--- title: "Piezoelectric transducer sensitivity and output voltage" subject: "Instrumentation I" chapter: 3 key: "semester_4/instrumentation_i:piezoelectric-transducer-output-voltage" source: https://www.bctengineer.com/q/semester_4/instrumentation_i/piezoelectric-transducer-output-voltage site: BCT Engineering --- # Piezoelectric transducer sensitivity and output voltage Instrumentation I · Chapter 3: Transducer - Asked in: year not given, 8 marks - Video: [piezoelectric transducer | piezoelectric transducer working principle | piezoelectric effect |sensor](https://www.youtube.com/watch?v=oYTI48RkV4I) — Mind of Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/piezoelectric-transducer-output-voltage - Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-3 ## Question What is piezoelectric transducer? Define voltage sensitivity and charge sensitivity. Give the equivalent circuit of piezoelectric transducer. Derive the expression for the output voltage by making suitable assumptions. ## Short answer A piezoelectric transducer generates charge proportional to applied force. Charge sensitivity d = q/F; voltage sensitivity g = d/ε. Equivalent circuit: current source jωdF in parallel with C and R. Output voltage Eo = d·F·ωR/√(1+ω²R²C²); at high frequencies Eo ≈ d·F/C, at low frequencies falls with ω. ## Answer 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 ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does a piezoelectric transducer produce when the crystal is mechanically stressed? A. A change in resistance B. A change in inductance C. A change in capacitor gap only D. Electric charge and voltage across the crystal faces Answer: D. It is an active transducer generating charge when stressed. (Key point: Piezoelectric effect) 2. Charge sensitivity d is defined as: A. Electric field per unit applied stress B. Voltage per unit force C. Charge produced per unit force (C/N) D. Capacitance per unit area Answer: C. d = q/F; voltage sensitivity g is field per unit stress, g = d/epsilon. (Key point: Sensitivities) 3. At high frequency (omega R C much greater than 1), the output voltage is approximately: A. Zero B. Proportional to omega C. Eo = d·F/C, independent of frequency D. Infinite Answer: C. Eo = dF·omega R/sqrt(1 + omega^2R^2C^2) tends to dF/C for large omega RC. (Key point: Output voltage at high frequency) 4. Why can a piezoelectric transducer not measure a static force, and why is a charge amplifier preferred? A. Output is zero at DC; and sensitivity d/(Ca+Cc+Ci) changes with the cable capacitance B. Output is maximum at DC; cable has no effect C. It has no capacitance; amplifier lowers gain D. It is passive; amplifier supplies charge Answer: A. Eo falls to zero at low frequency/DC, and since sensitivity depends on total C including the cable, a charge amplifier removes this dependence. (Key point: Static force and cable) ## Related in Instrumentation I - [Instrumentation amplifier and its gain](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Ainstrumentation-amplifier-gain) — Instrumentation I, chapter 4 - [Single-phase electrodynamometer wattmeter](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Aelectrodynamometer-wattmeter) — Instrumentation I, chapter 7 ## Related in other subjects - [Piezoelectric load cell and charge amplifier](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Apiezoelectric-load-cell-charge-numerical) — Instrumentation and Measurement, chapter 7 - [250/500 V test; equivalent circuit, max efficiency output](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machines%3Atf-250-500-max-efficiency-output) — Electrical Machines, chapter 2 - [Zener circuit output voltage and current](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_and_electronics_engineering%3Azener-circuit-120v-5k) — Basic Electrical and Electronics Engineering, chapter 3 - [IM equivalent circuit; starting/running torque](https://www.bctengineer.com/question.md?key=semester_4%2Felectrical_machine%3Aim-equivalent-circuit-starting-running-torque) — Electrical Machine, chapter 5