Analysis of the frequency and sensitivity of crystal-controlled oscillators

The equations for the frequency of a crystal oscillator, and the sensitivity of frequency to the circuit parameters are here extended to consider: (1) the effects of motional capacitance, tuning capacitance, circuit capacitors, and the active element impedance; and (2) the Z-type oscillator. A precise normalized frequency difference between the frequency of oscillation and the series resonant frequency of the crystal is defined and derived for both the Y-oscillator and Z-oscillator. A precise normalized frequency difference is likewise derived for the parallel resonant frequency of the crystal. From these, the sensitivity of oscillation frequency to various parameters of the crystal and the circuit is derived. The conditions of validity and practical significance of the expressions are discussed. Experimental results obtained with the Pierce oscillator verify the applicable equations.