# Analytical Modeling of the Maximum Power Point with Series Resistance
## 3 The Maximum Power Point
### Maximum Power Point Voltage  
The Taylor expansion of the principal branch of [PERSON]'s W function, W\({}_{0}\), is given by:  
\[\rm W_{0}(x)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}x^{n}=x-x^{2}+\frac{3}{2 }x^{3}-\frac{8}{3}x^{4}\ldots \tag{9}\]  
which converges as long as \(x\leq 1/e\). The series expansion in Equation (9) can be used to find an approximate analytical solution to Equation (8). To do so, notice that \(z_{\rm mpp}\) may be small for typical solar cells. To see this, note that \(i_{0}\) is given by \(i_{0}\approx qCV_{\rm t}\exp[-\varepsilon_{s}/q_{\rm t}]E_{g}^{2}\), where \(E_{g}\) is the bandgap of the semiconductor and \(C\) is a constant involving the speed of light, [PERSON]'s constant, and the external radiative efficiency (ERE) [17]. The latter is used to account for nonradiative recombination. \(z(V)\) can then be written as:  
\[z(V)=rCE_{g}^{2}\exp\bigg{[}\frac{V}{V_{\rm t}}-\frac{E_{g}/q}{V_{\rm t}}+ \frac{i_{\rm G}r}{V_{\rm t}}\bigg{]}. \tag{10}\]  
The voltage is limited by the bandgap, and therefore, it holds that \(V-\varepsilon_{s}/q<0\). As long as the value of \(r\) is not excessively large, the exponent in Equation (10) will be negative, resulting in a small \(z\). Assuming a reasonable cell quality, the series expansion in Equation (9) can be used to approximate W\((z_{\rm mpp})\) by \(z_{\rm mpp}\) in Equation (8). To show this, Figure 1 displays the values of W\((z_{\rm mpp})\) (continuous lines) and \(z_{\rm mpp}\) (crosses) as a function of the bandgap for various values of \(r\). There, it is confirmed that, for large bandgaps,\(\text{W}(z_{\text{mpp}})\) is well approximated by \(z_{\text{mpp}}\), even for values of \(r\) up to \(5\)\(\Omega\cdot\text{cm}^{2}\). As expected from Equation (10), Figure 1 shows that the approximation is more accurate for lower values of \(r\). In this context, it is worth noting that typical values for area-normalized series resistance usually are below \(2\)\(\Omega\cdot\text{cm}^{2}\) for both laboratory and commercial solar cells [18]. Therefore, \(\text{W}(z_{\text{mpp}})\) should be well approximated by \(z_{\text{mpp}}\) for most solar cells. Figure 1 is obtained assuming an AM1.5G spectrum and ERE value of \(10^{-4}\). The displayed values correspond to values of \(V_{\text{mpp}}\) that were obtained by solving Equation (8) numerically for various values of \(r\) and \(E_{\text{g}}\). Approximating \(\text{W}(z_{\text{mpp}})\) by \(z_{\text{mpp}}\) in Equation (8) results in:  
\[\frac{1}{z_{\text{mpp}}}\frac{V_{\text{G}}}{V_{\text{t}}}=\frac{1+z_{\text{mpp }}+\frac{V_{\text{mpp}}}{V_{\text{t}}}}{1+z_{\text{mpp}}}. \tag{11}\]  
Focusing now on the right-hand side of Equation (11), since \(z_{\text{mpp}}\) needs to be small so that the approximation \(\text{W}(z_{\text{mpp}})\approx z_{\text{mpp}}\) is accurate, the term \(V_{\text{mpp}}/V_{\text{t}}\) will dominate over \(z_{\text{mpp}}\) in the numerator. Equation (11) can then be simplified to:  
\[\frac{1}{z_{\text{mpp}}}\frac{V_{\text{G}}}{V_{\text{t}}}=\frac{1+\frac{V_{ \text{mpp}}}{V_{\text{t}}}}{1+z_{\text{mpp}}}, \tag{12}\]  
which is analytically solvable. The first step in finding the solution is to insert \(z_{\text{mpp}}\). After a bit of manipulation, Equation (12) becomes:  
\[\left(1+\frac{V_{\text{mpp}}-V_{\text{G}}}{V_{\text{t}}}\right)\exp\left[1+ \frac{V_{\text{mpp}}-V_{\text{G}}}{V_{\text{t}}}\right]=\frac{V_{\text{G}}}{ V_{0}}\exp\left[1-2\frac{V_{\text{G}}}{V_{\text{t}}}\right], \tag{13}\]  
which has the closed-form solution:  
\[V_{\text{mpp}}=V_{\text{G}}+V_{\text{t}}\left(\text{W}\left[\frac{V_{\text{G }}}{V_{0}}\exp\left[1-2\frac{V_{\text{G}}}{V_{\text{t}}}\right]\right]-1 \right). \tag{14}\]  
Finally, inserting the definitions for \(V_{\text{G}}\) and \(V_{0}\) yields:  
Figure 1: \(\text{W}(z_{\text{mpp}})\) (continuous lines) and \(z_{\text{mpp}}\) (crosses) as a function of the energy gap, \(E_{\text{g}}\), for various values of \(r\). The graph is obtained assuming an AM 1.5G spectrum and ERE = \(10^{-4}\).