- #1
whatisreality
- 290
- 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.
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!
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!
Last edited: