1408 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



We may now obtain expressions for the admittance and gain of the 

 4-pole when the nonhnear element consists of a nonUnear resistor and 

 a nonlinear capacitor in parallel. We shall do this for the case where a 

 conjugate match exists at the terminals by letting ?// = ^i* and y^ = 

 1)1*. Equations (11) and (12) may thus be written 



(Fii - 2/i*)(F22 + ?/2) = (Fn + VxWii - y2*) = F12F21 (16) 



When this is multiplied out, letting ¥„,„ = Gmn + jBmn , and the real 

 and imaginary parts set equal as indicated by the first equality we ob- 

 tain Gng2 = G-iigx and giiBn + &i) = QiiB-n + 62). In (8) and (9) it is 

 seen that Gn = (722 = (?o and that ^22 is positive in equations (8) and 

 negative in equations (9). We thus obtain 



gi = g2 bi + wiCo = 62 ± C02C0 (17) 



where the upper symbol of the ± sign is used in the noninverting case 

 and the lower symbol in the inverting case. When the real and imaginary 

 parts are set equal as indicated by the second equality in (16) we obtain, 

 using the results in (17), 



g' = Go' - Gi ± C01C02C1' - B' (18) 



where 



g = gi = g2 (19) 



and 



B = bl + COiCo = 62 ± COsCo = ± ^ (C02 ± C0l)(7l (20) 



2G-0 



These results may be put in (14) to obtain the modulator gain. Since a 

 conjugate match exists at the terminals of the 4-pole, this is the maxi- 

 mum available gain. The result is 



MAO. = ,^::^^% (2.) 



For the converter, using equation (15) we obtain 



These results are valid only when a conjugate match exists at the ter- 

 minals. For this to be possible, the right side of (18) must be positive. 

 If it is negative no combination of values of gi and ^2 will result in a 

 match. 



