# Measuring Torque and Temperature in a Rotating Shaft Using Commercial SAW Sensors
## 3 Results
### Calibration of Temperature Sensor  
The dependence of the resonant frequency \(f_{R_{i}}\) of each resonator with respect to temperature is given by the following expression:  
\[f_{R_{i}}(T)=f_{R_{i}}(T_{0})\cdot[1+\ C_{1i}\cdot(T-T_{0})+C_{2i}\cdot(T-T_{0} )^{2}], \tag{3}\]  
where \(f_{R_{i}}(T_{0})\) is the resonant frequency at 25 \({}^{\circ}\)C and \(C_{1i}\) and \(C_{2i}\) are the first and second order temperature coefficients, respectively, of each individual resonator (\(i=1,2\)). The temperature \(T\) is related with the difference between the two relative resonance frequencies \(\Delta f_{r}\) by:  
\[\Delta f_{r}=\frac{f_{\text{R}_{1}}(T)}{f_{\text{R}_{1}}(T_{0})}-\frac{f_{ \text{R}_{2}}(T)}{f_{\text{R}_{2}}(T_{0})}=(C_{11}-C_{12})\cdot\Delta T+(C_{21 }-C_{22})\cdot\Delta T^{2}, \tag{4}\]  
where \(\Delta T=T-T_{0}\) and \(T_{0}=25\)\({}^{\circ}\)C. Since \(C_{11}\) and \(C_{12}\) have opposite signs and \(C_{21}\) and \(C_{22}\) have the same sign, the differential measurement enhances the whole sensor sensibility to temperature and the output linearity significantly while the quadratic dependence is decreased.  
Once \(\Delta f_{r}\) for a given temperature is known, the calculation of \(T\) with Equation (4) is straightforward. However, the behavior of the temperature sensor mounted in the load cell may not be accurately described using the temperature coefficients taken from the sensor's datasheet and, as such, the calibration of the temperature sensor was required. The instrumented load cell was placed inside the thermal chamber (as described in Section 2.1), the sensors and the interrogation unit were connected to dipole antennas and the response of the temperature sensor was recorded at different temperatures. The difference between the relative resonant frequencies with temperature is plotted in Figure 7a. The data were fit to a second order polynomial, allowing for the determination of the first and second order temperature coefficients \(C_{1}=5.06\times 10^{-6}/^{\circ}\)C and \(C_{2}=18.9\times 10^{-9}/^{\circ}\)C\({}^{2}\) (Equation (5))--Table 3 (the values obtained with Equation (4) and the theoretical parameters taken from the manufacturer datasheet are included for comparison purposes). Once the coefficients \(C_{1}\) and \(C_{2}\) are determined, the temperature can be directly calculated applying the quadratic formula to solve Equation (5):  
\[\Delta f_{r}=C_{1}\cdot\Delta T+C_{2}\cdot\Delta T^{2}. \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c} \hline
**Coefficients** & **Theoretical Values** & **Dipole Antennas** & **RF Couplers** \\ \hline \(C_{1}\) (\(\times 10^{-6}/^{\circ}\)C) & 6.2 & 5.06 & 5.47 \\ \(C_{2}\) (\(\times 10^{-9}/^{\circ}\)C\({}^{2}\)) & 8.5 & 18.9 & 12.9 \\ \hline \end{tabular}
\end{table}
Table 3: Coefficients \(C_{1}\) and \(C_{2}\) obtained with dipole antennas and RF couplers. The theoretical values, obtained by replacing the coefficients taken from the sensor datasheet in Equation (4), are included for comparison.  
The temperature readings obtained by the thermocouple and the SAW sensor are plotted as a function of the temperature read by the thermocouple in Figure 7b. As temperature decreases or increases below or above ambient temperature, respectively, there in an increasing error between the values measured by the SAW sensor and the thermocouple. This effect is related with the difficulty in obtaining the accurate value of the temperature at the surface of the cell with the thermocouple.  
Following this calibration, the sensors and the interrogation unit were connected to the couplers and the calibration procedure was repeated. The extracted coefficients can be seen in Table 3. The theoretical coefficients, obtained by replacing the theoretical coefficients given in the sensor datasheet (Table 1) in Equation (4), are included for comparison.