SYSTEMS WITH NON-LINEAR REACTANCE 431 



If the third term of (14) is not neglected we must replace Zm in (19), 

 by the first and third terms of (17), that is. by the impedance of the 

 mechanical fre(|uency mesh, as modified by its coupling with the 

 difTerence frequency mesh. 



Similarly, when the measuring generator is of the difference fre- 

 quency, we get 



JC2T 2 _ /? _ jX 



Z,' = (R, + iX,) + -^ % , '^- , (20) 



where Zm is to be replaced by the first and second terms of (17), if 

 the second term of (14) is not neglected. 



The active impedance at the difTerence frequency (20) contains a 

 negative resistance similar to that which appeared at the mechanical 

 frequency (17). In fact, if the passive impedance, Zj, at the sum 

 frequency is infinite, the expressions for the two active impedances 

 are symmetrical. The active impedance at the sum frequency contains 

 only positive resistances, except in so far as the resistance of the 

 mechanical mesh is made negative by its coupling with the difference 

 mesh. This serves to emphasize the fact that the presence of current 

 of the difference frequency is essential to the oscillations, while that of 

 current of the sum frequency tends to make their production more 

 difficult. 



Free Oscillations 



In the above considerations it was assumed that the amplitudes at 

 all of the new frequencies were small compared with that at the genera- 

 tor frequency. While this assumption permits us to compute the 

 threshold conditions for the starting of free oscillations, it is violated as 

 soon as the oscillations become appreciable. In order to find out what 

 happens once the threshold is passed it is necessary to solve the second 

 degree equations (13) to (16) w^hen Fm is made zero. The presentation 

 of this solution will be simplified by considering first the case where 

 the sum frequency is eliminated and then the effect of its presence on 

 the simpler solution. 



The elimination of the sum frequency is accomplished by making Z^ 

 infinite and Is zero. This makes the second terms of (13) and (14) 

 zero, and makes (15) indeterminate. We are left then with (13), (14), 

 as modified, and (16). The equations for the mechanical and difference 

 frequencies are now symmetrical. In order to solve these equations 

 we express the exponentials in terms of sines and cosines and equate the 

 real and imaginary parts separately. In the equations derived from 

 (14) and (16) we transpose the second term in each equation to the right 

 member. For each pair we divide the equation containing sines by 



