CHAPTER XXVII. 

BALANCED AND UNBALANCED POLYPHASE SYSTEMS. 

267. If an alternating E.M.F. : 

e = E V2 sin (3, 
produces a current : 

* = 7V2sin (/? — a), 

where u> is the angle of lag, the power is : 

p = ei = 2 £Ssin ft sin (ft — S) 

= £S(cos a — cos (2 £ — a)), 

and the average value of power : 


Substituting this, the instantaneous value of power is 
found as : 


Hence the power, or the flow of energy, in an ordinary 
single-phase alternating-current circuit is fluctuating, and 
varies with twice the frequency of E.M.F. and current, 
unlike the power of a continuous-current circuit, which is 

constant : 

/-** 

If the angle of lag £ = 0 it is : 

p = P (1 — cos 2 0) ; 

hence the flow of power varies between zero and 2 Pt where 
P is the average flow of energy or the effective power of 
the circuit. 


BALANCED POLYPHASE SYSTEMS. 441 

If the current lags or leads the E.M.F. by angle £ the 
power varies between 

and 


cos u> 

that is, becomes negative for a certain part of each half- 
wave. That is, for a time during each half-wave, energy 
flows back into the generator, while during the other part 
of the half-wave the generator sends out energy, and the 
difference between both is the effective power of the circuit. 
If £ = 90°, it is : 

O rt , 

" p > 


that is, the effective power : P = 0, and the energy flows 
to and fro between generator and receiving circuit. 

Under any circumstances, however, the flow of energy in 
the single-phase system is fluctuating at least between zero 
and a maximum value, frequently even reversing. 

268. If in a polyphase system 

*D ez> *s> • • • • = instantaneous values of E.M.F. ; 
h) *2, t'a, • • • • = instantaneous values of current pro- 
duced thereby ; 

the total flow of power in the system is : 

p = glt\ -f <?2/2 -j- e,j, -f . . . . 
The average flow of power is : 

P = £i /i cos £>! -(- E<i /2 cos w2 -f- . . . . 

The polyphase system is called a balanced system, if the 
flow of energy : 

/ = e\i\ + <V2 + W, +.'.'.;.. 

is constant, and it is called an unbalanced system if the 
flow of energy varies periodically, as in the single-phase sys- 
tem ; and the ratio of the minimum value to the maximum 
value of power is called the balance factor of the system. 


442 ALTERNATING-CURRENT PHENOMENA. 

Hence in a single-phase system on non-inductive circuit, 
that is, at no-phase displacement, the balance factor is zero ; 
and it is negative in a single-phase system with lagging or 
leading current, and becomes = — 1, if the phase displace- 
ment is 90° — that is, the circuit is wattless. 

269. Obviously, in a polyphase system the balance of 
the system is a function of the distribution of load between 
the different branch circuits. 

A balanced system in particular is called a polyphase 
system, whose flow of Energy is constant, if all the circuits 
are loaded equally with a load of the same character, that 
is, the same phase displacement. 

270. All the symmetrical systems from the three-phase 
system upward are balanced systems. Many unsymmetrical 
systems are balanced systems also. 

1.) Three-phase system : 
Let 

^ = E V2 sin ft, and t\ = I V2 sin (ft — w) ; 

ez = E V2 sin (ft - 120), /2 = / V2 sin (0 - « - 120) ; 

ez = E V2 sin (ft - 240), /3 = / V2 sin (ft - & - 240) ; 

be the E.M.Fs. of a three-phase system, and the currents 
produced thereby. 

Then the total flow of power is : 

/ = 2 .57 (sin {3 sin (ft — fi) + sin ((3 — 120) sin (ft — & — 120) 

+ sin (ft — 240) sin ($ — <* — 240)) 
= 3 .£7 cos w = T5, or constant. 

Hence the symmetrical three-phase system is a balanced 
system. 

2.) Quarter-phase system : 

Let £l = £^2s\nft, t\ = I \/2 sin (ft - 5) ; 

e2 = E V2 cos ft, 4 = 7 V2 cos (ft - £) ; 


BALANCED POLYPHASE SYSTEMS. 443 

be the E.M.Fs. of the quarter-phase system, and the cur- 
rents produced thereby. 

This is an unsymmetrical" system, but the instantaneous 
flow of power is : 

/ = 2 £I(sm J3 sin (/? — 5) + cos ft cos (0 — £>)) 
= 2 £Scos w = P, or constant. 

Hence the quarter-phase system is an unsymmetrical bal- 
anced system. 

3.) The symmetrical «-phase system, with equal load 
and equal phase displacement in all n branches, is a bal- 
anced system. For, let : 

e( = E V2 sin ( ft - — "\ = E.M.F. ; 
V » / 

/ 2 IT A 

*',- = 7V2 sin O — S — = current 

V » V 

the instantaneous flow of power is : 


l V « 7 \ » 

EI \ yr cos a -57-035^2 /?-£- — 


or p = n E I cos w = T7, or constant. 

271. An unbalanced polyphase system is the so-called 
inverted three-phase system,* derived from two branches of 
a three-phase system by transformation by means of two 
transformers, whose secondaries are connected in opposite 
direction with respect to their primaries. Such a system 
takes an intermediate position between the Edison three- 
wire system and the three-phase system. It shares with 
the latter the polyphase feature, and with the Edison three- 

* Also called "polyphase monocyclic system," since the E.M.F. triangle is similar to 
that usual in the single-phase monocyclic system. 


444 ALTERNATING-CURRENT PHENOMENA. 

wire system the feature that the potential difference be- 
tween the outside wires is higher than between middle 
wire and outside wire. 

By such a pair of transformers the two primary E.M.Fs. 
of 120° displacement of phase are transformed into two 
secondary E.M.Fs. differing from each other by 60°. Thus 
in the secondary circuit the difference of potential between 
the outside wires is V3 times the difference of potential 
between middle wire and outside wire. At equal load on 
the two branches, the three currents are equal, and differ 
from each other by 120°, that is, have the same relative 
proportion as in a three-phase system. If the load on 
one branch is maintained constant, while the load of the 
other branch is reduced from equality with that in the 
first branch down to zero, the current in the middle wire 
first decreases, reaches a minimum value of 87 per cent of 
its original value, and then increases again, reaching at no 
load the same value as at full load. 

The balance factor of the inverted three-phase system 
on non-inductive load is .333. 

272. In Figs. 185 to 192 are shown the E.M.Fs. as 
e and currents as i in drawn lines, and the power as / in 
dotted lines, for : 


Fig. 185. Single-phase System on Non-inductive Load. 

Balance Factor, 0. 


BALANCED POLYPHASE SYSTEMS. 445 


Fig. 186. Single-phase System on Inductiue Load of 60° Lag. 

Balance Factor, - .333. 


Fig. 187. Quarter-phase System on Non-inductiui Load. 

Balance Factor, + 1. 


Fig. 183. Quarter-phase System on Inductiue Lozd of 60° Lag. 
Balance Factor, + 1. 


446 ALTERNATING-CURRENT PHENOMENA. 


Fig. 189. Three-phase System on Non-induct'we Load. 

Balance Factor, + 1. 


Fig. 190. Three-phase System on Inductive Load of 60° Lag. 

Balance Factor, + 1. 


Fig. 191. Inverted Three-phase System 
on Non-inductive Load. 


Balance Factor, + .333 


BALANCED POLYPHASE SYSTEMS. 


447 


Fig. 174. Inverted Three-phase System on 

Inductive Load of 60° Lag. 

Balance Factor, 0. 

273. The flow of power in an alternating-current system 
is a most important and characteristic feature of the system, 
and by its nature the systems may be classified into : 

Monocyclic systems, or systems with a balance factor zero 
or negative. 

Polycyclic systems, with a positive balance factor. 

Balance factor — 1 corresponds to a wattless circuit, 
balance factor zero to a non-inductive single-phase circuit, 
balance factor + 1 to a balanced polyphase system. 

274. In polar coordinates, the flow of power of an 
alternating-current system is represented by using the in- 
stantaneous flow of power as radius vector, with the angle 
($ corresponding to the time as amplitude, one complete 
period being represented by one revolution. 

In this way the power of an alternating-current system 
is represented by a closed symmetrical curve, having the 
zero point as quadruple point. In the monocyclic systems 
the zero point is quadruple nodal point ; in the polycyclic 
system quadruple isolated point. 

Thus these curves are sextics. « 


448 ALTERNATING-CURRENT PHENOMENA. 

Since the flow of power in any single-phase branch of 
the alternating-current system can be represented by a sine 
wave of double frequency : 


the total flow of power of the system as derived by the 
addition of the powers of the branch circuits can be rep- 
resented in the form : 

/ = />(! + « sin (2 £- a.)) 

This is a wave of double frequency also, with c as ampli- 
tude of fluctuation of power. 

This is the equation of the power characteristics of the 
system in polar coordinates. 

275. To derive the equation in rectangular coordinates 
we introduce a substitution which revolves the system of 
coordinates by an angle o>o/2, so as to make the symmetry 
axes of the power characteristic the coordinate axes. 


hence, sin (2 ft - S>0) = 2 sin ^ - ^ ) cos (/? - ^ j = 
substituted, 

^M' + ^j. 

or, expanded : 

— P2 (x2 + /* + 2 e A:^)2 = 0, 


the sextic equation of the power characteristic. 
Introducing : 

a = (! + «)/'= maximum value of power, 
b = (1 — c) P'= minimum value of power; 


BALANCED POLYPHASE SYSTEMS. 449 

it is **?> 


a + b 
hence, substituted, and expanded : 

(*»+/)» - \{a (x + j)2 + b (x -X>T> = 0 

the equation of the power characteristic, with the main 
power axes a and b, and the balance factor: b I a. 
It is thus : 

Single-phase non-inductive circuit : / = /> (1 + sin 2 <£), 
b = 0, a = 2P 


Single-phase circuit, 60° lag : / = P (1 + 2 sin 2 <£), 

i*.~+" 


Single-phase circuit, 90° lag :/ = ^ /sin 2 <£, b = — E I, 

a = + El 


2/, &/a= -1. 
Three-phase non-inductive circuit : p = P, ^ = 1, a = 

x^+y* — P2 = 0: circle. & / a = + 1. 
Three-phase circuit, 60° lag : / = P, 6 = 1, a = 1 

a? +/- 7>a = 0 : circle. £/«= + !. 
Quarter-phase non-inductive circuit :p = P,b = ]-) a = 

x* _|_ y» _ ^2 = o . circlei ^ / ^ = _|_ i. 

Quarter-phase circuit, 60° lag : p = P, b = 1, tf = 1 


450 ALTERNATING-CURRENT PHENOMENA. 

Inverted three-phase non-inductive circuit : 


Inverted three-phase circuit 60° lag :/ = f (1 -\- sin 2 <£), 
b = 0, a = 2 P 

(y? + /)3 _ />2 (• x _|_ yy = 0< fila = Qf 

a and <5 are called the main power axes of the alternating- 
current system, and the ratio b [a is the balance factor of 
the system. 


Figs. 193 and 104. Power Characteristic of Single-phase System, at 60° and 0° Lag. 

276. As seen, the flow of power of an alternating-cur- 
rent system is completely characterized by its two main 
power axes a and b. 

The power characteristics in polar coordinates, corre- 


BALANCED POLYPHASE SYSTEM. 


451 


spending to the Figs. 185, 186, 191, and 192 are shown in 
Figs. 193, 194, 195, and 196. 


Figs. 195 and 196. Power Characteristic of Inverted Three-phase System, at 0° and 
60° Lag. 

The balanced quarter-phase and three-phase systems give 
as polar characteristics concentric circles. 


452 ALTERNATING-CURRENT PHENOMENA.