430 BELL SYSTEM TECHNICAL JOURNAL 



applit'd force, fm, actinjj on the plate, with the voltage of the electric 

 generator zero. The velocity of the p\aU' vibration is then determined 

 by its passive impedance alone. Let us assume for the time being 

 that the impedance, Z,, of the electric circuit at the sum frequency is 

 infinite, so that its effect on the active impedance of the plate dis- 

 appears. Let us make (pd equal to (pm, and gradually increase the 

 generator voltage. As // increases, the negative impedance increases, 

 the total impedance decreases and the velocity, Vm, increases. This 

 condition is analogous with the behavior of the input impedance of a 

 regeneratively connected amplifier when the plate current is progres- 

 sively increased from zero. At a threshold value of Ig, the net im- 

 pedance becomes zero and the velocity infinite. This means that a 

 finite velocity can exist for an infinitesimal driving force, that is, the 

 oscillations, once started, are self-sustaining, even in the absence of 

 any sustained driving force, /«, at the mechanical frequency. 



If we make the electric impedance, Zd, at the difference frequency 

 infinite, all the resistances are positive; so sustained oscillations cannot 

 occur, in a dissipative system, in the absence of current at the difference 

 frequency. If both side frequencies are present, so that Zg and Zd 

 are both finite, sustained oscillations are still possible provided the 

 impedance at the sum frequency is not too small compared wath that 

 at the difference frequency. The presence of current at the sum 

 frequency always increases the critical value of the current at the 

 generator frequency. 



We may also compute the active impedance of the electric circuit at 

 the side frequencies, on the same assumption as to the constancy of 

 the current of generator frequency as was made in deriving (17). 

 To do this, we remove the mechanical generator, making the right 

 member of (14) zero, and insert low measuring voltages of frequencies 

 coj and cod in the right members of (15) and (16), in turn. In each 

 case we compute the ratio of this voltage to the accompanying current. 

 If we think of each frequency as being the analog of a mesh in an 

 electric circuit, we note that the mesh corresponding to the mechanical 

 frequency is coupled to both of the side frequencies; but the latter are 

 not directly coupled to each other. If the mutual impedances, which 

 depend on /„, are small enough, we may, for a generator at the sum 

 frequency, neglect the third term of (14), which represents the effect 

 of the loosely coupled difference frequency mesh, compared with the 

 first. The active impedance at the sum frequency then becomes 



Z/ = (7?, + iX..) + -^-^^ • '"'" „ , " • (19) 



CO,, COmW., Am 



