THE BIASED IDEAL RECTIFIER 141 



frequency of the network, we can calculate the output wave, which is the 

 one having most practical importance. The frequency selectivity may in 

 many cases be an inherent part of the rectifying or limiting action so that 

 discrete separation of the non-linear and linear features may not actually 

 be possible, but even then independent treatment of the two processes 

 often yields valuable information. 



The formulation of the analytical problem is very simple. The standard 

 theory of Fourier series may be used to obtain expressions for the amplitudes 

 of the harmonics in the rectifier output in the case of a single applied fre- 

 quency, or for the amplitudes of combination tones in the output when two 

 or more frequencies are applied. These expressions are definite integrals 

 involving nothing more compUcated than trigonometric functions and the 

 functions defining the conducting law of the rectifier. If we were content 

 to make calculations from these integrals directly by numerical or mechanical 

 methods, the complete solutions could readily be written down for a variety 

 of cases covering most communication needs, and straightforward though 

 often laborious computations could then be based on these to accumulate 

 eventually a suflficient volume of data to make further calculations un- 

 necessary. 



Such a procedure however falls short of being satisfactory to those who 

 would like to know more about the functions defined by these integrals 

 without making extensive numerical calculations. A question of consider- 

 able interest is that of determining under what conditions the integrals may 

 be evaluated in terms of tabulated functions or in terms of any other func- 

 tions about which something is already known. Information of this sort 

 would at least save numerical computing and could be a valuable aid in 

 studying the more general aspects of the communication system of which 

 the rectifier may be only one part. It is the purpose of this paper to present 

 some of these relationships that have been worked out over a considerable 

 period of time. These results have been found useful in a variety of prob- 

 lems, such as distortion and cross-modulation in overloaded ampUfiers, 

 the performance of modulators and detectors, and efifects of saturation in 

 magnetic materials. It is hoped that their publication will not only make 

 them available to more people, but also stimulate further investigations of 

 the functions encountered in biased rectifier problems. 



The general forms of the integrals defining the amplitudes of harmonics 

 and side frequencies when one or two frequencies are applied to a biased 

 rectifier are written down in Section I. These results are based on the 

 standard theory of Fourier series in one or more variables. Some general 

 relationships between positive and negative bias, and between limiters and 

 biased rectifiers are also set down for further reference. Some discussion is 

 given of the modifications necessary when reactive elements are used in the 

 circuit. 



