The physical model for the incident angle modifier is based on Snell’s and Bougher’s laws, and was published by De Soto et al. (2006).  Our presentation here includes correction to a few errors present in that paper.  The first step is to calculate the angle of refraction ($\theta&space;_{r}$) using Snell’s law.

$\theta&space;_{r}=\arcsin&space;\left&space;(&space;\frac{1}{n}\sin&space;\left&space;(&space;\theta&space;\right&space;)&space;\right&space;)$, where $n$ is the index of refraction of the cover glass and the 1 in the numerator is the index of refraction of the air.

The incident angle modifier at an angle, $\theta$, is a ratio between the transmittance $\tau&space;\left&space;(&space;\theta&space;\right&space;)$ at that angle and the transmittance when normal to the sun $\tau&space;\left&space;(&space;0&space;\right&space;)$:

$IAM_{B}=\frac{\tau&space;\left&space;(&space;\theta&space;\right&space;)}{\tau&space;\left&space;(&space;0&space;\right&space;)}$.

A good approximation of the transmittance of the module cover is given by:

$\tau&space;\left&space;(&space;\theta&space;\right&space;)=e^{-\left&space;(\frac{KL}{\cos&space;\left&space;(&space;\theta&space;_{r}&space;\right&space;)}&space;\right&space;)}\left&space;[&space;1-\frac{1}{2}\left&space;(&space;\frac{\sin&space;^{2}\left&space;(&space;\theta&space;_{r}-\theta&space;\right&space;)}{\sin&space;^{2}\left&space;(&space;\theta&space;_{r}+\theta&space;\right&space;)}&space;+\frac{\tan&space;^{2}\left&space;(&space;\theta&space;_{r}-\theta&space;\right&space;)}{\tan&space;^{2}\left&space;(&space;\theta&space;_{r}+\theta&space;\right&space;)}\right&space;)&space;\right&space;]$,

where $K$ is the glazing extinction coefficient (1/meters) and $L$ is the glazing thickness (meters).

$\tau&space;\left&space;(&space;0&space;\right&space;)$ can be determined by $\tau&space;\left&space;(&space;0&space;\right&space;)&space;=&space;\lim_{\theta&space;\rightarrow&space;0}&space;\tau&space;\left&space;(&space;\theta&space;\right&space;)&space;=&space;\exp\left&space;(&space;-KL&space;\right&space;)\left&space;[&space;1-\left&space;(\frac{1-n}{1+n}&space;\right&space;)^2&space;\right&space;]$ .

DeSoto et. al lists the following typical input parameters for PV modules:

$n=1.526$  for glass

$K=4\;&space;\textrm{m}^{-1}$ and

$L=0.002&space;\:&space;\textrm{m}$

The resulting IAM function is plotted below:

### References

De Soto, W., S. A. Klein and W. A. Beckman (2006). “Improvement and validation of a model for photovoltaic array performance.” Solar Energy 80(1): 78-88.