522-525] Pair of Circuits 455 



The difference of phase of the two currents 



= arg i 2 - arg i, 

 = arg (iz/ij 



( Nip \ 



523. The analysis is of practical importance in connection with the 

 theory of transformers. In such applications, the current usually is of very 

 high frequency, so that p is large, and we find that approximately the ratio 



of the amplitudes (cf. expression (452)) is -^, while the difference of phase 



(cf. expression (453)) is TT. These limiting results, for the case of p infinite, 

 can be obtained at a glance from equation (451). The right-hand member, 



Si 2 , is finite, so that ^- (Mi^ + Ni^) is finite in spite of the infinitely rapid 



variations in ^ and i 2 separately. In other words, we must have approxi- 

 mately M^ + Ni 2 constant, and clearly the value of this constant must be 

 zero, giving at once the two results just obtained. 



524. Whatever the value ofp, the result expressed in equation (452) can be 

 deduced at once from the principle of energy. The current in the primary 

 is the same as it would be if the secondary circuit were removed and R, L 

 changed to R, L'. Thus the rate at which the generator performs work is 

 R'ii, or averaged over a great number of periods (since ^ is a simply-harmonic 

 function of the time) is %R | % 2 . Of this an amount ^R \ ^ | 2 is consumed in 

 the primary, so that the rate at which work is performed in the secondary is 



$ (R - R) 



or 



SM*p* 



This rate of performing work is also known to be i$|i' 2 | 2 , and on 

 equating these two expressions we obtain at once the result expressed 

 by equation (452). 



Case in which LN M* is small. 

 525. The energy of currents i lt i z in the two circuits is 



KLiS+ZMi^ + Nif) ........................ (454), 



and since this must always be positive, it follows that LN M z rriust neces- 

 sarily be positive. The results obtained in the special case in which LN M* 

 is so small as to be negligible in comparison with the other quantities in- 

 volved are of special interest, so that we shall now examine what special 

 features are introduced into the problems when LN M* is very small. 



