MAGNETIC GENERATION OF A GROUP OF HARMONICS 453 



The effect of this shift on the harmonics produced may be found by 

 straightforward means in which the ampHtude of any harmonic is 

 expressed in terms of the bias. Hence when the extraneous component 

 or components vary with time, the sidebands produced may be evalu- 

 ated when the bias is expressed by the appropriate time function. 



If the bias is held constant, the wave is found to include both odd and 

 even harmonics, the amplitudes of which are given by 



In ^ I(n) \cosnb/Hi\, 

 = I{n) |sin nhlH,\, 



(n odd) , 1 

 (n even),] 



(5) 



I(n) being the harmonic distribution in the absence of bias as given 

 by eq. (2). 



If the extraneous input component is sinusoicjal, we have 



b = Q sin (qt + ^). 



(6) 



Substituting this expression for b in the equation for the harmonic com- 

 ponents yields odd harmonics of the fundamental, and modulation 

 products with the angular frequencies mp ± Iq, which may be grouped 

 as side-frequencies about the odd harmonics. The amplitude of the 

 wth (odd) harmonic is 



In = Hn) 



nQ 



and the amplitude of the modulation product mp ± Iq is 



'm, ±i 



I(m) 



Ji 



H, 



,{m-^l odd). 



(7) 



(8) 



where Ji{x) is the Bessel function of order /. 



Considering the side-frequencies about the wth harmonic, the largest 

 and nearest of these are {n -\- \)p — q and {n — \)p -\- q, n being 

 odd. The ratio of the amplitudes of either side-frequency to the nth 

 harmonic is 



In 



±1. Tl 



In 



J,\{n ± \)QIH{\ 



JoinQIH,) 



(9) 



on the assumption that the harmonic distribution in the neighborhood 

 of n is uniform so that I{n ± 1) = I{n). If the arguments of the 

 Bessel functions are less than four-tenths, a good approximation to the 

 right member of eq. (9) is (« ± l)Q/2IIi. Hence with sufficiently 

 small values of interference, the sidebands produced are proportional 



