# 填充因子

$$FF= \frac{P_{MP}}{V_{OC}\times I_{SC}}$$

## Fill Factor

$FF=\frac{{V}_{MP}{I}_{MP}}{{V}_{OC}{I}_{SC}}$

$$\frac{d(I V)}{d V}=0$$

$$V_{MP} = V_{OC} - \frac{nkT}{q}ln(\frac{q V_{MP}}{nkT}+1)$$

## Fill Factor - Empirical

$FF=\frac{{v}_{oc}-\mathrm{ln}\left({v}_{oc}+0.72\right)}{{v}_{oc}+1}$

## Voc normalized

${v}_{oc}=\frac{q}{nkT}{V}_{oc}$

Fill Factor Calculator 1

###### Results

Normalized VOC, voc (units):X
Fill Factor, FF:X

## Fill Factor

$FF=\frac{{V}_{MP}{I}_{MP}}{{V}_{OC}{I}_{SC}}$

Fill Factor Calculator 2

###### Results

Resulting fill factor, FF: X

### VMP的精确表达

$$I = I_L-I_0\left[\exp\left(\frac{V}{nV_t}\right)-1\right]$$

$$Power = V \times I$$ 此外，-1 项对 VMP 没有影响

$$P = V I_L- V I_0\exp\left(\frac{V}{nV_t}\right)$$

VMP 是当功率相对于 V 的导数为零时：

$$0 = I_L- I_0\exp\left(\frac{V_{MP}}{nV_t}\right)\left(1+\frac{V_{MP}}{nV_t}\right)$$

V> > Vt 并通过整理得出：

$$\frac{I_L}{I_0} =\exp\left(\frac{V_{MP}}{nV_t}\right)\left(\frac{V_{MP}}{nV_t}\right)$$

$$Y = Xe^x \Leftrightarrow X = W (Y)$$

$$\frac{V_{MP}}{nV_t} =W\left(\frac{I_L}{I_0}\right)$$

$$V_{MP} = nV_t W\left(\frac{I_L}{I_0}\right)$$

$$V_{MP} = nV_t W\left(\exp\left(\frac{V_{OC}}{nV_t}\right)\right)$$