Some Analyses of Wave Shapes Used in Harmonic Producers 



By F. R. STANSEL 



Analyses by Fourier's Series have been made of waves consisting of 

 sinusoidal, rectangular and trapezoidal pulses and also waves of the type 

 found in multivibrator circuits. The method of increasing harmonic 

 content by modulating a wave with a submultiple is treated mathe- 

 matically. 



'* I '^HE heterodyne method of frequency comparison requires, except in 

 -^ the case of the comparison of nearly identical frequencies, the genera- 

 tion of harmonics of either the unknown, or of the standard frequency or 

 of both. These harmonics may be generated directly in the modulator 

 which produces the difference frequency, or "beat note", or may be gener- 

 ated in an entirely separate circuit before the frequency is applied to the 

 modulator. An example of the latter is the multivibrator circuit often used 

 in connection with a frequency standard to produce a series of harmonics of 

 this standard frequency. 



The design of harmonic generators for frequency measuring equipment 

 presents a different problem from the design of equipment for producing a 

 single harmonic such as doubler or tripler stage in a radio transmitter. In 

 the latter case the amplitude of the one harmonic and the efficiency are of 

 primary importance. In frequency measuring equipment, although a large 

 amplitude of each harmonic is desirable, it is of greater importance that each 

 harmonic within the range to be used, which may be up to the 100th or 150th 

 harmonic or even higher, be present and that the amplitude of nearby 

 harmonics be of the same order of magnitude. Unless the latter conditions 

 are met, there is a danger that the beats obtained with a weak harmonic 

 will either be entirely overlooked or mistaken for a higher order modulation 

 product. 



The generation of harmonics is usually accomplished by the distortion of 

 the wave shape in some nonlinear circuit element such as a vacuum tube. 

 One such harmonic generator consists of a vacuum tube biased so that there 

 is no output for a portion of the cycle. The plate current of such a tube may 

 be approximated by a sine wave shaped pulse such as shown in Fig. 1. Any 

 such periodic wave can be resolved into its harmonic components^ and in 

 the case of this wave the amplitude of the nth harmonic is found to be 



^ This and the subsequent analyses were made by application of Fourier's Series. See 

 I. S. Sokolnikoff and E, S. Sokolnikoff, "Higher Mathematics for Engineers and Physi- 

 cists," Chapter VI. 



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