# Explanation of 5 Parameter Model for Solar Cells

I am having a hard time understanding this equation, the 5 Parameter model for photovoltaic cells.

$$\I = I_{ph} - I_o[\exp(\frac{V + R_sI}{nV_t})-1] - (\frac{V+R_sI}{R_{sh}})\$$

My goal is to apply a regression algorithm on this to extract the 5 parameters

($$\I_{ph}, I_o, R_s, R_{sh}, n\$$)

How do we interpret this equation?

If I have all the five parameters defined how will I get the value of current with just voltage. This seems confusing as $$\I\$$ appears on both sides of the equation.

• What are your goals in doing this? You can probably get what you need from I/V curves published by the manufacturer. I am not sure why you need the actual 5 parameters. – mkeith Jan 22 at 10:04
• Web searches such as this and this probably provide good guidance. – Russell McMahon Jan 22 at 10:36
• @mkeith My goal is to run a regression algorithm on the data that I have and extract these parameters from them. The problem with using the values provided by the manufacturer is that they do not account for mismatch error(they are the best case scenario). Leaving that aside for a minute, if you had the 5 parameters, how would you find the amount of current produced given voltage and temperature(for thermal voltage)? – Aakash Sasikumar Jan 23 at 6:07
• @RussellMcMahon thank you, but those papers don't really talk about extracting the parameters(I have gone through a few of them). I am currently looking at this paper, but even this one doesn't clearly explain how current and voltage are got from the trained model. – Aakash Sasikumar Jan 23 at 6:13
• Well, if you are sure it won't converge, then you could be right. But my thinking is that if your initial estimate is reasonably close (which it should be since you have empirical data already), then it SHOULD converge unless the function is very odd in the region of interest (contains inflection points or something). But it appears to me that your math knowledge is probably greater than mine, so you may well be right. Just don't give up to easily or without trying it in an excel sheet or something. – mkeith Jan 27 at 1:38

Well, this is silly. The answer I was looking for was in a Wikipedia post on the Theory of Solar Cells. Basically these types of equations are solved using the Lambert W function.

The original equation

$$\I = I_{ph} - I_o[\exp(\frac{V + R_sI}{nV_t})-1] - (\frac{V+R_sI}{R_{sh}})\$$

becomes

$$\I=\frac{(I_{ph}+I_o)-V/R_{sh}}{1+R_s/R_{sh}}-\frac{nV_t}{R_s}W(\frac{I_oR_s}{nV_t(1+R_s/R_{sh})}\exp(\frac{V}{nV_t}(1-\frac{R_s}{R_s+R_{sh}})+\frac{(I_{ph}+I_o)R_s}{nV_t(1+R_s/R_{sh})}))\$$

Now its very simple to calculate the current and power.