582 BELL SYSTEM TECHNICAL JOURNAL 



These expressions show, as did the vector diagrams, that for optimum 

 stability the amplifier phase shift should be made approximately zero, 

 the crystal should have as large a value of Q (as low a decrement) as 

 possible, and the amplifier should have high gain. Linearity in the 

 amplifier is also desirable, to minimize the modulation effects described 

 by Llewellyn.' When present, these effects appear as variations in 

 ImI and d. 



One of the significant differences between the bridge oscillator and 

 other oscillator circuits is the fact that its frequency stability is roughly 

 proportional to \ix\. This relationship holds at least for amounts of 

 gain that can be dealt with conveniently. Although increased gain is 

 generally accompanied by larger variations in phase, the two are not 

 necessarily proportional. For example, if greater stability were re- 

 quired for some precision application than could be achieved with a 

 single-tube bridge oscillator, and if the constancy of the crystal itself 

 warranted further circuit stabilization, it could be obtained by adding 

 another stage. The phase fluctuations in the amplifier might possibly 

 be doubled, but the value of ] ju I would be multiplied by the amplifica- 

 tion of the added tube, giving an overall improvement. 



To illustrate the high order of stability provided by a bridge oscil- 

 lator, let us consider a model composed of a single-tube amplifier and 

 a bridge in which all the fixed resistances are made equal to that of the 

 crystal. We will assume the crystal to have a reasonably high ^ Q 

 of 100,000. The amplifier phase, let us say, is normally zero, but may 

 possibly vary ± 0.1 radian (±6 degrees), and the value of |/x|, 

 nominally 400, maychange ±10 per cent. From (18) and (19) we find 



A/ 

 /o 



_ (8)(0-l) - ± 2 17 X 10- 



r ^ (100,000) (360 + 8) " ^ ^-'^ ^ '^ 



and (when Q has its maximum value of 0.1 radian) 



A/ 

 /o 



(8) (0.1) (40) _ ^ 2 36 X 10-« 



^ (100,000) (360 -f 8)2 " ^ ^-^^ ^ ^^ • 



This example represents the degree of stabilization against circuit 

 fluctuations that can be obtained with a simple form of the oscillator 

 operating under poorly controlled conditions. By stabilizing the power 

 supply and other factors affecting | ^ | and d, and by increasing the gain, 

 the frequency variations arising in the driving circuit may be made 

 negligible compared to the variations found at present in the properties 

 even of the best mounted crystals. 



* For crystals in vacuum, values of Q as great as 300,000 have been obtained. 



