CHAPTER IV. 

DISTRIBUTED CAPACITY OF HIGH-POTENTIAL 
TRANSFORMERS. 

40. In the high-potential coils of transformers designed for 
very high voltages phenomena resulting from distributed 
capacity occur. 

In transformers for very high voltages — 100;000 volts and 
more, or even considerably less in small transformers — the high- 
potential coil contains a large number of turns, a great length of 
conductor, and therefore its electrostatic capacity is appreciable, 
and such a coil thus represents a circuit of distributed resistance, 
inductance, and capacity somewhat similar to a transmission 
line. 

The same applies to reactive coils, etc., wound for very high 
voltages, and even in smaller reactive coils at very high frequency. 

This capacity effect is more marked in smaller transformers, 
where the size of the iron core and therewith the voltage per 
turn is less, and therefore the number of turns greater than in 
very large transformers, and at the same time the exciting cur- 
rent and the full-load current are less; that is, the charging 
current of the conductor more comparable with the load current 
of the transformer or reactive coil. 

However, even in large transformers and at moderately high 
voltages, capacity effects occur in transformers, if the frequency 
is sufficiently high, as is the case with the currents produced in 
overhead lines by lightning discharges, or by arcing grounds 
resulting from spark discharges between conductor and ground, 
or in starting or disconnecting the transformer. With such 
frequencies, of many thousand cycles, the internal capacity of 
the transformer becomes very marked in its effect on the dis- 
tribution of voltage and current, and may produce dangerous 
high-voltage points in the transformer. 

The distributed capacity of the transformer, however, is differ- 
ent from that of a transmission line. 

342 


HIGH-POTENTIAL TRANSFORMERS 


343 


In a transmission line the distributed capacity is shunted 
capacity, that is, can be represented diagrammatically by con- 
densers shunted across the circuit from line to line, or, what 
amounts to the same thing, from line to ground and from ground 
to return line, as shown diagrammatically in Fig. 88. 


ll.lllUllll-UllJ- 111 II I 
TTTTTTTTTTTTTTTTTTTTTT 


Fig. 88. Distributed capacity of a transmission line. 

0 

The high-potential coil of the transformer also contains shunted 
capacity, or capacity from the conductor to ground, and so each 
coil element consumes a charging current proportional to its 
potential difference against ground. Assuming the circuit as insu- 
lated, and the middle of the transformer coil at ground potential, 
the charge consumed by unit length of the coil increases from 
zero at the center to a maximum at the ends. If one terminal 
of the circuit is grounded, the charge consumed by the coil 
increases from zero at the grounded terminal to a maximum at 
the ungrounded terminal. 

In addition thereto, however, the transformer coil also con- 
tains a capacity between successive turns and between successive 
layers. Starting from one point of the conductor, after a certain 


C3 

HP 

r 

Hh 

nr 

ii 

II 

IL 

II 
Hi- 

ll 

II 

II 

ii 

ft 

n 

ii 

II 

±* 

-IHHh 

ii 

Tf 

HhHh 

ii 

Hh 

Hi 

-u 

ii 

-ih 

c, 

Hh 

-\[ 

II 
-•tt 

Hh 

Hh 

Hh 

Hh 

Hh 

Hh 

t 

_ci 

111111111. 
ITTTTTTTr 

Till 

•TTTCT 

Fig. 80. Distributed capacity of a high -potential transformer coil. 


length, the length of one turn, the conductor reapproaches the 
first point in the next adjacent turn. It again approaches the 


344 TRANSIENT PHENOMENA 

first point at a different and greater distance in the next adjacent 
layer. 

A transformer high-potential coil can be represented dia- 
grammatically as a conductor, Fig. 89. Cl represents the capacity 
against ground, C2 represents the capacity between adjacent 
turns, and C3 the capacity between adjacent layers of the coil. 

The capacities C2 and Ca are not uniformly distributed but 
more or less irregularly, depending upon the number and arrange- 
ment of the transformer coils and the number and arrangement 
of turns in the coil. As approximation, however, the capacities 
C2 and (73 can be assumed as uniformly distributed capacity 
between successive conductor elements. If I = length of con- 
ductor, they may be assumed as a capacity between I and / + dl, 
or as a capacity across the conductor element dl. 

This approximation is permissible in investigating the general 
effect of the distributed capacity, but omits the effect of the 
irregular distribution of C2 and C3, which leads to local oscilla- 
tions of higher frequencies, extending over sections of the circuit, 
and of lesser power. 

41. Let then, in the high-potential coil of a high- voltage trans- 
former, e = the e.m.f. generated per unit length of conductor, 
as, for instance, per turn; Z = r — ' jx = the impedance per unit 
length; Y = g — jb = the capacity admittance against ground 
per unit length of conductor, and Y' = pY= the capacity 
admittance, per unit length of conductor, between conductor 
elements distant from each other by unit length, as admittance 
between successive turns. Y' is assumed to represent the total 
effective admittance representing the capacity between successive 
turns, successive layers, and successive coils, as represented by 
the condensers C2 and C3 in Fig. 89. 

The charging current of a conductor element dl, due to the 
admittance Y', is made up of the charging currents against the 
next following and that against the preceding conductor element. 

Let 1Q = length of conductor; I = distance along conductor; 
E = potential at point I, or conductor element dl, and I = cur- 
rent in conductor element dl] then 

dE 

dE = — dl = the potential difference between successive 

Q/i 

conductor elements or turns. 


HIGH-POTENTIAL TRANSFORMERS 345 

Y7 — dl = the charging current between one conductor ele- 
ment and the next conductor element or turn. 

— Y' — — 77 — dl = the charging current between one con- 
dl 

ductor element and the preceding conductor element or turn, 
hence, 

dl = the charging current of one conductor element due 


to capacity between adjacent conductors or turns. 

If now the distance I is counted from the point of the con- 
ductor, which is at ground potential, YEdl = the charging cur- 
rent of one conductor element against ground, and 


^Idl 


is the total current consumed by a conductor element. 

However, the e.m.f. consumed by impedance equals the e.m.f. 
consumed per conductor element; thus 

dE = Zldl 


This gives the two differential equations : 

and e - - = ZI. (2) 

Differentiating (2) and substituting in (1) gives 


transposing, 

- E 


dP 1 ' (3) 

P ZY 

ffE 

or — = - a*E, (4) 


346 

where 


TRANSIENT PHENOMENA 

J_ 


a" = 


/y-j 

P~ZY 


(5) 


1 


If -== is small compared with p, we have, approximately, 


«2 » - (6) 

and E = A cos aZ -f B sin aZ, (7) 

and since, for Z = 0, E = 0, if the distance Z is counted from the 
point of zero potential, we have 

E = B sin aZ, 

and the current is given by equation (2) as 

1 ( dE ) 


(8) 
(9) 


substituting (8) in (9) gives 


I = — \ e — aB cos aZ 


(10) 


42. If now 1 1 = the current at the transformer terminals, 
I = Z0, we have, from (10), 


and 


ZI j = e — aB cos aZ0 


B - 


a cos a 


substituting in (8) and (10), 


E =(e - ZI,) 


sin al 
a cos aL 


and 


L \ cos al 

f -1 " Z C • - ' l cosaZ, 


(12) 


for 7j = 0, or open circuit of the transformer, this gives 

sin al 


E = e 


and 


a cos aZ( 
cos aZ 


/ = e f cos 
Z \cos 


(13) 


HIGH-POTENTIAL TRANSFORMERS 347 

The e.m.f., E, thus is a maximum at the terminals, 


the current a maximum at the zero point of potential, I = 0, 
where