PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 



175 



tween plate and grid may be of any type and oscillations are maintained 

 when the total gain through the circuit is unity (gain of tubes = attenuation 

 through circuit) and the phase relation between the induced plate voltage 

 (nVg) and the grid voltage (Vg) is 180° (the phase shift is zero when fi is 

 considered negative). The expression jujS = 1 defines these requirements. 

 Llewellyn applies this method to oscillator circuits in general and Koga^^ 

 uses it to study the crystal oscillator in particular. 



The equations are developed on the assumption that the grid-voltage vs. 

 plate-current characteristic of the tube is linear. The fundamental equa- 

 tion of fjLjS is given by the ratio of the voltage developed across the grid 



Fig. 12.12 — Circuit diagrams of crystal oscillators with crystal connected from grid 

 to cathode (A) and grid to plate (B) 



circuit by the fictitious driving voltage nVg to the voltage Vg. For the 

 general circuit, Fig. 12.11, it is 



I2 Z2 — M^l ^2 



^^ Vg R^Z, + Zy{Z2 + Zz) 



where 



Za = Zi-\r Z2-\- Zz 

 It is more convenient to write this in the reciprocal form 



1 ^ RpZ, + Zi(Z2 + Zz) ^ ^ 

 n^ —fiZiZz 



(12.20) 



(12.21) 



In applying this to the crystal oscillator, the additional assumptions made 

 are that the grid current is negligible and the resistance in the plate im- 

 pedance Zi is zero. 



12.21 Crystal Grid to Cathode 



With the assumptions made above and the crystal connected from grid 

 to cathode of the tube according to Fig. 12. 12 A, the impedances are 



Zl = jXl Zi = Reg + jXcg 



j^z 



