Piezoelectric transducer sensitivity and output voltage

A piezoelectric transducer produces an electric charge (and hence a voltage) across the faces of a crystal (quartz, Rochelle salt, barium titanate, PZT) when the crystal is mechanically stressed. It is an active transducer used to measure force, pressure and acceleration.

Sensitivities

• Charge sensitivity (d): charge produced per unit applied force along a given axis, d = q/F (C/N).
• Voltage sensitivity (g): electric field produced per unit applied stress, g = E/p (V·m/N) = d/ε (ε = permittivity of the crystal). For a crystal of thickness t, the open-circuit voltage is Eo = g·t·p.

Equivalent circuit

  Charge-generator form:
   +------+------+------+------+------+
   |      |      |      |      |      |
  (q)    Ca     Ra     Cc     Ci     Ri   --> to amplifier
   |      |      |      |      |      |
   +------+------+------+------+------+
   crystal       cable    amplifier input

• The crystal acts as a current (charge) source in parallel with its capacitance Ca and leakage resistance Ra.
• Cc = cable capacitance; Ci, Ri = amplifier input capacitance and resistance.

Equivalent total: C = Ca + Cc + Ci; R = Ra‖Ri.

Output voltage derivation

Assumptions: sinusoidal force F = Fm sin ωt, a linear crystal, and q = dF.

The charge is q = d·Fm sin ωt, so the source current is

i = dq/dt = d·ω·Fm cos ωt (phasor: I = jωdF)

This current flows through R ‖ C, whose impedance is Z = R/(1 + jωRC):

Eo = I·Z = jωdF·R / (1 + jωRC)

Magnitude: Eo = d·F·ωR / √(1 + ω²R²C²)

• High frequency (ωRC ≫ 1): Eo ≈ d·F/C, which is constant and independent of frequency. For the crystal alone (C = Ca = εA/t), this becomes Eo = d·F·t/(εA) = g·t·p.
• Low frequency: Eo falls in proportion to ω; the response at DC is zero, so a static force cannot be measured.

The sensitivity Eo/F = d/(Ca + Cc + Ci), so changing the cable changes the sensitivity; this is why a charge amplifier is preferred.