FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1407 



Going through the same steps for the hivertmg conversion transducer 

 where /2 = /o — /i we obtain 



/i = [Go + icoiCo]Fi + [G, + iciCilFo* 

 h* = [G, - jwoCiW, + [Go - MC0W2* 

 Equations (8) and (9) are in the form 



/i = FnFi + F12F2 



I2 = ^21'''l ~l~ F22F2 



(9) 



(10) 



A heterodyne conversion transducer may thus be represented by a 

 linear 4-pole, and the admittance and gain of the 4-pole may be expressed 

 in terms of the admittance coefficients. In Fig. 1 we see that the admit- 

 tance of the 4-pole yi at the terminals 1-2 is eciual to Ii/Vi and the 

 admittance 2/2 connected to terminals 3-4 is — /2/l^ . Putting these in 

 (10) we find 



YuY,, (11) 



yi 



Yn 



F22 + Vi 



Similarly the admittance of the 4-pole 2/2' at the terminals 3-4 is 72/ F2 

 and the admittance yi connected to terminals 1-2 is —Ii/Vi . Putting 

 these in (10) gives 



2/2 



F 



22 



i 12^ 21 

 Yn + 2/1 



(12) 



To compute the gain of the 4-pole when 7/1 contains the generator and 

 y-i the load, it is convenient to assume a current source connected across 

 ?/i . If the current from this source is Jo we have Ii = lo — yiVi . I2 

 equals —y2V2 as before. Putting these in (10) gives 



-'0 _ T^ 



(Fn + 2/l)(F22 + ?/2) 



F 



(13) 



21 



If we let yi = Qi -{- jbi and ?/2 = ^2 + i&2 , the power in the load isF2 ^2 

 and the power available from the generator is /oV4^i . Therefore the 

 transducer gain ri2 defined as the ratio of the power in y2 to that avail- 

 able from 2/1 becomes 



F2' 

 Tu =4gig2j-^ = "igig 



1 n" 



21 



(14) 



F12F21 - (Fn + 2/1) (F22 + 2/2) 



When ?/2 contains the generator and yi the load, we may proceed in the 

 same way (letting 7o flow in terminal 4) and obtain 



2 



r2i = 4^-1^2 



F 



12 



F12F21 - (Fu + ?/0(F22 + 2/2) 



(15) 



