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

AI Thread Summary
The discussion revolves around finding the maximum power of a photovoltaic (PV) cell, focusing on the relationship between short circuit current (Isc), reverse saturation current (Irs), and voltage (V). The user attempts to derive the maximum power condition but realizes they have made an error in their calculations, particularly in the signs and assumptions regarding maximum current and voltage. They clarify that maximum power occurs when both Vmax and Imax can be achieved together, which initially seemed impossible. After receiving feedback, the user identifies their mistake as a rearrangement error in their equations. Ultimately, they express gratitude for the assistance in resolving their confusion.
whatisreality
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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!
 
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whatisreality said:
For maximum power ##P_{max} = I_{max}V_{max}##
Only if Vmax and Imax can occur together, which does seem impossible.
 
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haruspex said:
Only if Vmax and Imax 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.
 
whatisreality said:
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 Irs and do dP/dV.
Is IL a constant?
 
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haruspex said:
Write out P in terms of V and Irs and do dP/dV.
Is IL 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!
 
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