Alliance for Lighting InformationAlliance for Lighting InformationThe relationships between the fundamental photometric units in lighting can be confusing.
The basic SI unit is luminous intensity, which is measured in candela (cd) (sometimes called candlepower), one of the six fundamental units in that system, like length measured in meters (m). Luminous intensity is a complex concept that can be described as luminous flux within a specified solid angle.
Flux in photometric terms is measured in lumens (lm), which is a flow rate of energy that has been "weighted" by the appropriate visual sensitivity function, assumed to be human and photopic unless noted otherwise. There are "parallel" units in radiometric terms, expressed in the SI system using the units of power (e.g. Watts), area (e.g. square meters) and solid angle. The sole difference between photometric and radiometric units is that photometric units are developed from radiometric units combined with an appropriate visual sensitivity function.
Solid angle relates to the human eye view of the universe. When we look at the world, we see an image of what is in front of us that is focused on a point, like the tip of a cone located "at our eyes". (The concept of visual size is the same as solid angle. The overlap of objects and different distances to visually adjacent objects within the image are not actually shown in the image, but instead something we discover by comparing several different images.) Such an image can be considered as a display on the surface of a sphere observed from its center. It makes no difference how big the sphere is considered to be, say a radius of one meter or one inch - the solid angle that an object "occupies" on that image sphere is the same.
A solid angle can be considered as a portion of the surface of any sphere, a share that stays the same regardless of the sphere's size, because of solid angle's intrinsic relationship to the center of the sphere. Solid angle is the three-dimensional version of angle, the two-dimensional concept describing the portion of a circle that is "occupied" by two lines that intersect at the circle's center. Solid angle is measured in steradians, like angles are measured in radians. Solid angle is related to area on the surface of a sphere the same way radians can be considered as lengths along a circle. The solid angle of any entire sphere is 4pi steradians (approx. 12.57 str) like the angle of any full circle is 2pi radians (approx. 6.28 rad).
Typically the human visual field covers a bit more than four steradians.
As an example of steradians, the cone that corresponds to one steradian extends out from the centerline of the cone (or the line of sight) by an angle of 32.77 degrees in all directions (i.e. a "diameter" around 65 deg.). A cone that extends out two degrees from its centerline - corresponding to human foveal vision - is 0.004 steradians (or 4 "millisteradians") while a cone that extends ten degrees - about the size of human parafoveal vision - is about 0.1 steradians. The cone of a 1/3rd degree luminance meter is about 0.03 millisteradians (note that the 1/3rd degree refers to the diameter of the sensor field, not its radius). The mathematical relationship is that the solid angle for any cone is equal to 2pi * (1-cosine(angle)).
A more general equation - used to determine the solid angle between two cones with the same centerline - is 2pi * (cos(small angle) - cos(large angle)), which produces the values known as zonal multipliers (e.g. in the IESNA Lighting Handbook), used to convert luminous intensity to luminous flux.
This is consistent with the definition of intensity as flux per solid angle. One candela equals one lumen per steradian. Also, one candela equals half a lumen in half a steradian, or one thousandth of a lumen in one thousandth of a steradian, or four lumens in four steradians. Similarly, one hundred lumens in one steradian equals one hundred candela, and one hundred lumens in one-tenth of a steradian equals one thousand candela.
While this relationships between intensity and flux is simple, the numerical correspondence becomes less direct when applied to more complex solid angles than cones. Most luminaires have photometric information reported in candela, and the presentation of intensity data on polar plots is commonly used to indicate light distribution. While this may be a valid partial representation in terms of intensity, such diagrams tend to misrepresent the distribution of flux, because they are merely showing a slice - a two dimensional representation - of the three-dimensional distribution.
For example, using conventional (Type C) photometric spherical orientation, elevation is measured from nadir (straight down from the center of the sphere) to zenith (straight up), with values between 0 and 180 degrees. Horizontal corresponds to 90 degrees elevation. The azimuth angle is measured from an arbitrarily selected direction, with values between 0 and 360 degrees (although most photometric reports will use symmetry to reduce the amount of data required to describe a complete distribution.)
For the discussion above about cones, nadir corresponds to the cone centerline, the angle from the centerline corresponds to the elevation angle, and the solid angle corresponding to the cone extends around the entire 360-degree-azimuth circle. The zonal multipliers (used to calculate zonal lumens from intensity) relate to "nested" cones, or the solid angle between two cones that have different angles from nadir. However, as the elevation increases from nadir toward horizontal, the solid angle corresponding to a fixed azimuth angle - one degree or the entire 360-degree circle - increases.
As an example, consider the size of a piece of paper needed to cover different sections of a ball. The area of the "band" that covers ten degrees of elevation next to nadir (i.e. a ten-degree cone that is a circle on the ball) is smaller than the band that covers ten degrees at horizontal.
This means that the solid angle subtended (i.e. "occupied") by each degree of elevation is dependent on the elevation. In a way this is not true for azimuth angles, since at the same elevation, any one section of azimuth subtends the same solid angle as any other of equal angular extent. However, as the associated elevation changes the solid angle of a fixed section of azimuth chages, as demonstrated on the ball with pieces of paper. In terms of solid angle, going from nadir to horizontal, each degree of elevation is different from every other degree of elevation, but all degrees of azimuth at the same elevation are the same.
Over the complete 360-degree-azimuth, the solid angle between 0 and 10 degrees elevation (i.e. a ten-degree cone) is 0.10 steradians. For the complete 360-degree-azimuth, the solid angle between 85 and 95 degrees elevation is 1.1 steradians while the solid angle between 89.5 and 90.5 degrees is 0.11 steradians. This means the solid angle of a ring at horizontal that is one degree "tall" is about the same as a cone of ten degrees.
One 360th of that "one-degree tall" ring - a quadrilateral extending over one degree of elevation at horizontal and any one degree of azimuth - is about 0.3 millisteradians, while a one degree cone is 1 millisteradian (in fact the difference is a factor of pi, as it should be.)
As seen from these examples, to convert any intensity value to its corresponding flux, it is necessary to use the angles that correspond to the intensity measurement. The IESNA Lighting Handbook (9th ed., page 2-25) states "The product of the midzone intensity and the zonal constant gives the zonal lumens." This means that every value in a photometric report has to be assigned a zone that extends halfway from its reported position to the next reported intensity value, in all four directions - higher and lower elevation, higher and lower azimuth. Typically, the intervals used in photometric reports are constant (although elevation intervals at nadir are often one half of the typical interval.) Therefore the determination of an intensity value's corresponding zone may be straightforward.
For example, a (Type C) photometric report might indicate 100 candelas at elevation of 30 degrees on the 45-degree azimuth plane. The corrersponding lumens are entirely dependent on the size of the zone that this measurement covers. If the report has ten-degree intervals of elevation - i.e. 20, 30, 40 - then the extent of this zone would be from 25 to 35 degrees of elevation. If the report uses five-degree intervals - i.e. 25, 30, 35 - then the extent of this zone would be from 27.5 to 32.5 degrees elevation. Similarly if the azimuth is reported at 0, 45 and 90, then the extent of the zone is from 22.5 to 67.5 degrees azimuth, while if the azimuth plane interval is different, the extent is correspondingly different.
For the entire 360 azimuth, the solid angle for a zone from 25 to 35 degrees elevation is equal to 0.548 str - while the solid angle for 27.5 to 32.5 is only 0.274 str. The portion of the azimuth is proportional to the entire circle - at the same elevation, one section of azimuth subtends the same solid angle as any other of equal angular extent. So if the azimuth interval is 45 degrees, the portion is 45 * 2pi/180 or one-eighth of the entire circle - 0.068 or 0.034 str for the two different elevation intervals.
So for 100 candelas at 30 deg elevation and 45 deg azimuth, we get 6.8 lumens when the elevation interval is 10 degrees and the azimuth interval is 45 degrees, and we get 3.4 lumens when the elevation interval is only 5 degrees. Simlarly if the azimuth interval were 90 degrees, we would get 17.6 and 6.8 lumens respectively.
Once solid angle is understood, the relationship between intensity and flux - or candela and lumens - should be clear. However, the numerical relationships - expressed in zonal multipliers - can be complex when the solid angle associated with an intensity value is complex.
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