436 

 where 



BELL SYSTEM TECHNICAL JOURNAL 



Z I 



COS a = -|r-^ sin (6 + <py)\ 



+ (?,/ = a' + TT ± 2 + (5 - ^>^ ' 



(42) 

 (43) 



e„, - a' ± ^ + (5 - <ps 



(44) 



where the sign of 7r/2 is again to be chosen so as to satisfy (27). 



Corresponding to (21) we have (32). If the mechanical motion 

 involves any dissipation, the mechanical resistance, Z„, cos (^,„, must be 

 positive, and since Z,„ is positive by definition, cos (p.,,, must be positive. 

 This means that (32) can be satisfied only if the denominator of {3S) is 

 positive. Hence oscillations can occur only if 



Z,i03d cos (fd 

 Zs<Jis cos (Ps 



(45^ 



This relation can hold when Z,/, the impedance at the difference fre- 

 quency, is infinite, only if <ps is ± 7r/2, that is, if there is no dissipation 

 at the sum frequency. 



To investigate the relative rates of dissipation at the sum and differ- 

 ence frequencies, we find the ratio of the powers Ps and Pa, associated 

 with them. 



Ps 



Pd 



Zd cos (Ps Ws 



Zs COS ifd 03d 



(46) 



Thus the ratio is always less than the ratio of the frequencies and ap- 

 proaches it only as the limiting condition for oscillations is approached. 

 A discussion of all possible values of impedance and phase angle at 

 the two side frequencies would be too involved to go into here. The 

 special case of resonance at both frequencies is, however, of some 

 interest since a given current is then accompanied by a maximum of 

 dissipation. It also provides that co,„ coincides with the mechanical 

 resonance, where Z„( is much smaller than for nearby frequencies. 

 Since Zm enters into the expression for the threshold force, this condi- 

 tion is particularly favorable for the occurrence of oscillations. When 

 we make ipd and (ps zero we see from (45) that the impedances, now pure 

 resistances, must be such that 



ZdCOd 



< 1. 



(■i7) 



