Different Views on Quality Factor and Equivalent Circuit Modeling

For ease of communication, scientific nomenclature is often named as short as possible. Unfortunately, sometimes this convention leads to different terms sharing the same name, which may lead to confusion. Taking piezoelectric impedance and dielectric impedance as examples in this blog post, we will clarify 2 ambiguous terms, the quality factor (Q), and equivalent circuit modeling (ECM).

Bandwidth Q (frequency-response perspective) vs. loss Q (energy-conservation perspective)

The quality factor Q can be used to describe both electrical (we disregard other types of resonances here for simplicity) resonances and device/materials losses.

In piezoelectric impedance measurements, the bandwidth Q (\(Q_{BW}\)), originating from a frequency-response perspective, is more often used. It is defined as the resonance frequency (\(f_{res}\)) dividing the full-width half maximum (\(f_{FWHM}\)). Simplified R+L+C and R||L||C circuit models are often used to represent such a resonance, but other real-world models are also used. If we assume R+L+C, we have:

\(Q_{BW} = {f_{res}/ f_{FWHM}} = {1\over R} \sqrt{{L\over C}}\)  (1)

\(Q_{BW}\) is theoretically constant, but real-world devices are usually more complicated. To get \(Q_{BW}\) we can selectively fit near the resonance peak, using the math tool of the LabOne Sweeper module as shown in Figure 1 (a).  The fitting relies on the bandwidth so it does not generate the RLC parameters other than \(Q_{BW}\). However, according to Equation (1), we see that a small R leads to a high \(Q_{BW}\).   

In contrast, the loss Q (\(Q_{loss}\)) describes how much energy can be stored within a charging cycle, normalized to the dissipated energy, or mathematically, the imaginary part of the impedance dividing the real part. It is often used in characterizing dielectrics. As resonance is not necessary, for simplicity, let's just assume an R||C circuit, we will then have: 

\(Q_{loss} = Z_{imag}/ Z_{real}= 2{\pi}fRC\)  (2)

In such a conventional R||C circuitry, different from the previous R+C case, a large R contributes to a high \(Q_{loss}\). In addition, \(Q_{loss}\) increases with frequency, as displayed in the LabOne Sweeper module in Figure 1 (b) below 10 kHz. Higher than 10 kHz, parasitic inductance can play a role, so we see \(Q_{loss}\) starts to saturate. Hence, we note that \(Q_{loss}\) is only meaningful when the measurement frequency is specified. The same also applies when referring to the inverse \(Q_{loss}\), known as the dissipation factor D (DF, loss angle, or loss tangent).

Real-time circuit model (device perspective) vs. post-processed circuit model (materials perspective)

To fully elucidate the story, we now extend our discussion on the quality factor to a higher level: equivalent circuit modeling (ECM). ECM is also a good example where the perspectives differ in research fields. Here we categorize ECM into 2 classes: real-time and post-processed.

In the LabOne software, we offer real-time models based on 2 circuit elements. This is because at a fixed frequency, we measure the impedance in 2 degrees of freedom: the real part and imaginary part (in the Cartesian system), or equivalently the amplitude and phase (in the Polar system). As such, it is only possible to obtain 2 unknown circuit elements at maximum. Modeling beyond 2 circuit elements in real-time, such as an R+L+C resonance, is mathematically impossible. 

Thanks to the real-time modeling in LabOne, we can confidently monitor the extracted parameters such as C from an R||C model, and correlate them to external perturbations without waiting. This can be extremely useful for sensing applications, particularly when a fast measurement speed is demanded. In Figure 2 (b), we see how the capacitance of a semiconducting device responds to a DC step voltage in a 10 μs time resolution.

It is important to note that the circuit elements from real-time ECM are scaled rather than fitted. In other words, we can always get a result even without choosing the best suitable circuit model. Such a simplification is helpful to have a quick overview. But when the model is oversimplified, for instance, when we use an R||C model instead of the true R+L+C model, the extracted C can be negative in the inductive region, hence broken on a log scale in Figure 1 (a).

Different from real-time ECM, post-processed ECM speaks for material property characterization. In terms of piezoelectric materials, in order to correctly get R, L, and C from an R+L+C model (or more complicated), we must increase the degree of freedom from 2, by performing a sweep to get the frequency-dependent impedance, also known as the electrical impedance spectrum (EIS) as in Figure 1 (a).  Using the LabOne software, we can conveniently export the measured data in various formats including .csv, .mat, .txt, and .h5, and then run a third-party fitting program. An example program is shown in a recent blog post. The fitted parameters in this approach are determined by the locally least square minimum, in other words, they are a solution consisting of frequency-independent constants. Consequently, the derived dielectric permittivity from a constant C, becomes really a material constant.

Conclusion

We hope this short blog post can help you better understand the different views on the quality factor and equivalent circuit modeling (ECM), and find the one that suits your applications the best. In particular, the differences between \(Q_{loss}\) and \(Q_{BW}\) can be seen as a specific example of real-time and post-processed ECM.

For your convenience, we summarize the points in the table below:

If you are interested to know more, please get in touch to set up a demo and discussion.