LECTURE IX. 
OSCILLATIONS OF THE COMPOUND CIRCUIT. 

38. The most interesting and most important application of 
the traveling wave is that of the stationary oscillation of a com- 
pound circuit, as industrial circuits are never uniform, but consist 
of sections of different characteristics, as the generating system, 
transformer, line, load, etc. Oscillograms of such circuits have 
been shown in the previous lecture. 

If we have a circuit consisting of sections 1, 2, 3 . . . , of the 
respective lengths (in velocity measure) Xi, X2, X3 . . . , this 
entire circuit, when left to itself, gradually dissipates its stored 
energy by a transient. As function of the time, this transient 
must decrease at the same rate u0 throughout the entire circuit. 
Thus the time decrement of all the sections must be 

6-**. 

Every section, however, has a power-dissipation constant, u\t Uz, 
u3 . . . , which represents the rate at which the stored energy 
of the section would be dissipated by the losses of power in the 
section, 

€-"»', €-«*', €-"*' . . . 

But since as part of the whole circuit each section must die 
down at the same rate e~Uot, in addition to its power-dissipation 
decrement e~Ul*, e~"2' . . . , each section must still have a second 
time decrement, €-(«*-*J*, e-(u0-u,)t t t t This latter does not 
represent power dissipation, and thus represents power transfer. 
That is, 

51 = U0 — Ui, 

52 = UQ — Uz, (1) 


It thus follows that in a compound circuit, if u0 is the average 
exponential time decrement of the complete circuit, or the average 

108 


OSCILLATIONS OF THE COMPOUND CIRCUIT. 109 

power-dissipation constant of the circuit, and u that of any section, 
this section must have a second exponential time decrement, 

S = UQ — U, (2) 

which represents power transfer from the section to other sections, 
or, if s is negative, power received from other sections. The oscil- 
lation of every individual section thus is a traveling wave, with a 
power-transfer constant s. 

As UQ is the average dissipation constant, that is, an average of 
the power-dissipation constants u of all the sections, and s = UQ — u 
the power-transfer constant, some of the s must be positive, some 
negative. 

In any section in which the power-dissipation constant u is less 
than the average UQ of the entire circuit, the power-transfer con- 
stant s is positive ; that is, the wave, passing over this section, in- 
creases in intensity, builds up, or in other words, gathers energy, 
which it carries away from this section into other sections. In 
any section in which the power- dissipation constant u is greater 
than the average UQ of the entire circuit, the power-transfer con- 
stant s is negative; that is, the wave, passing over this section, 
decreases in intensity and thus in energy, or in other words, leaves 
some of its energy in this section, that is, supplies energy to the 
section, which energy it brought from the other sections. 

By the power-transfer constant s, sections of low energy dissi- 
pation supply power to sections of high energy dissipation. 

39. Let for instance in Fig. 43 be represented a circuit consist- 
ing of step-up transformer, transmission line, and load. (The 
load, consisting of step-down transformer and its secondary cir- 
cuit, may for convenience be considered as one circuit section.) 
Assume now that the circuit is disconnected from the power sup- 
ply by low-tension switches, at A. This leaves transformer, line, 
and load as a compound oscillating circuit, consisting of four 
sections: the high-tension coil of the step-up transformer, the two 
lines, and the load. 

Let then Xi = length of line, X2 = length of transformer circuit, 
and Xs = length of load circuit, in velocity measure.* If then 

* If Zi = length of circuit section in any measure, and L0 = inductance, 
Co = capacity per unit of length Zi, then the length of the circuit in velocity 
measure is Xi = o-oZi, where <TO = v L0Co. 

Thus, if L = inductance, C = capacity per transformer coil, n = number of 
transformer coils, for the transformer the unit of length is the coil; hence the 


110 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

HI = 900 = power- dissipation constant of the line, u* = 100 = 
power-dissipation constant of transformer, and uz = 1600 = power- 
dissipation constant of the load, and the respective lengths of the 
circuit sections are 

Xi = 1.5 X 10-3; X2 = 1 X 10~3; \3 = 0.5 X 10~3, 
it is: 

Line. Transformer. Line. Load. Sum. 

Length: X = 1.5X10-3 1X10~3 1.5X10~3 .5X10~3 4.5X1Q-3 
Power-dissipa- 

tion constant: u = 900 100 900 1600 

«X = 1.35 .1 1.35 .8 3.6 

hence, Wo= average, u = — — = 800, and: 

ZA 

Power-transfer 
constant: S = MO-M= -100 +700 -100 -800 

The transformer thus dissipates power at the rate u2 = 100, 
while it sends out power into the other sections at the rate of 
s2 = 700, or seven times as much as it dissipates. That is, it sup- 
plies seven-eighths of its stored energy to other sections. The load 
dissipates power at the rate Uz = 1600, and receives power at the 
rate —s = 800; that is, half of the power which it dissipates is 
supplied from the other sections, in this case the transformer. 

The transmission line dissipates power at the rate HI = 900, 
that is, only a little faster than the average power dissipation of 
the entire circuit, u0 = 800; and the line thus receives power only 
at the rate —s= 100, that is, receives only one-ninth of its power 
from the transformer; the other eight-ninths come from its stored 
energy. 

The traveling wave passing along the circuit section thus 
increases or decreases in its power at the rate e+2*x; that is, 
in the line: 

p = pie~200X, the energy of the wave decreases slowly; 
in the transformer: 

p = 7?2C+1400X, the energy of the wave increases rapidly; 


length li = n, and the length in velocity measure, X = aQn = n VLC. Or, if 
L = inductance, C = capacity of the entire transformer, its length in velocity 
measure is X = v LC. 

Thus, the reduction to velocity measure of distance is very simple. 


OSCILLATIONS OF THE COMPOUND CIRCUIT, 


111 


in the load: 


p — p3e~l600X, the energy of the wave decreases rapidly. 

Here the coefficients of pi, p2, PS must be such that the wave at the 
beginning of one section has the same value as at the end of the 
preceding section. 

In general, two traveling waves run around the circuit in 
opposite direction. 

Each of the two waves reaches its maximum intensity in this 
circuit at the point where it leaves the transformer and enters 
the line, since in the transformer it increases, while in the line it 
again decreases, in intensity. 


Fig. 54. — Energy Distribution in Compound Oscillation of Closed Circuit; 

High Line Loss. 

Assuming that the maximum value of the one wave is 6, that of 
the opposite wave 4 megawatts, the power values of the two waves 
then are plotted in the upper part of Fig. 54, and their difference, 
that is, the resultant flow of power, in the lower part of Fig. 54. 
As seen from the latter, there are two power nodes, or points over 
which no power flows, one in the transformer and one in the load, 
and the power flows from the transformer over the line into the 
load; the transformer acts as generator of the power, and of this 


112 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

power a fraction is consumed in the line, the rest supplied to the 
load. 

40. The diagram of this transient power transfer of the system 
thus is very similar to that of the permanent power transmis- 
sion by alternating currents: a source of power, a partial con- 
sumption in the line, and the rest of the power consumed in 
the load. 

However, this transient power-transfer diagram does not repre- 
sent the entire power which is being consumed in the circuit, as 
power is also supplied from the stored energy of the circuit; and 
the case may thus arise — which cannot exist in a permanent 
power transmission — that the power dissipation of the line is 
less than corresponds to its stored energy, and the line also supplies 
power to the load, that is, acts as generator, and in this case the 
power would not be a maximum at the transformer terminals, 
but would still further increase in the line, reaching its maxi- 
mum at the load terminals. This obviously is possible only 
with transient power, where the line has a store of energy from 
which it can draw in supplying power. In permanent condition 
the line could not add to the power, but must consume, that is, 
the permanent power-transmission diagram must always be like 
Fig. 54. 

Not so, as seen, with the transient of the stationary oscillation. 
Assume, for instance, that we reduce the power dissipation in 
the line by doubling the conductor section, that is, reducing the 
resistance to one-half. As L thereby also slightly decreases, 
C increases, and g possibly changes, the change brought about in 

the constant u = =lj ~^7>) *s no^ necessarily a reduction to 

half, but depends upon the dimensions of the line. Assuming 
therefore, that the power-dissipation constant of the line is by the 
doubling of the line section reduced from u\ = 900 to HI = 500, 
this gives the constants: 

Line. Transformer. Line. Load. Sum. 

X= 1.5X10-3 1X10-3 1.5X10-3 .5X10-3 4.5X1Q-3 
u= 500 100 500 1600 

wX= .75 .1 .75 .8 2.4 

SwX 
hence, MO = average, u = — - = 533, and: 

2/A 

s= +33 +433 +33 -1067 


one- 


OSCILLATIONS OF THE COMPOUND CIRCUIT. 


113 


That is, the power-transfer constant of the line has become posi- 
tive, si = 33, and the line now assists the transformer in supplying 
power to the load. Assuming again the values of the two travel- 
ing waves, where they leave the transformer (which now are not 
the maximum values, since the waves still further increase in 
intensity in passing over the lines), as 6 and 4 megawatts respec- 
tively, the power diagram of the two waves, and the power dia- 
gram of their resultant, are given in Fig. 55. 


Fig. 55. — Energy Distribution in Compound Oscillation of Closed Circuit; 

Low Line Loss. 

In a closed circuit, as here discussed, the relative intensity of 
the two component waves of opposite direction is not definite, 
but depends on the circuit condition at the starting moment of 
the transient. 

In an oscillation of an open compound circuit, the relative 
intensities of the two component waves are fixed by the condition 
that at the open ends of the circuit the power transfer must be 
zero. 

As illustration may be considered a circuit comprising the high- 
potential coil of the step-up transformer, and the two lines, which 
are assumed as open at the step-down end, as illustrated diagram- 
matically in Fig. 56. 


114 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 


Choosing the same lengths and the same power-dissipation 
constants as in the previous illustrations, this gives: 


Line. Transformer. Line. Sum. 

1.5X10-3 1X10-3 1.5X10-3 4X10-3 

900 100 900 

1.35 .1 1.35 2.8 

SwX 


x= 

u\ = 
hence, w0 = average, u = ^^ = 700, and: 

s= -200 +600 -200 

Line 


Transformer 


Line 


Fig. 56. 

The diagram of the power of the two waves of opposite direc- 
tions, and of the resultant power, is shown in Fig. 57, assuming 
6 megawatts as the maximum power of each wave, which is reached 
at the point where it leaves the transformer. 


Transmission Line Transformer Transmission Line 


U =900 


U =100 


U=900 


Fig. 57. — Energy Distribution in Compound Oscillation of Open Circuit. 

In this case the two waves must be of the same intensity, so 
as to give 0 as resultant at the open ends of the line. A power 
node then appears in the center of the transformer. 

41. A stationary oscillation of a compound circuit consists of 
two traveling waves, traversing the circuit in opposite direction, 
and transferring power between the circuit section in such a manner 


OSCILLATIONS OF THE COMPOUND CIRCUIT. 115 

as to give the same rate of energy dissipation in all circuit sections. 
As the result of this power transfer, the stored energy of the 
system must be uniformly distributed throughout the entire 
circuit, and if it is not so in the beginning of the transient, local 
traveling waves redistribute the energy throughout the oscillat- 
ing circuit, as stated before. Such local oscillations are usually 
of very high frequency, but sometimes come within the range of 
the oscillograph, as in Fig. 47. 

During the oscillation of the complex circuit, every circuit 
element d\ (in velocity measure), or every wave length or equal 
part of the wave length, therefore contains the same amount of 
stored energy. That is, if e0 = maximum voltage, i0 = maximum 

current, and X0 = wave length, the average energy ° ° Q must 

be constant throughout the entire circuit. Since, however, in 
velocity measure, Xo is constant and equal to the period TO through- 
out all the sections of the circuit, the product of maximum voltage 
and of maximum current, e0^o, thus must be constant throughout 
the entire circuit. 

The same applies to an ordinary traveling wave or impulse. 
Since it is the same energy which moves along the circuit at a 
constant rate, the energy contents for equal sections of the circuit 
must be the same except for the factor e~ 2 "*, by which the energy 
decreases with the time, and thus with the distance traversed 
during this time. 

Maximum voltage e0 and maximum current i'o, however, are 
related to each other by the condition_ 

e° i /^° fo\ 

— = ZQ = y -FT , (3) 

and as the relation of L0 and <70 is different in the different sections, 
and that very much so, ZQ, and with it the ratio of maximum 
voltage to maximum current, differ for the different sections of 
the circuit. 

If then ei and ii are maximum voltage and maximum current 

respectively of one section, and z\ = y -^ is the "natural imped- 
ance " of this section, and ez, 12, and z2 — V/TT are the correspond- 

V 02 

ing values for another section, it is 

• _ • / ' A\ 


116 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

and since 

^ — ei — 
iz ' ii 

substituting e2 = i'2z2,l 

= 'z I ^ 

into (4) gives 

or 


and 

z2 
or 

fz» 


/ON 

That is, in the same oscillating circuit, the maximum voltages 
60 in the different sections are proportional to, and the maximum 
currents i0 inversely proportional to, the square root of the natural 
impedances z0 of the sections, that is, to the fourth root of the 

ratios of inductance to capacity -^ • 

to 

At every transition point between successive sections traversed 
by a traveling wave, as those of an oscillating system, a trans- 
formation of voltage and of current occurs, by a transformation 
ratio which is the square root of the ratio of the natural imped- 
ances, ZQ = V TT > of the two respective sections. 
* Co 

When passing from a section of high capacity and low induc- 
tance, that is, low impedance z0, to a section of low capacity and 
high inductance, that is, high impedance z0, as when passing from 
a transmission line into a transformer, or from a cable into a trans- 
mission line, the voltage thus is transformed up, and the current 
transformed down, and inversely, with a wave passing in opposite 
direction. 

A low-voltage high-current wave in a transmission line thus 
becomes a high-voltage low-current wave in a transformer, and 
inversely, and thus, while it may be harmless in the line, may 
become destructive in the transformer, etc. 


OSCILLATIONS OF THE COMPOUND CIRCUIT. 


117 


42. At the transition point between two successive sections, 
the current and voltage respectively must be the same in the 
two sections. Since the maximum values of current and voltage 
respectively are different in the two sections, the phase angles of 
the waves of the two sections must be different at the transition 
point; that is, a change of phase angle occurs at the transition 
point. 

This is illustrated in Fig. 58. Let z0 = 200 in the first section 
(transmission line), ZQ = 800 in the second section (transformer). 

/800 

The transformation ratio between the sections then is V onn=2; 

» ^Uu 

that is, the maximum voltage of the second section is twice, and 
the maximum current half, that of the first section, and the waves 
of current and of voltage in the two sections thus may be as 
illustrated for the voltage in Fig. 58, by e\e^. 


Fig. 58. — Effect of Transition Point on Traveling Wave. 


If then ef and if are the values of voltage and current respec- 
tively at the transition point between two sections 1 and 2, and 
e\ and ii the maximum voltage and maximum current respec- 
tively of the first, e% and iz of the second, section, the voltage phase 
and current phase at the transition point are, respectively: 


For the wave of the first section: 

_ = cos 71 and -r = cos 5i. 

For the wave of the second section: 

e' i' 

— — cos 72 and — = cos §2. 


(9) 


118 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 
Dividing the two pairs of equations of (9) gives 

cos 72 _ 61 
cos 71 62 


= ii = Jz* f 
iz V ?! 


cos <5i 
hence, multiplied, 

cos 72 cos 82 

cos 71 cos di 
or (11) 

cos 72 _ cos di 

COS 7i COS £2 

or 

cos 71 cos 5i = cos 72 cos 52; 

that is, the ratio of the cosines of the current phases at the tran- 
sition point is the reciprocal of the ratio of the cosines of the 
voltage phases at this point. 

Since at the transition point between two sections the voltage 
and current change, from ei, ii to 62, is, by the transformation ratio 

— , this change can also be represented as a partial reflection. 

That is, the current i\ can be considered as consisting of a compo- 
nent z'2, which passes over the transition point, is " transmitted " 
current, and a component i\ = i\ — iz, which is " reflected " 
current, etc. The greater then the change of circuit constants 
at the transition point, the greater is the difference between the 
currents and voltages of the two sections; that is, the more of 
current and voltage are reflected, the less transmitted, and if the 
change of constants is very great, as when entering from a trans- 
mission line a reactance of very low capacity, almost all the 
current is reflected, and very little passes into and through the 
reactance, but a high voltage is produced in the reactance. 


v/