COPPER OXIDE MODULATORS IN CARRIER SYSTEMS 327 



sired by 



i = aie + a^e^ + • • • + «„«". (1) 



If a carrier and signal voltage are applied to this non-linear element, 

 each term beyond the first will independently produce currents of 

 new frequencies composed of the intermodulation products of these 

 two voltages. If in turn the external impedances at these new fre- 

 quencies are not zero, new voltages will appear across the non-linear 

 element to produce still more new frequencies. In this case the 

 simplest consideration is to minimize the number of voltages by 

 presenting zero impedance to the modulation products. If the carrier 

 voltage is C cos ct and the input signal voltage 5 cos st, the current 

 flow in the wth term is 



i = a„{C cos ct -\- S cos st)". (2) 



In the binomial expansion of this expression it is obvious that 

 linear response to the signal and freedom from distortion result when 

 the ratio of carrier to signal is made sufficiently large so that only 

 the first two terms are important. 



i « a„[(C cos c/)" + n{C cos ct)"-^-S cos 5/]. (3) 



The first term in equation (3) is the current flow at d-c. and har- 

 monics of the carrier; it has no effect on the input signal and output 

 signal except in so far as impedance termination presented across the 

 non-linear element at the carrier harmonic frequencies may alter the 

 carrier voltage harmonic content. The signal input current and 

 the signal output current, as well as the unwanted modulation products 

 of the signal, result from the second term. The even values of n 

 produce the even order sidebands, second order being the output 

 signal, while the odd values of n produce the input signal current and 

 the odd order sidebands. These currents can be evaluated from 



2 , ^-- 2 



COS" ^ = -:r;r + L -^r;:iT ^^^ ^^> 

 in which iiC"_^ is equal to the combination of n things taken — - — at 



2 



r n — m . , ,. , r^ — ^ 'j. 1 



a time for — - — mtegral and is equal to zero tor — r — non-integral. 



