THE TRANSFORMER. 209 



During each twenty-fifth of a second that the battery voltage 

 continues to act steadily on the primary coil the core flux must 

 change at the constant rate of fifty million lines per second accord- 

 ing to equation (i) above, the resistance of the primary coil being 

 ignored. Therefore (a) the curve of core flux curve must be an 

 inclined straight line during each twenty-fifth of a second, and 

 (b) the total change of flux during each twenty-fifth of a second 

 must be ^ x 50,000,000 or two million lines. In every case, 

 however, the core flux of a transformer pulsates between equal 

 positive and negative maximum values, so that the total change 

 of flux in a half-cycle is 2<l> where <> is the maximum value of 

 the core flux, positive or negative. Therefore, in the case here 

 considered, the maximum value of the core flux is one million 

 lines as shown by the dotted curve in Fig. 1 76. 



Example of core flux for harmonic electromotive force. The 

 relation between core flux and primary voltage in this case is 

 most easily established by starting with a given harmonically 

 varying core flux $ and finding the corresponding primary 



voltage. Therefore let 



< = < sin ft)/ (ii) 



Differentiating this expression with respect to time we have 



d$ 



-~- = ft)3> COS ft)/ (ill) 



and substituting this value of d^jdt in equation (i) we have 



e' a)N f 3> cos a)/ (iv) 



from which it is evident : (a) That e' is harmonic ; that is to 

 say, a harmonically varying core flux induces a harmonic electro- 

 motive force in the primary coil, or a harmonic primary electro- 

 motive force always produces a harmonically varying core flux ; 

 (b) That the primary voltage is 90 ahead of the harmonically 

 varying core flux in phase, as shown in Figs. 172, 173 and 174; 

 and (c) that the maximum value of e f , namely E', is equal to 

 that is : 



E' = 



15 



