Calculator
Example data table
These examples use the south-based azimuth reference.
| Scenario | β (°) | γp (°) | Sun input | θi (°) |
|---|---|---|---|---|
| Zenith+Azimuth | 30 | 0 | θz=45°, γs=0° | 15.000 |
| Zenith+Azimuth | 25 | -20 | θz=60°, γs=40° | 44.407 |
| Elevation+Azimuth | 35 | 0 | α=30°, γs=90° | 69.296 |
| Latitude+Day+Time | 30 | 0 | φ=33.7°, n=172, ST=13.5 | 28.387 |
| Latitude+Day+Time | 20 | 15 | φ=40°, n=355, ST=10.0 | 41.802 |
Formula used
- Declination: δ = 23.45° · sin(360°·(284+n)/365)
- Hour angle: ω = 15° · (ST − 12)
- Zenith cosine: cos θz = sinδ sinφ + cosδ cosφ cosω
- Incidence cosine: cos θi = sinδ sinφ cosβ − sinδ cosφ sinβ cosγp + cosδ cosφ cosβ cosω + cosδ sinφ sinβ cosγp cosω + cosδ sinβ sinγp sinω
How to use this calculator
- Select an input method that matches your available data.
- Choose the azimuth reference used by your values.
- Enter panel tilt and panel azimuth for the surface.
- Enter sun data (zenith/elevation and azimuth) or site-time inputs.
- Click Calculate to see results above the form.
- Use the export buttons to download CSV or PDF.
Angle of incidence in practical solar design
1) Why this angle matters for energy yield
For direct beam sunlight, the useful component on a tilted surface scales with cos(θi). When θi is 0°, the beam hits straight on and projection loss is 0%. At 30°, cos(30°)=0.866 (about 13.4% less beam). At 60°, cos(60°)=0.5 (50% less). This calculator reports both θi and cos(θi) to show that projection effect instantly.
2) Distinguish zenith and incidence angles
Solar zenith (θz) describes the sun relative to the vertical; solar elevation is α=90°−θz. Incidence (θi) is different: it is measured relative to the panel’s normal. A low θz does not guarantee a low θi if the panel faces the wrong direction.
3) Tilt β: from quick rules to detailed checks
A common fixed-array starting point is β≈latitude for annual energy, with seasonal bias using latitude±10–15°. The calculator lets you test these rules: increase β for winter (lower sun) and decrease β for summer. The “beam ratio” output compares cos(θi) to cos(θz) when the sun is above the horizon.
4) Azimuth conventions can flip results
Some tools use compass azimuth (0°=North, clockwise), while solar engineering texts often use a south-based convention (0°=South, +West, −East). A 180° mismatch can turn a south-facing array into a north-facing one. Choose the reference selector before entering azimuth values.
5) Day-of-year, declination, and seasonal swing
Declination δ describes the sun’s seasonal north–south shift. Typical extremes are about ±23.45°. In the site-time mode, δ is estimated from day-of-year n, then combined with latitude φ and hour angle ω to compute θz and θi. This provides a consistent physics-based sun position for planning scenarios.
6) Tracking versus fixed mounting
Single-axis trackers reduce θi during much of the day, raising average cos(θi). Two-axis tracking can keep θi near 0° when unobstructed, pushing the beam multiplier close to 1. Use the direct-input modes to test tracker targets if you already know sun azimuth and elevation.
7) When cos(θi) is negative
If θi exceeds 90°, the sun is behind the plane and the direct beam contribution should be zero for that surface. The calculator flags this case and uses max(0, cos θi) as the beam multiplier so reports stay physically realistic.
8) Connecting incidence to real irradiance numbers
If you have direct normal irradiance (DNI) from weather data, approximate beam-on-plane as Gb,tilt ≈ DNI × max(0, cos θi). For example, with DNI=800 W/m² and θi=25°, Gb,tilt≈800×0.906=725 W/m². Add diffuse and ground-reflected components separately for full POA irradiance.
FAQs
1) What is the angle of incidence for a solar panel?
It is the angle between the incoming sun ray and the perpendicular (normal) to the panel surface. Smaller angles mean the sun hits more “straight on,” increasing the direct beam component on the panel.
2) Is incidence angle the same as solar zenith?
No. Solar zenith is relative to vertical. Incidence angle is relative to the panel’s normal, so it depends on tilt and panel direction as well as the sun’s position.
3) Why does power drop when θi increases?
The projected beam on the panel is proportional to cos(θi). As θi rises, cos(θi) falls, reducing plane-of-array beam irradiance even if DNI is unchanged.
4) Which azimuth reference should I choose?
Use “Compass” if your azimuth is 0°=North and increases clockwise. Use “South-based” if 0°=South with west positive. Keep the same convention for both solar and panel azimuth inputs.
5) What does “beam ratio” mean here?
When the sun is above the horizon, beam ratio compares the tilted-surface beam multiplier to the horizontal-surface beam multiplier: (max(0, cos θi)) / cos(θz). It’s a quick indicator of tilt advantage.
6) Do I need solar time or clock time?
Site-time mode uses solar time. Clock time must be corrected for longitude, time zone, and equation-of-time to become solar time. If you already know sun azimuth/elevation, use the direct-input modes instead.
7) Does this include diffuse sky radiation?
No. The calculator focuses on geometric incidence for direct beam projection. For full plane-of-array irradiance, add diffuse and ground-reflected models and then apply module and inverter performance factors.