SYSTEMS WITH NON-LINEAR REACTANCE 



435 



art- the jiowcrs corresponding to the two frequencies we have 





HZd COS ipd 



03d 



(31) 



Vm^Zm COS (p„ 



Thus the rate of energy dissipation is in the ratio of the frequencies. 



Effect of Sum Frequency 



The more general case where the sum frequency is also present calls 

 for the solution of (13) to (16) as they stand except for F^ being zero. 

 This may be done by substituting the values of Is and 6^ from (15), and 

 those of Id and dd, from (16), in (13) and (14), and proceeding in a 

 manner similar to that used above. The results take the form 



where 



tan 7 = 



ZsCOs sin ipd -\- ZdOid sin <^s 



ZgWs cos (fd — ZdOiid cos (fs ' 



(32) 

 {33) 



and the signs of sin y and cos y are determined by the numerator and 

 denominator of {33) respectively; 



T - ^ 



" ~ K 



where 



nCOmZdO^d 



1/2 



a = 



Vm = 



where 



and 



> I f ZdWd 1 ^ T ZdU)d , . N 

 1 + ^ — [ - 2 -^ — cos {^d 4- <^.) 



K 



ZdO^dZfjUfj 



( - cos (5 + <^„) 



P \2 



^O-i (I 



sin^ (5 + sr„) 



1/2 



1/2 



1 + 



Z.jCOd 



+ 2 



ZdCOd 



COS ( 



<fd 



tan 8 — 



Id = 



Zgiiif. I \ ZsCOs 



ZsOOs sin (pd + ZdOOd sin ips 



fs) 



ZsCOs COS <Pd + ZdOOd COS (Ps ' 



.COrf 



ZdOOmd 



V„r, 



T -^'^ T • 



Is-^Id, 



?„ = a + a' ± 



2' 



(34) 

 (35) 



(36) 



(37) 



(38) 



(39) 



(40) 

 (41) 



