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LEDs, Lumens and Brightness

Steradians, Candelas and Lumens, LED Light Output

New products such as the LED based PowerSURE® Power Failure Light improve safety in the dark by providing both power failure lighting and night light functionality. Here we continue with some of the more technical aspects of determining LED brightness.

LEDs and Lumens - Part 2


Part 1:
   Bulb Brightness in Lumens
   Light Measurement Units
   Light Measurement Relationships
   Units in LED specifications

Part 2:

  Steradians Explained

  Calculating Candelas and Lumens

  Lumen Calculator

  LED Radiation Diagrams

Part 3:
   Comparing real bulbs to LEDs
   Illuminating a room with an LED
   LED based power failure lights
   LEDs and Watts and Light Bulb Watts - How to Compare

LEDs are becoming more widely accepted as a light source for illumination of living spaces and many questions have been raised by individuals attempting to understand how much light is produced by a High Brightness (HB) LED when compared to an incandescent light bulb. Here we review how many Lumens are produced and how it can be calculated.

Steradians - Solid Geometry Lessons Anyone?

By definition, 1 lumen is the amount of light produced by a 1 candela source radiating out through 1 steradian (a specific cone shaped solid unit angle of 65.54°) within an imaginary sphere surrounding the light source. One candela illuminates the entire surface of a 1 meter radius sphere at an average 1 lumen for each sq. meter of surface in the 360° sphere. There are 4 π, or 12.57 steradians in a sphere. Thus, the standard candle at 1 candela intensity produces 12.57 lumens of total visible light radiated in all directions. Lumens are the total quanty of light flowing out in all directions.

a steradian 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 source.

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.

Lumen Calculator

Half Angle
Total Light Output

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

Radiation diagram from typical LED spec sheet

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 lumen output.

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.

So how does all this compare to real light bulbs?

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