Theory and Calculation of Transient Electric Phenomena and Oscillations — part 5 of 20

90 TRANSIENT PHENOMENA and, substituting af) in (7), and rearranging, the potential difference at the condenser terminals is ros _ atte e, = sin(0-0,- tt Ag Pe Sa a) Zo 2 2 The two integration constants A, and A, are given by the terminal conditions of the problem. Let, at the moment of start, 0=0, ? = 1, = instantaneous value of current and e€, = €, = instantaneous value of condenser potential (15) ~ difference. Substituting in (13) and (14), . £E ty = 5 cos (0, + ) +A,+A4, 0 and Ex, . ° r+s r—8s é = — >, sin @,+%)— a A - = As Therefore . . ££ A,+A,=t)—- 7 008 (6, + ¥) 0 and (16) lo +2 E , . A,—A,= _ Tote, — {r cos (6+) —2 z,sin (0,+%)}, 8 Sq or, rs te “2 °°? B(r—s . A,= oF? + , 008 (G+) — Xe sin (6,+) and (17) r+s., zi +e 2 ° ° EF (r+s . A,=+———- - -— +S cos (6, +) —2-8in (8, +) . 8 sz, ( 2

RESISTANCE, INDUCTANCE, AND CAPACITY 91 Substituting (7) in (13) and (14) gives the integral equations of the problem. The current is E\ -3<¢rr- . i= A cos (0—0,—7) + — ‘e a [Fos 0,+7)—2,8in 6,+9| Zz, 82, 2 -err+s : —<¢ 2 [75 "es (0,+7) —x, sin (6+) 1( -Se[r—-s. -Holrts . | -*Ne ? [ 5 i,te,|—« ? [ ite}? (18) and the potential difference at the condenser terminals is sO / owe sin (6—0,—1) | zy . -Sefr—s . | _ =} (r+s)e es cos (0,.+%)—« sin 0,49] 2 sz, 2

• Fefrss : —(r—s)e “* [= cos (0, + Y) — x, sin +m] 1 4 ~ "He fr-s. -¥Feprts. | +35) (r+s)e [ "9 ut ef (r—s)e > i, +64] : | (19) | where a= VE + (eR 2), tan y =", (10) | r and | s=VP_— dro... | The expressions of 7 and e, consist of three terms each: | (1) The permanent term, which is the only one remaining | efter some time; | (2) A transient term depending upon the constants of the | circuit, 7, 8, Ze Zp, x, the impressed e.m.f., H, and its phase 6, at the moment of starting, but independent of the conditions | axisting in the circuit before the start; and |

92 TRANSIENT PHENOMENA , (3) A term depending, besides upon the constants of the circuit, upon the instantaneous values of current and potential difference, 7, and e,, at the moment of starting the circuit, and thereby upon the electrical conditions of the circuit before impressing the e.m.f., e. This term disappears if the circuit is dead before the start. Equations (18) and (19) contain the term s=Vr — 472, = yr -—4 7 ; hence apply only when 7 > 4-.472,, but become indeterminate if r?=4 xz,, and imaginary if r’< 42z,; in the . latter cases they have to be rearranged so as to appear in real . form, in manner similar to that in Chapter V. 56. In the critical case, °.=4 22, and s = 0, equation (18), rearranged, assumes the form . £E E -3° t =—cos(9@—0,—¥)+—« * % (9 — 4,—1%) m : a +e 6 - a? : r . € —e r HE cos (8,+ Y) — 2, sin @,+7)| = 08 (@.+ ¥) 4 ay) _ +6 _e(fr. . 28 —¢ 22 . 6 {[5%e + |; —*— -4f. | However, developing in a series, and canceling all but the first term as infinitely small, we have & 8 | . ae be” 8. | s a ’ | hence the current is E -s i = cos (0 — 0-1) +8 22 Z 2 . 0 1B cos (@.+ Y) — z, sin (6, + ”|: — cos (A+ ¥) aoet - [5 + al: +e 0 2° ° At (20)

RESISTANCE, INDUCTANCE, AND CAPACITY 93 and in the same manner the potential difference at condenser terminals is Ex, . E -x¢ é, =z ane _ 6,—%) ~_- 22, r . 6 . [5 cos (6, + ¥) — z,rsin (8, + ”|- — 22, sin 6+ ¥)

• he SEs tre]e + 2a (21) 2 2° “jn S$ Here again three terms exist, namely: a permanent term, a transient term depending only on £ and @,, and a transient . term depending on 7, and e,.

57. In the trigonometric or oscillatory case, 7? < 42 2,, 8 be- comes imaginary, and equations (18) and (19) therefore contain complex imaginary exponents, which have to be eliminated, since the complex imaginary form of the equation obviously is only apparent, the phenomenon being real. Substituting q=V4zrz,—7 = js (22), in equations (13) and (14), and also substituting the trigono- | metric expressions . | +igy? q q € = cos 579 + 7 sin 9 i and (23) | -j-Le 132 a _ se gq ; & cos 59 jsin 59, and separating the imaginary and the real terms, gives _6 | ' i= cos 0 —0,—y) +e ® 2 (A + A,) cos 0 + j (A — A, sin £0} | 1 2. 2 zr 1 2 2 zr . | |

94 TRANSIENT PHENOMENA and . r --?é . e, = sin (0 -0,- 1) —6 2 ‘ A pe 1-4 | . + wey ft . Ai eA cos (O,+ Y)— qsin @, + 2 ee | Ce . [g cos (0, + Y) + rsin O@,+ V)]c: | . . . | then substituting herein the equations (16) and (22) the imagi- nary disappears, and we have the current, | EB -ge- | t= E eos (@-0.—")- -e ~” zy Zo “S qd 2 r, . r le cos (9, +) Cos 5 — O+ [Zsa (0,47) a 0,49] sin t 3 ~pet. q 2etri,. g il | +e §iscon st 0 — 280+ sin 94? (4) and the potential difference at the condenser terminals, Ex, . Ex, -x%¢ \ @ = Zo sin 0 — O)— ¥) +> 2 “ q r. 2x . 4 } sin (0+ M008 $-0+ | Esin(0,+9)— = c0s(0,+9)| sin on aa q 2re, +422, . 4 _ 2@g4 2 ot Treo, 4 gf. ; +e GCOS 5 + 24 sin 5,9 (28) Here the three component terms are seen also. 58. As examples are shown in Figs. 20 and 21, the starting of the current 2, its permanent term 2,, and the two transient terms 7, and 7,, and their difference, for the constants Z = 1000 volts = maximum value of impressed e.m.f.; 7 = 200 ohms =resistance; x = 75 ohms = inductive reactance, and z, = 75 ohms = condensive reactance. We have 4r2x, = 22,500 and r? = 40,000; therefore rP>dre,,

_ RESISTANCE, INDUCTANCE, AND CAPACITY 95

• that is, the start is logarithmic, and z, = 200, s = 132, and y= 0.

ReSeTE oH Ne nah rt polohtas ; NNER aan | mas pales ‘ BXNe JET Boe Pepe HONS SCE a Da A SS AS SE SD wee es <L|| sim NA EEE SS “EEE EEE EH PSS

• 0 20 40 60 sO 100 p 1) 140 160 «610 =—(200 Degrees Bg. 20. Starting of an alternating-current circuit, having capacity, inductance and resistance in series. Logarithmic start. ;

In Fig. 20 the circuit is closed at the moment 0, = 0, that is, at the maximum value of the impressed e.m.f., giving from the equations (18) and (19), since 2, = 0, e, = 0,

t = 5 {cos 0 — 1.26 e-? 2 + 0.26 e—0-4828 } and e, = 375 {sin 8 + 0.57 (e779 — e~ 0-820) } ERED =a Ti Be E F100 folts | | CEH eee Ret TEE Bake Amn 402 ak lot eae ROP CEE é, Zr SS < ARERR we: Hee EEE EEE AS Set ttl) Smee ee SS PI | 0 20 40 60 80 100 «6000 60180 00 Degrees Fig. 21. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Logarithmic start.

. In Fig. 21 the circuit is closed at the moment 9, = 90°, that is, at the zero value of the impressed e.m.f., giving the equa- tions ;

t= 5 {sin @ + 0.57 (6-779 — e-o-4sre)} and e, = — 375 {cos 6 + 0.26 e-7° — 1.26 e040) } , o| |

96 ; TRANSIENT PHENOMENA

There exists no value of 6, which does not give rise to a transient term.

Pre | TT age porte | tt ‘COCO rb imieis Poo ‘ | eT EN BESS) 41.1 BERR

Pe el Ss ERB SREE peso SS

— as ee a “CH SARA

RE Pret Frye | ert COCO SSC cig

a CCC SSE

0 20 40 60 = 1) ip 160 180 «62200 220

Fig. 22. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Critical start.

In Fig. 22 the start of a circuit is shown, with the inductive - reactance increased so as to give the critical condition,’

rP=4272,, but otherwise the constants are the same as in Figs. 20 and 21,_- that is, # = 1000 volts; r = 200 ohms; z = 133.3 ohms, and Z, = 75 ohms; therefore 2, = 208.3, 58.3 tan Y = 300 = 0.2915, or Y= 16°, assuming that the circuit is started at the moment 4, = 0, or at the maximum value of impressed e.m.f.

Then (20) and (21) give

1 = 4.78 cos (9 — 16°) + 6° (2.76 — 4.6)

and e, = 308 sin (9 — 16°) — es "(410 0 — 99).

Here also no value of 6, exists at which the transient term | disappears.

69. The most important is the oscillating case, 7 < 42z,, | since it is the most common in electrical circuits, as underground cable systems and overhead high potential circuits, and also is practically the only one in which excessive currents or excessive voltages, and thereby dangerous phenomena, may occur.

RESISTANCE, INDUCTANCE, AND CAPACITY 97 If the condensive reactance x, is high compared with the

resistance r and the inductive reactance x, the equations of :

start for the circuit from dead condition, that is, 7, = 0 and

é, = 0, are found by substitution into the general equations

(24) and (25), which give the current as

or, —— BE sin (9—0,) +e 7* [sin 0, cos / 70 Xe z Z, + z, ) .

• V2c0s asin V5]! (26) and the potential difference at the condenser terminals as a r = E S c08 (9—0,)—e 77 Ze r xz. f/f 9 +(,— cos 0,- V=sino,) sin /% 1;

[cose cos VE0 + (5 om ° 7% sin 7! » (27) _— Where q =2V7 ray %= to and y = — 90°. (28) In this case an oscillating term always exists whatever the

value of 0,, that is, the point of the wave, where the circuit is

started. The frequency of oscillation therefore is -4fe I f o” 2 x f vz- 4n or, approximately, (29) I, = VES, , x where f = fundamental frequency. ae _ 1 Substituting x = 2 7fL and z, = aft’ we have fe 1 / 1 P °" 2VoCL” LD’ or, approximately, (30) 1 f= 22/CL :

98 TRANSIENT PHENOMENA

60. The oscillating start, or, in general, change of circuit conditions, is the most important, since in circuits containing capacity the transient effect is almost always oscillating.

The most common examples of capacity are distributed capacity in transmission lines, cables, etc., and capacity in the

: form of electrostatic condensers for neutralizing lagging currents, for constant potential-constant current transformation, etc.

(a) In transmission lines or cables the charging current is a fraction of full-load current 7,, and the e.m.f. of self-inductance consumed by the line reactance is a fraction of the impressed e.m.f.e,. Since, however, the charging current is (approximately)

=< and the e.m.f. of self-inductance = r?,, we have c C6 0. . 7 < ty, tly < 3 hence, multiplying,

x

—<lande < r,.

x,

The resistance r is of the same magnitude as x; thus

4rzr,.>P.

For instance, with 10 per cent resistance drop, 30 per cent reactance voltage, and 20 per cent charging current in the line, assuming half the resistance and reactance as in series with the capacity (that is, representing the distributed capacity of the line by one condenser shunted across its center) and denoting

. _ Go P i, where e, = impressed voltage, 7, = full-load current, we have Dp - r,. = 02 =vOp, z= 0.5 X 0.3 p = 0.15 p, r = 0.5 X 0.1 p = 0.05 p, and r+xr+27,= 1+ 3 + 100, and 4xrz, +r =1200 +1. | |

: RESISTANCE, INDUCTANCE, AND CAPACITY 99 In this case, to make the start non-oscillating, we must have 1 wa , "a r< 00” or x < 0.000125 p, which is not possible; or r > V3 p, which can be done only by starting the circuit through a very large non-inductive resistance (of such size as to cut the starting current down to less than of full-load current). Even in this case, however, oscillations would appear by a change of . load, etc., after the start of the circuit. . (6) When using electrostatic condensers for producing watt- less leading currents, the resistance in series with the condensers is made as low as possible, for reasons of efficiency. Even with the extreme value of 10 per cent resistance, or r+z,= 1+ 10, the non-oscillating condition is z < ii r, or 0.23 per cent, which is not feasible. In general, if zconsumes........ 1 2 4 9 16 _ percent of the con- denser potential difference, r must consume > 20 28.3 40 60 80 _ per cent of the con- denser potential difference. ; That is, a very high non-inductive resistance is required to avoid oscillations. The frequency of oscillation is approximately f, = v= ft that is, is lower than the impressed frequency if z, < x (or the permanent current lags), and higher than the impressed fre- quency if x, > x (or the permanent current leads). In trans- mission lines and cables the latter is always the case. Since in a transmission line Fis approximately the charging ‘c | current, as fraction of full-load current, and , half the line | e.m.f. of self-inductance, or reactance voltage, as fraction of | impressed voltage, the following is approximately true: |

100 TRANSIENT PHENOMENA

The frequency of oscillation of a transmission line is the impressed frequency divided by the square root of the product of charging current and of half the reactance voltage of the line, given respectively as fractions of full-load current and of im- pressed voltage. For instance, 10 per cent charging current, 20 per cent reactance voltage, gives an oscillation frequency

f

= —-—-—_ = 10f.

Ie V0.1 x 0.1 f HAT | inl | HESRoe “AA Ava cee CCH [ \ L\ ar.Naa | ieee eee erat eA | A ae |_| A a DA a site anva rier Ad ol PET TT TT et [ag ohms or me

Fig. 28. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Oscillating start of transmission line.

61. In Figs. 23 and 24 is given as example the start of current in a circuit having the constants, EF = 35,000 cos (9 — 0,); r = 5 ohms; z = 10 ohms, and zx, = 1000 ohms.

In Fig. 23 for 6,= 0°, or approximately maximum oscilla- tion,

i= — 35 {sin @ — 10c~ °** * gin 10 0} and e, = 35,000 {cos 6 — «~ 5 * [cos 10 8 + 0.025 sin 10 6]}. In Fig. 24 for @, = 90°, or approximately minimum oscilla- tion, i = 35 {cos 0 — e~ 959 cos 10 0} and . e, = 35,000 {sin 6 + 0.1 e~ °5* sin 10 6}.

As seen, the frequency is 10 times the fundamental, and in

starting the potential difference nearly doubles.

RESISTANCE, INDUCTANCE, AND CAPACITY 101 As further example, Fig. 25 shows the start of a circuit of a frequency of oscillation of the same magnitude as the funda- mental, in resonance condition, z = z,, and of high resistance. ° Bs TERS RRMRS th. hoe || off CEPT ee |aeaties | Bee : o Ch Lio! | Loe Toe TS er pk i YS 1 EEE CCCP CR SR CSCI . . LL TAN u

• SOGSESSReeRamne coor Zoe . by 0 Sagan e Hse SER . “ aa ae ZEAL SO oT oe? = BAPE | ae EE | | . Fig. 24. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Oscillating start of transmission line. The circuit constants are EH = 1500 volts; r = 30 ohms; . x=20 ohms; z, = 20 ohms, and @, = — ¥; which give q = 26.46; z, = 30; ¥ = 0, and 4, = 0. oN oR el TT TT TE Ee folts HL | PNG TNE fe pone TTT TT ee ee ae + es | SAT ga . Meili ENNT Te Pty TE EANLT TT er Seer INSECT EE ptt SSH Eas meet tT TT tt NAAT TTT oe £505 8000 Fig. 25. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Oscillating start. High resistance. Substituting in equations (24) and (25) gives 1 = 50 {cos @ — e~ °75* [cos 0.661 6 — 1.14 sin 0.661 ” and e, = 1000 {sin @ — 1.51 e~ °75 ¢ sin 0.661 6}.

102 TRANSIENT PHENOMENA

As example of an oscillation of long wave, Fig. 26 represents

the start of a circuit having the constants E = 1500 volts; r = 10 ohms; zx = 62.5 ohms; x, = 10 ohms, and 6, = — 1; which give q = 49; z, = 53.4; ¥ = 79°, and 6, = — 79°.

Substituting in equations (24) and (25) gives

1 = 28 {eos 0 — e~ 98 8 [cos 0.39 6 — 0.2 sin 0.39 4)} and e, = 280 {sin 0 — 2.55 e~ °* ® sin 0.396 6}.

62. While in the preceding examples, Figs. 23 to 26, con- stants of transmission lines have been used, as will be shown in the following chapters, in the case of a transmission line

Pepa E = 1500 volts | } | e-CCCCCY CE epee : el ee | Ze= |10 ohms | ae | mh i i LTT] | Avot TT Ty 4 fin | ft Anew i i BRS Eee

• COCA HY ee sgh hea e/a gaane cE V/A = PERRET Fig. 26. Starting of an alternating-current circuit having capacity, inductance and resistance in series. Oscillating start of long period. with distributed capacity and inductance, the oscillation does not consist of one definite frequency but an infinite series of frequencies, and the preceding discussion thus ‘approximates only the fundamental frequency of the system. This, however, is the frequency which usually predominates in a high power low frequency surge of the system.

In an underground cable system the preceding discussion applies more closely, since in such a system capacity and induc- tance are more nearly localized: the capacity is in the under- ground cables, which are of low inductance, and the inductance is in the generating system, which has practically no capacity.

In an underground cable system the tendency therefore is

RESISTANCE, INDUCTANCE, AND CAPACITY 103 either towards a local, very high frequency oscillation, or travel- ing wave, of very limited power, in a part of the cables, or a low frequency high power surge, frequently of destructive magnitude, of the joint capacity of the cables, against the inductance of the generating system.

63. The physical meaning of the transient terms can best be understood by reviewing their origin.

In a circuit containing resistance and inductance only, but a single transient term appears of exponential nature. In such a circuit at any moment, and thus at the moment of start, the current should have a certain definite value, depending on the constants of the circuit. In the moment of start, however,the current may have a different value, depending on the preceding condition, as for instance the value zero if the circuit has been open before. The current thus adjusts itself from the initial value to the permanent value on an exponential curve, which disappears if the initial value happens to coincide with the final value, as for instance if the circuit is closed at the moment of the e.m.f. wave, when the permanent current should be zero. The approach of current to the permanent value is retarded by the inductance, accelerated by the resistance of the circuit.

In a circuit containing inductance and capacity, at any moment the current has a certain value and the condenser a certain charge, that is, potential difference. In the moment of start, current intensity and condenser charge have definite values, depending on the previous condition, as zero, if the circuit was open, and thus two transient terms must appear, depending upon the adjustment of current and of condenser e.m.f. to their permanent values. _,

Since at the moment when the current is zero the condenser e.m.f. is maximum, and inversely, in a circuit containing induc- tance and capacity, a change of circuit conditions always results in the appearance of a transient term.

If the circuit is closed at the moment when the condenser e.m.f. should be zero, that is, about the maximum value of cur- rent, the transient term of current cannot exceed in amplitude its final value, since its maximum or initial value equals the value which the current should have at this moment. If, however, the circuit is closed at the moment where the current should be zero and the condenser e.m.f. maximum, the condenser being

mi

104 _ TRANSIENT PHENOMENA

without charge acts in the first moment like a short circuit, that is, the current begins at a value corresponding to the impressed e.m.f. divided by the line impedance. Thus if we neglect the resistance and if the condenser reactance equals n? times line reactance, the current starts at n? times its final rate; thus it would, in a half wave, give n? times the full charge of the con- denser, or in other words, charge the condenser in of the time of a half wave. That is, the period of the starting current is 1 . . , a and the amplitude n times that of the final current. How- ever, as soon as the condenser is charged, in - of a period of the impressed e.m.f., the magnetic field of the charging current produces a return current, discharging the condenser again at the same rate.

Thus the normal condition of start is an oscillation of such a frequency as to give the full condenser charge at a rate which when continued up to full frequency would give an amplitude equal to the impressed e.m.f. divided by the line reactance. The effect of the line resistance is to consume e.m.f. and thus dampen the oscillation, until the resistance consumes during the condenser charge as much energy as the magnetic field would store up, and then the oscillation disappears and the start becomes exponential.

Analytically the double transient term appears as the result of the two roots of a quadratic equation, as seen above.

. » CHAPTER VIII.

LOW FREQUENCY SURGES IN HIGH POTENTIAL SYSTEMS.

64. In electric circuits of considerable capacity, that is, in extended high potential systems, as long distance transmission lines and underground cable systems, occasionally destructive high potential low frequency surges occur; that is, oscillations of the whole system, of the same character as in the case of localized capacity and inductance discussed in the preceding chapter.

While a system of distributed capacity has an infinite number of frequencies, which usually are the odd multiples of a funda- mental frequency of oscillation, in those cases where the fundamental frequency predominates and the effect of the higher frequencies is negligible, the oscillation can be approxi- mated by the equations of oscillation given in Chapters V and VII, which are far simpler than the equations of an oscillation of a system of distributed capacity.

Such low frequency surges comprise the total system, not only the transmission lines but also the step-up transformers, gen- erators, etc., and in an underground cable system in such an oscillation the capacity and inductance are indeed localized to a certain extent, the one in the cables, the other in the generating system. In an underground cable system, therefore, of the infinite series of frequencies of oscillations which theoretically exist, only the fundamental frequency and those very high harmonics which represent local oscillations of sections of cables can be pronounced, and the first higher harmonics of the fundamental frequency must be practically absent. That is, oscillations of an underground cable system are either

(a) Low frequency high power surges of the whole system, of a frequency of.a few hundred cycles, frequently of destructive character, or,

(b) Very high frequency low power oscillations, local in character, so called “static,’”’ probably of frequencies of hundred

105 .

: 106 TRANSIENT PHENOMENA thousands of cycles, rarely directly destructive, but indirectly | harmful in their weakening action on the insulation and the possibility of their starting a low frequency surge.

The former ones only are considered in the present chapter. Their causes may be manifold, — changes of circuit conditions, as starting, opening a short circuit, existence of a flaring arc onthe system, etc. |

In the circuit from the generating system to the capacity of | the transmission line or the underground cables, we have always | r< at that is, the phenomenon is always oscillatory, and | equations (24) and (25), Chapter VII, apply, and for the current we have’ |

| . E -ce(f. E q | =— -§7.— 22 -— —_

) 7, 080 6,.-—Y) +e {li = 008 0,+7) eos x0 | Qe,+ri, E . J q | [ 7 +o, 22, sin (6,+yY) reos (0,+))| ain 5 rat (1) | and for the condenser potential we have | | Ex, . - 6 Ez, . q |

= —* _ _ 22 —s — e; z sin(@—0,—y) +e {leat z sin @,+7)| cos, 8 | re,t4 221, =| . )] - q |

oo TA eho we 0 _ Ss

+° 24 +e, r sin (9,+y)—2 z cos (8,+y) sin 50 | | Q

65. These equations (1) and (2) can be essentially simplified by neglecting terms of secondary magnitude.

z, is in high potential transmission lines or cables always very large compared with 7 and z.

The full-load resistance and reactance voltage may vary from less than 5 per cent to about 20 per cent of the impressed e.m.f., the charging current of the line from 5 per cent to about 20 per cent of full-load current, at normal voltage and

frequency. |

In this case, x, is from 25 to more than 400 times as large as r or x, and r and z thus negligible compared with z,.

HIGH POTENTIAL SYSTEMS 107 It is then, in close approximation : 2 = 2, q=2VE5, Ca n Y=- 27 —90°. Substituting these values in equations (1) and (2) gives the current as _ -E. _ (f. E. V% =~ zm (0—6,) + €~2z if 70 0, | 208 =o eo tr, E ( r. )]sia v4 —| —2*—* — ——__(2 cos 6, + — sin 6,) |sin / 6 4 Fess 2Vz22, ° 2 ° zy’ ®) | and the potential difference at the condenser as e, = Ecos (6 — 6.) + ¢ 22° Se - E cos 0, cos / 6

• [Peto Bet te E . 4Verxz, 4V22, ) (2 r cos 6, + mt Aeon 0,) [sin V6 . Le x These equations consist of three terms: . 4 = v + yw + uw . ‘ Mm” wt (5) e=e te +e; v= — sino - 6); Te (6) e,/ = Ecos (6 — 9,);

. : |

108 TRANSIENT PHENOMENA .

. r )

ye Ee {sin 6, cost) 0 - [VZc0s 6, |

Le x £

• sin] sin 0 . : e/’= — Be" e050, cos /20 +|—7—00s 0, z 2V2z x,
• mt ate ain 0,| siny/*9! ; 4x, Vz 2, x $ J . | or, by dropping terms of secondary order, | E -fe x . uv = ——e 22 cos 0 sin 220 we a 6 ~s6 x, | e,’ = — Ee 7+ cos 0, cos /29: and: 7 ime Fi, cooy/Z0 — Pe siny/ 20, | zr 2Vrz, zx m 73a! [% 2ret+(P +407), . (a 04. | e, € Je, cos) not awe Viz siny/ 9 | or, by dropping terms of secondary order, |

• • 6 x é £ ; ' a BN; cos / 76 -—_ siny/"9/

: .° x Vro, zy’ | . _— 6 yr r (10) i e "=e 2 Je, cos V9 + i, Vil, sin} |

Thus the ¢otal current is approximately ° . E. -fo(. ‘ | ¢ =——sin (0—0,) +¢ 77 |, cos y/o | | t, 0 ' | e, — Ecos6, . Zo | — %— sin 9 . | Vor L, “ ve S | |

and the difference of potential at the condenser is (11) -7e6 a e, = Ecos(@—4,) +e 7” \e.- B e080.) cosy * 9

—— [% | :

• ty Vrw, sin i 4: | . | | | | |

HIGH POTENTIAL SYSTEMS 109 Of the three terms: 7’, e,’; 7”, ¢,/”; 1”, e,’”, the first obviously represents the stationary condition of charging current and con- denser potential, since the two other terms disappear for t = ». The second term, 7”, e,’”, represents that component of oscilla- tion which depends upon the phase of impressed e.m.f., or the
point of the impressed e.m.f. wave, at which the oscillation begins, while the third term, 7’”, e,’”, represents the component of oscillation which depends upon the instantaneous values of current and e.m.f. respectively, at the moment at which the -—6 oscillation begins. « 7? is the decrement of the oscillation. 66. The frequency of oscillation is Z, | n=Ves, where f is the impressed frequency. That is, the frequency of oscillation equals the impressed frequency times the square root of the ratio of condensive reactance and inductive reactance of the circuit, or is the impressed frequency divided by the square root of inductance voltage and capacity current, as fraction of impressed voltage and full-load current. Since - - 1 x, ~ 2afC and x = 2 afL, the frequency of oscillation is i-—.; . * 22VCL’ that is, is independent of the frequency of the impressed e.m.f. Substituting 1 . 0=2 aft, t= 2afC and r=2 afL in equations (8), (10), and (11), we have C - a! t 7} at = a E 2L 0 . _
a Ve é cos 0, sin Jeu’ (12) -57! t e,” =— Ee *” cos6,cos——; , ° ACL

110 TRANSIENT PHENOMENA uw = <i! fi cos oe e visn tt, ] ° MCL ONL VEL (18) -35! t L t efv=e 7% je cos —— + iE cin I; mm MCL oN Co VELS’ | t =— 2af/CE sin (6 — 6,)+ rs i cos f 0 0 ‘CL Cc t

• • 6 Ve i +} (e, — E cos 8,) Z sin or , (14) -— t t e, = Ecos(9@—0,)+e 7% XC — E cos 6,) cos ——— 1 0) ° 0. VCL +1 ve sin a . °YC MCL

The oscillating terms of these equations are independent of the impressed frequency. That is, the oscillating currents and potential differences, caused by a change of circuit conditions i (as starting, change of load, or opening circuit), are independent . of the impressed frequency, and thus also of the wave shape of . the impressed e.m.f., or its higher harmonics (except as regards terms of secondary order). .

The first component of oscillation, equation (12), depends not only upon the line constants and the impressed e.m.f., but principally upon the phase, or the point of the impressed e.m_f. wave, at which the oscillation starts; however, it does not depend upon the previous condition of the circuit. Therefore this component of oscillation is the same as the oscillation produced in starting the transmission line, that is, connecting it, unexcited, to the generator terminals.

There exists no point of the impressed e.m.f. wave where no oscillation occurs (while, when starting a circuit containing resistance and inductance only, at the point of the impressed e.m.f. wave where the final current passes zero the stationary condition is instantly reached).

With capacity in circuit, any change of circuit conditions involves an electric oscillation.

I ) HIGH POTENTIAL SYSTEMS 111 | | The maximum intensities of the starting oscillation occur near the value 0, = 0, and are E -5¢°. £ ji =; * sin VEZ V2 2, z and (15) e”=-E 2 cos Vo. L Since . E ., vo zn (6 — 6,) is the stationary value of charging current, it follows that the maximum intensity which the oscillating current, produced in starting a transmission line, may reach is ve times the sta- tionary charging current, or the.initial current bears to the stationary value the same ratio as the frequency of oscillation to the impressed frequency. . The maximum oscillating e.m.f. generated in starting a trans- - mission line is of the same value as the impressed e.m.f. Thus the maximum value of potential difference occurring in a trans- | mission line at starting is less than twice the impressed e.m.f. and no excessive voltages can be generated in starting a circuit. The minimum values of the starting oscillation occur near 6, = 90°, and are, from equations .(22), wa EB ony Ly z and (16) -r6 ef = viz e 7* sin V = 9; that is, the oscillating current is of the same intensity as the charging current, and the maximum rush of current thus is less than twice the stationary value. The potential difference in the circuit rises only little above the impressed e.m.f. The second component of the oscillation, equation (13), does not depend upon the point of the impressed e.m.f. wave at

112 TRANSIENT PHENOMENA

which the oscillation starts, 0,, nor upon the impressed e.m.f. as

a whole, H, but, besides upon the constants of the circuit, it

depends only upon the instantaneous values of current and of

potential difference in the circuit at the moment when the . oscillation starts, 7, and e,.

Thus, if 7, = 0, e, = 0, or in starting a transmission line, unexcited, by connecting it to the impressed e.m.f., this term disappears. It is this component which may cause excessive potential differences. Two cases shall more fully be discussed, namely :

(a) Opening the circuit of a transmission line under load, and (b) rupturing a short-circuit on the transmission line.

_ 67. (a) If 2, is the instantaneous value of full-load current, e, the instantaneous value of difference of potential at the condenser, 77 is small compared with e,, and Vz zx, 1, is of the same magnitude as e,.

Writing

ey ; | tan 0 i, Vix and substituting in equations (10), we have 2 fF uv ~ Vira 2 22" cog (V0 +0) ; LX, x and : ty (17) a!” =Ver+ijrae ** sin (vz a+ a); 2 that is, the amplitude of oscillation isl) 2,” +2 forthecurrent, and Ve? +7i2zz, for the e.m.f. Thus the generated e.m.f. can be larger than the impressed e.m.f., but is, as a rule, still of the same magnitude, except when zr, is very large.

In the expressions of the total current and potential difference at condenser, in equations (11), (e, — E cos 9,) is the difference between the potential difference at the condenser and the impressed e.m.f., at the instant of starting of the oscillation, or the voltage consumed by the line impedance, and this is small

HIGH POTENTIAL SYSTEMS 118 if the current is not excessive. Thus, neglecting the terms with (e, — Ecos 6,), equations (11) assume the form

a E in (@ _- 6.) + ie ?* cos Vo

Le z and ; _ (18) --—96

e,= E cos(0—0,)+i,V az, ** sin 20; that is, the oscillation of current is of the amplitude of full-load current, and the oscillation of condenser potential difference is of the amplitude i,V’z z,-

zz, is the ratio of inductance voltage to condenser current, in fractions of full-load voltage and current. We have, therefore,

• 5 — ../Lb ; | Thus in circuits of very high inductance L and relatively low . capacity C,7,/xz, may be much higher than the impressed _ e.m.f., and a serious rise of potential occur when opening the circuit under load, while in low inductance cables of high capacity i,Vzz, is moderate; that is, the inductance, by tending to maintain the current, generates an e.m.f., producing a rise in potential, while capacity exerts a cushioning effect. Low inductance and high capacity thus are of advantage when breaking full-load current in a circuit.

68. (6) If a transmission line containing resistance, induc- tance, and capacity is short-circuited, and the short-circuit suddenly opened at time ¢ = 0, we have, fort < 0,

e=0

• and i = 2 cos (0 — 0, — 9), ——. (19) where z=VP4+2 and tany = z. r

114 TRANSIENT PHENOMENA thus, at time ¢ = 0, | . ££ ty = 7 608 (6, + 7). (20) Substituting these values of e, and 2, in equations (9) gives E -3° T, r - ” =—cos(O,ty)e 7” \ cos W/89 — "sin y/¥e 0 , z (,+ 7) zt Van z, z and E -y°Pt+4en, . Vi me 2z G “ec e, cos (6, + y)e Viz sin z 0, or, neglecting terms of secondary magnitude, . E -3?¢ T. vw" = sf = cos (0, + ¥) cos/% g and (21) Vzz, -x° t. e!” = EV 2% e ** cos (6,+7) sin/ % 9; z . L that is, 2’” is of the magnitude of short-circuit current, and e,/” of higher magnitude than the impressed e.m.f., since z is small compared with Vrz,. - The total values of current and condenser potential difference, from equation (11), are -~¢6 . i= — Zino — 0) + Be" {fee Gut) Le z : _ =| cos v= A+ 08 0 sin V2 of 2 x V 2x, Zz and (22)

• a? rT ‘ e, = E cos (6 — 6,)— He “* } cos 6, cos 7! Vrzx, _ Vrx, cos (6, + Y sin y/o,

2 x | |

=

HIGH POTENTIAL SYSTEMS 115 or approximately, since all terms are negligible compared with eh and e”, .

. E -3¢ L, tse ?= cos (0, + 7) cos V/ 0 and (23) Ser. -¢ : e, = ENT: 5 22° cog (0,+ 7) sin 0. z x These values are a maximum, if the circuit is opened at the moment 6 = — 7, that is, at the maximum value of the short- circuit current, and are then _-¢@ t= E. ad con 20 z x and (24) V2z.. - 5° e=——*Re ** siny/= 0. 2 x The amplitude of oscillation of the condenser potential dif- ference is VIL, E,

z or, neglecting the line resistance, as rough approximation, Z=z2,

Z,

vee: :

that is, the potential difference at the condenser is increased above the impressed e.m.f. in the proportion of the square root of the ratio of condensive reactance to inductive reactance, or inversely proportional to the square root of inductance voltage times capacity current, as fraction of the impressed voltage and the full-load current. Thus, in this’case, the rise of voltage is excessive.

The minimum intensity of the oscillation due to rupturing short-circuit occurs if the circuit is broken at the moment .

| | 116 TRANSIENT PHENOMENA 6 = 90° — 7, that is, at the zero value of short-circuit current. Then we have

• se a) i= E nos (+7)— Ee 7? {2°87 coay/Zo Ly Ly L sin; . ve
• T= sn — Vix, x 4 (25) and . rae Te e,=Esin(6+7) — Ee sin 7 cos W/% 0; that is, the potential difference at the condenser is less than twice the impressed e.m.f.; therefore is moderate. Hence, a short- | circuit can be opened safely only at or near the zero value of the | short-circuit current. The phenomenon ceases to be oscillating, and becomes an ordinary logarithmic discharge, if V?— 4 rz, is real, or r>2V rz, | Some examples may illustrate the phenomena discussed in the | preceding paragraphs.

69. Let, in a transmission line carrying 100 amperes at full load, under an impressed e.m.f. of 20,000 volts, the resistance | drop = 8 per cent, the inductance voltage = 15 per cent of the | impressed voltage, and the charging current =8 per cent of full- | load current. Assuming 1 per cent resistance drop in the | step-up transformers, and a reactance voltage of 24 per cent, | the resistance drop between the constant potential generator terminals and the middle of the transmission line is then 5 per cent, or r = 10 ohms, and the inductance voltage is 10 per cent, or z = 20 ohms. The charging current of the line is 8 amperes, thus the condensive reactance x, = 2500 ohms. Then, assuming a sine wave of impressed e.m.f., we have E = 20,000 V2 = 28,280 volts; . v =— 11.3 sin @ — 6,); . e,/ = 28,280 cos (8 — 4,); i” =— 11.3 "(sin 8, cos 11.2 6 — 11.2 cos 6, sin 11.2 6],

• and . — e,/” = — 28,280 <~°**[cos 6, cos 11.2 6—0.089 sin 8, sin 11.2 6]. yo

HIGH POTENTIAL SYSTEMS 117 . Therefore the oscillations produced in starting the trans- mission line are 4 = — 11.3 [sin (6 — 6,) + -°* (Gin 6, cos 11.2 0 — 11.2 cos 4, sin 11.2 6)] and e, = 28,280 [cos (@ — 0,) — 58 (aog 6, cos 11.2 6 — 0.089 sin 4, sin 11.2 4)). »»—— : ww Sy i cet i . o_alt Ki Nal TAT T lenfzodume TTT TT o—wli lt foie HAN A 0-3-0 a oe LRT oivty it AIA RE EEIAAIA TY Poo Sod 74 a A 0 a 0-~9 SEL ‘5 —m NP ee BERRA ‘owt tt tT eT TE AT

• att tt hf gw TP Ta EA a TT ee TTT
• LLL TAT ET ee | . » » » © © © © © w 10 : Fig. 27. Starting of a transmission line. x — ae ee | "COOL eS SS CA eer ST 8C ORD 2 : 30 lA TT NT TT LTT loots |_| és 57_aeSe aoe ner eeees <, er _t tie tt IN te | Tt ' oe eae wwe LL TTT Tit tt tt tt tt TSH | 0 10 r x w 60 0 70 8 9 109 Degrees Fig. 28. Starting of a transmission line. Hence the maximum values for 6, = 0, are _ ¢=— 11.8 (gin 6 — 11.2 e7°® sin 11.2 8) and e, = 28,280 (cos 6 — «~°** cos 11.26), and the minimum values, for 0, = 90°, are 7 = 11.3 (cos @ — e~™** cog 11.2 4) and =e, = 28,280 (sin 8 + 0.089 «~°**sin 11.2 6). . . !

118 TRANSIENT PHENOMENA These values are plotted in Figs. 27 and 28, with the current, 7, in dotted and the potential difference, e,, in drawn line. The stationary values are plotted also, in thin lines, 7 and e’, respec- tively. , (a) Opening the circuit under full load, we have i == 11.3sin (6 — 0,) + -°** (i, — 11.3 sin 8,) cos 11.28 and e, = 28,280 cos (9 — 8,) + 224 i,e-°"**sin 11.2 6. a BACCARAT » LARA ee eee o—_» PTAC INE Tt NT TT Te TT TT wo—olptie\ | tATY TT PET TT TaN »—ol tit Wee AE Ta AL A ET gost til INT fit | Vere AT TT Pd 208 LZ ERT -o— Seno ft ett te to Tot ned o—-ot tty | tT tT TERT TPT TAT TNT TT w—o ttt ttt tt atta tpi Tey TT ott th ET TT AT Ta TT TT colt tite EET TT TT Rt TTT ott tila TTT eT TTT TNT TT SERRE ) w” » 30 0 50 0 i] 1) 90 100

• Degrees Fig. 29. Opening a loaded transmission line. These values are maximum for 6, = 0 and non-inductive circuit, or 7, = 141.4, and are 4 =—11.3 sin 6 + 141.4 e~*** cos 11.20 and €, = 28,280 cos @ + 31,600 <— °° sin 11.26. These values are plotted, in Fig. 29, in the same manner as Figs. 27 and 28. (b) Rupturing the line under short-circuit, we have z= 22.4 and 4, = 1265 cos (6, + 7); and therefore at =— 11.3 sin (0 — 8,) + 1265 «~°** f[cos (8, + 7)
• — 0,0089 sin @,] cos 11.2 6 + 0.1 cos 6, sin 11.263

HIGH POTENTIAL SYSTEMS 119 and ¢, = 28,280 {cos (6 + 6,) — «~°[cos 0, cos 11.2 4 — 10 cos (6, + 7) sin 11.2 6)}. These values are a maximum for #@, = — 7 = — 63°, thus i =— 11.3 sin (6 + 63°) + 1260 ~* (cog 11.2 0

• 0.044 sin 11.2 6) and e, = 28,280 cos (@ + 63°)— 282,800 «~°*? (0.044 cos 11.2 0 . — sin 11.28); | that is, the potential difference rises about tenfold, to 282,800 volts. These values are plotted in Fig. 30. 20—300 : RAL a papi cee eal Neel LL AAA frctanctid ty feo | TTT eS NCCE WAC 20 He A Ag Pol ETT PTA AT Tey AT Te [A POC EBOE Era ‘wool NT TTT Vie 0-00 CONE owl PPT TT AAT TT A TAY TA mol | AT VI7 | Ty AT PTT ATA TT TT ~~ Hf ERE Lf nn nn eee 0 w” 2 30 w@ 50 6 7 i) 9 00 Degrees Fig. 30. Opening a short-circuited transmission line. |

70. On an experimental 10,000-volt, 40-cycle line, when a destructive e.m.f. was produced by a short-circuiting arc, the author observed a drop in generator e.m.f. to about 5000 volts, | due to the limited machine capacity. The resistance of the i system was very low, about r = 1 ohm, while the inductive . reactance may be estimated as x = 10 ohms, and the condensive reactance as zx, = 20,000 ohms. Therefore tan 7 = 10, or approximately, 7 = 90°. : . Herefrom it follows that t = 707 «°° cos 44.7 6 and e, = 316,000 e~°** sin 44.7 6; __]

120 TRANSIENT PHENOMENA that is, the oscillation has a frequency of about 1800 cycles per second and a maximum e.m.f. of nearly one-third million volts, which fully accounts for its disruptive effects.

71. As conclusion, it follows herefrom:

72. A most important source of destructive high voltage