THERMIONIC VACUUM TUBES 71 



Let Fig. 37, C, represent a high frequency wave modulated by a 

 sinusoidal low frequency. The wave C can be represented by 



y = a(b + cos qt) cos pt, (10) 



in which £- is the high (carrier) frequency and -~ the low (signal) fre- 

 2ir 2ir 



quency. Equation 8 can be rewritten in the form 



y = ab cos pt + -^[cos (p-q)t + cos (p + q)t], (11) 



which brings out the fact that the modulated wave C contains, in 

 general, three distinct frequencies — the carrier frequency £-, a differ- 



2lT 



ence frequency , and a summation frequency ¥— — -. These latter 



2-k 2ir 



frequencies represent the so-called "side bands" of the modulated wave. 

 Two special cases of the wave represented by Equations 10 and 11 

 are represented graphically at D and E of Fig. 37 and correspond to 

 b = l and b = o respectively. When b = l it is evident that the ampli- 

 tude of each side band is half the amplitude of the carrier frequency; 

 such a wave is said to be "completely modulated"; when b = the 



carrier frequency ~- is absent altogether. 31 



2tt 



35. Means for Producing Modulation. Perhaps the simplest case 

 of modulation is that illustrated by continuous-wave radio telegraphy, 

 in which the intermittent radiation of a uniform wave is accomplished 

 by means of a telegraph key. In most cases, however, modulation 

 requires a gradual change in the amplitude of the high frequency 

 wave. For effecting this the vacuum tube possesses two properties 

 which make it particularly useful — (a) the E g ,I p characteristic is 

 very nearly parabolic (Sec. 8) ; (b) the current in the plate circuit 

 is a function of the grid potential (Fig. 10). 



Circuits, by means of which modulation may be effected by each of 

 these properties, are described in the following paragraphs. 



36. Modulation by Curved Characteristic. Considering the circuit 

 of the type illustrated in Fig. 38, let it be assumed that a voltage 



e = A cos pt-\-B cos qt 



is applied to the input of the tube. The result is shown graphically 

 in Fig. 39. When this value of e is substituted in Equation 4 we 



11 For a more complete discussion of modulation and the nature of the side bands, 

 see R. V. L. Hartley, Procd. Inst, of Radio Engrs., Feb., 1923, or Bell System Technical 

 Journal, Apr., 1923. 



