Exercise set

Piezoelectric Sensing, Actuation, and Harvesting Exercises

Worked piezoelectric sensing, actuation and harvesting exercises for charge, amplifiers, leakage, resonance, drive current, heating and release checks.

Branch: Engineering Physics
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises focus on piezoelectric devices used as sensors, actuators, resonant elements and vibration energy harvesters. The goal is to compute charge, voltage, displacement, force, bandwidth, driver current, heating or stored energy, then decide whether the installed electrical and mechanical interface can support a release decision.

Use these exercises as device-level screens. Final design still requires material coefficients over temperature, preload records, mounting evidence, cable and amplifier data, dielectric limits, frequency sweeps, environmental qualification and calibration uncertainty.

Release Evidence Notes

A credible piezoelectric release package keeps the mechanical load path and electrical interface together. Charge sensitivity, capacitance, leakage resistance, cable capacitance, feedback components, preload, resonance, drive waveform and temperature must describe the same installed device. A free-stack stroke, ideal material coefficient or unloaded resonance is not enough evidence for a fielded sensor or actuator.

Engineering Boundary Notes

Piezoelectric sensors are strongest for dynamic measurements. Slowly varying loads require leakage and time-constant checks. Piezoelectric actuators can be precise, fast and stiff, but stroke under load, hysteresis, creep, resonant response, driver current and dielectric heating often control the design.

Common Release Mistakes

using free displacement when the actuator is loaded by a stiff structure;
ignoring charge-amplifier saturation after a large impact;
treating an accelerometer as a static force sensor;
checking reactive current without checking dielectric heating;
using unloaded resonance in a mounted or fluid-loaded assembly;
omitting cable noise, leakage, preload and mounting-torque evidence.

Scenario Map

Scenario	Main calculation	Release decision
Dynamic sensing	$Q=dF$ and $V_o=-Q/C_f$	Check range, noise and droop.
Leakage path	$\tau=RC$	Decide whether low-frequency content is valid.
Actuation	$\delta_0=dV$ and load-line correction	Verify usable stroke under load.
Resonance	$Q_m=f_0/\Delta f$	Check bandwidth, ringing and mounted frequency.
Energy harvesting	energy per cycle and storage reserve	Decide whether duty cycle closes.
Driver heating	reactive current and dielectric loss	Prevent thermal failure at high frequency.

Validation Package Checklist

sensor or actuator orientation and preload record;
cable capacitance, insulation resistance and shielding evidence;
charge-amplifier feedback capacitor, resistor, range and recovery time;
mounted resonance or impedance sweep;
driver current, reactive VA and dielectric-heating calculation;
calibration or functional test at the intended frequency, load and temperature.

Exercise 1: Piezoelectric Charge From Force

A piezoelectric force sensor has:

d=12\ \text{pC/N}

It is loaded by a dynamic force peak:

F=350\ \text{N}

Compute generated charge.

Solution

Q=dF=12(350)=4200\ \text{pC}=4.2\ \text{nC}

Engineering Comment

The charge is large enough for a charge amplifier, but the result applies only along the calibrated force axis and within the dynamic frequency range.

Plausibility Check

A few nanocoulombs for hundreds of newtons is plausible for a piezoelectric force sensor.

Exercise 2: Charge-Amplifier Output

The charge from Exercise 1 is connected to a charge amplifier with:

C_f=2.2\ \text{nF}

Compute output magnitude.

Solution

\displaystyle |V_o|=\frac{Q}{C_f}=\frac{4.2\ \text{nC}}{2.2\ \text{nF}}=1.91\ \text{V}

Engineering Comment

This is within a typical +/-10 V amplifier range. A larger impact should still be checked for saturation and recovery.

Plausibility Check

Nanocoulombs divided by nanofarads gives volts, so the unit scale is correct.

Exercise 3: Leakage Time Constant

A piezoelectric sensor and cable have total capacitance:

C=8.0\ \text{nF}

The effective leakage resistance is:

R=250\ \text{M}\Omega

Compute the time constant and high-pass corner.

Solution

\tau=RC=(250\times10^6)(8.0\times10^{-9})=2.0\ \text{s}

\displaystyle f_c=\frac{1}{2\pi\tau}=\frac{1}{2\pi(2.0)}=0.0796\ \text{Hz}

Engineering Comment

The setup can measure dynamic events above this very low corner, but it should not be treated as a static load measurement system.

Plausibility Check

A larger resistance or capacitance increases the time constant and lowers the corner frequency.

Exercise 4: Accelerometer Saturation

A piezoelectric accelerometer has sensitivity:

S=100\ \text{mV/g}

The signal conditioner clips at $8\ \text{V}$ . Compute the maximum acceleration before clipping.

Solution

\displaystyle a_{max}=\frac{8}{0.100}=80\ g

Engineering Comment

If shocks above $80g$ are credible, the channel can saturate and hide the real peak. A lower sensitivity range or shock-rated conditioner is needed.

Plausibility Check

At $10g$ , the output would be $1\ \text{V}$ , so $80g$ gives $8\ \text{V}$ .

Exercise 5: Free Stroke of a Stack Actuator

A piezo stack has effective coefficient:

d_{eff}=35\ \text{nm/V}

It is driven with:

V=120\ \text{V}

Compute free displacement.

Solution

\delta_0=d_{eff}V=35(120)=4200\ \text{nm}=4.2\ \mu\text{m}

Engineering Comment

The value is free stroke. A real flexure, preload spring or workpiece stiffness will reduce usable displacement.

Plausibility Check

Micrometre stroke is typical for compact piezo stack actuators.

Exercise 6: Actuator Load Line

The stack free displacement is $4.2\ \mu\text{m}$ and actuator stiffness is:

k_a=25\ \text{N}/\mu\text{m}

It drives a load stiffness:

k_L=15\ \text{N}/\mu\text{m}

Estimate loaded displacement:

\displaystyle \delta=\delta_0\frac{k_a}{k_a+k_L}

Solution

\displaystyle \delta=4.2\frac{25}{25+15}=2.625\ \mu\text{m}

Engineering Comment

The load consumes about $38\%$ of free stroke. If the required motion is $3\ \mu\text{m}$ , the actuator does not pass under this stiffness.

Plausibility Check

Loaded displacement is smaller than free displacement and larger than zero.

Exercise 7: Electric Field Limit

A piezo layer is $0.50\ \text{mm}$ thick and is driven at $160\ \text{V}$ . The design limit is $0.45\ \text{kV/mm}$ . Compute electric field and decide.

Solution

\displaystyle E=\frac{V}{t}=\frac{160}{0.50}=320\ \text{V/mm}=0.320\ \text{kV/mm}

The field is below $0.45\ \text{kV/mm}$ .

Engineering Comment

The voltage passes the simplified field limit. Temperature, duty cycle, humidity and transients still need qualification.

Plausibility Check

Increasing layer thickness lowers field for the same voltage.

Exercise 8: Hysteresis Position Error

An open-loop piezo stage has commanded travel $20\ \mu\text{m}$ and hysteresis equal to $12\%$ of command. Compute possible hysteresis error.

Solution

e_h=0.12(20)=2.4\ \mu\text{m}

Engineering Comment

Open-loop control is not acceptable if the positioning tolerance is below $2.4\ \mu\text{m}$ . Closed-loop feedback or calibration by direction is required.

Plausibility Check

The error is a fraction of commanded travel, not a fixed offset.

Exercise 9: Creep During Hold

An actuator reaches $10\ \mu\text{m}$ displacement. Creep after one decade of time is estimated as $1.5\%$ of displacement. Compute drift.

Solution

\Delta \delta=0.015(10)=0.15\ \mu\text{m}

Engineering Comment

This drift may be negligible for a coarse shutter and unacceptable for precision optics. The hold-time requirement decides the release.

Plausibility Check

Creep is much smaller than total stroke but not zero.

Exercise 10: Mounted Resonance Bandwidth

A mounted piezo device has resonant frequency:

f_0=18\ \text{kHz}

Its measured $Q_m$ is $30$ . Estimate half-power bandwidth.

Solution

\displaystyle \Delta f=\frac{f_0}{Q_m}=\frac{18000}{30}=600\ \text{Hz}

Engineering Comment

The device is selective. If the application requires broadband actuation, this resonance may ring and distort the command.

Plausibility Check

Higher Q gives narrower bandwidth; $600\ \text{Hz}$ is narrow compared with $18\ \text{kHz}$ .

Exercise 11: Ringdown Time Screen

Use an approximate ringdown time:

\displaystyle t_s\approx\frac{Q_m}{\pi f_0}

For $Q_m=30$ and $f_0=18\ \text{kHz}$ , estimate $t_s$ .

Solution

\displaystyle t_s=\frac{30}{\pi(18000)}=5.31\times10^{-4}\ \text{s}=0.531\ \text{ms}

Engineering Comment

The ringdown is fast in absolute time, but it can still matter in pulsed ultrasonic, impact or scanning applications.

Plausibility Check

The time is many cycles but still below one millisecond because the frequency is high.

Exercise 12: Vibration Harvester Average Power

A piezoelectric harvester produces $2.0\ \text{mJ}$ per vibration event. Events occur $90$ times per hour. Compute average power.

Solution

Energy per hour:

E_h=2.0\ \text{mJ}(90)=180\ \text{mJ}=0.180\ \text{J}

Average power:

\displaystyle P=\frac{0.180}{3600}=50\ \mu\text{W}

Engineering Comment

This can support intermittent sensing, not a high-duty radio. The duty cycle must be built around stored energy.

Plausibility Check

Millijoule events at sparse frequency naturally produce microwatt average power.

Exercise 13: Storage-Capacitor Reserve

A harvester storage capacitor is:

C=22\ \text{mF}

It can discharge from $3.3\ \text{V}$ to $2.7\ \text{V}$ . Compute usable stored energy.

Solution

\displaystyle E=\frac{1}{2}C(V_1^2-V_2^2)

E=0.5(0.022)(3.3^2-2.7^2)=0.0396\ \text{J}

Engineering Comment

Only about $40\ \text{mJ}$ is available over this voltage window. A transmit burst must be shorter or the capacitance must increase.

Plausibility Check

The energy is small because the capacitor is small and the voltage window is narrow.

Exercise 14: Driver Reactive Current

A piezo actuator has capacitance:

C=1.8\ \mu\text{F}

It is driven by a sinusoid with:

V_{rms}=70\ \text{V},\quad f=600\ \text{Hz}

Compute RMS current:

I_{rms}=2\pi fCV_{rms}

Solution

I_{rms}=2\pi(600)(1.8\times10^{-6})(70)=0.475\ \text{A}

Engineering Comment

The driver must handle nearly half an ampere of reactive current. Current capacity can fail even when average mechanical power is modest.

Plausibility Check

Current rises linearly with frequency, capacitance and voltage.

Exercise 15: Dielectric Heating

Use the actuator from Exercise 14 with loss tangent:

\tan\delta=0.025

Estimate dielectric loss:

P_{loss}=2\pi fCV_{rms}^2\tan\delta

Solution

P_{loss}=2\pi(600)(1.8\times10^{-6})(70^2)(0.025)=0.831\ \text{W}

Engineering Comment

If the stack thermal resistance is high, this loss can create a temperature gate even when the driver current passes.

Plausibility Check

Loss rises with voltage squared, so reducing voltage is powerful for thermal control.

Exercise 16: Temperature Rise From Dielectric Loss

The piezo stack dissipates $0.831\ \text{W}$ and has thermal resistance:

R_{th}=18\ \text{K/W}

Compute temperature rise.

Solution

\Delta T=P_{loss}R_{th}=0.831(18)=15.0\ \text{K}

Engineering Comment

A $15\ \text{K}$ internal rise may shift sensitivity, reduce lifetime or exceed a touch-temperature guard. Drive frequency and voltage need thermal validation.

Plausibility Check

One watt through tens of kelvin per watt gives a temperature rise of order tens of kelvin.

Exercise 17: Cable Capacitance Effect on Voltage Mode

A piezo element has capacitance $4.0\ \text{nF}$ and generates $Q=800\ \text{pC}$ . Cable capacitance adds $6.0\ \text{nF}$ . Compute voltage before and after cable capacitance if measured in voltage mode.

Solution

Without cable:

\displaystyle V_1=\frac{800\ \text{pC}}{4.0\ \text{nF}}=0.200\ \text{V}

With cable:

\displaystyle V_2=\frac{800\ \text{pC}}{10.0\ \text{nF}}=0.080\ \text{V}

Engineering Comment

The cable cuts sensitivity by $60\%$ . Charge mode or cable-specific calibration is needed if cable length can change.

Plausibility Check

More capacitance for the same charge means lower voltage.

Exercise 18: Piezoelectric Release Gate

A piezo actuator must provide at least $2.0\ \mu\text{m}$ loaded stroke. Calculated loaded stroke is $2.45\ \mu\text{m}$ with expanded uncertainty $0.28\ \mu\text{m}$ . Driver current limit is $0.60\ \text{A}$ and predicted RMS current is $0.475\ \text{A}$ . Temperature rise limit is $20\ \text{K}$ and predicted rise is $15.0\ \text{K}$ . Decide release status using guarded stroke:

\delta_{guarded}=\delta-U

Solution

\delta_{guarded}=2.45-0.28=2.17\ \mu\text{m}

Stroke margin:

2.17-2.0=0.17\ \mu\text{m}

Current and temperature margins are:

0.60-0.475=0.125\ \text{A}

20-15.0=5.0\ \text{K}

Engineering Comment

The actuator passes all simplified gates, but stroke margin is narrow. Release should require mounted displacement evidence, driver-current test and thermal soak at the intended waveform.

Plausibility Check

All guarded margins are positive. The smallest margin controls the confidence level, so the correct decision is conditional release rather than broad approval.

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