398 Mr. E. C. Rimington on an Air-Core Transformer 

 or 



p = l 1 2_^ 2 { y(2LN-M 2 ) -2 V3 [. . . (12) 



Equation (12) shows that I will only be less than I, when 

 the quantity inside the brackets is positive; so that, if 

 2r 1 y 2 >p 2 (2LN — M 2 ), P is greater than I x *, and hence I>I„ 

 or the impedance of the primary is increased on closing the 

 secondary. -^ p ^ 



For convenience let ai= — , and a 2 = — • a x is of course 



Tl r 2 



the tangent of the angle of lag of the primary current behind 



the P.D. when the secondary is unclosed, while — + tan _1 « 2 



is the phase-angle between the primary and secondary currents 

 when the secondary is closed. 



Let M = /3 VLNj so that ft represents the ratio of the 

 magnetic induction passing through the secondary to that 

 through the primary, and is of course less than unity, also 

 100(1 —ft) is the percentage magnetic leakage*. Substituting 



* This is only the case when the two coils are equal in dimensions 

 and similar in shape : otherwise the ratio of the total lines of magnetic 

 induction through the secondary to those through the primary, when a 

 current flows in the primary, will not be the same as the ratio of the lines 

 through the primary to those through the secondary when there is a 

 current in the latter. 



/3 is the geometrical mean of these two ratios. Thus : let n l be the 

 number of turns in the primary and G a some constant depending on 

 its shape and size, then the magnetic induction through the primary 

 = G 1 w 1 , and L=G l ^ 1 2 . The induction through the secondary = G 2 n 21 

 and N=G 2 w 3 2 , where the constant G 2 depends on the shape and size of the 

 secondary. Let /3, be the fraction of the primary induction that threads 

 the secondary, and /3 2 the fraction of the secondary induction that threads 

 the primary. Then 



M = Q l n l ft l n 2 — G 2 n 2 j3 2 n r 



Hence M 2 = j8 i j8 2 G 1 G a » 1 2 » 2 2 , 



or M=Vjp~ 2 VLN; 



so that 0= s/WK- 



Also G^sGjjjS,, or & = & 



So that for coils of the same shape and size 



/3 1= =/3 2 — 13, since G! = G 2 . 

 When the coils are of different shapes or sizes, 



a G, L " \nj ' 



also if the number of turns in the primary and secondary is equal, 



& L' 



