LECTURE X. 

INDUCTANCE AND CAPACITY OF ROUND 
PARALLEL CONDUCTORS. 

A. Inductance and capacity. 

43. 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, 

£ = ?- (i) 

i/ 

where <i> = 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, 


where \f/ 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 

119 


'iJBLtiGTRIC DISCHARGES, WAVES AND IMPULSES. 


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. 59. — 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. 59. 

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. 59, 
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- 


ROUND PARALLEL CONDUCTORS. 


121 


ductors out of phase with each other, as in a three-phase circuit), 
only the lines 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. 

44. If r is the radius of the conductor, s the distance of the 
return conductor, in Fig. 60, 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. 60. — Inductance Calculation of Circuit. 

At distance x from the conductor center, the length of the mag 
netic circuit is 2 irx, and if F = m.m.f. of the conductor, the mag- 
netizing force is 


and the field intensity 

hence the magnetic density 

(B 


2F 

x 


(4) 
(5) 


122 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 
and the magnetic flux in the zone dx thus is 

d^=^fdx, I (6) 

and the magnetic flux interlinked with the conductor thus is 


X 

hence the total magnetic flux between the distances x\ and z2 is 

rx*2 

thus the inductance 


X 


1. External magnetic flux, xi = r; xz = s; jP = i, as this flux 
surrounds the total current; and n = 1, as each line of magnetic 
force surrounds the conductor once, ju = 1 in air, thus: 

?-""-:- <»> 

2. Internal magnetic flux. Assuming uniform current ^density 
throughout the conductor section, it is 


Cx\2 
-J , 

as each line of magnetic force surrounds only a part of the con- 
ductor 


and the total inductance of the conductor thus is 

C 9 // i 
L = LI + L2 = 2 j log- +T( per cm. length of conductor, (11) 

or, if the conductor consists of nonmagnetic material, ju = 1 : 

(12) 


ROUND PARALLEL CONDUCTORS. 


123 


This is in absolute units, and, reduced to henry s, = 109 absolute 
units : 


= 2 j log ? + 1 1 10-9 h per cm. 


(13) 
(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. 

45. 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. 


Fig. 61. — 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. 61 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. 61. 

* 0.4343 = log10*, 


124 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

The flux then consists of three parts: 

3>i, between the conductors, giving the inductance 


and shown shaded in Fig. 61. 
$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 

J5, and thus has a m.m.f. less than i, that is, gives - < 1. 

% 

That is, a line of magnetic force at distance s — r<x<s + r 
incloses the part q of the conductor B, thus incloses the fraction 

-y- of the return current, and thus has the m.m.f. 

F -1 q 

1 ' ~7v" 

An exact calculation of the flux <£3, and the component inductance 
LS 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 permits. 

The magnetic flux $3 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 $1, 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 $1 exists, which is shown shaded in Fig. 5; but <J>2 
and $3 are zero, and the inductance is 

. (15) 


ROUND PARALLEL CONDUCTORS. 


125 


That is, in other words, with small conductors and moderate 
currents, the total inductance in Fig. 61 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. 62, that is, from 
s — r to s + f the m.m.f. varies uniformly. 


B 


Fig. 62. Fig. 63. 

Inductance Calculation of Cable. 

(6) The return conductor is of the shape Fig. 63, 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 + r. 

(a) For s — r < x < s + r, it is 


-f r — x 
2r 


2r 2r 


hence by (8), 


/»s+rg_|_r fa r*+*dx 
J8_r r x Ja_r r 


s — r 


by the approximation 

log (1 ± x) = 


(16) 

(17) 
(18) 


126 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

it is 

, s + r . s + r , s — r , L . r\ , /., r\ rtr 
log— = log — - log — = log (l + -) - log(l - -) = 2-g, 

hence 


r 
(6) For s — r < x < s, it is 


f-l-sl^^r^h (20) 


and for s < x < s + r, it is 

:' '- 

hence, 


and integrated this gives 

fc-aiog^ + ^log'-f'- ii^log^-3, (23) 

o — if o / o — / 

and by the approximation (18) this reduces to 

L,-^, (24) 

O 

that is, the same value as (19); and as the actual case, Fig. 60, 
should lie between Figs. 61 and 62, 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 = L! + L2 + L3 

(25) 


However, - can be considered as the approximation of — log 
s 

( 1 -- )= log - , and substituting this in (25) gives, by com- 
\ s/ s — r _ 


o _ y o o 

bining log -- h log - = log - : 

T S T T 


(26) 


ROUND PARALLEL CONDUCTORS. 


127 


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. 

46. The lines of dielectric force of the conductor A are straight 
radial lines, shown dotted in Fig. 64, and the dielectric equipoten- 
tial lines are concentric circles, shown drawn in Fig. 64. 


Fig. 64. — Electric Field of Conductor. 

If 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 = - = potential gradient, or electrifying force, (27) 


128 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

and K = - — 2 = - — ^ = dielectric field intensity, (28) 

4 Trf 4 irV L 

where v2 is the reduction factor from the electrostatic to the 
electromagnetic system of units, and 

v = 3 X 1010 cm. sec. = velocity of light; (29) 

the dielectric density then is 


where K = specific capacity of medium, = 1 in air. The dielectric 
flux then is 


where A = section of dielectric flux. Or inversely: 

-IS?* : || (32) 

If then ^ = dielectric flux, in Fig. 60, at a distance x from the 
conductor A, in a zone of thickness dx, and section 2 TTZ, the voltage 

is, by (32), 

, 
de 


and the voltage consumed between distances x\ and x2 thus is 

/»*2 2v2^ Xz 
ei2 = / de = —^-Llog-, (34) 

hence the capacity of this space : 

C2 r K /'Q^^ 

i — —5 — * (po) 


The capacity of the conductor A against the return conductor B 
then is the capacity of the space from the distance Zi = r to the 
distance x^ = s, hence is, by (35), 

C = — per cm. (36) 

2t;2log- 


ROUND PARALLEL CONDUCTORS. 129 

in absolute units, hence, reduced to farads, 

C= *1Q9 /per cm., (37) 

2z;2log- 

and in air, for K = 1 : 

1H9 

(38) 


Immediately it follows: the external inductance was, by (9), 

Li = 2 log- 10~9 h per cm., 
and multiplying this with (38) gives 


or 


CL> = £' 


(39) 


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 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 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 

CLt = ^, (40) 

where K = specific capacity or permittivity, /* = permeability of 
the medium. 


130 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

E. Conductor with ground return. 

47. 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. 59, 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 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 s 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, 


ROUND PARALLEL CONDUCTORS. 131 

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 = i"; s = 2 X 25' = 600", hence - = 2400, 


L = 2 ] log - + 10~9 = 16.066 X 10~9 h. 

T Z \ 


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- 


132 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

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. 

48. 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 

Lim — ~ -- -r i 
ll li 

where $2 is the magnetic flux interlinked with the second circuit, 
which is produced by current i\ in the first circuit. 

. In the same manner as the self-inductance L, 

the mutual inductance Lm between two circuits is 

calculated; while the (external) self-inductance cor- 

° B responds to the magnetic flux between the dis- 

tances r and s, the mutual inductance of a conductor 

k a A upon a circuit ab corresponds to the magnetic flux 

0 ° produced by the conductor A and passing between 

Fig- 65' the distances Aa and Ab, Fig. 65. 

Thus the mutual inductance between a circuit AB and a circuit 

ab is mutual inductance of A upon ab, 


Jiutual inductance of B upon ab, 


hence mutual inductance between circuits AB and ab, 
Lm = Lm" — Lm , 


where A a, Ab, Ba, Bb are the distances between the respective 
conductors, as shown in Fig. 66. 


ROUND PARALLEL CONDUCTORS. 


133 


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 ab is great compared to the dis- 
tance S between the conductors of circuit 
A B, and the distance s between the con- 
ductors of circuit ab, and 0 = angle which 
the plane of circuit AB makes with the 
distance D, ty the corresponding angle of 
shown in Fig. 66, it is 


circuit a&, as 
approximately 


Fig. 66. 


Aa = D -f- £ cos 0 + - cos 
Ab = D + — cos 0 — - cos 

A A 

Ba = D — — cos 0 -{- ~ cos 

2i 2i 

Bb = D — - cos 0 — ^ cos 


(42) 


hence 


m = 21og- 


n , 

D+ 


2 log 


D2 - I- cos 0 - ~ cos 

D2- (7:COS0 -fxCOS 


= 2 


log 


2 COS 0 — jz COS 


- log 1 - 


x io~s /?, 


hence by ( 

T __ rt 

18) 

PC s 

s \2 AS s 

)2 

D2 

134 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

thus 

2 **!()-.*. (43) 


For 0 = 90 degrees or ty = 90 degrees, Lm is a minimum,, and 
the approximation (43) vanishes. 

G. Mutual capacity between circuits. 

49. 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 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), 

t = Ce = - , (44) 


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 A b, the potential difference 

Aa 

g' 


and the dielectric flux of conductor B produces the potential 
difference 

2v2-fr, Ba. /A0* 

e = - — l°g^r> (4w 

K no 

hence the total potential difference between a and b is 

2v2iK AbBa. 


substituting (44) into (47), 

e Ab Ba 


ROUND PARALLEL CONDUCTORS. 135 

and the dielectric flux produced by the potential difference e" — ef 
between the conductors a and b is 

. K€ , Ab Ba 


2 v2 log- log ^ 


hence the mutual capacity 

K 


2 v2 log - log — 


or, by approximation (18), as in (43), 

Cm= «&«*»*«»* 1(y ,. (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, e\ and e2, where e\ + e2 = 2 e, as 
for instance one conductor grounded: 

ei = 0, 62 = e, (50) 

the dielectric fluxes of the two conductors are different, and that 
of A is: crt/r; that of B is: c2^, where 


= f2. 

2 e ' 

and 

d + c2 = 2, 

the equations (45) to (49) assume the forms; 

Aa 


K AO 

2 v2^ , Ba 


(52) 
(53) 

// / — ( Y1" i i j s'< -, ./I rt / 

e" - e' = - -^ j c2 log BT - ci log^r [ , (54) 

ir r*\r\ /\ r\ \ % - <f 


136 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

Cm = -, j C2 log^ - c, log^ | lO-9/ 

« -i- *i • » I Bo Ab \ 


2 v2 log - log -5 


(55) 


y £42): 


Ba Aa 

~5l -cilogT7 


COS 0 — S COS 


+ SCOS' 


+ 


ir^-)- 


COS 0 — S COS 


2D 


and this gives : 

fe — Ci) s cos 


D 


+ 


+ Cfc) /Ss COS 0 


(56) 


hence 

c.=- 


x . S COS ^ , / N 

fe-cO—^+fe + d) 2D, 


and for 61 = 0, and thus c\ = 0, c2 = 2: 


COS 


+ 


cos <t> 


9 n 

^ LJ 


io9/, 


(57) 
(58) 


hence very much larger than (49). However, equation (58) 
applies only, if the ground is at a distance very large compared 
with Z), as it does not consider the ground as the static return of 
the conductor B. 

H. The three-phase circuit. 

50. The equations of the inductance and the capacity of a 
conductor 

(26) 


109/ 


(37) 


ROUND PARALLEL CONDUCTORS. 137 

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: 

Li' 

(59) 


and of the inductance voltage e' and capacity current i', per 
conductor: 

' = (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, —T=- the 

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 c 
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 o^ 

three-phase conductors above each other, or beside 
each other, as shown in Fig. 67, if s is the distance 
between middle conductor and outside conductors, the OQ 
distance between the two outside conductors is 2 s. Fig. 67. 

The inductance of the middle conductor then is: 

(61) 

The inductance of each of the outside conductors is, with respect 
to the middle conductor: 

* See discussion in paragraph 47. 


138 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

(62) 
With respect to the other outside conductor: 

L = 2Jlogy + ^jlO-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. 67. 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 2s 

log - and log -5- , that is: 
r o 


• 3 

In the same manner, with any other configuration of the line 
conductors, in case of transposition the inductance and capacity 

Q 

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 


ROUND PARALLEL CONDUCTORS. 139 

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. Q^ 

Thus, if A, B, C are the three three-phase con- 
ductors, and a and b the conductors of the second 
circuit, as shown in Fig. 68, and if ii, iz, i3 are °C OB 
the three currents, with their respective phase 
angles 71, 72, 73, and i the average current, b a 

denoting: 


o 

Fig. 68. 

1\ 12 ^3 


conductor A gives: 
conductor B: 

conductor C: 

Lm'" = 2 c3 cos (0 - 240° - 73) log^?>* 
hence, 

Lm = 2 ) ci cos 03 - 71) log 4r + C2 cos 08 - 120° - 72) log |?, 


Lm" = 2 c2 cos 08 - 120°- 72) log!?, 

no 


4- c3 cos (0 - 240°- 73) log ^ | 10-9 /i, 


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.