SYSTEMS wrni NON-LINEAR REACTANCE 425 



charge and mechanical (Hsplacement then give rise to currents of the 

 combination, or sideband frequencies. Among the properties of the 

 system which were studied was the reaction of the jilate on the me- 

 chanical "generator." This was expressed as a mechanical impedance, 

 i.e., the complex ratio of the alternating force to the alternating 

 velocity. 



The expression for this mechanical impedance was found to include 

 a negative resistance, which under certain conditions became equal to 

 the positive resistance representing the remainder of the system. It 

 was evident, therefore, that, under these conditions, oscillations of the 

 frequencies involved could persist in the absence of any external 

 driving force on the plate. The existence of such oscillations was first 

 verified experimentally by Mr. E. Peterson. This and a quantitative 

 experimental study of the phenomenon are described in an accom- 

 l)anying paper.^ Oscillations of the same general type, associated 

 with iron core coils, had been predicted much earlier by the writer 

 and discovered independently by Mr. E. T. Burton. ^ 



However, what happened once the threshold condition was passed, 

 was not apparent from this analysis. The answer to this question 

 was found by assuming the existence of the oscillations, computing 

 their values, and determining under what conditions the values are 

 real. Both methods will be employed in what follows. 



Representation of the System 



In the analysis it will be assumed that, except for the non-linearity 

 associated with the electromechanical coupling, the law of superposition 

 holds throughout. This means that all parts of the system other than 

 the coupling may be represented by linear impedances, of the form 



Z = i? + ?X = Ze^^. (1) 



"Linear," as here used, means that the impedance is independent of 

 the magnitudes of the oscillations. 



If then the plate has an alternating velocity of magnitude F,„ and 

 phase dm, we represent it by Vme'""". The resultant of all the linear 

 restoring forces may be represented by a force Z,„Fme'^*"""^*'"^- All of 

 the quantities involved will, in general, be functions of the frequency. 

 Similarly a current leC'^' will be accompanied by a counter electro- 

 motive force Z,/ee*('^^+*'\ where Ze is the impedance of the connected 

 electric circuit in series with that of the condenser with its movable 

 plate at rest in the position of zero displacement. 



^Hussey, L. W. and Wrathall, L. R.; "Oscillations in an Electromechanical 

 System" in this issue of the Bell Sys. Tech. Jour. 



* Peterson, E.; Bell Laboratories Record, P'eb., 1929, p. 231. 



