Achieving maximum efficiency in LED luminary and LCD backlight designs
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).