CHAPTER XXXI 
INTERLINKED POLYPHASE SYSTEMS 

283. In a polyphase system the different circuits of displaced 
phases, which constitute the system, may either be entirely 
separate and without electrical connection with each other, or 
they may be connected with each other electrically, so that a 
part of the electrical conductors are in common to the different 
phases, and in this case the system is called an interlinked poly- 
phase system. 

Thus, for instance, the quarter-phase system will be called an 
independent sj^stem if the two e.m.fs. in quadrature with each 
other are produced by two entirely separate coils of the same, 
or different, but rigidly connected, armatures, and are connected 
to four wires which energize independent circuits in motors or 
other receiving devices. If the quarter-phase system is derived 
by connecting four equidistant points of a closed-circuit drum 
or ring-wound armature to the four collector rings, the system is 
an interlinked quarter-phase system. 

Similarly in a three-phase system. Since each of the three 
currents which differ from each other by one-third of a period 
is equal to the resultant of the other two currents, it can be con- 
sidered as the return circuit of the other two currents, and an 
interlinked three-phase system thus consists of three wires con- 
veying currents differing by one-third of a period from each 
other, so that each of the three currents is a common return of 
the other two, and inversely. 

284. In an interlinked polyphase system two ways exist of 
connecting apparatus into the system. 

1. The star connection, represented diagrammatically in Fig. 
208. In this connection the n circuits, excited by currents differ- 
ing from each other by - of a period, are connected with their 

one end together into a neutral point or common connection, 
which may either be grounded, or connected with other corre- 
sponding neutral points, or insulated. 

415 


416 ALTERNATING-CURRENT PHENOMENA 

In a three-phase system this connection is usually called a Y 
connection, from a similarity of its diagrammatical representa- 
tion with the letter Y, as shown in Fig. 197. 


C4( 


01 


3 

) 

^°c.. 

n^fTfTr^ 

1 

\j \) \) \_) 

0 

^ 

Fig. 208. 


2. The ring connection, represented diagrammatically in Fig. 
209, where the n circuits of the apparatus are connected with 
each other in closed circuit, and the corners or points of connec- 
tion of adjacent circuits connected to the n lines of the polyphase 


5 

/ 

J 

4 

C4 

Cz 

3 

\. 

2 

hCe 

1 

Fig. 209. 


system. In a three-phase system this connection is called the 
delta (A) connection, from the similarity of its diagrammatic 
representation with the Greek letter delta, as shown in Fig. 193. 


INTERLINKED POLYPHASE SYSTEMS 417 

In consequence hereof we distinguish between star-connected 
and ring-connected generators, motors, etc., or in three-phase 
systems Y-connected and A-connected apparatus. 

285. Obviously, the polyphase system as a whole does not 
differ, whether star connection or ring connection is used in the 
generators or other apparatus; and the transmission line of a 
symmetrical n-phase system always consists of n wires carrying 
currents of equal strength, when balanced, differing from each 

other in phase by — of a period. Since the line wires radiate 

from the n terminals of the generator, the lines can be considered 
as being in star connection. 

The circuits of all the apparatus, generators, motors, etc., can 
either be connected in star connection, that is, between one line 
and a neutral point, or in ring connection, that is, between two 
adjacent lines. 

In general some of the apparatus will be arranged in star con- 
nection, some in ring connection, as the occasion may require. 

286. In the same way as we speak of star connection and ring 
connection of the circuits of the apparatus, the terms star voltage 
and ring voltage, star current and ring current, etc., are used, 
whereby as star voltage or in a three-phase circuit Y voltage, the 
potential difference between one of the lines and the neutral 
point, that is, a point having the same difference of potential 
against all the lines, is understood; that is, the voltage as meas- 
ured by a voltmeter connected into star or Y connection. By 
ring or delta voltage is understood the difference of potential 
between adjacent lines, as measured by a voltmeter connected 
between adjacent lines, in ring or delta connection. 

In the same way the star or Y current is the current in a cir- 
cuit from one line to a neutral point; the ring or delta current, 
the current in a circuit from one line to the next line. 

The current in the transmission line is always the star or Y 
current, and the potential difference between the line wires, the 
ring or delta voltage. 

Since the star voltage and the ring voltage differ from each 
other, apparatus requiring different voltages can be connected 
into the same polyphase mains, by using either star or ring 
connection. 

287. If in a generator with star-connected circuits, the e.m.f. 
per circuit = E, and the common connection or neutral point 

27 


418 ALTERNATING-CURRENT PHENOMENA 

is denoted by zero, the voltages of the n terminals are 

E,eE,e^E . . . . €^-'E; 
or in general, e^E, 

at the i^ terminal, where, 

„^_ 1 27r..27r n /— 

t = 0, 1, 2 . . . . n — 1, e = cos h J sin — = v 1. 

Hence the e.m.f. in the circuit from the i^^ to the k^^ terminal is 

Eki = 6* ^ - e^ ^ = (e* - e^E. 
The e.m.f. between adjacent terminals i and ^ + 1 is 

(e^+i _ e')E = €^6 - 1)E. 

In a generator with ring-connected circuits, the e.m.f. per 
circuit 

e^E, 

is the ring e.m.f., and takes the place of 

6^(6 - 1)^; 

while the e.m.f. between terminal and neutral point, or the star 
e.m.f., is 

Hence in a star-connected generator with the e.m.f. E per 
circuit, it is: 

star e.m.f., e* E, 

ring e.m.f., e' (e — 1)E, 

e.m.f. between terminal i and terminal k, (e^' — e^)E. 

In a ring-connected generator with the e.m.f., E, per circuit, 
it is 

star e.m.f., _ E, 

ring e.m.f., e^E, 

e^ — €* 

e.m.f. between terminals i and k, z-E. 

e — 1 . 

In a star-connected apparatus, the e.m.f. and the cun'ent per 


INTERLINKED POLYPHASE SYSTEMS 419 

circuit have to be the star e.m.f. and the star current. In a 
ring-connected apparatus the e.m.f. and current per circuit have 
to be the ring e.m.f. and ring current. 

In the generator of a symmetrical polyphase system, if 

e^E are the e.m.fs. between the n terminals and the neutral 
point, or star e.m.fs. 

li = the currents issuing from terminals / over a line of the 
impedance, Zi (including generator impedance in star connec- 
tion), we have 

voltage at end of line i, 

eE - Zi/., 

and difference of potential between terminals h and i 

(e^- - eOf - {Zuh - ZJi), 

where /» is the star current of the system, Zi the star impedance. 
The ring voltage at the end of the line between terminals i 
and k is Eik, and 

Eik = — Ekf 

If now lik denotes the current from terminal i to terminal k, 
and Zik impedance of the circuit between terminal i and ter- 
minal kf where 

lik = — Iki, 

Zik = Zki, 
we have Eik = Ziulik. 

If lio denotes the current in the circuit from terminal i to a 
ground or neutral point, and Z,o is the impedance of this circuit 
between terminal i and neutral point, it is 

E^o = e'E - Zili = Ziolio. 

288. We have thus, by Ohm's law and Kirchoff's law: 

If €»£' is the e.m.f. per circuit of the generator, between the 

terminal, i, and the neutral point of the generator, or the star 
e.m.f. 

li = the current at the terminal, i, of the generator, or the 
star current. 

Zi = the impedance of the line connected to a terminal, ?', of 
the generator, including generator impedance. 


420 ALTERNATING-CURRENT PHENOMENA 

Ei = the e.m.f. at the end of hne connected to a terminal, i, 
of the generator. 

Eik = the difference of potential between the ends of the lines, 
i and k. 

lik = the current from line i to line k. 

Zik = the impedance of the circuit between lines i and k. 

lio, lioo . . . . = the current from line i to neutral points 
0, 00, .... 

Zio, Zioo . . . . = the impedance of the circuits between 
line i and neutral points 0, 00, .... 

Then: 

1. Eilc = — Eki, -l ik — Iki Zik — Zki, lio = — lot, 

Zio = Zoi, etc. 
2. Ei = eE - Zili. 

O . Hi % /j iqJ- to ^ ioo-^ too .... 

4. Eik - Ek'- E, = (e* - e')E - {Zkh - ZJi). 

5. Eik — Ziklik. 

6. li = i^iik. 

0 • 
7. If the neutral point of the generator does not exist, as in 
ring connection, or is insulated from the other neutral points : 

n 

S^/i = 0 
1 • 

i'lio = 0; 
1 • 

n 

2* lioo = 0, etc 
1 • 

Where 0, 00, etc., are the different neutral points which are 
insulated from each other. 

If the neutral point of the generator and all the other neutral 
points are grounded or connected with each other, we have, 

hli= ii{Iio + I,oo+ . . . .) 

1 • 1 • 

n n 

= ^'' lio + S' lioo + . . . . 

1 • 1 • 


INTERLINKED POLYPHASE SYSTEMS 421 

If the neutral point of the generator or other neutral points 
are grounded, the system is called a grounded system. If the 
neutral points are not grounded, the system is an insulated poly- 
phase system, and an insulated polyphase system with equalizing 
return, if all the neutral points are connected with each other. 

8. The power of the polyphase system is 

n 

P = S* e'EIii cos di at the generator, 


P = 2' 2* Eiklik cos dik in the receiving circuits. 

0 i