# Piezo sensor at resonance - equivalent model

piezo materials are commonly modelled with the BVD model

When the piezo is used as a sensor, they are commonly modelled as a current source in parallel with a capacitor because à external force create a charge change in the piezo element (in the assumption that the frequency being sensed is less than the resonance).

Well it would then make complete sens, looking at those two model, that at resonance the sensor could be modelled as a current source in parallel with the BVD model. And we actually find this model in many books on piezo sensor before the author quickly reduce it to the "out of resonnance" simplification seen above.

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simulate this circuit – Schematic created using CircuitLab

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BUT, this circuit give incorrect results because, at resonance, the impedance of the BVD model drops and so does the voltage output at the piezo (U=RI correct right ? I'm not sure anymore). When in, real life, voltage output is maximal at resonance.

We find some other model out there, like this one (used mainly for energy harvesting):

simulate this circuit

With this model the voltage output is maximised at mechanical resonance but i don't see how it would simplify as the simple sensor model seen above (the current source in // with the capacitor).

Well, il my head it's a all mess and I don't know what is what. In brief, I understand the simple piezo sensor model no doubt about this. But I don't inderstand how transform it to take in account the resonance.

Sorry for this longue post, i'm new on the forum so if i broke some rules i'm happy to do the modifications. If one of you could sheed some light on this topic that would be very much appreciated.

• I believe the piezoelectric sensing circuit that includes resonance is more difficult than what you are showing above. There are two equivalent circuits that I'm aware of for piezoelectric materials, the Manson circuit and the KLM circuit (I'm sure there are more). Both have an electrical port (voltage and current) and a mechanical port (force and velocity), and both circuits contain transformers. The Mason circuit is the less complex of the two but requires a negative capacitor at the electrical port, and thus is not physically realizable. Commented Aug 3, 2023 at 14:46
• Thanks for your comment, You're sadly correct i guess... It's not as simple as that. I'll try to dig into those model but the entry ticket is expensive ! Commented Aug 4, 2023 at 8:10

An attempt to untangle the "mess"...
It may be useful to broaden the term "resonance". The parallel capacitor makes this term squirrelly, for the electrical realm. But in the mechanical realm, resonance is unambiguous: the series arm of L1 (42mH), C1 (282pf) and R1 (168 ohms) is key...C2 (5nf) is a real capacitor formed by the plates on either side of a ceramic (piezo-active) dielectric material.
That series-arm mechanical resonance (in this case, at 46245.65 Hz) might vary due to temperature, or age but otherwise is considered constant. At this frequency, series arm reduces to 168 ohms - at other frequencies, impedance is higher and takes on a reactive part. This arm isolated from other circuit elements is a simple series-resonant circuit: V(out1)
The parallel capacitor throws a monkey-wrench into this simple picture. V(out2)

Now there appears a few resonant frequencies:V(out2). The series resonant frequency appears to be similar to the (Vout1) case, but now with a phase shift of about 30 degrees instead of 0 degrees.
A second resonance appears at a higher frequency (47595 Hz) where voltage peaks near 3 V. At this peak, a phase shift of about 25 degrees exists: if it were a true parallel resonance, phase should be near zero degrees, so we know that a resistive component exists along with a reactive component. This resistive component is transformed R2. Since terminal voltage is 2.8V when 1ma is applied, this resistive component is a little above 2800 ohms.

One might ask, "How to extract or excite power from/to this piezo with maximum power transfer?". One might apply the principle that impedance of the source or load should match internal resistance of the piezo. But piezo impedance is highly dependent on frequency:

• at 46245 Hz, 168 ohms would be close to impedance match
• at 47595 Hz, about 3000 ohms would be close to impedance match

This electrical model of a piezo satisfies an electrical engineer because it is the simplest circuit that describes its electrical characteristics. But a closer look at how mechanical and electrical energy interact adds complication.
For example, the same procedure of finding a satisfactory electrical equivalent for a loudspeaker transducer can come up with a 4-element circuit equivalent, but this circuit blows up into uber-complexity when you consider how electrical current is transformed into the energy that moves the speaker cone against air resistance. I'd imagine that this same complexity exists for a piezo transducer.

• Thanks for your reply, It reassuring to see that we have the same understanding of resonance and of this model. Well I think my issue, as you stated at the end of your reply, is more link to the fact that this model is to simple to really show in detail the behaviour of the piezo. I will try to dig into the manson and KLM modem as @C.Dunn mentioned. Commented Aug 4, 2023 at 8:08
• @glen_geek The circuit you show is a good model for the input electrical impedance for a piezoelectric actuator at or below the first resonance (a single port network described in the IEEE 177 standard). For sensing you need both an electrical port and a mechanical port, which this model does not have. Commented Aug 7, 2023 at 20:17