112 BELL SYSTEM TECHNICAL JOURNAL 



three terms of eq. 1, and shall suppose two generated potentials, of 

 frequency pjlir and qjlir respectively, to be applied to a tube circuit 

 which includes a series impedance. This impedance Zk may be a 

 function of frequency as indicated by the subscript k, which refers to 

 the particular frequency at which the impedance is effective. The 

 variable part of eq. 1^ — the change in current produced by application 

 of the alternating potentials — is clearly 



/ = aiv + a-iV^. (5) 



The potential drop across the tube is that impressed minus the Zi 



drop or 



V =i:{Ek- ZuJk) (6) 



the summation extending over all current components. As a first 

 approximation to the fundamental currents we may neglect the non- 

 linear term {a2V^) in eq. 5 to obtain 



Jp = ai£p/(l + axZp), 

 Jq = aiEJ{\ + aiZg). 



Using these solutions we can obtain a second approximation ^ taking 

 into account the non-linear term which we neglected for the first 

 approximation. Thus 



-p ■p 



I -\- aiZp 1 -f- aiZg 



+ 



^ \ 1 -H aiZp 1 + aiZgJ 



(8) 



in which the subscript 2 indicates the second approximation. By 

 squaring the second member of eq. 8 it is observed that the second 

 approximation includes direct current, second harmonics of the two 

 impressed frequencies p/lir and q/lir, and the second order sidebands. 

 For these last we obtain the expression 



-^^ " (1 -f arZ^) (1 + a,Zp)il -f «iZ,) ' 



The sideband potential across the variable element is clearly Z^J+ or 

 Z-J- according to the particular sideband in which we are interested. 

 Thus 



V - -^ T T ^- = -^j J ^- (10) 



where 1/ai is equivalent to i^o, the plate resistance of the tube. 



3 Carson, "A Theoretical Study of the Tlirce Kleinent \acuiiin Tube," Proc. L R. 

 E., 1919. 



