Alternate Current Transformer. 69 



These transformers are wound in five or seven layers and their 

 regulation is of a very high order. The formula in § 54 would 

 only enable us to obtain a very rough approximation to x-^ + x^ for 

 this type. 



General Solution of the Transformer Problem by a 

 Vector Method. 



Explanatory, 



59. {a) If a be any vector representing e.m.f., current, or flux, 

 on the plane alternate current diagram (Fig. 2) and if we under- 

 stand by 



(.a 

 the vector got by rotating a through a right angle in the positive 

 direction, and hence if we understand by 



(Oos^ + tSin^)aor e'-^o. 

 the vector got by rotating a through the angle Q in the positive 



direction, then it is well-known that operators such as e^^ 

 can be manipulated as ordinary algebraic symbols, and that 

 t can be treated as if it were the algebraic imaginary \/ _ \* 

 (b) If rtj, a.2, «3 etc., be numerical multipliers, then the vector 



or the resultant or sum of the vectors 



a^e'-^ia, a,/-^-^a, a^e'-^^a, etc. 

 is ={2aCos(9 + t2aSin6'}a 



= A(Cosi/^-|-tSini/^)a=Ae'''/'a 

 where 



A2 = (2rtCos^f + (2aSin^f 

 = ^a^+'2%a^a.,Go^{d^-e.) 

 and 



hence the operator 



fli^'^i + a./'^2 -1- rtge'^s + etc. = A^"A 

 where A and i/^ are given by the above equations. 



*Lyle. Alternate Current Prohleim. "Electrician," 41, pp. 816-818; 42, pp. 72-74 

 and 148-151, 1898. 



