# A Thermopile-Based Colorimetric Temperature Measurement Method for Arbitrary Bandwidth
## 3 Mathematical Proof of Temperature Measurement Methods
### Thermopile-Based Colorimetric Thermometry with Arbitrary Bandwidth  
Thermopile sensors operate based on the Seebeck effect and infrared radiation principles. They convert the absorption of infrared radiation into thermal energy through the photothermal effect, detecting changes in resistance or electric potential to achieve accurate and efficient infrared sensing [17]. They are characterized by their broad absorption of infrared radiation wavelength, compact size, high sensitivity, no need for cooling, and low detection costs [18]. Building upon the established monotonicity, considering the many advantages of thermopiles as detectors for infrared radiation, this article extends the traditional colorimetric thermometry to propose an arbitrary bandwidth colorimetric thermometry based on thermopiles. This method employs thermopiles as infrared detectors, enabling the selection of infrared bands of any width rather than being confined to narrow bands at specific wavelengths. The thermopiles absorb infrared radiation within two selected bands and convert the received energies into two voltage or current signals. The temperature of the measured object can be determined by comparing the ratio of these signals. As long as the ratio of the electrical signals \(Q(T)\) remains strictly monotonic with respect to \(T\), the output ratio of electrical signals directly yields a uniquely determined temperature value. By calibrating with a standard radiation source (such as a blackbody furnace), the relationship curve between the thermopile electrical signal ratio and the measured temperature can be fitted. Ultimately, the temperature of the measured object can be determined based on the output electrical signal ratio. Here is the proof that \(Q(T)\) is strictly monotonic with respect to \(T\):  
Combining Equation (11), we can obtain the electric signals \(V_{1}(T)\) and \(V_{2}(T)\) generated by two thermopiles absorbing infrared radiation within two different spectral bands:  
\[ \begin{cases}V_{1}(T)=\int_{\lambda_{A}}^{\lambda_{b}}M(\lambda,T)S_{1}( \lambda)R_{1}(\lambda)d\lambda=\int_{\lambda_{A}}^{\lambda_{b}}M(\lambda,T)G _{1}(\lambda)d\lambda\\ V_{2}(T)=\int_{\lambda_{c}}^{\lambda_{d}}M(\lambda,T)S_{2}(\lambda)R_{2}( \lambda)d\lambda=\int_{\lambda_{c}}^{\lambda_{d}}M(\lambda,T)G_{2}(\lambda)d \lambda^{\prime}\end{cases}  \tag{18}\]  
in the equation, \(G_{1}(\lambda)=S_{1}(\lambda)R_{1}(\lambda)\), \(G_{2}(\lambda)=S_{2}(\lambda)R_{2}(\lambda)\), where \(S_{1}(\lambda)\) and \(S_{2}(\lambda)\) denote the spectral transmitances of the optical filters for two thermopiles, and \(R_{1}(\lambda)\) and \(R_{2}(\lambda)\) represent the responsivities of the respective thermopiles. Additionally, \(\lambda_{a}\), \(\lambda_{b}\), as well as \(\lambda_{c}\), \(\lambda_{d}\), correspond to the lower and upper wavelength limits of the infrared radiation bands absorbed by these thermopiles, furthermore, it holds that \(\lambda_{a}<\lambda_{b}\), \(\lambda_{c}<\lambda_{d}\). The ratio \(Q(T)\) between the two electric signals is given by:  
\[Q(T)=\frac{V_{1}(T)}{V_{2}(T)}=\frac{\int_{\lambda_{a}}^{\lambda_{b}}M( \lambda,T)G_{1}(\lambda)d\lambda}{\int_{\lambda_{c}}^{\lambda_{d}}M(\lambda,T) G_{2}(\lambda)d\lambda}=\frac{\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)G_{1}( \lambda)d\lambda}{\int_{\lambda_{c}}^{\lambda_{d}}M_{0}(\lambda,T)G_{2}( \lambda)d\lambda}. \tag{19}\]  
Applying the same procedure as in Equation (12) to Equation (19), we eventually obtain:  
\[ \begin{split}\frac{d\ln Q(T)}{dT}&=\frac{\frac{d \left(\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)G_{1}(\lambda)d\lambda \right)}{dT}}{K_{1}(T)}-\frac{d\left(\int_{\lambda_{c}}^{\lambda_{d}}M_{0}( \lambda,T)G_{2}(\lambda)d\lambda\right)}{dT}\\ &=\frac{K_{2}(T)\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T) G_{1}(\lambda)C_{2}\lambda^{-1}T^{-2}d\lambda-K_{1}(T)\int_{\lambda_{c}}^{ \lambda_{d}}M_{0}(\lambda,T)G_{2}(\lambda)C_{2}\lambda^{-1}T^{-2}d\lambda}{K _{1}(T)K_{2}(T)}\\ &=\frac{D(T)}{K_{1}(T)K_{2}(T)},\end{split}  \tag{20}\]in the equation,  
\[