118 BELL SYSTEM TECHNICAL JOURNAL 



frequencies as high as possible, and match the tube impedance at 

 sideband frequencies in order to develop maximum sideband power in 

 the load resistance. It will be obser\^ed that with these conditions 

 satisfied, the plate circuit of the tube acts substantially as an amplifier 

 of the sideband produced in the grid circuit, since none is developed in 

 the plate circuit. 



Reaction of Sideband Flow on Tube Impedance 

 In any non-linear system such as the grid circuit or the plate circuit 

 of a tube, the modulation products resulting from the lack of linearity 

 have amplitudes which depend upon one or more of the impressed 

 fundamentals, and react upon the fundamental amplitudes. It follows 

 that the amplitude of any modulation product depends in general upon 

 the amplitudes of all other modulation products, and that the im- 

 pedance offered to the fiow of fundamental depends upon the reaction 

 of the modulation products. Stated otherwise, the amplitude of any 

 one current component depends upon all other components. 



This may be demonstrated quantitatively by higher approximations 

 than the two which we have already obtained, in which expressions for 

 the currents are found to contain terms proportional to the sideband 

 voltage across the tube; in fact, if we went to the labor of including a 

 number of distorting components, terms in the fundamental current 

 equation due to their reaction would result. The effects found with a 

 single sideband are simply typical. If, for example, we put (7) and (9) 

 in (8), we get 



J j,= ax{Ep-Zi,J y)- a2{Eq - Z^Jq) 



a^EpEgZ^ 



(1 -i-aiZp){l +ai2,)(l +ai2+)' 



and a similar expression for Jq, as second approximations to the 

 fundamentals. These furnish us with a pair of simultaneous cubics in 

 J p and J (J. When we assume the reaction of the sideband flow on J ^ 

 to be small so that eq. 7 remains valid, the above equation becomes 

 linear in Jp, 



This shows that the impedance in the fundamental path has been 

 increased due to non-linearity by the amount of the last term which 

 may be denoted by AZp where 



^ " a,' 1 +axZ+ "• 



