THE BRIDGE STABILIZED OSCILLATOR 577 



could be used by merely exchanging its position in the bridge with 

 i?2 or Rz. 



The frequency control exerted by the crystal depends upon the fact 

 that the phase shift of the amplifier must be equal and opposite to that 

 through the bridge. In the notation of Black,^ applied to the circuit 

 of Fig. 1, 



E , ,,„ 

 ^=7 



= IMI 



The condition for oscillation is 



M/^ = 1 l_0, 



which implies that | /i|3 1 = 1 and d = — \]/. 



The vector diagrams of Fig. 2 illustrate the frequency-stabilizing 

 action of the bridge by showing the voltage relations therein for two 

 values of amplifier phase shift, 6. When d is zero, as in diagram A, 

 the unbalance vector e is in phase with the generated voltage E applied 

 to the bridge input, and thus all the vectors shown are parallel. They 

 are displaced vertically from each other merely to clarify the drawing. 

 The crystal is here constrained to operate at exact resonance. 



In diagram B, the amplifier is assumed to have changed its phase for 

 some reason by an amount far in excess of what would be anticipated 

 in practice, 6 hene having a value of + 45 degrees. The important point 

 to be observed is that the ratio of d to the resulting change in the phase 

 angle <^ of the crystal impedance Z4 is very large. That is, the crystal 

 is still operating close to resonance in spite of the exaggerated change in 

 the driving circuit. If the gain of the amplifier were greater, the action 

 of the thermally controlled resistance would keep the amplifier output 

 vector E practically the same in length, making the unbalance vector e 

 correspondingly shorter. The angle 4> would therefore have to be more 

 acute for the same value of 6, and it follows that with increased gain the 

 crystal is held closer to true resonance and the stability is improved. 



When 6 equals zero, changes in | /u | do not affect the crystal operating 

 phase, but for any other small value of d, gain variations cause slight 

 readjustments of the angles between vectors. The amplifier should 

 accordingly be designed for zero phase shift, and also, of course, should 

 have as much phase stability as possible. 



^ "Stabilized Feedback Amplifiers," H. S. Black, Bell System Technical Journal, 

 January 1934. 



