Imagine two neighbouring houses with identical rooftop solar installations. Both are exposed to nearly identical sunlight conditions throughout the day, yet one generates more electricity than the other. Why does this happen? The difference may not lie in the quality or capacity of the solar panels, but in the direction from which sunlight reaches their surfaces and the angle at which it strikes them.
In our previous articles, we explored several solar angles individually and tried to understand their significance. We learned how geographical latitude identifies the location of a place on Earth, how solar declination describes the seasonal movement of the Sun, and how the hour angle explains its changing position throughout the day. We also discussed the solar zenith and azimuth angles, which help us understand the Sun’s position in the sky. However, the real significance of these angles becomes clearer when we examine their combined effect on the solar radiation received by an inclined surface.
We generally assume that the more sunlight available at a location, the greater the electricity generation from solar panels. However, the amount of sunlight available at a location and the amount actually received by a panel are not necessarily the same. The difference depends significantly on how the panel is positioned relative to the incoming sunlight. This relationship brings us to another important angle, known as the angle of incidence, represented by θ (theta).
To understand this angle, imagine holding a book in bright sunlight. When the book faces the Sun directly, the sunlight strikes its surface almost perpendicularly. Now gradually tilt the book away from the Sun. Although the intensity of the incoming sunlight remains approximately unchanged, its rays begin to strike the book obliquely. The same incoming beam of sunlight is now spread over a larger surface area, reducing the amount of solar energy received per unit area. The same principle applies to rooftop solar panels.
Imagine a straight line emerging perpendicularly from the centre of a solar panel. This imaginary line is called the surface normal. The angle formed between this line and the incoming solar rays is known as the angle of incidence, θ. When sunlight strikes the panel perpendicularly, θ becomes zero, and the panel receives the maximum possible direct solar radiation for the prevailing sunlight intensity. As θ increases towards 90°, the direct solar radiation received per unit area decreases, assuming the intensity of incoming sunlight remains unchanged.
For example, assuming the intensity of incoming sunlight remains unchanged, when θ is 30°, the panel receives approximately 86.6% of the direct radiation it would receive under perpendicular illumination. At 45°, this reduces to approximately 70.7%, while at 60°, only 50% remains. When θ reaches 90°, the sunlight travels parallel to the surface, and the panel receives no direct beam radiation under ideal geometrical conditions.
The relationship between the angle of incidence and the direct solar radiation received by the panel can be expressed mathematically as:
Here, (G_{bn}) represents the direct normal solar irradiance, while (G_{b,T}) represents the direct beam irradiance received by the panel. The term (\cos\theta) represents the geometrical reduction factor that determines how much available direct solar radiation is received per unit area of the panel.
However, solar panels also receive diffuse radiation scattered by the atmosphere, as well as solar radiation reflected from the ground and other surrounding surfaces. Therefore, even when the angle of incidence is unfavourable, a panel may still receive some solar energy. Nevertheless, θ remains a fundamental factor in determining how effectively a panel captures direct sunlight.
We may now ask what determines the value of θ at a particular moment. The answer lies in the combined influence of the solar angles discussed in our earlier articles. Latitude, represented by φ, identifies the geographical position of installation. Solar declination, δ, describes the Sun’s seasonal position, while the hour angle, ω, indicates its position at a particular time. The surface tilt angle, β, describes how much the panel is inclined relative to the horizontal plane, and the surface azimuth angle, γ, describes the direction in which it faces.
Consider Delhi, located at approximately 28.5° north latitude. Its geographical position remains unchanged, but the Sun’s apparent position varies throughout the year. During summer, the Sun follows a higher path across the sky, than during winter, when its apparent path remains relatively lower. At the same time, the Earth’s rotation causes the Sun to appear to move from east to west. Consequently, even when the solar panels remain fixed, the direction from which sunlight approaches their surfaces changes continuously.
The combined influence of these angles can be expressed through the general equation for the angle of incidence:
Here, (G_{bn}) represents the direct normal solar irradiance, while (G_{b,T}) represents the direct beam irradiance received by the panel. The term (\cos\theta) represents the geometrical reduction factor that determines how much available direct solar radiation is received per unit area of the panel.
However, solar panels also receive diffuse radiation scattered by the atmosphere, as well as solar radiation reflected from the ground and other surrounding surfaces. Therefore, even when the angle of incidence is unfavourable, a panel may still receive some solar energy. Nevertheless, θ remains a fundamental factor in determining how effectively a panel captures direct sunlight.
We may now ask what determines the value of θ at a particular moment. The answer lies in the combined influence of the solar angles discussed in our earlier articles. Latitude, represented by φ, identifies the geographical position of installation. Solar declination, δ, describes the Sun’s seasonal position, while the hour angle, ω, indicates its position at a particular time. The surface tilt angle, β, describes how much the panel is inclined relative to the horizontal plane, and the surface azimuth angle, γ, describes the direction in which it faces.
Consider Delhi, located at approximately 28.5° north latitude. Its geographical position remains unchanged, but the Sun’s apparent position varies throughout the year. During summer, the Sun follows a higher path across the sky, than during winter, when its apparent path remains relatively lower. At the same time, the Earth’s rotation causes the Sun to appear to move from east to west. Consequently, even when the solar panels remain fixed, the direction from which sunlight approaches their surfaces changes continuously.
The combined influence of these angles can be expressed through the general equation for the angle of incidence:
Although this equation may appear complicated, its physical meaning is straightforward. It combines the geographical location, date, time, inclination, and orientation of a solar panel to determine to determine the angle at which sunlight strikes its surface. In this equation, the surface azimuth is measured from due south, with east represented by −90°, west by +90°, and north by 180°.
The importance of this relationship lies not in solving the equation manually but in understanding what it reveals. Even when a solar panel remains exposed to sunlight, the amount of direct radiation it receives varies throughout the day and across different season. This variation occurs partly because the angle of the incidence changes continuously with the Sun’s apparent movement across the sky.
This also explains why selecting the inclination and orientation of rooftop solar panels is an important part of solar system design. By calculating θ at different times of the day and throughout the year, we can estimate how effectively a particular surface captures the available direct solar radiation.
We have now reached an important understanding: the solar radiation available at a location and the radiation actually received by an inclined surface are not necessarily the same. The angle of incidence plays a significant role in determining how much available direct solar radiation reaches the surface.
But what happens when we change the inclination of the panel or turn it towards a different direction? Will a horizontal panel receive the same amount of solar radiation as a south-facing inclined panel? Why do north-facing and south-facing walls of a house experience different solar exposure?
These questions bring us to our next article, where we will explore how the same mathematical relationship explains the solar radiation received by horizontal, inclined, and vertical surfaces—and what these differences mean for the performance of our rooftop solar installations.
