GRID CURRENT MODULATION 111 



in which the coefficients ik involve the characteristic constants of the 

 tube, together with the applied potential amplitudes: 



io= ao + a2(P' + (2')/2 + •••, 



i^ = a,P + 3a3P(P2 + 2(22)/4 + • • •, 



i, = ayQ + 3a3(2((2^ + 2P2)/4 + . • . , 



«2« = ^2(272 + • • •, 



i+ = i- = aiPQ + • • • , 



The new frequencies produced are made up of sums and differences of 

 integral multiples of the two original frequencies, and an inspection of 

 eq. 3 shows that the frequency of any component may be put in the 

 form 



\mp ± nq\/2T m, n = 0, \, 2.- ■ • 



It is convenient to designate the sum of the two numbers m and n as 

 the order of a wave component, so that the frequencies 2p/2ir, 2g/27r, 

 and (p ± q)l2T are products of the second order. The last of these 

 serves as the basis for the operation of all present ^ carrier systems, and 

 the r61e of any modulator is therefore to produce one or both of these 

 components which are known as side frequencies, or as sidebands when 

 the signal wave is made up of a band of frequencies. Now by repeating 

 the modulating process, but this time with the frequencies {p + g)/2x 

 or {p — q)/2Tr, or both, together with the component of frequency 

 p/2Tr, designated as the carrier wave, it is well known that one of the 

 resultant second order products has the frequency of the original signal 

 q/2Tr. This second or receiving modulator, sometimes designated as a 

 demodulator or detector, is separated from the first one by a trans- 

 mitting medium and frequency-selective apparatus so that only the 

 desired components may be transmitted and received. The im- 

 pedance-frequency characteristics of these elements with which the 

 modulators are associated are of prime importance in determining the 

 modulation, as is best brought out by a discussion of some approximate 

 mathematical analyses which follow. 



Modulator Circuit, Small Alternating Potentials 



We shall consider the current-voltage characteristic of a vacuum 



tube to be given, to sufficient accuracy for our purposes, by the first 



2 Higher order products, such as {2p ± g)l2ir have been equally well employed 

 but we shall confine our attention here to the usual second order system. 



