106 ALTERNATING CURRENTS 



single-phase transformer would have the shape corresponding to 

 Fig. 81. In order to construct a three-phase transformer, three 

 single-phase cores would be piled on the top of each other as in 

 Fig. 86, only the top core being provided with a top yoke. 



The thickness of the core-sheets varies from - 014 inch to 0'020 

 inch. The insulation (consisting of thin paper) between them 

 occupies some 12 per cent, of the total thickness of the core. 



52. Calculation of e.m.f. induced in Trans- 

 former Winding 



The relation of the value of the maximum induction in the core 

 to the e.m.f. induced in either coil at a given frequency is easily 

 determined. Let B stand for the maximum value of the induction, 

 a for the cross-sectional area of the core, in sq. cms., n for the 

 frequency, and S for the number of turns in the coil. The maximum 

 flux through each turn is aB. This changes from -f aB to aB in 



?r- sec. The mean rate of change is thus 4.naB. the mean arith- 

 2n 



metical value of the e.m.f., in volts, induced in each turn is 

 4waB . 10 ~ 8 , and the mean arithmetical value of the total e.m.f. 

 4waSB. 10~ 8 . Hence, if /denote the form factor of the e.m.f. wave 

 ( 1), we have for the r.m.s. value E of the induced e.m.f. . 



E = 4rca/SB.10- 8 (1) 



For a sine wave of e.m.f.,/ = I'll ( 3), so that 



E = 4-44naSB.10- 8 (2) 



The usual values of B range from 2500 to 4000, the lower values 

 being used at higher frequencies. 



53. Losses in Transformer. Hysteresis Loss 



The losses taking place in a transformer may be divided into core 

 losses and copper losses. 



The core losses arise from hysteresis and eddy currents, and 

 depend on the maximum value of the induction in the core. This 

 maximum value (the amplitude of the induction wave) is frequently 

 referred to as the induction simply. 



We have seen that, owing to the smallness of the resistance drop, 

 the e.m.f. induced in the primary is nearly equal to the primary p.d. 

 Hence, if the latter be maintained constant, as is normally the case, 

 the induced e.m.f. will also remain nearly constant. The constancy 

 of the induced e.m.f., however, involves, in accordance with equation 



