THE BRIDGE STABILIZED OSCILLATOR 



579 



The effect of the bridge upon harmonics of the oscillation frequency 

 is of interest. Harmonics, being far from the resonant frequency of the 

 crystal, are passed through the bridge with little attenuation but with a 

 phase reversal approximating 180 degrees, as illustrated by the dotted 

 locus in Fig. 2. Thus if the amplifier were designed to cover a band 

 broad enough to include one or more harmonics and if care were taken 

 to avoid singing at some unwanted frequency, a considerable amount 

 of negative feedback could be applied to the suppression of the har- 

 monics in question. 



Circuit Analysis 



In the following section, expressions are derived for the frequency 

 of oscillation in terms of the gain and phase shift of the amplifier, the 

 Q of the crystal, and values of the bridge resistances. It is assumed 

 that the latter are constant and non-reactive, and therefore, as ex- 

 plained previously, that all sources of frequency fluctuations apart from 

 changes in the crystal itself appear as variations in | ;ti | or ^. Because 

 the bridge oscillator does not rely upon non-linearity in the ordinary 

 sense to limit its amplitude, the analysis can be based reasonably on 

 simple linear theory. 



In the near vicinity of series resonance the crystal may be repre- 

 sented accurately by a resistance Ra, inductance L and capacitance C, 

 connected in series. The reactive component of the crystal's im- 

 pedance is accordingly 



Solving for the frequency, 





'LC - 1 



(1) 



2L^ 

 1 



IL ^ LC 



\X, 

 LC[ 2 



C 



^Lc[ 



1 + 



L 



X, 



+ \1 + 



m- 



C 1/X_4 



L~^2\ 2 



C 

 L 



C\2 



_1 1/X_4 jC 



2' 4\ 2 \L 



+ 



(2) 



Near series resonance, (X4/2) ^{CIL) < < 1. We therefore disregard 

 powers higher than the first in the series expansion above and obtain 

 the close approximation. 



