326 BELL SYSTEM TECHNICAL JOURNAL 



Linear Modulator Theory 



The analytical studies that have been of most benefit in the develop- 

 ment of copper oxide modulators have made use of a variable resistance 

 characteristic controlled by the carrier. This assumption has made it 

 possible to investigate modulator performance ^ for a wide variety of 

 characteristics under a great many operating conditions. Copper 

 oxide modulator performance in particular cases as well as the effects 

 of the circuit elements on this performance can readily be inferred 

 from the data at hand about these idealized modulators. 



In limited space it is not possible to discuss the varieties of re- 

 sistance modulators that have been analyzed. However, certain view- 

 points will be discussed that have been very useful not only for obtain- 

 ing solutions for some of the hypothetical cases, but also in supple- 

 menting laboratory experiments on actual modulators. 



All of these analytical studies have assumed a signal sufficiently 

 small compared to the carrier, that it can be varied in magnitude 

 without noticeable changes in the signal impedance or in the linearity 

 between input and output signal amplitudes. This is in agreement 

 with design procedure, as the circuit impedances and losses are deter- 

 mined on such a linear basis. 



Superposition Principle 



All of the modulator circuits with which we have been dealing, 

 though composed of non-linear elements, have been resolved into the 

 equivalent of linear systems by virtue of using a large carrier and small 

 signal amplitude. We may simultaneously apply any number of 

 signal frequencies, but all have negligible efTect on the periodic changing 

 of the non-linear element resistance by the carrier. These frequencies 

 may be modulation product voltages, some applied at the output 

 terminals and some at the input terminals, but in all cases, even though 

 frequencies may coincide, it can be shown that the principle of super- 

 position will hold without interaction between the applied forces and 

 the responses. This permits a great simplification in the mathe- 

 matical approach to modulator analysis, because the modulation 

 product or signal voltages can be applied one at a time and the current 

 responses summed. The voltage-current ratios at each frequency can 

 then be replaced by equivalent impedances. 



Any non-linear resistance like copper oxide will have a current- 

 voltage characteristic that can be expressed as accurately as de- 



^ A physical picture of modulator performance in terms of linear networks is 

 developed in a paper published in the January 1939 Bell System Technical Journal, 

 "Equivalent Modulator Circuits" by E. Peterson and L. W. Hussey. 



