Often misunderstood, there are finite limits regarding how much light can be produced with a given amount of electrical power. Knowledge of these limits yields insights into LED luminary and LCD backlight design, with the ultimate goal of developing both a functional and optimal first-pass prototype.
Achieving an optimal design may involve investment in higher efficacy LEDs, improved switching regulator design, and/or compromises to the industrial design. Optimal designs feature reduced heat sink size and minimal heat output, yielding a desirable industrial design while sipping a bare minimum of electrical power.
The Commission Internationale De L’Eclairage (CIE) is the key international governing body of colorimetry. CIE has defined two sets of color-matching functions that are the cornerstone for the calculations used throughout this article. The CIE1931 color-matching functions define light in fields with two degrees of angular subtense to the viewer and are used when matching colors in a small area, such as accent lighting. The CIE1964 color-matching functions define light in fields with 10 degrees of angular subtense to the viewer. These supplementary functions are used when matching colors over a broader area, such as lights used to wash a wall.
Efficacy with respect to lighting typically refers to the amount of light (lumens) produced by a luminary (lamp, light bulb, LED, and so on), as a ratio of the amount of electrical power (watts) consumed to produce it. A lumen is defined to be unity for a radiant energy of 1/683 watt at a frequency of 540 THz. In air at standard temperature and pressure (STP), light with a frequency of 540 THz corresponds to a wavelength of 555.017 nm.
Depending on the color-matching functions used (CIE1931 or CIE1964), the maximum possible efficacy changes slightly. The peak of the CIE1931 luminosity function occurs at 555 nm; 555.017 nm on the CIE1931 luminosity curve corresponds to 0.999997, which equates to 683 / 0.999997 = 683.002 lm/W. The peak of the CIE1964 luminosity function is slightly offset from the CIE1931 peak at 557 nm. Whereas, 555.017 nm on the CIE1964 luminosity curve corresponds to 0.999122 or 683 / 0.999122 = 683.601 lm/W. These definitions only apply if the light source is monochromatic and green (555 nm or 557 nm accordingly). For simplicity, all calculations assume a maximum efficacy of 683 lm/W regardless of the chosen color-matching curves. Light sources with differing chromaticity coordinates and/or spectral distributions have lower maximum efficacies.
Optical efficiency is calculated by dividing the maximum possible efficacy at the corresponding chromaticity coordinates into the measured efficacy at the same coordinates. Since the maximum possible efficacy will change depending on the spectral distribution (and thus chromaticity coordinates), using 683 lm/W as a maximum efficacy for all colors will yield incorrect results. Exercise care to ensure that the spectral distribution of the light source equates to that used to calculate the maximum value.
Limits of the visible spectrum
Optical calculations, particularly ones dealing with maximum efficiency, are impacted greatly by the definition of the visible spectrum. Meaningful comparisons demand a consistent definition.
A wide variety of definitions are available from various sources. CIE has published 5 nm color-matching tables for monochromatic light that include wavelengths between 380 – 780 nm. Color-matching tables with 1 nm increments, including wavelengths between 360 – 830 nm, are also available. The 1988 Photopic Luminous Efficiency Function from CIE (Figure 1) shows visible light between the wavelengths of 380 – 780 nm. Additionally, a narrower spectrum of light between 400 – 700 nm is frequently used as 99.93% of the optical energy beneath the photopic curve falls between these wavelengths.
Calculations, particularly those dealing with ideal black body models, can change drastically depending on the definition of what wavelength of light is visible. Both the full photopic range (380 – 780 nm) and the narrower alternate range (400 – 700 nm) will be considered as standard throughout the rest of the discussion.
Figure 1: The 1988 photopic curve defines the visible spectrum as containing wavelengths between 380 – 780 nm. Alternate limits between 400 – 700 nm contain 99.93% of the total visible optical energy.
Calculating maximum efficacy from a spectral density curve
Chromaticity coordinates  and maximum efficacy can be calculated after normalizing the total energy beneath the spectral density curve. Once normalized, multiplying the Y coordinate by 683 lm/W yields the maximum efficacy of the light source.
The spectral density curve in Figure 2 can be digitized using a number of free software tools . Once digitized, the resulting data is then normalized (green curve) such that the sum of the power underneath the curve totals 1. Chromaticity coordinates are calculated using the normalized data.
Figure 2: Digitized data is indexed, smoothed, and normalized before calculating chromaticity coordinates and maximum efficacy.
Calculated chromaticity coordinates for this particular LED (Nichia NNSW208CT) are x = 0.2989 and y = 0.2952, which correlate closely with the published coordinates in the datasheet for the binned selection sbj26. The calculated maximum efficacy is 296.36 lm/W and is dependent on the shape of the spectral distribution.
The calculated typical efficacy of the LED at a specified operating point of 20 mA is 150.00 lm/W, which is obtained by using a few common datasheet parameters (VF, IF, and FF). Dividing the typical efficacy into the theoretical maximum efficacy yields an efficiency of 50.61%.
Just over half of the electrical power injected into the LED is converted into light energy within the visible spectrum. By mixing two monochromatic sources (dichromatic) it is possible to create a light source yielding the same chromaticity coordinates, but yielding a much higher efficacy. The maximum dichromatic efficacy at these chromaticity coordinates is 382.71 lm/W.
Calculating maximum efficacy of dichromatic light
In 1949 David MacAdam postulated that the maximum efficiency of any colored light could be created in only one way: by mixing two monochromatic sources at suitable intensities. MacAdam’s original data (Figure 3 ) was digitized from a copy of his original submission. While his theory and calculated curves are widely known, little is known on the method MacAdam used to obtain the data.
Figure 3: Maximum possible luminous efficiency (lumens per watt) shown on CIE 1931 chromaticity diagram (MacAdam)
Replicating MacAdam’s results is relatively easy using modern computer processing power and a simple brute force calculation method. CIE chromaticity coordinates can be calculated by using two monochromatic light sources at varying wavelengths and intensities. Ensuring that the intensities of the monochromatic sources sum to a value of one, an apples-to-apples comparison across the entire color loci can be ascertained. A sweep of all monochromatic wavelengths between 360 – 830 nm blankets the color loci with calculated xyY values. For each pair of monochromatic sources, the intensity of the first monochromatic source is swept from 0 to 1 in increments of 0.0001.
The intensity of the second monochromatic source is fixed at 1 minus the intensity of the first source. As an iterative process, xyY values are calculated, compared against previously calculated values (if any), and stored in a large matrix memory array. Calculated xy values are rounded to the nearest 0.0001 increment. The corresponding Y value is compared against the existing value. If the newly calculated Y value is larger, the xyY contents of that matrix cell are replaced.
Calculations generate xyY coordinates along a line between the two monochromatic sources used to generate the target color. Using monochromatic sources spaced at 5 nm increments (Figure 4) leaves a substantial number of uncalculated holes in the color loci.
Decreasing the spacing between monochromatics to 1 nm improves the calculated results substantially. The 5 nm CIE color-matching tables are available directly from CIE. However, 1 nm tables are more difficult to find. They are available in print , and are also available for download  in Excel format from various third-party websites.
The 1-nm monochromatic wavelength spacing produces results good enough to prove the concept, but to achieve results that rival MacAdam’s 1949 paper, smaller intervals are required. Interpolation is the key. CIE recommends linear interpolation. Results displayed in Figures 5 – 7 use monochromatics spaced at 0.01 nm increments.
Figure 4: Accurate results depend on adequate coverage by dichromatic calculation lines. Using 5 nm increments, as illustrated here, does not provide the required coverage. Note that 0.01 nm increments are required to produce comprehensive results.
Figure 5: Maximum possible luminous efficiency (lumens per watt) shown on CIE 1931 chromaticity diagram (Schelle).
Comparing dichromatic results
These calculations yielded similar results to what MacAdam obtained 65 years ago. Comparing the two datasets (Figure 6 )reveals a few noticeable differences. MacAdam’s dataset consistentlycalculated maximum efficacy values lower than the modern dataset. Thisis most noticeable at the knees of the 100, 150, and 200 lm/W contours.
Littleis known about how MacAdam calculated his dataset, though at the timeelectronic calculators did not exist and all mathematical computationswere done either by hand or by slide rule. When performing repetitivecomputations, researchers often built custom slide-rules that vastlyaccelerated calculations.
The number of calculations required toobtain the accurate plots shown in Figure 5 surpass 1.1 trillion. Moderncomputer processors solve trillions of calculations in a relativelyshort time. Performing these calculations on a modern PC with asingle-threaded calculation engine took approximately 11 days.
Moderncomputer processing power simply wasn’t available to MacAdam in 1949when he published his work. As a point comparison, the ENIAC representedthe pinnacle of computing in MacAdam’s day and it would have takendecades to complete these calculations under the best of circumstances.Calculating all possible unique dichromatic permutations was simplyimpossible in 1949.
Figure 6: Maximum possible luminous efficiency (lumens per watt) shown on CIE 1931 chromaticity diagram (Schelle vs MacAdam)
CIEdefines two sets of color-matching functions for use as standardobservers. The CIE1931 Standard Colorimetric Observer is based on afield of view constrained to a 2o angular subtense of the eye. The 2olimitation was imposed to constrain the image to the fovea within theeye. The fovea contains a dense concentration of cones (color receptors)and no rods.
The CIE 1964 Supplementary Standard ColorimetricObserver is based on experiments with matching fields constrained to 10oof angular subtense to the eye. CIE recommends using the CIE1964functions “whenever correlation with visual color matching of fields ofangular subtense greater than about four degrees at the eye of theobserver is desired.” The color loci, and the maximum efficacy curvesshown in Figure 7, are calculated using the same brute force method asthat used for Figures 5 – 6 .
The practical applicationshould determine which locus is used when calculating maximumdichromatic efficacy. Applications where the majority of the light iscontained to small point sources (mobile-devices, accent lighting, plusothers), should use the CIE1931 loci. Applications where the light isdispersed over a large area (LED down lighting, wall-washing, and so on)should use the CIE1964 loci.
Figure 7: Maximum possible luminous efficiency (lumens per watt) shown on CIE 1964 chromaticity diagram (Schelle)
Maximum efficacy and the black body model
Morerelevant to modern day electronics is the production of white lightthat falls along the black body curve. The algorithm used to create thecontours seen in the above figures also stores maximum efficacy data atevery point on the loci. Pulling data points off the curve thatcorrespond to a particular correlated color temperature (CCT) yieldsinteresting results (Figure 8 ).
Included in the plot areCIE point sources for Illuminant A, and the D65 white point. Thesesources are plotted according to their closest CCTs of 2857oK and6503oK, respectively. Also included is the Nichia LED (NNSW208CT with a CCT of 7924oK. The CCT for all point sources is calculated usingMcCamy’s  formula.
Click on image to enlarge.
Figure 8: Maximum efficacyplotted against correlated color temperature and compared against CIEstandard illuminants and real LEDs.
Table 1 summarizes key pieces of this data and compares them to real light sources.
Efficacy, efficiency, and LED heat sink design
Theinformation provided here can be used to more closely approximate theactual power losses within an LED luminary or backlight design thatmight be attributed as heat. Traditionally, the conservative approachwould be to assume that 100% of the power applied to the LED generatesheat. This approach may yield undesirable design requirements such as anLED luminary heat sink that is too large for the required enclosure.
Startby calculating the maximum efficacy of the LED using the given spectraldensity that typically is located in the manufacturer’s datasheet. Thecalculated maximum optical efficacy from our example LED (Figure 2 )is 296.36 lm/W. Dividing the typical efficacy (150.00 lm/W) obtainedfrom the manufacturer’s datasheet yields a net power efficiency of50.61% at 56 mW of electrical power input. These figures are calculatedusing an LED operating point of 2.8V@20 mA.
Assuming 10 of theseLEDs are used in a luminary design, total power can be calculated at560 mW. Of that, 50.61% or 283 mW is converted to light. The law ofconservation of energy dictates that no energy can be created anddestroyed – only transformed. A conservative assumption would be toattribute the remaining power (277 mW) as heat that is dissipated in theheat sink. Size the heat sink accordingly.
As with all designadvice, there is no substitute for a prototype. Thoroughly test theproduct in the end application environment before committing toproduction.
Figure9: In this example, of the 560 mW delivered to an LED, 283 mW isconverted to visible light. The conservation of energy law dictates thatthe remaining energy must be converted to something else. Aconservative approach would attribute the remaining energy to heat.
AnExcel spreadsheet  contains graphs, calculated data, and visualbasic code used to calculate maximum efficacies across the CIE 1931 and1964 loci. The included Excel/VBA code also generates maximum efficacyfigures for all points on the CIE loci and stores them in a separateExcel spreadsheet. CIE loci x/y granularity dictates the size of thespreadsheet. An x/y increment of 0.0001 (used for all graphs in thisarticle) generates a spreadsheet that is approximately 600 MB in size,making it unsuitable for general download. Recreate the spreadsheet byrunning the embedded visual basic code.
Donald Schelle is an Analog Field Applications Engineer for Texas Instruments power group and has more than a decade of engineering experience. Hereceived his Bachelor of Electrical Engineering from LakeheadUniversity, Thunder Bay, Ontario, Canada. Donald can be
reached at .
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