CHAPTER VII 


HIGHER HARMONICS IN INDUCTION MOTORS 

88. The usual theory and calculation of induction motors, 
.■is discussed in '* Theoretical Elements of Electrical Enginccr- 
ing" and in "Theory and Calculation of Alternating-current 
Phenomena," is based on the assumption of the sine wave. That 
U, it is assumed that the voltage impressed upon the motor 
per phase, and therefore the magnetic flux and the current, KM 
sine waves, and it is further assumed, that the distribution of 
the winding on the circumference of the armature or primary, 
is sinusoidal in space. While in most eases this is sufficicntly 
the ease, it is not always so, and especially the space or air-gap 
distribution of the magnetic flux may sufficiently differ from sine 
shape, to exert an appreciable effect on the torque at lower 
speeds, and require consideration where motor action and 
braking action with considerable power is required throughout 
the entire range of speed. 

Let then: 
r — iji cos * + e» cos (3 * — a,) + es cos (5 * — at) 4- e? cos 
(7* - a-) + e, cos (9 * - a„) + . . . (1) 

be the voltage impressed u|hjn one phase of the induction motor. 

If the motor is a quarter-pha.se motor, the voltage of the 

second motor phase, which lags 90° or behind the first motor 
phase, is: 


= e,eos^«- gj + c3 
+ 8) eos (? 0 


3*- 


cos(.5«- 


•(♦-$■ 


A3*- 

1 . cost 3 <t> - «■( + *}) + etcoalS <t> - 


+ «odb(7#-«t + 5) + *«*(&* -■•-£) + ■ • ■ W 

The magnetic flux produced by these (wo voltages thus con- 
sists of a series of component fluxes, corresponding respective]] 


HIGHER HARMONICS 145 

to the successive components. The secondary currents induced 
by these component fluxes, and the torque produced by the 
secondary currents, thus show the same components. 

Thus the motor* torque consists of the sum of a series of 
components: 

The main or fundamental torque of the motor, given by the 
usual sine-wave theory of the induction motor, and due to the 
fundamental voltage wave: 

ei cos 0 ] 

Iju *\ (3) 

d cos \<t> ~ 2) 


is shown as T\ in Fig. 55, of the usual shape, increasing from 
standstill, with increasing speed, upj to a maximum torque, and 
then decreasing again to zero at synchronism. 
The third harmonics of the voltage waves are : 

e3cos(3 0 — a3), j 

e3cos(3 0- «» + 5)-| (4) 

As seen, these also constitute a quarter-phase system of 
voltage, but the second wave, which is lagging in the funda- 
mental, is 90° leading in the third harmonic, or in other words, 
the third harmonic gives a backward rotation of the poles with 
triple frequency. It thus produces a torque in opposite direc- 
tion to the. fundamental, and would reach its synchronism, that 
is, zero torque, at one-third of synchronism in negative direction, 
or at the speed <S, = — J£, given in fraction of synchronous speed. 
For backward rotation above one-third synchronism, this triple 
harmonic then gives an induction generator torque, and the 
complete torque curve given by the third harmonics thus is as 
shown by curve T* of Fig. 55. 

The fifth harmonics: 

66 cos (5 0 — a6), | 

eb cos ^5 05 - a& - 2) 

give again phase rotation in the same direction as the funda- 
mental, that is, motor torque, and assist the fundamental. But 
synchronism is reached at one-fifth of the synchronous speed of 

the fundamental, or at: S = +}i} and above this speed, the 
10 


14(i ELECTRICAL APPARATUS 

fifth harmonic becomes induction, genera tor, due to overayn- 
chronous rotation, and retards. Its torque curve is shown as 
7\ in Fig. 55. 

The seventh harmonic again gives negative torque, due to 
backward phase rotation of the phases, and reaches synchronism 
at S = — J-j, that is, one-seventh speed in backward rotation, 
as shown by curve T-, in Fig. 55. 

„ 

DT( 

.- 

1 

N^ 

■-T 

OK 

= HASE 

3 

^ 

, I 

% 

s 

\f 

s 

b 

T', 

\t 

j 

\ 

I 

<T 

T, 

A' 

\- 

J3" 

— 

T, 

1 

* 

1 

Flo. 55. — Quarti'r-plmsr imliirtinii riintnr, I'diiipoiii-iil harmonica mid 
resultant torque. 

The ninth harmonic again gives positive motor torque up to 
its synchronism, 5 = %, and above this negative induction 
generator torque, etc. 

We then have the effects of the various harmonics on the 

QUAIITER-I'HASE INDUCTION MlJTOR 

1 ■( 

+ i -H 

+ll - 

5 

+ 

+ '. 

7 

9 

+ 

+ '» 

+J6 

11 is 

+ 
-H, + ',. 

- +K. 

Synchronous 

Torque positrt 

otherwise n 

peed: S -. . . . 

i'Uj.i tu; .-. = . 

galive. 

HIGHER HARMONICS 


147 


Adding now the torque curves of the various voltage harmonics, 
Tz, 7\, T7, to the fundamental torque curve, 7\, of the induction 
motor, gives the resultant torque curve, T. 

As seen from Fig. 55, if the voltage harmonics are consider- 
able, the torque curve of the motor at lower speeds, forward 
and backward, that is, when used as brake, is rather irregular, 
showing depressions or "dead points." 

89. Assume now, the general voltage wave (1) is one of the 
three-phase voltages, and is impressed upon one of the phases 
of a three-phase induction motor. The second and third 

phase then is lagging by -«- and -«- respectively behind the first 

phase (1): 

e' = ei cos \6 - 3~ J + e8 cos (3 0 £ - a8) 

+ eh cos (5 0 - ■-•„* - abJ + e7 cos \7 <t> - * - a7) 

18 IT 


(6) 


+ e9cos(90- Qir~a»)+ • 
= 61 cos (0 — «- J + e% cos (3 4> — a8) 

+ eb cos (5 0 - a5 + q~) + e7 cos ( 7 <t> - a7 - ^ J 

+ e9 cos (9 4> — ag) + . 
e" = 61 cos ( 0 — -s J + e8 cos (3 0 — a8) 

+ e& cos (5 0 — «* + ■ 3*) + e7 cos ( 7 0 - a7 - «- j 

+ e9 cos (9 4> — ag) + • 


Thus the voltage components of different frequency, impressed 
upon the three motor phases, are : 


ei cos * 

rj COR 

n cos 

e? cos 

r* cos 

(3 4> - aa) 

(5* - a4) 

(7* - a;) 

(9 * - a») 

ei cos 

/ 2t\ 

ft cos 

es cos 

/ 2t\ 

e 7 cos 
/ 

2f\ 

r» cos 

(♦-t) 

(3 0 - oa) 

^♦-« + --j 

17* - ai 

-t) 

(9 0 - a.) 

ei cos 

(-¥) 

f 3 cos 

(3 0 - ai) 

f 5 cos 

(5*-°'+V) 

ei 00s 

(7* - ai 

-t) 

n cos 

(9 0 - a») 

\ / 

Fundamental.... 

1 

3d 

\ / 

5th 

7th 

9th 

148 ELECTRICAL APPARATUS 

As spm, in this case of the three-phase motor, the third 
harmonics have no phase rotation, but are in phase with each 
other, or single-phase voltages. The fifth harmonic gives 
backward phase rotation, and thus negative torque, while the 
seventh harmonic has the same phase rotation, as the funda- 
menlal, thus adds its torque up to its synchronous speed, S = 
+ \i, and above this gives negative or generator torque. The 
ninth harmonic again is single-phase. 

Fig. 56 shows the Fundamental torque, 5ft, the higher harmonics 

,..!. 

•" 

s 

i 

T,, 

1 

THREE PHA 
INDUCTION UIC 

E 
TO 

1 

\ 

\i 

s 

*T 

n 

T 

v3 

^ 

Tt 

~\r 

T; 

T[ 

' 

Fig. 56. — HtrBC-ph;ini> inilii<-t.inn motor, component harmonica and 
resultant torque. 

of torque, T& ami 5T;, and the resultant torque, T. As seen, the 
distortion of the torque curve is materially less, due In 1 lie 
absence, in Fig. 50, of the third harmonic torque. 

However, while the third harmonic (and its multiples) in the 
three-phase system of voltages are in phase, thus give no phase 
rotation, they may give torque, as a single-phase induction motor 
has torque, at speed, though al standstill the torque is eero. 

Fig. 57 Ii shows diagrammatical ly, as T, the development of 
the air-gap distribution of a hue three-phase winding, such as 
used in synchronous converters, etc. Each phase 1, 2, 3, coi an 

one-third of the pitch of a pair of poles or -5-, of the upper layer, 

HIGHER HARMONICS 


149 


and its return, 1', 2', 3', covers another third of the circumference 
of two poles, in the lower layer of the armature winding, 180° 
away from 1, 2, 3. However, this type of true three-phase wind- 
ing is practically never used in induction or synchronous machines, 
but the type of winding is used, which is shown as S, in Fig. 
57 C. This is in reality a six-phase winding: each of the three 


e 


Uddt 


N 


i; 


2' 


2o 


r 


2' 


■ 


2' 


1o 3' 2'a 


I 


KCffl 


N 


2o 


1 


mm a 


r 


2' 


''^«a 


2 


3 


i' 


r 3; 2' 


3' 


30 2 


i; 3' 2i 


£ 


MtttMtt 


K 


^ B 


r 


W 


^_l 


r 3'ft 2' 


77- 


& c 


Fig. 57. — Current distribution at air gap of induction motor, fundamental 

and harmonics. 


phases, 1, 2, 3, covers only one-sixth of the pitch of a pair of 

poles, or ~ or 60°, and between the successive phases is placed 

the opposite phase, connected in the reverse direction. Thus 
the return conductors of phases 1,2, 3 of the upper layer, are 
shown in the lower layer as 1', 2', 3'; in the upper layer, above 
1', 2', 3', is placed again the phase 1, 2, 3, but connected in the 
reverse direction, and indicated as 10, 2o, 3o. As 10 is connected 
in the reverse direction to 1, and 1' is the return of 1, lo is in 


150 ELECTRICAL APPARATUS 

phase with I', and the return of I..: I'o, is in the lower layer, in 

phase with, and beneath I. Tims the phase rotation is: 1,-3, 
2, -1,3, -2, 1, etc. 

For comparison, Fig. 57 .4 shows the usual quarter-phase 
winding, Q, of the same general type as the winding, Fig.' 57 C. 

If then the three third harmonics of 1, 2 and 3 are in phase 
with each other, for these third harmonics the true three-phase 
winding, T, gives the phase diagram shown as 7*» in Fig. 57 D. 
As seen, the current flows in one direction, single-phase, through- 
out the entire upper layer, and in the opposite direction in the 
lower layer, anil thus its magnetizing action neutralizes, that is, 
there can be no third' harmonic flux in the true three-phase 
winding. 

The third harmonic diagram of the customary six-phase ar- 
rangement of three-phase winding, S, is shown as iS3 in Fig. 57 
E. As seen, in this case alternately the single-phase third har- 
monic current flows in one direction for 60° or „. and in the 

opposite direction for the next „. In other words, a single-phase 
m.m.f. and single -phase flux exists, of three times as many poles 
:is the fundamental flux. 

Thus, with the usual three-phase induction-motor winding, 
a third harmonic in the voltage wave produces a single-phase 
triple harmonic flux of three times the number of motor poles, 
and this gives a single-phase motor-torque curve, that is, a torque 
which, starting with zero at standstill, increases to a maximum 
in positive direction or assisting, and then decreases again to zero 
at its synchronous speed, and above this, becomes negative as 
single-phase induction-generator torque. Triple frequency with 
three times the number of poles gives a synchronous speed of 
S = +}(>. That is, the third harmonic in a three-phase vol- 
tage may give a single-phase motor torque with a synchronous 
speed of one-ninth that of the fundamental torque, and in cither 
direction, as shown as Tj in dotted lines, in Fig. 56. 

As usually the third harmonic is absent in three-phase vol- 
tages, such a triple harmonic single-phase torque, as shown 
dotted in Fig. 56, is of rare occurrence: it could occur only in a 
four-wire three-phase system, that is, system containing the 
three phase-wires and the neutral. 

90. All the torque components produced by the higher har- 
monics of the voltage wave have the same number of motor poles 


HIGHER HARMONICS 


151 


as the fundamental (except the single-phase third harmonic 
above discussed, and its multiples, which have three times as 


/ \_ ! /_ 8INE 


- Q-0 


- S-0 


T-o 


Q-fc 


r£Vis\ 


II^L^,— rJZI s-Vs 


Q-'/i . J. \ 


Q-'A L- 


iznzr 


ZL s-'/t 


£\ 


Fig. 58. — Current and flux distribution in induction-motor air gap, with 

different types of windings. 

many motor poles), but a lower synchronous speed, due to their 
higher frequency. 

Torque harmonics may also occur, having the fundamental 


152 


ELECTRICAL APPARATUS 


frequency, but higher number of pairs of poles than the Funda- 
mental, and thus lower synchronous speeds, doe to the deviation 
of the space distribution of the motor winding from sine, 

The fundamental motor torque, I\, of Figs. 55 and 56, is given 
by ft sine wave of voltage and thus of flux, if the winding of each 
phase is distributed around the circumference of the motor air 
gap in a sinusoidal manner, as shown as F under " Sine," in Fig. 
58, and the flux distribution of each phase around the circum- 
ference of the air gap is sinusoidal also, as shown as * under 
"Sine," in Fig. 58. 

This, however, is never the ease, but the winding is always 
distributed in a non-sinusoidal manner. 

The space distribution of magnetizing force and thus of flux 
of each phase, along the c i re u inference of the motor air gap, 
thus can in tin' general case lie represented by a trigonometw 
series, with to as space angle, in electrical degrees, that is, counting 
a pair of poles as 2jt or 3ti0°. It is then: 

The distribution of the conductors of one phase, in the motor 
air gap: 


= Fa J COS oj -\- flj cos 3 tu + i 


; COS 5 C 

+ 


+ Or eos 7 n 
a eos 9 to + . 


(8) 


hen (he assumption is made, that all the harmonics are in phase, 
that is, the magnetic distribution symmetrical. This is prac- 
tically always the case, and if it were not, it would simply add 
phase angle, a„, to the harmonics, the same as in paragraphs 88 
and 89, but would make no change in the result, as the component 
torque harmonics are independent of the phase relations between 
the harmonic and the fundamental, as seen below. 

In a quarter-phase motor, the second phase is located 90° 
or u) = - displaced in space, from the first phase, and thus 
represented by the expression: 

F'= PojoOfl(« - *) + »scos(3 w - 32") + U(.cos(5u - 52T) 


- a7cos(7 w - ■*) + 
" V\ + "' c°9 f 3 w + ^ 


+ «;COS(7 


cos 9u - " 


.(9.-IH 


HIGHER HARMONICS 


153 


Such a general or non-sinusoidal space distribution of magnetiz- 
ing force and thus of magnetic flux, as represented by F and F', 
can be considered as the superposition of a series of sinusoidal 
magnetizing forces and magnetic fluxes : 

as cos 5co 

m (5 « - D 


COS CO 

cos (• - J) 

CL? COS 7 CO 

a7cos 


a 3 cos 3 to 


The first component : 


a3 cos (3 co + 1 j 
09 cos 9 co 

a9 cos (9w- J 

cos CO, 

cos (co - p > 


a5 cos 


(10) 


(10) 


gives the fundamental torque of the motor, as calculated in the 
customary manner, and represented by 7\ in F*igs. 55 and 56. 

The second component of space distribution of magnetizing 
force : 

a3 cos 3 co, j 

(id 


a3 cos 


(3»+3 


gives a distribution, which makes three times as many cycles 
in the motor-gap circumference, than (10), that is, corresponds 
to a motor of three times as many poles. This component of 
space distribution of magnetizing force would thus, with the 
fundamental voltage and current wave, give a torque curve 
reaching synchronism as one-third speed; with the third harmonic 
of the voltage wave, (11) would reach synchronism at one-ninth, 
with the fifth harmonic of the voltage wave at one-fifteenth of 
the normal synchronous speed. 

In (11), the sign of the second term is reversed from that in 
(10), that is, in (11), the space rotation is backward from that 
of (10). In other words, (11) gives a synchronous speed of 
S = —% with the fundamental or full-frequency voltage wave. 

The third component of space distribution : 

as cos 5 co, 1 


a5 cos 


(»-3- 


(12) 


gives a motor of five times as many poles as (10), but with same 
space rotation as (10), and this component thus would give a 
torque, reaching synchronism at S = +^. 


1.54 


ELMCTRtCAL APPARATUS 


In the same manner, the seventh space harmonic gives 
S «* —%, the ninth apace harmonic S = + }■$, etc. 

91. As seen, the component torque curves of the harmonies 
of the space distribution of magnetizing force and magnetic 
flux in the motor air gap, have the same characteristics as the 
component torque due to the time harmonics of the impioawd 
voltage wave, and thus are represented by the same torque 
diagrams : 

Fig. 55 for a quarter-phase motor, 

Fig. 56 for a three-phase motor. 

Here again, we see that the three-phase motor is less liable 
to irregularities in the torque curve, caused by higher harmonics, 
than the quarter-phase motor is. 

Two classes of harmonics thus may occur in the induction 
motor, and give component torques of lower synchronous speed: 

Time harmonies, that is, harmonics of the voltage wave, 
which are of higher frequency, but the same number of motor 
poles, and 

Space harmonics, that is, harmonics in the air-gap distribu- 
tion, which are of fundamental frequency, but of a higher number 
of motor poles. 

Compound harmonics, that is, higher space harmonics of 
higher time harmonics, theoretically exist, but their torque 
necessarily is already so small, that they can be neglected, except 
where they are intentionally produced in the design. 

We thus get the two elates of harmonics, and their 
characteristics : 


Quarttr- phair motor .- 
PKue tul»tion . 
Synchro noun tpwil 

Tim- II I F*«™™>' 
I No of polr* 

sp.«H(£etrnr! 

I No. ol polp» 

Phup rotation .... 


" HIGHER HARMONICS 155 

• 

92. The space harmonics usually are more important than the 
time harmonics, as the space distribution of the winding in the 
motor usually materially differs from sinusoidal, while the devia- 
tion of the voltage wave from sine shape in modern electric power- 
supply systems is small, and the time harmonics thus usually 
negligible. 

The space harmonics can easily be calculated from the dis- 
tribution of the winding around the periphery of the motor air 
gap. (See "Engineering Mathematics," the chapter on the 
trigonometric series.) 

A number of the more common winding arrangements are 
shown in Fig. 58, in development. The arrangement of the 
conductors of one phase is shown to the left, under F, and the 
wave shape of the m.m.f. and thus the magnetic flux produced 
by it is shown under <& to the right. The pitch of a turn of the 
winding is indicated under F. 

Fig. 58 shows: 

Full-pitch quarter-phase winding: Q — 0. 

Full-pitch six-phase winding: S — 0. 

This is the three-phase winding almost always used in induction 
and synchronous machines. 

Full-pitch three-phase winding: T — 0. 

This is the true three-phase winding, as used in closed-circuit 
armatures, as synchronous converters, but of little importance 
in induction and synchronous motors. 
%> % and J^-pitch quarter-phase windings: 

Q - H; Q - «; Q - «. 

%l % and J^-pitch six-phase windings: 

S - Ve; S - }£; s-y2. 

%-pitch true three-phase windings: T — J^. 

As seen, the pitch deficiency, p, is denoted by the index. 
Denoting the winding, F, on the left side of Fig. 58, by the 
Fourier series: 

F = Fo (cos co + a3 cos 3 co + a6 cos 5 co + a7 cos 7 co + . . . ). (13) 
It is, in general : 


IT JO 


Foan = - I F cosncodw. 
njo 

If, then: p = pitch deficiency, 

q = number of phases 


(14) 


156 


ELECTRICAL APPARATUS 


(four with quarter-phase, Q, six with six-phase, 5, three with 
three-phase, T); 

any fractional pitch winding then consists of the superposition 
of two layers: 


and 


From w = 0 to co = + rt- > 

q 2 

from cj = 0 to co = — jr-t 

q 2 


and the integral (14) become: 


r , Pw v Vt 

4F| f« 2 A" 2 

f ofln = - I <*os wcorfco + I cos ncodco 

?r IJo Jo 


= - ■ ; sin n\ + ~ ) + sin /<( r~- J :■ 

717T I \(J 2 / \<7 2 / J 


8F . 717T »/?7T 

= — sin — cos ■ > 
q 2 


1VK 


(15) 


as for: n = 1; an = 1, it is, substituted in (15) 


8F 


.IT VTT ' 

sin cos rrt 
9 2 


hence, substituting (16) into (15) : 


o„ = 


. nir pur 

sin — cos n 

_ 9_ _. 2 

sin - cos - 
q 2 


For full-pitch winding: 


p = 0. 


It is, from (17): 


(16) 


(17) 


«»° = 


sin 


sin 


nir 


(18) 


HIGHER HARMONICS 157 

and for a fractional-pitch winding of pitch deficiency, p, it thus is : 

VWK 

cos 2 
a„ = a„° — • (19) 

cosT 

93. By substituting the values: q = 4, 6, 3 and p = 0, %, 
Hj %} into equation (17), we get the coefficients an of the 
trigonometric series: 

F = Fq { cos co + a3 cos 3 co + a6 cos 5 co + a7 cos 7w+ . . . } , 

(20) 

which represents the current distribution per phase through the 
air gap of the induction machine, shown by the diagrams F of 
Fig. 58. 

The corresponding flux distribution, $, in Fig. 58, expressed by 
a trignometric series: 

<fr = $o {sin o> + 63 sin 3 w + 65 sin 5 co + 67 sin 7 co + . . . | 

(21) 

could be calculated in the same manner, from the constructive 
characteristics of $ in Fig. 58. 

It can, however, be derived immediately from the consideration, 
that $ is the. summation, that is, the integral of F: 


& 


= J>dco (22) 


and herefrom follows: 


bn = * (23) 

and this gives the coefficients, bn, of the series, 4>. 

In the following tables are given the coefficients an and bn, 
for the winding arrangements of Fig. 58, up to the twenty-first 
harmonic. 

As seen, some of the lower harmonics are very considerable 
thus may exert an appreciable effect on the motor torque at low 
speeds, especially in the quarter-phase motor. 


158 


ELECTRICAL APPARATUS 


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