Formula
For a panel tilted β from horizontal and facing azimuth γ, with the sun at zenith angle θz and azimuth γs:
cos θ = cos θz · cos β + sin θz · sin β · cos(γs − γ)
A flat panel (β = 0) reduces this to θ = θz; a vertical wall (β = 90°) to cos θ = sin θz · cos(γs − γ).
The cosine loss
A beam of sunlight spreads over a larger area of panel as it comes in more obliquely. The direct light received falls with cos θ:
| Angle of incidence | Direct light received | Plus reflection loss at the glass |
|---|---|---|
| 0° | 100% | 100% |
| 20° | 94% | 94% |
| 30° | 87% | 86% |
| 45° | 71% | 69% |
| 60° | 50% | 48% |
| 75° | 26% | 22% |
The last column adds the extra reflection of anti-reflective glass at glancing angles (ASHRAE model, b0 = 0.05), which our yield model applies to direct, sky and ground light separately.
Designing with it
Because the loss is small near 0° and grows fast after 45°, a panel does not need to track the sun precisely — but it should not face it at grazing angles for long. That is the geometric core of the off-grid 70° argument: in December a 35° panel meets the noon sun at about 38°, a 70° panel at about 3°.
The Panel tab of the sun map shows the angle of incidence for any tilt, orientation, place and minute, together with the clear-sky light on the panel.