CHAPTER III. 

THE NATURAL PERIOD OF THE TRANSMISSION LINE. 

27. An interesting application of the equations of the long 
distance transmission line given in the preceding chapter can be 
made to the determination of the natural period of a transmis- 
sion line; that is, the frequency at which such a line discharges 
an accumulated charge of atmospheric electricity (lightning), or 
oscillates because of a sudden change of load, as a break of circuit, 
or in general a change of circuit conditions, as closing the circuit, 
etc. 

The discharge of a condenser through a circuit containing self- 
inductance and resistance is oscillating (provided the resistance 
does not exceed a certain critical value depending upon the 
capacity and the self-inductance) ; that is, the discharge current 
alternates with constantly decreasing intensity. The frequency 
of this oscillating discharge depends upon the capacity C and 
the self -inductance 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 frequency of oscillation can, by neglecting the 
resistance, be expressed with fair, or even close, approximation 
by the formula 


An electric transmission line represents a circuit having 
capacity as well as self-inductance ; and thus when charged to a 
certain potential, for instance, by atmospheric electricity, as by 
induction 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 ordinary 
condenser in that with the former the capacity and the self- 
inductance are distributed along the circuit. 

In determining the frequency of the oscillating discharge of 
such a transmission line, a sufficiently close approximation is 

320 


NATURAL PERIOD OF TRANSMISSION LINE 321 

obtained by neglecting the resistance of the line, which, at the 
relatively high frequency of oscillating discharges, is small com- 
pared 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 approximately the same frequency 
and the same intensity as the initial waves of the oscillating 
discharge current. By this means the problem is essentially 
simplified. 

28. Let 10 = total length of a transmission line; I = the dis- 
tance from the beginning of the line; r = resistance per unit 
length; x = reactance per unit length = 2 nfL, where L = 
inductance per unit length; g = conductance from line to return 
(leakage and discharge into the air) per unit length; b = capacity 
susceptance per unit length = 2 nfC, where C = capacity per 
unit length. 

Neglecting the line resistance and line conductance, 

r = 0 and g = 0, 

the line constants a and /?, by equations (14), Chapter II, then 
assume the form 

a = 0 and ft = Vxb, (1) 

and the line equations (17) of Chapter II become 
/ = (AA - A2) cos pi - j (Aj + A,) sin pi 
and 


E = V ^ (A, + A2)cos fl - / (4i - 42) sin pi 

or writing 

4i ~ 42 = <7i and 4i + 42 = Q» 

and substituting 


c 

we have 

and • • l * • 2 


(3) 


322 TRANSIENT PHENOMENA 

A free oscillation of a circuit implies that energy is neither 
supplied to the circuit nor abstracted from it. This means that 
at both ends of the circuit, I = 0 and I = 1Q, the power equals zero. 

If this is the case, the following conditions may exist: 

(1) The current is zero at one end, the voltage zero at the 
other end. 

(2) Either the current is zero at both ends or the voltage is 
zero at both ends. 

(3) The circuit has no end but is closed upon itself. 

(4) The current is in quadrature with the voltage. This case 
does not represent a free oscillation, since the frequency depends 
also on the connected circuit, but rather represents a line supply- 
ing a wattless or reactive load. 

In free oscillation the circuit thus must be either open or 
grounded at its ends or closed upon itself. 

(1) Circuit open at one end, grounded at other end. 

29. Assuming the circuit grounded at I = 0, open at / = Z0, 
we have for I = 0, 

# = #o = 0, 

and for I = Z0, 

! -I.i-D; 

hence, substituting in equations (3), at I — 0, 


hence, 

7 = C, cos 81 
and 


(4) 


and at I = Z0, 


! cos /?Z0 = 0, 


and since Ct cannot be zero without the oscillation disappearing 
altogether, 

cos#0 = 0; (5) 

hence, 

#0 = (2n-l), (6) 


NATURAL PERIOD OF TRANSMISSION LINE 323 

where n = 1, 2, 3 ... or any integer and 

£11 = (2n- l)~l. (7) 

Z L0 

Substituting (1) in (6) gives 

ft-V*;-V^, (8) 

z Lo 

or substituting for x and b, x = 2 Tr/L and 6 = 2 7r/<7, gives 


or 


is the frequency of oscillation of the circuit. 

The lowest frequency or fundamental frequency of oscillation 
is, for n = 1, 


and besides this fundamental frequency, all its odd multiples or 
higher harmonics may exist in the oscillation 

f** (2n-l)/r (11) 

Writing L0 = Z0L = total inductance, and C0 = 10C = total 
capacity of the circuit, equation (9) assumes the form 

(12) 


The fundamental frequency of oscillation of a transmission 
line open at one end and grounded at the other, and having a 
total inductance L0 and a total capacity (70, is, neglecting energy 
losses, 

fl = ~ rr-TT ' 


324 


TRANSIENT PHENOMENA 


while the frequency of oscillation of a localized inductance L0 
and localized capacity (70, that is, the frequency of discharge of 
a condenser CQ through an inductance L0, is 

/ = ^= • d3) 

The difference is due to the distributed character of L0 and C0 
in the transmission line and the resultant phase displacement 
between the elements of the line, which causes the inductance 
and capacity of the line elements, in their effect on the frequency, 
not to add but to combine to a resultant, which is the projection 

2 

of the elements of a quadrant, on the diameter, or - times the 

n 

sum, just as, for instance, the resultant m.m.f. of a distributed 

2 
armature winding of n turns of i amperes is not ni but - ni. 

7t 

Hence, the effective inductance of a transmission line in free 
oscillation is 

L> 2IL 

L0 •- - L0L 

n 


and the effective capacity is 


(14) 


and using the effective values L0' and C0', the fundamental 
frequency, equation (11), then appears in the form 


2* ^fLJc^ ' 

that is, the same value as found for the condenser discharge. 

In comparing with localized inductances and capacities, the 
distributed capacity and inductance, in free oscillation, thus are 
represented by their effective values (13) and (14). 

30. Substituting in equations (4), 

Cl = <>i + jcv (16) 

gives 

I = (cl + jc2) cos ftl 


and 


(17) 


NATURAL PERIOD OF TRANSMISSION LINE 


325 


By the definition of the complex quantity as vector represen- 
tation of an alternating wave the cosine component of the wave 
is represented by the real, the sine component by the imaginary 
term; that is, a wave of the form ct cos 2 nft + c2sin 2 nft is 
represented by cl + jc2J and inversely, the equations (17), in 
their analytic expression, are 

i = (ct cos 2 nft + c2 sin 2 xft) cos ftl 


and 


V/Z 
e -\n 


(c2 cos 2 nft - c1 sin 2 rft] sin pi. 


Substituting (7) and (11) in (18), and writing 
0 = 2 rfj, and r = ^~ 

2 *0 

gives 

i == {cjcos (2n- 1)6 + c2 sin (2n- 1)0} cos (2n -I) r 

= ccos (2n - 1) (0 - r) cos (2 n - l)r 
and 


(18) 


(19) 


e =• Vc2 cos (2 w-l)^-c1 


- \ - c sin (2 n - 1) (0 - r) sin (2 n - l)r, 


(20) 


where 


tan (2 n - 1) r = J and c = 


(21) 


In the denotation (19), 0 represents the time angle, with the 
complete cycle of the fundamental frequency of oscillation as 
one revolution or 360 degrees, and r represents the distance 
angle, with the length of the line as a quadrant or 90 degrees. 
That is, distances are represented by angles, and the whole line 
is a quarter wave of the fundamental frequency of oscillation. 
This form of free oscillation may be called quarter-wave oscillation. 

The fundamental or lowest discharge wave or oscillation of 
the circuit then is 

il = c cos (0 — fj) cos T 


and 


el = - \ — c sin (0 — 


sn r. 


(22) 


326 


TRANSIENT PHENOMENA 


With this wave the voltage is a maximum at the open end of 
the line, I = Z0, and gradually decreases to zero at the other end 
or beginning of the line, I = 0. 

The current is zero at the open end of the line, and gradually 
increases to a maximum at I = 0, or the grounded end of the 
line. 

Thus the relative intensities of current and potential along 
the line are as represented by Fig. 85, where the current is shown 
as 7, the voltage as E. 


Fig. 85. Discharge of current and e.in.f. along a transmission line open at 
one end. Fundamental discharge frequency. 


The next higher discharge frequency, for n = 2, gives 
i3 = c3 cos 3 (0 — 7-3) cos 3 r 


and 


= - c3 V - 


sin 3 (0 - r3) sin 3 T. 


(23) 


Here the voltage is again a maximum at the open end of the 
line, I = 10, or r = - = 90°, and gradually decreases, but reaches 


zero at two-thirds of the line, I = 


2L 


or 


-=60°, then 

o 


increases again in the opposite direction, reaches a second but 

opposite maximum at one-third of the line, I = -| , or r = = 30°, 

o o 

and decreases to zero at the beginning of the line. There is thus 
a node of voltage at a point situated at a distance of two-thirds 
of the length of the line. 

The current is zero at the end of the line, / = Z0, rises to a 
maximum at a distance of two-thirds of the length of the line, 
decreases to zero at a distance of one-third of the length of the 
line, and rises again to a second but opposite maximum at the 


NATURAL PERIOD OF TRANSMISSION LINE 


327 


beginning of the line, I = 0. The current thus has a node at a 
point situated at a distance of one-third of the length of the line. 


Fig. 86. Discharge of current and e.m.f. along a transmission 
line open at one end. 

The discharge waves, n = 2, are shown in Fig. 86, those with 
n = 3, with two nodal points, in Fig. 87. 


\ 


A 


/ 


Fig. 87. Discharge of current and e.m.f. along a transmission 
line open at one end. 

31. 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, lh= height of conductor 
above ground, and 10 = length of conductor, the capacity is 

1.11 X 10-8Z 


?;- 


4L 


. 
°,mmf. 


the self-inductance is 


inmh. 


(24) 


328 TRANSIENT PHENOMENA 

. The fundamental frequency of oscillation, by substituting (24) 
in (10), is 

1 7.5 X IP. 

' ' (25) 


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, /0, of the line, being 
inversely proportional thereto. 

We thus get the numerical values, 
Length of line 

100 miles 
16 X 106 cm. 

hence frequency, 

£ = 4700 2350 1570 1175 940 783 587 470 cycles per sec. 

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 represented 
in equation (20) can occur simultaneously in the oscillating dis- 
charge of a transmission line, and, in general, the oscillating 
discharge of a transmission line is thus of the form 


n cn cos (2 n — 1) (0 — pn) cos (2 n — 1) T, 


=-\/iix 


e = ~ VS 2>cn sin (2 n - 1) (6 - rj sin (2 n - 1) r. 


(26) 


A simple harmonic oscillation as a line discharge would require 
a sinoidal distribution of potential on the transmission line at the 
instant of discharge, which is not probable, 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 discharge, upon the atmospheric electrostatic field of 
force. 


NATURAL PERIOD OF TRANSMISSION LINE 329 

The fundamental frequency of the oscillating discharge of a 
transmission line is relatively low, and of not much higher mag- 
nitude than frequencies in commercial use in alternating-current 
circuits. Obviously, the more nearly sinoidal the distribution 
of potential before the discharge, tfye more the low harmonics 
predominate, while a very unequal distribution of potential, 
that is a very rapid change along the line, causes the higher har- 
monics to predominate. 

32. As an example the discharge of a transmission line may be 
investigated, the line having the following constants per mile : 
r - 0.21 ohm; L = 1.2 X 10~3 henry; C = 0.03 X 10~6 farad, 
and of the length Z0 = 200; hence, by equations (10), (19), 
/! = 208 cycles per sec.; 0 = 1315 t, and T = 0.00785 Z, when 
charged to a uniform voltage of e0 = 60,000 volts but with no 
current in the line before the discharge, and the line then 
grounded at one end, Z = 0, while open at the other end, Z = Z0. 

Then, for t = 0 or 6 = 0, i = 0 for all values of r except T = 0; 
hence, by (26), 

cos (2 n — 1) -jTn = 0, 

and thus 

(2»=-l)y.-? (27) 

and 

cos (2 n - 1) (e - rn) = sin (2 n - 1) 0, 
sin (2 n - 1) (0 - Tn) = - cos (2 n - 1) 0; 

hence, 

i = lL,n cn sin (2 n - 1) 0 cos (2 n - 1) T 


and 


e = \ - ^ncn cos (2 n - 1) 6 sin (2 n - 1) r. 


(28) 


Also for t = 0, or 0 = 0, e = e0 for all values of T except T = 0; 
hence, by (28), 

"c.ffln <2 » -1)'1- .(29) 


330 TRANSIENT PHENOMENA 

From equation (29), the coefficients cn are determined in the 
usual manner of evaluating a Fourier series, that is, by multiply- 
ing with sin (2 m — 1) r (or cos (2m - 1) r) and integrating: 

ren sin (2 m — 1) r dr = 
• 

\^2,ncn I sin (2n - l)rsin (2m- 1) r dr. 

C j JQ 

Since 

J sin (2 n — 1) r sin (2 m — 1) r dr 

rcos 2 (n - m) r - cos 2 (n + m — 1) r 
J — - dT' 

which is zero forn = ± m, while for m = n the term 

X17 cos 2 (rz. - m) r 7 T71" dr n 
— -*-J. ^ = 2 

and 

cos (2 n - 1) Tl* 2 e0 


we have 


and 

c = - 4 g° - V/~: (30) 

(2n - !)^VL? 

hence, 

4 . /C A sin (2 n - 1) 0 cos (2 TI - 1) r 

* - 


C ( . sin 3 6 cos 3 T sin 5 6 cos 5 r 


(31) 

_ ( . sin 3 6 cos 3 r sin 5 6 cos 5 r ) 

= 382 < sin 0 cos r H — - + - — — — + • • • > , 

in amperes, 


NATURAL PERIOD OF TRANSMISSION LINE 331 

and 

4 A cos (2 n - 1) 0 sin (2 n - 1) r 
TT ° i 2/1 — 1 

cos 3 # sin 3 r cos 5 # sin 5 r > 

cos 0 sm T -| 1 — h • • • J 

o 5 ) 

(32) 

_ ( cos 3 6 sin 3 r cos 5 0 sin 5 r ) 

= 76,400 ] cos 0 sm r + - — + [-•••{» 

o 5 ) 

in volts. 

33. As further example, assume now that this line is short- 
circuited at one end, I = 0, while supplied with 25-cycle alter- 
nating power at the other end, I = /0, and that the generator 
voltage drops, by the short circuit, to 30,000, and then the line 
cuts off from the generating system at about the maximum value 
of the short-circuit current, that is, at the moment of zero value 
of the impressed e.m.f. 

At a frequency of /0 = 25 cycles, the reactance per unit length 
of line or per mile is 

x = 2 TT/OL = 0.188 ohm 
and the impedance is 

z = Vr2 + x* = 0.283 ohm, 
or, for the total line, 

z0 = I0z = 56.6 ohms; 

hence, the approximate short-circuit current 
e 30,000 


and its maximum value is 

i0 - 530 X \/2 = 750 amp. 

Therefore, in equations (26), at time t = 0, or 0 = 0, e= 0 
for all values of T except T = — ; hence, 

Zi 

sin (2 n - 1) yn = 0, 
or, yn - 0, 


332 TRANSIENT PHENOMENA 

and thus 

i = ^n cn cos (2 n. — 1) 6 cos (2 n — 1) r 
and (33) 

e= -y5]£ncnsin(2n - 1)# sin (2 ri - l)r. 

However, at t = 0, or 6 = 0, for all values of r except T = ^, ' 


hence, substituting in (33), 

A 
t'0 - 2/n cn cos (2 n - 1) r. 


(34) 


From equation (34), the coefficients cn are determined in the 
same manner as in the preceding example, by multiplying with 
cos (2 n — 1) r and integrating, as 


hence, 
i = — 


(2n-l)*' 

cos (2n - l)0cos(2n - 1) r 
2n - 1 


(35) 


4 i' ( cos 3 6 cos 3 r cos 5 0 cos 5 r 

= — - cos 0 cos T — - — 


(36) 


( cos 3 6 cos 3 T cos 5 0 cos 5 r 
956 ) cos 0 cos T + - 


in amperes, 


and 


4i 


n - 1) 0 sin (2 n - 1) T 


4 t* [L ( . sin 3 /9 sin 3 T si 

-V- Utfmnr- -^ - + - 


2n - 1 

sin 5 (9 sin 5 T 


- + 


(37) 


( , sin 3 6 sin 3 r sin 5 6 sin 5 T ) 

= 191,200 ) sin 6 sin T— — — - - + — +•••{ 


in volts. 


NATURAL PERIOD OF TRANSMISSION LINE 333 


The maximum voltage is reached at time 6 = - , and is 


e = 


and since the series 


4 i IL . sin 3 r sin 5 r 


sin 3 T sin 5 r 
sm T + - 


the maximum voltage is 

e = i V/^ = 300,000 volts. 


As seen, very high voltages may be produced by the interrup- 
tion of the short-circuit current. 

(2a) Circuit grounded at both ends. 

34. The method of investigation is the same as in paragraph 
29; the terminal conditions are, for I = 0, 


E= 0, 


and for I = I, 


Substituting Z = 0 into equations (3) gives 


hence, 


/ = C, cos /M, 


sn 


Substituting Z = Z0 in (38) gives 

& - o - - jev 

hence, 

sin £HQ = 0, or /?£ = TITT, 


(38) 


(39) 


334 TRANSIENT PHENOMENA 

and, in the same manner as in (1), 

pi = n^l = nr; 


(40) 


that is, the length of the line, Z0, represents one half wave, or 
r = TT, or a multiple thereof. 


n 


210VLC 2VL0C0 


and the fundamental frequency of oscillation is 

1 


210VLC 


and 


(41) 

(42) 
(43) 


that is, the line can oscillate at a fundamental frequency fv for 
which the length, 1OJ of the line is a half wave, and at all multiples 
or higher harmonics thereof, the even ones as well as the odd ones. 

This kind of oscillation may be called a half-wave oscillation. 

35. Unlike the quarter-wave oscillation, which contains only 
the odd higher harmonics of the fundamental wave, the half- 
wave oscillation also contains the even harmonics of the funda- 
mental frequency of oscillation. 

Substituting Ot = cl + j'c2 into (38) gives 


and 


jc2) cos 


(44) 


and replacing the complex imaginary by the analytic expression, 
that is, the real term by cos 2 nft, the imaginary term by sin 2 nft, 
gives 

i — { cx cos 2 nft + c2 sin 2 nft] cos pi 
and 


e 


-s/i 


- {c2 cos 2 nft - cl sin 2 


sin pl} 


NATURAL PERIOD OF TRANSMISSION LINE 335 

and substituting 

we have (45) 

2 nft = nO] 

then (44) gives, by (40) : 

i = (c1 cos nO + c2 sin nO) cos nr 
and 


or writing 
and 


e = \ - (ca cos 7i# — ct sin nd) sin nr; 
* C 

ct = c cos nj \ 
c2 = c sin n/- J 

i = c cos n (0 — r) cos nr 


(46) 


(47) 


- c y — sin n (0 — 7-) s 


(48) 


gives 

and 

e = — c V — sin n (U — r) sin nr, 

and herefrom the general equations of this half-wave oscillation are 

co 

t = 7,7ic» cos n (0 — r«) cos 


and 


e = — y — X » cn sin n (0 — fn) sin 


(49) 


(26) Circuit open at both ends. 
36. For Z = 0 we have 


hence, 

and 

and 


0 


= - sn 


(50) 


while for I = L, 7=0; 


336 

hence, 


TRANSIENT PHENOMENA 


sin /?Z0 = 0, or /?Z 


(51) 


that is, the circuit performs a half-wave oscillation of funda- 
mental frequency, 

1 

(52) 


and all its higher harmonics, the even ones as well as the odd ones 
have a frequency 

/ - «A, (53) 

and the final equations are 

«>v 

i = — n c n sin n (0 — f) sin nr 


and 


where 


= y ~ Zjncn cos n (6 — f) 


cos 


0 


and T = -I. 

^0 


(54) 


(55) 


(3) Circuit closed upon itself. 

37. If a circuit of length 10 is closed upon itself, then the free 
oscillation of such a circuit is characterized by the condition that 
current and voltage at I = 10 are the same as at I = 0, since I = 10 
and I = 0 are the same point of the circuit. 

Substituting this condition in equations (3) gives 


= = cos - 


sn 


and 


- E = C, = C2 cos /?Z0 - j^! sin /?Z0; 
herefrom follows 


(56) 


- cos 


o, (i - cos 


0) = - yCa sin ^0, > 

= - jc, sin ^0, y 


hence, 
or 


- cos 


= - sn 


(57) 


(58) 


NATURAL PERIOD OF TRANSMISSION LINE 


337 


hence, 


= 2 


(59) 


that is, the circuit must be a complete wave or a multiple 
thereof. 

The free oscillation of a circuit which is closed upon itself is a 
full-wave oscillation , containing a fundamental wave of frequency 


and all the higher harmonics thereof, the even ones as well as the 
odd ones, 

/ - nfr (61) 

Substituting in (3), 


and 

gives 

and 


<7i = ci + K 


V 2 ~~ 


cos 


= «, + i 

= c' + je," J 


(c2" - yc2 


cos 


sn 


(62) 


Substituting the analytic expression, 

c/ + jc/' = c/ cos 2 TT/^ + c/' sin 2 TT/J^, etc., 
also 


and 
where 


27T 


(63) 


(64) 


that is, the length of the circuit, I = Z0, is represented by the 
angle r = 2 x, or a complete cycle, this gives 


338 


TRANSIENT PHENOMENA 


/ = (c/ cos nd + c/' sin nd) cos nr 

sin nr 


and 


= V n 


(c2" cos nO — cJ sin 


or writing 


:/ cos nO + c2" sin nff) cos n# 

cos nd - c/ sin n#) sin nr} , 

c/ = a cos n/- 
c/7 = a sin n/- 
c/ = 6 cos n% 
c" = 6 sin n^ 


(65) 


gives 
and 


' = a cos n (6 — -f) cos nr — b sin n (0 — %) sin nr 

\ - { b cos n (6 — %) cos nr - a sin n (6 - ;-) sin nr j . 
C 


(66) 


Thus in its most general form the full-wave oscillation gives 
the equations 


i = n { an cos n (6 — j-n) cos m: — bn sin n (0 —%n) sin nr 


^ 


where 


~n cos nr - 


0 = 


(0 - fn) sin nr}, 


(67) 


(68) 


and an, fn and 6n, /n are groups of four integration constants. 

38. With a short circuit at the end of a transmission line, the 
drop of potential along the line varies fairly gradually and 
uniformly, and the instantaneous rupture of a short circuit - 
as by a short-circuiting arc blowing itself out explosively — 


NATURAL PERIOD OF TRANSMISSION LINE 339 

causes an oscillation in which the lower frequencies predominate, 
that is, a low-frequency high-power surge. A spark discharge 
from the line, a sudden high voltage charge entering the line 
locally, as directly by a lightning stroke, or indirectly by induc- 
tion during a lightning discharge elsewhere, gives a distribution 
of potential which momentarily is very non-uniform, changes 
very abruptly along the line, and thus gives rise mainly to very 
high harmonics, but as a rule does not contain to any appre- 
ciable extent the lower frequencies; that is, it causes a high- 
frequency oscillation, more or less local in extent, and while of 
high voltage, of rather limited power, and therefore less destruc- 
tive than a low-frequency surge. 

At the frequencies of the high-frequency oscillation neither 
capacity nor inductance of the transmission line is perfectly 
constant: the inductance varies with the frequency, by the 
increasing screening effect or unequal current distribution in 
the conductor; the capacity increases by brush discharge over the 
insulator surface, by the increase of the effective conductor 
diameter due to corona effect, etc. The frequencies of the very 
high harmonics are therefore not definite but to some extent 
variable, and since they are close to each other they overlap; 
that is, at very high frequencies the transmission line has no 
definite frequency of oscillation, but can oscillate with any 
frequency. 

A long-distance transmission line has a definite natural period 
of oscillation, of a relatively low fundamental frequency and its 
overtones, but can also oscillate with any frequency whatever, 
provided that this frequency is very high. 

This is analogous to waves formed in a body of water of 
regular shape : large standing waves have a definite wave length, 
depending upon the dimensions of the body of water, but very 
short waves, ripples in the water, can have any wave length, and 
do not depend on the size of the body of water. 

A further investigation of oscillations in conductors with 
distributed capacity, inductance, and resistance requires, how- 
ever, the consideration of the resistance, and so leads to the 
investigation of phenomena transient in space as well as in time, 
which are discussed in Section IV. 

39. In the equations discussed in the preceding, of the free 
oscillations of a circuit containing uniformly distributed resist- 


340 TRANSIENT PHENOMENA 

ance, inductance, capacity , and conductance, the energy losses 
in the circuit have been neglected, and voltage and current 
therefore appear alternating instead of oscillating. That is, 
these equations represent only the initial or maximum values of 
the phenomenon, but to represent it completely an exponential 
function of time enters as factor, which, as will be seen in Section 
IV, is of the form 

(69) 

where u = ~ (7, + f ) mav be called the "time constant" of the 
2 \C LI 

circuit. 

While quarter-wave oscillations occasionally occur, and are of 
serious importance, the occurrence of half-wave oscillations and 
especially of full-wave oscillations of the character discussed 
before, that is, of a uniform circuit, is less frequent. 

When in a circuit, as a transmission line, a disturbance or 
oscillation occurs while this circuit is connected to other cir- 
cuits — as the generating system and the receiving apparatus - 
as is usually the case, the disturbance generally penetrates into 
the circuits connected to the circuit in which the disturbance 
originated, that is, the entire system oscillates, and this oscilla- 
tion usually is a full- wave oscillation; that is, the oscillation of 
a circuit closed upon itself; occasionally a half- wave oscillation. 
For instance, if in a transmission system comprising generators, 
step-up transformers, high-potential lines, step-down trans- 
formers, and load, a short circuit occurs in the line, the circuit 
comprising the load, the step-down transformers, and the lines 
from the step-down transformers to the short circuit is left 
closed upon itself without power supply, and its stored energy is, 
therefore, dissipated as a full-wave oscillation. Or, if in this 
system an excessive load, as the dropping out of step of a syn- 
chronous converter, causes the circuit to open at the generating 
station, the dissipation of the stored energy — in this case that 
of the excessive current in the system — occurs as a full-wave 
oscillation, if the line cuts off from the generating station on the 
low-tension side of the step-up transformers, and the oscillating 
circuit comprises the high-tension coils of the step-up trans- 
formers, the transmission line, step-down transformers, and load. 
If the line disconnects from the generating system on the high- 


NATURAL PERIOD OF TRANSMISSION LINE 341 

potential side of the step-up transformers, the oscillation is a 
half-wave oscillation, with the two ends of the oscillating circuit 
open. 

Such oscillating circuits, however, — representing the most 
frequent and most important case of high-potential disturbances 
in transmission systems, — cannot be represented by the preced- 
ing equations since they are not circuits of uniformly distributed 
constants but complex circuits comprising several sections of 
different constants, and therefore of different ratios of energy 

consumption and energy storage, -and ^- During the free 

Ju C 

oscillation of such circuits an energy transfer takes place be- 
tween the different sections of the circuit, and energy flows from 
those sections in which the energy consumption is small com- 
pared with the energy storage, as transformer coils and highly 
inductive loads, to those sections in which the energy consump- 
tion is large compared with the energy storage, as the more 
non-inductive parts of the system. This introduces into the equa- 
tions exponential functions of the distance as well as the time, 
and requires a study of the phenomenon as one transient in 
distance as well as in time. The investigation of the oscillation 
of a complex circuit, comprising sections of different constants, is 
treated in Section IV.