Extracting Resonator Model Parameters from Impedance Data

August 30, 2024 by Jim Phillips

Resonators form the bedrock of many precision measurements, from macroscopic mechanical devices at frequencies of a few Hz through MEMS and NEMS devices at frequencies in the GHz range.  Often when working with a resonator, we wish to determine the parameters of an underlying model.  Consider the model for a mechanical resonator such as a piezoelectric quartz crystal or a Surface Acoustic Wave (SAW) or Bulk Acoustic Wave (BAW) device as in Figure 1. Here, the parameters are the mechanical inductance \(L_S\), the mechanical capacitance \(C_S\), the mechanical resistance \(R_S\), and the parallel capacitance \(C_P\). We may be estimating these parameters in order to understand the fabrication process and to improve it, we may wish to construct an acceptance test for use on a production line, or we may simply want to have a model of the resonator so as to be able to design a circuit around it.  In this blog, we show how to derive the estimates.

Equivalent circuit diagram of a resonator

Figure 1: Electrical model for a mechanical resonator. The series inductor, capacitor, and resistor are not actual electrical components, but arise from the electrical-mechanical coupling.  The electrical impedance measured at the terminals is the same as if these did exist as electrical components.

We focus on resonators having a high quality factor, \(Q\). \(Q\) is the ratio of resonant frequency to linewidth. Resonators of high \(Q\) have low damping and can make oscillators of low phase noise or sensors that precisely measure parameters of their environment.

Such a resonator usually has a series resonance in which the impedance takes on a minimum absolute value, and a parallel resonance at a slightly higher frequency at which the impedance takes on a maximum value (Figure 2). These two resonances may overlap, leading to an asymmetric resonance shape. The parameters of the model can nonetheless be extracted by the approximations given here or with greater accuracy by a fitting procedure.

Figure 2: Magnitude and phase of resonator impedance acquired with LabOne Sweeper module.  The Math tool measures the series and parallel resonant frequencies, and the impedance at those frequencies.

The impedance of this resonator model, without approximation, is

\[Z = {{1 \: - \: \omega^2 L_S \, C_S \: + \: j\omega R_S \, C_S} \over {j\omega [C_S \: + \: C_P \: - \: \omega^2 L_S \, C_S \, C_P] \: - \: \omega^2 R_S \, C_S \, C_P}}\]

(1)

Estimates of The Parameters of The Model

Figure 2 shows a frequency sweep of the resonator taken with the LabOne Sweeper Module.  The two traces are absolute impedance (red) and the phase (blue). Using the math tools of the LabOne Sweeper Module, such as the peak and trough tool, we can easily measure four key parameters as follows:

  • the frequency of the series resonance,  \(\omega_S\:\)
  • the frequency of the parallel resonance,  \(\omega_P\)  
  • the impedance at the series resonance,  \(Z_S\)
  • the value of the impedance off-resonance,  \(Z_{off}\)

The values read off above allow us to estimate the four parameters of the model in equation (1).  These can be used directly or used as inputs to a least-squares fit that will give more precise estimates.
We first write equations relating the above observed quantities to the desired parameters. We then show the solution of these equations for the parameters.  The series resonance occurs when the magnitude of the numerator is at a minimum. For a high-Q resonator, the series-resonant frequency is approximately

 

\[ \omega_S \: = \: {1 \over {\sqrt { L_S \: C_S} } } \]

(2)

At the series resonance, we use the impedance which gives directly our estimate for the series resistance,

 

\[ R_S \: = \: |Z_S| \: . \]

(3)

The parallel resonance occurs when the magnitude of the denominator is at a minimum. The parallel resonance frequency is approximately

 

\[ \omega_P \: = \: { \omega_S {\sqrt { {C_S \: + \: C_P } \over {C_p} } } } \]

(4)

Finally, the impedance well off-resonance gives a value for the parallel capacitance

 

\[ C_P = { 1 \over {j \omega \: |Z_{off} }| } . \]

(5)

For greater accuracy, use the average of the off-resonance values measured at frequencies equally spaced above and below the mean of the series and parallel resonance frequencies.  Solving (4) for the series capacitance,

 

\[ C_S = 2 ( { {\omega_P - \omega_S } \over {\omega_S} } ) C_P . \]

(6)

Finally, solving (2) for the series inductance,

 

\[ L_S = {1 \over { \omega_S^2 \: C_S } } . \]

(7)

The values for our four parameters are now computed.  The values for mechanical inductance and capacitance will usually be far higher and lower, respectively, than is possible with actual electrical components.  This helps to explain the high \( Q \) of a mechanical resonator.  These values will be quite accurate in the case of high \( Q \) and well-separated resonances. For further-improved accuracy, these values can be used as starting values for a least-squares estimate. The least-squares can employ a larger set of data, typically a sweep. Using the more complete dataset, one can also plot the data and the model, which serves as an excellent test of the model.

For an empirical view of fitting resonator parameters, see this blog.  For a fitting approach using 3rd-party software, see this blog.  Another useful blog gives a fast and practical way to measure the \(Q\) of a resonator with very high \(Q\). 

Summary

This straightforward technique comprising a single impedance sweep over the dual resonances of your resonator provides you with swift and accurate knowledge of the mechanical inductance, capacitance, and resistance, and of the parallel capacitance. It gives you a fuller understanding of your resonator, whether you are looking to optimize the fabrication process of your resonators, to construct a production-line acceptance test, or simply to have a model of the resonator so that you can design a circuit around it. 

If you'd like to know more, please get in touch to set up a demo.

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