists of two terms, the main wave A in equations (272), (273), and its reflected wave B. .

The factor «~ “+** =e“ in equations (273) and (275) repre- sents the time decrement, or the decrease of the intensity of the wave with the time, and as such is the same throughout the entire circuit. In an isolated section, of time constant u, the time decrement, from Chapters III and V, is, however, e~“; that is, with the decrement «“ the wave dies out in the isolated sec- tion at the rate at which its stored energy is dissipated by the power lost in resistance and conductance. In a section of the circuit connected to other sections the time decrement « “* does not correspond to the power dissipation in the section; that is, the wave does not die out in each section at the rate as given by the power consumed in this section, or, in other words, power transfer occurs from section to section during the oscillation of a complex circuit.

If s is negative, u, is less than u, and the wave dies out in that

particular section at a lesser rate than corresponds to the power consumed in the section, or, in other words, in this section of the complex circuit more power is consumed by r and g than is sup- plied by the decrease of the stored energy, and this section, therefore, must receive energy from adjoining sections. Inversely, if sis positive, u, > u, and the wave dies out more rapidly in that section than its stored energy is consumed by r and g; that is, a part of the stored energy of this section is transferred to the adjoining sections, and only a part — occasionally a very small part — dissipated in the section, and this section acts as a store of energy for supplying the other sections of the system.

The constant s of the circuit, therefore, may be called energy transfer constant, and positive s means transfer of energy from the section to the rest of the circuit, and negative s means reception of energy from other sections. This explains the vanishing of s in a standing wave of a uniform circuit, due to the absence of energy transfer, and the presence of s in the equations of the traveling wave, due to the transfer of energy along the circuit, and in the general equations of alternating-current circuits. -

It immediately follows herefrom that in a complex circuit some of the s of the different sections must always be positive, some negative.

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508 TRANSIENT PHENOMENA

In addition to the time decrement ¢“ +! =e-“ the waves in equations (273) and (275) also contain the distance decrement e* for the main wave, «“ for the reflected wave. Negative s therefore means a decrease of the main wave for increasing 4, or in the direction of propagation, and a decrease of the reflected wave for decreasing A, that is, also in the direction of propagation; while positive s means increase of main wave as well as reflected wave in the direction of propagation along the circuit. In other words, if s is negative and the section consumes more power than is given by its stored energy, and therefore receives power from the adjoining sections, the electric wave decreases in the direction of its propagation, or builds down, showing the gradual dissipation of the power received from adjoining sections. Inversely, if s is positive and the section thus supplies power to adjoining sections, the electric wave increases in this section in the direction of its propagation, or builds up.

In other words, in a complex circuit, in sections of low power dissipation, the wave increases and transfers power to sections of high power dissipation, in which the wave decreases.

This can still better be seen from equations (272) and (274). Here the time decrement «“ represents the dissipation of stored energy by the power consumed in the section by r and g. The time distance decrement, ¢+*-® for the main wave, ¢°@~® for the reflected wave, represents the decrement of the wave for con-

_ stant (A — 2) or (A + ¢) respectively; that is, shows the change of wave intensity during its propagation. Thus for instance, following a wave crest, the wave decreases for negative s and increases for positive s, in addition to the uniform decrease by the time constant «~“; or, in other words, for positive s the

wave gathers intensity during its progress, for negative s it loses intensity in addition to the loss of intensity by the time con- stant of this particular section of the circuit.

47. Introducing the resultant time decrement u, of the com- plex circuit, the equations of any section, (273) and (275), can also be expressed by the resultant time decrement of the entire complex circuit, u,, and the energy transfer constant of the individual section; thus

s=U,—U, (288)

TRANSITION POINTS AND THE COMPLEX CIRCUIT 509 i =e“#{ Ae +4 cos [gq (A—t) — a] —Be~“ cos [¢(A+t) —5]} enon ae +4 cos[q(A—t) —a]+ Be“ cos[q(d +1) —A}}, or t= enue fete 4 cos q (A — t)+ Bsing (A — 0) — eG cosg (A +t) + Dsing (A + )} and 290 Z (290) . 0 = Vee fet [A cos g (A — 0+ Bsing (4 — 0]

• 4 [C cosg (A + t)+ Dsing (A + t)}}. The constants A, B, C, D are the integration constants, and are such as given by the terminal conditions of the problem, as by the distribution of current and e.m.f. in the circuit at the starting moment, for ¢ = 0, or at one particular point, as 4 = 0.

48. The constants u, and g depend upon the circuit conditions. If the circuit is closed upon itself — as usually is the case with an electrical transmission or distribution circuit — and A is the total length of the closed circuit, the equations must give for 4 = A the same values as for 4 = 0, and therefore g must be a complete cycle or a multiple thereof, 2 nz; that is, 2nz ’ =-—, 291 1-5 (291) and the least value of g, or the fundamental frequency of oscilla- tion, is 22 t= (292) and
    q = 14 (293) If the complex circuit is open at both ends, or grounded at both ~ ends, and thus performs a half-wave oscillation, and A, = total length of the circuit, q=— and = nq, (204) | At . | |

510 TRANSIENT PHENOMENA _ and if the circuit is open at one end, grounded at the other end, thus performing a quarter-wave oscillation, and A, = total length of circuit, it is ra to ~ OK, and q = (2n — 1)q, ( 295) ; while, if the length of the complex circuit is very great compared with the frequency of the oscillation, g, may have any value; that is, if the wave length of the oscillation is very short com- pared with the length of the circuit, any wave length, and there- fore any frequency, may occur. With uniform circuits, as trans- mission lines, this latter case, that is, the response of the line to any frequency, can occur only in the range of very high fre- quencies. Even in a transmission line of several hundred miles’ length the lowest frequency of free oscillation is fairly high, and frequencies which are so high compared with the fundamental frequency of the circuit that, considered as higher harmonics thereof, they overlap (as discussed in the above), must be extremely high — of the magnitude of million cycles. In a com- plex circuit, however, the fundainental frequency may be very much lower, and below machine frequencies, as the velocity of propagation Fe may be quite low in some sections of the cir- cuit, as in the high-potential coils of large transformers, and the presence of iron increases the inconstancy of L for high frequen- cies, so that in such a complex circuit, even at fairly moderate frequencies, of the magnitude of 10,000 cycles, the circuit may respond to any frequency. 49. The constant u, is also determined by the circuit constants. Upon u, depends the energy transfer constant of the circuit sec- tion, and therewith the rate of building up in a section of low power consumption, or building down in a section of high power ‘consumption. In a closed circuit, however, passing around the entire circuit, the same values of e and 7 must again be reached, and the rates of building up and building down of the wave in the different sections must therefore be such as to neutralize each other when carried through the entire circuit; that is, the total building up through the entire complex circuit must be zero. This gives an equation from which w, is determined.

TRANSITION POINTS AND THE COMPLEX ciRcUIT 511 In a complex circuit having 7 sections of different constants and therefore n transition points, at the distances ; Ay Age An (296) where 2n4, = 4, + A, and A = the total length of the circuit, the equations of 7 and e of any section 7 are given by equations (290) containing the constants A,, B;, C;, D,. The constants A, B, C, D of any section are determined by the constants of the preceding section by equations (285) to _ (287). The constants of the second section thus are determined . by those of the first section, the constants of the third section by those of the second section, and thereby, by substituting for the latter, by the constants of the first section, and in this manner, by successive substitutions, the constants of any section 7 can be expressed by the constants of the first section as linear func- . tions thereof. Ultimately thereby the constants of section (n + 1) are expressed as linear functions of the constants of the first section: Antt _ a’A, 4 a”’B, 4+ aC, 4+ aD, : Buss = vA, + b”’B, + b’'C, + b’'"D,, (297) Cat = CA, + CB, + CC, + ""D,, Das; = d’A, + as, + dc, + dD, where a’, a’, a’”’, a’’””, b’, b’’, etc., are functions of s,; and 4,. The (n + 1)st section, however, is again the first section, and it is thereby, by equations (290) and (296), Ang, = Ayo", Bn, = Byeo*", (298) Cn41 = Cyt, Day = Diet", and substituting (298) into (297) gives four symmetrical linear equations in A,, B,, C,, D,, from which these four constants can be eliminated, as n symmetrical linear equations with

512 TRANSIENT PHENOMENA n variables are dependent equations, containing an identity, thus: (a’ — eA) A, + a’B, + ac" + a’”’”"D, = 0; vA, + (b” _ e~%4) B, + aC, + b”’”"D, = 0; , (299) CA, + cB, + (C” — e*™) C, + oD, = 0: dA, + a”B, + dC, + (a’’”’ _ etd) D, = 0, and herefrom , (a’ _ e~*4) a’ a’! ql!” b’ (0” —_ e—*4) vo” oy” = 0. (300) td Cc (c” _ et) (hid d’ dq” qd” (@ _ et) Substituting in this determinant equation for s; the values from (276) N 8; = Uy — uy (301) gives an exponential equation in u,, thus:

F (uy, Uy As om) = 0, (302) from which the value u,, or the resultant time decrement of the circuit, is determined.

In general, this equation (302) can be solved only by approxi- mation, except in special cases.

CHAPTER VII. POWER AND ENERGY OF THE COMPLEX CIRCUIT. 60. The free oscillation of a complex circuit differs from that of the uniform circuit in that the former contains exponential functions of the distance 4 which represent the shifting or transfer of power between the sections of the circuit. Thus the general expression of one term or frequency of current and voltage in a section of a complex circuit is given by equations (290); t =e {e+[A cosg (A — t) + Bsing (A — t)] —«e-"(Ccosg (A+ t) + Dsing (A+ d)]} and e= Ve eterna cos g (A—t)+ B sin g (A—2)] .

• e~"[C cos g (A+t)+Dsing (A+d)]}, where q = 7q,, q, = = , A = total length of circuit, expressed in the distance co6rdinate A = al, 1 being the distance coérdinate of the circuit section in any measure, as miles, turns, etc., and r, L, g, C the circuit constants per unit length of 1, ao=VIL, =i , = C’ “= 1(2 + ay = time constant of circuit secti Ate section, . u,= u +s = resultant time decrement of complex circuit, $ =u, — u = energy transfer constant of circuit section. 518 |

514 TRANSIENT PHENOMENA

The instantaneous value of power at any point 4 of the circuit at any time ¢ is p=e .

= Veet {e+?*4TA cosg (A — t) + Bsing (A — t)P

— e~41C cosg (A + t) + Dsing (A + t)P} _ Vem {[e +44 (A? 4 B?) _ e720 (C? + D?)) +[e +?" (A?— B?) cos 2q (A—t) —e7 24 (C? — D) cos 2g (A+-1)] +2[ABet® sin 2g (A—t) —CDe~?" sin 2q (A+)]}; (803) ’ that is, the instantaneous value of power consists of a constant term and terms of double frequency in (A — t) and (A + 2) or in distance A and time ¢.

Integrating (303) over a complete period in time gives the effective or mean value of power at any point A as P =i Ze fet? (427 4B) — 2-24 (C74 D)}; (804)

that is, the effective power at any point of the circuit is the difference between the effective power of the main wave and that of the reflected wave, and also, the instantaneous power at any time and any point of the circuit is the difference between the instantaneous power of the main wave and that of the reflected wave. ’