## 2 Materials and Methods
Finally, a \(100\) k\(\Omega\) negative temperature coefficient (NTC) thermistor is positioned immediately below the aluminum plate for monitoring and controlling the heater using a PID algorithm implemented on the microcontroller firmware, while a second thermistor of the same type is used to measure the temperature of the heat sink, in order to know and check the temperature difference between the two sides of the cell. The output of the cell is then connected to the programmable DC electronic load BK8600 by B&K Precision, which is controlled by the LabVIEW software and emulates the equivalent output resistive load, performing a resistive sweep and providing instantaneous output voltage and current of the cell under test. The final implementation of the system is illustrated in Figure 2.  
Figure 1: Single couple of P-type and N-type materials constituting a thermoelectric generator. The test system was then employed, to characterize the cell at three values of temperature difference between the aluminum bed and the heatsink  
\[\Delta\mathrm{T1}=\mathrm{T_{bed,1}-T_{heatsink,1}}=4\ ^{\circ}\mathrm{C};\]  
\[\Delta\mathrm{T2}=\mathrm{T_{bed,2}-T_{heatsink,2}}=8\ ^{\circ}\mathrm{C};\]  
\[\Delta\mathrm{T3}=\mathrm{T_{bed,3}-T_{heatsink,3}}=16\ ^{\circ}\mathrm{C}.\]  
A temperature difference lower than \(4\ ^{\circ}\mathrm{C}\) was not considered, since the thermal conductance \(K\) is not infinite and the temperature between hot and cold side tends to equalize, leading to a system collapse, while temperature differences higher than \(16\ ^{\circ}\mathrm{C}\) were not investigated, according to the real scenario of low temperature, low power energy harvesting as from human body heat. For the SPICE model identification, a simple electrical description has been considered here to represent the TEG cell [40], which consists of an ideal generator \(V_{TEG}\) in series with an internal resistance \(R_{TEG}\) (Figure 3). The equivalent voltage supplied by the internal generator relative to this model is given by the temperature gradient applied between the two faces of the cell multiplied by the Seebeck coefficient \(S\), which should be obtained from the measurements. The internal resistance of the model is also temperature-dependent, as is the resistivity of each material, so it is necessary to calculate its thermal coefficient, that could be positive or negative. The expressions of \(V_{TEG}\) can be identified from the measured data by comparing the measured open-circuit voltages with the values of the chosen temperature differences  
\[ \begin{split} S_{TEG,1}=\frac{V_{oc1}}{\Delta T_{1}},\quad S_{TEG,2 }=\frac{V_{oc2}}{\Delta T_{2}},\ S_{TEG,3}=\frac{V_{oc3}}{\Delta T_{3}}\\ S_{TEG}=\ \frac{1}{n}\ \sum_{i=1}^{n}S_{TEG,i},\ n=3\end{split}  \tag{3}\]  
Figure 3: Adopted electrical description of the TEG cell. Figure 2: Block scheme of the test platform. where \(V_{OCI}\) is the measured open-circuit voltage at the temperature difference \(i\). The internal resistance \(R_{TEG}\) can be computed, at each temperature gradient of analysis, by means of the voltage divider rule, where \(V_{out}\) is the output voltage of the cell over the equivalent load  
\[R_{TEG}=\frac{V_{TEG,\Delta Ti}-V_{out,\Delta Ti}}{V_{out,\Delta Ti}}.R_{Load,i },\ i=1,2,3 \tag{4}\]  
Therefore, measurements are used to obtain a polynomial expression of the first order  
\[ \begin{array}{l}R_{TEG,i}=\ \alpha_{i}\Delta T_{i}+\beta_{i}\\ R_{TEG}=\ \frac{1}{n}\sum\limits_{i=1}^{n}\alpha_{i}\Delta T+\frac{1}{n}\sum \limits_{i=1}^{n}\beta_{i}=\alpha\cdot\Delta T+\beta\ \ \ \ \ n=3\end{array}  \tag{5}\]  
where \(\alpha\) represents the thermal coefficient while \(\beta\) is a fixed resistive term. Once the SPICE model has been obtained by means of the above-described method, parametric load sweep simulations can be conducted and data can be compared with measurements, in order to validate the model at different temperature differences.