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Mechanical engineer needing some electrical help. So electrically piezo actuators are modeled as capacitors.

Power(P) = C*U*dU/dt, where U equals voltage.

I'm seeing several sites like here: http://www.piezo.ws/piezoelectric_actuator_tutorial/Piezo_Design_part3.php

They say for sinusoidal operation Pavg ≈ C*U^2*f and Pmax ≈ ∏*C*U^2*f (assuming you start from 0 Volts).

So my questions are:

1. How can you substitute U*f for dU/dt to get average power from the main power equation?

2. How is Pmax ≈ ∏*Pavg?

3. What does average power mean for a cycling device? Wouldn't RMS make more sense?

Hopefully someone with some familiarity and/or better math skills can help me out.

Thanks!