# Thermal Control of CubeSat Electronics Using Thermoelectrics
## 2 Methodology
### Calculating CPU Temperature  
For the baseline example with no thermoelectric module, the increase in temperature of the CPU under hypothetical operation is calculated by multiplying the emitted CPU power (in W) by the thermal resistance of the system (in \({}^{\circ}\)C/W). In the analysis, the CPU power was set and varied between 0 and 60 W, in 0.1 W steps. Likewise, the supposed thermal resistance was varied between 0 and 1.2 \({}^{\circ}\)C/W, in 0.01 \({}^{\circ}\)C/W steps. The CPU temperature was then calculated by adding this increase to the assumed initial temperature inside the payload, prior to the electronics becoming operational. The internal payload temperature would vary throughout orbit as a function of attitude, but for low-Earth orbit is typically in the range \(-\)20 \({}^{\circ}\)C to +70 \({}^{\circ}\)C [14]. In this study, this initial temperature  
Figure 1: Empirical models for the total heat dissipated at the hot side of the thermoelectric module (black lines) and the temperature difference (\(\Delta T\)) across the module (red lines), as a function of the heat pumped at the cold-side [12]. The legend shows the average operating temperature of the module when the performance testing was carried out, in \({}^{\circ}\)C.  
was considered to be either 25 \({}^{\circ}\)C or 75 \({}^{\circ}\)C, since thermoelectric performance data was available for these temperatures. It was important that both cases were considered, since the module's performance varied as a function of its average temperature.  
For the thermoelectric-cooled comparison, the cold-side temperature of the thermoelectric module was considered equivalent to the CPU temperature from the baseline example. This cold-side temperature was calculated in two steps: First, a temperature was calculated by multiplying the total heat dissipated at the hot-side of the thermoelectric device according to the set thermal resistance of the system. Note that this total included both the emitted CPU power as before, as well as the additional Joule heat generated in the thermoelectric device. The total heat dissipated was found by reference to the appropriate empirical model (black line) in Figure 1. For example, if the initial temperature was 25 \({}^{\circ}\)C and 40 W was to be removed from the CPU, the total heat removed at the hot side was (approximately) 75 W, as indicated by the short-dashed line in Figure 1. Likewise, by referring to the appropriate \(\Delta T\) empirical model in Figure 1 (red lines), the imposed temperature difference across the thermoelectric device was then identified and subtracted to give the cold-side temperature. Comparison of the cold-side temperature and the baseline case allowed the benefit, or detriment, enacted by the thermoelectric device to be calculated. MATLAB was used to carry out all combinations of calculations and steps used in the calculations are shown in Figure 2.  
Figure 2: Steps involved in the calculation of CPU temperature for the baseline case (radiator only) and in the case where a thermoelectric cooler is used. In each calculation the CPU power was set and varied between 0 and 60 W, in 0.1 W steps, the thermal resistance was set and varied between 0 and 1.2 \({}^{\circ}\)C/W, in 0.01 \({}^{\circ}\)C/W steps, and the initial temperature was set as either 25 \({}^{\circ}\)C or 75 \({}^{\circ}\)C. For thermoelectric cooling the impact of Joule heating and imposed temperature difference were incorporated using linear fits to the manufacturer’s empirical data from Figure 1. Diagrams at the bottom show the schematic structure of the simple thermoelectric module (TEM) cooling geometry and the baseline set-up.