The Piezo-Electric Resonator and Its Equivalent Network (1928)

The theory of the piezo-electric and the mechanical behavior of a quartz resonator is stated following Voight and Cady. The functions of the quartz as dielectric and as vibrator are shown to be separable and replaceable by a condenser in parallel with an electrical resonator, i.e., a series chain of inductance, resistance, and capacity. For a Curie-cut quartz rod excited lengthwise through the transverse piezo-electric effect into compressional vibration at the fundamental frequency the series elements become L=M/4ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ; R= N/4ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ; C=4b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /g. This mode of vibration may be termed the "fundamental normal mode" since the vibration direction is normal to the electric field. For a Curie-cut quartz plate excited through the longitudinal piezo-electric effect into compressional vibration at the fundamental thickness frequency, the series elements become L =e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> M/4ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ; R=e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N/4ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ; C=4ϵ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> b <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> /e <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> g. This mode of vibration may be termed the "fundamental parallel mode" since the vibration direction is parallel to the electric field. M, N, and g are respectively the half-mass, mechanical resistance factor, and "equivalent stiffness" of the rod or plate whose thickness along the field is e, and dimensions normal to the field are l and b, the latter being along the optic axis. The parallel capacity is shown to be less than that for a quartz dielectric which is free to assume piezo-electric strain and to be equal to that for unstrained quartz. Phase and amplitude variations of current to the resonator are shown as obtained with the cathode ray oscillograph.