For wide beam patterns and/or concentrated arrays of LEDs, one can estimate lux (lumens/meter^{2}) levels using the inverse square law. For narrow beams and distributed LED arrays, it is better to use optical software such as DIAlux and LightTools.

To calculate the lux level at any distance, we need to specify the lumen level of the source and the FWHM beam angle. A previous article discusses the different types of beam angles and here we just assume the inverse square law is applicable.

Calculation procedure:

- The solid angle in steradians of a cone whose cross-section subtends the angle 2θ (FWHM) is:

So, the luminous intensity (lumens/sr or candela) is the luminous flux divided by Ω.

- The identity,

can be used to relate candela, which is easy to calculate, and lux which needs one known value and distance pair to use the inverse square law for lux.

- The inverse square law is:

- So, the lux at any distance I(x) can be calculated using

Where the “LED candela” is a constant, independent of distance, for a given FWHM beam angle and lumen output of an LED. Strictly speaking, there should be a unit constant to handle the unit conversions, but we are leaving it out. We are using photometric units for this discussion, but these techniques also apply to radiometric units.

As an example, we will use the Luminus SFT-40-WxS LED datasheet to do some calculations.

Calculate Ω: From the Product Characteristics table on page 6, the FWHM beam angle is 120°. The cosine of (120/2)° is 1/2 so the value of Ω for this LED is π steradians.

The lumen output of an LED depends on the operating current and temperature. Datasheets provide plots to determine the relative changes for current and temperature for each intensity bin available for a given part number. The SFT-40-WxS datasheet has a convenient table on page 3 (below). The relative change plots are on page 7.

The reliability for an LED with a junction temperature of 85°C is very good and this value is commonly for “hot binning” tests for modern LEDs.

Using bin P3 and a drive current of 8 A at 85°C, the minimum expected luminous flux is 2555 lumens. Using these as inputs, we can calculate the table and figure below.

Semilog plot of Illuminance (lux) versus distance for the SFT-40-WxS driven at 8A at 85°C.

We have highlighted three interesting distances in the results above. They are 200 mm, 1000 mm, and, in this case, 1,275 mm. The 1000 mm distance is the equivalence point where we can jump from candela to lux.

The other two distances are related to IEC 62778 and IEC/EN 62471 testing. 200 mm is the standard test distance for IEC 62778. 1,275 mm is the distance (for this particular set of conditions) where the LED has a 500-lux illuminance which is the test condition for “GLS” white LEDs in IEC/EN 62471.

Some LED applications such as torches use the 1 lux distance as a metric. We have included the 1 lux distance in the table, but it is not generally used for such a wide beam pattern.

This method is an approximation technique and is most valid for large beam angles and reasonably long distances where the LED source can be considered effectively a point source. More details about beam patterns for different FWHM angles are in the Help Center article “Optical - What do the Radiation Plots in LED datasheets mean and how do I calculate Lux?”

An interesting extension of this technique is calculating the number of LEDs needed for a light fixture at distances where the point source condition is valid. Arrays of LEDs in the far field have very similar beam angles as the individual sources. An example of this effect is shown in “Horticulture - How do I relate PPF to PPFD and DLI?” (Towards the end where we talk about using Dialux to model arrays of LEDs).

Light cones – In the far field all light sources can be approximated as cones with a uniform intensity at the calculation distance.

An example might be a high bay application where we want to provide 500 lux of task plane light at a

10-meter distance from a single light fixture. We can calculate that there are 8.1 lux from a single LED at 10,000 mm when using the same drive conditions as above. We can then use the percentage of the 500-lux target value at that distance to calculate that we need 62 LEDs in our fixture (or ~ 158,410 lumens).

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