EQUIVALENT MODULATOR CIRCUITS 37 



Thus an expansion for the Hnear rectifier includes only those coefficients 

 for which n is odd, whereas one for a modulator exhibiting odd sym- 

 metry in its current-voltage relation, such as thyrite, includes only 

 coefficients for which n is even. 



The choice between (5a) and {Sh) in any given case is usually a 

 matter of convenience.^ This will be made clear by the forthcoming 

 examples. In every case use will be made of Ohm's law in one of 

 the two forms 



V = Ri 

 or 



i = Gv. 



For simplicity, we select the relation which leads to the smallest 

 number of terms in the expansion. Thus if, from the form of the 

 terminating impedance, we know that i involves only a small number 

 of significant frequency components, whereas the voltage involves a 

 large number, (5a) will be used. If the potential, v, across the modu- 

 lating element is known to be the simpler, {5b) will be used. 



In the practical application of modulators to carrier systems the 

 impedance characteristics of the connected selective circuits for taking 

 out the desired sideband energy provide, to a good approximation, 

 just such simplification. Thus a filter is substantially resistive in its 

 pass band and the suppression regions may be designed to have either 

 a very high or a very low impedance. If very high, no currents flow 

 in these frequency regions and (5a) applies ; if very low no potentials 

 appear across it in these frequency regions so that (5b) applies. 



II. Single Sideband — High Impedance Outside Band 

 We will first consider a single sideband modulator involving any 

 variable resistance which can be expressed in the form (5a). The 

 terminating resistance is Rq to signal and Ri+ to the upper second 

 order sideband. Because of the high terminating impedance which 

 we assume to all other products, all current components other than 

 signal (Q) and sideband (/i+) are negligibly small. 

 The total current flowing in the circuit is then 



i = Q cos qt -f /i+ cos (p + q)t. (6) 



The potential across the non-linear element (v = Ri) is obtained 

 from (5a) and (6) as 



fo + '^2rn. cos npt 

 1 



[_Q cos qt + /i+ cos (p -f g)0. (7) 



^ Except for those cases in which the occurrence of an infinity in any one of 

 these two quantities prohibits its use. 



