1892.] 



Transformers. 



459 



allowable, and if such a transformer had one primary and many- 

 secondary circuits, we might at once write 



I =N 1 V 1 /B lff j. (2), 



-C^^N^/R^ I (3). 



Of course, with any law of magnetisation these are the 1 answers if 

 A may be neglected in equation (1). 



Taking the third assumption, the simplest hysteresis cycle, which 

 when / = is also the second assumption. If I = Ao* sin (1) gives 

 for Vi the value 



AoRx 



Vi = „ {cos /sin M-\-(siii. f+qak) cos kt— b sin Sht+ m sin 5kt}. 



JNi . . •. 



If h = 600 or a frequency of 95 per second, the lowest value of 

 qah for the above-mentioned transformer is 1411, sothat taking / of 

 any value whatever, and b and m even much larger than the values 

 given above, it is obvious that for all practical purposes, certainly in 

 the case where it is the effective value of Vi, which is important, the 

 above expression for Vi is the same as if /, &, and m were zero. No 

 doubt Vi does possess small traces of the higher harmonics, but 

 taking the above values for b and m, and takiDg/ = 20°, I find that 

 the error in neglecting /, b, and m, in calculating the effective volt- 

 age, is utterly insignificant. I have applied to this problem the 

 most complex law of magnetisation which I could formulate which 

 was at all likely to be true, and in all cases I have arrived at the 

 same result : — Given the voltage V x at the terminals of a primary 

 €oil of a transformer with many secondary coils, the induction and 

 the secondary currents may be calculated from 2 and 3, which were 

 worked out on the assumption of constant permeability, and in all 

 cases the error in the effective values is exceedingly small. 



V. Magnetic Leakage. — If all the induction due to the current in 

 any coil does not thread through all the other coils, the effect of 

 leakage is obtained by assuming that each N is really less than the 

 number of turns, and that there is some self-induction in each circuit 

 in addition to NV. If the additional inductances are Z b l 2 , &c, then, 

 in our expression R b R 2 , &c, will be replaced by B^ + Zifl, R 2 + Z 2 0, 

 &c. Thus, in the above-mentioned 1500-watt transformer, let I 

 represent the induction when there is no load on the transformer. 

 Assume that 1 per cent, of the induction due to the primary current 

 escapes the secondary, and that 1 per cent, of the induction due to 

 the secondary current escapes the primary, or that l x = IQT^^g = 

 0-6348, k = 10 -a N 2 2 ff = 1*738 X 10~ 3 . Let I be the induction at full 

 load, or when R 2 == 6 - 8 ohms. When there is no magnetic leakage 

 I = 0'99I ; when there is 1 per cent, magnetic leakage I/I =r 



VOL. LI. 2 I 



