# Physical IAM Model

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 ($$\theta_r$$) using Snell’s law.

$$\theta_{r}=\arcsin \left(\frac{1}{n} \sin \left( \theta\right)\right)$$, 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 $$\tau \left( \theta \right)$$ at that angle and the transmittance when normal to the sun $$\tau \left( 0\right)$$:

$$IAM_B= \frac{\tau \left(\theta \right)}{\tau \left(0\right)}$$.

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

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

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

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

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

$$n=1.526$$  for glass

$$K=4\; \textrm{m}^{-1}$$ and

$$L=0.002 \: \textrm{m}$$

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.