OPER.rnON Of THERMIONIC VACUUM TUBE CIRCUITS 441 



General Analysis 



Coming back lo the detailed problem in hand, we follow out the 

 method illustrated in the special case, but must use the notation 

 developed in the preceding section to take care of a general impedance, 

 z, in the plate circuit, and a general impedance, q, on the grid circuit. 

 Fig. 1 as before, shows the skeleton circuit, where, however, lower 

 case z and g must be substituted for the capitals. Then 



e = eih+'eih + e\k-\-eik-{- +ei„+^i„ 



Analogous to (11) and (12) : 



ip = auieg\h-\-aih^g\h-\-a-ikeg\k-Vaikegik-\- - 



1 



-\-a2{h-k)eg2at-k)-\-a2{h-k)eg2{h-k)-\- J 



is = b\he\h-[-~hike\h-\-b\keik^~b\keik-\- 



-\-b2(h-k)S2(h-k)-\-b2(h-k)^2(h-k)-\- 



Hence, analogous to (13) and (17) : 



^p^^ -^\0- In^lnegin ~rfll);3li;?gl« ~r'22OT'^mfg2m 'J' a im'->neg2m\ 



(27) 



(28) 



(29) 



eg = '^[(l — binqn)ein-\-{l — bi„q„)ei„ — b2mqme2m — b2,nqme2m] (30) 



where the vsummation refers to terms of different frequencies but of 

 similar form. 



From this point on, the procedure is exactly the same as that given 

 in the special case. Coefificients of terms of like order and frequency 

 are equated, and the final results are: 



ip = '^aih{l — bihqh)eih 



+ ^[(1 — ^i;;9/j)"fl2(2/0~Cri(2/,)(Z(2;0^2(2/,)]f2(2/;) 



-'r'^[(l — bihqh) (l — bikqk)a2{h+k) — aHh+k)q{i,+k)b2{h+k)\e2{h+k) 

 -\-'^[{l — bihqh)(l — bikqk)a2o,-k) — ai{i!-k)q(h-k)b2{h~k)]e2{h~k) 

 -\-'^[{i — b\hqh) C^ — bihqh)a2{0h) — <ii{0h)q(0h)b2m)]^2(0h) 

 + . . . 



where the summation refers to terms of different frequencies but of 

 similar form. Note that, having fl2(/!-Jt) and &2(/i-fe). we may readily 

 write the appropriate expressions for the other a2S and 62's by refer- 

 ence to the formation in equation (31). 



(31) 



