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

 Assume *, then, 

 c i — Oi sin pt, c 2 = C 2 sin (pt + 0), and e = E sin (pt + <f>) , 



jl =pCi cos pt and -p = — p?Ci sin pt ; 



de 

 also -7- =pE cos (pt + (f>). 



Inserting these values in equation (4) gives 

 (^[{n^— i> 2 (LN — M 2 )j- sin^+^^i + L^) cospf) 



= Er 2 sin (^ + $) + EpN cos (jctf + (/>). . . . (6) 



For shortness, let 



a denote r x r 2 — ^ a (LN — M 2 ), and 

 5 denote jt? (Nr x + Lr 2 ). 

 Then (6) may be written 



Oj V« 2 + ^ 2 sin (^ + >/r) = EI 2 sin Qt?£ + <£ + %), . (7) 

 where 



tan \/r = - and tan y = - — . 

 r a ** r 2 



As (7) must hold for all values of t, it follows that 



Ci Va^TP=m 2) 



or p EI 2 m 



and that ^=^> + ^, or 0=^—^. 



* This assumption will evidently give a particular solution to equation 

 (4), viz. c 1 = C l sin pt. 



The complete solution is obtained by adding to this the solution of 

 equation (4), assuming the right-hand member zero. So that the com- 

 plete solution to (4) is 



k+h Tc-h 



e 1 = C 1 sin^+Ae 2 + Be 2 , 



where N^+L^ 



K "" LN-M 2 ' 



and ,. 



V(Nr 1 -Lr 2 ) 2 +4r 1 r 2 M 2 



h ~ LN-M 2 



The constants A and B depend on the phase of the P.D. at the instant 

 the coil is switched on. The exponential terms (since they are both real 

 and negative) rapidly die away, so that practically c 1 = C 1 sinpt after a 

 short/ time has elapsed. The same remarks apply to the value of c 2 . 



