User:Th5th/MPP derivation

The maximum power point of a solar cell occurs when $$\frac{dP}{dV}=0$$ where P = VI.


 * $$I = I_{L} - I_{0} \left[\exp\left(\frac{qV}{nkT}\right) - 1\right]$$

And so P is given by:


 * $$P = VI = VI_{L} - VI_{0} \left[\exp\left(\frac{qV}{nkT}\right) - 1\right]$$

Differentiating yields:



\begin{align} \frac{dP}{dV} & = I_{L} - \frac{d}{dV}\left[VI_{0}\exp\left(\frac{qV}{nkT}\right)\right] + I_{0} \\ & = I_{L} - I_{0}\exp\left(\frac{qV}{nkT}\right) - \frac{qVI_{0}}{nkT}\exp\left(\frac{qV}{nkT}\right) + I_{0} \\ & = 0 \end{align} $$