1887.J 



On Induction Coils or " Transformers" 



165 



(4.) nE = nRx— m(r + p)y ; 



substituting from (1), 



(5.) x{n*R+m\r+p)} = w 2 E + (lotl4ar)m(r+p) , 



(6.) 2/{^ 2 R+m 2 (r+/3)} = - wmE + (7a/4/r>R ; 



_ + />)mE ZaR(r + p) 



- n *R + m *(r + p) ^ 47r{n 2 R + m 2 (r + />)}' 



We may now advantageously make a first approximation, neglect 

 Za in comparison with ^mx, that is, assume the permeability to be 

 very large, we have 



(r+p)mB sin(2^/T) . 

 (r+/))mB cos(2^/T) 



Ad - |^ 2R + m 2( r + /0 )} ( 2^/T 



For practical purposes these equations are really sufficient. 



We see firstly that the transformer transforms the potential in the 

 ratio n/m, and adds to the external resistance of the secondary 

 circuit p a resistance (w 2 R/m 2 )+r. This at once gives us the varia- 

 tion of potential caused by varying the number of lamps used. The 

 phase of the secondary current is exactly opposite to that of the 

 primary. 



In designing a transformer it is particularly necessary to take 

 note of equation (9), for the assumption is that a is limited so that la. 

 maybe neglected. The greatest value of a is B/{(27r/T)mA}, and this 

 must not exceed a chosen value. We observe that B varies as the 

 number of reversals of the primary current per unit of time. 



But this first approximation, though enough for practical work, gives 

 no account of what happens when transformers are worked so that 

 the iron is nearly saturated, or how energy is wasted in the iron core 

 by the continual reversal of its magnetism. The amount of such 

 waste is easily estimated from Ewing's results when the extreme 

 value of a is known, but it is more instructive to proceed to a second 

 approximation, and see how the magnetic properties of the iron affect 

 the value and phase of x and y. We shall as a second approximation 

 substitute in equations (5) (6) (7) values of a. deduced from the 

 value of a furnished by the first approximation in equation (9). 



In the accompanying diagram Ox represents a, Oy represents a, 

 and Oz the time t. 



The curves ABCD represent the relations of a and a. EFGr 

 the induction a as a function of the time, and HIK the deduced 

 relation between a. and t. We may substitute the values of a obtained 



