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

  1. Dec 6, 2017 #1
    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.

    2. Relevant equations


    3. 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!
     
    Last edited: Dec 6, 2017
  2. jcsd
  3. Dec 6, 2017 #2

    haruspex

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    Only if Vmax and Imax can occur together, which does seem impossible.
     
  4. Dec 7, 2017 #3
    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.
     
  5. Dec 7, 2017 #4

    haruspex

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    Write out P in terms of V and Irs and do dP/dV.
    Is IL a constant?
     
  6. Dec 7, 2017 #5
    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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