# A Thermopile-Based Colorimetric Temperature Measurement Method for Arbitrary Bandwidth
## 4 Simulation Results and Discussion  
### Simulation Based on Thermopile Product Parameters  
Firstly, several thermopiles from well-known manufacturers are selected for simulation. The selected thermopiles and their parameters are as follows:  
1. Hamamatsu T15770 (Hamamatsu Photonics, Hamamatsu, Japan): responsivity of 50 V/W. The spectral transmittance of the optical filter is shown in Figure 1a, with a passband wavelength range of 4.2\(\sim\)4.9 \(\upmu\)m [19];
2. TE TS318-1B0814 (TE Connectivity, Schaffhausen, Switzerland): responsivity of 50 V/W. The spectral transmittance of the optical filter is shown in Figure 1b, with a passband wavelength range of 7.5\(\sim\)15 \(\upmu\)m [20];
3. TE TS305-11C55 (TE Connectivity, Schaffhausen, Switzerland): responsivity of 84 V/W. The spectral transmittance of the optical filter is shown in Figure 1c. Although the manufacturer does not provide the transmittance curve for wavelength greater than 25 \(\upmu\)m, this does not affect the simulation results. It is assumed that the passband wavelength range is 6\(\sim\)25 \(\upmu\)m [20].  
Using MATLAB R2022b (9.13.0.2049777) for simulation, the aim is to derive the \(Q\)-\(T\) relationship curve for the electric signal ratio \(Q\) and temperature \(T\). The specific simulation steps are as follows:  
1. Import the spectral transmittance curves of the two selected thermopiles into MATLAB, convert them to binary format, and extract the raw data of the curves from the binary images for further processing and saving;
2. Import the responsivity data of the two selected thermopiles and the raw data of the spectral transmittance curves obtained in step 1 into MATLAB. Integrate them according to Equation (18). For each temperature value \(T\), compute \(V_{1}\) and \(V_{2}\). Then, using Equation (19), calculate the electric signal ratio \(Q\) at the current temperature \(T\);
3. Within the range of 273\(\sim\)1000 K, obtain a \(T\) value at regular step intervals and repeat step 2. This will yield a \(Q\) value for each corresponding \(T\) value, ultimately resulting in the \(Q\)-\(T\) relationship curve.  
First, T15770 and TS318-1B0814 are selected for simulation. The wavelength ranges of the infrared bands absorbed by these two thermopiles do not overlap. The \(Q\)-\(T\) relationship curve obtained from the simulation is shown in Figure 2a. Next, TS318-1B0814 and TS305-11C55 are selected for simulation. The wavelength ranges of the infrared bands absorbed by these two thermopiles do overlap. The resulting \(Q\)-\(T\) relationship curve is shown in Figure 2b.  
According to the results shown in Figure 2, when the wavelength ranges of the infrared bands absorbed by the selected thermopiles do not overlap, there is a strictly monotonic relationship between temperature \(T\) and the electric signal ratio \(Q\). This allows for temperature measurement based on the output \(Q\) value. However, when the wavelength ranges of the infrared bands absorbed by the selected thermopiles do overlap, the relationship between temperature \(T\) and the electric signal ratio \(Q\) is not strictly monotonic. In this case, a single \(Q\) value corresponds to two different \(T\) values, making it impossible to determine the temperature solely based on the electric signal ratio. The simulation results are consistent with the mathematical proof.  
Figure 1: Thermopile spectral transmitances. (**a**) Spectral transmittance of T15770; (**b**) spectral transmittance of TS318-1B0814; and (**c**) spectral transmittance of TS305-11C55.