426 BELL SYSTEM TECHNICAL JOURNAL 



To evaluate the non-linear forces, consider a parallel plate, air 

 condenser of area. A, and normal separation, xq, one plate of which is 

 fixed and the other of which is free to move under the action of linear 

 restoring forces. Let the movable plate be displaced a distance, x, in 

 a direction to increase the separation, and let a charge, q, be put on 

 the plates. Then it can easily be shown that the static forces tending 

 to oppose the displacements are 



= Sx + Kq\ (2) 



e = I + 2Kxq, (3) 



where 5 is the stiffness of the constraints on the plate; C is the capaci- 

 tance, in electrostatic units, when x is zero; and 



i^ = ^ (4) 



is a quantity which will be referred to as the constant of non-linearity. 



The first terms of (2) and (3) represent components of the forces 

 which were represented above by the mechanical and electrical 

 impedances respectively. Hence only the last terms need be used in 

 expressing the electromechanical coupling. 



We shall assume that there is connected in series with the condenser 

 and its associated electric impedance, a generator of negligible internal 

 impedance, which provides an alternating electromotive force, Cg, of 

 amplitude. Eg, and frequency, oig, in radians per second. The phase of 

 this generator will arbitrarily be taken as zero. 



For the first part of the analysis, we shall assume that there is an 

 alternating force, /„, exerted on the plate by a "mechanical generator," 

 which has an amplitude, Fm, frequency, a)„, and phase, \pm. We shall 

 investigate the impedance offered to this force in the resulting condition 

 of forced oscillation. In the second part, the mechanical generator 

 will be omitted, and the free oscillations investigated. It is first 

 necessary, however, to determine what frequencies need be considered. 



Possible Frequencies 



With the system just described there will be developed oscillations, 

 the frequencies of which constitute an infinite series. It will therefore 

 be necessary to introduce limiting assumptions. First let us consider 

 what frequencies may be present in the system. In doing this it must 

 be recognized that the conventional use of complex quantities is not 

 justified when the system is non-linear. This difficulty is avoided 

 and the advantages of the complex exponential notation are retained 



