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I am trying to understand the fundamental principles of piezoelectrics, but my searches haven't helped me answer this question. As I understand it, squeezing a piezoelectric causes a mechanical segregation of charges in the crystal structure of the piezoelectric material. Now there is a potential difference between the two collections of charges, (and that, I imagine, should cause the crystal to spontaneously return to its original unstrained position once the stress is removed).

Everything from this point on is my ill-informed conjecture: Now suppose we coat the opposite faces with electrode paste or something, and then join the two faces the long way around with a loop of wire, featuring an ammeter in the circuit. My guess is that the negative charges in the wire would be drawn to the positive field face of the crystal and vice-versa, so while no electrons "flow" through the crystal, we still have a displacement current (if that is the right terminology, as I recall from learning about capacitors) and the ammeter will read a net flow of charges passing by.

If I am correct so far, now I wonder what the current vs time graph would look like. My intuition tells me that, as long as we maintain a constant strain on the piezoelectric material, the charges will be drawn to one face or the other but at some point they will slow down and approach equilibrium; the negative charges in the wire will move to and cancel out the positive charges on the face of the crystal, and at that point there is no more difference to drive any more current. If we want more current, we could push harder on the crystal to further segregate the charges within it and that would put us out of equilibrium, or we could release the strain on the crystal and the current would flow in the reverse direction until we were back where we started. A periodic stress cycle on the piezoelectric would then generate an alternating current.

Is that at all how this works? Or would maintaining a constant strain on the crystal somehow drive a constant direct current? All the articles I read say simply "stressing the piezoelectric generates a voltage" and that is insufficient information for me to understand the effect.

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You are almost correct. the sticky details are as follows:

A slab of piezo crystal (or plastic), when flexed, generates a significant voltage with almost no current behind it: it is a high-impedance transducer. To measure the current output of a piezo transducer requires an extremely sensitive ammeter with extremely fast response time. For that matter, an ordinary voltmeter will probably not respond quickly or sensitively enough to catch the flexural signal from the piezo device either, unless its input impedance is of order ~10 megohms.

A useful way to model a piezo disc in your mind is to think of it as a small capacitor bearing a small voltage. Because the voltage and capacitance are small, the energy stored is correspondingly small, and measurements taken on that capacitor require fast and sensitive instruments.

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