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:

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