Max power of a photovoltaic cell -- where did I go wrong?

[whatisreality]

1. The problem statement, all variables and given/known da
Show that for a PV cell finding a maximum of power leads to the following equation:
$(1 +\frac{qV_{max}}{kT})\exp\left(\frac{qV_{max}}{kT}\right) = 1 + \frac{I_{sc}}{I_{rs}}$
$I_{sc}$ is short circuit current and $I_{rs}$ is reverse saturation current.

Homework Equations

The Attempt at a Solution

I get pretty close, but I've missed something out. The current is given by:
$I = I_{rs}(\exp(V/V_t)-1)-I_L = I_{rs}(e^{qV/kT}-1)-I_L$
For maximum power $P_{max} = I_{max}V_{max}$ and $I_{max}$ is the short circuit current, which occurs at $V=0$. Subbing $V=0$ into the current equation gives $I_{sc} = -I_L$, so

$I = I_{rs}(e^{qV/kT}-1)+I_{sc}$

Max power is at $\frac{dP}{dV}=0$ so given $P=IV$:

$\frac{dP}{dV} = I_{rs}(e^{\frac{qV_m}{kT}}-1)+\frac{qI_{rs}V_m}{kT}(e^{\frac{qV_m}{kT}})+I_{sc}$

$\left(1+\frac{qV_m}{kT}\right)e^{\frac{qV_m}{kT}} = -\frac{I_{sc}}{I_{rs}}$

That's very close to what I'm looking for, I'm missing a $1$ on the RHS and the sign of the fraction is wrong, have I gone wrong somewhere? I've looked and really can't spot it, thanks for any help!

 

[haruspex]

For maximum power $P_{max} = I_{max}V_{max}$

Only if V_max and I_max can occur together, which does seem impossible.

 

[whatisreality]

Only if V_max and I_max can occur together, which does seem impossible.

Yes, that's wrong too, I saw $I_m$ and $V_m$ in my notes and assumed they meant maximum current and voltage rather than current and voltage at max power, I should have thought that through.

 

[haruspex]

Yes, that's wrong too, I saw $I_m$ and $V_m$ in my notes and assumed they meant maximum current and voltage rather than current and voltage at max power, I should have thought that through.

Write out P in terms of V and I_rs and do dP/dV.
Is I_L a constant?

 

[whatisreality]

Write out P in terms of V and I_rs and do dP/dV.
Is I_L a constant?

Thanks for your help, spotted my mistake, I just rearranged wrong. My last two lines don't follow from each other. Silly mistake!