The Piezo-Electric Resonator and Its Equivalent Network

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
²
b
²
; R= N/4ϵ
²
b
²
; C=4b
²
ϵ
²
/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
²
M/4ϵ
²
l
²
b
²
; R=e
²
N/4ϵ
²
l
²
b
²
; C=4ϵ
²
l
²
b
²
/e
²
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.