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The direct and the inverse effect
In a crystal without a centre of symmetry, mechanical stress separates the centres of positive and negative charge and produces a polarisation: electric charge appears on the faces (direct effect). Conversely an electric field strains the crystal (inverse effect). In the linear regime, with piezoelectric constant d:
D = d T + εT E, S = sE T + d E so Q = d F (short circuit), Δt = d Vd is the same number in both: pC per newton and pm per volt. An open-circuit plate of capacitance C0 = εA/t gives Voc = Q/C0 = d F / C0. Because ε is small for quartz and PVDF, they give high voltages; PZT ceramics have d two orders larger and are used for actuators and energy harvesting.
Energy harvesting into a load
For a sinusoidal force F0 sin ωt the plate acts as a current source i = d dF/dt in parallel with C0, loaded by R. The page integrates C dv/dt + v/R = d dF/dt (RK4) and compares with the phasor result:
Vpk = d F0 ω R / √(1 + (ωRC)2), P = Vpk2/2R, Pmax at R = 1/(ωC): Pmax = (d F0 ω)2 / (4 ω C)Resonance
Near the thickness-mode resonance fs = v/(2t), v = √(Y/ρ), the plate behaves like the Butterworth-Van Dyke circuit: a series LmCmRm branch in parallel with C0. The impedance has a minimum at fs and a maximum at fp, and the effective coupling is k2 = 1 − (fs/fp)2. This is how quartz oscillators keep time.
Honest simplifications
- Linear, lossless-in-the-dielectric model: real ceramics show hysteresis, creep, ageing, depolarisation above the Curie temperature and non-linearity at high fields.
- Material constants are typical catalogue values for the direction used (d33 for the ceramics, an effective d for quartz and PVDF), not data for a specific grade; the sign of d is ignored.
- Quasi-static drive (up to 1 kHz, far below resonance); the resonance graph uses a single thickness mode with the plate clamped laterally, k and Q from tables.
- The lattice drawing is a cartoon with exaggerated displacement; real displacements are picometres.
- Frequencies above 2 Hz are animated in slow motion; the numbers shown are for real time.
What to try
- Run PZT lighter igniter: a 200 N squeeze of a 3 mm ceramic gives kilovolts at open circuit; with a load resistor the voltage across it is much less.
- Run Quartz at the optimum load and move the load slider: the power graph peaks exactly at R = 1/(wC).
- Change the frequency: the optimum load moves as 1/f; the available power per period is the same.
- Run Piezo actuator: 100 V on a 1 mm PZT changes its thickness by tens of nanometres (drawn 100 times larger).
- Compare materials in the bar chart of d on a log axis and in the impedance curve (fs, fp).