# Development of a Thermal Analysis Tool for Linear Machines
## 4 Validation of the Thermal Model  
Two different linear machines have been used to validate the thermal analysis tool: a linear induction machine, and a linear switched-flux machine. The idea behind this double validation was to demonstrate the generic nature of the tool. The main parameters of the machines can be found in Table 7. Figure 8 shows the prototypes that were tested. The LIM has a double-layer distributed winding, and 3 slots at each side of the machine are half-filled. On the other hand, the LSFPM has a double-layer concentrated winding. The width of the edge slots is half of that of the rest of slots. The end slots are opened, so there are no teeth at the ends of the machine. The validation of the thermal model was done via DC thermal tests. In these tests, the machine was supplied with a constant DC power, with the phase windings connected in series. This way, the magnetic losses and mechanical friction losses were avoided, and the overall heat flow of the machine was controlled directly with a power supply. The main advantage of DC thermal tests is that the only voltage in the phase coils is generated because of the resistance of the armature winding. Thus, if the current and the voltage are measured, the value of the resistance can be calculated with Ohms law,  
\[R=\frac{V}{I}. \tag{13}\]  
Then, the resistance can be used to obtain the average temperature of the copper of the machine. \begin{table}
\begin{tabular}{l c} \hline
**Element** & **Correlation** \\ \hline Top of the Yorke Surface & Vertical plate \\ Bottom of the Machine & Vertical plate \\ Front Side of the Yorke Surface & Vertical plate \\ Rear Side of the Yorke Surface & Vertical plate \\ Left End of the Machine & Hot horizontal plate, bottom \\ Right End of the Machine & Hot horizontal plate, top \\ Top of the End Winding Block & Vertical plate \\ Bottom of the End Winding Block & Vertical plate \\ Front side of the End Winding Block & Vertical plate \\ Rear side of the End Winding Block & Vertical plate \\ Left End of the End Winding Block & Hot horizontal plate, bottom \\ Right End of the End Winding Block & Hot horizontal plate, top \\ Aluminium Plate & Vertical plate \\ \hline \end{tabular}
\end{table} 
Table 6: Correlations for the vertical layout with the vehicle structure. \begin{table}
\begin{tabular}{l c c} \hline
**Parameter** & **Unit** & **LIM** & **LSFPM** \\ \hline Number of Slots & mm & 15 & 7 \\ Yoke Thickness & mm & 15 & 2.5 \\ Slot Height & mm & 35 & 11.9 \\ Primary Slot Width & mm & 12 & 12.8 \\ Primary Tooth Width & mm & 9 & 4.1 \\ Stack Length & mm & 130 & 30 \\ Width of the PMs & mm & - & 3 \\ Height of the PMs & mm & - & 14.4 \\ Slot Liner Thickness & mm & 0.3 & 1 \\ Wedge Thickness & mm & 1.7 & 1.4 \\ Fill Factor & - & 0.35 & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 7: Main parameters of the prototypes for the thermal tests. The resistance of the phase windings varies according to  
\[R=R_{0}\cdot[1+\alpha_{cut}(T-T_{0})], \tag{14}\]  
where \(R_{0}\) is the resistance of the series connected phase windings, \(\alpha_{cut}\) is the temperature coefficient of the copper, 0.00393, and \(T_{0}\) is the temperature at which \(R_{0}\) was measured. Assuming that the machine is at ambient temperature at the beginning of the tests, \(T_{0}\) can be considered to be the same as the ambient temperature. It can be deduced from (14) that the first step when testing the machines is to measure the initial resistance of the phase windings and the ambient temperature when the machine is cold. Then, the data from the tests can be used to calculate the temperature of the copper with  
\[T=\frac{\frac{V/I-1}{R_{0}}-1}{\alpha_{cut}}+T_{0}. \tag{15}\]  
Thermocouples were used in the rest of the points where the temperature was wanted to be known. The placement of the thermocouples in the thermal tests to the LIM and the LSFPM is shown in Figure 9. And the temperatures of the machines under different conditions are given in Figures 10 and 11.