Piezoelectric Impedance: Measurement and Modeling

April 13, 2023 by Meng Li

Piezoelectric materials can produce electricity under mechanical stresses, and often find uses in both sensing and transducing applications. In this blog post, we focus on the electrical impedance measurement of piezoelectric materials, using an MFIA Impedance Analyzer (measuring the electrical response of piezoelectric devices often requires a charge integrator, which won't be discussed here). We will also explain how the stress condition affects the impedance, and how to properly fit it thereafter.

Figure 1. Sketch showing the setup of an MFIA Impedance Analyzer connected to a piezoelectric material-under-test (MUT). 3 testing conditions are: (a) in red, hanging in the air; (b) in blue, sitting on the desk; and (c) in green, freestanding on the desk.

Measurement: Pay attention to the stress condition

Piezoelectric materials such as lead zirconium titanate (PZT) are crystallographically anisotropic such that the piezoelectric response differs in directions. Further, since the stress and electrical impedance are electromechanically coupled, the stress condition affects the impedance measurement, too. A loudspeaker, for instance, demonstrates a different impedance with and without an enclosure.

In Figure 2, we see the results when the PZT material is hanging in the air (red), sitting on the desk (blue), and freestanding on the desk (green). When the material-under-test (MUT) is sitting on the desk, the resonance frequency shifts slightly from 2.75 kHz, and the bandwidth quality factor decreases, as compared with the hanging scenario (red). Also interestingly, there is a difference between when the MUT is freestanding or not. Clearly, the freestanding scenario (green) shows a lot more resonances, due to a different electromechanical coupling to the desk underneath.

Figure 2. LabOne Sweeper showing the impedance and derived capacitance using an oversimplified R||C model. The color indicates different stress loading directions. Red: hanging in the air; Blue: sitting on the desk; and Green: freestanding on the desk. If we assume an (Rs+Ls+Cs)||Cp model, at lower frequencies, the derived capacitance approximates to Cs+Cp. Whereas at higher frequencies, it approximates to Cp.

Modeling: An empirical approach

Equivalent circuit modeling (ECM) is a complex topic, but we hope to shine a light on it by fitting the impedance spectrum of PZT material in Figure 2, allowing the readers to understand the challenges and how to improve their own fitting later. Independent from any fitting program or software, the workflow is as follows:

1. Choose a proper model based on the trend of the impedance spectrum

We know a resonance can often be described by an Rs+Ls+Cs circuit. We also notice that in Figure 2, the impedance decreases linearly with respect to frequency. This cannot be described by Rs+Ls+Cs, where the impedance should keep increasing with frequency. So it immediately hints that we should add a significantly large capacitance in parallel (Cp). As such, we have (Rs+Ls+Cs)||Cp.

2. Determine suitable initial conditions using a simplified model

The (Rs+Ls+Cs)||Cp model is mostly determined by Cp at high enough frequencies (for instance, 3.5 kHz in Figure 2). So using the built-in R||C model in LabOne, we get the initial value of Cp as roughly 21.6 nF. Similarly, one may also consider, at low enough frequencies, both Cs and Cp contribute. So the total C (30.9 nF) extracted from R||C model would be the sum of Cs and Cp. Cs could then be around 9.3 nF. The resonance frequency at 2.75 kHz is given by \({1\over 2\pi\sqrt{L_{s}C_{s}}}\) , so Ls might be around 0.36 H.

3. Optimize the fitting by narrowing the boundary conditions

Despite the presence of multiple resonances, it is better to focus only on fitting the primary one. In Figure 2 for instance, we observe a satellite peak at around 4.5 kHz, but there is no way that a circuit of (Rs+Ls+Cs)||Cp can produce it. Therefore, we must narrow our fitting frequency range to roughly between 2 kHz and 3.6 kHz to exclude this satellite peak. We should also keep narrowing the boundary conditions to help the fitting to converge. This step is critical, as there are very few 'valid' data points near the resonance.

The fitted impedance spectrum is displayed in Figure 3, where we see a slight mismatch at the anti-resonance (2.6 kHz). To further improve, we need more circuit elements than 4 [1], which is beyond the scope of this blog post. Nonetheless, to highlight the difference between the initial conditions from empirical modeling and the fitted results afterward, we list the results in the table below.

Parameter	Initial condition	Fitted result
Rs	n.a.	162 Ohm
Ls	0.36 H	1.13 H
Cs	9.3 nF	3.36 nF
Cp	21.6 nF	25.2 nF

Figure 3. Bode plot showing the measured electric impedance (blue open circle) and fitting (orange solid line) of a piezoelectric MUT.

Conclusion

We explained in this blog post that piezoelectric impedance should be better measured and modeled in a stress-free condition. This simplified case shows fewer resonances and is easy to start with an (Rs+Ls+Cs)||Cp model. Of course, attention is still necessary when fitting equivalent circuit parameters. A tight boundary condition is almost compulsory for a sharp resonance peak. In real-world scenarios, we most likely need to add more fitting circuit elements.

Please stay tuned for more piezoelectric impedance measurement topics on our blog channel.

Reference

[1] Gogoi, Niharika, et al. "Dependence of piezoelectric discs electrical impedance on mechanical loading condition." Sensors 22.5 (2022): 1710.