Getting the Most Power from a Solar Panel

A solar panel's power depends on the voltage you run it at. Too low and the current can't make up for it; too high and the current collapses. Somewhere between is a peak, a critical point, and charge controllers hunt for it constantly.

How it works

Critical points and absolute extrema

In one sentence

A critical point of $f$ is where $f'(c) = 0$ or $f'(c)$ is undefined. On a closed interval, a continuous $f$ attains its absolute extrema at critical points or endpoints (Extreme Value Theorem).

General formula

{\htmlData{sym=0}{P(V)}} = {\htmlData{sym=1}{V}}\left({\htmlData{sym=2}{I_{sc}}} - {\htmlData{sym=3}{I_0}}\left(e^{{\htmlData{sym=1}{V}}/{\htmlData{sym=4}{V_t}}} - 1\right)\right), \qquad P'({\htmlData{sym=5}{V^*}}) = 0

At the top of a hill the ground is level. Power is zero at both ends of the range, so the highest point must be a level spot in between.

What the symbols mean

$P(V)$
Power
Watts the panel delivers at voltage V.
$V$
Operating voltage
The voltage the charge controller holds the panel at.
$I_{sc}$
Short-circuit current
The current with the panel shorted: proportional to sunlight.
$I_0$
Saturation current
A tiny diode current that grows exponentially with voltage.
$V_t$
Thermal voltage
About 26 mV per cell; sets how sharply the current falls.
$V^*$
Maximum power point
The critical point where P′ = 0.

The panel's current falls as voltage rises, slowly at first, then sharply near the open-circuit voltage.
Power is voltage times current, zero at both ends of $[0, V_{oc}]$ .
A continuous function on a closed interval has an absolute maximum, here at a critical point where $P'(V) = 0$ .
Check the endpoints too: $P = 0$ there, so the critical point is the maximum power point.

Learn the math

Critical PointsCalc I · Lesson 27, Applications of Derivatives Minimum and Maximum ValuesCalc I · Lesson 28, Applications of Derivatives Finding Absolute ExtremaCalc I · Lesson 29, Applications of Derivatives More Optimization ProblemsCalc I · Lesson 34, Applications of Derivatives