De Soto › Published

Method

The accepted fraction of the direct sunlight is the transmittance of the module's glass cover at the incidence angle divided by its transmittance at perpendicular incidence. The glass refracts the light by Snell's law. It reflects the average of the two polarisations given by Fresnel's equations. It absorbs the light along the refracted path by Bouguer's law. The cover has no anti-reflective coating, and the angle has no limits. At 90° the fraction is zero. The sky and ground light take the same fraction at the angles the diffuse incidence-angle step gives them.

Settings

Setting	Value
Refractive index of the glass $n_g$	1.526
Thickness of the glass $\ell_g$	2 mm
Extinction coefficient $K$	4 /m

The math

1. Refraction into the glass
2. Transmittance of the cover
3. Transmittance at perpendicular incidence
4. Accepted fraction of the direct light

$$\sin\theta_{gl} = \frac{\sin\theta}{n_g} \tag{1}$$ $$\tau(\theta) = e^{-K \ell_g/\cos\theta_{gl}}\left[1 - \tfrac{1}{2}\left(\frac{\sin^2(\theta_{gl} - \theta)}{\sin^2(\theta_{gl} + \theta)} + \frac{\tan^2(\theta_{gl} - \theta)}{\tan^2(\theta_{gl} + \theta)}\right)\right] \tag{2}$$ $$\tau(0) = \left[1 - \left(\frac{n_g - 1}{n_g + 1}\right)^2\right] e^{-K \ell_g} \tag{3}$$ $$F(\theta) = \frac{\tau(\theta)}{\tau(0)}\ \text{for}\ \theta < 90°,\qquad F(90°) = 0 \tag{4}$$

where

Symbol	Meaning	Unit
$\theta$	the incidence angle: between the light's direction and the perpendicular to the module plane. Of the direct light	degrees
$\theta_{gl}$	the angle refracted into the glass	degrees
$n_g$	the refractive index of the glass	—
$\tau(\theta)$	the transmittance of the cover at incidence angle $\theta$	—
$K$	the extinction coefficient. Of the glass	1/m
$\ell_g$	the thickness of the glass	m
$\tau(0)$	the transmittance of the cover at perpendicular incidence	—
$F(\theta)$	the incidence-angle modifier: the accepted fraction of light at the incidence angle	—

References

1. De Soto, Klein & Beckman 2006, Improvement and validation of a model for photovoltaic array performance, Solar Energy 80(1) 78–88

API

iam.kind = desoto_published

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