# Decrease of internal resistance of a solar cell while increasing the load resistor?

Imagine to model a solar cell as a power supply $V_p$ in series with an internal resistance $R_p$. This solar cell is connected to a variable load resistance $R_{load}$.

Suppose that I keep the solar cell at fixed distance from a fixed light source (say, a light bulb). The power incident on the solar cell should be a constant, as the distance or the source do not change.

Now I change the value of $R_{load}$ many times and everytime I measure the current $I$ flowing in $R_{load}$ and the voltage across load, $V_{load}$.

Before taking all the measurements I measure, one time only (as it should be a constant), the voltage across the cell once i disconnected the load $V_{open \, circuit}$: that should be $V_p$ and it should not change, as said.

I tried all this procedure experimentally but I noticed one strange thing: Everytime I calculated $R_p$ as $$R_p=\frac{V_{open \, circuit}-V_{load}}{I}$$ and I noticed that $R_p$ changed while $R_{load}$ was being varied! $R_p$ decreased as $R_{load}$ increased!

Is there an explanation for that?

The most strange thing is that, since we are at constant distance and source $P_{incident}$ is constant and we have $P_{produced}=\frac{V^2_p}{R_p}=\eta P_{incident}$, where $\eta$ is the efficiency: if both $\eta$ and $V_p$ are constant, $R_p$ should be a constant too!!

Therefore, is it wrong to assume that $V_p$ is a constant? Or maybe is just $\eta$ changing?

• What range of load resistance did you measure? Mar 5, 2017 at 4:08

The solar cell can only produce an amount of current proportional to the incident light. If the load draws less current than the cell can produce then its output voltage doesn't drop much, indicating a low internal resistance. In this region resistance is dominated by series resistances of the bulk and sheet silicon, surface contacts and interconnects.

As load resistance is reduced it draws more current until getting to the 'knee' where cell output is current limited. Then the voltage drops sharply, which corresponds to a sharp increase in internal resistance. Here the primary internal resistance contributor is the limited current supply. Your formula will only give a sliding increase after the knee because you are using the total current and voltage (relative to open-circuit) instead of the incremental change. To get the true (dynamic) resistance you should use the differences between voltages and currents before and after each decrease in load resistance.

• Thanks a lot for the answer! If I may ask: since $R_{p}$ varies with $R_{load}$, I cannot use maximum power transfer theorem to say that $P_{load}$ is max for $R_{load}=R_{p}$ (the equality does not make sense). I tried to plot measured $P_{load}$ vs $R_{load}$ and indeed the maximum is near the resistance $R_{p}$ calculated when $R_{load}$ was $0 \, \Omega$ but it's not exactly there. So is there a relation that describes for what $R_{load}$ I have the maximum $P_{load}$? Is it necessary to know the function $R_{p}=f(R_{load})$ (assuming that such function exists)? Mar 6, 2017 at 0:04
• Maximum power transfer should occur close to the 'knee' because this is where the product of voltage and current is highest. The cell can be modeled as a diode (regulating the open circuit voltage) in parallel with a current source (causing the steep slope), in series with a small fixed resistance (causing the shallow slope). en.wikipedia.org/wiki/… Mar 6, 2017 at 1:43

Most internal resistances vary with varying load current. Temperature coefficient as well as other factors effect the value.

If I drain a battery at 1mA vs 1A the internal resistance can vary significantly. I assume the same principal applies to a solar cell.