## 3 Simulations
### Irradiance Distributions  
Next, we analyzed the light distribution of a hybrid CPV module with a pyramidal SOE (\(\theta\) = 30\({}^{\circ}\), h = 15 mm). Figure 4 shows the simulation results for the six silicon solar cells using the layout from Figure 1a. Since the hybrid CPV module is symmetric, we only needed to identify six of the 12 silicon solar cells. The irradiances on the Si #7-12 were 10.8 W/m\({}^{2}\), 13.8 W/m\({}^{2}\), 11.0 W/m\({}^{2}\), 9.2 W/m\({}^{2}\), 9.2 W/m\({}^{2}\), and 9.3 W/m\({}^{2}\), respectively. We analyzed the irradiances and irradiance distributions using the effective wavelength range. Although single crystal silicon modules are more efficient than smaller polycrystalline silicon solar cells, we simulated polycrystalline silicon solar cells to better match the experimental measurements to follow. Figure 4b shows that, although the silicon solar cell placed next to the SOE was affected by the shadow of the SOE, that solar cell still collected the highest power (13.9 W/m\({}^{2}\)). In addition, the silicon solar cells placed away from the CPV also showed irradiance of at least 9.2 W/m\({}^{2}\) by absorbing diffuse light. Considering that hybrid CPV modules are constructed as arrays, we used a square Fresnel lens, which seems to have affected the irradiance distribution.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**SOE Height (mm)** & **ETJ,eff (W/m\({}^{2}\))** & **ETJ,loss (W/m\({}^{2}\))** & **ETJ,loss/EIJ,eff Ratio (\%)** & **ESi,eff (W/m\({}^{2}\))** & **ESi,loss (W/m\({}^{2}\))** \\ \hline
0 (No SOE) & 590.8 & 20.4 & 3.5 & 131.2 & 7.1 \\
5 & 601.8 & 26.1 & 4.3 & 131.5 & 9.8 \\
10 & 605.2 & 30.9 & 5.1 & 130.3 & 10.4 \\
15 & 606.2 & 34.5 & 5.7 & 127.4 & 9.9 \\
20 & 608.3 & 37.9 & 6.2 & 124.8 & 9.3 \\
100 & 609.7 & 47.4 & 7.8 & 85.4 & 1.6 \\ \hline \end{tabular}
\end{table}
Table 2: Irradiance on the triple-junction and silicon solar cells of the hybrid CPV module with different mirror heights, as calculated using TracePro. We again separated the results into the effective (0.3–1.299 μm) and lossy (1.3–2.5 μm) wavelength regions.  
Figure 3: SOE designs with different mirror heights of (a) 0 mm (no SOE), (b) 5 mm, (c) 10 mm, (d) 15 mm, (e) 20 mm, and (f) 100 mm.  
To identify the irradiance distribution of the TJ and silicon solar cells according to the design of the SOE, we examined the three cases shown in Figure 5. In the perpendicular SOE, the concentrated power on the TJ solar cell was the smallest (326.8 W/m\({}^{2}\)), while the irradiance on the silicon solar cells was the largest (135.8 W/m\({}^{2}\)). Without an SOE, the irradiance on the TJ and silicon solar cells were 590.8 W/m\({}^{2}\) and 131.2 W/m\({}^{2}\), respectively. When using the pyramidal SOE optimized above, we saw the largest power (606.2 W/m\({}^{2}\)) concentrated on the TJ solar cell, and the smallest power (127.4 W/m\({}^{2}\)) on the silicon solar cell. The solar cell conversion efficiency relation is:  
\[\eta\ =\ \frac{P_{MAX}}{P_{input}}\times 100\ (\%) \tag{1}\]  
where \(\eta\) is the conversion efficiency, P\({}_{MAX}\) is the maximum output power, and \(P_{input}\) is the input optical power. Therefore, assuming a TJ solar cell conversion efficiency of 38.5% when using a pyramidal SOE, we calculate \(P_{MAX}\) as 233.4 W/m\({}^{2}\). In addition, assuming an 18% conversion efficiency for the polycrystalline silicon solar cell, we expect to generate 22.9 W/m\({}^{2}\) of electric power. Using the same assumption, we predict maximum output powers of 125.8 W/m\({}^{2}\) and 24.4 W/m\({}^{2}\) for the perpendicular SOE, and maximum output powers of 227.5 W/m\({}^{2}\) and 23.6 W/m\({}^{2}\) for the no SOE case. We will compare these predictions with experimental measurements in the next section.  
Figure 4: Irradiance distributions of the six silicon solar cells for a pyramidal SOE with \(\theta\) = 30\({}^{\circ}\) and h = 15 mm: (**a**) Si #1, (**b**) Si #2, (**c**) Si #3, (**d**) Si #4, (**e**) Si #5, and (**f**) Si #6. The size of each cell was 80 \(\times\) 55 mm\({}^{2}\).