User:Th5th/MPP derivation
The maximum power point of a solar cell occurs when where P = VI.
And so P is given by:
Differentiating yields:
The maximum power point of a solar cell occurs when 
![{\displaystyle I=I_{L}-I_{0}\left[\exp \left({\frac {qV}{nkT}}\right)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef60e53812c493bfa5ac3bb42c1bb5a316865ca)
And so P is given by:
![{\displaystyle P=VI=VI_{L}-VI_{0}\left[\exp \left({\frac {qV}{nkT}}\right)-1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5eaac8309284c67f0b3883022bbe638b769cfb89)
Differentiating yields:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72e49ac9032ec673bbe27b62ab1a2cc526631bc)