# Equivalent Circuit Model Extraction for a SAW Resonator: Below and above Room Temperature

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{11}and Y

_{22}) rather than a real Lorentzian function to fit only the real part of them [34,35,36], thereby allowing an improvement in the determination of the resonant parameters, which are used for the extraction of the equivalent-circuit elements.

## 2. Materials and Methods

#### 2.1. Measurement Setup

- Measurement chamber: It includes a Peltier cell and a Pt100 thermoresistance, used for the temperature actuation and control, and a polarity-reversal relay. It is also equipped with all the connections needed by the device under test.
- Measurement instruments: These include all the instrumentation involved in the measurement process. The Agilent E3631A power supply is used to provide the supply voltage to the Peltier cell and its cooling fan. The Agilent 34401A digital multimeter is used to measure the Pt100 resistance and convert it into a temperature unit as a feedback signal for the control system. Finally, the Agilent 8753ES VNA is employed for the electrical characterization, in terms of Y- parameters, of the microwave device under test.
- Control system unit: Each measurement instrument included in the control chain loop is connected to a desktop PC that acts as a central control unit. It represents the third and last part of the measurement system. Through the IEEE 488.2 GPIB interface, the PC is able to set the right power supply for the Peltier cell and acquire the Pt100 resistance value and the S- parameters from the VNA. In addition, a custom-developed software allows not only the real-time data acquisition and processing, but also allows saving measurements on files for post-processing analysis.

^{2}with a thickness of 4 mm. The measuring system adopted can be used to carry out measurements at temperatures both above and below room temperature. In the latter case, the surface in contact with the SAW becomes cooler, while the opposite one tends to heat up accordingly. To prevent the cell from being damaged by excessive overheating and to improve its performance, a passive heat sink was placed in contact with the opposite side of the cell. This heat sink, placed outside the measurement chamber, allows the dispersion of the excess heat with the aid of a fan that, via software, automatically comes into operation when measurements at temperatures below the ambient temperature are carried out.

#### 2.2. Equivalent-Circuit Model

_{01}and C

_{02}, by an R

_{m}L

_{m}C

_{m}series network, and by an ideal transformer. The model was developed considering the physical behavior of the SAW resonator: C

_{01}and C

_{02}represent the static capacitances at each port of the device; R

_{m}, L

_{m}, and C

_{m}are associated with the contributions of damping, inertia, and elasticity, respectively; and, finally, the transformer is meant to represent the conversion between mechanical and electrical energy. The transformer is considered to be ideal and its effect consists of producing a 180° phase shift in Y

_{21}and Y

_{12}.

_{r}can be expressed as:

_{m}, C

_{01}, and C

_{02}can be straightforwardly extracted from Re(Y

_{11}), Im(Y

_{11}), and Im(Y

_{22}) at the resonance, respectively. The other parameters, i.e., L

_{m}and C

_{m}, can be calculated from the SAW resonant frequency and Q factor, the analytical expressions of which are reported in Equations (1) and (2), respectively. The description of the parameters’ extraction methodology is reported in the next section.

_{11}= Y

_{22}and Y

_{21}= Y

_{12}. This property was verified through the measurements carried out on the device. Only few deviations were observed on Y

_{11}and Y

_{22}, while the reciprocity condition (i.e., Y

_{21}= Y

_{12}) was completely fulfilled. For this reason, in this work only the Y

_{11}, Y

_{12}and Y

_{22}parameters are taken into account for the device characterization.

#### 2.3. Equivalent-Circuit Parameter Extraction

_{m}and C

_{01}can be straightforwardly extracted from Re(Y

_{11}) and Im(Y

_{11}), respectively. Considering the expressions previously reported of the Y- parameters at resonance (Equations (6)–(8)), the values of the concerned parameters can be calculated as:

_{02}can be estimated using Im(Y

_{22}):

_{r}and Q values from the measurements carried out on the SAW under test, the values for L

_{m}and C

_{m}can be derived from Equations (1) and (2), and can be written as:

_{11r}), Im(Y

_{11r}), Im(Y

_{22r}), f

_{r}, and Q are obtained from the measurements carried out on the device under test. In particular, f

_{r}is the frequency in which the Re(Y

_{11}) is maximum. Re(Y

_{11r}) and Im(Y

_{11r}) are the values of Re(Y

_{11}) and Im(Y

_{11}) at f = f

_{r}, respectively. Finally, Q is estimated from measurements as the ratio between the resonant frequency and the half-power bandwidth.

_{11r}), Im(Y

_{11r}), Im(Y

_{22r}), f

_{r}, and Q from the acquired measurements is not trivial. In particular, because of the presence of noise in measurements or because of limited points of the acquired spectrum, the estimation of such quantities could be inaccurate or, in other words, the measurement uncertainty may not fit the project requirements. In the literature, there are different strategies in order to increase the accuracy of the determination of f

_{r}and Q

_{r}[42,43,44]. For instance, a Lorentzian fitting can be performed on the acquired data points so that f

_{r}, Q, and the other parameters can be derived from the fitted equation analytically and with higher accuracy [45,46].

_{11}and Y

_{22}parameters.

_{0}and f

_{0}are two complex coefficients estimated by the selected best fit algorithm. The term f

_{0}can be expressed as:

_{r}is the SAW resonant frequency and 2g is the half-power bandwidth of the peak. For the selected Lorentzian function, the Q factor can be written as:

_{n}are complex coefficients, and f

_{c}is a real quantity. In the present work, experimental measurements proved that a good description of the background signal is possible with N = 1. The final function used to model the generic admittance parameter (Y

_{ij}) is:

_{11r}), Im(Y

_{11r}), Im(Y

_{22r}), f

_{r}, and Q), and extract the equivalent-circuit model element values. For this purpose, the scikit-rf package and the non-linear least-squares minimization and curve-fitting (lmfit) library were used. Scikit-rf is an Open Source BSD-licensed package for python designed for RF/Microwave engineering. It was used to manage the acquired S- parameters, making all the conversions and calculations needed. This tool provides a modern and object-oriented library, very useful for data management in RF measurements. The lmfit library was used for the fitting procedure. It implements many optimization methods including the least square and the Levenberg–Marquardt method. The library is free and is released using an open source license.

_{11}and Y

_{22}is obtained by the fitting procedure, the SAW resonant frequency is explicitly given by the coefficient f

_{r}, calculated by the best-fit algorithm. The Q factor is calculated using the Equation (16). The quantities Re(Y

_{11r}), Im(Y

_{11r}), and Im(Y

_{22r}) can be derived from the fitted functions at f = f

_{r}. The equivalent-circuit elements values are thus calculated using the Equations (9)–(13).

_{22}and its fitted function is shown at 0 °C and 100 °C, respectively. Since Equation (14) describes a complex Lorentzian function, both real and imaginary parts are reported in Figure 5 and Figure 7, respectively. As can be observed, the Lorentzian function fits very well the acquired points of the resonant peak, as proof of the reliability of the fitting procedure. The residuals of the Lorentzian fitting and their probability distribution function at 0 °C and 100 °C are depicted in Figure 6 and Figure 8, respectively. Residuals are relatively small (<40 µS) for frequencies close to f

_{r}. The probability distribution functions can be considered normal as additional evidence of the goodness of fitting procedure.

_{r}, Q, and the other parameters with higher accuracy.

## 3. Results and Discussion

_{11}, Y

_{21}, and Y

_{22}at 0 °C, 60 °C, and 100 °C are reported. Plots at other temperatures have been omitted here.

_{r}. This proves that the model validity is not limited to cryogenic temperatures [34], since the extracted model is also able to describe very well the behavior of a two-port SAW resonator at temperatures above the ambient temperature (at least up to 100 °C).

_{r}and Q with the temperature. While the Q factor decreases almost linearly with increasing temperature, the resonant frequency has a parabolic trend with a maximum close to 40 °C. This behavior is not atypical, since it characterizes all the Murata SAW devices that belong to the SAR series [38].

_{m}, L

_{m}, and C

_{m}, is depicted in Figure 13. When the temperature increases, both the inductance and resistance values increase while the capacitance C

_{m}decreases. It is worth noting how the resistance value is related to the resonator Q factor. As described in Section 2.2, from a physical point of view, the resistance R

_{m}represents the damping effect. In other words, it is related to the resonator losses. As the temperature increases, the series resistance R

_{m}increases, thus lowering the resonator Q factor (see Figure 12a and Figure 13c).

_{01}are very close to those of C

_{02}. Except for an abrupt change in the values of the two capacitances when decreasing the temperature from 20 °C to 0 °C, C

_{01}and C

_{02}are almost constant and equal to 2.00 pF and 2.13 pF, respectively. Such behavior was not observed in [34], where a similar SAW was characterized using a cryogenic experimental system. In this case, the abrupt change of C

_{01}and C

_{02}at 0 °C may be ascribed to water vapor condensation on the SAW package affecting the equivalent parasitic capacitances of the case.

## 4. Conclusions

_{m}, L

_{m}, C

_{m}, C

_{01}, and C

_{02}) was described. Moreover, the variation in the SAW resonant frequency and Q factor with the temperature was reported and discussed. By increasing the temperature, the resonant frequency shows a parabolic trend and the Q factor decreases almost linearly.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) DUT connected to the test fixture. (

**b**) Measurement chamber with the test fixture mounted on the Peltier cell.

**Figure 5.**Comparison between measurements and Lorentzian fitting on the (

**a**) Re(Y

_{22}) and (

**b**) Im(Y

_{22}) for the tested SAW resonator at 0 °C.

**Figure 6.**(

**a**) Residuals of the Lorentzian fitting at 0 °C and (

**b**) their probability distribution function that can be considered normal, thereby providing proof of good fitting.

**Figure 7.**Comparison between measurements and Lorentzian fitting on the (

**a**) Re(Y

_{22}) and (

**b**) Im(Y

_{22}) for the tested SAW resonator at 100 °C.

**Figure 8.**(

**a**) Residuals of the Lorentzian fitting at 100 °C and (

**b**) their probability distribution function that can be considered normal, thereby providing proof of good fitting.

**Figure 9.**Comparison between measurements and simulations of the real and imaginary parts of (

**a**) Y

_{11}, (

**b**) Y

_{21}, and (

**c**) Y

_{22}versus the frequency for the tested SAW resonator at 0 °C.

**Figure 10.**Comparison between measurements and simulations of the real and imaginary parts of (

**a**) Y

_{11}, (

**b**) Y

_{21}, and (

**c**) Y

_{22}versus the frequency for the tested SAW resonator at 60 °C.

**Figure 11.**Comparison between measurements and simulations of the real and imaginary parts of (

**a**) Y

_{11}, (

**b**) Y

_{21}, and (

**c**) Y

_{22}versus the frequency for the tested SAW resonator at 100 °C.

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**MDPI and ACS Style**

Gugliandolo, G.; Marinković, Z.; Crupi, G.; Campobello, G.; Donato, N. Equivalent Circuit Model Extraction for a SAW Resonator: Below and above Room Temperature. *Sensors* **2022**, *22*, 2546.
https://doi.org/10.3390/s22072546

**AMA Style**

Gugliandolo G, Marinković Z, Crupi G, Campobello G, Donato N. Equivalent Circuit Model Extraction for a SAW Resonator: Below and above Room Temperature. *Sensors*. 2022; 22(7):2546.
https://doi.org/10.3390/s22072546

**Chicago/Turabian Style**

Gugliandolo, Giovanni, Zlatica Marinković, Giovanni Crupi, Giuseppe Campobello, and Nicola Donato. 2022. "Equivalent Circuit Model Extraction for a SAW Resonator: Below and above Room Temperature" *Sensors* 22, no. 7: 2546.
https://doi.org/10.3390/s22072546