1404 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



that of the converter was several decibels greater. In one instance the 

 loss in a modulator used to convert a 70 mc signal to one at 11,130 mc 

 was found to be only 2.3 db but when the direction of transmission 

 through it was reversed and it was used as a converter, the loss was 7.8 

 decibels. 



h Similar effects were observed several years ago in conversion trans- 

 ducers using welded contact germanium rectifiers. In these early experi- 

 ments substantial converter gain and negative conductance at the inter- 

 mediate frequency terminals were also observed. These results were 

 accounted for by assuming the presence of a nonlinear capacitance at the 

 point contact in parallel with the nonlinear resistance. At that time at- 

 tention was devoted mainly to the behavior of converters where noise 

 is a vital factor. It was found that although the conversion loss could 

 be reduced, the noise temperature increased and no improvement in 

 noise figure resulted. However, the noise temperature requirements in 

 modulators are much less severe and the nonlinear capacitance effect is 

 useful and can substantially improve the performance. 



THEORY 



The mathematical analysis given here was undertaken in order to 

 clarify the effect of the nonlinear capacitance in the frequency conversion 

 process and to obtain an estimate of the usefulness of modulators ex- 

 hibiting gain. The analysis is restricted to the simplest case in which 

 signal voltages are allowed to develop across the nonlinear elements at 

 the input and output frequencies only. This is not an unrealistic restric- 

 tion since the conversion transducers used in microwave relay systems 

 have filters associated with them which suppress the modulation products 

 outside the signal band. The final results will be given only for those con- 

 ditions which permit a conjugate match at the input and output of the 

 transducer. 



The procedure used to obtain expressions for the admittance and 

 gain of conversion transducers utilizing a nonlinear element made up 

 of a nonlinear resistance and a nonlinear capacitance in parallel follows 

 the commonly used method of treating the nonlinear elements as local 

 oscillator controlled linear time varying elements. The current through 

 the nonlinear resistor is a function of the applied voltage. The derivative 

 of this function is the conductance as a function of the applied voltage. 

 Thus when the local oscillator is applied, the conductance varies at the 

 local oscillator frequency and the conductance as a function of time 

 may be obtained. This is periodic and may be expressed as a Fourier 

 series. The conductance is real and if we make the usual assumption that 



