. The Bell System technical journal . pvg Fig. —Equivalent circuit of Pierce and Miller types of oscillators shown in Fig. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 175 tween plate and grid may be of any type and oscillations are maintainedwhen the total gain through the circuit is unity (gain of tubes = attenuationthrough 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 isconsidered negative). The expression jujS = 1 defines these applies this method to oscillator cir
. The Bell System technical journal . pvg Fig. —Equivalent circuit of Pierce and Miller types of oscillators shown in Fig. PIEZOELECTRIC CRYSTALS IN OSCILLATOR CIRCUITS 175 tween plate and grid may be of any type and oscillations are maintainedwhen the total gain through the circuit is unity (gain of tubes = attenuationthrough 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 isconsidered negative). The expression jujS = 1 defines these 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 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. —Circuit diagrams of crystal oscillators with crystal connected from gridto cathode (A) and grid to plate (B) circuit by the fictitious driving voltage nVg to the voltage Vg. For thegeneral circuit, Fig. , it is I2 Z2 — M^l ^2 ^^ Vg R^Z, + Zy{Z2 +Zz) where Za = Zi-\r Z2-\- ZzIt is more convenient to write this in the reciprocal form 1 ^ RpZ, + Zi(Z2 + Zz) ^ ^n^ —fiZiZz () () In applying this to the crystal oscillator, the additional assumptions madeare that the grid current is negligible and the resistance in the plate im-pedance Zi is zero. Crystal Grid to Cathode With the assumptions made above and the crystal connected from gridto cathode of the tube according to Fig. , the impedances are Zl = jXl Zi = Reg + jXcg j^z 176 BELL SYSTEM TECHNICAL JOURNAL where Reg is the effective resistance and Xcg the effective reactance of thecrystal, the grid resistance Rg and the circuit capacitance Cg in parallel atthe oscillating frequency. Upon substit
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