# Finger Resistance

To provide higher conductivity, the top of a cell has a series of regularly spaced fingers. While tapered fingers theoretically provide lower losses, technology limitations mean that fingers are usually uniform in width. The technology (screen printing, photolithography, etc.) dictates the finger width. The finger spacing is optimized for the lowest power loss as there is a trade-off between shading by the fingers and resistive losses.

### Calculation of Power Loss in the Fingers

Calculation of the power loss in a single finger. The width is assumed constant. It is assumed that the current is uniformly generated and flows perpendicularly into the finger, i.e., no current flows directly into the busbar.

Consider an element dx at a distance x from the end of the finger.

The current through the element dx is: $$x J_{MP} S_{f}$$,

where JMP is the current at the maximum power point, and Sf is the finger spacing.

The resistance of the element dx is: $$\frac{d x \rho_{f}}{w_{f} d_{f}}$$,

where wf is the finger width, df is the finger depth (or height) and ρf is the effective resistivity of the metal.

The power loss in the element dx is: $$I^{2} R=\frac{d x \rho_{f}}{w_{f} d_{f}}\left(x J_{MP} S_{f}\right)^{2}$$

Integrating x from 0 to L gives the power loss in the finger:

$$\int_{0}^{L} \frac{\left(x J_{M P} S_{f}\right)^{2} \rho_{f}}{w_{f} d_{f}} d x=\frac{1}{3} L^{3} J_{M P}^{2} S_{f}^{2} \frac{\rho_{f}}{w_{f} d_{f}}$$

Multiplying the above expression by the number of fingers gives the power loss in the whole cell.

To calculate the fractional power loss, divide the power loss in the finger by the power generated by the area of the finger, which is VMP × JMP × L × Sf.

$$P_{loss.fraction}=\frac{ L^{2} J_{M P} S_{f} \rho_{f}}{3 V_{MP} w_{f} d_{f}}$$

The contribution to the cell series resistance is the fractional power loss multiplied by the cell characteristic resistance.

$$R_{series.finger}=\frac{ L^{2} S_{f} \rho_{f}}{3 w_{f} d_{f}}$$