CHAPTER XXVI. 

SYMMETRICAL POLYPHASE SYSTEMS. 

263. If all the E.M.Fs. of a polyphase system are equal 
in intensity, and differ from each other by the same angle 
of difference of phase, the system is called a symmetrical 
polyphase system. 

Hence, a symmetrical w-phase system is a system of n 
E.M.Fs. of equal intensity, differing from each other in 
phase by 1 / n of a period : 

*i = E sin (3 ; 
e2=£sm((3-^L\', 


en = E sin ( ft - L V* ~ - 
\ 

The next E.M.F. is again : 

^ = E sin (ft — 2 TT) = E sin ft. 

In the polar diagram the n E.M.Fs. of the symmetrical 
0-phase system are represented by n equal vectors, follow- 
ing each other under equal angles. 

Since in symbolic writing, rotation by l/« of a period, 
or angle 2ir/n, is represented by multiplication with : 


the E.M.Fs. of the symmetrical polyphase system are: 


SYMMETRICAL POLYPHASE SYSTEMS. 435 


/ 9 T- ? -rr 

E( cos — + / sin — = 
• ' 


n 

„ f 2 (n — 1) TT . . . 2 (« — 1) 
^ f cos — -i - L -- \-j sm — ^ - ^ 

' V » 

The next E.M.F. is again : 

E ( cos 2 -n- +j sin 2 TT) = .£ e" = .£. 
Hence, it is 

27T . • . 27T n/? 

e = cos - - -f J sm - = V 1. 
;z « 

Or in other words : 

In a symmetrical «-phase system any E.M.F. of the 
system is expressed by : 

e'-Ej 

where : e = -y/1. 

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

*E\ 

, n/T 2 7T . . 2 7T 

where, e = vl = cos -- \-j sin — • . 

n n 

1.) « = 1 e = 1 c«'^ = .£, 
the ordinary single-phase system. 

2.) « = 2 e = - 1 J £ = £ and - £. 

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


3 
-1-/V3 


436 ALTERNATING-CURRENT PHENOMENA. 

The three E.M.Fs. of the three-phase system are : 


-i-yV3 


Consequently the three-phase system is the lowest sym- 
metrical polyphase system. 

4.) n = 4, c = cos — +/ sin — =/, £2 = — 1, e3 = - /. 
4 4 

The four E.M.Fs. of the four-phase system are: 

*£ = £, J£, -E, -JE. 
They are in pairs opposite to each other : 
E and — E • j E and —JE. 

Hence 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 transformation 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. 

265. A characteristic feature of the symmetrical »- 
phase system is that under certain conditions it can pro- 
duce a M.M.F. of constant intensity. 

If « equal magnetizing coils act upon a point under 
equal angular displacements in space, and are excited by the 
n E.M.Fs. of a symmetrical w-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. 


SYMMETRICAL POLYPHASE SYSTEMS. 437 

E= effective value of impressed E.M.F. 
/ = effective value of current. 

Hence, 
& =n'f= 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 2 «•*'/» is : 


The two rectangular space components of this M.M.F. are ; 


and 


Hence the M.M.F. of this coil can be expressed by the 
symbolic formula : 


fi 

n \ n 

Thus the total or resultant M.M.F. of the n coils dis- 
placed under the n equal angles is : 


or, expanded : 


n 


438 ALTERNATING-CURRENT PHENOMENA. 

It is, however : 


cos'2 — + / sin — cos — = £ ( 1 + cos — +/ sin —] 

n n n V w w / 

\ / 

sin 2=1 cos ?Z£+ysin«2=£= ^Yl - cos i^'-ysin4^' 

« » • « z y « « 

_ ^ /I _ ,2A X 


2(1-^ 

and, since: 

5t<2< = 0, 

it is, /= nn'f^ (-sin ft _ y cos ft), 

or, 


the symbolic expression of the M.M.F. produced by the 
« circuits of the symmetrical «-phase system, when exciting 
n equal magnetizing coils displaced in space under equal 
angles. 

The absolute value of this M.M.F. is : 

nn' I n"S n <5 


V2 V2 2 

Hence constant and equal w/V2 times the effective 
M.M.F. of each coil or «/2 times the maximum M.M.F. 
of each coil. 

The phase of the resultant M.M.F. at the time repre- 
sented by the angle ft is : 

tan w = — cot /8 ; hence w = /? — ^ 

That is, the M.M.F. produced by a symmetrical «-phase 
system revolves with constant intensity : 


SYMMETRICAL POLYPHASE SYSTEMS. 439 

F= — • 

V25 

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 symmetri- 
cally by the n M.M.Fs. of the w-phase system. 

This is a characteristic feature of the symmetrical poly- 
phase system. 

266. In the three-phase system, n = 3, F= 1.5 <5max 
where $max is the maximum M.M.F. of each of the magne- 
tizing coils. 

In a symmetrical quarter-phase system, n = 4, F = 2 
^tnax, where $maje is the maximum M.M.F. of each of the 
four magnetizing coils, or, if only two coils are used, since 
the four-phase M.M.Fs. are opposite in phase by two, F = 
&max> where ^max is the maximum M.M.F. of each of the 
two magnetizing coils of the quarter-phase system. 

While the quarter-phase system, consisting of two E.M.Fs. 
displaced by one-quarter of a period, is by its nature an 
unsymmetrical system, it shares a number of features — 
as, for instance, the ability of producing a constant result- 
ant 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. 


440 ALTERNATING-CURRENT PHENOMENA.