CHAPTER XIII. 

DISTRIBUTED CAPACITY, INDUCTANCE, RESISTANCE, AND 
LEAKAGE. 

107. As far as capacity has been considered in the 
foregoing chapters, the assumption has been made that the 
condenser or other source of negative reactance is shunted 
across the circuit at a definite point. In many cases, how- 
ever, the capacity is distributed over the whole length of the 
conductor, so that the circuit can be considered as shunted 
by an infinite number of infinitely small condensers infi 
nitely near together, as diagrammatically shown in Fig. 83. 


iiiimiiiiumiiiT 

TTTTTTTTTT.TTTTTTTTTT 


i 


Fig. 83. Distributed Capacity. 

In this case the intensity as well as phase of the current, 
and consequently of the counter E.M.F. of inductance and 
resistance, vary from point to point ; and it is no longer 
possible to treat the circuit in the usual manner by the 
vector diagram. 

This phenomenon is especially noticeable in long-distance 
lines, in underground cables, and to a certain degree in the 
high-potential coils of alternating-current transformers for 
very high voltage. It has the effect that not only the 
E.M.Fs., but also the currents, at the beginning, end, and 
different points of the conductor, are different in intensity 
and in phase. 

Where the capacity effect of the line is small, it may 
with sufficient approximation be represented by one con- 


DISTRIBUTED CAPACITY. 159 

denser of the same capacity as the line, shunted across the 
line. Frequently it makes no difference either, whether 
this condenser is considered as connected across the line at 
the generator end, or at the receiver end, or at the middle. 

The best approximation is to consider the line as shunted 
at the generator and at the motor end, by two condensers 
of \ the line capacity each, and in the middle by a con- 
denser of | the line capacity. This approximation, based 
on Simpson's rule, assumes the variation of the electric 
quantities in the line as parabolic. If, however, the capacity 
of the line is considerable, and the condenser current is of 
the same magnitude as the main current, such an approxi- 
mation is not permissible, but each line element has to be 
considered as an infinitely small condenser, and the differ- 
ential equations based thereon integrated. Or the pheno- 
mena occurring in the circuit can be investigated graphically 
by the method given in Chapter VI. § 37, by dividing the 
circuit into a sufficiently large number of sections or line 
elements, and then passing from line element to line element, 
to construct the topographic circuit characteristics. 

108. It is thus desirable to first investigate the limits 
of applicability of the approximate representation of the line 
by one or by three condensers. 

Assuming, for instance, that the line conductors are of 
1 cm. diameter, and at a distance from each other of 50 cm., 
and that the length of transmission is 50 km., we get the 
capacity of the transmission line from the formula — 

C = 1.11 X 10 -«K/ -=- 4 loge 2 d/ 8 microfarads, 
where 

K = dielectric constant of the surrounding medium = 1 in air ; 

/ = length of conductor = 5 x 106 cm. ; 

d = distance of conductors from each other = 50 cm. ; 

8 = diameter of conductor = 1 cm. 

Since C = .3 microfarads, 

the capacity reactance is x — 106 / 2 TT NC ohms, 


160 ALTERNATING-CURRENT PHENOMENA. 

where N '= frequency; hence, at N = 60 cycles, 

x = 8,900 ohms ; 

and the charging current of the line, at E = 20,000 volts, 
becomes, ^ = E / x = 2.25 amperes. 

The resistance of 100 km of line of 1 cm diameter is 22 
ohms ; therefore, at 10 per cent = 2,000 volts loss in the 
line, the main current transmitted over the line is 

2,000 
/ = -^- = 91 amperes, 

representing about 1,800 kw. 

In this case, the condenser current thus amounts to less 
than 2^ per cent., and hence can still be represented by the 
approximation of one condenser shunted across the line. 

If the length of transmission is 150 km., and the voltage, 
30,000, 

capacity reactance at 60 cycles, x = 2,970 ohms ; 

charging current, i0 = 10.1 amperes ; 

line resistance, r = 66 ohms ; 

main current at 10 per cent loss, 7= 45.5 amperes. 

The condenser current is thus about 22 per cent of the 
main current, and the approximate calculation of the effect 
of line capacity still fairly accurate. 

At 300 km length of transmission it will, at 10 per cent, 
loss and with the same size of conductor, rise to nearly 90 
per cent, of the main current, thus making a more explicit 
investigation of the phenomena in the line necessary. 

In most cases of practical engineering, however, the ca- 
pacity effect is small enough to be represented by the approx- 
imation of one ; viz., three condensers shunted across the line. 

109. A.} Line capacity represented by one condenser 
shunted across middle of line. 

Let — 

Y = g + j b = admittance of receiving circuit ; 
z = r — j x = impedance of line ; 
be = condenser susceptance of line. 


DISTRIBUTED CAPACITY. 


161 


Denoting, in Fig. 84, 

the E.M.F., viz., current in receiving circuit by £, It 

the E.M.F. at middle of line by £', 

the E.M.F., viz., current at generator by E0)I0\ 


If 


We have, 


Fig. 84. Capacity Shunted across Middle of Line. 


. = I-jbcE' 


E\\ \ (r 


Jbe(r-Jx) ., (r-jxy( 

~~ 


or, expanding, 


[(* - bc} - (rg+ 


-jx) 


I (r-jx)(g+jt)-\} 
2 Jf 


110. ^.) Z«W capacity represented by three condensers^ 
in the middle and at the ends of the line. 
Denoting, in Fig. 85, 

the E.M.F. and current in receiving circuit by £, 7, 

the E.M.F. at middle of line by £' ', 


162 


ALTERNATING-CURRENT PHENOMENA. 


the current on receiving side of line by /', 
the current on generator side of line by 7", 
the E.M.F., viz., current at generator by £0, f0, 


Iff 


_L I 


85. Distributed Capacity. 


otherwise retaining the same denotations as in A.), 
We have, 
7 = 


2" = 1' - 


As will be seen, the first terms in the expression of E0 
and of I0 are the same in A.) and in B.). 


DISTRIBUTED CAPACITY. 163 

111. C.) Complete investigation of distributed capacity, 
inductance, leakage, and resistance. 

In some cases, especially in very long circuits, as in 
lines conveying alternating power currents at high potential 
over extremely long distances by overhead conductors or un- 
derground cables, or with very feeble currents at extremely 
high frequency, such as telephone currents, the consideration 
of the line resistance — which consumes E.M.Fs. in phase 
with the current — and of the line reactance — which con- 
sumes E.M.Fs. in quadrature with the current — is not 
sufficient for the explanation of the phenomena taking place 
in the line, but several other factors have to be taken into 
account. 

In long lines, especially at high potentials, the electro- 
static capacity of the line is sufficient to consume noticeable 
currents. The charging current of the line condenser is 
proportional to the difference of potential, and is one-fourth 
period ahead of the E.M.F. Hence, it will either increase 
or decrease the main current, according to the relative phase 
of the main current and the E.M.F. 

As a consequence, the current will change in intensity 
as well as in phase, in the line from point to point ; and the 
E.M.Fs. consumed by the resistance and inductance will 
therefore also change in phase and intensity from point 
to point, being dependent upon the current. 

Since no insulator has an infinite resistance, and as at 
high potentials not only leakage, but even direct escape of 
electricity into the air, takes place by " silent discharge," we 
have to recognize the existence of a current approximately 
proportional and in phase with the E.M.F. of the line. 
This current represents consumption of energy, and is 
therefore analogous to the E.M.F. consumed by resistance, 
while the condenser current and the E.M.F. of inductance 
are wattless. 

Furthermore, the alternate current passing over the line 
induces in all neighboring conductors secondary currents, 


164 ALTERNATING-CURRENT PHENOMENA. 

which react upon the primary current, and thereby intro- 
duce E.M.Fs. of mutual inductance into the primary circuit. 
Mutual inductance is neither in phase nor in quadrature 
with the current, and can therefore be resolved into an 
energy component of mutual inductance in phase with the 
current, which acts as an increase of resistance, and into 
a wattless component in quadrature with the current, which 
decreases the self-inductance. 

This mutual inductance is not always negligible, as, 
for instance, its disturbing influence in telephone circuits 
shows. 

The alternating potential of the line induces, by electro- 
static influence, electric charges in neighboring conductors 
outside of the circuit, which retain corresponding opposite 
charges on the line wires. This electrostatic influence re- 
quires the expenditure of a current proportional to the 
E.M.F., and consisting of an energy component, in phase 
with the E.M.F., and a wattless component, in quadrature 
thereto. 

The alternating electromagnetic field of force set up by 
the line current produces in some materials a loss of energy 
by magnetic hysteresis, or an expenditure of E.M'.F. in 
phase with the current, which acts as an increase of re- 
sistance. This electromagnetic hysteretic loss may take 
place in the conductor proper if iron wires are used, and 
will then be very serious at high frequencies, such as those 
of telephone currents. 

The effect of eddy currents has already been referred 
to under "mutual inductance," of which it is an energy 
component. 

The alternating electrostatic field of force expends 
energy in dielectrics by what is called dielectric hysteresis. 
In concentric cables, where the electrostatic gradient in the 
dielectric is comparatively large, the dielectric hysteresis 
may at high potentials consume considerable amounts of 
energy. The dielectric hysteresis appears in the circuit 


DISTRIBUTED CAPACITY. 165 

as consumption of a current, whose component in phase 
with the E.M.F. is the dielectric energy current, which 
may be considered as the power component of the capacity 
current. 

Besides this, there is the increase of ohmic resistance 
due to unequal distribution of current, which, however, is 
usually not large enough to be noticeable. 

112. This gives, as the most general case, and per unit 
length of line : 

E.M.Fs. consumed in phase with the current I, and = rl, 
representing consumption of energy, and due to : 
Resistance, and its increase by unequal current distri- 
tribution ; to the energy component of mutual 
inductance; to induced currents ; to the energy 
component of self-inductance ; or to electromag- 
netic hysteresis. 
E.M.Fs. consumed in quadrature with the current I, and 

= x I, wattless, and due to : 
Self-inductance, and Mutual inductance. 
Currents consumed in phase with the E.M.F., E, and 
= gE, representing consumption of energy, and 
due to : 

Leakage through the insulating material, including 
silent discharge; energy component of electro- 
static influence ; energy component of capacity, or 
of dielectric hysteresis. 
Currents consumed in quadrature to the E.M.F., E, and 

= bE, being wattless, and due to : 
Capacity and Electrostatic influence. 

Hence we get fo'ur constants : — 

Effective resistance, r, 
Effective reactance, x, 
Effective conductance, g, 
Effective susceptance, b — — bc, 


1GG ALTERNATING-CURRENT PHENOMENA. 

per unit length of line, which represent the coefficients, per 
unit length of line, of 

E.M.F. consumed in phase with current ; 
E.M.F. consumed in quadrature with current ; 
Current consumed in phase with E.M.F. ; 
Current consumed in quadrature with E.M.F. 

113. This line we may assume now as feeding into a 
receiver circuit of any description, and determine the current 
and E.M.F. at any point of the circuit. 

That is, an E.M.F, and current (differing in phase by any 
desired angle) may be given at the terminals of receiving cir- 
cuit. To be determined are the E.M.F. and current at any 
point of the line ; for instance, at the generator terminals. 
Or, Zl=rl— JXl ; 

the impedance of receiver circuit, or admittance, 


and E.M.F., E0, at generator terminals are given. Current 
and E.M.F. at any point of circuit to be determined, etc. 

114. Counting now the distance, x, from a point, 0, of 
the line which has the E.M.F., 

•Ei = e\ + Je\i and the current : /i = i\ +///, 

and counting x positive in the direction of rising energy, 
and negative in the direction of decreasing energy, we have 
at any point, X, in the line differential, dx : 

Leakage current : JEgdx', 
Capacity current : — j E bc d x ; 

hence, the total current consumed by the line element, dx, 
is dl= E(g-jbc}d*, or, 

d-t=E(g-jbc\ (1) 

E.M.F. consumed by resistance, Ird*\ 
E.M.F. consumed by reactance, — j 


DISTRIBUTED CAPACITY. 107 

hence, the total E.M.F. consumed in the line element, ^/x, is 
dE = I (r — j'x) </x, or, 
ffi. -I(f-jx). (2) 

These fundamental differential equations : 

*L-E(g-jt,),\ (1) 


(2) 


are symmetrical with respect to / and E. 
Differentiating these equations : 
d*I dE , 


and substituting (1) and (2) in (3), we get : 

(4) 


(5) 
the differential equations of E and L 

115. These differential equations are identical, and con- 
sequently I and E are functions differing by their limiting 
conditions only. 

These equations, (4) and (5), are of the form : 

(6) 


and are integrated by 

W = tf 6rx, 

where e is the basis of natural logarithms ; for, differen- 
tiating this, we get, 


168 ALTERNATING-CURRENT PHENOMENA. 

hence, z>2 = (g — j bc) (r — jx) ; 


(7) 


or, v = ± V (g - Jbe) (r — joe) \ 

hence, the general integral is : 

tr*.«e+«-M«r«« (8) 

where a and b are the two constants of integration ; 
Substituting 

r-«-/0 (9) 

into (7), we have, 

(a -JP)* = (g - jbc) (r - jx) ; 
or, 


therefore, _ f 

);-' (10) 


Vl/2 6 - e 


/3= Vl/2 
substituting (9) into (8) : 


= a-cax (cos/3x — /sin^Sx) + ^c~ax (cos/3x +y sin/3x) ; 
«/ = (a£«x + /5>e~ax) cos)8x — y(aeax — ^«-ax) sin /3x (12) 

which is the general solution of differential equations (4) 
and (5) 

Differentiating (8) gives : 


hence, substituting, (9) : 
(a —JP) {(a 


x}. (13) 

Substituting now / for w, and substituting (13) in (1), 
and writing, 


DISTRIBUTED CAPACITY. 


169 


we get, 


/• \( Jfax. _i_ > 

?e-«)cosj8x-y(y 
?«-«)cos/8x-y(y 

-• 

* — •/_> < \ ' 
a — 7/5 

sin /2x} ; 

'** 1 K^" i 

S — J^c 
sin ySxf ; 

where ^4 and ^ are the constants of integration. 
Transformed, we get, 

/= J Aea* (cos )8x — j sin 0x) + Bf.~™ 

a — JP ( ' 

(cos /?x +/ sin /8x) > 
1 


^4eax (cos /8x — y sin 


^-. 

(cos /3x +y sin y8x) 

Thus the waves consist of two components, one, with 
factor ^eax, increasing in amplitude toward the generator, 
the other, with factor ^e-ax, decreasing toward the genera- 
tor. The latter may be considered as a reflected wave. 

At the point x = 0. 


a-j/3 
A-B 


n 

Thus m (cos to — j sin G) = -— 

and, 

m = amplitude. 

w = angle of reflection. 

These are the general integral equations of the problem. 


116. If — 

/! = /! + /// is the current 
{ is the E.M.F. 


at point, x 


(15) 


170 ALTERNATING-CURRENT PHENOMENA. 

by substituting (15) in (14), we get : 
2 A = {(a t\ + ft //) + (gev + bc ^') 

(16) 


2 B = {(a /! + /? //) - (ge, + /;c ,/)} 
+ /{(«//- 0/0 -(^I'-^ 
a and ft being determined by equations (11). 

117. H Z — R — j X is the impedance of the receiver 
circuit, E0 = e0 + j >0' is the E.M.F. at dynamo terminals 
(17), and / = length of line, we get 
at 


hence 


g — jbc 
or 


a-; ft 
At X = /, 


E0 


sin/?/}. (19) 


Equations (18) and (19) determine the constants A and B, 
which, substituted in (14), give the final integral equations. 

The length, X0 = 2 TT / ft is a complete wave length (20), 
,vhich means, that in the distance 2 IT / ft the phases of the 
components of current and E.M.F. repeat, and that in half 
this distance, they are just opposite. 

Hence the remarkable condition exists that, in a very 
long line, at different points the currents at the same time 
flow in opposite directions, and the E.M.Fs. are opposite. 

118. The difference of phase, w, between current, /, and 
E.M.F., Ey at any point, x, of the line, is determined by 


DISTRIBUTED CAPACITY. 171 


the equation, 

Z?(cos«+/sin£) =y, : \j JsTI71 

where Z> is a constant. 

Hence, w varies from point to point, oscillating around a 
medium position, wx, which it approaches at infinity. 

This difference of phase, C>x, towards which current and 
E.M.F. tend at infinity, is determined by the expression, 


^(cos . .. , (/ 

or, substituting for E and /their values, and since e~a* = 0, 
and A eax (cos ft x — j sin ft x), cancels, and 

D (cos tow +/sin oioc) = — 2-p- 


hence, tan ^ = ~a° c + ^ • (21) 

This angle, Stx, = 0 ; that is, current and E.M.F. come 
more and more in phase with each other, when 

abc — fig — 0 ; that is, 

a -T- ft — g -r- bc , or, 

2a/3 !*2^*/ 5 
substituting (10), gives, 


hence, expanding, r -4- ^ = ^ -f- ^c ; (22) 

that is, tJie ratio of resistance to inductance equals the ratio 
of leakage to capacity. 

This angle, wx, = 45° ; that is, current and E.M.F. differ 
by £th period, if — a bc + fig = a.g + pbc ; or, 


which gives : rg + x bc = 0. (23) 


172 


ALTERNA TING-CURRENT PHENOMENA. 


That is, two of the four line constants must be zero; cither 
g and x, or g and bc. 

The case where g = 0 = x, that is a line having only 
resistance and distributed capacity, but no self-induction is 
approximately realized in concentric or multiple conductor 
cables, and in these the phase angle tends towards 45° lead 
for infinite length. 

119. As an instance, in Fig. 86 a line diagram is shown, 
with the distances from the receiver end as abscissae. 
The diagram represents one and one-half complete waves, 
and gives total effective current, total E.M.F., and differ- 


<± 

"o^ 

+ 30 

'»sr 
I 

\ 

OLT» 
.0,000 

»20 

i 

\ 

8()0{ 

1 
1 

\ 

/ 

• 

*\ 

£ 

j 

ja 

u 

i 

\ 

X 

• 

•*> 

V 

u 

( 

\ 

^ - 

+'' 

/ 

-20 

\ 

/ 

/ 

-30 

/ 

"*"•> 

/ 

• 

-40. 

: 

.„, 

•us 
Kl 

s 

I 
-50 

/ 

/.o 

7,0 

/ 

/ 

z;o 

/ 

/ 

p. 

„. 

, — 

•j^, 

/ 

u-j 

c 

/ 

'' 

/ 

.- 

S 

«.ooo 

/ 

? ' 

N 

/ 

/ 

' 000 

/ 

\ 

^~ 

/ 

V 

100 

0,00, 

X 

/ 

N>, 

.s 

x = ' 

60 

9 000 

/ 
/ 

g=i 

bc= 

XI 
'X| 

rj-4 

»0 

«.ooo 

\ 

\ 

/ 

4,000 

\, 

-"* 

JO 

J.OOO 

0 

i 

3 

- 

J 
\ 

L 

5 
1 

L 

i 

Fig. 86. 


DISTRIBUTED CAPACITY. 173 

ence of phase between both as function of the distance from 
receiver circuit ; under the conditions, 

E.M.F. at receiving end, 10,000 volts; hence, Ev =el = 10,000; 
current at receiving end, 65 amperes, with a power factor 
of .385. 

that is, / = t\ + j // = 25 + 60 j ; 

line constants per unit length, 

r = 1, g = 2 X 10-5, 

hence, 

a = 4.95 x 10-3, ] 
13 = 28.36 x 10 -3, j- 


length of line corresponding to 
one complete period of the wave 


x0 = L = — = 221.5 = 

(^ of propagation. 
A = 1.012 - 1.206 y ) 
B = .812 + .794 / j 

These values, substituted, give, 

/= {£«x (47.3 cos /?x + 27.4 sin fix) — e-«* 

(22.3 cos ftx + 32.6 sin fix)} 
+ y (e«x (27.4 cos ftx — 47.3 sin ftx) + €-«x 

(32.6 cos y3x — 22.3 sin /3x)}; 
E = {eox (6450 cos /3x + 4410 sin j8x) + c-ax 

(3530 cos fix + 4410 sin /?x)} 
+ y (eox (4410 cos /3x — 6450 sin £x) — e~ax 
(4410 cos ft x- 3530 sin /3x)}; 

tan 5, = ~ °-ljc + PS = _ .073, JJ« = - 4.2°. 

120. As a further instance are shown the characteristic 
curves of a transmission line of the relative constants, 

r\x\g>.b = % : 32 : 1.25 X 10 ~4 : 25 X 10 ~4, and e 
= 25,000, i = 200 at the receiving circuit, for the con- 
ditions, 

a, non-inductive load in the receiving circuit, Fig. 87. 


174 


ALTERNATING-CURRENT PHENOMENA. 


b, wattless receiving circuit of 90° lag, Fig. 88. 

c, wattless receiving circuit of 90° lead, Fig. 89. 
These curves are determined graphically by constructing 

the topographic circuit characteristics in polar coordinates 
as explained in Chapter VI., paragraphs 36 and 37, and de- 
riving corresponding values of current, potential difference 
and phase angle therefrom. 

As seen from these diagrams, for wattless receiving cir- 
cuit, current and E.M.F. oscillate in intensity inversely to 


ZJ 

7 

6sa 


7 


rig. 87. 


DISTRIBUTED CAPACITY. 


175 


each other, with an amplitude of oscillation gradually de- 
creasing when passing from the receiving circuit towards 
the generator, while the phase angle between current and 
E.M.F. oscillates between lag and lead with decreasing am- 
plitude. Approximately maxima and minima of current co- 
incide with minima and maxima of E.M.F. and zero phase 
angles. 


\ 


V 


Fig. 88. 


176 


AL TERNA TING-CURRENT PHENOMENA. 


For such graphical constructions, polar coordinate paper 
and two angles a and 8 are desirable, the angle a being the 

angle between current and change of E.M.F., tan a = - = 4, 
and the angle 8 the angle between E.M.F. and change of 

current, tan 8 = - = 20 in above instance. 
g 


\ 


Fig. 89. 


DISTRIBUTED CAPACITY. 


177 


With non-inductive load, Fig. 87, these oscillations of 
intensity have almost disappeared, and only traces of them 
are noticeable in the fluctuations of the phase angle and the 
relative values of current and E.M.F. along the line. 

Towards the generator end of the line, that is towards 
rising power, the curves can be extended indefinitely, ap- 
proaching more and more the conditions of non-inductive 
circuit, towards decreasing power, however, all curves ulti- 
mately reach the conditions of a wattless receiving circuit, 
as Figs. 88 and 89, at the point where the total energy in- 


t 


a +120 


ISSION LINE 


V 


Fig. 90. 

put into the line has been consumed therein, and at this 
point the two curves for lead and for lag join each other as 
shown in Fig. 90, the one being a prolongation of the other, 
and the flow of power in the line reverses. Thus in Fig. 90 
power flows from both sides of the line towards the point of 
zero power marked by 0, where current and E.M.F. are in 
quadrature with each other, the current being leading with 
regard to the flow of power from the left, and lagging with 
regard to the flow of power from the right side of the 
diagram. 


178 DISTRIBUTED CAPACITY. 

121. The following are some particular cases : 
A.) Open circuit at end of lines : 
x = 0 : /! = 0. 

hence, 

E = i-r— ^{(eax + e-ax) cos/3x — y(cax — c-ax)sin/3x}; 


.£?.) Line grounded at end: 


A — (a/\ -J- /?//) +/ (a// — ^zi) = -? 
-^-T-^{(eax — c-ax) cos/?x — >(eax + c~ax) sin)8x}; 


(T.) Infinitely long conductor : 

Replacing x by — x, that is, counting the distance posi- 
tive in the direction of decreasing energy, we have, 

x = oo : 7= 0, E = 0; 
hence 


and 


I = — - — ^£-°x(cos/Sx +ysin/3x), 

' 


revolving decay of the electric wave, that is the reflected 
wave does not exist. 

The total impedance of the infinitely long conductor is 


(q-yff) (g+M 


+ b? g* + b* 


ALTERNATING-CURRENT PHENOMENA. 179 

The infinitely long conductor acts like an impedance 

7 _ °-K + P ?>c _ • fig — Q-bc 

f*+v g* + K' 

that is, like a resistance 


combined with a reactance 


We thus get the difference of phase between E.M.F. 
and current, 


which is constant at all points of the line. 
If g = 0, x = 0, we have, 


hence, 

tan to = 1, or, 

£ = 45° ; 

that is, current and E.M.F. differ by £th period. 
D.) Generator feeding into closed circuit : 
Let x = 0 be the center of cable ; then, 

hence : E — 0 at x = 0 ; 


which equations are the same as in B, where the line is 
grounded at x = 0. 

E.) Let the length of a line be one-quarter wave length; 


and assume the resistance r and conductance g as negligible 


180 AL TERN A TING-CURRENT PHENOMENA. 

compared with x and bc. 

r=0=g 
These values substituted in (11) give 

a=0. 

(3= V^ 

Let the E.M.F. at the receiving end of the line be 
assumed zero vector 

£l = ei = E.M.F. and 

fi — i'i + ji\. — current at end of line x = 0 
£0 = E.M.F. and 

S0 = current at beginning of line 


Substituting in (16) these values of El and 7: and also r = 0 
= g, we have 


From these equations it follows that 


which values, together with the foregoing values of Ev Iv r, 
g, a, and /8, substituted in (14) reduce these equations to 


— j (i\ +jiC) \~r s^ 


ALTERNATING-CURRENT TRANSFORMER, 181 

Then at x 


Hence also 


•£"„ and 70 are both in quadrature ahead of <?x and 7j 
respectively. 

Il = EQ y — = constant, if 7f0 = constant. That is, at 

constant impressed E.M.F. E& the current 7X in the receiv- 
ing circuit of a line of one-quarter wave length is constant, 
and inversely (constant potential — constant current trans- 
formation by inductive line). In this case, the current 70 at 
the beginning of the line is proportional to the load el at the 
end of the line. 

If XQ = lx = total reactance, 

b0 = lbc = total susceptance of line, then 

*<A> = 4- 

Instance* = 4, bc = 20 X 10 ~5, E0 = 10,000 V. Hence 
/ = 55.5, *0 = 222, b0 = .0111, 7j = 70.7, 70 = .00707 e. 

122. An interesting application of this method is the 
determination of the natural period of a transmission line ; 
that is the frequency at which such a line discharges an 
accumulated charge of atmospheric electricity (lightning), 
or oscillates at a sudden change of load, as a break of cir- 
cuit. 


182 ALTERNATING-CURRENT PHENOMENA. 

The discharge of a condenser through a circuit contain- 
ing self-induction and resistance is oscillating (provided that 
the resistance does not exceed a certain critical value de- 
pending upon the capacity and the self-induction). That is, 
the discharge current alternates with constantly decreasing 
intensity. The frequency of this oscillating discharge de- 
pends upon the capacity, C, and the self-induction, L, of the 
circuit, and to a much lesser extent upon the resistance, so 
that if the resistance of the circuit is not excessive the fre- 
quency of oscillation can, by neglecting the resistance, be 
expressed with fair, or even close, approximation by the 
formula - 


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

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

In determining the frequency of the oscillating discharge 
of such a transmission line, a sufficiently close approximation 
is obtained by neglecting the resistance of the line, which, 
at the relatively high frequency of oscillating discharges, 
is small compared with the reactance. This assumption 
means that the dying out of the discharge current through 
the influence of the resistance of the circuit is neglected, 
and the current assumed as an alternating current of ap- 
proximately the same frequency and the same intensity as 
the initial waves of the oscillating discharge current. By 
this means the problem is essentially simplified. 

Let / = total length of a transmission line, 
r = resistance per unit length, 
x = reactance per unit length = 2 ?r NL. 


DISTRIBUTED CAPACITY. 183 

where L = coefficient of self-induction or inductance per unit 

length ; 

g = conductance from line to return (leakage and dis- 
charge into the air) per unit length ; 
b = capacity susceptance per unit length = 2 TT NC 
where C = capacity per unit length. 

x = the distance from the beginning of the line, 

We have then the equations : 
The E.M.F., 


(^eax _ ^e-ax) CQS £x _j (4€ 

g — jb I + ^e~ax) sin /3x 
the current, 

1 ^ (Aea* + ^e~ax) COS /3x — y (^4e 


where, 


,(14.) 


(r1 + ^c2) + (^r - 

' (11.) 


c = base of the natural logarithms, and A and B integration 
constants. 

Neglecting the line resistance, r = 0, and the conduc- 
tance (leakage, etc.), g=0, gives, 


These values substituted in (14) give, 
J- 


= J-\(A - B} cos ^fbx^ -j (A + H) sin 


/ = -4= J (^ + -ff) cos V^x — y (<4 - B) sin 

; J 


184 ALTERNATING-CURRENT PHENOMENA. 

If the discharge takes place at the point : x = 0, that is, 
if the distance is counted from the discharge point to the 
end of the line ; x = /, hence : 

At x = 0, E = 0, 
Atx=/, 7=0. 

Substituting these values in (25) gives, 

For x = 0, 

^-7^ = 0 A = B 

which reduces these equations to, 

E = — — sin Nbx x 

b \ 

7= -^4^= cos V&t: x 

VA* I 

and at x = 0, 


At x = /, / = 0, thus, substituted in (26), 

cos V^/ = 0 (28.) 

hence : 

V^/^2**1)", 1 = 0,1, 2,... (29.) 

that is, *Jbx I is an odd multiple of ^ • And at x = /, 

2t 

O A 


Substituting in (29) the values, 

we have, 

hence, 

^=M + l (31.) 

4/VCZ 


DISTRIBUTED CAPACITY. 185 

the frequency of the oscillating discharge, 
where k = 0, 1, 2. . . . 

That is, the oscillating discharge of a transmission line 
of distributed capacity does not occur at one definite fre- 
quency (as that of a condenser), but the line can discharge 
at any one of an infinite number of frequencies, which are 
the odd multiples of the fundamental discharge frequency, 

*-I7^z (32'> 

Since 

C0 = 1C = total capacity of transmission line, ) 

L0 = IL = total self-inductance of transmission line, J ^ '' 

we have, 

2,£ + 1 

-= the frequency of oscillation, (34.) 


or natural period of the line, and 

NI — - - the fundamental, 
- 


or lowest natural period of the line. 
From (30), (33), and (34), 

b = 2irNC= 2/ \T0 (36-) 

and from (29), 

V^ = (2^2f/)7r- <37') 

These substituted in (26) give, 

f- (38.) 
4/7 (2£ + l)7rx 

/=(2TTi)-^cosL^H 

The oscillating discharge of a line can thus follow any of 
the forms given by making k — 0, 1, 2, 3 . . .in equation 
(38). 

Reduced from symbolic representation to absolute values 


186 ALTERNATING-CURRENT PHENOMENA. 

by multiplying E with cos 2 * Nt and / with sin 2 TT A7/ and 
omitting j, and substituting A7" from equation (34), we have, 

(2£+l)7rx 

— sin — JT— - — -cos 
2/ 


where ^4 is an integration constant, depending upon the 
initial distribution of voltage, before the discharge, and 
/ = time after discharge. 

123. The fundamental discharge wave is thus, for k = 0, 


47. Lo . . 7TX 7T/ 

-^ A sin 7^ 
C0 2/ 


. o . . 7TX 

= — \ -^ A sin 7^— cos 

TT V 


4 / - _, 7T X 7T/ 

fi = — A cos 7n- sin - 


With this wave the current is a maximum at the begin- 
ning of the line : x = 0, and gradually decreases to zero at 
the end of the line : x = /. 

The voltage is zero at the beginning of the line, and 
rises to a maximum at the end of the line. 

Thus the relative intensities of current and potential 
along the line are as represented by Fig. 91, where the cur- 
is shown as /, the potential as E. 

The next higher discharge frequency, for : k — 1, gives : 

47. [Ln . . 3v_ 

(41.) 

4/ " " - ' 

/, = o- A cos 


n 7 
27 


DISTRIBUTED CAPACITY. 


187 


Here the current is again a maximum at the beginning 
of the line : x = 0, and gradually decreases, but reaches 

zero at one-third of the line : x = _, then increases again, in 

o 


Fig. 


H----0 


Fig. 


\ 


\ 


\ 


\1 


Figs. 91-93. 


188 ALTERNATING CURRENT-PHENOMENA. 

the opposite direction, reaches a second but opposite maxi- 

2/ 

mum at two-thirds of the line : x = ^— , and decreases to 

o 

zero at the end of the line. There is thus a nodal point of 
current at one-third of the line. 

The E.M.F. is zero at the beginning of the line : x = 0, 

rises to a maximum at one-third of the line : x = - , de- 

2/ 3 
creases to zero at two-thirds of the line : x = IT > and rises 

again to a second but opposite maximum at the end of the 
line: x = /. The E.M.F. thus has a nodal point at two- 
thirds of the line. 

The discharge waves : k = 1, are shown in Fig. 92, 
those with k = 2, with two nodal points, in Fig. 93. 

Thus k is the number of nodal points or zero points of 
current and of E.M.F. existing in the line (not counting 
zero points at the ends of the line, which of course are not 
nodes). 

In case of a lightning discharge the capacity C0 is the 
capacity of the line against ground, and thus has no direct 
relation to the capacity of the line conductor against its 
return. The same applies to the inductance L0. 

If d = diameter of line conductor, 

D = distance of conductor above ground, 
and / = length of conductor, 

the capacity is, 


1.11 x 10-6/ ,. 

~ 


the self-inductance, 


The fundamental frequency of oscillation is thus, by 
substituting (42) in (35), 


DISTRIBUTED CAPACITY. 189 

That is, the frequency of oscillation of a line discharging 
to ground is independent of the size of line wire and its 
distance from the ground, and merely depends upon the 
length / of the line, being inversely proportional thereto. 

We thus get the numerical values, 

Length of line 

10 20 30 40 50 60 80 100 miles. 
= 1.6 3.2 4.8 6.4 8 9.6 12.8 16 x 106 cm.. 

hence frequency, 

N-i = 4680 2340 1560 1170 937.5 780 585 475 cycles-.. 

As seen, these frequencies are comparatively low, and 
especially with very long lines almost approach alternator 
frequencies. 

The higher harmonics of the oscillation are the odd! 
multiples of these frequencies. 

Obviously all these waves of different frequencies repre- 
sented in equation (39) can occur simultaneously in the 
oscillating discharge of a transmission line, and in general 
the oscillating discharge of a transmission line is thus of 
the form, 

(by substituting: ak = * j 


where a^ as ay . . . are constants depending upon the 
initial distribution of potential in the transmission line, at 
the moment of discharge, or at / = 0, and calculated there- 
from. 


190 AL TERN A TING-CURRENT PHENOMENA . 

124. As an instance the following discharge equation 
of a line charged to a uniform potential e over, its entire 
length, and then discharging at x = 0, has been calculated. 

The harmonics are determined up to the 11 — that is, av 

•a& #5> av a9> an- 

These six unknown quantities require six equations, which 

/ 2/ 3/ 4/ 5/ 6/ 
are given by assuming E = e for x = g, _,_,_,_,_. 

At / = 0, E = e, equation (44) assumes the form 

4 / HT ( . TTX , . 3 TTX 
e = — V £? j «i sm 27 + *3 sm~27 + ' ' ' ' + *u 

(45.) 

/ 2/ 6/ 

Substituting herein for x the values : - , — , . . . — 

gives six equations for the determination of av <73 . . . an. 
These equations solved give, 


E = e (1.26 sin w cos $ + .40 sin 3 w cos 3 <f» + .22 sin ^ 
5 w cos 5 <^ + .12 sin 7 o> cos 7 <£ + .07 sin 9 co 
cos 9 ^ + .02 sin 11 o> cos 11 ^> 

5 

L0 

cos 5 <o sin 5 <^ + .12 cos 7 o> sin 7 <£ + .07 cos 
9 to sin 9 <£ + .02 cos 11 o> sin 11 </> 


7 = e i/5 (1.26 cos o> sin <£ + .40 cos 3 w sin 3 <£ + .22 

V 7rt 


,(46.) 


where, 

"-57 1 

„ r<47') 


Instance, . 

Length of line, / = 25 miles = 4 x 106 cm. 
Size of wire : No. 000 B. & S. G., thus : d = 1 cm. 
Height above ground : D — 18 feet = 550 cm. 
Let e = 25,000 volts = potential of line in the moment of 
-discharge. 


DISTRIBUTED CAPACITY. 191 

We then have, 

E = 31,500 sin w cos <fr + 10,000 sin 3 <o cos 3 <£ + 5500 sin 

5 o> cos 5 <J> + 3000 sin 7 o> cos 7 <£ -j- 1750 sin 9 o> cos 

9 <£ + 500 sin 11 w cos 11 <£. 
/= 61.7 cos w sin <£ + 19.6 cos 3 o> sin 3 <£ + 10.8 cos 5 « sin 

5 <£ + 5.9 cos 7 CD sin 7 </> + 3.4 cos 9 to sin 9 <£ + 1.0 

cos 11 <o sin 11 <J>. 

<o= .39 .r 10 -6 
</> = 1.18/ 10+4 

A simple harmonic oscillation as a line discharge would 
require a sinoidal distribution of potential on the trans- 
mission line at the instant of discharge, which is not proba- 
ble, so that probably all lightning discharges of transmission 
lines or oscillations produced by sudden changes of circuit 
conditions are complex waves of many harmonics, which in 
their relative magnitude depend upon the initial charge and 
its distribution — that is, in the case of the lightning dis- 
charge, upon the atmospheric electrostatic field of force. 

The fundamental frequency of the oscillating discharge 
of a transmission line is relatively low, and of not much 
higher magnitude than frequencies in commercial use in 
alternating current circuits. Obviously, the more nearly 
sinusoidal the distribution of potential before the discharge, 
the more the low harmonics predominate, while a very un- 
equal distribution of potential, that is a very rapid change 
along the line, as caused for instance by a sudden short 
circuit rupturing itself instantly, causes the higher harmo- 
nics to predominate, which as a rule are more liable to cause 
excessive rises of voltage by resonance. 

125. As has been shown, the electric distribution in a 
transmission line containing distributed capacity, self-induc- 
tion, etc., can be represented either by a polar diagram 
with the phase as amplitude, and the intensity as radius 
vector, as in Fig. 34, or by a rectangular diagram with the 


192 ALTERNATING-CURRENT PHENOMENA. 

distance as abscissae, and the intensity as ordinate, as in 
Fig. 35 and in the preceding paragraphs. 

In the former case, the consecutive points of the circuit 
characteristic refer to consecutive points along the trans- 
mission line, and thus to give a complete representation of 
the phenomenon, should not be plotted in one plane but in 
front of each other by their distance along the transmission 
line. That is, if 0, 1, 2, etc., are the polar vectors in Fig. 
34, corresponding to equi-distant points of the transmission 
line, 1 should be in a plane vertically in front of the plane 
of 0, 2 by the same distance in front of 1, etc. 

In Fig. 35 the consecutive points of the circuit charac- 
teristic represent vectors of different phase, and thus should 
be rotated out of the plane around the zero axis by the 
angles of phase difference, and then give a length view of 
the same space diagram, of which Fig. 34 gives a view along 
the axis. 

Thus, the electric distribution in a transmission line can 
be represented completely only by a space diagram, and as 
complete circuit characteristic we get for each of the lines 
a screw shaped space curve, of which the distance along the 
axis of the screw represents the distance along the transmis- 
sion line, and the distance of each point from the axis rep- 
resents by its direction the phase, and by its length the 
intensity. 

Hence the electric distribution in a transmission line 
leads to a space problem of which Figs. 34 and 35 are par- 
tial views. The single-phase line is represented by a double 
screw, the three-phase line by a triple screw, and the quarter- 
phase four-wire line by a quadruple screw. In the symbolic 
expression of the electric distribution in the transmission 
line, the real part of the symbolic equation represents a pro- 
jection on a plane passing through the axis of the screw, 
and the imaginary part a projection on a plane perpendicular 
to the first, and also passing through the axis of the screw. 


ALTERNATING-CURRENT TRANSFORMER. 193