CHAPTER I. 

GENERAL EQUATIONS. 

1. The energy relations of an electric circuit can be charac- 
terized, as discussed in Section III, by the four constants, 
namely : 

r = effective resistance, representing the power or rate of 
energy consumption depending upon the current, tfr; or the 
power component of the e.m.f. consumed in the circuit, that is, 
with an alternating current, the voltage, ir, in phase with the 
current. 

L = effective inductance, representing the energy storage 

i2L 

depending upon the current, - — , as electromagnetic component 

& 

of the electric field; or the voltage generated due to the change 

of the current, L — , that is, with an alternating current, the 
at 

reactive voltage consumed in the circuit - jxi, where x = 2 nfL 
and / = frequency. 

g = effective (shunted) conductance, representing the power 
or rate of energy consumption depending upon the voltage, e*g; 
or the power component of the current consumed in the circuit, 
that is, with an alternating voltage, the current, eg, in phase with 
the voltage. 

C = effective capacity, representing the energy storage 

e*C 
depending upon the voltage, — , as electrostatic component of 

the electric field; or the current consumed by a change of the 

de 

voltage, C — , that is, with an alternating voltage, the (leading) 
dt 

reactive current consumed in the circuit - jbe, where 6 = 2 
and / = frequency. 

417 


418 TRANSIENT PHENOMENA 

In the investigation of electric circuits, these four constants, 
r, L, g, C, usually are assumed as located separately from each 
other, or localized. Although this assumption can never be per- 
fectly correct, — for instance, every resistor has some inductance 
and every reactor has some resistance, — nevertheless in most 
cases it is permissible and necessary, and only in some classes of 
phenomena, and in some kinds of circuits, such as high-frequency 
phenomena, voltage and current distribution in long-distance, 
high-potential circuits, cables, telephone circuits, etc., this 
assumption is not permissible, but r, L, g, C must be treated as 
distributed throughout the circuit. 

In the case of a circuit with distributed resistance, inductance, 
conductance, and capacity, as r, L, g, C, are denoted the effec- 
tive resistance, inductance, conductance, and capacity, respec- 
tively, per unit length of circuit. The unit of length of the circuit- 
may be chosen as is convenient, thus : the centimeters in the high- 
frequency oscillation over the multigap lightning arrester circuit, 
or a mile in a long-distance transmission circuit or high-potential 
cable, or the distance of the velocity of light, 300,000 km., etc. 

The permanent values of current and e.m.f. in such circuits 
of distributed constants have, for alternating-current circuits, 
been investigated in Section III, where it was shown that they 
can be treated as transient phenomena in space, of the complex 
variables, current / and e.m.f. E. 

Transient phenomena in circuits with distributed constants, 
and, therefore, the general investigation of such circuits, leads to 
transient phenomena of two independent variables, time t and 
space or distance /; that is, these phenomena are transient in 
time and in space. 

The difficulty met in studying such phenomena is that they 
are not alternating functions of time, and therefore can no longer 
be represented by the complex quantity. 

It is possible, however, to derive from the constants of the 
circuit, r, L, g, C, and without any assumption whatever regard- 
ing current, voltage, etc., general equations of the electric cir- 
cuits, and to derive some results and conclusions from such 
equations. 

These general equations of the electric circuit are based on the 
single assumption that the constants r, L, g, C remain constant 
with the time t and distance I, that is, are the same for every unit 


GENERAL EQUATIONS 419 

length of the circuit or of the section of the circuit to which the 
equations apply. Where the circuit constants change, as where 
another circuit joins the circuit in question, the integration con- 
stants in the equations also change correspondingly. 

Special cases of these general equations then are all the phe- 
nomena of direct currents, alternating currents, discharges of 
reactive coils, high-frequency oscillations, etc., and the difference 
between these different circuits is due merely to different values 
of the integration constants. 

2. In a circuit or a section of a circuit containing distributed 
resistance, inductance, conductance, and capacity, as a trans- 
mission line, cable, high-potential coil of a transformer, telephone 
or telegraph circuit, etc., let r = the effective resistance per unit 
length of circuit; L = the effective inductance per unit length 
of circuit; g = the effective shunted conductance per unit 
length of circuit; C = the effective capacity per unit length of 
circuit; t = the time, I = the distance, from some starting 
point; e =•- the voltage, 'and i = the current at any point I and 
at any time t\ then e and i are functions of the time t and the dis- 
tance I. 

In an element dl of the circuit, the voltage e changes, by de, 
by the voltage consumed by the resistance of the circuit element, 
ri dl, and by the voltage consumed by the inductance of the cir- 

cuit element, L — dl. Hence, 

de _ di 


In this circuit element dl the current i changes, by di, by the 
current consumed by the conductance of the circuit element, 
gedl, and by the current consumed by the capacity of the circuit 

de 
element, C — dl. Hence, 


Differentiating (1) with respect to t and (2) with respect to I, 
and substituting then (1) into (2), gives 


420 TRANSIENT PHENOMENA 

and in the same manner, 

+K7. (4) 


These differential equations, of the second order, of current i 
and voltage e are identical; that is, in an electric circuit current 
and e.m.f. are represented by the same equations, which differ 
by the integration constants only, which are derived from the 
terminal conditions of the problem. 

Equation (3) is integrated by terms of the form 

i = Ae~al~bt. (5) 

Substituting (5) in (3) gives the identity 

a2 = rg - (rC + gL) b + LCb2 

= (bL - r) (bC - g). (6) 

In the terms of the form (5) the relation (6) thus must exist 
between the coefficients of I and t. 
Substituting (5) into (1) gives 


L«- -»', (7) 

tit 

and, integrated, 


a 

The integration constant of (8) would be a function of t} and 
since it must fulfill equation (4), must also have the form (5) 

T O 

for the special value a = 0, hence, by (6), b = - or b = -JJand 

L C 

therefore can be dropped. 

In their most general form the equations of the electric circuit 
are 

Ane-fl"|-w}, (9) 

\bnL ~rAne-^l-^\f (10) 


an2 - (bnL - r] (bnC - g) = 0, (11) 


GENERAL EQUATIONS 421 

where An and an and bn are integration constants, the last two 
being related to each other by the equation (11). 

3. These pairs of integration constants, An and (an, 6n), are 
determinated by the terminal conditions of the problem. 

Some such terminal conditions, for instance, are : 

Current i and voltage e given as a function of time at one 
point 1Q of the circuit — at the generating station feeding into 
the circuit or at the receiving end of the transmission line. 

Current i given at one point, voltage e at another point — 
as voltage at the generator end, current at the receiving end of 
the line. 

Voltage given at one point and the impedance, that is, the 

complex ratio - - — , at another point — voltage at the gen- 
amperes 

erator end, load at the receiving end of the circuit. 

Current and voltage given at one time t0 as function of the 
distance I — distribution of voltage and current in the circuit 
at the starting moment of an oscillation, etc. 

Other frequent terminal conditions are: 

Current zero at all times at one point Z0 — the open end of the 
circuit. 

Voltage zero at all times at one point 10 — the grounded or the 
short-circuited end of the circuit. 

Current and voltage, at all times, at one point Z0 of the circuit, 
equal to current and voltage at one point of another circuit — 
connecting point of one circuit with another one. 

As illustration, some of these cases will be discussed below. 

The quantities i and e must always be real; but since an and 
bn appear in the exponent of the exponential function, an and 
bn may be complex quantities, in which case the integration 
constants An must be such complex quantities that by com- 
bining the different exponential terms of the same index n, that 
is, corresponding to the different pairs of a and b derived from 
the same equation (10), the imaginary terms in An and 

bnL - r 

An cancel. 


an 
In the exponential function 


-al-bt 


422 TRANSIENT PHENOMENA 

writing 

a = h + jk and b = p + jq, (12) 

we have 

£-al-bt = e-hl-pt£-j(kl+qt) 

and the latter term resolves into trigonometric functions of the 
angle 

kl + qt. 

kl + qt = constant (13) 

therefore gives the relation between I and t for constant phase 
of the oscillation or alternation of the current or voltage. 

With change of time t the phase thus changes in position I 
in the circuit, that is, moves along the circuit. 

Differentiating (13) with respect to t gives 

**+•••*> 

or 


that is, the phase of the oscillation or alternation moves along 
the circuit with the speed — *., or, in other words, 


(15) 


is the speed of propagation of the electric phenomenon in the cir- 
cuit. 

(If no energy losses occur, r = 0, g = 0, in a straight con- 
ductor in a medium of unit magnetic and dielectric constant, 
that is, unit permeability and unit inductive capacity, S is the 
velocity of light.) 

4. Since (11) is a quadratic equation, several pairs or corre- 
sponding values of a and b exist, which, in the most general case, 
are complex imaginary. The terms with conjugate complex 
imaginary values of a and b then have to be combined for the 
elimination of their imaginary form, and thereby trigonometric 
functions appear; that is, several terms in the equations (9) and 


GENERAL EQUATIONS 


423 


(10), which correspond to the same equation (11), and thus can 
be said to form a group, can be combined with each other. 

Such a group of terms, of the same index n, is defined by the 
equation (11), 

a«2 = (bnL - r) (bnC - g). 

For convenience the index n may be dropped in the investiga- 
tion of a group of terms of current and voltage, thus : 


a2 = (bL - r) (bC - g), 
and the following substitutions may be made : 

a = 


a = h + jk, 
al = hl + jkv 
b = p + jq, 
from which 

h = h} VLC and 
k = k, VLC. 

Substituting (18) in (16), 

(h, + jk,)2 = \(p + jq) - y] \(p + jq) - £l. 

L/J oj 


(16) 

(17) 
(18) 


(19) 


(20) 


Carrying out and separating the real and the imaginary terms, 
equation (20) resolves into the two equations thus : 


and 


Substituting 


-.*(»".-£-$ 


(21) 


-C--A 

2VL C/' 


(22) 
(23) 


424 

and 

into (21) gives 

or 
and 

or 


TRANSIENT PHENOMENA 

p = S + u 

n u* — O/Y 

riiKi 6*/> 

s2 - (f = h* — k? + m2, 
sg = /i1A:1. 


| 
J 


(24) 
(25) 

(26) 


Adding four times the square of the second equation to the square 
of the first equation of (25) and (26) respectively, gives 


2 - (f - 7ft2)2 + 4 «V 

= V(s2 + cf - m2)2 + 4 £2m2 


(27) 


and 


+ q2 = (h* - k* + m2)2 + 4 V&i2 


x2 + A?!2 f m2)2 - 4 A;,2™2 


and substituting (19), gives, by (25), (26) and (27), (28) 


(28) 


k = 


? = V(si + <f - TO2)2 + 4 <fm\ 


(29) 


or 


V(h* + It? + LCntf --4 


(30) 


GENERAL EQUATIONS 


425 


If, however (+ h + jk) and (u + s + jq) satisfy equation (16), 
then any other one of the expressions 

(± h ± jk) and (u ± s ± jq) 

also satisfies equation (16), providing also the second equation of 
(25) or (26) is satisfied, 

hk = 


sq] 


(31) 


that is, if s and q have the same sign, h and k must have the same 
sign, and inversely, if s and q have opposite signs, h and k must 
have opposite signs. 

This then gives the corresponding values of a and b : 


(1) a = + h + jk 

+ h - jk 

(2) a = -h - jk 

- h + jk 

(3) a = - h + jk 

- h - jk 

(4) a = • + h - jk 

+ h + jk 


b = u — s — jq 

u — s + jq 

b = u — s — jq 

u - s + jq 

b = u + s — jq 

u + s + jq 

b = u + s — jq 

u 4- s + ia 


(32) 


or eight pairs of corresponding values of a and b. 

5. Substituting the values (1) of (32) into one group of 
terms of equations (9) and (10), 

i = Ae-al~bi 
and 


bL -r 


-al-bt 


a 


(33) 


gives 


4- A '£-(*- 


and substituting for the exponential functions with imaginary 
exponents their trigonometric expressions by the equation 


cos x 


426 

gives 


TRANSIENT PHENOMENA 


il = s ' (u s)t\A1 [cos (qt — kl) + j sin (qt - 
+ A / [cos (qt — kl) — j sin (qt - 
=€-w-(M-a)<| (A^ + AI) cos (qt — kl)+j (Al — A/) sin (g£ — , 

hence, Ax and A/ must be conjugate complex imaginary quan- 
tities, and writing 

C, = A, + A/ 
and 

C/ = j (^ - A/) 
gives 

Substituting in the same manner in the equation of e, in (33), 
gives 

(u — s — jq) L — r 


(34) 


jk 


- j A; 


-M-^-rtt ([^ -s-jq)L- r] (h - jk) +i(at_kl) 

. ( h2 + k2 

[(u-8 + jq) L-r](h + jk) ,,„_ 


l 


hence expanding, and substituting the trigonometric expressions 

f(u — s) L — r] h — 
h? + k* 

, — s}L — r\h — qkL 


_ .[(M-S) L-r] k + qhL\ 
tf + k* I 


.[(u-s)L-r] k + qkL 


A /[cos (qt - kl) - j sin (qt - kl)]l , (36) 


and introducing the denotations 

= qkL + h [r - (u - s) L] _ qk + h (m + s) 

h2 + k2 h2 -f k2 

, = k [r - (u - s) L] - ghL _ k (m + s) - qh 
°1= h2 + tf h2 -f k2 


(37) 


GENERAL EQUATIONS 


427 


and substituting (37) in (36), gives 
ei = £-hl-(u-s)t{(-cl + jc/) AJcos ($ -/c/) + /sin 

+ ( - ct - jc/) A/ [cos ($ - &) - j sin ($ - kl}} } 
= €-«-(u-.)*{[_ Cj (4i + AI/)+ jCi'(A1 - A/)] cos (qt - kl) 
+ [- /c^- A/)- c/(At+ A/)]sin(^ - kl)\. (38) 
Substituting the denotations (34) into (38) gives 
e, = €-"-<«-•>'{ (c/C/ - C/7J cos (qt- kl) 

-(c/Y + c/CJ sin (0 -A;/)}. (39) 

The second group of values of a and b in equation (32) differs 
from the first one merely by the reversal of the signs of h and k, 
and the values i2 and e2 thus are derived from those of ^ and el 
by reversing the signs of h and k. 

Leaving then the same denotations ct and c/ would reverse 
the sign of e2, or, by reversing the sign of the integration con- 
stants C, that is, substituting 

C2=- (A, + A,') 
and (40) 


, } 

~~ ^-2 /> ^ 


the sign of ^2 reverses; that is, 
ta - - £+w-(w-s)/{C2 cos ( 
and 


kl) +C2' sin 


(41) 


cos 


(42) 

The third group in equation (32) differs from the first one by 
the reversal of the signs of h and s, and its values i3 and e3 there- 
fore are derived from il and el by reversing the signs of h and s. 

Introducing the denotations 

qk - h (m - 


c, = 


h2 + k2 

k (m — s) + qh 
h2 + tf 


L, 


and 


(43) 


Ct' - j (At- 


(44) 


TRANSIENT PHENOMENA 


i3 = fi +«-<«+«>< {C3 cos (qt - kl) + C3' sin (qt - kl) } (45) 


428 
gives 

and 

e3 = e +*-<«+«>< { (ca'Cs' - c2C3) cos (qt - kl) 

~ (c2C8' + c2'C3) sin (qt - kl)}. (46) 

The fourth group in (32) follows from the third group by the 
reversal of the signs h and k} and retaining the denotations c2 
and c2'; but introducing the integration constants, 

C4 = - (A. + A/) } 
and L (47) 

gives 

i4 = - e~hl-(u+s}i {C4 cos (qt + kl) + Ct' sin (qt + kl)} (48) 
and 


e = 


cos 


- (c2C4' + ca7C4) sin (^ + 


(49) 


6. This then gives as the general expression of the equations of 
the electric circuit: 


cos 


sn _ 


sn 


and 


cos (qt - kl) + C/ sin (qt - kl) } (i3) 
cos (^ + kl) + C/ sin (qt + M) }] (g 


i(ciGi - CjC/i) cos (gf - /c/) 

- (c/Ct + c^/) sin (qt - kl) } (e,) 

- (Ci'C2 + c^/) sin ($ + A;Z) } (e2) 

[c2'C3 — c2C3) cos (qt — kl) 

'c2C4' — c2C4) cos (qt + A;Z) 


(50) 


(51) 


GENERAL EQUATIONS 


429 


where Cv C/, Cv C{, Cv C,', C4> C/ and two of the four values 
s, g, hj k are integration constants, depending on the terminal 
conditions, and 

qk + h (m + s) _ 


, _ k(m + s) - qh 
i : # + # 

<?& — /& (m — s) 


(m - s) + g/t 


T 


and 


-.(£- 1) 


and /i, A; and s, </ are related by the equations 

h = VLC \/i j/2,2 + s2 - (f - m2}, 

A; = 
and 


m2}, 


R* = V(s2 + q2 - m2)2 + 4 <fm2\ 
hence, h2 + # = LC/2^, 

or 


s = 


and 
hence, 


- 4 


(52) 


(53) 


(54) 


(55) 


(56) 


(57) 


430 
Writing 


TRANSIENT PHENOMENA 


D (qt ± kl) = C cos (qt ± kl) + C' sin (qt ± kl) (58) 


and 


H (qt ± kl) = (c'C" - cC) cos (qt ± kl) 

- (c'O + cC') sin (qt ± kl), 


equations (50) and (51) can be written thus: 


(qt-kl) - 


and 


(59) 


(60) 


(ef) 


(61)