Quartz Piezoelectric Resonator: Cut Angle, Thickness & BVD Circuit (2D)

This 2D companion drives the same piezoelectric physics as the 3D quartz lab through a plain canvas view built for reading the mechanism rather than orbiting a scene. Quartz's trigonal point-group symmetry makes its piezoelectric coefficient repeat every 120° and flip sign every 60° as the cut angle changes, so a cut-angle slider directly scales the effective coefficient d_eff = d11·cos(3θ) and, with it, the static polarization charge produced under load. A thickness slider sets the wafer's thickness-shear resonant frequency (f_s = N/t, the same relation real AT-cut resonators are manufactured against), and a Butterworth–Van Dyke equivalent-circuit model — the standard way electronics engineers represent a quartz crystal — turns those two numbers into a live admittance curve you can sweep a drive frequency across, watching the sharp peak at series resonance and the antiresonance dip just above it, exactly as a real crystal's frequency response behaves.