# Voltage at the maximum power point - Vmp

To gain the maximum amount of power from the solar cell it should operate at the manximum power voltage. The maximum power voltage is further described by VMP, the maximum power voltage and IMP, the current at the maximum power point. The maximum power voltage occurs when the differential of the power produced by the cell is zero.

Starting with the IV equation for a solar cell:

$I=IL-I0eVVt$

${V}_{t}=\frac{nkT}{q}$ to simplify the notation in the derivation, where kT/q ~ 0.026 volts and n is the ideality factor. The ideality factor varies with operating point. For these equations the correct value to use is the average from VMP to VOC.

Power  produced by the cell is the product of the voltage and the current, i.e., P = IV.

$P=VIL-VI0eVVt$

Using differentiation by parts on the second term: $u=VI0$, , $v=eVVt$, $v'=1VteVVt$

The differential of power respect to voltage:

$dPdV=IL-VI01VteVVt+I0eVVt$

Vmp occurs when $dPdV=0$

$VmpI01VteVmpVt-I0eVmpVt=IL$

Detailed steps for rearranging and simplifying:

$VmpVteVmpVt-eVmpVt=ILI0$

$eVmpVtVmpVt-1=ILI0$

$VmpVt+ln⁡VmpVt-1=lnILI0$

$Vmp=VtlnILI0-ln⁡VmpVt-1$

Using $Voc=VtlnILI0$

$Vmp=Voc-ln⁡VmpVt-1$The implicit equation above does not have a simple solution but it converges quickly with iteration. An initial guess of VMP = 0.9 VOC gives an accurate solution in two iterations.

### Using Lambert Functions

An exact solution to finding the maximum power voltage is with lambert functions. These are transcendetal functions much like e or trigonometric functions. Lambert is available in most advanced math packages such as Maple, Mathematica and in Python with SciPy, but they are not on most handheld calculators.

As above we need to solve for the voltage when the derivative of the power is zero

$P=VIL-VI0eVVt$

$0=IL-I0exp⁡VVt1+VVt$

Since V/Vt is much great than 1 we can remove the +1 term: $ILI0=exp⁡VVt×VVt$

The Lambert provides a solution of the form:$Y=XeX⇔X=W(Y). Using this relation the previous equatoin becomes:$

$VVt=W⁡JLJ0$

so that we get a simple expression for Vmp.

$Vmp=VtW⁡JLJ0$

Alternatively, using the full equation above for dP/dV = 0 and not dropping the +1 term the solution becomes:

$Vmp=VtW⁡eJLJ0-1$

# Exact determination of Vmp

Math partly verified using Wolfram Alpha

$J = J_{L} - J_{0}\left\lbrack \exp\left( \frac{V}{V_{t}} \right) - 1 \right\rbrack$

Power = V x J

$P=V{J}_{L}-V{J}_{0}\left[exp\left(\frac{V}{{V}_{t}}\right)-1\right]$

Lose the -1 since we are above 100 mV