CHAPTER XXVII 

SYMBOLIC REPRESENTATION OF GENERAL 
ALTERNATING WAVES 

259. The vector representation, 

A — a'^ -{- ja^'^ = a (cos d -\- j sin 6) 
of the alternating wave, 

A = tto cos {(f) — 6) 

apphes to the sine wave only. 

The general alternating wave, however, contains an infinite 
series of terms, of odd frequencies, 

A = Aicos( 0- ^i) + ^3 cos (3 (^ - 63) + As cos (5 <^ - ^5) + 
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 accurate in many 
cases, completely fails in other cases, especially in circuits con- 
taining capacity, or in circuits containing periodically (and in 
synchronism with the wave) varying resistance or reactance (as 
alternating arcs, reaction machines, synchronous induction 
motors, oversaturated magnetic circuits, etc.). 

Since, however, the individual harmonics of the general alter- 
nating 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 individual harmonics. 

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

A = 22n-i(a„i+i„a„ii),^ 
1 

1 The index 2n — 1 in the S sign denotes that only the odd values of 
n are considered. If the wave contained even harmonics, the even vahies 
of n would also be considered, and the index in the 2 sign would be n. 

379 


380 ALTERNATING-CURRENT PHENOMENA 
where 

jn = V - 1, 

and the index of the j„ merely denotes that the j's of differ- 
ent indices, n, while algebraically identical, physically represent 
different frequencies, and thus cannot be combined. 
The general wave of e.m.f. is thus represented by 

E = 2:2n-i(e„i4-j„e„ii), 
1 

the general wave of current by 

1 
If 

Zi = r -^ j (x„, -{- xo -\- Xc) 

is the impedance of the fundamental harmonic, where 

Xm is that part of the reactance which is proportional to the 
frequency (inductance, etc.), ^ 

Xo is that part of the reactance which is independent of the 
frequency (mutual inductance, synchronous motion, etc.). 

Xc is that part of the reactance which is inversely propor- 
tional to the frequency (capacity, etc.). 

The impedance for the nth harmonic is 


+ jn (nxr^ + a^o + -^ ) 


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

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

7 = ^or, S2n-i(t„i +j„^„ii) = Ssn-i 


E 


r -{- Jn [^nxm + Xo + — j 
= IZ or, 22n-i (e„i -\-jner}^) = S2n-i l^r + jn {rixm + 3^0 + ^) ] 

Kj'n "T" Jnt'n ) , 
1 \ n I ?„' -\-Jnln^ 

The symbols of multiplication and division of the terms, E, I, Z, 


GENERAL ALTERNATING WAVES 381 

thus represent, not algebraic operation, but multiplication and 
division of corresponding terms of E, I, Z, that is, terms of the 
same index, ??., or, in algebraic multiplication and division of the 
series, E, I, all compound terms, that is, terms containing two 
different n's, vanish. 

260. The effective value of the general wave, 
a = A 1 cos (0 - ^i) + A3 cos (3 0 - ^3) + ^5 cos (5 0 - ^2) +• • 
is the square root of the sum of mean squares of individual har- 
monics, 

A = VmITTaT+^TTTTT}". 

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


A Vo 1 Q» ~T" Jnttn 

1 On ^ JnOn 


is thus. 


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

Thus, the absolute value of the e.m.f., 


E = S2.-i(e„i+j„e„u)^ 
1 
is 


E = ^/22n-i(e/ + e„u2)^ 


the absolute value of the current, 

1 
is 


/ = ^/S2.-1(^/ + ZV^'). 


261. The double frequency power (torque, etc.) equation of 
the general alternating wave has the same symbolic expression 
as with the sine wave, 


382 ALTERNATING-CURRENT PHENOMENA 

P = [EI] 
= Pi + jP^ 

= [Ely -{- j[Ei]j 

1 1 

where 

Pi = [Ely = i:2n-i(e„ii„i + e^iHV^), 
1 

1 J 

The jn enters under the summation sign of the reactive or 
"wattless power," P', so that the wattless powers of the different 
harmonics cannot be algebraically added. 

Thus, 

The total "true power'' of a general alternating-cihrrent circuit 
is the algebraic sum of the poioers of the individual harmonics. 

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

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

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


P, = EI = JX2n-i{ej" + e„i022n-i(4i' + *V^'). 
\ 1 1 

The power-factor of the circuit is, 

P^ 1 

P =w = 


\ 1 1 

The term "inductance factor," however, has no meaning any 
more, since the reactive powers of the different harmonics are not 
directly comparable. 

The quantity 

go = Vl — p^ 

, ,.,..„ , . reactive power 

has no physical sigmncance, and is not . , , : 

total apparent power 


GENERAL ALTERNATING WAVES 383 

The term 

pi " IP iiy 1 p 1I7 1 

M=f''-7 ^7 

1 J 

where 

^" = ^ EI ' 

consists of a series of inductance factors, q„, of the individual 
harmonics. 

As a rule, if g^ = S2n-ig(„2^ 

1 

p' + 9' < 1, 

for the general alternating wave, that is, q differs from 


qo = Vl - p'. 
The complex quantity, 

V = S. = ^-^ JiiL±MIl 

^ Pa EI EI 

_ J 1 

Ji2n-i (en'' + e„ii') 22 n- 1 (^•„l' + ^•„ll') 
\ 1 1 

= P -\- 22n-lj„g„^ 

1 

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

p 

y = 77 may be called the "circuit-factor." 

■la 

It consists of a real term, p, the power-factor, and a series of 
imaginary terms, in^n, the inductance factors of the individual 
harmonics. 

The absolute value of the circuit-factor. 


as a rule, is < 1. 


384 ALTERNATING-CURRENT PHENOMENA - 

262. Some applications of this symbolism will explain its 
mechanism and its usefulness more fully. 
First Example. — Let the e.m.f., 

5 

1 
be impressed upon a circuit of the impedance, 

Z = r^jr. [nxr. - ^) = 10+ i„(lOn- ^) 

that is, containing resistance, r, inductive reactance Xm and con- 

densive reactance Xc in series. 

Let 

eii = 720 ei^i = - 540 

egi = 283 63^^ = 283 

e,' = - 104 65^' = - 138 

or, 

ei = 900 tan Oi = 0.75 

63 = 400 tan 03 = -1.0 

65 = 173 tan 05 = - 1-33 

It is thus in symbolic expression, 

Zi = 10 - 80 ji zi = 80.6 

Zz =10 23 = 10.0 

Z, = 10 + 32 is 25 = 33.5, 
and e.m.f., 

E = (720 - 540 JO + (283 + 283 js) + ( - 104 - 138 J5), 

or, absolute, 

E = 1000, 
and current, 

J _E _ 720 - 540 ii 283 + 283 jz - 104 - 138 js 
l~ Z~ 10 ~ 80 ii "*" 10 "^ 10 + 32 J5 

= (7.76 + 8.04ii) + (28.3 + 28.3i3) + ( - 4.86 + 1.73 js) 

or, absolute, 

7 = 41.85, 

of which is of fundamental frequency, 

7i = 11.15 
of triple frequency, 

/3 = 40 


GENERAL ALTERNATING WAVES 385 

of quintuple frequency, 

U = 5.17. 

The total apparent power of the circuit is 

Pa = EI = 41,850. 

The true power of the circuit is, 

pi = [Ely = 1240 + 16,000 + 270, 
= 17,510, 

the reactive power, 

jP^ = j[EI]> = - 10,000 ji + 850 is; 

thus, the total power, 

P = 17,510 - 10,000 ji + 850 ja. 

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

17,510 = Pr, as obvious. 

The circuit-factor is, 

., P [EI] 
. Pa EI 

= 0.418 - 0.239 ji + 0.0203 i5, 
or, absolute, 

V = VO.4182 + 0.2392 + 0.02032. 

= 0.482. 

The power-factor is 

p = 0.418. 

The inductance factor of the first harmonic is qi = — 0.239, 
that of the third harmonic 53 = 0, and of the fifth harmonic 
55 = + 0.0203. 

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

p^ + qo^ = 1, 

the inductance factor would be, 

Qo = 0.914, 
and the phase angle, 

tan d = ^ = ^-^I^ = 2.8, 6 = 65.4°, 

p 0.418 ' 

25 


386 ALTERNATING-CURRENT PHENOMENA 

giving apparently a very great phase displacement, while in 
reality, of the 41.85 amp. total current, 40 amp. (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 harmonic 
waves which cannot be represented by their equivalent 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 displacement, the circuit-factor being less 
than one-half. 

Such circuits, for instance, are those including alternating 
arcs, reaction machines, synchronous induction motors, react- 
ances with over-saturated magnetic circuit, high potential lines 
in which the maximum difference of potential exceeds the corona 
voltage, polarization cells and in general electrolytic conductors 
above the dissociation voltage of the electrolyte, etc. Such cir- 
cuits cannot correctly, and in many cases not even approximately, 
be treated by the theory of the equivalent sine waves, but re- 
quire the symbolism of the complex harmonic wave. 

263. Second Example. — A condenser of capacity, Co = 20 mf. is 
connected into the circuit of a 60-cycle alternator giving a wave 
of the form, 

e = E{cos 4> - 0.10 cos 3 0 - 0.08 cos 5 0 + 0.06 cos 7 4>), 

or, in symbolic expression, 

E = e(li - O.IO3 - O.O85 + O.O67). 

The synchronous impedance of the alternator is 

Zo = n -\- jntixo = 0.3 + 5 njn. 

What is the apparent capacity, C, of the condenser (as calcu- 
lated from its terminal volts and amperes) when connected 
directly with the alternator terminals, and when connected 
thereto through various amounts of resistance and inductive 
reactance? 

The condensive reactance of the condenser is 

10'' 
^c = o fn = 132 ohms, 

^TTjCo 

or, in symbolic expression, 

_ . a:^ _ _ 132 . 


GENERAL ALTERNATING WAVES 387 

Let Zi = r -\- j„nx = impedance inserted in series with the 
condenser. 

The total impedance of the circuit is then 


Z 


132\ 


Xr , s . /r ■, 132\ 

= Zo + Zi-i„^ = (0.3 +r) +Jn([5 + a:]n---^) 


The current in the circuit is 

_ E _ V 1 QA 

~ Z~ ^ L(0.3 + r) +j (x - 127) (0.3 + r) + js (3 x - 29) 

0.08 0.06 


0.0(3 -| 

"^ (0.3 + r) -77(7a; + 16.1)J' 


(0.3 + r) +i5(5a;-1.4) ^ (0.3 + r) - jr (7 a; + 16.1). 
and the e.m.f. at the condenser terminals, 


xj r 132 Ji 4.4i 


(0.3 + r) +ii(a;- 127) (0.3+r)+i3(3a;-29) 

2.11i5 , 1.13i7 -] 

"^ fO.3 4- r) + 77 (7 a; 4- 16.1) J' 


(0.3 + r) -\-j,{6x - 1.4) ^ (0.3 + r) +i7(7a;+ 16.1) 
thus the apparent condensive reactance of the condenser is 

E, 

Xi = 
and the apparent capacity, 


Xi = -^' 


c= "> 


2irfXi 

(a) X = 0: Resistance, r, in series with the condenser. Re- 
duced to absolute values it is 

1 0.01 0.0064 0.0036 


J_ _ (0.3+r) + 16129^(0.3+0 + 841 ^ (0.3+r) + 1.96 ^ (0.3+r) + 259 
xi^ ~ 16129 19.4 4.45 1.28 

(0.3+r)2+16129 + (0.3+r) + 841 + (0.3+r) + 1.96 "^ (0.3+r)2+259 

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

1 , 0-01 , 0.0064 


1 0.09+(x-127)''^0.09+(3x-29)''^0.09+(5j-1.4)'' 

xi2 ~ 16129 19.4 4.45 

0.09+(x -127)2 + 0.09+(3x -29)2 +0.09+ (5 X- 1.4)''"^ 

0.0036 


0.09+ (7 x+ 16.1)2 


1.28 


0.09+ (7 .1+16.1)= 


388 


AL TERN A TING-C URREN T PHENOMENA 


From — ^ are derived the values of apparent capacity, 


Xi 


C 


and plotted in Fig. 189 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 mf., or 4.2 times the true capacity, and gradually de- 
creases with increasing series resistance, to C = 27 m.f. = 1.35 
times the true capacity at r = 13.2 ohms, or one-tenth the true 
capacity reactance. With r = 132 ohms, or with an additional 
resistance equal to the condensive reactance, C = 20.2 mf. or only 


CAPACITY Co 

= 20mf 

IN CIRCUIT OF GENERATOR 

£"=£ (1-0.l-O.O8-U.UCi) OF IMPEDANCE 
Zo-=0.3-5njn WITH RESISTANCE T (I) 
OR REACTANCE X (II) IN SERIES 

< 
r 

C: 

100 

h 

90 

\ 

o 

80 

V 

s. 

< 
I 

70 

\ 

^ 

\, 

c 

60 

^ 

K 

/ 

/ 

\ 

50 

\ 

N 

/ 

\ 

40 

V 

II 

■*-v 

^ 

^ 

V 

=^>^ 

30 

■"* 

Co 


RESI 

STA> 

CE 7 

^ 

REAC 

TANC 

EX 

_ 

20 
10 

; 

II 

2 3 4. 

I 

i 

1 

9 10 1 

1 1 

2 1 

i 1 

4 15 1 

S 1 

7 1 

8 1 

3 2 

0 

0 

Fig. 180. 

one per cent, in excess of the true capacity, Co, and at r = co , 
C = 20 mf. or the true capacity. 

With reactance, x, but no additional resistance, r, in series, the 
apparent capacity, C, rises from 4.2 times the true capacity at 
a; = 0, to a maximum of 5.03 times the true capacity, or C ~ 
100. G mf. at a; = 0.28, the condition of resonance of the fifth 
harmonic, then decreases to a minimum of 27 mf., or 35 per cent, 
in excess of the true capacity, rises again to 60.2 mf., or 3.01 
times the true capacity at a: = 9.67, the condition of resonance 
with the third harmonic, and finally decreases, reaching 20 mf., 
or the true capacity at re = 132, or an inductive reactance equal 
to the condensive reactance. 


GENERAL ALTERNATING WAVES 389 

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 resistance or reactance 
in series to the condenser. 

264. Third Exam/pie. — An alternating-current generator of the 
wave, 

^0 = 2000 [li + 0.123 - 0.235 - O.ISt], 
and of synchronous impedance, 

Zo = 0.3 + 5 njn, 
feeds over a line of impedance, 

Zi = 2 + 4 njn, 
a synchronous motor of the wave, 

El = 2250 [(cos e - ji sin 6) + 0.24 (cos ^ 6 - jz sin 3 6)], 
and of synchronous impedance, 

Z2 = 0.3 + 6 njn. 
The total impedance of the system is then, 

Z = Zo + Zi + Z2 = 2.6 + 15 njn, 
thus the current, 

J _ Eg — El 

. ~ z 

_ 2000 - 2250 cos g+ 2250 ji sing 240 -540 cos 3^+540^3 sin 3 g 
2.6+15ji + 2.6-45J3 

_ " 460 _ 260 

2.6 +75 is ~ 2.6 + 105^7 
= (fli^ - iitti") + {a,^ - jsa,'') + (agi - j,a,'') + (a^i - jyay^; 

where 

ai^ = 22.5 - 25.2 cos d + 146 sin 6, 

agi = 0.306 - 0.69 cos 3 0 + 11.9 sin 3 d, 

at' = 0.213, 

a^i = - 0.061, 

ai^i = 130 - 146 cos 9 - 25.2 sin 6, 

11 _ 

i — 

as" = - 6.12, 
07" = - 2.48, 

or, absolute, 


390 ALTERNATING-CURRENT PHENOMENA 

first harmonic, 

third harmonic, 

^3 = vWM- as , 
fifth harmonic, 

as = 6.12, 
seventh harmonic, 

a-! = 2.48, 

I = Vai' + as^ + as^ + ay^; 
while the total current of higher harmonics is 
Iq = Vas^ + ae^ + ay^ 
The true input of the synchronous motor is 
Pi = [EJY 
= (2250 ail cos ^+2250 ai^^sin 6) + (540 a^^ cos 3 ^ + 540 as^sin 3 6 

= Pi' + Ps' 
Pi^ = 2250 (ail cos 6 + ai" sin d), 
is the power of the fundamental wave, 

Ps^ = 540 {as' cos 3 ^ + 03" sin 3 6), 

the power of the third harmonic. 

The fifth and seventh harmonics do not give any power, since 
they are not contained in the synchronous motor wave. Sub- 
stituting now different numerical values for 6, the phase angle 
between generator e.m.f. and synchronous motor counter e.m.f., 
corresponding values of the. currents, /, Jo, and the powers, P^, 
Pi^, Ps', are derived. These are plotted in Fig. 190 with the 
total current, I, as abscissas. 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, 7o, Curva 
IV the difference between the total current and the current of 
fundamental frequency, I — 7i, in percentage of the total current, 
/, and V the power of the third harmonic, Ps', in percentage of 
the total power, P^. 

Curves III, IV, and V correspond to the positive or synchron- 
ous motor part of the power curve, P^. As seen, the increase of 


GENERAL ALTERNATING WAVES 


391 


current due to the higher harmonics is small, and entirely dis- 
appears at about 180 amp. The power of the third harmonic 
is positive, that is, adds to the power of the synchronous motor 


% 

I 

PI 
240 

7 

r 

> 

N 

210 

6 

\ 

y 

y 

\ 

180 

5 

\ 

:a 

/ 

> 

\ 

150 

4 

\ 

/ 

/ 

120 

3 > 

V 

/ 

\ 

90 

2 

s 

::n 

lo 

\ 

20 

60 

1 

/ 

•^ 

H 

^ 

i^ 

10 

30 

"•^^ 

* 

0 1 

H 

^ 

■n~ 

■^ 

■^ 

7l 

0 

0 

X 

V^ 

A 

/ 

-30 

\ 

^v. 

"~T~ 

/ 

-60 

1 2 

\ 

0 6 

) f 

0 1 

AMPERES/ 
» 120 1' 

0 1 

)0 \i 

ffl 2 

M 220 2 

to 2 

30 i 

iO 

-90 

\ 

-120 

N 

\ 

-150 

\ 

v. 

-180 

\, 

/ 

-210 

\ 

/ 

-240 

v^ 

/ 

-270 

E, 

SYNCHRONOUS MOTOR 
= 2250 (COS. 0 + ji 3in.f?)H- 

^ 

\ 

/ 

-300 

0.24(cos. 35+ja sin. 3d 
OPERATED FROM GENERATOR 

^^0=2000 ( 1 + 0.12-0.23-0.13 

OVER TOTAL IMPEDANCE 

\ 

\ 

J 

-330 

^s 

Jl^ 

-^ 

/ 

-360 

Z„ = 5 

.6+1 

5U 

-.. 

Fig. 190. — Synchronous motor. 


up to about 140 amp. 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 represent a mere 
waste of current, but may contribute more or less to the output of 


392 ALTERNATING-CURRENT PHENOMENA 

the motor. Thus at 75 amp. 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. 

265. Fourth Example. — In a small three-phase induction motor, 
the constants per delta circuit are 

primary admittance Y = 0.002 — 0.03 j, 

self-inductive impedance Zo = Zi = 0.6 + 2.4 j,: 

and a sine wave of e.m.f., eo = 110 volts, is impressed upon the 
motor. 

The power output, P, current input, Is, and power-factor, p, as 
function of the slip, s, are given in the first columns of the follow- 
ing 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 amp. leading current at 110 volts, that is, neglecting 
the power loss in the condenser, capacity susceptance 

4.28 


110 


0.039. 


In this case, let Is = current input into the motor per delta cir- 
cuit 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 

/' = 7,-1-4.28 i, 

power current . . 

and hereirom the power-iactor = . , . -, given m the 

^ total current 

second columns of the table. 

If the impressed e.m.f. is not a sine wave but a wave of the 

shape, 

^0 = eo(li + 0.123 - 0.235 - 0.1347), 

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

Eo = llOi -1- 13.23 - 25.35 - 14.77 = eo + Eo\ 


GENERAL ALTERNATING WAVES 393 

where E^^ = 13.23 - 25.35 - 14.77 

= component of impressed e.m.f. of higher frequency. 

The effective value is 

E(, = 114.5 volts. 

The condenser admittance for the general alternating wave is 

Yc = 0.039 7ljn. 

Since the frequency of rotation of the motor is small com- 
pared 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 resist- 
ance we have 

Z^ = nj„(a;o + Xi) = 4.8 njn 

The exciting admittance of the motor, for these higher har- 
monics, is, by neglecting the conductance, 

71 = _ ^ = - ^M 
n n 

and the higher harmonics of counter e.m.f., 

E^ = — ^• 
. 2 

Thus we have, 

current input in the condenser, 

L - EoYc = + 4.28ii + 1.54 i3 - 4.93^5 - 4.02i7; 

high-frequency component of motor-impedance current, 

W ^ 

■^ = - 0.92 i3 + 1.06 i5 + 0.44 Jt; 

high-frequency component of motor-exciting current, 
^lyi = ---— = _ 0.07 i3 + O.OSis + 0.03 J7: 
thus, total high-frequency component of motor current, 
7oi =|l -I- E^Y^ = - 0.99 i3 + 1.14 J5 + 0.47 iv, 


394 ALTERNATING-CURRENT PHENOMENA 

and total current, without condenser, 

Io= Is + U = Is- 0.99 is + 1.14i5 + 0.47 jj, 

with condenser, 

I = h-i- lo' - Ic = Is + 4.28ii + 0.55 is - 3.79 J5 + 3.55 jV; 

and herefrom the power-factor. 

In the following table and in Fig. 191 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 110 volts, and with condenser. 

III. With distorted wave of e.m.f., of 114.5 volts, and no condenser. 

IV. With distorted wave of e.m.f., of 114.5 volts, and with condenser. 

TABLE 
I. II. III. IV. 


s 

P 

Is 

/, P 

l' 

P 

h P 

/ p 

0.0 

0 

0.24+ d.lOj 

3.1 7.8 

1.2 

20.0 

3.5 6.6 

5.2 4.4 

0.01 

160 

1.73+ 3.16J 

3.6 48.0 

2.1 

84.0 

3.9 43.0 

5.5 31.0 

0.02 

320 

3.32+ 3.47i 

4.8 69.0 

3.4 

97.2 

5.1 64.0 

6.1 54.0 

0.035 

500 

5.16+ 4.28i 

6.7 77.0 

5.2 

100.0 

6.9 72.5 

7.2 68.0 

0.05 

660 

6.95+ 5.4i 

8.8 79.0 

7.0 

98.7 

8.9 76.0 

8.6 77.0 

0.07 

810 

8.77+ 7.3i 

11.4 77.0 

9.3 

94.5 

11.5 73.5 

10.6 80.0 

0.10 

885 

10.1 + 9.85i 

14.1 71.5 

11.5 

87.0 

14.2 68.0 

12.6 77.0 

0.13 

900 

10. 45 + 11. 45i 

15.5 67.5 

12.7 

82.0 

15.6 64.5 

13 7 73.0 

0.15 

890 

10.75 + 12.9 i 

16.8 64.0 

13.8 

78.0 

16.9 61.0 

14.7 70.0 

The curves II and IV with condenser are plotted in dotted lines 
in Fig. 191. As seen, even with such a distorted 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 con- 
denser, is greater than the decrease, due to the compensation of 
lagging currents, and the power-factor is actually lowered by the 
condenser, over the total range of load up to overload, and espe- 
cially at light load. 

Where a compensator or transformer is used for feeding the 
condenser, due to the internal self-inductance of the compensa- 
tor, 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 compen- 
sator and that due to the higher harmonics of current in the motor 


GENERAL ALTERNATING WAVES 


395 


has been neglected. The effect of this energy loss is a slight 
decrease of efficiency and corresponding increase of power-factor. 
The power produced by the higher harmonics has also been 
neglected; it may be positive or negative, according to the index 


AMPERES 

thre'e-ph'ase induction motor 

y 

14 

1 110 V 

1 

5LT SINE 
1 

WAVE, NO CONDENSER 
1 1 1 1 

/) 

13 

II 
III 

10 VOLT SINE 

1 1 
-14.6' VOLrS D 

'/VAVE,CO^ 

1 

STORTED- 

DENSER 1 

IDEN 
SER 

3Efl_ 

/ 

// 

12 

IV 

14.6 

VOL 

rs Di 

STOF 

TED 

WAV! 

. cc 

NDEh 

0 

■/ 

U 

8PE 

ED 

_<^ 

/ 

/ 

10 

100 

,-■ 

--■ 


—^ 

b; 

^ 

!) 

90 

^^ 

■\\ 

^ 

f 

/ 

^\ 

8 

80 

/ 


^ 

^ 

->= 

y 

^ 

i::^ 

^: 

7 

70 

/ 

Vx- 

^ 

::C^ 

:^ 

■'^ 

■y 

^"^ 

X 

G 

CO 


/ 
r 

rv 

'/ 

A 

t 

^ 

■>^ 

^^ 

5 

50 

/ 

i — 

^ 

^ 

\ii. 

^ 

<\ 

/^ 

^ 

4 

40 

4\ 

^^' 

'^v 

3 

30 

r 

// 

,-''' 

.^■^ 

2 

20 

A 

^' 

'"^ 

OUTPUT PER PHASE, 
1 1 

WA- 

TS. 

1 

10 

(L 

1( 

0 

2 

K) 

3 

)0 

400 500 1 6C 

0 

7 

X) 

8C 

« 

9( 

K) 

0 

0 

Fig. 191. 


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.; alto- 
gether, however, the effect of these harmonics is usually small. 


SECTION VII 
POLYPHASE SYSTEMS