AN1889. Adjustable Color Temperature Lighting INTRODUCTION COLOR TEMPERATURE AND COLORIMETRY. CIE 1931 xy CHROMATICITY DIAGRAM 1931 PLANE

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1 Adjustable Color Temperature Lighting AN1889 Author: Mihai Cuciuc Microchip Technology Inc. FIGURE 1: CIE 1931 xy CHROMATICITY DIAGRAM INTRODUCTION This document provides an example of obtaining custom color temperature lighting by mixing the outputs of four individual white LEDs with different parameters. The application allows the user to have a dimmable lamp with a particular color temperature or use a dynamic model which simulates an incandescent lamp whose color temperature changes as it is dimmed. The methodology presented in this note can be used as-is to produce adjustable color temperature lights using LEDs with known parameters as well as part of an automated setup to yield consistent lighting sources by measuring and calibrating unknown LEDs. COLOR TEMPERATURE AND COLORIMETRY The color temperature of a light source is the temperature of a black body that radiates light which is close to the color of the light source. In order to define distance between two colors, one needs a parametric way of defining colors and the 1931 CIE xy color space, depicted in Figure 1, has been chosen for this purpose. This color space also provides a simple mechanism that aids in color mixing. Figure 1 also shows the Planckian locus defined by the possible colors of a black body as its temperature changes. The colors that this application tries to achieve lie on the Planckian locus. The coordinates of a color corresponding to temperature T in this plane have the following values (see Equation 1). EQUATION 1: xt yt COORDINATES IN THE CIE 1931 PLANE XT = XT + YT + ZT YT = XT + YT + ZT X(T), Y(T) and Z(T) are the tristimulus values and can be computed from the spectral power distribution of the source, I(,T). The functions x(, y(, z( are the CIE color matching functions Microchip Technology Inc. DS A-page 1

2 EQUATION 2: XT YT ZT TRISTIMULUS VALUES FOR A GIVEN SPECTRAL POWER DISTRIBUTION, I () The use case for this application is that the light source should be configured starting from a color temperature. Equation 3 gives Planck s law which describes the spectral power distribution of a black body of a specific temperature. EQUATION 3: I T = = = PLANCK S LAW Here, is the wavelength, h is Planck s constant, c is the speed of light, k B is the Boltzmann constant and T is the temperature that the color needs to be computed for. Plugging the temperature into Equation 3 and using the resulting expression to carry out the integrals in Equation 2, yields the tristimulus values for a specific temperature. These can then be used in conjunction with Equation 1 to obtain the coordinates in the CIE 1931 color space of the color that the lamp should have. Using these coordinates, the color mixing can be performed as described in Section Mixing Colors to obtain the combination of LED currents which yields that color. LIGHT OUTPUT I Txd I Tyd I Tzd 2hc 2 1 = Since this application allows the configuration of the color temperature, it would be desirable to have the same luminous flux range for any color temperature chosen, for consistency. This requires an extra normalization step at the end of the color mixing process to bring the maximum luminous flux to the same value for all possible color temperatures. e hc k B T 1 Simulating an incandescent light requires yet an additional scaling factor for each color temperature that needs to be computed. The chosen dimming mechanism models a black body whose power output is being controlled by a slider. This allows linking the color temperature with the luminous flux output in a natural manner, using the Stefan-Boltzmann law which states that the power radiated by a black body is proportional to the fourth power of its temperature (see Equation 4). EQUATION 4: STEFAN-BOLTZMANN LAW For this application, the power is linearly varied between two arbitrary values, P MIN and P MAX and the corresponding temperature is bounded by T MIN and T MAX, such that their color temperatures can be rendered by the combination of LEDs (see Equation 5). EQUATION 5: P = a T 4 COLOR TEMPERATURE AS A FUNCTION OF POWER 4 P P TP = T min + T max T min min Note: The powers in Equation 5 are dimensionless, since the proportionality constant between P and T 4 is ignored. While a dimension could be harmlessly added since it would cancel out, its omission is more suggestive of the virtuality of the role this value plays in this application. Computing the coordinates in the CIE 1931 color space that correspond to each temperature can be carried out using the mechanism in Section Color Temperature and Colorimetry. The tristimulus value Y(T) is proportional to the luminous flux for a given power emitted by the LED. Since the color mixing algorithm produces relative intensities of the individual LEDs, this proportionality allows computing a scaling factor for each color temperature that applies to all LEDs. 4 4 P max 4 P min DS A-page Microchip Technology Inc.

3 MIXING COLORS Starting with a desired color defined by its coordinates in the 1931 CIE color space, the LED contributions are computed using the following mechanism (see Equation 6): Checking if the point lies within any of the triangles defined in the 1931 CIE color space by the coordinates of three of the available LEDs. For all triangles that contain the point, compute the individual LED contributions, as described in Equation 6. Select the configuration that yields the maximum output for the chosen LED. EQUATION 6: RELATIVE FLUX VALUES REQUIRED FOR YIELDING THE DESIRED OUTPUT COLOR Here, indices A, B and C can refer to any combination of the four available LEDs in the application. Equation 7 gives the LED dimming values that can be obtained from the flux values under the following conditions: Flux output approximated to be proportional to LED die current All maximum luminous flux values are given at the same die current (from LED measurements) Application only requires relative flux (the color is defined by the proportions of fluxes, not their absolute values). EQUATION 7: LED DIMMING VALUES k i = Flux i Y i HARDWARE The present application targets a modified Microchip Lighting Communications Main Board using four CREE LEDs whose color coordinates in the CIE 1931 color space have been measured. The parameters for these LEDs are shown in Table 1, measured at 350 ma. The Microchip Lighting Communications Main Board can supply a maximum current through each LED on the order of 200 ma. The color coordinates are assumed constant with varying currents and the luminous flux values proportional to the current. Within these assumptions, the currents needed to mix the outputs from the LEDs into a particular color temperature become easily computable. The last step is to scale these dimming values such that they lie between 0 and the maximum value usable for the LED PWM. The values obtained after this step can be used to yield a constant luminous flux of any color temperature. Further scaling can be done in order to do dimming on a single color temperature, or these values can be used in conjunction with the scaling factors computed in the Section Light Output to simulate an incandescent lamp Microchip Technology Inc. DS A-page 3

4 TABLE 1: LED COORDINATES IN THE 1931 CIE COLOR SPACE AND THEIR LUMINOUS FLUX VALUES x y Y (Luminous flux [lm]) LED LED LED LED Using these four LEDs, just a small subset of colors from the CIE 1931 color space are available. This subset is the bounded area delimited by the coordinates of the individual LEDs. Figure 2 shows this area as well as the part of the Planckian locus that can be reproduced using this setup. FIGURE 2: ACCESSIBLE GAMUT USING THE FOUR WHITE LEDs AND THE SECTION OF THE COLOR TEMPERATURE CURVE THAT CAN BE RENDERED The application can reproduce light with the color temperature in the range 2100K 5700K with constant luminous flux. The intensities for each LED to achieve this are given in Figure 3. FIGURE 3: LED INTENSITIES USED TO REPRODUCE EACH COLOR TEMPERATURE When simulating an incandescent light, each color temperature is associated with a relative luminous flux and this relationship is depicted in Figure 4. The color temperature is dictated by the power of a virtual black body (Figure 5), configured using the slider available on the board. The mechanism for obtaining these values is described in Section Light Output. FIGURE 4: SCALING FACTOR USED FOR SIMULATING AN INCANDESCENT LAMP DS A-page Microchip Technology Inc.

5 FIGURE 5: COLOR TEMPERATURE OF THE SOURCE AS A FUNCTION OF SLIDER POSITION The lamp has three modes of operation that the user can switch through: Simulating an incandescent lamp this both dims and changes the color temperature according to the proposed model. Constant luminous flux this mode allows the user to pick a color temperature using the slider while keeping the luminous flux at the maximum. Dimming with constant color temperature this mode performs dimming while using the color temperature chosen in the previous mode. PC APPLICATION The simulation of an incandescent lamp starts with the user selecting a slider position. This is translated into a color temperature (Figure 5) which can be used to pick both a scaling factor for all LEDs (Figure 4), as well as individual scaling factors for each one (Figure 4). Controlling the individual LEDs to simulate an incandescent lamp requires a considerable amount of computation which can be performed offline and stored in the PIC device at compile time. A PC application is provided which computes these tables, based on user configuration (e.g., LED parameters). A screenshot of this application is shown in Figure 6. FIGURE 6: PC APPLICATION THAT PERFORMS MOST OF THE COMPUTATION FOR CONFIGURING THE LAMP The PC application s output consists of a number of constants grouped in six tables. These need to be copied from the PC application into the application.c source file for the firmware. The application can be used with other LED parameters since these are fully adjustable, as is the color temperature range Microchip Technology Inc. DS A-page 5

6 REFERENCES (10-deg XYZ CMFs transformed from the CIE (2006) 2-deg LMS cone fundamentals) 3. Microchip AN1562 High Resolution RGB LED Color Mixing Application Note DS A-page Microchip Technology Inc.

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