FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1409 



It may be shown that if the slope of the voltage-current characteristic 

 of the nonlinear resistor is always positive, then Gi/Go can never be 

 greater than unity. (Reference 1, p. 410.) It is therefore convenient to 

 normalize the above results with respect to Go . If we let 



— = p, 



CO-) 



COlCl 



= px, 



CO' 



2C1 



Go Go 



equations (18) through (22) become 



= X, 



Gi 



Go 



7T = y, 



Go _ 

 Gx ^ 



If ± px- - 



XIJ 



(l±p)i^ 



Go 



= ± 



:i±p)f 



pxz. 



MAG12 = 



MAGn = 



62 ^ 

 Jf + X- 



XIJ 



±(1 ± p) '-^ ± xz 



1 + 



Go 



+ 



(1 ±p) 



xii_ 



y' + {9xY 



1 + 



Q_ 



Go 



+ 



(1±P)^ 



(23) 



(24) 



(25) 



(26) 



(27) 



In these equations, p is less than \ in the noninverting case and less than 1 

 in the inverting case. Ordinarily it will be very much less than 1. The 

 value of z will be determined by the shape of the nonlinear capacitor 

 characteristic. However z appears only in (25) where it influences the 

 values of the matching susceptances so that it does not affect the con- 

 ductance or gain. While we can be certain that y will have values be- 

 tween and 1, limitations on the value of x will depend on the particular 

 device used. We will therefore assume that x may have any value. 



EFFECT OF NONLINEAR CAPACITOR 



We may now examine, in a general way, the manner in A\hich the non- 

 linear capacitor influences the behavior of the 4-pole. Consider first the 

 case where the nonlinear capacitor is absent. It is well known, and can 

 be seen in the above equations by letting Go = Gi = 0, that the non- 

 inverting and inverting cases are alike, that the 4-pole can always be 

 matched and that the gain is the same in both directions and can never 

 be greater than unity. In addition, the matching susceptances are zero 

 and the gain is independent of frequency so that there is no limitation 

 to the bandwidth. When the nonlinear capacitor is added, all but one of 

 these conditions are changed. Equations (8) and (9) show that the non- 



