What does piezoelectric analysis calculate?

Piezoelectric materials generate charge under force and deform under voltage. Piezoelectric analysis couples the structural equation of motion with the charge equation (Gauss's law) to compute deformation from voltage, voltage from force and resonance frequencies. It's used to design actuators, sensors, ultrasonic transducers and energy harvesters.

How much stroke and force does a stack actuator give?

With PZT-5H (d33 = 593 pC/N), 100 layers of 0.1 mm at 150 V give free stroke d33·V·n = 8.9 μm. A 5 mm square, 10 mm long element has stiffness of about 1.2×10⁸ N/m, so blocking force (to hold displacement at zero) is about 1,070 N. Against a load as stiff as the actuator, stroke halves to 4.4 μm.

What output voltage does a piezo sensor give?

The voltage constant g33 = d33/ε33 is about 19.7×10⁻³ V·m/N for PZT-5H. A 1 mm disc under 1 MPa gives an open-circuit voltage g33·t·σ ≈ 19.7 V, and 0.2 V at 10 kPa. Real output is divided by cable and amplifier capacitance, so the circuit must be included.

Piezoelectric Analysis

Category: Coupled Analysis – Electromagnetic-Structural | Revised 2026-10-01

Theory: a material linking force and electricity

Overview

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How does piezo analysis differ from ordinary structural analysis?

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In piezoelectric materials strain is coupled to electric field and stress to electric displacement: apply voltage and it extends, press it and charge appears. So deformation and electrical state must be solved together. Electrical boundary conditions matter: the same element has different stiffness with electrodes shorted or open. When open, charge produced by deformation creates a field that opposes the deformation, stiffening the element. For actuators the load's stiffness sets actual stroke; for sensors circuit capacitance sets output. Let's first get magnitudes from 1D formulas.

Piezoelectric constitutive equations

$$ S = s^E T + d^{\mathsf T} E,\qquad D = d\,T + \varepsilon^T E $$

S is strain, T stress, E electric field and D electric displacement. s^E is compliance at constant field (short circuit), d the piezoelectric constants and ε^T permittivity at constant stress (free).

Actuator and sensor formulas

$$ x_0 = d_{33} V n,\qquad k_a = \frac{A}{s_{33}^E L},\qquad F_b = k_a x_0,\qquad x = x_0\frac{k_a}{k_a + k_L},\qquad V_{oc} = g_{33}\,t\,\sigma,\quad g_{33} = \frac{d_{33}}{\varepsilon_{33}^T} $$

PZT-5H values: d33 = 593 pC/N, s33^E = 20.7×10⁻¹² m²/N (Young's modulus 48 GPa), relative permittivity ε33^T/ε₀ = 3,400, coupling factor k33 = 0.75.

Coffee Break Trivia

The Curie brothers' discovery

The Curie brothers, Pierre and Jacques, discovered the piezoelectric effect in quartz and other crystals in 1880. The converse effect — deformation under applied voltage — was predicted theoretically the next year and soon confirmed. During the First World War Langevin built underwater sound transducers with quartz, the beginning of sonar. Lead zirconate titanate (PZT) ceramics, found in the 1950s, gave large piezoelectric constants and spread into igniters, buzzers, ultrasound imaging and inkjet printing.

Worked examples

Example 1: stack actuator (100 × 0.1 mm layers, 5 mm square, 150 V)

Item	Value
Length	10 mm
Field	1.5 kV/mm
Free stroke	8.89 μm
Stiffness	1.21×10⁸ N/m
Blocking force	1,074 N
Capacitance	0.75 μF
Peak current, 1 kHz 150 V sine	0.71 A

Load stiffness (× actuator)	Actual stroke
0.1	8.09 μm
1	4.45 μm
10	0.81 μm

Example 2: sensor and electrical boundary conditions

Item	Value
Voltage constant g33	19.7×10⁻³ V·m/N
Open-circuit voltage, 1 mm, 1 MPa	19.7 V
Open-circuit voltage, 1 mm, 10 kPa	0.20 V
Open/short compliance ratio (1 − k33²)	0.438 (2.3× stiffer open)

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Simply opening the electrodes makes it more than twice as stiff — that seems easy to get wrong in analysis.

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Exactly — a common mistake. In materials with large coupling like PZT-5H (k33 = 0.75), electrical state greatly changes mechanical properties. Shorted, charge escapes and the material shows its intrinsic compliance; open, charge accumulates and the field opposes deformation. Resonance shifts too, and the difference between short-circuit resonance and open-circuit antiresonance is how coupling factor is measured. Analysis must set electrodes as fixed voltage, floating conductors or connected to a circuit to match real use. Computing a sensor with grounded electrodes gives zero voltage and underestimates stiffness.

Analysis workflow

Enter material constants (s^E, d, ε^T or transformed forms) aligned with the anisotropy (poling direction).
Define electrodes as equipotential surfaces, choosing fixed voltage, floating or circuit-connected.
Check stroke, force and voltage magnitudes with 1D formulas.
Find resonance and antiresonance by modal analysis to check coupling.
Check operating range, since nonlinearity and hysteresis appear at high fields.

Coffee Break Trivia

“The positioner moved only half as far”

On a precision positioning stage, a piezo stack at rated voltage moved the stage only about half the catalogue free stroke. The specification's free stroke was the unloaded value. Calculating stiffness of the stage's flexure springs and structure showed it about equal to the actuator's, so the stroke should indeed be halved. Softer springs and a longer actuator secured the needed travel.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Using free stroke directly	Stroke overestimated	Include load stiffness
Grounded electrodes	Zero sensor voltage	Floating or circuit
Wrong poling direction	Wrong sign or magnitude	Check coordinates
Mixing short/open constants	Wrong stiffness and resonance	Distinguish constant E and D
Not estimating drive current	Supply inadequate	Compute from capacitance

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I'd like to learn related topics.

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