CHAPTER XXVII. 

TBANSFOBMATION OF POLYPHASE SYSTEMS. 

255. In transforming a polyphase system into another 
polyphase system, it is obvious that the primary system 
must have the same flow of power as the secondary system, 
neglecting losses in transformation, and that consequently 
a balanced system will be transformed again in a balanced 
system, and an unbalanced system into an unbalanced sys- 
tem of the same balance factor, since the transformer is an 
apparatus not able to store energy, and thereby to change 
the nature of the flow of power. The energy stored as 
magnetism, amounts in a well-designed transformer only to 
a very small percentage of the total energy. This shows 
the futility of producing symmetrical balanced polyphase 
systems by transformation from the unbalanced single-phase 
system without additional apparatus able to store energy 
efficiently, as revolving machinery. 

Since any E.M.F. can be resolved into, or produced by, 
two components of given directions, the E.M.Fs. of any 
polyphase system can be resolved into components or pro- 
duced from components of two given directions. This en- 
ables the transformation of any polyphase system into any 
other polyphase system of the same balance factor by two 
transformers only. 

266. Let £*,, ^2, ^3 .... be the E.M.Fs. of the 
primary system which shall be transformed into — 

E(, E^, E^ , , , , the E.M.Fs. of the secondary 
system. 

Choosing two magnetic fluxes, </» and 5» ^^ different 


§256] TRANSFORMATION OF POLYPHASE SYSTEMS, 377 

phases, as magnetic circuits of the two transformers, which 
induce the E.M.Fs., e and ?, per turn, by the law of paral- 
lelogram the E.M.Fs., E^y E^, .... can be dissolved into 

two components, E<^^ and E^ , E^ and E^, , , , . oi the phases, 
e and 7, 
Then, — 

El, E^y . . . . are the counter KM.Fs. which have to be 
induced in the primary polyphase circuits of the first 
transformer ; 

hence 

Ei/e^E^I^* . . . are the numbers of turns of the primary 
coils of the first transformer. 

Analogously 

El 1 7 E^ 17 . , . , are the number of turns of the primary' coils 
* in the second transformer. 

In the same manner as the E.M.Fs. of the primary 
system have been resolved into components in phase with 
e and T, the E.M.Fs. of the secondary system, E^y E^, .... 

are produced from components, E^ and E^y E^ and E^y 
.... in phase with c and F, and give as numbers of second- 
ary turns, — 

El I Cy E^ / r, .... in the first transformer ; 
El jly E^ 17, .... in the second transformer.* 

That means each of the two transformers hi and vi con- 
tains in general primary turns of each of the primary 
phases, and secondary turns of each of the secondary 
phases. Loading now the secondary polyphase system in 
any desired manner, corresponding to the secondary cur- 
rents, primary currents will flow in such a manner that the 
total flow of power in the primary' polyphase system is the 
same as the total flow of power in the secondary system, 
plus the loss of power in the transformers. 


878 AL TERNA TING-CURRENT PHENOMENA, [§ 257 

257. As an instance may be considered the transforma- 
tion of the symmetrical balanced three-phase system 

E sin )3, E sin (^ — 120), E sin (fi — 240), 
in an unsymmetrical balanced quarter-phase system : 

E' sin )3, E' sin {fi - 90). 
Let the magnetic flux of the two transformers be 

<^ cos fi and <^ cos ()3 — 90). 
Then the E.M.Fs. induced per turn in the transformers 
^^^ ^sin)3 and ^sin()3-90); 

hence, in the primary circuit the first phase, E sin /8, will 
give, in the first transformer, Efe primary turns; in the 
second transformer, primary turns. 

The second phase, E sin (fi — 120), will give, in the 
first transformer, —Ej^c primary turns; in the second 

. ^xV3 . 
transformer, primary turns. 

The third phase, E sin ifi — 240), will give, in the first 
transformer, — Ej^e primary turns; in the second trans- 

former, primary turns. 

In the secondary circuit the first phase E* sin /8 will give 
in the first transformer: E' j e secondary turns; in the 
second transformer : secondary turns. 

The second phase : £*' sin ()3 — 90) will give in the first 
transformer : secondary turns ; in the second transformer, 
E' j e secondary turns. 

Or, if : 

E = 5,000 E' = 100, e = 10. 

PRIMARY. SECONDARY, 

lat. Sd. 3d. lit. Sd. Phase. 

first transformer + 500 - 250 - 250 10 

second transformer -+- 433 — 433 10 turns. 


$258] TRANSFORMATION OF POLYPHASE SYSTEMS. 379 

That means : 

Any balanced polyphase system can be transformed by two 
trafisformers only, without storage of energy, into any other 
balanced polypliase system, 

258. Transformation with a change of the balance 
factor of the system is possible only by means of apparatus 
able to store energy, since the difference of power between 
primary and secondary circuit has to be stored at the time 
when the secondary power is below the primary, and re- 
turned during the time when the primary power is below 
the secondary. The most efficient storing device of electric 
energy is mechanical momentum in revolving machinery. 
It has, however, the disadvantage of requiring attendance. 


380 ALTERNATING-CURRENT PHENOMENA. [§259