Theory and Calculation of Transient Electric Phenomena and Oscillations — part 12 of 20

304 TRANSIENT PHENOMENA 17.’ Let the circuit be grounded or connected to the return conductor at one end, / = 0, and supplied by a constant impressed e.m.f. Z, at the other end, / = 1,. Then for Z = 0, - E=E,=0 and for / = l,, . E=E,; hence, substituting in (17), 0=A,+A4, and ‘ Z . . E,= v2i A,e***(cos Bl, —j sin Al,) +.A,~ “(cos Al,+} sin Al,) ; hence, A, = — A, =A, and eve A= (ete — ¢-%) cos Bl, — j (et + e-%) sin Bl,’ and, substituting in (17), I=E.\ /Y (e** + e~“) cos fl — j (¢*4#—e~%) sin Al 7 TEN ZF (ete _ ¢-h) cog Bl, — 7 (e+ + ¢%) sin Bl, 7 EoR (e+ — e™) cos Bl — j (e+™ + €~%) sin Al (48) v1 (gts — —) cos Bl, — j (e+ + &-%) sin Bl, At the grounded end, l= 0, yz ore 2ENG (49) ; 70” “(etree — 2-4) cos Bl, —j (et + e-) sin Al, - . | Substituting (49) in (48) gives | . I =41,{(et% + e-%) cos pl — j (e+% — e-%) sin Bl} , and Zz (50) E=} 1V2 (e+ —e-) cos Bl — j (e+ + e-%) sin Al.

LONG-DISTANCE TRANSMISSION LINE 305 In this case nodes of voltage and crests of current appear at l = 0 and at the even quadrants, fl =2n 5) and nodes of current and crests of voltage appear at the odd quadrants, fl = (2n — 1) ° (C) Infinitely long conductor. . , 18. If an e.m.f. E, is impressed upon an infinitely long con- ductor, that is, a conductor of such length that of the power input no appreciable part reaches the end, we have, for / = 0, E = E, and for 1 = o, E =0Oand / = 0; hence, substituting in (23) gives A,= 0 and Z E, = v2 A; hence, Y _4 . 1=E, Zé (cos Bl + j sin Bl) (51) and ; E = E,% (cos Bl + jsin Bl). From (51) it follows that E /Z ! TV y that is, an infinitely long conductor acts like an impedance, ‘ [Z ; Z,.=Vy5r1 ity and the current at every point of the conductor thus has the same space-phase angle to the voltage, tan a, = fa ‘ un

806 * TRANSIENT PHENOMENA The equivalent impedance of the infinite conductor is Z. = V2 V_a«—ib 1 Yy Y g-jb ag+ fb . fg — ab =H - GS (52) . yi and the space-phase angle is _ §9 — ab tan a= ag + Bb ° (53) If g = 0 and z = 0, we have \ lor . e=BRVE and tan a, = 1, or a, = 45°; that is, current and e.m.f. differ by one-eighth period.

This is approximately the case in cables, in which the dielectric losses and the inductance are small.

An infinitely long conductor therefore shows the wave propa- gation in its simplest and most perspicuous form, since the reflected wave is absent. ;

(D) Generator feeding into a closed circuit.

19. Let i = 0 be the center of the circuit; then

E, = -E_, and T,=T_y hence, E=0Oatl=0, and the equations are the same as those of a line grounded at the end I = 0, which have been discussed under (B).

(E) Line of quarter wave length.

20. Interesting is the case of a line of quarter wave length.

Let the length /, of the line be one quarter wave of the im- pressed e.m.f.

4 Bl, = 3° (54)

LONG-DISTANCE TRANSMISSION LINE 807 To illustrate the general character of the phenomena, we may as first approximation neglect the energy losses in the circuit, , that is, assume the resistance r and the conductance g as neg- ligible compared with z and b, ; r=0=g. These values substituted in (14) give a =0 and B = V2. (55) Counting the distance / from the end of the line J, we have for L=0, Ey = € + Jey and (56) : To =% + jr, and at the beginning of the line for / = l,, E, = &, + je,’ and - (57) L=%y+7/, and by (54) and (55), Zz =—=: 58 1 = oa 8) Substituting (56), (57), and (54) in (17) gives I,=4A,-A, and _ yz / x E, = 7 (Ar + 4a) = 5 (Ar + Ad), or I 1= 7-7] (A, + A,) and | 4/Z fe B,-- iV, - Ay == z Ai ~ Ad; | hence, eliminating A, and A, gives the relations between the | electric quantities at the generator end of the quarter-wave line, . E,, I,, and at the receiving end, E,, I ,: | |

808 TRANSIENT PHENOMENA ) . Lt E,=j vi I, and (59) 1/6 . I om] I, E vy . and the absolute values are ! x E,= vir 1 and. _ (60) b I,= v2 E » which means that if the supply voltage F, is constant, the output current J, is constant and lags 90 space-degrees behind the input voltage; if the supply current /, is constant, the output voltage E, is constant, and lags 90 space-degrees, and inversely. A quarter-wave line of negligible losses thus converts from constant potential to constant current, or from constant cur- rent to constant voltage. (Constant-potential constant-current transformation.) Multiplying (60) gives Eyl o=l E ” El or E, = T,. ; hence, if 7, = 0, that is, the line is open at the end, Z, = ©, and with a finite voltage supply to a line of quarter-wave length, an infinite (extremely high) voltage is produced at the other end. | Such a circuit thus may be used to produce very high voltages. Since x, = 1,2 = total reactance and b, = 1,b = total sus- ceptance of the circuit, by (58) we have | 2 , ! typ, “7 , (61) | | or the condition of quarter-wave length. |

LONG-DISTANCE TRANSMISSION LINE 3809 Substituting z, = 2 2/L, and b, = 2 xfC,, we have 1 LC, = 16f? , (62) 4VLC, the condition of quarter-wave transmission. . 21. If the resistance, r, and the conductance, g, of a quarter- wave circuit are not negligible, substituting (56), (54) and (57) in (17) we have, for 1 = 0, 1,=A4,-A, and yz (64) . E, = (Ai + A,), and for 1 = 1, [= i }4er + Ae} and _ (65) B= iV Eh aete—aye f From (64) it follows that 1 Y | . A, = (1+ viz) | and . (66) 1 \ /Y A, = ~(1- zh), and substituting in (65) and rearranging we have Ark [Y ete 4 gab ete _ gay I= - i\E, a + 1.5} and (67) . . Z ete 4 eae ete — gale Q E,= 4] {,yZetees py stae ,

310 TRANSIENT PHENOMENA or, ; yz ete 4 —-ah ett _ .-eb (Le iENG Sra el meer} and (68) . Z ete 4 e— elo ete _ —— E, ST ee ea or, analogous to equation (59), . ; ve 2 jE, E, (e+ — e-%) LoVe pe pe and (69) ; E V2 2 jl, I, (e*™ — 67%) BV ep ee ef

In these equations the second term is usually small, due to the factor («+** — e~*), and the first term represents constant potential-constant current transformation. .

22. In a quarter-wave line, at constant impressed e.m.f. E,, the current output J, is approximately constant and lagging 90 degrees behind £,; it falls off slightly, however, with increasing load, that is, increasing J ,, due to the second term in equation (68); the voltage at the end of the line, E,, at constant impressed voltage, is approximately proportional to the load, but does not reach infinity at open circuit, but a finite, though high, limiting value. .

Inversely, at constant current input the voltage output is approximately constant and the output current proportional to the load.

The deviation from constancy, at constant E,, of Z,, or at constant J,, of E,, therefore, is due to the second term, with factor (e **%— ¢- %), l

Substituting (54), |

1-27, a, = 2 2 ) | hence, al, is usually a very small quantity, and ¢**® = e 2? thus can be represented by the first terms of the series: ,

| LONG-DISTANCE TRANSMISSION LINE 811 #531 pot llezy soa (55) _ ee 1455 +a(g5) *axalps +e an hence, ete 4 o— eh : a nn and et to _ ¢— ah ae 2 B 2’ and, by (69), VY a x/¥ I,=] Zi 33 zg bo : Vi1,-25V3 ° . an and E,=j ys 32 ple . If r and g are small compared with x and b, B= V¥ (ey — 19 + 2b) = Vad and a = V4 (ey + 1g — 2b); substituting, by the binomial theorem, 3 t -verararn ~ aft + JTC] uy =VEPPE) GTS = 2b 1+(=) 1+( Lory Lig om }145 (+5 (+... gives vist zy, hay ca ° sta (3) +2) +72 bz bz /r £) vEC+9) a lyr g and 5-3 (5 +5} |

312 TRANSIENT PHENOMENA The quantity

a1 g9\ _

2 =9 (; + 4 =U (7 1) may be called the time constant of the circuit.

The equations of quarter-wave transmission thus are

Y(., wz I, -y¥ Vik, - pt F (72) . ur and p= V2 Sin, -2 1), and the maximum voltage E,, at the open end of the circuit, at constant impressed e.m.f. F,,, is 1,=0 27K and E, = hs, (73) and the current input is

• /Y L=-j Z Be 2, V/ Y ua YZ 7) where, approximately, Z_,/L vz “VG (75)

28. Consider as an example a high potential coil of a trans- former with one of its terminals connected to a source of high potential, for testing its insulation to ground, while the other terminal is open. |

. Assume the following constants per unit length of circuit: r = 0.1 ohm, L = 0.02 mh.,C = 0.01 x 10-*farad, and g =0; then, with a length of circuit J, = 100, the quarter-wave fre- quency is, by (47), | 1 = ———__— = 177 cycles per sec., Pa4 WLC yor Pe

LONG-DISTANCE TRANSMISSION LINE 313 or very close to the third harmonic of a 60-cycle impressed voltage. If, therefore, the testing frequency is low, 59 cycles, the circuit’ , is a quarter wave of the third harmonic. Assuming an impressed e.m.f. of 50,000 volts and 59 cycles, containing a third harmonic of 10 per cent, or E, = 5000 volts at 177 cycles, for this harmonic, we have x = 22.2 ohms and b = 11.1 X 10-° ohm; hence, u = 0.00225 and BE, =. B, = 283;; o7 un 1 ie therefore at E, = 5000 volts, E, = 1,415,000 volts; that is, infinity, as far as insulation strength is concerned. . Quarter-wave circuits thus may be used, and are used, to pro- ~ duce extremely high voltages, and if a sufficiently high frequency

"is used — 100,000 cycles and more, as in wireless telegraphy, etc.

| —the length of the circuit is moderate.

| This method of producing high voltages has the disadvantage ‘ that it does not give constant potential, but the high voltage is due to the tendency of the circuit to regulate for constant current, which means infinite voltage at infinite resistance or open circuit,

' but as soon as current is taken off the high potential point the

• -voltage falls. The great advantage of the quarter-wave method of producing high voltage is its simplicity and ease of insula- tion; as the voltage gradually builds up along the circuit,

_ the high voltage point or end of circuit may be any distance away from the power supply, and thus can easily be made safe.

| 24. As a quarter-wave circuit converts from constant poten- tial to constant current, it is not possible, with constant voltage

| impressed upon a circuit of approximately a quarter-wave length,

| to get constant voltage at the other or receiving end of the circuit.

Long before the circuit approaches quarter-wave length, and as

| soon as it becomes an appreciable part of a quarter wave, this

' tendency to constant current regulation makes itself felt by great

; Variations of voltage with changes of load at the receiving end of —_

' the circuit, constant voltage being impressed upon the generator

| _ end; that is, with increasing length of transmission lines the volt-

| age regulation at the receiving end becomes seriously impaired

|

314 TRANSIENT PHENOMENA

hereby, even if the line resistance is moderate, and the operation of apparatus which require approximate constancy of voltage but do not operate on constant current — as most synchronous apparatus — becomes difficult.

Hence, at the end of very long transmission lines the voltage regulation becomes poor, and synchronous machines tend to instability and have to be provided with powerful steadying devices, giving induction motor features, and with a line approaching quarter-wave length, voltage regulation at the receiving end ceases.

In this case the constant potential-constant current trans- formation may be used to produce constant or appfoximately constant voltage at the load, by supplying constant current to the line; that is, the transmission line is made a quarter-wave

“ length by modifying its constants, or choosing the proper fre- quency, the generators are designed to regulate for constant current and thus give a voltage varying with the load, and are -

' connected in series (with constant current generators series con- nection is stable, parallel connection unstable) and feed constant ’ current, at variable voltage, into the quarter-wave line. At the receiving end of the line, constant voltage then exists with varying load, or rather a voltage, which slightly falls off with the load, due to the power loss in the line. To maintain constant receiver voltage at all loads, then, would require a slight increase of generator current with increase of load, that is, increase of generator voltage, which can be produced by compounding

regulated by the voltage.

In such a quarter-wave transmission the voltage at the receiv- ing end then remains constant, while the current output from the line increases from nothing at no load. At the generator end the current remains approximately constant, increasing from no load to full load by the amount required to take care of the line loss, while the voltage at the generators increases from nearly nothing at no load, with increasing load, approximately proportional thereto. |

25. There is, however, a serious limitation imposed upon | quarter-wave transmission by considerations of voltage; to use the transmission line economically the voltage throughout it should not differ much, since the insulation of the line depends on the maximum, the efficiency of transmission, however, on the

LONG-DISTANCE TRANSMISSION LINE 315 average voltage, and a line in which the voltage at the two ends is very different is uneconomical. .

To use line copper and line insulation economically, in a quarter-wave transmission, the voltages at the two ends should be approximately equal at maximum load. These voltages are related to each other and to the current by the line constants, by equations (72).

By these equations (neglecting the term’ with wu), reduced to absolute values, we have approximately

. y y= Vite and _ YE , ey = y? and if e, = é,, _

• aly | ty = VE ei hence, the power is _ . y Po = oho = ve ey or 2 e.' = Py vi (76) hence, the voltage e, required to transmit the power p, without great potential differences in the line depends on the power p, . and the line constants, and inversely.

26. As an example of a quarter-wave transmission may be | considered the transmission of 60,000 kilowatts over a distance of 700 miles, for the supply of a general three-phase distribution ' system, of 95 per cent power factor, lag. The design of the transmission line is based on a compromise | between different and conflicting requirements: economy in _ first cost requires the highest possible voltage and smallest con- : ductor section, or high power loss in the line; economy of opera- . tion requires high voltage and large conductor section, or low | power loss; reliability of operation of the line requires lowest

‘ 316 TRANSIENT PHENOMENA permissible voltage and therefore large conductor section or high power loss; reliability of operation of the receiving system requires good voltage regulation and thus low line resistance, etc., etc.

Assume that the maximum effective voltage between the line conductors is limited to 120,000, and that there are two sepa- rate pole lines, each carrying three wires of 500,000 circular mils cross section, placed 6 feet between wires, and _ provided with a grounded neutral.

If there were no energy losses in the line and no increase of capacity due to insulators, etc., the speed of propagation would

be the velocity of light, S = 188,000 miles per second, and the quarter-wave frequency of a line of J, = 700 miles would be SS, tS = = = 67 cycles per sec. ; 41, hence, fairly close to the standard frequency of 60 cycles.

The loss of power in the line, and thus the increase of induc- tance by the magnetic field inside of the conductor (which would not exist in a conductor of perfect conductivity or zero resistance loss’, the increase of capacity by insulators, poles, etc., lowers the frequency below that corresponding to the velocity of light and | brings it nearer to 60 cycles. |

In a line as above assumed the constants per mile of double conductor are: r = 0.055 ohm: L = 0.001 henry, and C = 0.032 x 107° farad, and, neglecting the conductance, g = 0, the quarter-wave frequency is |

1 : I 11, Vie 63 cycles per sec.

Either then the frequency of 63 cycles per second, or slightly above standard, may be chosen, or the line inductance or line capacity increased, to bring the frequency down to 60 cycles.

Assuming the inductance increased to LZ = 0.0011 henry gives f = 60 cycles per second, and the line constants then are l, = 700 miles; f = 60 cycles per second; r = 0.055 ohm: L = | 0.0011 henry: C = 0.032 x 107* farad, and g = 0: hence, x= 0.415 ohm; z= 042 ohm; Z = 0.055 — 0.415] ohm;

LONG-DISTANCE TRANSMISSION LINE 317 6 = 12.1 x 10~* mho; y = 12.1 x 10~* mho, and Y = — 712.1 X 10-* mho, and _ : /[Z ; ; \ y~ 186 + 12 j, vi ~ 19, y l/r g u=5(-+%)- 0.066, B = 2.247 x 10° a = uf = 0.148 x 10>.

At 60,000 kilowatts total input, or 20,000 kilowatts per line, and 120,000 volts between lines, or ea = 69,000 volts per line, and about 95 per cent power factor, the current input at full load is 306 amp. per line (of two conductors in multiple).

_ + To get at full load p = 20 x 10° watts, approximately the same voltage at both ends of the line, by equation (67), we must have _

Z e = Vz ) P y or ' @ = 61,000 volts.

Assuming therefore at the receiving end the voltage of 110,000 between the lines, or, 63,500 volts per line, and choosing the output current as zero vector, and counting the distance from the receiving end towards the generator, we have for / = 0,

: T=1,=% and the voltage, at 95 per cent power factor, or V1 — 0.95 = 0.312 inductance factor, is E = E, = e, (0.95 — 0.312 7) = 60,300 — 19,800 j.

| Substituting these values in equations (72) gives ; |

. a {jE, — 0.104 (60,300 — 19,800 j)}

| a 186 + 12 , !

60,300 — 19,800 j = (186 + 127) {7 1, — 0.104 i,);

| .

318 TRANSIENT PHENOMENA hence, jE, = (186 + 123) 7, + (6250 — 20607) and . 60,300 — 19,8007 . w= ~ 186 +12]; + 0.104 2, = 317 — 1287 + 0.1042, and the absolute values are e, = V (1862, + 6250)? + (121, — 2060)? ands i, = V@IT F OAOEa) + Le, a3 pt tt ttt tt tt tt a PPA al pba Ep /_) Ht Ale! pitt AY I —— LEAL os im Av ard a whl aff pea aan FF UL oe es oe act A Kb bh te ta PIALV YM | fel Td. PV YY ie TT Td AAT La tn Azer VAZAZER REE ZZAR EEE ERE Aro Power Onypat por Phase Po Fig. 84. Long-distance quarter-wave transmission. herefrom the power output and input, efficiency, power factor, etc., can be obtained. In Fig. 84, with the power output per phase as abscissas, are shown the following quantities: voltage input e, and output ¢,

LONG-DISTANCE TRANSMISSION LINE 319 in drawn lines; amperes input 7, and output 7,, in dotted lines; power input p, and output p,, in dash-dotted lines, and efficiency and power factor in dashed lines. | As seen, the power factor at the generator is above 93 per cent leading, and the efficiency reaches nearly 85 per cent. . At full load input of 20,000 kilowatts per phase, and 95 per cent power factor, lagging, of the output, the generator voltage . is 58,500, or still 8 per cent below the output voltage of 63,500. The generator voltage equals the output voltage at 10 per cent overload, and exceeds it by 14 per cent at 25 per cent overload. To maintain constant voltage at the output side of the line, 7 the generator current has to be increased from 342 amperes at no load to 370 amperes at full load, or by 8.2 per cent, and inversely, at constant-current input, the output voltage would drop off, from no load to full load, by about 8 per cent. This, with a line of 15 per cent resistance drop, is a far closer voltage regulation than can be produced by constant potential supply, except by the use of synchronous machines for phase control. | | . |

| | | | | i | | | | CHAPTER III. ; . ‘ THE NATURAL PERIOD OF THE TRANSMISSION LINE. 27. An interesting application of the equations of the long : . distance transmission line given in the preceding chapter can be | made to the determination of the natural period of a transmis- ston line; that is, the frequency at which such a line discharges an accumulated charge of atmospheric electricity (lightning), or . oscillates because of a sudden change of load, as a break of circuit, or in general a change of circuit conditions, as closing the circuit, etc.

The discharge of a condenser through a circuit containing self- inductance and resistance is oscillating (provided the resistance does not exceed a certain critical value depending upon the capacity and the self-inductance) ; that is, the discharge current alternates with constantly decreasing intensity. The frequency | of this oscillating discharge depends upon the capacity C and the self-inductance L of the circuit, and to a much lesser extent upon the resistance, so that, if the resistance of the circuit is not excessive, the frequency of oscillation can, by neglecting the resistance, be expressed with fair, or even close, approximation by the formula :

; 1 f 22VCL

An electric transmission line represents a circuit having capacity as well as self-inductance; and thus when charged to a certain potential, for instance, by atmospheric electricity, as by induction from a thunder-cloud passing over or near the line, the transmission line discharges by an oscillating current.

Such a transmission line differs, however, from an ordinary condenser in that with the former the capacity and the self- inductance are distributed along the circuit.

In determining the frequency of the oscillating discharge of such a transmission line, a sufficiently close approximation is

820

NATURAL PERIOD OF TRANSMISSION LINE 821 obtained by neglecting the resistance of the line, which, at the relatively high frequency of oscillating discharges, is small com- pared with the reactance. This assumption means that the dying out of the discharge current through the influence of the resistance of the circuit is neglected, and the current assumed ‘ as an alternating current of approximately the same frequency . and the same intensity as the initial waves of the oscillating ! discharge current. By this means the problem is essentially | simplified. 28. Let J, = total length of a transmission line; / = the dis- ° tance from the beginning of the line; r = resistance per unit length; z+ = reactance per unit length = 2 2fZ, where L = inductance per unit length; g = conductance from line to return (leakage and discharge into the air) per unit length; b = capacity susceptance per unit length = 2 2fC, where C = capacity per unit length. Neglecting the line resistance and line conductance, r=0 and g = 0, the line constants a and #, by equations (14), Chapter II, then assume the form ; a = 0 and B = V2, (1) and the line equations (17) of Chapter II become I = (A, - A,) cos Bl — i] (A, + A,) sin fl and | va | B= VES (A, + A,)00s fl (4, - 44) sin alt, or writing A,-A,=(C, and A, + A, = Cy and substituting - V2 yz L ! 7-V5-VE °) | we have and I =C, cos fl — 7C, sin pl | ; ve ys (3) | E= } Cs cos fl — 70, sin fl. : 7 |

822 - TRANSIENT PHENOMENA A free oscillation of a circuit implies that energy is neither supplied to the circuit nor abstracted from it. This means that at both ends of the circuit,2 = 0 and/ = 1,, the power equals zero. If this is the case, the following conditions may exist: : (1) The current is zero at one end, the voltage zero at the other end. | (2) Either the current is zero at both ends or the voltage is zero at both ends. (3) The circuit has no end but is closed upon itself. (4) The current is in quadrature with the voltage. This case _ does not represent a free oscillation, since the frequency depends also on the connected circuit, but rather represents a line supply- ing a wattless or reactive load. In free oscillation the circuit thus must be either open or grounded at its ends or closed upon itself. (1) Circuit open at one end, grounded at other end. 29. Assuming the circuit grounded at / = 0, open at / = l,, . we have for 1 = 0, E=E,=9, and for / = 1,, I=1,=0; hence, substituting in equations (3), at / = 0, C, = 9; hence, and I = C, cos fl Ve (4) E= —7) oo: sin fl, , and at 1 = 1,, C, cos Bl, = 0, and since C’, cannot be zero without the oscillation disappearing . altogether, - | cos fl, = 0; (5) | hence, | Al, = 2n-1)5, 6) |

NATURAL PERIOD OF TRANSMISSION LINE 323 ; where n = 1, 2,3... or any integer and nT Bl = (2n — 1) a7" (7) Substituting (1) in (6) gives B =Vxb = Qn-Ya (8) 21, or substituting for x and b, x = 2 2fL and b = 2 2fC, gives aa VIC =P Ue, | 21, or (2n — 1) = ——= 9 ve (9) is the frequency of oscillation of the circuit.

The lowest frequency or fundamental frequency of oscillation

is, for n = 1, 1 f 1 4 l, VLC ’ (10) and besides this fundamental frequency, all its odd multiples or higher harmonics may exist in the oscillation f= (2n-1)f, (11) Writing L, = 1, = total inductance, and C, = 1,C = total capacity of the circuit, equation (9) assumes the form 1 =——— . (12) f 4VLC,

The fundamental frequency of oscillation of a transmission line open at one end and grounded at the other, and having a total inductance L, and a total capacity C,, is, neglecting energy losses,

f- 1

• 4VZG,

— 824 TRANSIENT PHENOMENA while the frequency of oscillation of a localized inductance L, and localized capacity C’,, that is, the frequency of discharge of a condenser C’,, through an inductance L,, is ; 1 = ——_——.. 13) f 2rzVLC, a)

The difference is due to the distributed character of L, and C, in the transmission line and the resultant phase displacement between the elements of the line, which causes the inductance and capacity of the line elements, in their effect‘on the frequency,

• not to add but to combine to a resultant, which is the projection of the elements of a quadrant, on the diameter, or times the sum, just as, for instance, the resultant m.m.f. of a distributed armature winding of n turns of 7 amperes is not nz but : nt.

Hence, the effective inductance of a transmission line in free oscillation is

2 L,’ = a lol and the effective capacity is (14) 2 Col == LC, and using the effective values L,’ and C,’, the fundamental frequency, equation (11), then appears in the form 1 = ts; (15) hoy LC, that is, the same value as found for the condenser discharge.

In comparing with localized inductances and capacities, the distributed capacity and inductance, in free oscillation, thus are represented by their effective values (13) and (14).

30. Substituting in equations (4), .

C, = ¢, + je, (16) gives I = (¢, + je,) cos Bl | and L vs (17) : E= vi (, — je,) sin Al. |

NATURAL PERIOD OF TRANSMISSION LINE 825 By the definition of the complex quantity as vector represen- tation of an alternating wave the cosine component of the wave is represented by the real, the sine component by the imaginary term; that is, a wave of the form c, cos22/t + c, sin 2 xt is represented by c, + jc,, and inversely, the equations (17), in their analytic expression, are L = (¢, cos 2 zft + ¢, sin 2 zt) cos fl i L (18) anc e= Vz (c, cos 2 xft — c, sin 2 xft) sin Al. | Substituting (7) and (11) in (18), and writing 0=2-f,t and = = = (19) coded |) gives t= {c, cos (2n— 1) + c, sin (2n— 1)6} cos (2n —-1) t = ccos (2n — 1) (9 — 7) cos (2n — 1)r . and , L . . e= vis c, cos (2 n—1)4—c, sin (2 n—1)ot sin (2n—1)t (20) | /L a. ; | = — Vaesin (22 — 1) @ — 7) sin Qn — 1)r, | where | tan (2n — )y=2 and ¢ = Vc. 4+ ¢,%. (21) 1 In the denotation (19), 6 represents the time angle, with the complete cycle of the fundamental frequency of oscillation as one revolution or 360 degrees, and rt represents the distance angle, with the length of the line as a quadrant or 90 degrees. That is, distances are represented by angles, and the whole line is a quarter wave of the fundamental frequency of oscillation. This form of free oscillation may be called quarter-wave oscillation. The fundamental or lowest discharge waye or oscillation of the circuit then is | 2, = ccos (0 — 7,) cost . 7 (22) and éa=- vez sin (6 — 7,) sinc.

326 TRANSIENT PHENOMENA

With this wave the voltage is a maximum at the open end of the line, 7 = 1,, and gradually decreases to zero at the other end or beginning of the line, J = 0.

The current is zero at the open end of the line, and gradually increases to a maximum at / = 0, or the grounded end of the line.

Thus the relative intensities of current and potential along the line are as represented by Fig. 85, where the current is shown | as I, the voltage as E.

a = Pt Teer FT | pt opeer | Pe 3 | See Fig. 85. Discharge of current and e.m.f. along a transmission line open at one end. Fundamental discharge frequency. The next higher discharge frequency, for n = 2, gives i, = c, cos 3 (0 — 7,) cos3t (28) and a= — 6, VE sin 3. 7) sind

Here the voltage is again a maximum at the open end of the

line, / = 1,, ort = 5 = 90°, and gradually decreases, but reaches l

; zero at two-thirds of the line, 1 = =, ort = 5 = 60°, then increases again in the opposite direction, reaches a second but opposite maximum at one-third of the line, 7 = 2 ore = 5 = 30°, and decreases to zero at the beginning of the line. There is thus a node of voltage at a point situated at a distance of two-thirds of the length of the line.

The current is zero at the end of the line, / = 1,, rises to a maximum at a distance of two-thirds of the length of the line, decreases to zero at a distance of one-third of the length of the

line, and rises again to a second but opposite maximum at the

NATURAL PERIOD OF TRANSMISSION LINE 827 | beginning of the line, ? = 0. The current thus has a node at a point situated at a distance of one-third of the length of the line.

ANETEN EE EE

BAKEGEEXGEEEE

AEE NEEENEEE

BREEN ENEEY

SRR ERNEEENE

Pt | tT | | NV LeN

Fig. 86. Discharge of current and e.m.f. along a transmission line open at one end.

The discharge waves, n = 2, are shown in Fig. 86, those with n = 3, with two nodal points, in Fig. 87.

PNZTFAT Et LN

AN TRE TTAL VN

ABNENE EARP EK

NEARER :

COP RAZ eo 3

EREARVUONEAREE

Fig. 87. Discharge of current and e.m.f. along a transmission ; line open at one end.

31. In case of a lightning discharge the capacity C, is the capacity of the line against ground, and thus has no direct relation to the capacity of the line conductor against its return. The same applies to the inductance L,.

If d = diameter of line conductor, /,= height of conductor above ground, and /, = length of conductor, the capacity is

C, = 1.11 x 107"), in mf. 2 log the self-inductance is : (24) L, =2 x 10-°l, log Ah, in mh.

328 TRANSIENT PHENOMENA The fundamental frequency of oscillation, by substituting (24) in (10), is 1 7.5 X 10° . = — _ = — 25 SVL. @) that is, the frequency of oscillation of a line discharging to ground is independent of the size of line wire and its distance from the ground, and merely depends upon the length, J,, of the line, being inversely proportional thereto. We thus get the numerical values, Length of line _§ 10 20 30 40 50 60 80 100 miles 1.6 3.2 4.8 6.4 8 9.6 12.8 16 x 10cm. hente frequency, Jf, = 4700 2350 1570 1175 940 783 587 470 cycles per sec. As seen, these frequencies are comparatively low, and especially with very long lines almost approach alternator frequencies. The higher harmonics of the oscillation are the odd multiples of these frequencies. Obviously all these waves of different frequencies represented in equation (20) can occur simultaneously in the oscillating dis- ‘ charge of a transmission line, and, in general, the oscillating discharge of a transmission line is thus of the form . t= Sac, cos (2 — 1) (0 — 7,) cos (2n — 1) t, ; = (26) | e=- VE Sncsin @n 1) @- 7) sin @n- 1). | 24 | | A simple harmonic oscillation as a line discharge would require a sinoidal distribution of potential on the transmission line at the instant of discharge, which is not probable, so that probably all lightning discharges of transmission lines or oscillations produced by sudden changes of circuit conditions are complex waves of many harmonics, which in their relative magnitude depend upon the initial charge and its distribution — that is, in the case of the lightning discharge, upon the atmospheric electrostatic field of force. |

NATURAL PERIOD OF TRANSMISSION LINE 329 © The fundamental frequency of the oscillating discharge of a transmission line is relatively low, and of not much higher mag-_ - nitude than frequencies in commercial use in alternating-current circuits. Obviously, the more nearly sinoidal the distribution of potential before the discharge, the more the low harmonics predominate, while a very unequal distribution of potential, that is a very rapid change along the line, causes the higher har- monics to predominate. $2. As an example the discharge of a transmission line may be investigated, the line having the following constants per mile: r = 0.21 ohm; L = 1.2 x 107 henry; C = 0.03 x 10° farad, and of the length 7, = 200; hence, by equations (10), (19), J, = 208 cycles per sec.; 9 = 1315 ¢, and t = 0.00785 J, when charged to a uniform voltage of e, = 60,000 volts but with no current in the line before the discharge, and the line then grounded at one end, / = 0, while open at the other end, J = J,. Then, for? = Ooré = 0,7 = 0 for all values of t except t = 0; hence, by (26), cos (2n — 1) 7, = O, , and thus . Qn-1)% =5 (27) and . cos (2m — 1) (@ —7,) = sin (2n — 1)8, sin (27 — 1) (@ — 7,) = — cos (2n — 1) 8; hence, t= Ss c, sin (2m — 1) 6 cos (2n — 1) t 1 and (28) e= VE See, cos (2m — 1) Asin (2n — 1) r. 1 Also for t = 0, or 0 = 0, e = e, for all values of ¢ except + = 0; hence, by (28), : : Le . &= vi 2 Cy Sin (2n—1) t. (29)

330 TRANSIENT PHENOMENA From equation (29), the coefficients c, are determined in the

• usual manner of evaluating a Fourier series, that is, by multiply- ing with sin,(2m — 1)z (or cos (2m — 1)r) and integrating: Joecsin @m — 1) ede = yz : oan , a Dre J sin (2m — 1)tsin (2m-— 1) 7 dz. Cy ° Since ° fain (2m — 1)tsin (2m — 1) tdr ° f cos 2 (n — m)t — cos2(n +m —1)t rr which is zero for n = + m, while for m = n the term ** cos 2(n — m)t - feat J 2 d= J, 373 and : , _ cos On =O) - 2e, Jo esin @n— de = es nal Lt on 1 we have 2% _. 7 vz 2n-1 "2°C and _ de, Ve; 30) on Qn—-DxVL’ ¢ hence, ; . 4 vi 2. sin (2n — 1) 6 cos (2n — 1)t i= TaVE Se 2n-1 4 v5} . sin30cos3zt _ sin50cos5t = 3% Z sin 6 cos t + 3 + 5 +-- (31) . 50 — 382 }sin 4 cos + sin $0 cos3e one eos 5s pee , in amperes, |

NATURAL PERIOD OF TRANSMISSION LINE 381 and ; _ 4) &. cos (2n — 1) Asin (2n — 1)t em Fe 2" 2n—1 4 . cos30sin37 cos5@sindr ; = =e, }cos 0 sin t + ————- + —— + - - - r 3 5 (32) = 76,400 }cos 9 sin cee 80sin Se cos SO sim be . , in volts. ‘ 83. As further example, assume now that this line is short- circuited at one end, / = 0, while supplied with 25-cycle alter- nating power at the other end, / = /,, and that the generator voltage drops, by the short circuit, to 30,000, and then the line cuts off from the generating system at about the maximum value of the short-circuit current, that is, at the moment of zero value of the impressed e.m.f. At a frequency of f, = 25,cycles, the reactance per unit length of line or per mile is z= 2nf,L = 0.188 ohm and the impedance is ; z2=Vr+2 = 0.283 ohm, or, for the total line, 2, = Lz = 56.6 ohms; . hence, the approximate short-circuit current : ._ @ 30,000 | v= z, = 366. = 530 amp., and its maximum value is . i, = 530 x V2 = 750 amp. Therefore, in equations (26), at time ¢ = 0, or 6 = 0,e=0 for all values of + except t = 5 ; hence, sin (2n — 1) % = 0, , or, Yn, = 0,

3382 TRANSIENT PHENOMENA and thus i= >)" en cos (2n — 1) @cos(2n —1)t 1 . and . (33) , Le ; ; e= - VES ncusin (2n — 1) Asin (2n — 1) cr. However, at ¢ = 0, or 6 = 0, for all values of + except t = 5 t = 15; ’ hence, substituting in (33), i, = nc, cos (2 — 1) t. (84) 1 From equation (34), the coefficients c, are determined in the same manner as in the preceding example, by multiplying with cos (2 — 1) ct and integrating, as n 4% ¢, = — (- 1) @n—-Dr (35) hence, . . 4%< 008 (2n — 1) Acos(2n —1)t ; = pn (= 2n—1 4%, cos36cos3t cos5@cosbt = —j cos 8 cos t — ———,——— + ———_ - +: ': m 3 5 (36) s 34 cos 350 _ 956 eos # eos + _ cos 3 cose, ome eset 4 of, in amperes, and \ — 4% (LS. , in (22 — 1) Asin (2n —1)t , es VES 0) 2n-1 _ Sty /E in Oain p23 Asin 3 z | sin 50sin 5 == at sin ¢— + — of (37) 191,200 sin @sin Sms Pan St Spo rent _ 4 . 4, : in volts. |

NATURAL PERIOD OF TRANSMISSION LINE 333 The maximum voltage is reached at time 0 = 5 , and is ~ tin /E . sin3ct sindt e=— G jain + SS + + t, and since the series : . sn3dt sindt " sin t + —3— + eee =9’ the maximum voltage is e=1, Vz = 300,000 volts. As seen, very high voltages may be produced by the interrup- tion of the short-circuit current. (2a) Circwt grounded at both ends. | 34. The method of investigation is the same as in paragraph 29; the terminal conditions are, for / = 0, E=0, and for / = 1,, E =0. Substituting / = 0 into equations (3) gives C, = 0; . . hence, I = C, cos fl, 38 Be- iC, E sin a (38) | ° C ; Substituting 2 = /, in (38) gives | . L. £,=0=— 7, Vesin Al,; hence, sin fl, = 0, or Bl = na, (39) | wo

334 TRANSIENT PHENOMENA and, in the same manner as in (1), ; Bl = n=l = NT; (40) (') that is, the length of the line, 7,, represents one half wave, or | t = z, or a multiple thereof. . fee 41 21VLC 2VLC, (41) and the fundamental frequency of oscillation is 1 = ee 42 f, 1 2 i, Te ( ) and f=nf, re (48) that is, the line can oscillate at a fundamental frequency f,, for which the length, /,, of the line is a half wave, and at all multiples or higher harmonics thereof, the even ones as well as the odd ones. | This kind of oscillation may be called a half-wave oscillation. | 35. Unlike the quarter-wave oscillation, which contains only the odd higher harmonics of the fundamental wave, the half- wave oscillation also contains the even harmonics of the funda- mental frequency of oscillation. Substituting C, = c, + jc, into (38) gives | I = (c, + je,) cos pl and 7 _\ fk. (44) E = (¢, — je,) ve sin fl, and replacing the complex imaginary by the analytic expression, that is, the real term by cos 2 aft, the imaginary term by sin 2 zt, gives i = {c, cos2 aft + c, sin 2 xft} cos Al and L . | @= Ve {c, cos 2 xft — c, sin 2 xft} sin Al, !

, ; a NATURAL PERIOD OF TRANSMISSION LINE 335 and substituting

• Qaft = 8, we have (45) 2 aft = nb; then (44) gives, by (40): . t = (¢, cos nO + c, sin n8) cos nt and y (46) e= vz (c, cos nO —c, sin n8) sin nr; or writing c, = ccos ny and | (47) c, =csinny gives 4 =ccosn (9 — 7) cosnt and Z (48) | e=-6 VE sinn 6-7) sin mr , C 2 and herefrom the general.equations of this half-wave oscillation are t = Src, cos n (0 — 7,) cos nt 1 and 7 (49) Le . . | 6 =- V3 Seq sin n — 7) sin ms (2b) Circuit open at both ends.

36. For / = 0 we have | I =0; | hence, C, =0 and I =— jC, sin Bl . and 7 (60) E= ve C, cos fl, . while for 2 =1,, I = 0; .

336 TRANSIENT PHENOMENA . hence, sin fl, = 0, or Bl, = nz; (1) that is, the circuit performs a half-wave oscillation of funda- | | mental frequency, | ,-—2 - | 21,VIC (82) | and all its higher harmonics, the even ones as well as the odd ones have a frequency | f=nf, ry (53) and the final equations are | i= — Sacasinn (@ — 7) sin ne | 1 1 and 4) . 0 = VE Sncn cos 0 ~ 7) conn, | where 6 =2nft and t= a (85) | 0 . i (3) Circuit closed upon itself. . 37. Ifa circuit of length J, is closed upon itself, then the free oscillation of such a circuit is characterized by the condition that current and voltage at / = /, are the same as at / = 0, since! = 1, and | = 0 are the same point of the circuit. . Substituting this condition in equations (3) gives I =C, = C, cos fl, — jC, sin Al, and : | Ci . , (56) Vex = C, = C, cos Al, — jC, sin fl,; herefrom follows | C, (1 — cos fl,) = — 7C, sin Al,, ' (57) C, (1 — cos fl,) =— jC, sin fl,, | hence, _ (1 — cos fl,» =— __ sin’ Al, or cos fl, = 1; (58) | | | | | |

NATURAL PERIOD OF TRANSMISSION LINE 887 hence,

Bl, = 2nz; . (59) that is, the circuit must be a ‘complete wave or a multiple thereof.