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Summary

The angle of incidence is the angle between incoming solar rays and the normal (perpendicular) to the module surface. Accurate calculation of the angle of incidence is essential for determining direct beam irradiance on the plane of array and applying incidence angle modifiers.

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Inputs

NameSymbolUnitsDescription
Solar Zenith Angleθz\theta_zdegreesAngle between sun and local vertical
Solar Azimuth Angleγs\gamma_sdegreesSolar azimuth, measured clockwise from north
Surface Tilt Angleβm\beta_mdegreesAngle of module surface from horizontal
Surface Azimuth Angleγm\gamma_mdegreesAzimuth angle of surface normal, measured clockwise from North

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Outputs

NameSymbolUnitsDescription
Angle of IncidenceθAOI\theta_{AOI}degreesAngle between sun’s rays and surface normal

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Detailed Description

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Vector Formulation

The angle of incidence is calculated using the dot product of the solar position vector and the surface normal vector. Solar position unit vector: s⃗=[sin⁡(θz)sin⁡(γs)cos⁡(θz)sin⁡(θz)cos⁡(γs)]\vec{s} = \begin{bmatrix} \sin(\theta_z) \sin(\gamma_s) \\ \cos(\theta_z) \\ \sin(\theta_z) \cos(\gamma_s) \end{bmatrix} Surface normal unit vector: n⃗=[sin⁡(βm)sin⁡(γm)cos⁡(βm)sin⁡(βm)cos⁡(γm)]\vec{n} = \begin{bmatrix} \sin(\beta_m) \sin(\gamma_m) \\ \cos(\beta_m) \\ \sin(\beta_m) \cos(\gamma_m) \end{bmatrix} Angle of incidence: cos⁡(θAOI)=s⃗⋅n⃗\cos(\theta_{AOI}) = \vec{s} \cdot \vec{n} θAOI=arccos⁡(sin⁡(βm)sin⁡(θz)cos⁡(γs−γm)+cos⁡(βm)cos⁡(θz))\theta_{AOI} = \arccos\left( \sin(\beta_m) \sin(\theta_z) \cos(\gamma_s - \gamma_m) + \cos(\beta_m) \cos(\theta_z) \right)

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References

  • Iqbal, M. (1983). An Introduction to Solar Radiation. Academic Press, p. 23.