An LED specification sheet shows the luminous intensity (mcd)
of the LED for a specific viewing angle of the LED. The viewing angle is the
angle of the beam of light produced by the LED and lens. The angle is bounded
by the edges where the intensity falls to 50% of the max intensity usually found
at the optical center of the beam. Thus, a 25,000 mcd LED with a viewing angle
of 20° can provide 25 Candelas of light intensity within the 20° viewing
angle. The question most often posed is how many lumens do you get from this
type of LED light?
We can find this out by understanding the relationship of candelas, steradians and spheres
using the diagram below. Note that the sphere segment has a 1 meter radius.
Calculating Candelas and Lumens
A steradian is a solid cone having 1 square meter of surface area on a sphere having a 1
meter radius. There are 4π (12.57) steradians in a sphere. The light grey
area is known as a spherical cap that we will illuminate with a point light
The surface area of a spherical cap is calculated using the formula S=2πRh where h,
the height of the cap, is completely dependent on the viewing angle (apex of
the cone). A smaller angle, results in a smaller area on the surface and a
smaller height of the center of the cap.
If you remember your trigonometry, we can calculate
the height of the cap h, in a 20° cone (the viewing angle) by finding the
length of a 10° right triangle. A 1 Meter radius simplifies the calculation
and the height is found using the formula 1-Cos(10°). Notice we use half of the
viewing angle for the calculation. LED specs will show either viewing angle or
half angle so it is necessary to distinguish between the two when reading the
spec. Viewing angle is denoted by "2∅
" in most LED specs.
At 1 meter, a 25,000 mcd LED with a 20° viewing angle covers a spherical
cap area of 2*π*1*0.015 = 0.095 square meters with 25 candelas of
intensity. In order to determine total lumens flowing, we must determine how
bright the intensity would be if the same amount of light flow were covering an entire
1 square meter area on the surface of that 1 meter radius sphere. We know that
the 20° cone covers 0.095 M2
and multiplying that small area times
the candela value tells us how many lumens the device delivers into a 1 M2
area. This half angle cosine formula 2 π*(1-Cos(∅))*25 Candelas yields [6.283*(1-cos(10°))*25]
= 2.39 lumens.
Now that we've standardized our light output in lumens per square meter, the same LED
die, when used with a wider beam lens (i.e. 40°), will have a lower candela
value. Using the above calculation, a 25,000 mcd 20° viewing angle LED has the
same luminous flux as a 6,250 mcd 40° viewing angle LED. This is the same LED
die with a different lens. Recent LED specs have begun to include lumen values
to help in this determination of total light output.
This mathematical method is a close approximation of lumens using the candela value
and the viewing angle of the LED. Other factors in the construction of the LED
can change the luminous flux relationship with the luminous intensity of the die.
For example, some LEDs will have less light lost to the sides and rear of the
die and reflect more of the intensity out through the lens.
This is a simple mathematical model to calculate lumens from candelas when you know the
viewing angle of the LED. It is easy to use but it is really only an
approximation because it does not consider the spectral qualities of the light.
LED Radiation Diagrams
A radiation diagram for a specific LED shows how the viewing angle is distinguished from
all the other light emitted by the LED. The red arrows indicate the 60° half
angle or 120° viewing angle where the light drops to 50% of max intensity. This
is the forward radiation. Note that the most light is usually on the 0° X axis and this represents
the maximum candela output. Note also that as the light spreads out across the viewing angle,
the relative candela output drops to zero (in this particular case) at 90°.
Any light that is absorbed into the substrate of the
LED base is not seen or measured. You can see why altering the construction of
the LED to reduce light lost in other directions would cause a change in the
When an LED spec sheet specifies the lumen output rather than candela or millicandela,
the Total Luminous Flux (lm) number represents all the light that is emitted from the device
as measured by an integration sphere. The radiation diagram is used to see what the relative level of light
is as it radiates out in any given direction.