Theory of the Transformer. 55 



the value of the number of lines of induction will then 

 represent the effect very well. But the current is not in 

 general a simple sine curve, and so we must write 



y = a x sin (bt -f <?,) + a 2 sin (2 bt + e 2 ) 4- a 3 sin (3 bt + e z ) + . 



In this case it is much more difficult to express the hysteresis 

 empirically. In most cases the first term in the value of y is 

 the largest. A term of the same nature as before will, in 

 this case, suffice to express the hysteresis approximately. 

 We can then write for the total flux of magnetic induction, 



p = A cos {bt + e x ) + Buy + On* if -f My' + &c. 



Problem 1. — To find the electromotive force necessary to 

 make the electric current a sine curve in a transformer without 

 secondary. Let the resistance be R, and make ?/ = csin (bt). 

 Then Maxwell's equation becomes 



■ -*+.$. 



Substituting the value of y we have 



E = (Re + Abn) sin (bt) + Bncb cos (bt) + 3 Qn z sin 2 (bt) cos bt + &c. 



But 



Sin 2 bt cos bt= \ (cos bt— cos 3 bt), 



Sin 4 ^ cos^^j^ (cos 5 bt — 3 cos 3 bt+2 cos bt), 



Sin 6 focos^ = &c. 



Hence the electromotive force that must be given to the 

 circuit must contain not only the frequency of the current 

 but also frequencies of 3, 5, 7, &c. times as many. In other 

 words, the odd harmonics. 



Problem 2. — Transformer without secondary, the electro- 

 motive force being a sine curve. 



E $mbt = Hy+n-¥. 



First it is to be noted that when we place in this equation 

 the general value of y and make the coefficients of like 

 functions, of bt zero, all the even harmonics will strike out. 



Hence the value of the electric current will be 



y == a x sin (bt + e x ) + a 3 sin (3 bt + <? 3 ) -f a- sin (bt + e 5 ) + . 



Substituting this value in the value for p, the equation is 

 theoretically sufficient to determine a lf a 3 , &c, and e l7 e s , &c. 

 The equations are cubic or of higher order and the solution 

 can only be approximate, and I have not thought it worth 

 while to go further with the calculation. However, it is easy 

 to draw the following conclusions : — ■ 



