ivhen the Frequency is below a certain Critical Value. 395 



apparent impedance of the primary is diminished on closing 

 the secondary. Under certain conditions, however, this is 

 not the case, as the following investigation will show. 



Let r r be the resistance of the primary circuit ; 

 L its inductance ; 



r 2 the resistance of the secondary circuit ; 

 N its inductance ; 

 M the mutual inductance between the two coils. 



The coefficients of induction are assumed constant in the 

 following investigation, a result that can only be obtained in 

 practice when coils not containing iron cores are employed. 

 A pure sine-function alternating P.D. is also assumed. 



Let p = 2rrrn ) where n is the frequency of alternation. 



Let e be the value of the P.D. at any instant t, and E its 

 maximum value. 



Let Ci and c 2 be the currents in the primary and secondary 

 circuits respectively, C x and C 2 being their maxima. 



Let Ii= V?'i 2 +jp 2 L 2 , the impedance of the primary ; 

 and I 2 = Vr 2 2 +// 2 N 2 , the impedance of the secondary. 



We have the well-known equations, 



*&+*S? +**-.; • • • • Ci) 



N J 2 + M & + ^=° « 



Differentiate (1) with respect to t, and multiply by N ; 

 differentiate (2) and multiply by M ; then on subtraction we 

 obtain 



(LN _ M2 )§ +N , 1 5-M, 2 5= N J . (3) 

 Multiply (1) by r 2 and add to (3). This gives 



(LN - M«) § + <F?t + *"*) 7% + ' Wl = ** + N If (4) 

 Similarly we obtain 



(LN-M*)^f + (Nn+L^J +'«= - M '|- ( 5 ) 



Now it is obvious, if the P.D. be a pure sine function and 

 the coefficients constants, that the currents must also be 

 pure sine functions differing only in phase from the P.D. 



