434 BELL SYSTEM TECHNICAL JOURNAL 



above which the new oscillations are possible. (It is found to agree 

 with that obtained by the negative resistance method.) Moreover, 

 this value is that which is just necessary to maintain an electric cur- 

 rent, of the generator frequency, in the absence of the new frequencies, 

 with an amplitude equal to the constant amplitude that exists in the 

 presence of the new frequencies. For values of Eg large compared with 

 the threshold value, the amplitudes of the new frequencies increase 

 nearly as the square root of the amplitude of the generator voltage. 



In the special case of resonance at both co„ and wd, Z„ and Zd tend 

 to be small and so from (24) the threshold voltage is correspondingly 

 small. This therefore is a particularly favorable condition for the 

 production of the oscillations. 



The case where cos (^m + ^g) is negative occurs when all of the 

 three impedances are predominantly reactive, the reactances being all 

 of the same sign. The first term of (23) is then positive and Vm will 

 be real if the second term is positive, as it will be if 



Eg > Zglg\sm {ip„ + ^„) I . (30) 



For this case, then, the threshold amplitude of the generator voltage 

 may be much less than that required to maintain the current at the 

 constant amplitude, Ig, in the absence of the new frequencies. 



In the extreme case where there is no dissipation and the phase 

 angles of the impedances are all ± 7r/2, the threshold voltage reduces 

 to zero and so sustained oscillations are possible in the absence of any 

 generator. (23) and (24) then reduce to forms symmetrical with (22). 

 This means that for such a system the frequencies would be determined 

 by the constants of the system and the amount of energy present, 

 since this would limit the possible amplitudes. 



There is some question as to the sign to be given to the inner radical 

 in (23). When cos {(pm + ^Pg) is positive the plus sign must be used. 

 When it is negative the plus sign must be used if Eg is greater than Zglg. 

 If Eg is between this and the threshold given by (30), either sign gives a 

 real value for the amplitude. When the sign is negative the amplitude 

 decreases with increasing voltage, which appears to be an unstable 

 condition. 



Regarding the phases, the condition represented by (27) is imposed 

 because the energy flow must be from the generator to the circuit. 

 Only the sum of the phases of the new oscillations is determined. 

 Their individual values depend on the starting conditions, just as does 

 the phase of a pendulum clock. 



One more result may be of interest. This is the relative rates at 

 which energy is dissipated at the two new frequencies. If Pm and Pd 



