CHAPTER XXIV. 

SYMBOLIC REPRESENTATION OF GENERAL 
ALTERNATING WAVES. 

253. The vector representation, 

A = a1 +y<zu = a (cos a -\-j sin d) 
of the alternating wave, 

A — a0 cos (<£ — a) 

applies to the sine wave only. 

The general alternating wave, however, contains an in- 
finite series of terms, of odd frequencies, 

A = Al cos (<£ — #1) 4- Az cos (3 <£ — #3) + A& cos (5 <£ — #5) -f 

thus cannot be directly represented by one complex vector 
quantity. 

The replacement of the general wave by its equivalent 
sine wave, as before discussed, that is a sine wave of equal 
effective intensity and equal power, while sufficiently accu- 
rate in many cases, completely fails in other cases, espe- 
cially in circuits containing capacity, or in circuits containing 
periodically (and in synchronism with the wave) varying 
resistance or reactance (as alternating arcs, reaction ma- 
chines, synchronous induction motors, oversaturated mag- 
netic circuits, etc.). 

Since, however, the individual harmonics of the general 
alternating wave are independent of each other, that is, all 
products of different harmonics vanish, each term can be 
represented by a complex symbol, and the equations of the 
general wave then are the resultants of those of the indi- 
vidual harmonics. 


REPRESENTATION OF ALTERNATING WAVES. 411 

This can be represented symbolically by combining in 
one formula symbolic representations of different frequen- 
cies, thus, 

00 

A = £.»-i (a* +jn */) 

i 
where, 

and the index of the/M merely denotes that the/s of differ- 

entindices n, while algebraically identical, physically rep- 

resent different frequencies, and thus cannot be combined. 

The general wave of E.M.F. is thus represented by, 


the general wave of current by, 


if, 


is the impedance of the fundamental harmonic, where 

xm is that part of the reactance which is proportional to 

the frequency (inductance, etc.). 

x0 is that part of the reactance which is independent of 

the frequency (mutual induction, synchronous motion, etc.). 
xc is that part of the reactance which is inversely pro- 

portional to the frequency (capacity, etc.). 

The impedance for the nth harmonic is, 


r —Jnn xm 


This term can be considered as the general symbolic 
expression of the impedance of a circuit of general wave 
shape. 


412 ALTERNATING-CURRENT PHENOMENA. 

Ohm's law, in symbolic expression, assumes for the 
general alternating wave the form, 

/-Jo, 


E = IZ or, 


Z = £or, 


Z = r -n 


The symbols of multiplication and division of the terms 
E, /, ^f, thus represent not algebraic operation, but multi- 
plication and division of corresponding terms of E, T, Z, 
that is, terms of the same index «, or, in algebraic multipli- 
cation and division of the series E, /, all compound terms, 
that is terms containing two different w's, vanish. 

254. The effective value of the general wave : 
a = AI cos (<£ — «,) + As cos (3 <£ — a8) +^5 cos (5 <f> — #6) +. . 

is the square root of the sum of mean squares of individual 
harmonics, 

A= V i { A? + A82 + A? + . . . | 

Since, as discussed above, the compound terms, of two 
different indices «, vanish, the absolute value of the general 
alternating wave, 


REPRESENTATION OF ALTERNATING WAVES. 413 

is thus, 

A 


which offers an easy means of reduction from symbolic to 
absolute values. 

Thus, the absolute value of the E.M.F. 


s, 


the absolute value of the current, 


is, 


255. The double frequency power (torque, etc.) equa- 
tion of the general alternating wave has the same symbolic 
expression as with the sine wave : 


= Pl +JPJ 


1 

where, 


41-4 ALTERNATING-CURRENT PHENOMENA. 

The jn enters under the summation sign of the " watt- 
less power " 1$, so that the wattless powers of the different 
harmonics cannot be algebraically added. 

i Thus, 

The total " true power" of a general alternating current 
circuit is the algebraic sum of the powers of the individual 
harmonics. 

The total "wattless power" of a general alternating 
current circuit is not the algebraic, but the absolute sum of 
the wattless powers of the individual harmonics. 

Thus, regarding the wattless power as a whole, in the 
general alternating circuit no distinction can be made be- 
tween lead and lag, since some harmonics may be leading, 
others lagging. 

The apparent power, or total volt-amperes, of the circuit 
is, 


The power factor of the circuit is, 


The term "inductance factor," however, has no mean- 
ing any more, since the wattless powers of the different 
harmonics are not directly comparable. 

The quantity, 


,...._ ... wattless power 

has no physical significance, and is not = 

total apparent power 


REPRESENTATION OF ALTERNATING WAVES. 4] > 

The term, /#. 

El 

= 2/n~17 


where, 


consists of a series of inductance factors qn of the individual 
harmonics. 

As a rule, if <f = 2^-1 ^n2, 


for the general alternating wave, that is q differs from 

fo=vr^72 

The complex quantity, 


Q El ~ El 


1 

takes in the circuit of the general alternating wave the 
same position as power factor and inductance factor with 
the sine wave. 

p 

17= -~ may be called the " circuit factor " 

It consists of a real term /, the power factor, and a 
series of imaginary terms jn qn, the inductance factors of 
the individual harmonics. 


416 ALTERNATING-CURRENT PHENOMENA. 

The absolute value of the circuit factor : 


as a rule, is < 1. 

256. Some applications of this symbolism will explain 
its mechanism and its usefulness more fully. 

\st Instance : Let the E.M.F., 


be impressed upon a circuit of the impedance, 

7 • ( *CN 

Z = *•—./„ \nxm -- 


that is, containing resistance r, inductive reactance xm and 
capacity reactance xc in series. 

Let 

e? = 720 ef = 540 

V = 283 4" = - 283 

e£ = - 104 *6" = 138 

or, 

^ = 900 tan e^ = .75 

*, = 400 tan o)3 = - 1 

^5 = 173 tan w5 = - 1.33 

It is thus in symbolic expression, 

Zj = 10 + 80/; *! = 80.6 

Z3 = 10 zz = 10 

ZB = 10 - 32/; 25 = 33.5 

and, E.M.F., 

^ = (720 + 540/0 + (283 - 283y;) + (- 104 + 138/5) 

or absolute, 

E = 1000 


REPRESENTATION OF ALTERNATING WAVES. 417 


and current, 

_ £ _ 720 + 540/t 283 - 283/8 - 104 + 138./; 
Z~~ 10 + 80/i " 10 10-32y5 


= (7.76 - 8.04/i) + (28.3 - 28.3/8) + (- 4.86 - 1.73 A) 

or, absolute, 

7=41.85 

of which is of fundamental frequency, ll = 11.15 
" " " " triple " I3 = 40 

« « « quintuple " I5 = 5.17 

The total apparent power of the circuit is, 

Q = £7=41,850 
The true power of the circuit is : 

/» = [7i 7]1 = 1240 + 16,000 + 270 

= 17,510 
the wattless power, 

j PJ =/ [7i 7]J = 10,000^ - 850/6 
thus, the total power, 

P= 17,510 + 10,000/; - 850y5 

That is, the wattless power of the first harmonic is 
leading, that of the third harmonic zero, and that of the fifth 
harmonic lagging. 

17,510 = I2 r, as obvious. 
The circuit factor is, 


• Q El 

= .418 + .239 j\ - .0203/5 

or, absolute, 


u = V.4182+ .2392 + .02032 
= .482 


The power factor is, 

p = .418 


418 ALTERNATING-CURRENT PHENOMENA. 

The inductance factor of the first harmonic is : ql = .239, 
that of the third harmonic ft = 0, and of the fifth harmonic 
ft = - -0203. 

Considering the waves as replaced by their equivalent 
sine waves, from the sine wave formula, 

f + qf = 1 
the inductance factor would be, 

ft = -914 
and the phase angle, 

tan a, = ^= '-^=2.8 « = 65.4° 

p .41o 

giving apparently a very great phase displacement, while in 
reality, of the 41.85 amperes total current, 40 amperes (the 
current of the third harmonic) are in phase with their 
E.M.F. 

We thus have here a case of a circuit with complex har- 
monic waves which cannot be represented by their equiva- 
lent sine waves. The relative magnitudes of the different 
harmonics in the wave of current and of E.M.F. differ 
essentially, and the circuit has simultaneously a very low 
power factor and a very low inductance factor; that is, a low 
power factor exists without corresponding phase displace- 
ment, the circuit factor being less than one-half. 

Such circuits, for instance, are those including alternat- 
ing arcs, reaction machines, synchronous induction motors, 
reactances with over-saturated magnetic circuit, high poten- 
tial lines in which the maximum difference of potential ex- 
ceeds the voltage at which brush discharges begin, polariza- 
tion cells, and in general electrolytic conductors above the 
dissociation voltage of the electrolyte, etc. Such circuits 
cannot correctly, and in many cases not even approxi- 
mately, be treated by the theory of the equivalent sine 
waves, but require the symbolism of the complex harmonic 
wave. 


REPRESENTATION OF ALTERNATING WAVES. 419 

257. 2d instance: A condenser of capacity C0 = 20 
m.f. is connected into the circuit of a 60-cycle alternator 
giving a wave of the form, 

e = E (cos <£ - .10 cos 3 <£ - .08 cos 5 <f> + .06 cos 7 <£) 
or, in symbolic expression, 

£ = e(!1- .10, - .085 + .067) 
The synchronous impedance of the alternator is, 
ZQ = r0 —jnnx0 = .3 — 5 njn 

What is the apparent capacity C of the condenser (as cal- 
culated from its terminal volts and amperes) when connected 
directly with the alternator terminals, and when connected 
thereto through various amounts of resistance and induc- 
tive reactance. 

The capacity reactance of the condenser is, 
106 


or, in symbolic expression, 

Let 

Z^ =.r — jn nv = impedance inserted in series with the 
condenser. 

The total impedance of the circuit is then, 

n 
The current in the circuit is, 


(.3 + r) - j (x - 132) (.3 + r) -j3 (3 x - 29) 

^8 ^6 -j 

(.3 + r) -j, (5x- 1.4) (.3 + r) -j\(7x + 16.1)J 


420 ALTERNATING-CURRENT PHENOMENA. 

and the E.M.F. at the condenser terminals, 


; 
Jn V 

4.4 js 


(.3 + r) -A (x - 132) (.3 + r) - jz (3 * - 29) 

__ 2.iiy5 1.13;; -i 

(.3 + r) -j6 (5x- 1.4) ^ (.3 + r) -/7 (7 x + 16.1) J 
thus the apparent capacity reactance of the condenser is, 


and the apparent capacity, 

106 


^.) ^r = 0 : Resistance r in series with the condenser. 
Reduced to absolute values, it is, 

1 .01 .0064 .0036 


17424 19.4 


(.8+r)a+ 17424 (.3 +r)2 + 841 (.3 + r)2 + 1.96 (.3 -f r)2 +2 

(£.) r = 0 : Inductive reactance x in series with the 
condenser. Reduced to absolute values, it is, 

1 .01 .0064 __ .0036 

— 1.42 "*". 


1.4)2 .09+(7;r-f 16. 


— 132)2 . 

From —g are derived the values of apparent capacity, 


c= 


and plotted in Fig. 179 for values of r and x respectively 
varying from 0 to 22 ohms. 

As seen, with neither additional resistance nor reactance 
in series to the condenser, the apparent capacity with this 
generator wave is 84 m.f., or 4.2 times the true capacity, 


REPRESENTATION OF ALTERNATING WAVES. 421 

and gradually decreases with increasing series resistance, to 
C= 27.5 m.f. = 1.375 times the true capacity at r= 13.2 
ohms, or TV the true capacity reactance, with r = 132 ohms, 
or with an additional resistance equal to the capacity reac- 
tance, C = 20.5 m.f. or only 2.5% in excess of the true 
capacity C0, and at r = oo , C = 20,3 m.f. or 1.5% in excess 
of the true capacity. 

With reactances, but no additional resistance r in series, 
the apparent capacity C rises from 4.2 times the true 
capacity at x = 0, to a maximum of 5,03 times the true 
capacity, or C= 100.6 m.f. at x = .28, the condition of res- 
onance of the fifth harmonic, then decreases to a minimum 
of 27 m.f., or 35 % in excess of the true capacity, rises again 
to 60.2 m.f., or 3.01 times the true capacity at x = 9.67, 
the condition of resonance with the third harmonic, and 
finally decreases, reaching 20 m.f., or the true capacity at 
x = 132, or an inductive reactance equal to the capacity 
reactance, then increases again to 20.2 m.f. at x = oo . 

This rise and fall of the apparent capacity is within cer- 
tain limits independent of the magnitude of the higher 
harmonics of the generator wave of E.M.F., but merely de- 
pends upon their presence. That is, with such a reactance 
connected in series as to cause resonance with one of the 
higher harmonics, the increase of apparent capacity is ap- 
proximately the same, whatever the value of the harmonic, 
whether it equals 25% of the fundamental or less than 5%, 
provided the resistance in the circuit is negligible. The 
only effect of the amplitude of the higher harmonic is that 
when it is small, a lower resistance makes itself felt by re- 
ducing the increase of apparent capacity below the value it 
would have were the amplitude greater. 

It thus follows that the true capacity of a condenser 
cannot even approximately be determined by measuring 
volts and amperes if there are any higher harmonics present 
in the generator wave, except by inserting a very large re- 
sistance or reactance in series to the condenser. 


422 


ALTERNATING-CURRENT PHENOMENA. 


258. §d instance : An alternating current generator 

of the wave, 

E. = 2000 [lt + .12, - .23B - .13,] 

and of synchronous impedance, 

Z0 = .3-5*/; 
feeds over a line of impedance, 


C4PJ 

CITV 

Co = 

= 20 

mf i 

CM 

CL'IT 

OF 

r,E\ 

HAT 

R 

1 

8 

= EI O-J--I.L— .ya-t-uc/ OF 

Zo^S-S), n WITH RESIS 

fASC 

DANCE 

k r(I) 

! 

c 

R RE 

ACT 

NCE 

*^ 

I) 1 

SE 

!ES 

C: 

£ 

100 

/\ 

0 

90 

J 

i 

^ft 

I 

k 

5 

rn 

I 

\ 

\ 

i 

H 

^ 

/ 

\ 

.w 

\ 

\ 

/ 

X 

10 

REE 

X 

STAC 

ii 

^=^~ 

CE r 

= 

;=" 

^ 

^ 

= 

REA( 

— - 
TAN!1 

X 

•- 

^S 

•^ 

* , 

; 

— 

= 

^= 

_» 

=3<F 

10 

I ; 

, 

i !'o ! 

1 

2 1 

1 

1 

1 

1 

- 1 

-t 1 

r, 2, 

1 

0 

a synchronous motor of the wave, 

EI = 2250 [(cos oj +/i sin «) + .24 (cos 3 w -(-y's sin 3 o>)] 
and of synchronous impedance, 

Z2 = .3 - C «/; 

The total impedance of the system is then, 
Z = ZQ + Zl + Z2 
= 2.6-15«/n 


REPRESENTATION OF ALTERNATING WAVES. 423 

thus the current, 


_ 2000 - 2250 cos o> - 2250/\ sin o> 240 - 540 cos 3a> - 540/; sin 3a> 
2.6 - 15/i 2.6 - 45y8 

460 260 

~~ 2.6 - 75 j\ 2.6 - 105 jj 

= « 

where, 

aj1 = 22.5 - 25.2 cos co + 146 sin a> 

ag1 = .306 - .69 cos 3 to + 11.9 sin 3 

a,1 = - .213 

«7i = - .061 

V1 = 130 - 146 cos w - 25.2 sin a> 

^8« = 5.3 - 11.9 cos 3 o> - .69 sin 3 o> 

a* = - 6.12 

a7u = - 2.48 

or, absolute, 

1st harmonic, 

3d harmonic, 

5th harmonic, 

a6 = 6.12 
7th harmonic, 

«7 = 2.48 


/= V 
while the total current of higher harmonics is, 


424 ALTERNATING-CURRENT PHENOMENA. 

The true input of the synchronous motor is, 


= ( 2250 a£ cos o> + 2250 a? sin o> ) + ( 540 a? cos 3o> + 540 asn sin 3o>) 

= /V + /'s1 
^ = 2250 (a? cos <o + af sin o>) 


. 780. Synchronous Motor, 


REPRESENTATION OF ALTERNATING WAVES. 425 

is the power of the fundamental wave, 

P£ = 540 (a,,1 cos 3 w + as11 sin 3 o>) 

the power of the third harmonic. 

The 5th and 7th harmonics do not give any power, 
since they are not contained in the synchronous motor 
wave. Substituting now different numerical values for u> 
the phase angle between generator E.M.F. and synchronous 
motor counter E.M.F., corresponding values of the currents 
/ 70, and the powers P\ P*, /Y are derived. These are 
plotted in Fig. 180 with the total current /as abcissae. To 
each value of the total current / correspond two values of 
the total power P\ a positive value plotted as Curve I. — 
synchronous motor — and a negative value plotted as 
Curve II. — alternating current generator — . Curve III. 
gives the total current of higher frequency I0, Curve IV., 
the difference between the total current and the current of 
fundamental frequency, / — alt in percentage of the total 
current /, and V the power of the third harmonic, Pj, in 
percentage of the total power P1. 

Curves III., IV. and V. correspond to the positive or 
synchronous motor part of the power curve P\ As seen, 
the increase of current due to the higher harmonics is 
small, and entirely disappears at about 180 amperes. The 
power of the third harmonic is positive, that is, adds to the 
work of the synchronous motor up to about 140 amperes, 
or near the maximum output of the motor, and then becomes 
negative. 

It follows herefrom that higher harmonics in the E.M.F. 
waves of generators and synchronous motors do not repre- 
sent a mere waste of current, but may contribute more or 
less to the output of the motor. Thus at 75 amperes total 
current, the percentage of increase of power due to the 
higher harmonic is equal to the increase of current, or in 
other words the higher harmonics of current do work with 
the same efficiency as the fundamental wave. 


426 ALTERNATING-CURRENT PHENOMENA. 

259. kth Instance: In a small three-phase induction 
motor, the constants per delta circuit are 

Primary admittance Y= .002 + .03/ 

Self-inductive impedance ZQ = Zl = .6 — 2.4/ 

and a sine wave of E.M.F. e0 = 110 volts is impressed upon 
the motor. 

The power output P, current input 7S, and power factor 
/, as function of the slip s are given in the first columns of 
the following table, calculated in the manner as described in 
the chapter on Induction Motors. 

To improve the power factor of the motor and bring it 
to unity at an output of 500 watts, a condenser capacity is 
required giving 4.28 amperes leading current at 110 volts, 
that is, neglecting the energy loss in the condenser, capacity 
susceptance 


In this case, let Is = current input into the motor per 
delta circuit at slip s, as given in the following table. 

The total current supplied by the circuit with a sine 
wave of impressed E.M.F., is 

/i = ls - 4.28/ 

energy current 
and heref rom the power factor = - ; — — , given in 

total current 
the second columns of the table. 

If the impressed E.M.F. is not a sine wave but a wave 
of the shape 

E, = e, (lx + .12. - .235 - .134,) 

to give the same output, the fundamental wave must be the 
same : e0 = 110 volts, when assuming the higher harmonics 
in the motor as wattless, that is 

£0 = 110, + 13.2, - 25.3B - 14.7, 

= *o + £<? 
where £0l = 13.2, - 25.3B - 14.7T 

= component of impressed E.M.F. of higher frequency- 


REPRESENTATION^ Of ALTERNATING WAVES. 427 

The effective value is : 

EQ = 114.5 volts. 

The condenser admittance for the general alternating 
wave is 

Yc= -.039«/; 

Since the frequency of rotation of the motor is very 
small compared with the frequency of the higher harmonics, 
as total impedance of the motor for these higher harmonics 
can be assumed the stationary impedance, and by neglecting 
the resistance it is 

Z1 = - njn (XQ + XJ 

= - 4.8 njn 

The exciting admittance of the motor, for these higher 
harmonics, is, by neglecting the conductance, 


n 
and the higher harmonics of counter E.M.F. 


Thus we have, 


Current input in the condenser, 

fc = E, Yc 

= - 4.28/i - 1.54/3 + 4.93/5 + 4.02/7 

High frequency component of motor impedance current, 

|£ = .92/3 - 1.06y5 - .44/7 
High frequency component of motor exciting current, 


= .07/3 - -08/5 - . 


428 


AL TERN A TING-CURRENT PHEA'OAIENA. 


thus, total high frequency component of motor current, 

/o1 = |f + & y1 

= .99y3 - 1.14,; - .47/7 
and total current, 

without condenser, 

4 = 4 + 41 

= Is + .99/3 - 1.14,; - .47/7 

with condenser, 


= 4 - 4.28,i - . 
and herefrom the power factor. 


3.79,; + 3.55/7 


T PER PHASE 


In the following table and in Fig. 181 are given the 
values of current and power factor : — 

I. With sine wave of E.M.F., of 110 volts, and no condenser. 

II. With sine wave of E.M.F , of 1 10 volts, and with condenser. 

III. With distorted wave of E.M.F., of 114.6 volts, and no condenser. 

IV. With distorted wave of E.M.F., of 114.5 volts, and with condenser. 


REPRESENTATION OF ALTERNATING WAVES. 429 


f 
0 
.01 
.02 
.035 
.05 
.07 
.10 
.13 
.15 

P 

0 
160 
320 
500 
660 
810 
885 
900 
890 

I, 
.24+ 3.10/ 
1.73+ 3.16/ 
3.32+ 3.47> 
5.16+ 4.28/ 
6.95+ 5.4/ 
8.77+ 7.3; 
10.1 + 9.85/ 
10.45 + 11.45/ 
10.75 + 12.9/ 

It 
3.1 
3.6 
4.8 
6.7 
8.8 
11.4 
14.1 
15.5 
16.8 

7.8 
48 
69 
77 
79 
77 
71.5 
67.5 
64 

f 

1.2 
2.1 
3.4 
5.2 
7.0 
9.3 
11.5 
12.7 
13.8 

— •> 
P 
20 
84 
97.2 
100 
98.7 
94.5 
87 
82 
78 

i — 

3.5 
3.9 
5.1 
6.9 
8.9 
11.5 
14.2 
15.6 
16.9 

1 — \ 

6.6 
43 
64 
72.5 
76 
73.5 
68 
64.5 
61 

/ 
I 

5.2 
5.5 
6.1 
7.2 
8.6 
10.6 
12.6 
13.7 
14.7 

— i 

4. 
81 
64 

(18 
7T 
80 
7T 
73: 
7Q/ 

The curves II. and IV. with condenser are plotted in 
dotted lines in Fig. 181. As seen, even with such a dis- 
torted wave the current input and power factor of the motor 
are not much changed if no condenser is used. When using 
a condenser in shunt to the motor, however, with such a 
wave of impressed E.M.F. the increase of the total current, 
due to higher frequency currents in the condenser, is greater 
than the decrease, due to the compensation of lagging cur- 
rents, and the power factor is actually lowered by the con- 
denser, over the total range of load up to overloads, and 
especially at light loads. 

Where a compensator or transformer is used for feeding- 
the condenser, due to the internal self-induction of the com- 
pensator, the higher harmonics of current are still more 
accentuated, that is the power factor still more lowered. 

In the preceding the energy loss in the condenser and 
compensator and that due to the higher harmonics of cur- 
rent in the motor has been neglected. The effect of this 
energy loss is a slight decrease of efficiency and correspond- 
ing increase of power factor. The power produced by the 
higher harmonics has also been neglected ; it may be posi- 
tive or negative, according to the index of the harmonic, 
and the winding of the motor primary. Thus for instance, 
the effect of the triple harmonic is negative in the quarter- 
phase motor, zero in the three-phase motor, etc., altogether,, 
however, the effect of these harmonics is very small. 


430 ALTERNATING-CURRENT PHENOMENA.