Piezo Actuator Power Consumption

[emosier]

Hi all,

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!

 

[anorlunda]

For a pure capacitor, no resistance. Average power over an entire cycle is zero. The relevant equations are

I=C dV/dt
P=VI instantaneous
P=0 average if I and V are 90 degrees out of phase.

 

[Henryk]

electrically piezo actuators are modeled as capacitors.

This is a good approximation at low frequencies, that is well below any of the resonant frequencies of the piezo element.
As a mechanical engineer, you realize that each piezo element will have its own modes of vibrations. The exact frequency of the vibrational modes depends on the shape, dimension, stiffness and density of the piezo element, etc.
In the simplest case, the lowest resonant frequency will be much lower than higher modes. In this case, the electrical equivalent would consist of a large capacitor in parallel with a series LCR resonant circuit with low capacitance and high equivalent inductance. The equivalent resistance would depend on the properties of the material as well as on the applied load, if any.

The net result is that electrically, the piezo element has a complex, frequency-dependent impedance. Usually, manufacturers would provide the impedance vs frequency curve for their product.
At the end of the day, there is apparent and real power consumed by the element.
The apparent power is calculated as $P = U_{rms}^2 /|Z|$, where Z is the complex input impedance of the element while the actual power is given by $P_a = U_{rms}^2 /Re(Z)$,
I know, I'm not specific enough but that's the whole picture. Details really depend on an actual piezo actuator.