The zero-angle quartz crystal, having electrodes in direct contact with its surface, is used to control the output of a typical oscillating crystal circuit. The crystal is first considered as a simple mechanical oscillator, and the required plate circuit adjustment for sustained vibrations of the crystal obtained by Miler's method on the basis of an assumed electrically equivalent crystal circuit. The electrical equivalent of the crystal is then considered as the grid circuit of an oscillating vacuum-tube circuit and the equations for the frequency and condition for oscillation derived. The effects of the tube and circuit upon the frequency of a crystal-controlled oscillator are then shown.
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J.W. Wright (1929) studied this question.