CHAPTER XXXI. 

THREE-PHASE SYSTEM. 

292. With equal load of the same phase displacement 
in all three branches, the symmetrical three-phase system 
offers no special features over those of three equally loaded 
single-phase systems, and can be treated as such ; since the 
mutual reactions between the three phases balance at equal 
distribution of load, that is, since each phase is acted upon 
by the preceding phase in an equal but opposite manner 
as by the following phase. 

With unequal distribution of load between the different 
branches, the voltages and phase differences become more or 
less unequal. These unbalancing effects are obviously maxi- 
mum, if some of the phases are fully loaded, others unloaded, 

Let: 

E — E.M.F. between branches 1 and 2 of a three-phaser. 
Then: 

« E = E.M.F. between 2 and 3, 
(*£= E.M.F. between 3 and 1, 


where, e= ^1= ~ - 

Let 

ZD Z2, Zs = impedances of the lines issuing from genera- 

tor terminals 1, 2, 3, 
and Yl} Y2, Ys = admittances of the consumer circuits con- 

nected between lines 2 and 3, 3 and 1, 1 and 2. 
Jf then, 

ID It, /8, are the currents issuing from the generator termi- 
nals into the lines, it is, 

/I + /2 + /3 = 0. (1) 


THREE-PHASE SYSTEM. 479 

If //, 72', 7/ = currents flowing through the admittances Y1, 
F2, F3, from 2 to 3, 3 to 1, 1 to 2, it is, 

/! = /,'-/,', or, /1 + /2'_/3' = Ol 
>,->/-/.', or, /2 + /3'-7/ = o[ (2) 

>3 = //->/, or, /3 + >1/-// = OJ 

These three equations (2) added, give (1) as dependent 
equation. 

At the ends of the lines 1, 2, 3, it is : 


(3) 
Il + ztIt) • 

the differences of potential, and 

ti 

(4) 


the currents in the receiver circuits. 

These nine equations (2), (3), (4), determine the nine 
quantities : flt 72, /3, //, 7a', 73', ^', Ti^ £&• 

Equations (4) substituted in (2) give : 


(5) 


These equations (5) substituted in (3), and transposed, 
give, 

since £l = c E 

Ez = £ E \ as E.M.Fs. at the generator terminals. 


480 AL TERNA TING-CURRENT PHENOMENA. 

as three linear equations with the three quantities 2T/, 

Substituting the abbreviations : 

a I \7 7 I I/" 7 \ I/" 7 ~\7 7 i 

~T * 1^2 ~T *1^3)> -tZ^S) •*8^'2 I 

7 V 7 /1_1_V7_1_V7N>/ 

^zt y 2-^D — V*1 ~r -^s^i T *»^V / 


A 


c, F2Z3, F3Z2 

a, - (1 + ^^3 + 

, Y,Zlt -(1 + F3Z1+F3Z2) 

- (1 + Y,Z2 + FiZ,), c, F3Z2 
F.Z3, c2, YtZ, 
Y.Z,, 1, - (1 + F3ZX + F3Z2) 

(i + ^iz. + yiz,), F2z3, £ 


A = / FIZS, - (i 

FaZ2, F2ZX, 


it is: 


D 


72 = i 


__ F2Z>2- 


hence, 


(8) 


(9) 


(10) 


(11) 


THREE-PHASE SYSTEM. 

293. SPECIAL CASES. 

A. Balanced System 

Y, = F2 = F8 = F 
Z, = Z2 = Z3 = Z. 

Substituting this in (6), and transposing : 


481 


c E 


£s = £ 


EI = 


3FZ 


1 + 3FZ 


1 + 3YZ 
EY 


1 + 3KZJ 


3FZ 


3FZ 


3 YZ 


(12) 


The equations of the symmetrical balanced three-phase 
system. 

B. One circuit loaded, two unloaded: 

F! = F2 = 0, F8 = F 
Zj = Z2 = Z3 = Z. 

Substituted in equations (6) : 

= ( unloaded branches. 
E — E3'(l + 2 FZ) = 0, loaded branch. 


hence : 
r./ 

, 


2KZ 


2FZ 


1 + 2 FZ 


unloaded ; 


loaded ; 


all three 

KM.F.'s 

unequal, and (13) 
of unequal 
phase angles. 


482 


AL TERNA TING-CURRENT PHENOMENA. 


(13) 


(13) 


C. Two circuits loaded, one tinloaded. 

F! = F2 = F, F8 = 0, 
Zt = Z2 = Z3 = Z. 

Substituting this in equations (6), it is : 

e E — E{ (1 + 2 FZ) + .£/ FZ = 0) 
£E — El (1 + 2 FZ) + E{ FZ = 0 J 

E — £s' + (,£,' + ^2') FZ = 0 unloaded branch, 
or, since : 

E — Ez''— EZ'Y2 :'= 0, 

E1 = ? 

\ + FZ 

thus: 


1 + 4 FZ + 3 F2Z2 


1 + 4 FZ + 3 F2Z2 
E 

I+'FZ 


loaded branches. 


unloaded branch. 


(14) 


As seen, with unsymmetrical distribution of load, all 
three branches become more or less unequal, and the phase 
displacement between them unequal also. 


QUARTER-PHASE SYSTEM. 483