Alternate Current Transformer. 13 



2Sin(8 + <^) / Cos<^> 



+ 2 {xS^n<t> + ) Sin8 | + ^^Cos^.^ 



,1 1 2 / „ SinS\ / 1 \ 1 



^1 ^2 ^1 i-l ~2 ' 



To find the value of 6, for which t; is a maximum when ^ is 

 constant, we note that rf is of the form 



a + ljO + cO'' 

 which is a maximum when O.^^a/c^Oa^ (say), 

 and its maximum value is 

 1 



Hence the value of 6 for maximum efiiciency is given by 



SinS+ - 



^^Cos^ = I^ 



1 1 2/ ^ „ Sin8\ / 1\ 



- H |--{x,Cos8+ ) + ( AV + ^JSinS 



which for all practical purposes may be reduced to 

 ., SinS 



- + - 



and the maximum efficiency is given to a sufficient approxima- 

 tion by 



7;(max) = 



1 +_^ A/|i + i )sin8-f 2.T2tan</.Sin8 

 Cos^ ^ Iti tJ 



Note. — It is obvious that all the formulae we have obtained 



will apply to non-inductive loads when we make ^ =0, and to 



loads having capacity when we make ^ negative. 



16. From §15 we find that the ratio of the copper losses Hi + Hj 

 to the iron loss H3 is 

 A" g^CosV 



_ Tl T2 



M^SinS 

 Putting in tliis expression for A and M their values given in 

 § 12 and then substituting for B its value at maximum efficiency, 

 we find that this ratio is 



