LECTURE X. CONTINUAL AND CUMULATIVE OSCILLATIONS. 43. A transient is the phenomenon by which the stored energy readjusts itself to a change of circuit conditions. In an oscilla- tory transient, the difference of stored energy of the previous and the after condition of the circuit, at a circuit change, oscillates between magnetic and dielectric energy. As there always must be some energy dissipation in the circuit, the oscillating energy of the transient must steadily decline, that is, the transient must die out, at a rate depending on the energy dissipation in the cir- cuit. Thus, the oscillation resulting from a change of circuit condi- tions can become continual, that is, of constant amplitude, or cumulative, that is, of increasing am^plitude, only if a steady supply of oscillating energy occurs. Continual and cumulative oscillations thus involve a con- tinual energy supply to the oscillating system, therefore cannot be mere readjustments of circuit conditions by the dissipation of stored energy. If the continual energy supply is less than the energy dissipa- tion in the circuit, the oscillation dies out, that is, is transient, but A\nth a lowered attenuation constant. This for instance is the case with the transient in those sections of a compound circuit, in which the energy transfer constant is negative. If the con- tinual energy supply equals the energy dissipation, the oscillation is continual, and if the energy supply is greater than the energy dissipation, the oscillation becomes cumulative, that is, increas- ing in amplitude, until either the system breaks down or, by the increase of the energy dissipation, it becomes equal to the energy supply, and the oscillation becomes continual. A continual or cumulative oscillation thus involves an energy and frequency transformation, from the low-frequency or con- tinuous-current energy of the power supply of the system to the high-frequency energy of the oscillation. 119 120 ELECTRICAL DISCHARGES, WAVES AND IMPULSES This energy transformation may be brought about by the transient of energy readjustment, resulting from a change of circuit conditions, producing again a change of circuit conditions and thereby an energy readjustment by transient, etc. For instance, if in an isolated high-potential transmission line, the ground is brought within striking distance of one of the line conductors — as by the puncture of an insulator. A spark dis- charge then occurs to the ground, and the arc following the spark discharges the line by a transient oscillation, that is, brings it down to ground potential (and the other two lines, in a three- phase sj^stem, then correspondingly rise in voltage to ground, from the Y to the delta voltage). As soon as the line is dis- charged the arc ceases, that is, the spark gap to ground opens, and the line then charges again, from the power supply of the system, and its voltage to ground rises, until sufficient to jump to ground again and start a second transient oscillation, and so on continual transient oscillations follow each other, as a ^'con- tinual transient," or ''arcing ground." Oscillograms, Figs. 59 and 60, show such a series of successive or "recurrent" oscilla- tions, resulting from an arcing ground on one phase of an isolated system consisting of 10 and 21.5 miles respectively of 22,000-volt three-phase 40-cycle circuit. In this case, three transients occur during each half wave of voltage. Oscillogram, Fig. 61, shows an arcing ground oscillations on 32 miles of the same system. Here each transient oscillation persists up to the beginning of the next transient of the same half wave of circuit voltage, and the recur- rent oscihations thus tend to run into each other and approach a continuous oscillation. When successive transients run into each other, they naturally synchronize, as the first oscillation of the second transient would start at a maximum voltage point, that is, at an oscillation max- imum of the preceding transient. Oscillograms cd15073 and cd15087, Figs. 62 and 63, show the formation of a continuous oscillation by the overlapping of suc- cessive recurrent oscillations, and finally oscillograms cd 15091 and 15034, Figs. 64 and 65, show a continuous oscillation. In Figs. 62 and 63, the component recurrent oscillations are still noticeable by a periodical rise and fall of the amplitude of the wave, while in Figs. 64 and 65, the amplitude has become constant. These oscillo- CONTINUAL AND CUMULATIVE OSCILLATIONS 121 Fig. 59. — Recurrent Oscillation of Arcing Ground, in 10 Miles of 22,000- volt Three-phase 40-cycle Transmission Line. Fig. 60. — Recurrent Oscillation of Arcing Ground, in 21.5 Miles of 22,000- volt Three-phase 40-cycle Transmission Line. Fig. 61. — Change from Recurrent to Continuous Oscillation of Arcing Ground, in 32 Miles of 22,000-volt Three-phase 40-cycle Transmission Line. 122 ELECTRICAL DISCHARGES, WAVES ANDJMPULSES grams, Figs. 62 to 65, were taken on an artificial transmission line.* Oscillations of the type 64 and 65 are industrially used, as ''sing- ing arc, " in wireless telegraphy, and are produced by shunting a suitable arc by a circuit containing capacity and inductance in series with each other. Fig. 62. — Semi -continuous Recurrent Oscillation of Arcing Ground in Transmission Line. Fig. 63. — Semi-continuous Hecurrent Oscillation of Arcing Ground in Transmission Lino. * "Design, Construction and Test of an Artificial Transmission Line," J. H. Cunningham, A. I. E. E. Transact., 19U. CONTINUAL AND CUMULATIVE OSCILLATIONS 123 Fig. 64. — Continuous Oscillation of Arcing Ground in Transmission Line. Fig. 65. — Continuous Oscillation of Arcing Ground in Transmission Line. 44. However, the formation of continuous oscillations, Figs. 61 to 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 and 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 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 arc, as illustrated by Fig. 66: the transient volt-ampere charac- teristic of a short high-temperature metal arc, between titanium and carbon. In this figure, the stationary arc characteristic, that is, the relation between arc voltage and arc current in stationary conditions, is shown in dotted lines, and the drawn line shows the cycle existing between arc current and arc voltage during a cyclic change of current, from zero to 4.1 amperes and back to zero, within y^o of ^ second, with the current varying approximately as a sine function of the time. As seen, for rising current, the arc 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 hysteresis 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 cycle, thus both increase together. Such cumulative oscillations are represented for instance by Fig. 46, page 99. * "Instability of Electric Circuits," A. I. E. E. Proceedings, Jan., 1914. CONTINUAL AND CUMULATIVE OSCILLATIONS 125 Transient Volt-Ampere Characteristic of High Temperature Metal Arc Ti - C. 1 1 en \\ \ ior> ■ \ \ \ \ ^ Vi A ■;^ 100 ' \ \ \ K N :^ ^^ ZTtZn ^ ' nA c ] Ami 3 eres i V 0 Fig. 66. 126 ELECTRICAL DISCHARGES, WAVES AND IMPULSES 45. The frequency of the oscillations usually is the natural frequency of the oscillating circuit or section of circuit; but it may be some of the higher harmonics of the generator wave, where such harmonic is near the natural frequency of the system. The latter seems to be the case in the continual arcing ground oscillation in Figs. 44 and 45, page 98. In Fig. 44, the beginning of the disturbance, apparently a harmonic of the generator wave builds up by the energy supply through a beginning arc, and then builds down again, by being slightly out of resonance with a multiple of the natural frequency of the circuit. In Fig. 45, the arc has completely developed, and one of the harmonics of the generator wave appears as a steady continuous oscillation. Continual and cumulative oscillations naturally are the most dangerous phenomena in an electric circuit, for the reasons: 1. While each individual wave or wave train may not be sufficient in energy and in over-voltage to do harm, the recurrence through seconds, minutes or hours finally destroys the insulation, either by the sustained over-voltage, or by the decrease of dielec- tric strength resulting from the heating of the insulation caused by the high frequency. The over-voltage protective devices fre- quently do not offer protection, as the over-voltage of the oscilla- tion is insufficient to cause a discharge over the lightning arrester. The only effective protection seems to be a continuous dissipa- tion of the oscillating energy by a resistance closing the oscillat- ing circuit. In general, a moderate capacity would be connected in series with such damping resistance, and would be chosen so as to allow the high frequency to pass practically unobstructed, while practically stopping the passage of the machine frequency, and the waste of power, incident thereto. 2. A continual oscillation involves an energy transformation from the power supply of the system to the oscillation frequency. The energy of the oscillation which gives its destructiveness thus is not limited to the small amount of the stored magnetic and dielectric energy of the system, but is supplied continuously from the engine or turbine power. 3. The continual oscillation is not a resonance phenomenon which depends on the frequency of the exciting disturbance just coinciding with one of the natural frequencies of the oscillating system, and which thus can occur only very rarely. The dis- CONTINUAL AND CUMULATIVE OSCILLATIONS 127 turbance in the system, as lightning, change of load, etc., is only the exciting cause which starts the energy transformation to the oscillating frequency by the arc, etc., and the frequency with which the oscillation occurs 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 Hues 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. LECTUEE 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, L^f, (1) v>^here $ = magnetic flux or number of lines of magnetic force, and n the number of times which each line of magnetic force interlinks with the current i. The capacity is the ratio of the dielectric flux to the voltage, C=l (2) where ^ 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 antennae) 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, page 11. 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 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. ' Fig. 67. — Magnetic Field of Circuit. 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 force, 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- 130 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ductors out of phase with each other, as in a three-phase circuit), only the hues of magnetic force within the space from conductor to return conductor need to be considered. Thus, the resultant magnetic flux of a circuit consisting of conductor A and return conductor B, at distance s from each other, consists of the lines of magnetic force surrounding A up to the distance s, and the lines of magnetic force surrounding B up to the distance s. The former is attributed to the inductance of conductor A, the latter to the inductance of conductor B. If both conductors have the same size, they give equal inductances; if of unequal size, the smaller conductor has the higher inductance. In the same manner in a three-phase circuit, the inductance of each of the three con- ductors is that corresponding to the lines of magnetic force sur- rounding the respective conductor, up to the distance of the return conductor. B. Calculation of inductance, 47. If r is the radius of the conductor, s the distance of the return conductor, in Fig. 68, the magnetic flux consists of that external to the conductor, from r to s, and that internal to the conductor, from 0 to r. Fig. 68. — Inductance Calculation of Circuit. At distance x from the conductor center, the length of the mag- netic circuit is 2 irx, and ii F = m.m.f. of the conductor, the mag- netizing force is F 2 7ra;' and the field intensity OC = 4 tt/ = hence the magnetic density 2F ■ — J X 2iiF (4) (5) ROUND PARALLEL CONDUCTORS. 131 and the magnetic flux in the zone dx thus is d^ ='^dx, (6) X and the magnetic flux interhnked with the conductor thus is nd* = h^dx, (7) X ^ ^ hence the total magnetic flux between the distances Xi and X2 is '^2 2 finF dx . thus the inductance ^ p 2 lint L]f = 'i*li= pa^'^'^- (8) I X 1. External magnetic flux. Xi = r; X2 = s; F = i, sls this flux surrounds the total current; and n = 1, as each line of magnetic force surrounds the conductor once, ^t = 1 in air, thus: (9) 2. Internal magnetic flux. Assuming uniform current density throughout the conductor section, it is xi = 0; X2 = r; -. = (^-j , as the flux is produced by a part of the current only; and n =[-] ? as each line of magnetic force surrounds only a part of the con- ductor 24*^ = ^, (10) and the total inductance of the conductor thus is =x L = Li + L2 = 2 j log - + ^ > per cm. length of conductor, (11) or, if the conductor consists of nonmagnetic material, /x = 1 : L = 2Jlog^ + ^j. (12) 132 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This is in absolute units, and, reduced to henry s, = 10^ absolute units : L = 2Jlog^+^|lO-9/iper cm. (13) = 2 hog ^ + 1 1 10-9 h per cm. (14) ( r 4) 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. A 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 = logioe. ROUND PARALLEL CONDUCTORS. 133 The flux then consists of three parts: $1, between the conductors, giving the inductance Li = 2 log , r and shown shaded in Fig. 69. 2, inside of conductor A, giving the inductance ^3, the flux external to A, which passes through conductor B and thereby incloses the conductor A and part of the conductor F B, and thus has a m.m.f. less than i, that is, gives - < 1. 1 That is, a line of magnetic force at distance s — ri exists, which is shown shaded in Fig. 5; but $2 and $3 are zero, and the inductance is L = 2\og^-^^10-'h. (15) 184 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 increase 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- ductance as determining factors. The exact determination of the inductance of round parallel conductors at short distances from each other thus is only of theoretical, but rarely of practical, importance. An approximate estimate of the inductance L3 is given by con- sidering two extreme cases: (a) The return conductor is of the shape Fig. 70, that is, from s — r to s -\- r the m.m.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 + ^. (a) For s — r < X < s -]- r, it IS hence by (8), F_s-\-r — x_s-i-r x i ~ 27 " ~27 27' _ 1*'+' s -\-r dx r+'dx J s—r ?' ^ U s—r ^ s + r s-\-r ^ — — log 2. s — r by the approximation log (1 dz x) = ± a; + (16) (17) (18) ROUND PARALLEL CONDUCTORS. 135 it is log^; = >og^^^ - log?^^ = log(l + r)- ,og(l - ^) = 2^ hence i3 = i±Ix?-^-2 = 2r. (19) r s s hence, (b) For s — r < X < s, it is and for s < a: < s + r, it is and integrated this gives L3 = 21og^-+^i±^^ogi+-^-^^^^og-^ 3, (23) s — r r^ s r^ s — r a^id by the approximation (18) this reduces to L^ = — , (24) o 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 = Li -\- L2 -{- Ls = 2|log^ + ^+^jlO-»A. (25) T However, - can be considered as the approximation of — log o t\ s 1 )= log , and substituting this in (25) gives, by S/ S T com- bining log h log = log - r s — r r L = 2|log^ + ^j 10-^ h, (26) 136 ELECTRIC DISCHARGES, WAVES AND IMPULSES. where s = distance between conductor centers, as the closest approximation in the case where the distance between the con- ductors is small. This is the same expression as (13). In view of the secondary phenomena unavoidable in the con- ductors, equation (26) appears sufficiently accurate for all practi- cal purposes, except when taking into consideration the secondary phenomena, as unequal current distribution, etc., in which case the frequency, conductivity, etc., are required. D. Calculation of capacity. 49. The lines of dielectric force of the conductor A are straight radial lines, shown dotted in Fig. 72, and the dielectric equipoten- tial lines are concentric circles, shown drawn in Fig. 72. Fig. 72. — Electric Field of Conductor. li e = voltage between conductor A and return conductor B, and s the distance between the conductors, the potential difference between the equipotential line at the surface of A, and the equi- potential line which traverses B, must be e. If e = potential difference or voltage, and I = distance, over which this potential difference acts. G -J = potential gradient, or electrifying force, (27) ROUND PARALLEL CONDUCTORS. 137 G e and K = - — -„ = - — -, = dielectric field intensity, (28) where v- is the reduction factor from the electrostatic to the electromagnetic system of units, and y = 3 X 10^0 cm. sec. = velocity of light; (29) the dielectric density then is D = kK = -^j, (30) 4 TTVH where k = specific capacity of medium, = 1 in air. The dielectric flux then is where A = section of dielectric flux. Or inversely: e = i^^^. (32) If then "^ = dielectric flux, in Fig. 68, at a distance x from the conductor A, in a zone of thickness dx, and section 2 ttx, the voltage is, by (32), 2'irXK = ^V*^, (33) K X and the voltage consumed between distances Xi and x^ thus is e] = I de = log -J (34) Jx: K Xi hence the capacity of this space : ^^ 2z;Mog-^ ^Xi The capacity of the conductor A with the return conductor at distance s then is the capacity of the space from the distance 0^1 = r to the distance x^. = s, hence is, by (35), C = per cm. ■ (36) 2 v'^ log - 138 ELECTRIC DISCHARGES, WAVES AND IMPULSES. in absolute units, hence, reduced to farads, C= ^'^ ^ /per cm., (37) r and in air, for k = 1: C = — — — / per cm. (38) 2z;2 1og- r is the capacity, per conductor, or " capacity to neutral," as often stated. Immediately it follows: the external inductance was, by (9), Li = 2 log - 10~^ h per cm., and multiplying this with (38) gives (39) CLi — — : ^2 or V"Li that is, the capacity equals the reciprocal of the external inductance Li times the velocity square of light. The external inductance Li would be the inductance of a conductor which had perfect con- ductivity, or zero losses of power. It is Vlc = velocity of propagation of the electric field, and this velocity is less than the velocity of light, due to the retardacion 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 Li, 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 CLi = '^, (40) v where k = specific capacity or permittivity, jjl = permeability of the medium. ROUND PARALLEL CONDUCTORS. 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 Hne 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 same 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 applied to the inductance and capacity of a con- ductor with ground return, by using as distance 5 twice the dis- 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 image 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, 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 J inch diameter and 25 feet above ground, then, assuming perfect conductivity of the ground surface, the inductance would be and r = 1''; s = 2 X 25' = 600'', hence - = 2400, L = 2 Hog - + J ^ 10-9 = 16.066 X 10-^ h. r 4 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 X 75' = 1800", hence - = 7200, and L = 18.264 X 10-9 h, or only 13.7 per cent higher. In this case, however, the ground sec- 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 hetioeeji 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, J _ 2 _ $1 J^m — — — — J ^l tj where <^2 is the magnetic flux interlinked with the second circuit, which is produced by current ii in the first circuit. A In the same manner as the self-inductance L, the mutual inductance L„j between two circuits is calculated; while the (external) self-inductance cor- o ^ responds to the magnetic flux between the dis- tances r and s, the mutual inductance of a conductor Y^ ^ A upon a circuit ah corresponds to the magnetic flux ° ° produced by the conductor A and passing between Fig. 73. ^j^g distances Aa and Ah, Fig. 73. Thus the mutual inductance between a circuit AB and a circuit ah is mutual inductance of A upon ah, L^' = 21og^ XlO-9/i, mutual inductance of B upon ah, hence mutual inductance between circuits AB and ab, T — T " — T ' where Aa, Ah, Ba, Bb are the distances between the respective conductors, as shown in Fig. 74. 142 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If one or both circuits have ground return, they are replaced by the circuit of the overhead conductor and its image conductor below ground, as discussed before. If the distance D between the circuits AB and ah is great compared to the dis- tance S between the conductors of circuit AB, and the distance s between the con- ductors of circuit ah, and 0 = angle which the plane of circuit AB makes with the distance D, '^ the corresponding angle of circuit ah, as shown in Fig. 74, it is approximately S s ^a = D + - cos 0 + - cos -yjr, S s Ah = D -}- — cos 0 — ^ cos -^j S s Ba = Z) — - cos 0 + - cos yjr, Bh = D — -^Goscf) — -cos t/^, Fig. 74. (42) hence 2 log 7 (D-{- -^coQcf>-- cos ylrjlu- ^cos0+-cosyl = 2 log iD-{--cos(f) +^cos -^jf D— ^cos(/)— -cos-^/^j f - cos 0 - 2 COS -f 1 10-9/i D »-(! 10-9 /i COS 0+0 COS = 2 log 'S COS0 ^cos-v/^ D - log hence by (18) - cos 0 + - COS ir D Lm = 2 (^cos0+|cost)^-g COS0 COS ^1^] w 10-9 /i; ROUND PARALLEL CONDUCTORS. 143 thus ^^^2g.cos^^costio_,^_ (43) For (j) = 90 degrees or i/^ = 90 degrees, Lm is a 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- tric flux produced by this voltage between the conductors of the other circuit. That is where "^2 is the dielectric flux produced between the conductors of the second circuit by the voltage Ci between the conductors of the first circuit. If e = voltage between conductors A and B, the dielectric flux of conductor A is, by (36), ^ = Ce = — ^^ ^ (44) 2 v^ log ^ where R is the radius of these conductors and S their distance from each other. This dielectric flux produces, by (32), between the distances Aa and Ah, the potential difference , 2z;2\p" Aa ..„. and the dielectric flux of conductor B produces the potential difference hence the total potential difference between a and b is 2v^'¥, AbBa, K substituting (44) into (47), ^"-^' = — '°^I-ai' ^''^ ,, , e , Ab Ba e — e = -; log -r- -7^> , .S "^AaBb log R 144 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and the dielectric flux produced by the potential difference e'' — e' between the conductors a and b is ^. = KG log AhBa 2z;2 1og-log^ hence the mutual capacity 2 ?;2 log - log r r or, by approximation (18), as in (43), K^T) kSs cos cf) cos '^ 2 D^y^log - log - / £1/ 10'/. (48) (49) 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, ei and 62, where ei -{- 62 = 2 e, as for instance one conductor grounded: ei = 0, 62 (50) the dielectric fluxes of the two conductors are different, and that oi A is: Ci"^; that of 5 is: 02'^, where ei e ^2 C2 = -, e (51) and ci + C2 = 2, the equations (45) to (49) assume the forms: , 2z;2ci^, Aa „ 2 z;2c2^ . Ba e" - e' = \ C2 log 7. (52) (53) jclogg-clog^^j, (54) ROUND PARALLEL CONDUCTORS. U5 ^m — 2 D^ log - log r r 2f2 1og^log^ ti («^'°sg-^''°glf fio-V •-(S'(sr-'^' (55) and by (42) ; + C2 |log(] C2l0g Ba cilog Aa S coscj) — s cos yjr -s COS yjrX , /. iScos^+scos-^/rX ) D j-'°4l 2Z^ -)\ ( , /^ , aS COS 0 + S COS -^X , / 1 + S COS 4> — s COS '>/^ 2^D and this gives: (C2 — Ci) S cos -^/r (Ci + C2) >Ss COS (/) COS yjr Z) 2Z)2 hence 2t;2 1og^log^ , sScos-v/r . Ss COS (f) COS -yjr D 2D' and for ei = 0, and thus Ci = 0, C2 = 2: KS COS ^ 1 1 1 '^ ^^^ ^ jDz^Mog^log^ lOV, (57) (58) hence very much larger than (49). However, equation (58) appHes 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 i = 2Jlog^ + ^jlO-/., C = — - — Wf (26) (37) 2 t;2 log 146 ELECTRIC DISCHARGES, WAVES AND IMPULSES. apply equally to the two-wire single-phase circuit, the single wire circuit with ground return, or the three-phase circuit. In the expression of the energy per conductor: Ce' (59) and of the inductance voltage e' and capacity current i\ per conductor: i' = 2 irfCe, ) (60) i is the current in the conductor, thus in a three-phase system the Y or star current, and e is the voltage per conductor, that is, the voltage from conductor to ground, which is one-half the voltage between the conductors of a single-phase two-wire circuit, —7^ the Vs voltage between the conductors of a three-phase circuit (that is, it is the Y or star voltage), and is the voltage of the circuit in a conductor to ground, s is the distance between the conductors, and is twice the distance from conductor to ground in a single con- ductor with ground return.* If the conductors of a three-phase system are arranged in a triangle, s is the same for all three conductors; otherwise the different conductors have different values of s, and A B 0 the same conductor may have two different values of ^ "^ o s, for its two return conductors or phases. For instance, in the common arrangement of the qA three-phase conductors above each other, or beside each other, as shown in Fig. 75, if s is the distance between middle conductor and outside conductors, the O Q distance between the two outside conductors is 2 s. Fig. 75. The inductance of the middle conductor then is: L = 2Jlog^+^jlO-A. (61) The inductance of each of the outside conductors is, with respect to the middle conductor: * See discussion in paragraph 50. ROUND PARALLEL CONDUCTORS. 147 L = 2hog^+^|l0-9/i. (62) With respect to the other outside conductor: L = 2|logy + ^|l0-9/i. (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 can be calculated by using the average of the three distances s, s, 2 s between the conductors, 4 . s that is, ^ s, or more accurately, by using the average of the log - ? o r s 2 s log - and log — , that is : r r 21og- +log — ~ ^^^^ In the same manner, with any other configuration of the line conductors, in case of transposition the inductance and capacity can 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 ELECTRIC 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, a A, B, C are the three three-phase con- ductors, and a and b the conductors of the second circuit, as shown in Fig. 76, and if ii, 2*2, is are ^^ oB the three currents, with their respective phase angles 71, 72, 73, and i the average current, b a denoting: o O Fig. 76. ^1 t2 H . conductor A gives : Aa LJ = 2ci cos {^ - 7i) log TI' conductor B: LJ' = 2c2 cos (/3 - 120°- 72) log 1^. conductor C: hence, LJ^' = 2 C3 cos (/3 - 240° - 73) log^ ; L^ = 2 I ci cos (/3 - 71) log ^ + C2 cos (/3 - 120° - 72) log |^, -I- C3 cos (^ - 240° - 73) log ^ ( 10-^ h, and in analogous manner the capacity Cm is derived. In these expressions, the trigonometric functions represent a rotation of the inductance combined with a pulsation.