Definitions

# Luminous efficacy

Luminous efficacy is a property of light sources, which indicates what portion of the emitted electromagnetic radiation is usable for human vision. It is the ratio of emitted luminous flux to radiant flux. Luminous efficacy is related to the overall efficiency of a light source for illumination, but the overall lighting efficiency also depends on how much of the input energy is converted into electromagnetic waves (whether visible or not).

## Explanation

Wavelengths of light outside of the visible spectrum are not useful for illumination because they cannot be seen by the human eye. Furthermore, the eye responds more to some wavelengths of light than others, even within the visible spectrum. This response of the eye is represented by the luminosity function. This is a standardized function which represents the response of a "typical" eye under bright conditions (Photopic vision). One can also define a similar curve for dim conditions (Scotopic vision). When not specified, photopic conditions are generally assumed.

Luminous efficacy measures the fraction of electromagnetic power which is useful for lighting. It is obtained by dividing the luminous flux by the radiant flux. Light with wavelengths outside the visible spectrum reduces luminous efficacy, because it contributes to the radiant flux while the luminous flux of such light is zero. Wavelengths near the peak of the eye's response contribute more strongly than those near the edges.

In SI, luminous efficacy has units of lumens per watt (lm/W). Photopic luminous efficacy has a maximum possible value of 683 lm/W, for the case of monochromatic light at a wavelength of 555 nm (green). Scotopic luminous efficacy reaches a maximum of 1700 lm/W for narrowband light of wavelength 507 nm.

## Efficacy and efficiency

In some other systems of units, luminous flux has the same units as radiant flux. The luminous efficacy is then dimensionless. In this case, it is often instead called the luminous efficiency or luminous coefficient and may be expressed as a percentage. For example, it is common to express the luminous efficiency in units where the maximum possible efficacy, 683 lm/W, corresponds to an efficiency of 100%. The distinction between efficacy and efficiency is not always carefully maintained in published sources, so it is not uncommon to see "efficiencies" expressed in lumens per watt, or "efficacies" expressed as a percentage.

### Mathematical definition

The dimensionless luminous efficiency measures the integrated fraction of the radiant power that contributes to its luminous properties as evaluated by means of the standard luminosity function. The luminous coefficient is

$frac\left\{ int^infin_0 y_lambda J_lambda dlambda \right\} \left\{ int^infin_0 J_lambda dlambda \right\},$

where

yλ is the standard luminosity function,
Jλ is the spectral power distribution of the radiant intensity.

The luminous coefficient is unity for a narrow band of wavelengths at 555 nanometres.

Note that $int^infin_0 y_lambda J_lambda dlambda$ is an inner product between $y_lambda$ and $J_lambda$ and that $int^infin_0 J_lambda dlambda$ is the one-norm of $J_lambda$.

## Examples

 Type Luminous efficacy(lm/W) Luminous efficiency Class M star (Antares, Betelgeuse), 3000 K 30 4% ideal black-body radiator at 4000 K 47.5 7.0% Class G star (Sun, Capella), 5800 K 80 12% natural sunlight 93 14% ideal black-body radiator at 7000 K 95 14% ideal white light source 242.5 35.5% ideal monochromatic 555 nm source 683 100%

## Lighting efficiency

Artificial light sources are usually evaluated in terms of a related quantity, the overall luminous efficacy. This is the ratio between the total luminous flux emitted by a device and the total amount of input power (electrical, etc.) it consumes. This is often simply called “luminous efficacy”, which can be confusing as it also has units of lm/W. It is also sometimes referred to as the wall-plug luminous efficacy or simply wall-plug efficacy. The overall luminous efficacy is a measure of the efficiency of the device with the output adjusted to account for the spectral response curve (the “luminosity function”). When expressed in dimensionless form (for example, as a fraction of the maximum possible luminous efficacy), this value may be called overall luminous efficiency, wall-plug luminous efficiency, or simply the lighting efficiency.

The main difference between the regular and “overall” efficacies is that the latter account for input energy that is lost as heat or otherwise exits the source as something other than electromagnetic radiation. True luminous efficacy is a property of the radiation emitted by a source. Overall luminous efficacy is a property of the source as a whole.

### Examples

The following table lists overall luminous efficacy and efficiency for various light sources:

 Category Type Overallluminous efficacy (lm/W) Overallluminous efficiency Combustion candle 0.3 0.04% gas mantle 2 0.3% Incandescent 5 W tungsten incandescent (120 V) 5 0.7% 40 W tungsten incandescent (120 V) 12.6 1.9% 100 W tungsten incandescent (120 V) 16.8 2.5% 100 W tungsten incandescent (220 V) 13.8 2.0% 100 W tungsten glass halogen (220 V) 16.7 2.4% 2.6 W tungsten glass halogen (5.2 V) 19.2 2.8% quartz halogen (12–24 V) 24 3.5% high-temperature incandescent 35 5.1% Fluorescent 5–24 W compact fluorescent 45–60 6.6–8.8% T12 tube with magnetic ballast 60 9% T8 tube with electronic ballast 80–100 12–15% T5 tube 70–100 10–15% Light-emitting diode white LED 10 to 90 1.5–13% white OLED 102 15% Prototype LEDs up to 150 up to 22% Arc lamp xenon arc lamp 30–50 4.4–7.3% mercury-xenon arc lamp 50–55 7.3–8.0% Gas discharge high pressure sodium lamp 150 22% low pressure sodium lamp 183 up to 200 27–29% Metal halide 65–115 9.5–17% 1400 W sulfur lamp 100 15% Theoretical maximum 683.002 100%

Sources that depend on thermal emission from a solid filament, such as incandescent light bulbs, tend to have low overall efficacy compared to an ideal blackbody source because, as explained by Donald L. Klipstein, “An ideal thermal radiator produces visible light most efficiently at temperatures around 6300 °C (6600 K or 11,500 °F). Even at this high temperature, a lot of the radiation is either infrared or ultraviolet, and the theoretical luminous [efficacy] is 95 lumens per watt. Of course, nothing known to any humans is solid and usable as a light bulb filament at temperatures anywhere close to this. The surface of the sun is not quite that hot.” At temperatures where the tungsten filament of an ordinary light bulb remains solid (below 3683 kelvins), most of its emission is in the infrared.