Elementary Lectures on Electric Discharges, Waves and Impulses, and Other Transients — part 6 of 7

CONTINUAL AND CUMULATIVE OSCILLATIONS 121 , pote ao oe _ LO fee, parca! peent: aan 1 hos suite peeoeenn ' wT -_ : OTT ORNS PL ES ta To ar es ST en en eg SSE Fig. 59.— Recurrent Oscillation of Arcing Ground, in 10 Miles of 22,000- volt Three-phase 40-cycle Transmission Line. Wee tls es Ba ae ln. 2M Ree al, “ + i Pees a Cece ees ene nena 1 eee < a peat Met SSS WE ose fh DL et OE a pee i OS ed EE Re CP Se : PE pa Se aS OEE ED A wr FA ’ bid!» cenameemmemmrenll'de aa a reremreennresrenans ni ' ae : “ a a ° u : = Wehee Yt - : a a ‘ ee : 8 RE . a ae RT BANE Py lg sath retin eta aa 7 ; _ : a aa oh ; os By ee Wohl pe nan: EL i ot Je aaa aon : : agte tes at, as a were Rg re ESE ty AE ee ettag wb. . Fig. 60. — Recurrent Oscillation of Arcing Ground, in 21.5 Miles of 22,000- volt Three-phase 40-cycle Transimission Line. (Oe Tg he ae ued tet nae nd wage eR ee ae ehh. Aa Ra aS eeaaicaca ieee Ne E eee Sade 'a "ind yee ™ _ be, Lan . . _ ma if! Vw vee Tey Rigen? jit a ae an oe Pon a “ vot. a ye _ * ara ari oe eee Peet pe a Nye ED en A ate EE “oo A SY a Mp ec y mae Eas . . oa ae pe EE sey fe OE TLR heey gt Fig. 61.—Change from Recurrent to Continuous Oscillation of Arcing Ground, in 32 Miles of 22,000-volt Three-phase 40-cycle Transmission ~ Line.

1220 ELECTRICAL DISCHARGES, WAVES AND IMPULSES

• grams, Figs. 62 to 65, were taken on an artificial transmission line.*

Oscillations of the type 64 and 65 are industrially used, as “‘sing-

; 2 5 WW ” ra hd © , . bd

ing are,” in wireless telegraphy, and are produced by shunting a

suitable are by a circuit containing capacity and inductance in

series with each other.

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Ey le ice WE ate Sg 0 SERRE OR AEE Dig tine ak aia aie B02

SED OR ea Pe et ER eh te a UM Ps FEE ge et Tet,

: Vig. 62.—Semi-continuous Reeurrent Oscillation of Arcing Ground in ‘Transmission Line.

BER SOS ee PA ae Seb OR phe aad gr RR es a eee eee

OR Ec EONS A fie Ne EEE a EQ A Me a Fhe aes

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REE el ST RY ve A ag Be RRS Og he ie

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tpt le Aa PE Peete TS eet ge

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BU Ree PE Te a ee Pe hag RES

a a a ade Ng et Cee gh me ee OH ae oe Sete

ee Te Sg Ae Tet Pa a is be fg EE

ee ee eg Th UL a bag et oo gee ea .

Fig. 63. —Semi-continuous Recurrent Oscillation of Arcing Ground in Transmission Line.

• “Design, Construction and Test of an Artificial Transmission Line,’’ J. H. Cunningham, A. I. E. LE. Transact., 1911.

CONTINUAL AND CUMULATIVE OSCILLATIONS 123 NEE Rag 2555 So ae acy Reg TE Si fgg SE igh SPEER gE SNE RS nC SE Ae fr rcs Aa Wes ta rece Beee nay tty Caren ha tas Vn: . re Aer Ve ee Pe Mee NAS ES: Ee Re OE ge Taek OS tad tate eae Sa ER SUN tee Bilge cay Ce Ee eT eae ei PSA nn ES | Ag oda oped flamt tae os ee, i wig Ps nM NINE oc WA cc chia RC here Le a RRO LST ocd TEN retin SSN DA cern LR ceo CR coer Fcc TE Rasen TET ale aT BT eres es fe cad Aoneae “pte eB webengtinne vege en ee oY odecee Sta? Ma PR ORO gd CO RIDIES , OM SPICS ol TRLLAY ANSEL INT RE ESR, LNT Rae PN ae oe f. Set geW he he Veoh 1 ARSE 2 Yee ee Bediidat Ete fer y AS Sha Ria gs ete lat SER Pe Ree sen Sapa etm oe bl Meat LO Ve Beh ah ect aa ea Eo ek agai Berea ai te Ce TSG Ras Pe ake Fa re ag ee eee pare neg Mabe gf tent Sas) om pan hee Sees BERS. ton Rees aciemersocn fet wae Soe =. AI ari BORED aK girs PRN CD RL SOME Eat bs SOR TEES OREO Lae RRR TR, ORR TE Tee EF AAD EMO RESS AP LESIY , UTES, OLDE ne Eo e A ROTARY STAR ECM AN aa ae See er eee SoA . Fig. 64. — Continuous Oscillation of Arcing Ground in Transmission Line. Apt OP BE Pe ek gi SM cet i Se ll et eR * Sue Sere on eet Tea eee PE By BOS TST ee EE ie a aise gee eee oak ge igh So cape act nade MOE BESS OB oe ese PREPS a" tage Mev oeee Bet 8) ABS Aa See ah ge oe aes ener SE Te Nae sehen fie REE oe og? ESS U ee Oa 9 SE A ae BR safes iy te oda, Ogee SOE ETE CAT OP Sea POE Ca ES teats ee pam Ete BRE ee ee coe * ra TRS fect eS, Fae Deb ek nears > nes ai WSS ees ee tebe ae Be cae Bi en Ra ie tin, See Le Chaar A OnE eA Seco TET oe gee ee Wal ey TES a ee Bey, eagle ies eo a Sey WR a ee gs “hey: agent DTT gets PT Ra AE Pe a ee EP ge re eR Sie Te ee en, Re BREST OF 2 gol, Rial ORME BEE TET BSE ER EL Tig. 65. — Continuous Oscillation of Arcing Ground in Transmission Line. . 44. Lowever, the formation of continuous oscillations, Figs. 61 io 65, from the recurrent oscillations, Figs. 59 and 60, is not a mere running together and overlapping of successive wave trains. In Fig. 59, the succeeding oscillation cannot start, until the pre- ceding oscillation has died out anc a sufficient time elapsed, for the line to charge again to a voltage which is high enough to dis- charge to ground and so start the next oscillation, that is, to store the energy for the next oscillation. If then, with an overlap of successive oscillations, no dead period occurs, during which the energy, which oscillates during the next wave train, is supplied to the line, this energy must be supplied during the oscillation, that is, there must be such a phase displacement or lag within the oscil- lation, which gives a negative energy cycle, or reversed hysteresis loop. “Thus, essential for such a continual oscillation is’ the

CONTINUAL AND CUMULATIVE OSCILLATIONS 127 turbance in the system, as lightning, change of load, ete., is only the exciting cause which starts the energy transformation to the oscillating frequency by the are, etc., and the frequency with which the oscillation oecurs then is determined by the circuit constants. Or, as is often stated: the electric arc has no fre- quency of its own, but oscillates with whatever frequency the circuit is able to oscillate. Thus such oscillations are not un- common, and have in the last years been observed, measured and recorded in| numerous instances, and experimentally produced in lines and high-potential transformer windings. The continual oscillations in transmission lines usually seem to be recurrent oscillations, as in Figs. 59 and 60, while in high-potential trans- former windings, due to their much lesser damping, continuous oscillations seem to be more common, as in Fig. 46. Our knowl- edge of these phenomena is however still extremely incomplete.

ROUND PARALLEL CONDUCTORS. 129 . . lines of force is still more complicated, and varies during the cyclic change of current. The calculation of such more complex magnetic and dielectric fields becomes simple, however, by the method of superposition of fields. As long as the magnetic and the dielectric flux are pro- _ portional respectively to the current and the voltage, — which is the case with the former in nonmagnetic materials, with the latter . _ for all densities below the dielectric strength of the material, — the resultant field of any number of conductors at any point in space is the combination of the component fields of the individual conductors. : / _ a “\ -
/ / \ " y f <> A (RS
i WGN
{ ENO. WSZUSZ | | CONS | \ OG
\ . J / \ SY Yo / . \ ss, a 7 . Fig. 67. — Magnetic Field of Cireuit. Thus the field of conductor A and return conductor B is the combination of the field of A, of the shape Fig. 8, and the field of B, of the same shape, but in opposite direction, as shown for the magnetic fields in Fig. 67. All the lines of magnetic force of the resultant magnetic field must pass between the two conductors, since a line of magnetic foree, which surrounds both conductors, would have no m.m.f., and thus could not exist. That is, the lines of magnetic force of A beyond B, and those of B beyond A, shown dotted in Fig. 67, neutralize each other and thereby vanish; thus, in determining the resultant magnetic flux of conductor and return conductor . (whether the latter is a single conductor, or divided into two con-

124 ELECTRICAL DISCHARGES,.WAVES AND IMPULSES | existence of a hysteresis loop, formed by the lag of the effect be- -hind the cause. Such a hysteresis loop exists in the transient are, as illustrated by Fig. 66: the transient volt-ampere charac- teristic of a short high-temperature inetal arc, between titanium and carbon. In this figure, the stationary are characteristic, that is, the relation between are voltage and are current in stationary conditions, is shown in dotted lines, and the drawn line shows the cycle existing between are current and are voltage during a cyclic change of current, from zero to 4.1 amperes and back to zero, within 745 of a second, with the current varying approximately as a sine function of the time. As seen, for rising current, the are voltage is materially higher than for decreasing current. Close to zero current, the arc has ceased, and Geissler tube con- duction passes the current through the residual vapor stream. _ Other hysteresis cycles than those of the arc are instrumental in the energy supply to other systems of continual oscillation. Thus, for instance, the hysteresis cycle -between synchronizing force and position displacement supplies the energy of the con- tinual or cumulative oscillation, called hunting, in synchronous machines, as alternators, synchronous motors and converters. . The mechanism, by which the hystcresis cycle supplies the energy ; of continual oscillations, has been investigated in the case of the . hunting of synchronous machines,* but is still practically un- known in the case of continual oscillations between magnetic and dielectric energy in electric circuits. Recurrent oscillations, as in Fig. 59, must be or very soon be- ‘come continual, that is, the successive wave trains are of approx- imately constant amplitude, since each starts with the same energy, the stored energy of the supply system. Continual oscillations, however, in which the energy supply is through a hysteresis cycle, may be cumulative: the area of the hysteresis cycle, that is, the energy supply, depends on and increases with the voltage and current of the oscillation, and the voltage and current, that is, the intensity of the oscillation, depends on and increases with the energy supply, that is, the area of the hysteresis , eycle, thus both increase together. Such cumulative oscillations are represented for instance by Fig. 46, page 99.

• “Instability of Electric Circuits,” A. I. IE. IX. Proceedings, Jan., 1914.

ROUND PARALLEL CONDUCTORS. 131 ; and the magnetic flux in the zone dz thus is “2uF . db =" dr, (6) x and the magnetic flux interlinked with the conductor thus is 2 unk . ndb = 2 unl dz, (7) x hence the total magnetic flux between the distances 2; and 22 is’ nag = fan, x x thus the inductance 2 np]i _ “2 nF dx La ME = [Pout (8)

1. External magnetic flux. 2, = r; 2% = s; F = 17, as this flux surrounds the total current; and n = 1, as each line of magnetic foree surrounds the conductor once. yw = 1 in air, thus: 92 dx 8 L, = | “xr =2 log = (9)
2. Internal magnetic flux. Assuming uniform current density throughout the conductor section, it is x = 0; m=T7; += (3) . . Z r as the flux is produced by a part of the current only; and » = (=) as each line of magnetic force surrounds only a part of the con- ductor rQ2uedz wu In= [Pee a8, 2 0 rl 2 (10) and the total inductance of the conductor thus is ’ L=Ih+L, =2 log + ri per cm. length of conductor, (11) or, if the conductor consists of nonmagnetic material, » = 1: | L=2 loge +3. (12) or 4

ROUND PARALLEL CONDUCTORS. 185 | it is str_ str_ sa T_ r\ —t)=3% . | log —, = log a log a log (1 + “) log (3 ; 20 | hence r s 8 (6) Fors —r<2z<-s, itis F- 1/x—s+r? pri-3( a), (20) and fors < xz < s +7, itis F_ifs+r-—<2x? 773 a) , (21) hence, nd . 1 td s x—-s+r x str/s+r— z? dx tam f° [2-4 Ae and integrated this gives | _ Ss (s +r)? str _ (s—r)? So Lz =2 log —,. + Ps) log 3s re log 5 a 3, (23) and by the approximation (18) this reduces to Ls = ar (24) s that is, the same value as (19); and as the actual case, Fig. 69, should lie between Figs. 70 and 71, the common approximation of the latter two cases should be a close approximation of case 4. That is, for conductors close together it is L = Ly + L, + Ls = STTr ye? -9 2 | log . +47} 10 h. (25) | However, , can be considered as the approximation of — log (1 _ “)= log 7 , and substituting this in (25) gives, by com- . os s—T s Ss. bining log —— + log —— = log-: | L= 2} log: +4} 10-2, (26)

ROUND PARALLEL CONDUCTORS. 133 The flux then consists of three parts: ©,, between the conductors, giving the inductance Ly = 2 log-—, and shown shaded in Tig. 69. be, inside of conductor A, giving the inductance _#, . Ly = 5 $3, the flux external to A, which passes through conductor B and thereby incloses the conductor A and part of the conductor B, and thus has a in.m.f. less than 7, that is, gives . <1. That is, a line of magnetic force at distances ~-r<2<s+r incloses the part q of the conductor B, thus incloses the fraction — of the return current, and thus has the m.m.f. { Faye i. . z 7 An exact calculation of the flux $3, and the component inductance L; resulting from it, is complicated, and, due to the nature of the phenomenon, the result could not be accurate; and an approxima- tion is sufficient in giving an accuracy as great as the variability of the phenomenon perinits. The magnetic flux &; does not merely give an inductance, but, if alternating, produces a potential difference between the two sides of conductor B, and thereby a higher current density on the side of B toward A; and as this effect depends on the conduc- tivity of the conductor material, and on the frequency of the current, it cannot be determined without having the frequency, etc., given. The same applies for the flux ,, which is reduced by unequal current density due to its screening effect, so that in the limiting case, for conductors of perfect conductivity, that is, zero resistance, or for infinite, that is, very high frequency, only the magnetic flux @, exists, which is shown shaded in Fig. 5; but 2 and 4; are zero, and the inductance is . | L = 2log-—" 10h. (15)

LECTURE XI. INDUCTANCE AND CAPACITY OF ROUND PARALLEL CONDUCTORS.

A. Inductance and capacity.

46. As inductance and capacity are the two circuit constants which represent the energy storage, and which therefore are of fundamental importance in the study of transients, their calcula- tion is discussed in the following.

The inductance is the ratio of the interlinkages of the mag- netic flux to the current,

nP L= >? (1) where ® = magnetic flux or number of lines of magnetic force, ; and n the number of times which eaeh line of magnetic force interlinks with the current 7. The capacity is the ratio of the dielectric flux to the voltage, Vv C=-, (2) where W is the dielectric flux, or number of lines of dielectric force, and e the voltage which produces it.

With a single round conductor without return conductor (as wireless antenne) or with the return conductor at infinite dis- tance, the lines of magnetic force are concentric circles, shown by drawn lines in Fig. 8, page 10, and the lines of dielectric force

; are straight lines radiating from the conductor, shown dotted in Fig. 8.

Due to the return conductor, in a two-wire circuit, the lines of — magnetic and dielectric force are crowded together between the conductors, and the former become eccentric circles, the latter circles intersecting in two points (the foci) inside of the con- ductors, as shown in Fig. 9, page11. With more than one return conductor, and with phase displacement between the return currents, as in a three-phase three-wire circuit, the path of the

128

132 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This is in absolute units, and, reduced to henrys, = 10° absolute : units: L= 2hlog? +f 10~ h per cm. (13) 5 sl 9 = - = ~ 2 . 4 2) log + Gg 10 h per cm (14) In these equations the logarithm is the natural logarithm, which is most conveniently derived by dividing the common or 10 logarithm by 0.4343.* . C. Discussion of inductance.

48. In equations (11) to (14) s is the distance between the con- ductors. If s is large compared with r, it is immaterial whether as s is considered the distance between the conductor centers, or between the insides, or outsides, etc.; and, in calculating the in- ductance of transmission-line conductors, this is the case, and it . therefore is immaterial which distance is chosen as s; and usually, in speaking of the “distance between the line conductors,” no attention is paid to the meaning of s.

UN 4 yee EY, x —4—-— §---—}}-- . Fig. 69.— Inductance Calculation of Cable. ;

However, if s is of the same magnitude as r, as with the con- ductors of cables, the meaning of s has to be specified.

Let then in Fig. 69 r = radius of conductors, and s = distance

| between conductor centers. Assuming uniform current density in the conductors, the flux distribution of conductor A then is as indicated diagrammatically in Fig. 69.

• 0.4343 = logice.

ROUND PARALLEL CONDUCTORS. 13T and k= ae = — °_ = dielectric field intensity (28) 4nv? Axvl , . where v? is the reduction factor from the electrostatic to the electromagnetic system of units, and v= 3 X 10" em. sec. = velocity of light; (29) the dielectric density then is .

• Ke D=x«KkK = Dave’ (30) where x = specific capacity of medium, = 1 in air. The dielectric flux then is KeA w=4D= 4, (31) where A = section of dielectric flux. Or inversely: . 4 xvl é= A Y, (32) If then W = dielectric flux, in F ig. 68, at a distance z from the conductor A, in a zone of thickness dz, and section 2 rz, the voltage is, by (82), 2 de = 4 av? dz v ; 2 wxK ; 2 _ 2 pdt , (33) K x and the voltage consumed between distances x; and a, thus is ty 2y2V, x. , él =f de = log, (34) hence the capacity of this space: Vv CH =F —; . (35)

• 2v log= . U1 The capacity of the conductor A with the return conductor at . distance s then is the capacity of the space from the distance xz, =r to the distance x, = s, hence is, by (35), C = ——per cm. (36) s 2 v log =

1384 ELECTRIC DISCHARGES, WAVES AND IMPULSES.

That is, in other words, with small conductors and moderate currents, the total inductance in Fig. 69 is so small compared with the inductances in the other parts of the electric circuit - that no very great accuracy of its calculation is required; with large conductors and large currents, however, the unequal current distribution and resultant inerease of resistance become so con- siderable, with round conductors, as to make their use uneconom- ical, and leads to the use of flat conductors. With flat conductors, — - however, conductivity and frequency enter into the value of in- duetanee as deterniining factors.

The exact determination of the inductance of round parallel conductors at short distances from cach other thus is only of theoretical, but rarely of practical, importance.

An approximate estimate of the inductance Zg is given by con- sidering two extreme cases:

(a) The return conductor is of the shape Fig. 70, that is, from s—rtos+r the mm.f. varies uniformly.

Fig. 70.. Fig. 71. Inductance Calculation of Cable. , . (b) The return conductor is of the shape Fig. 71, that is, the , m.m.f. of the return conductor increases uniformly from s — r to s, and then decreases again from s to s + 7. (a) Fors—-r<ax<s-+y, itis F str-x s+r x Sn; Pomdienier, -aieet- wt (16) ; 1 27 2r 2r hence by (8), tetrs ter de str dx a aan amr ? x ar «(OT ; str, str = eT op SET LE , are 2, (17) by the approximation 2 log(ltz)=t2+5+ we (18)

ROUND PARALLEL CONDUCTORES. | 139

E. Conductor with ground return. .

50. As seen in the preceding, in the electric field of conductor A and return conductor B, at distance s from each other, Fig. 9, the lines of magnetic force from conductor A to the center line CC’ are equal in number and in magnetic energy to the lines of mag- netic force which surround the conductor in Fig. 67, in concentric circles up to the distance s, and give the inductance L of conductor A. The lines of dielectric force which radiate from conductor A up to the center line CC’, Fig. 9, are equal in number and in dielec- tric energy to the lines of dielectric force which issue as straight lines from the conductor, Fig. 8, up to the distance s, and repre- sent the capacity C of the conductor A. The center line CC’ is a dielectric equipotential line, and a line of magnetic force, and there- fore, if it were replaced by a conducting plane of perfect conduc- tivity, this would exert no effect on the magnetic or the dielectric field between the conductors A and B.

If then, in the electric field between overhead conductor and ground, we consider the ground as a plane of perfect conductivity, we get the sanie electric field as between conductor A and central plane CC’ in Fig. 9. That is, the equations of inductance and capacity of a conductor with return conductor at distance s can be immediately apphed to the inductance and capacity of a con- ductor with ground return, by using as distance s twice the cis- tance of the conductor from the ground return. That is, the -— inductance and capacity of a conductor with ground return are

_ the same as the inductance and capacity of the conductor against its tmage conductor, that is, against a conductor at the same dis- tance below the ground as the conductor is above ground.

As the distance s between conductor and image conductor in the case of ground return is very much larger — usually 10 and more times — than the distance between conductor and overhead return conductor, the inductance of a conductor with ground return is much larger, and the capacity smaller, than that of the same conductor with overhead return. In the former case, how-

ever, this inductance and capacity are those of the entire circuit, : since the ground return, as conducting plane, has no inductance and capacity; while in the case of overhead return, the inductance of the entire circuit of conductor and return conductor is twice, the capacity half, that of a single conductor, and therefore the total inductance of a circuit of two overhead conductors is greater,

188 ELECTRIC DISCHARGES, WAVES AND IMPULSES. . in absolute units, hence, reduced to farads, . 1Q° C=——— fper cm., (37) 2v* log - r and in air, for « = 1: 10° C= ——f per cm. (38) 2 ° 2v 108; is the capacity, per conductor, or ‘‘ capacity to neutral,’’ as often stated. Immnediately it follows: the external inductance was, by (9), I, = 2 log = 10-9 h per cm., and multiplying this with (38) gives Cl, = 5; ye or (39) 1 C — vl, ’ that is, the capacity equals the reciprocal of the external inductance L, times the velocity square of light. The external inductance L; would be the inductance of a conductor which had perfect con- . ductivity, or zero losses of power. It is , 1 80F= ee== . VLC = velocity of propagation of the electric field, and this velocity is ; less than the velocity of light, due to the retardation by the power dissipation in the conductor, and becomes equal to the velocity of light v if there is no power dissipation, and, in the latter case, L : would be equal to Z;, the external inductance. The equation (39) is the most convenient to calculate capacities in complex systems of circuits from the inductances, or inversely, to determine the inductance of cables from the measured capacity, etc. More complete, this equation is - oe CL, = ‘A ? (40) where x = specific capacity or permittivity, » = permeability of the medium.

ROUND PARALLEL CONDUCTORS. 141 tion available for the return current, assuming its effective width | as 800 feet, would be 80,000 square feet, or 60 million times greater than the section of the overhead conductor. Thus only with very high resistance soil, as very dry sandy soil, or rock, can a considerable increase of the inductance of the over- head conductor be expected over that calculated by the assump- tion of the ground as perfect conductor. ; F. Mutual induction between circuits. 51. The mutual inductance between two circuits is the ratio of the current in one circuit into the magnetic flux produced by this current and interlinked with the second circuit. That is, In ==, V1 Ve where ®, is the magnetic flux interlinked with the second circuit, which is produced by current 7 in the first circuit. oA In the same manner as the self-inductance JD, the mutual inductance LZ, between two circuits is calculated; while the (external) self-inductance cor- oB responds to the magnetic flux between the dis- tances r and s, the mutual inductance of a conductor b a 4 upona circuit ab corresponds to the magnetic flux _ _° © produced by the conductor A and passing between Fig. 73. the distances Aa and Ab, Fig. 73. Thus the mutual inductance between a circuit AB and a circuit _ ab is mutual inductance of A upon ab, a Aa —9 Lin = 210g 75 X 107% h, mutual inductance of B upon ab, Ba Ln!’ = 2 log 7, X 10-%h, hence mutual inductance between circuits AB and ab, Lin = 1,” - Ln’, . Ba X Ab,., = 2 log SBS x. Bb 107 h, (41) where Aa, Ab, Ba, Bb are the distances between the respective conductors, as shown in Fig. 74. .

ROUND PARALLEL CONDUCTORS. 143 thus 5 Lm = 2OS LOS b COS ¥ ant C08 F197, (43) For ¢ = 90 degrees or YW = 90 degrees, Lm is &@ minimum, and the approximation (43) vanishes. G. Mutual capacity between circuits. - 52. The mutual capacity between two circuits is the ratio of the voltage between the conductors of one circuit into the dielec- trie flux produced by this voltage between the conductors of the other circuit. That-is c, = 2. ™ ei @ ? where W,2 is the dielectric flux produced between the conductors of the second circuit by the voltage e, between the conductors of the first circuit. If e = voltage between conductors A and B, the dielectric flux of conductor A is, by (36), v= Ce=—2, (4a) 2v* log R where # is the radius of these conductors and S their distance , from each other. This dielectric flux produces, by (32), between the distances Aa and Ab, the potential difference ,_2vW, Aa e = —— log 7 (45) and the dielectric flux of conductor B produces the potential difference 2 B el = aoe log He (46) hence the total potential difference between a and 6 is ng 2 20, Ab Ba. ; en et = 108 77 Be’ (47) substituting (44) into (47), e — el = lo Ab Ba lop 3 Aa BB’ eR

140 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the capacity less, than that of a single conductor with ground _ . return.

The conception of the image conductor is based on that of the ground as a conducting plane of perfect conductivity, and assumes that the return is by a current sheet at the ground surface. As regards the capacity, this is probably almost always the case, as even dry sandy soil or rock has sufficient conductivity to carry, distributed over its wide surface, a current equal to the capacity current of the overhead conductor. With the magnetic field, and thus with the inductance, this is not always the case, but the con- ductivity of the soil may be much below that required to conduct the return current as a surface current sheet. If the return cur- rent penetrates to a considerable depth into the ground, it may be represented approximately as a current sheet at some distance below the ground, and the “image conductor” then is not the - image of the overhead conductor below ground, but much lower; that is, the distance s in the equation of the inductance is more, and often much more, than twice the distance of the overhead conductor above ground. However, even if the ground is of relatively low conductivity, and the return current thus has to penetrate to a considerable distance into the ground, the induc- tance of the overhead conductor usually is not very much increased, as it varies only little with the distance s. For instance, if the overhead conductor is 3 inch diameter and 25 feet above ground, then, assuming perfect conductivity of the ground surface, the inductance would be .

r=1"; s§=2 X 25’ = 600”, hence = = 2400, and L= 2} tog? +i 107 = 16.066 X 10h. If, however, the ground were of such high resistance that the cur- rent would have to penetrate to a depth of over a hundred feet, and the mean depth of the ground current were at 50 feet, this would give s = 2 X75’ = 1800’, hence ° = 7200, and L = 18.264 X 107A,

or only 13.7 per cent higher. In this case, however, the ground sec-

1440 BLECTRIC DISCUARGES, WAVES AND IMPULSES. — . and the dielectric flux produced by the potential difference e’’ — e’ between the conductors a and b is Ab Ba Win = ——_< | T= Br? 2 v? log * lo 8 © da Bb 8708 B hence the mutual capacity K Ab Ba 2 = a log 2 S10" C Je loz lo 8 8 Fa Bh 10°f, (48) B 7108 or, by approximation (18), as in (43), C= KSS COS @ COS a 10°f. (49) 2 D*v"log s log = r od This value applies only if conductors A and B have the same voltage against ground, in opposite direction, as is the case if their neutral is grounded. If the voltages are different, e: and e2, where e + e = 2e, as for instance one conductor grounded: '“ @) = 0, €2 = €, (50) the dielectric fluxes of the two conductors are different, and that of A is: qV; that of B is: eV, where . a1 | qQ= @’ (51) €2 C2 =-) | . e€ and Cy + Co = 2, the equations (45) to (49) assume the forms: | 2vcVP,. Aa e’ = — log Zp" (52) . 20°, Ba e” = —~ log B” (53) 2v°V Ba Aa ed ==) log gy alos se | , (54)

ROUND PARALLEL CONDUCTORS. 145 | ~~ * § Ba _ Aa} 19-5 Ca = ; B ) 2 lee Bp clog SF | 10 f 2 v? log log > K Ab\ /Ba* , (55) “tel Cs 2 0? log? log = Aa} \Bb r ode . and by (42): l Ba l Aa) Co °8 Bb Cy °8 Ab § _ — Reese — sees v) :( nse = + ce} log(1 sp log {1 aH _ Scosg +005 ¥) _ ( Sovs ¢ — scos¥') cx) log (1 + Seog S008 Y log {1 +S and this gives: — 108 C2) Ss cos ‘ _ (ce as cos v 4 (e, + Ce) ase ¢ COs Y (56) hence C, = K atc — 6) £0 cane (co + cy) Ss cos¢ cos +] 10° f 20? low Slow D 2D? v* log «log R| | (57) and for e; = 0, and thus ¢ = 0, c. = 2: C,, = KOS vv F 4 S cos 4] 10°f, (58) Dv? log? log? 2D dl | | hence very much larger than (49). However, equation (58) applies only, if the ground is at a distance very large compared _ with D, as it does not consider the ground as the static return of . the conductor B. . H. The three-phase circuit. 53. The equations of the inductance and the capacity of a conductor = f ,> fa 9 . L=2)log +] 10h, (26) C=—— 10°F | (37) s 2 v? log -

ROUND PARALLEL CONDUCTORS. 147 . =~ Jog? +# 19-9 32 L 2} log? + 5 { 10 h. (62) With respect to the other outside conductor: $y 28 Blige Lp = Q- oo ~ {107 R. 63 L 2) log + 44 10 u (63) |

The inductance (62) applies to the component of current, which returns over the middle conductor, the inductance (63), which is larger, to the component of current which returns over

’ the other outside conductor. These two currents are 60 degrees displaced in phase from each other. The inductance voltages, which are 90 degrees ahead of the current, thus also are 60 degrees displaced from each other. As they are unequal, their resultant is not 90 degrees ahead of the resultant current, but more in the one, less in the other outside conductor. The inductance voltage of the two outside conductors thus contains an energy component,

which is positive in the one, negative in the other outside conductor. That is, a power transfer by mutual inductance occurs between the outside conductors of the three-phase circuit arranged as in Fig. 75. The investigation of this phenomenon is given by C. M. Davis in the Electrical Review and Western Electrician for April 1, 1911.

If the line conductors are transposed sufficiently often to average their inductances, the inductances of all three conductors, and also their capacities, become equal, and ean be calculated by using the average of the three distances s, s, 2s between the conductors, that is, : s, or more accurately, by using the average of the log °,

s 25s .

• and log —, that is: log » and og —, tha 2 2 log? + log = ———--———' 64 3 (64)

In the same manner, with any other configuration of the line conductors, in case of transposition the inductance and capacity ean be calculated by using the average value of the log ° between the three conductors.

The calculation of the mutual inductance and mutual capacity between the three-phase circuit and a two-wire circuit is made

148 £LECTRIC DISCHARGES, WAVES AND IMPULSES. in the same manner as in equation (41), except that three terms appear, and the phases of the three currents have to be con- . sidered. oA Thus, if A, B, C are the three three-phase con- ductors, and a and 6 the conductors of the second circuit, as shown in Fig. 76, and if 2, 2, i; areQG OB the three currents, with their respective phase angles yi, Ye, v3, and 7 the average current, b a . ° ° denoting: Fig. 76. a4.,28.. 8. Cc = Fi > C2 2 , C3 Fi ’ conductor A gives: Ly,’ = 2c, cos (8 — 71) log 42, Ab . conductor B: " ° Ba Lm’’ = 2 ce cos (8 — 120°— ye) log conductor C: ; mr ° Ca, Lm’”” = 23 cos (8 — 240° — 3) log Gi hence, Aa Ba Lim = 24 ¢1 cos (8 — log —— — 120° ~ log =- } 1 cos (8 — ¥;) log Ab + 2008 (8 — 120° — 2) 08 By’ | ° , Ca — ; + ¢3 cos (8 — 240°— 3) log Cb 107° hk, and in analogous manner the capacity C, is derived. In these expressions, the trigonometric functions represent a rotation of the inductance combined with a pulsation.

INDEX. PAGE Acceleration as mechanical transient... 02.0... cee cee eee ceca eee 4 single-energy transient... 00... ee eee eee ee este neee 8 Admittance, natural or surge, of circuit... eee eee eee eee eee OL, 84 Alternating current in Hine as undaniped oscillation. ...........0.252. 97 phenotnena as transients... cee een eee ee eens 9 reduction to pormanents. 6.0.0.0. eee eee eee eee 9 Alternators, momentary short-cireuit currents......0.....0 00002000 e. BF CONSTPUCHION. 60. cece ence tent eeeseeeeeseese 40 calculation... 0 eee eee eee eter eeceereee = 44 ; Areing grounds... 2.0.0.2 eee ee teen ee eeeeteenene OF as cause of cuimulative oscillations. .............00eeeeeee ee 120 Armature transient of alternator short circuit... ........ 2c eee eee eee = 4h Attenuation of transient, see Duration. Cable inductance, calculation. 2.00.0. ccc cc cee cece ne ceeesese Ol, 132 Capacity... cece teen eee eee ene ete eee eteceeecereceses 18 Caleulation. . 0... cece cnet ee ere teeta eteeeeeee 136 of eiventit, definition. 0.0. ec eee cece tenn eeees 12 CUITONL. 0 ee en eee en ee ee eee ene e eee aeeee 13 Gefinition. 0. cece ee ete ee enetereree 198 efeetive, of line transient... 00.0000 ee ee eens 7d . CQUAON. ee tect c eee ewetee eee 138 ; and inductance caleulation of Ciree-pliase eireuit............. 146 CAVPACTPY AND INDUCTANCIS OF ROUND PARALLEL CONDUCTORS. 00002 cece eee ene cecnes 128 Capacity, mutual, caleulation...0. 0006006 cee ee ee ee eee LAB, 147 Specific. eee eee eee este eee LG, 17, 18 Charge, electric, of conductor... 0.60 e enc eeeeeeeeee 14 Charging current... 0.0.0. eee ee eee eee cee cee ce eeeseeecee 4B Circuit, diclectrie. 0. eee eee eee ee eee LA, 17, 18 of distributed capacity and inductance, also see Line. ClOCUIG. cece t eee t nee e eee eee e 17, 18 INAGNCTIO. oe eee ee eee ene tree eeeees ld, 17, 18 Closed compound-cireuit transient... 0.0.0.2 eee eee ee eee eee LL, 12 Combination of effective and reactive power. 2.2.0... 6.00 ee eee eee 100 tYQUSIONE. Ce cee ee eee eee eee eeceees es 100 of standing and traveling waves. ..........000e000e02.--2... 100 COMPOUND CERCUIT OSCILLATION........................ 108 Conipound eireuit, power flow... 00.000 cece ee eee eee ee cece = 90 velocity unit of length... ccc eee ee eee eee es = M2 oscillution of closed circuit. 00.0.0... cee ee ee eee eee eee TEE, 112 of open circuit. 2... cee ere reer eeteeeeeeees LIE Condenser current... 0... ce nen e nen eee teen ene 13 Conduetance. 0.0 ce ccc e ect e eee et cetentectcecees 18 effeetive, of Jine transient... 00.00.0000. eee cease = 18 Conduetivity, cleetrie. 00.000 ec cee eee ens 18 , CONTINUAL AND CUMULATIVE OSCILLATIONS. .......... 119 Continual energy supply in continual and cuniulative oseillations....119, 123 oscillation of hunting synchronous machines................. 124 Continuous oscillation. 60.0600 0 0 ec eee een eect veces 120 149