CHAPTER XXIX. 

THBKB-FHASE SYSTEM. 

263. 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-phasen 
Then : 

c ^s = E.M.F. between 2 and 3, 
^ E= E.M.F. between 3 and 1, 

where, e = "v^i = — — "^ — - . 

Let 

Zi, Z2, Zj = impedances of the lines issuing from genera- 
tor terminals 1, 2, 3, 

and Fj, Y^, Yz = admittances of the consumer circuits con- 
nected between lines 2 and 3, 3 and 1, 1 and 2. 

If then, 

Ii, /a, /j, are the currents issuing from the generator termi- 
nals into the lines, it is, 

/i+/a + /8 = 0. (1) 


§263] TIJKEE-P/IASE SYSTEM, 391 


If I{,hyU = 


i'= currents flowing through the admittances Yiy 
5, la, from 2 to 3, 3 to 1, 1 to 2, it is, 

I^^U-J^. or, /i + /2'-// = 0] 

/o = //-//, or, /« + /3'-// = 0" (2) 

/, = //_//, or, /8 + y/~/3' = oJ 

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

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

E\ "= -El — <^2 A + ^z Jz I 

E^ ==E^-Z^h + Z,l\ (3) 

E^ ^E^-Z,h + Z^h] 
the differences of potential, and 

j:-^e:y: 

U = E^y^ (4> 

j: ^e'y 


the currents in the receiver circuits. 

These nine equations (2), (3), (4), determine the nine 
quantities : A, /j, /a, //, A', A', E{, E.^, E^, 

Equations (4) substituted in (2) give : 

/, = /^:,'y,-e:./yA 

/, = E,'V^-E,'Y,\ (5) 

A = E,'y2-E/Y,j 

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

since Ei = € £ 1 

£2 = €^E V as E.M.P"s. at the generator terminals. 
£» = £ j 

c£- ^/(l + }\X. + }\Z^) + A/ i;Za + 7?,' j-.z, = 0] 

€*£ - £,' a + y,z, + y,Z:) + ^/ j:z, + Ay r, z, = o\ (6) 

^ - £t' (1 + I'aZ, + JaZa,) + Hi/iiZ., + A/ J's Zi = J 


892 


AL TERN A TING-CURREXT PHENOMENA, [ § 263 


as three linear equations with the three quantities J?/, 
F ' F' 

2 ♦ 8 • 

Substituting the abbreviations : 


D = 


A = 


A = 


A = 


it IS : 


hence, 


- (1 + y^Zr + y,z^), y^Zt, y^z, 
y,z», -(i + r,z, + y,z,), r.z. 
K,z„ r,z„ - (1 + r.z, + r.z,) 

£', _ (1 + i;z, + r,Zi), r.z, 
1, r,z„ -(i + r,z, + r,z,) 


- (1 + r,z, + }',z,), €, r.z, 

K,z„ 1, _(i+ i'.z, + };z,) 

-(i + r,z, + }',z,), i',z., c 
y,z^, _ a + r.z, + y-jZ,), £» 


_ r. A ^ 


D 


y,B, 


D 


y.D, 


D . 


j; /?. - 


j;a^ 


D 


r, /?, - 

D 


.^»A 


y.D. 


— 


r.A 


x> 


yr/ + 7t/ + k: = 
/i + /» + /, =0 


• (") 


(8) 


(9) 


(10) 


(11) 


§264] 


THREE-PHASE SYSTEM. 


893 


264. Special Cases. 

A. Baiaficcd System 

y, = i; = r, = K 

^1 ^^ ^2 ~~^ 8 ""~ ^' 

Substituting this in (6), and transposing : 

€E 


Er 


= iE " 


Et 


= ^E 


► 


Et 


= E 


li 


l + SVZ 


U 


1 + 3 VZ 1 


/»' 


E] 
1 + 3 


r 
f 

vz] 


1 + 3KZ 


(12) 


1 + 3KZ 
' l + SVZ 
' l + SVZ 

J _ (e-i)^r 

"^ l + SVZ 

J _ e(c-l)^r 
" 1 + 3 KZ 


The equations of the symmetrical balanced three-phase 
system. 

B. (7«t' cirmit loadcdy two unloaded: 
F, = r, = 0, r3 = F 

^j ^ ^2 ^^ ^8 ^^ ^' 

Substituted in equations (6) : 

t E- E( + E^VZ = ^\ y ... . 

E - E^\l + 2 VZ) = 0, loaded branch, 
hence : 


* 1 4- 2 KZ 


+ 

E ' = ^{c^+ (l + 2c'0 KZ} 
' 1 + 21'Z 


^ unloaded ; 


^,' = 


E 
1 + 2VZ 


loaded ; 


all three 
KM.Fs. 
unequal. 


(13) 


894 


AL TERN A TING-CUR RENT PHENOMENA. [ § 264 


/»' = 
/i = 


// = 
EY 

EY 

1 + 2 YZ 

EY 


1+2YZ 


= 


(13) 


(13) 


C. Two circuits loaded, one unloaded. 

Y, = y^ = y, li = 0, 

/jt\ = ^% = Zz = Z» 

Substituting this in equations (6), it is : 

tE^ E,'{1 + 2YZ) + E.'YZ = 0\ , , ,, , 
.^E - E^ (1 + 2 YZ) + E{YZ = 0/ ^"^^^^ ^'"^^^^^• 

E - E^ + {E^ + E^) YZ = unloaded branch. 

or, since : 

(^Z + E,') = -E^) 
E-- E^ - Ei YZ = 0, 

E 


^8' = 


thus 


1 -^YZ 


^._ ige(l + (2 + e)rZ} ^ 
' 1 + 4 P^Z + 3 K^z-* 

^,,_^ ig€'{i + (2 + €^rz} 


a 7*2 


^/ = 


1 _|_ 4 >'Z + 3 K^Z 


^ loaded branches. 


1 + YZ 


unloaded branch. 


(14) 


As seen, with un symmetrical distribution of load, all 
three branches become more or less unequal. 


§ 266] QUARTER-PHASE SYSTEM, 395