CHAPTER XXIX 
SYMMETRICAL POLYPHASE SYSTEMS 

269. If all the e.m.fs. of a polyphase system are equal in 
intensity and differ from each other by the same angle of differ- 
ence of phase, the system is called a symmetrical polyphase 
system. 

Hence, a symmetrical n-phase system is a system of n e.m.fs. 

of equal intensity, differing from each other in phase by - of a 

period : 

e\ = E sin /3; 

€2 = E sin (/3 — —j ; 

63 = ^_sin(^ - -^y, 

' . I ^ 2{n - l)x\ 

The next e.m.f. is, again, 

ei = E sin (/3 - 2 x) = E sin /3. 

In the vector diagram the n e.m.fs. of the symmetrical n-phase 
system are represented by n equal vectors, following each other 
under equal angles. 

Since in symbolic writing rotation by - of a period, or angle 

2 TT . ..... 

— , IS represented by multiplication with 

27r , . . 27r 
COS h 7 sin — = e, 

n 71 

the e.m.fs. of the symmetrical polyphase system are 

E; 

27r . . . 2 


( 27r , . . 27r\ 

(cos \- J sin — = Eel 

\ n n I ' 


E 

•■ \ 11 ' " 

399 


400 ALTERNATING-CURRENT PHENOMENA 

E ( cos — + j sin — ) = Ee'-, 

^ I 2 (t? - 1) TT , . . 2 (n - 1) 7r\ ^ n-\ 

E { cos --^ ~ + J sm -^ —-) - E^-^. 

The next e.m.f . is again, 

^(cos 2 X + i sin 2 x) = J^e" = E. 
Hence, it is 

2 TT , . . 2 TT „ /— 

e = cos — + 7 sin — = V 1 • 
n n 

Or in other words: 

In a symmetrical n-phase system any e.m.f. of the system is 

expressed by 

eE^ 

where 

e = VT. 

270. Substituting now for n different values, we get the 
different symmetrical polyphase systems, represented by 

eE, 
where 

„ /— 2 TT , . . 2 TT 

6 = VI = cos — + .7 sm — • 

1. n = 1, e = 1, eE = E, 

the ordinary single-phase system. 

2. n = 2, c = - 1, e^E = E and - E. 

Since — E i^ the return oi E, n = 2 gives again the single- 
phase system. 

„ . 27r^ . . 27r -H-i\/3 

3. n = 3, € = cos "o -T J sin -^ = « ■ 

' 2 

The three e.m.fs. of the three-phase system are 

,E = E, ^-i^ E, =^^^E. ■ 

Consequently the three-phase system is the lowest symmetrical 
polyphase system. 


SYMMETRICAL POLYPHASE SYSTEMS 401 

A A 27r . 27r . 1 , 

4. n = 4, e = COS ^ + ^ Sin ^ = J, e^ = - 1, e^ = - j. 

The four e.m.fs. of the four-phase system are, 

e^E = E, jE, -E, -jE. 
They are in pairs opposite to each other, 

E and — E; jE and — jE. 

H^nce can be produced by two coils in quadrature with each 
other, analogous as the two-phase system, or ordinary alternating 
current system, can be produced by one coil. 

Thus the symmetrical quarter-phase system is a four-phase 
system. 

Higher systems than the quarter-phase or four-phase system 
have not been very extensively used, and are thus of less practical 
interest. A symmetrical six-phase system, derived by trans- 
formation from a three-phase system, has found application in 
synchronous converters, as offering a higher output from these 
machines, and a symmetrical eight-phase system proposed for 
the same purpose. 

271. A characteristic feature of the symmetrical n-phase sys- 
tem is that under certain conditions it can produce a rotating 
m.m.f. of constant intensity. 

If n equal magnetizing coils act upon a point under equal 
angular displacements in space, and are excited by the 7i e.m.fs. 
of a symmetrical n-phase system, a m.m.f. of constant intensity 
is produced at this point, whose direction revolves synchronously 
with uniform velocity. 

Let 

n' = number of turns of each magnetizing coil. 
E — effective value of impressed e.m.f. 
7 = effective value of current. 
Hence, 

F = n'l = effective m.m.f. of one of the magnetizing coils. 

Then the instantaneous value of the m.m.f. of the coil acting 

in the direction, , is 

n 

f. = FV2sin (^-^■) 
= n'l \/2 sin (& ^) * 

26 


402 ALTERNATING-CURRENT PHENOMENA 
The two rectangular space components of this m.m.f. are 


/T r- Zirl . [^ ZTn\ 

= n L \/2 COS sin 1/3 I 

n \ n / 

and 


ft, f ■ 2 7ri 
h = }i Sin — - 


,^^.27rt. /^ 2ti\ 

= n i v2 sm sin \B ) • 

n \ n I 

Hence the m.m.f. of this coil can be expressed by the symbolic 
formula 

,r A- • / 2 7rA / 2 7rt , . . 2 7rA 

/. = n/V2 sm (^ - --) ( cos— +jsm~-j- 

Thus the total or resultant m.m.f. of the n coils displaced under 
the n equal angles is 

/= Si/, = n7V2S^ sinf^-?^Vcos^"+jsin^![!V 
1 \ \ n l\ n n ) 

or, expanded, 

£■ IT /7^ \ ■ a ^-^ ■ I 9 2 TTi, . . 2 TTZ 2 TT A 

/ = n i V 2 sm /3 2/* I cos-' + 7 sin cos I 

( 1 \ n ' n n / 

„ ^ . / • 2 XI 2 Tri , . . „ 2 TTZ 

— cos p 2/» I sin cos 1- j sm^ - — 

\ \ n 71 n 

It is, however, 

„ 2x1. . . 27rt 2x2 , /. 4xi . . 4xi\ 

cos^ — + J sm ^ COS ^ = ^ (1 + cos-^ + J sm -^j 

. 2xt 2iri , . . „2x?' 7'/^ 4xf . . 4x'i\ 

sin cos \- J sin-^ = o ( 1 ~ cos ? sin 

n n n 2 \ n n I 


= |(i-."): 


and, since 


1 1 

it is, 

nn'I\/2 , . ^ 
f = 2 ^^^^ ^ ~ ^ ^^^ ^^ ' 


SYMMETRICAL POLYPHASE SYSTEMS 403 

or, 

. nn'I , . . . 

/ = — ^ (sin (8 - J cos /?) 

nF 
= --^ (sm 13 - j cos /3) ; 

the symbolic expression of the m.m.f. produced by the n circuits 
of the symmetrical n-phase system, when exciting n equal mag- 
netizing coils displaced in space under equal angles. 
The absolute value of this m.m.f. is 

_ nn'I _ nF _ nF^ax 

n 
Hence constant and equal — 7=: times the effective m.m.f. of 

each coil or ^ times the maximum m.m.f. of each coil. 

The phase of the resultant m.m.f. at the time represented by 
the angle /3 is 

tan d ^ — cot jS; hence d = ~ ^ o' 

That is, the m.m.f. produced by a symmetrical ?i-phase system 
revolves with constant intensity, 

V2 

and constant speed, in synchronism with the frequency of the 
system; and, if the reluctance of the magnetic circuit is constant, 
the magnetism revolves with constant intensity and constant 
speed also, at the point acted upon symmetrically by the n 
m.m.fs. of the n-phase system. 

This is a characteristic feature of the symmetrical polyphase 
system. 

272. In the three-phase system, ti = 3, Fo = 1-5 Fmax, where 
Fmax is- the maximum m.m.f, of each of the magnetizing coils. 

In a symmetrical quarter-phase system, n = 4, Fo = 2 Fmax, 
where F^ax is the maximum m.m.f, of each of the four magnet- 
izing coils, or, if only two coils are used, since the four-phase 
m.m.fs, are opposite in phase by two, Fo = Fmax, where Fmax is 
the maximum m.m.f, of each of the two magnetizing coils of the 
quarter-phase system. 


404 ALTERNATING-CURRENT PHENOMENA 

While the quarter-phase system, consisting of two e.m.fs. dis- 
placed by one-quarter of a period, is by its nature an unsym- 
metrical system, it shares a number of features — as, for instance, 
the ability of producing a constant-resultant m.m.f. — with the 
symmetrical system, and may be considered as one-half of a 
symmetrical four-phase system. 

Such systems, consisting of one-half of a symmetrical system, 
are called hemisymmetrical systems.