520 BELL SYSTEM TECHNICAL JOURNAL 



We may represent these three transducers as in Fig. Al-2a, where the 

 input and output transducers have each been replaced by an ideal trans- 

 former of turns ratio N and a shunt admittance Yp . This representation 

 is general enough for present purposes, provided that Yp and N are 

 allowed to be complex functions of frequency and provided that terminals 

 0-0' and 3-3' are chosen so that a potential minimum occurs at those 

 points when points 1-1' and 2-2' are shorted. 



The short-circuit admittances for the whole transducer as seen at ter- 

 minals 0-0' and 3-3' are then 



Y*n = N\ (Fn + Yp,) 



F*22 = Nl (F,2 + F,2) (Al-1) 



F*2i = A^A^2F2i 



F*io = i\^A^2Fi2 



where the Y a are the short-circuit admittances of the electron transducer 

 alone as seen at terminals 1-1' and 2-2'. 



If the feedback admittance F12 is assumed negligible the insertion volt- 

 age gain may be written as 



^, . 2N,A\Yn 



Go(l + ^i)(l + CT2) 

 where the sigmas are admittance-matching factors: 





iV?(Fu + Fpi) Y*n Al(F2o -|- Yp,) F22* ,,. .>, 

 T^i = "TT- , o'2 — ^ p7- V^i-'i; 



The gain is maximum when a, , o-o are minimum, i.e., when tube and 

 circuits are resonant and losses are minimum. 



We may rewrite this in terms of the total F*,> as follows: 



tro(l + CTi)(l + 0-2J 



Many practical cases are well approximated by the more special repre- 

 sentation of Fig. Al-2b, where the turns ratios of the ideal transformers 

 are real and independent of frequency, and the shunt admittance consists 

 of ordinary lumped constant circuit elements. The feedback admittance 

 F12 is neglected. 



This representation as simple, lumped-constant elements holds very 

 well for any admittance, even a distributed, cavity-type microwave 

 circuit, or an electronic admittance, provided that the combined circuit 

 has no series and only one shunt resonance near the frequency band in 



