Piezoelectric Cantilever Energy Harvester Calculator

Piezoelectric Cantilever Energy Harvesting Principles

Piezoelectric cantilever harvesters convert ambient mechanical vibrations (from industrial motors, pumps, HVAC compressors, or bridges) into AC electricity via the direct piezoelectric effect, powering wireless IoT telemetry nodes without batteries.

1. Resonant Tuning with Tip Proof Mass

Cantilever harvesters behave as second-order mechanical oscillators. Power transfer collapses by over 90% if the harvester's natural frequency does not match the driving vibration frequency. The fundamental natural frequency (f_n) is tuned by adjusting the tip proof mass (M_t):

$$f_n = \frac{1}{2\pi} \sqrt{\frac{k_{\text{eff}}}{M_{\text{eff}} + M_t}}$$

Where (k_{\text{eff}} = \frac{3 E I}{L^3}) is the composite beam bending stiffness, and (M_{\text{eff}} \approx 0.24 \cdot M_{\text{beam}}).

2. Optimal Electrical Load Matching

The piezoelectric element behaves electrically as an AC current source in parallel with its internal capacitance (C_p). Maximum continuous power is transferred to a resistive load when the load resistance (R_{\text{opt}}) matches the internal capacitive reactance at the driving vibration frequency (\omega = 2\pi f_n):

$$R_{\text{opt}} = \frac{1}{\omega \cdot C_p} = \frac{1}{2\pi \cdot f_n \cdot C_p}$$

3. Maximum Power Formula

When operating at resonance under matched impedance, the maximum harvestable electrical power is given by:

$$P_{\text{max}} = \frac{1}{2} \cdot \frac{V_{\text{oc}}^2}{4 R_{\text{opt}}} = \frac{\omega C_p V_{\text{oc}}^2}{8}$$