# Influence of Peak Wavelengths on Properties of Mixed-LED White-Light Sources

Authors: Snjezana Soltic; Andrew N. Chalmers
Journal: Advances in OptoElectronics
 (2010)

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## Abstract

The purpose of this investigation is to quantify the influence of the peak wavelength shifts in commercially available LEDs on the characteristics of the mixed-LED white-light sources. For this purpose, a tetrachromatic spectrum was optimized and then subjected to deviations in the peak wavelengths. A total of 882 combinations of peak wavelength values were evaluated, and the results are reported in terms of correlated colour temperature, colour-rendering properties, and radiant luminous efficacy. The results show that there can be significant changes in the characteristics of the source under these conditions. Such changes are highly likely to present problems when dealing with applications where an effective and accurate white-light source is important.


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## Body

## 1. Introduction

White light can be produced by additively combining the outputs of multiple monochromatic light-emitting diodes (LEDs) [1–6]. Thus, by fine-tuning the spectral intensity of individual LEDs, which emit different narrow bands of radiation, a white-light source characterized by a good colour-rendering index, Ra, and high luminous efficacy of radiation, ηrad, can be designed.Creating a stable white light using multiple LEDs is a complex task since both colour rendering and luminous efficacy of the mixture depend on the emitted spectrum of the individual LEDs. Therefore, any change in the LED parameters (peak wavelengths, spectral widths, lumen outputs, etc.), for example, due to variations in junction temperature of the LEDs, causes a change in the spectrum of the LED devices and consequently a change in the spectrum of the white-light source [7, 8]. The amount of change will depend on the magnitude of the deviations for each LED in the mixture. Keeping the mixture stable is further complicated by the fact that different types of LEDs are affected differently even when they are working under the same conditions [9]. Therefore, designing a simple yet accurate control system to maintain the white point within acceptable tolerances is one of the key challenges [7, 8, 10, 11].We demonstrated that white-light sources based on LEDs have the potential of becoming the optimum choice for high-colour-rendering tasks [1], and in [2] we introduced our approach to intelligent spectral design. However, implementing a stable mixed-LED white-light source based on a theoretical (optimized) spectrum is an exacting task since there is no guarantee that a given set of real LEDs will exactly match the parameters used during the spectrum design. Hence, we have investigated the sensitivity of an optimized 4-band LED mixture to possible changes in the peak wavelengths of any one or more of the four LEDs. The aim was to quantify the influence of peak wavelength shifts in commercially available LEDs in terms of changes in correlated colour temperature, colour rendering, and luminous efficacy.


## 2. Background

### 2.1. Correlated Colour Temperature

Correlated colour temperature (CCT) of a light source is defined as the temperature of a Planckian (blackbody) radiator with chromaticity nearest to the chromaticity coordinates of the source on the CIE 1960 (u,v) diagram [12–14]. As the temperature of a blackbody radiator increases from 2000 K to 20000 K, the perceived colour of the white light changes from “warm” (very reddish) to “cool” (very bluish). For example, incandescent lamps (circa 3000 K) are characterized as being yellowish white, while fluorescent lights (4000 K–7000 K) are bluish white. In the text the correlated colour temperature is symbolized as Tc. Since a given change in colour temperature expressed in kelvins (K) at different temperatures results in unequal changes in chromaticity [13], the colour temperature (or correlated colour temperature) is often expressed in reciprocal megakelvins, MK−1, and given the symbol TcM, where TcM=106/Tc[K], which provides more uniform chromaticity differences. Reciprocal megakelvins were formally known as “microreciprocal degrees”, abbreviated to “mireds.” In the text the correlated colour temperature when expressed in Kelvin degrees is symbolized as Tc and as TcM when is expressed in MK−1.


### 2.2. Colour-Rendering Indices (CRI)

Colour rendering is the characteristic of light sources that describes the visual effect of a light source on the colour of an object. Typically, at the present time, the colour-rendering properties of light sources are evaluated using the colour-rendering indexRa [15]. In general, the higher the Ra value of a light source, the more “natural” the colours of objects look under this source. Sunlight and incandescent lamps have Ra=100, which is the maximum value a light source can have. The acceptable value of CRI depends on the application of the light source. Values of Ra above 80 are considered sufficient for most social and commercial indoor lighting applications, while Ra above 90 or 95 is desirable in colour-matching tasks.The fundamental idea behind theRa calculation is a comparison of the colours of eight standard test colour samples, which have low to moderate chromatic saturation (Table 1, samples 1–8), illuminated, in turn, by the test and reference light sources. The reference source has to be a source with the same CCT as the test source, either a Planckian radiator for test sources having CCT below 5000 K or a phase of daylight for test sources having CCT at or above 5000 K. After accounting for chromatic adaptation with a Von Kries-type chromatic adaptation transform [13], the special colour-rendering index Ri for each test colour sample is calculated as(1)Ri=100-4.6ΔEi,i=1,2,…,8,
where ΔEi is the colour difference for each sample between the two light sources, as computed in CIE 1964 W*U*V* uniform colour space [13].Table 1
CIE test colours [15].
No.Approximate munsell notationColour appearance under daylight17.5 R 6/4Light greyish red25 Y 6/4Dark greyish yellow35 GY 6/8Strong yellow green42.5 G 6/6Moderate yellowish green510 BG 6/4Light bluish green65 PB 6/8Light Blue72.5 P 6/8Light violet810 P 6/8Light reddish purple94.5 R 4/13Strong red105 Y 8/10Strong yellow114.5 G 5/8Strong green123 PB 3/11Strong blue135 YR 8/4Light human complexion145 GY 4/4Moderate olive green (leaf green)Averaging the eightRi values results in the general rendering index Ra:(2)Ra=18∑i=18Ri.The CRI metric was originally designed to assess the quality of traditional fluorescent lamps and is found deficient when applied to assessing the colour-rendering quality of narrow-band light sources, such as the LED-based white-light sources [16–20]. Hence, optimizing mixed-LED white-light spectra using only eight colour samples of low to medium saturation can result in spectra with good computed Ra but actually providing poor rendering of saturated colours. Therefore, in an attempt to improve the descriptive power of colour-rendering index, the number of test samples was extended to 14 by the addition of six additional test samples (Table 1, samples 9–14) representing saturated red, yellow, green, and blue, plus light human complexion and leaf green. Averaging the additional six Ri values gives what we term index Rb, and averaging all 14 Ri values results in an “overall” index Rc.In order to provide additional data on the colour-rendering properties of each source, we have chosen also to quote the lowestRi (Rimin) and the corresponding sample number (imin) for the colour yielding Rimin.


### 2.3. Radiant Luminous Efficacy

Radiant luminous efficacy,ηrad lm/W, compares the amounts of luminous flux and radiant flux emitted by the source:(3)ηrad=Km∫λV(λ)S(λ)dλ∫λS(λ)dλ,
where S(λ) is the spectral distribution of the light source, V(λ) is the CIE spectral sensitivity function for photopic vision, and Km is the maximum luminous efficacy of radiation (Km=683 lm/W). The peak in the V(λ) curve occurs for monochromatic radiation at 555 nm.


## 2.1. Correlated Colour Temperature

Correlated colour temperature (CCT) of a light source is defined as the temperature of a Planckian (blackbody) radiator with chromaticity nearest to the chromaticity coordinates of the source on the CIE 1960 (u,v) diagram [12–14]. As the temperature of a blackbody radiator increases from 2000 K to 20000 K, the perceived colour of the white light changes from “warm” (very reddish) to “cool” (very bluish). For example, incandescent lamps (circa 3000 K) are characterized as being yellowish white, while fluorescent lights (4000 K–7000 K) are bluish white. In the text the correlated colour temperature is symbolized as Tc. Since a given change in colour temperature expressed in kelvins (K) at different temperatures results in unequal changes in chromaticity [13], the colour temperature (or correlated colour temperature) is often expressed in reciprocal megakelvins, MK−1, and given the symbol TcM, where TcM=106/Tc[K], which provides more uniform chromaticity differences. Reciprocal megakelvins were formally known as “microreciprocal degrees”, abbreviated to “mireds.” In the text the correlated colour temperature when expressed in Kelvin degrees is symbolized as Tc and as TcM when is expressed in MK−1.


## 2.2. Colour-Rendering Indices (CRI)

Colour rendering is the characteristic of light sources that describes the visual effect of a light source on the colour of an object. Typically, at the present time, the colour-rendering properties of light sources are evaluated using the colour-rendering indexRa [15]. In general, the higher the Ra value of a light source, the more “natural” the colours of objects look under this source. Sunlight and incandescent lamps have Ra=100, which is the maximum value a light source can have. The acceptable value of CRI depends on the application of the light source. Values of Ra above 80 are considered sufficient for most social and commercial indoor lighting applications, while Ra above 90 or 95 is desirable in colour-matching tasks.The fundamental idea behind theRa calculation is a comparison of the colours of eight standard test colour samples, which have low to moderate chromatic saturation (Table 1, samples 1–8), illuminated, in turn, by the test and reference light sources. The reference source has to be a source with the same CCT as the test source, either a Planckian radiator for test sources having CCT below 5000 K or a phase of daylight for test sources having CCT at or above 5000 K. After accounting for chromatic adaptation with a Von Kries-type chromatic adaptation transform [13], the special colour-rendering index Ri for each test colour sample is calculated as(1)Ri=100-4.6ΔEi,i=1,2,…,8,
where ΔEi is the colour difference for each sample between the two light sources, as computed in CIE 1964 W*U*V* uniform colour space [13].Table 1
CIE test colours [15].
No.Approximate munsell notationColour appearance under daylight17.5 R 6/4Light greyish red25 Y 6/4Dark greyish yellow35 GY 6/8Strong yellow green42.5 G 6/6Moderate yellowish green510 BG 6/4Light bluish green65 PB 6/8Light Blue72.5 P 6/8Light violet810 P 6/8Light reddish purple94.5 R 4/13Strong red105 Y 8/10Strong yellow114.5 G 5/8Strong green123 PB 3/11Strong blue135 YR 8/4Light human complexion145 GY 4/4Moderate olive green (leaf green)Averaging the eightRi values results in the general rendering index Ra:(2)Ra=18∑i=18Ri.The CRI metric was originally designed to assess the quality of traditional fluorescent lamps and is found deficient when applied to assessing the colour-rendering quality of narrow-band light sources, such as the LED-based white-light sources [16–20]. Hence, optimizing mixed-LED white-light spectra using only eight colour samples of low to medium saturation can result in spectra with good computed Ra but actually providing poor rendering of saturated colours. Therefore, in an attempt to improve the descriptive power of colour-rendering index, the number of test samples was extended to 14 by the addition of six additional test samples (Table 1, samples 9–14) representing saturated red, yellow, green, and blue, plus light human complexion and leaf green. Averaging the additional six Ri values gives what we term index Rb, and averaging all 14 Ri values results in an “overall” index Rc.In order to provide additional data on the colour-rendering properties of each source, we have chosen also to quote the lowestRi (Rimin) and the corresponding sample number (imin) for the colour yielding Rimin.


## 2.3. Radiant Luminous Efficacy

Radiant luminous efficacy,ηrad lm/W, compares the amounts of luminous flux and radiant flux emitted by the source:(3)ηrad=Km∫λV(λ)S(λ)dλ∫λS(λ)dλ,
where S(λ) is the spectral distribution of the light source, V(λ) is the CIE spectral sensitivity function for photopic vision, and Km is the maximum luminous efficacy of radiation (Km=683 lm/W). The peak in the V(λ) curve occurs for monochromatic radiation at 555 nm.


## 3. Method

Evaluation of the influence of peak wavelengths on the characteristics of LED-based light sources was performed on a tetrachromatic spectrum of commercially available LEDs, chosen from the Luxeon range, with a peak wavelength combination of 470, 530, 590, and 625 nm. The mixture was optimized using a differential evolution algorithm [2]. The peak wavelengths were taken from the Luxeon datasheet [21] and are shown in Table 2 as λi(typ). The bracketed figures indicate the nearest available values in our data analysis which was based on a 5 nm wavelength interval. The optimized spectrum (Figure 1) has a correlated colour temperature Tc of approximately 2919 K, good colour-rendering properties (Ra=88), and high luminous efficacy (ηrad=390 lm/W). This spectrum is the reference against which all later comparisons are made.Table 2
Ranges of peak wavelengths for four LEDs used to optimize the theoretical white-light source. The table is an extract from the Luxeon datasheet [21].
LEDλi(min)(nm)λi(typ)(nm)λi(max)(nm)Red (λR)620.5 (620)627 (625)645Amber (λA)584.5 (585)590597 (595)Green (λG)520530550Blue (λB)460470490Figure 1
The spectrum of the optimized light source (used as  reference for comparisons).The Luxeon data specifies wavelength rangesλi for the four LEDs in the mixture. The peak wavelength of the red LED can be anywhere between 620.5 and 645 nm, the amber LED between 584.5 and 597 nm, the green LED between 520 and 550 nm, and 460 to 490 nm for the blue LED (Table 2). For the purpose of quantifying the influences of λi values on the characteristics of the mixture, these ranges were divided into 5 nm bins λij (Table 3) and then the Tc, the colour-rendering indices (Ri,  Ra,  Rb,  and Rc), and the ηrad values were calculated for all 882 (6×3×7×7) possible combinations of λij. The first set of values was calculated at the λij(min)=λi1 values and the last at the λi(max) values (i.e., λR6, λA3,  λG7,  λB7).Table 3
The ranges of possibleλij values divided into 5 nm bins, where i designates one of four LEDs studied in this paper (red λR, amber λA, green λG, blue λB) and j one of the possible 5 nm bins.
LEDλi1 (nm)λi2 (nm)λi3 (nm)λi4 (nm)λi5 (nm)λi6 (nm)λi7 (nm)λR620625630635640645—λA585590595————λG520525530535540545550λB460465470475480485490


## 4. Results

### 4.1. Summary of Results forTc

Table4 summarizes the influences of each individual LED on the correlated colour temperature of the optimized tetrachromatic spectrum over the range of λij, showing Tcij values in kelvins calculated for all λij bin values of one LED at a time, while the λkj of other three LEDs were kept at their typical values, λk(typ), k≠i. For example, the TcRj values show the influence of changes in the red LED and were calculated with λAj=λA(typ)=λA2, λGj=λG(typ)=λG3, and λBj=λB(typ)=λB3 and λRj in the range from λR1 to λR6. Note that the typical peak wavelength of the red LED is λR(typ)=λR2. Thus, TcRj shows the dependence of Tc on λRj, while other LEDs are at λk(typ),TcGj shows the dependence of Tc on λGj, while other LEDs are at λk(typ), and so on. In addition, Table 4 lists the average values for each dependency of Tc, Tciavg, and the difference between minimum and maximum values ΔTci=Tcimax-Tcimin, the percentage change ΔTci[%]=ΔTci/Tciavg×100, and the differences in MK−1 defined as ΔTciM=106/Tcimin-106/Tcimax and rounded to nearest integer.Table 4
The summary of influences of each individual LED on correlated colour temperature of the optimized tetrachromatic spectrum,λij changes from bin to bin, while λkj=λk(typ), k≠i.
Tcij/λijλi1 (nm)λi2 (nm)λi3 (nm)λi4 (nm)λi5 (nm)λi6 (nm)λi7 (nm)Tciavg (K)ΔTci (K)ΔTci[%]ΔTciM (MK−1)TcRj282629193056322934283641—31838152679TcAj304829192813————2927235827TcGj297229642919284127332604245927855131870TcBj291129152919292329282935294429253314As seen in Table4, the changes in λRj and λGj result in the biggest Tc variations (i.e., ΔTcRM=79 MK−1 and ΔTcGM=70 MK−1, resp.), while the blue LED causes the smallest variations (ΔTcBM=4 MK−1). The correlated colour temperature of the tetrachromatic spectrum increases with λRj and λBj and decreases with increasing λGj and λAj. It is evident that significant correlated colour temperature errors occur due to peak wavelength shifts in the red and green spectra and that the error caused by the shifts in the blue LED is one order of magnitude smaller.TheTc values are plotted in Figure 2. The five step-changes in the Tc plot are caused by a change of λRj from bin to bin (j∈[1,6]) together with “reset” of other LEDs’ peak wavelengths to their minimum values, that is, λk(max)⇒λk(min)=λk1, k≠i. There are six distinctive patterns in the Tc plot, one for each λRj shift (bin). A λRj pattern is influenced by seven green shifts (λG1-λG7), and each green pattern is influenced by the seven blue peak wavelength shifts (λB1-λB7). There are three λAj shifts per a λBj bin. As seen in Figure 2, the wavelength shifts of the red LED to a higher value increase the correlated colour temperature of the spectrum. Close inspection of the plot reveals that the increases in Tc are larger at the higher λRj values; the change from λR1 (A) to λR2 (B) resulted in ΔTc=TcR2-TcR1=757 K (104 MK−1), while the change from λR5 (C) to λR6 (D) resulted in ΔTc=TcR6-TcR5=1256 K (116 MK−1). The influences of λij values on the Tc of the mixture are further explored in Figures 3–5 and Table 5.Table 5
The influences ofλRj and λGj on ΔTcM(A)[MK−1]. The values are rounded to 2d.p.
λRj/λGjλR1λR2λR3λR4λR5λR6λG11.661.651.641.701.571.27λG21.621.611.521.531.381.28λG31.401.411.461.621.401.19λG41.341.391.461.561.391.28λG51.451.391.521.561.521.43λG61.301.421.611.571.471.64λG71.131.461.561.451.541.57Figure 2
Tc values calculated for the 882 possible combinations of λij values.Figure 3
ΔTcM(A)=f(λRj,λGj,λBj).Figure 4
Influence ofλGj on Tc of the optimized tetrachromatic spectrum; λGj↑⇒Tc↓.Figure 5
Influence ofλBj on Tc of the optimized tetrachromatic spectrum. The circled portion of the graph shows that some values of λRj and λGj  change the typical influence of λBj (typical:  λBj↑⇒Tc↑).Keeping the red, green, and blue LED at theirλk(typ) while changing λAj from bin to bin (j=1,2,3) resulted in Tc having a maximum value (TcA1=3048 K) at λA1 which decreases to its minimum (TcA3=2813 K) at λA3 (Table 4), a change of 235 K (27 MK−1). Figure 3 shows the dependency of the correlated colour temperature change caused by the λAj shifts from λA1 to λA3, ΔTcM(A)=TcMA1-TcMA3, and Table 5 summarizes the influence of λRj and λGj on ΔTcM(A).TheΔTcM(A) values are calculated as the red peak wavelengths change from λRj to λRj+1 with the peak wavelength of the green LED being at a λGj value, j∈[1,7]. The process was repeated for all λGj and λRj shifts. In Figure 3  λR2, λR3, λR4, and λG1 to λG7 are labelled for reference. It is observed that the ΔTcM(A) magnitudes of the discontinuities at the λRj to λRj+1 increase at higher λRj values (e.g., compare A and B). Table 5 reveals that the ΔTcM(A) values at λG7 are lower than at λG1 for λRj={λR1,…,λR5}.The influence ofλGj and λBj on Tc is shown in Figures 4 and 5, respectively. The Tc values shown in Figure 4 were calculated with peak wavelengths at λRj, λA1, λB1, and λGj, and those in Figure 5 with peak wavelengths at λRj, λA1, λBj, and λGj. The correlated colour temperature of the optimized tetrachromatic spectrum Tc decreases as the λGj value increases. This is most easily observed by inspecting the plot in Figure 4. Seven Tc values are calculated (TcG1-TcG7) for each of the six red LED bins. For the sake of clarity, only one set of TcG1 to TcG7 are labeled on the plot (at λR4). It is observed that Tc is highest at the λG1 wavelength (TcG1) and lowest at λG7 wavelength (TcG7).The influence of theλBj on the correlated colour temperature is not so obvious. In Figure 5 the Tc values are plotted for all possible combinations of λBj. The general trend is that Tc increases with λBj. However, close inspection of the plot reveals that, for some combination of λij (at the higher values of λRj (i.e., λR5 and λR6) and lower values of λGj (i.e., λG1 and λG2)), the Tc values exhibit a small drop in values instead of the increase observed elsewhere. This can be seen inspecting the circled part in Figure 5.The collated results show that a change inλij could cause the correlated colour temperature, Tc, of the optimized mixture to be as high as 3976 K when λRj=λR6, λAj=λA1, λGj=λG1, and λBj=λB1 and as low as 2311 K at minimum peak wavelengths for the red and blue LEDs and maximum peak wavelength values for the green and amber LEDs (λR1,  λA3,  λG7,  λB1) or a difference of 1665 K (181 MK−1). The average value of Tc was 3039 K with a standard deviation of 359 K. Using the specified set of LEDs, Tc changes −21% to +36% of its value in the optimized mixture.


### 4.2. Summary of Results for Colour-Rendering Indices

Table6 provides a summary of the effects of peak wavelength shifts λij on the colour characteristics of the optimized tetrachromatic spectrum, expressed in terms of colour-rendering indices Ra, Rb, and Rc. The errors for each colour-rendering index, shown in Table 6, are calculated as ΔR=R(max)-R(min) and ΔR[%]=ΔR/R(avg)×100. It is seen that variations in colour-rendering characteristics are significant. In particular, the Rb index is the most affected with Rbmax=84 (λR2,  λA1,  λG2,  λB1) and Rbmin=31 (λR1,  λA1,   λG7,  λB1). These rather high variations (ΔRb=53) mean that the colour rendering of the saturated colour samples will be poor. The most affected samples were Samples 9 (strong red) and 12 (strong blue); R12 had the lowest Ri 397 times (in 45 % of combinations) and R9 301 times (in 34% of combinations). Four more samples had the lowest Ri: sample 11 (strong green) 142 times (in 16% of combinations), sample 10 (strong yellow) 38 times (4%), sample 3 (strong yellow green) three times, and sample 4 (moderate yellowish green) once. Significant changes were observed in Rimin=min{Ri}, with Rimin being always below its optimized value Rimin(opt)=73.Table 6
Summary of the influences ofλij on the colour-rendering characteristics (Ra, Rb, Rc) of the optimized tetrachromatic spectrum (Ra(opt)=88, Rb(opt)=79, Rc(opt)=84,  Rimin(opt)=73).
R(avg)R(max)R(min)ΔRΔR[%]Ra7692573546Rb6484315383Rc7187474057Rimin3073−45118387It is also interesting to note the effect of the peak wavelengths of four LEDs on the colour-rendering characteristics as shown in Figure6. For the sake of clarity, the Rc values (Rc=Ra+Rb) are not plotted. As expected, there are five discontinuities in the plot, caused by a change of λRj from bin to bin together with “reset” of other LEDs’ peak wavelengths to their minimum values (λkj(min)=λk1,k≠R). The largest Rb change is when λRj changes from λR1 (A) to λR2(B),  ΔRb=45.5. The size of ΔRb decreases to ΔRb=31.2 at the λR4 (C) to λR5 (D) change, before increasing to 35.6 at the λR5 (E) to λR6 (F) change.Figure 6
Ra and Rb values calculated for 882 possible combinations of λij values.


### 4.3. Summary of Results for Radiant Luminous Efficacy

The effect ofλij changes on ηrad is evident in Figure 7. The radiant efficacy has a maximum ηrad(max)=438 lm/W (λR1, λG7, λB7, λA1) at A and a minimum ηrad(min)=308 lm/W (λR6, λG1, λB1, λA3) at B (which is +12% and −21% from the value in the optimized mixture ηrad(opt)). The ηrad>ηrad(opt) in 292 combinations of λij (in 33% of combinations) and was below ηrad(opt) 590 times (in 67% of combinations). Figure 7 also shows the step changes in ηrad that result from λRj changing from bin to bin (e.g., at A in the graph).Figure 7
ηrad values calculated for 882 possible combinations of λij values.The influences of the green, blue, and amber LEDs are further explored in Figure8, where ηrad is plotted for λR1 and λij of the other three LEDs changing from bin to bin. There are seven plots, one per each of seven λGj bins and 21 ηrad values (for different λAj and λBj combinations) per each λGj bin. For the sake of clarity, the ηrad points were marked only on the λG1 plot. It is evident that the wavelength shifts of the green LED to a higher value λGj result in a ηrad increase; however, the increases are smaller at the higher values of λGj. It is also evident that, as λAj changes from its lowest value (at λA1) to its highest value (at λA3), one observes reductions in ηrad. Again for the sake of clarity, only one set of λA1 to λA3 values are labeled on the λG1 plot. Increasing λBj leads to improvements in ηrad. The ηrad values at λB1 wavelength are lower than the ηrad values at λB7. It was interesting to observe that the difference between the ηrad(min) (A) and ηrad(max) (B) values was constant (67 lm/W) across all λGj plots.Figure 8
ηrad=f(λR1,λij). There are seven plots, one per each λGj bin, and 21 ηrad values for all possible combinations of λBj and λAj bins.


## 4.1. Summary of Results forTc

Table4 summarizes the influences of each individual LED on the correlated colour temperature of the optimized tetrachromatic spectrum over the range of λij, showing Tcij values in kelvins calculated for all λij bin values of one LED at a time, while the λkj of other three LEDs were kept at their typical values, λk(typ), k≠i. For example, the TcRj values show the influence of changes in the red LED and were calculated with λAj=λA(typ)=λA2, λGj=λG(typ)=λG3, and λBj=λB(typ)=λB3 and λRj in the range from λR1 to λR6. Note that the typical peak wavelength of the red LED is λR(typ)=λR2. Thus, TcRj shows the dependence of Tc on λRj, while other LEDs are at λk(typ),TcGj shows the dependence of Tc on λGj, while other LEDs are at λk(typ), and so on. In addition, Table 4 lists the average values for each dependency of Tc, Tciavg, and the difference between minimum and maximum values ΔTci=Tcimax-Tcimin, the percentage change ΔTci[%]=ΔTci/Tciavg×100, and the differences in MK−1 defined as ΔTciM=106/Tcimin-106/Tcimax and rounded to nearest integer.Table 4
The summary of influences of each individual LED on correlated colour temperature of the optimized tetrachromatic spectrum,λij changes from bin to bin, while λkj=λk(typ), k≠i.
Tcij/λijλi1 (nm)λi2 (nm)λi3 (nm)λi4 (nm)λi5 (nm)λi6 (nm)λi7 (nm)Tciavg (K)ΔTci (K)ΔTci[%]ΔTciM (MK−1)TcRj282629193056322934283641—31838152679TcAj304829192813————2927235827TcGj297229642919284127332604245927855131870TcBj291129152919292329282935294429253314As seen in Table4, the changes in λRj and λGj result in the biggest Tc variations (i.e., ΔTcRM=79 MK−1 and ΔTcGM=70 MK−1, resp.), while the blue LED causes the smallest variations (ΔTcBM=4 MK−1). The correlated colour temperature of the tetrachromatic spectrum increases with λRj and λBj and decreases with increasing λGj and λAj. It is evident that significant correlated colour temperature errors occur due to peak wavelength shifts in the red and green spectra and that the error caused by the shifts in the blue LED is one order of magnitude smaller.TheTc values are plotted in Figure 2. The five step-changes in the Tc plot are caused by a change of λRj from bin to bin (j∈[1,6]) together with “reset” of other LEDs’ peak wavelengths to their minimum values, that is, λk(max)⇒λk(min)=λk1, k≠i. There are six distinctive patterns in the Tc plot, one for each λRj shift (bin). A λRj pattern is influenced by seven green shifts (λG1-λG7), and each green pattern is influenced by the seven blue peak wavelength shifts (λB1-λB7). There are three λAj shifts per a λBj bin. As seen in Figure 2, the wavelength shifts of the red LED to a higher value increase the correlated colour temperature of the spectrum. Close inspection of the plot reveals that the increases in Tc are larger at the higher λRj values; the change from λR1 (A) to λR2 (B) resulted in ΔTc=TcR2-TcR1=757 K (104 MK−1), while the change from λR5 (C) to λR6 (D) resulted in ΔTc=TcR6-TcR5=1256 K (116 MK−1). The influences of λij values on the Tc of the mixture are further explored in Figures 3–5 and Table 5.Table 5
The influences ofλRj and λGj on ΔTcM(A)[MK−1]. The values are rounded to 2d.p.
λRj/λGjλR1λR2λR3λR4λR5λR6λG11.661.651.641.701.571.27λG21.621.611.521.531.381.28λG31.401.411.461.621.401.19λG41.341.391.461.561.391.28λG51.451.391.521.561.521.43λG61.301.421.611.571.471.64λG71.131.461.561.451.541.57Figure 2
Tc values calculated for the 882 possible combinations of λij values.Figure 3
ΔTcM(A)=f(λRj,λGj,λBj).Figure 4
Influence ofλGj on Tc of the optimized tetrachromatic spectrum; λGj↑⇒Tc↓.Figure 5
Influence ofλBj on Tc of the optimized tetrachromatic spectrum. The circled portion of the graph shows that some values of λRj and λGj  change the typical influence of λBj (typical:  λBj↑⇒Tc↑).Keeping the red, green, and blue LED at theirλk(typ) while changing λAj from bin to bin (j=1,2,3) resulted in Tc having a maximum value (TcA1=3048 K) at λA1 which decreases to its minimum (TcA3=2813 K) at λA3 (Table 4), a change of 235 K (27 MK−1). Figure 3 shows the dependency of the correlated colour temperature change caused by the λAj shifts from λA1 to λA3, ΔTcM(A)=TcMA1-TcMA3, and Table 5 summarizes the influence of λRj and λGj on ΔTcM(A).TheΔTcM(A) values are calculated as the red peak wavelengths change from λRj to λRj+1 with the peak wavelength of the green LED being at a λGj value, j∈[1,7]. The process was repeated for all λGj and λRj shifts. In Figure 3  λR2, λR3, λR4, and λG1 to λG7 are labelled for reference. It is observed that the ΔTcM(A) magnitudes of the discontinuities at the λRj to λRj+1 increase at higher λRj values (e.g., compare A and B). Table 5 reveals that the ΔTcM(A) values at λG7 are lower than at λG1 for λRj={λR1,…,λR5}.The influence ofλGj and λBj on Tc is shown in Figures 4 and 5, respectively. The Tc values shown in Figure 4 were calculated with peak wavelengths at λRj, λA1, λB1, and λGj, and those in Figure 5 with peak wavelengths at λRj, λA1, λBj, and λGj. The correlated colour temperature of the optimized tetrachromatic spectrum Tc decreases as the λGj value increases. This is most easily observed by inspecting the plot in Figure 4. Seven Tc values are calculated (TcG1-TcG7) for each of the six red LED bins. For the sake of clarity, only one set of TcG1 to TcG7 are labeled on the plot (at λR4). It is observed that Tc is highest at the λG1 wavelength (TcG1) and lowest at λG7 wavelength (TcG7).The influence of theλBj on the correlated colour temperature is not so obvious. In Figure 5 the Tc values are plotted for all possible combinations of λBj. The general trend is that Tc increases with λBj. However, close inspection of the plot reveals that, for some combination of λij (at the higher values of λRj (i.e., λR5 and λR6) and lower values of λGj (i.e., λG1 and λG2)), the Tc values exhibit a small drop in values instead of the increase observed elsewhere. This can be seen inspecting the circled part in Figure 5.The collated results show that a change inλij could cause the correlated colour temperature, Tc, of the optimized mixture to be as high as 3976 K when λRj=λR6, λAj=λA1, λGj=λG1, and λBj=λB1 and as low as 2311 K at minimum peak wavelengths for the red and blue LEDs and maximum peak wavelength values for the green and amber LEDs (λR1,  λA3,  λG7,  λB1) or a difference of 1665 K (181 MK−1). The average value of Tc was 3039 K with a standard deviation of 359 K. Using the specified set of LEDs, Tc changes −21% to +36% of its value in the optimized mixture.


## 4.2. Summary of Results for Colour-Rendering Indices

Table6 provides a summary of the effects of peak wavelength shifts λij on the colour characteristics of the optimized tetrachromatic spectrum, expressed in terms of colour-rendering indices Ra, Rb, and Rc. The errors for each colour-rendering index, shown in Table 6, are calculated as ΔR=R(max)-R(min) and ΔR[%]=ΔR/R(avg)×100. It is seen that variations in colour-rendering characteristics are significant. In particular, the Rb index is the most affected with Rbmax=84 (λR2,  λA1,  λG2,  λB1) and Rbmin=31 (λR1,  λA1,   λG7,  λB1). These rather high variations (ΔRb=53) mean that the colour rendering of the saturated colour samples will be poor. The most affected samples were Samples 9 (strong red) and 12 (strong blue); R12 had the lowest Ri 397 times (in 45 % of combinations) and R9 301 times (in 34% of combinations). Four more samples had the lowest Ri: sample 11 (strong green) 142 times (in 16% of combinations), sample 10 (strong yellow) 38 times (4%), sample 3 (strong yellow green) three times, and sample 4 (moderate yellowish green) once. Significant changes were observed in Rimin=min{Ri}, with Rimin being always below its optimized value Rimin(opt)=73.Table 6
Summary of the influences ofλij on the colour-rendering characteristics (Ra, Rb, Rc) of the optimized tetrachromatic spectrum (Ra(opt)=88, Rb(opt)=79, Rc(opt)=84,  Rimin(opt)=73).
R(avg)R(max)R(min)ΔRΔR[%]Ra7692573546Rb6484315383Rc7187474057Rimin3073−45118387It is also interesting to note the effect of the peak wavelengths of four LEDs on the colour-rendering characteristics as shown in Figure6. For the sake of clarity, the Rc values (Rc=Ra+Rb) are not plotted. As expected, there are five discontinuities in the plot, caused by a change of λRj from bin to bin together with “reset” of other LEDs’ peak wavelengths to their minimum values (λkj(min)=λk1,k≠R). The largest Rb change is when λRj changes from λR1 (A) to λR2(B),  ΔRb=45.5. The size of ΔRb decreases to ΔRb=31.2 at the λR4 (C) to λR5 (D) change, before increasing to 35.6 at the λR5 (E) to λR6 (F) change.Figure 6
Ra and Rb values calculated for 882 possible combinations of λij values.


## 4.3. Summary of Results for Radiant Luminous Efficacy

The effect ofλij changes on ηrad is evident in Figure 7. The radiant efficacy has a maximum ηrad(max)=438 lm/W (λR1, λG7, λB7, λA1) at A and a minimum ηrad(min)=308 lm/W (λR6, λG1, λB1, λA3) at B (which is +12% and −21% from the value in the optimized mixture ηrad(opt)). The ηrad>ηrad(opt) in 292 combinations of λij (in 33% of combinations) and was below ηrad(opt) 590 times (in 67% of combinations). Figure 7 also shows the step changes in ηrad that result from λRj changing from bin to bin (e.g., at A in the graph).Figure 7
ηrad values calculated for 882 possible combinations of λij values.The influences of the green, blue, and amber LEDs are further explored in Figure8, where ηrad is plotted for λR1 and λij of the other three LEDs changing from bin to bin. There are seven plots, one per each of seven λGj bins and 21 ηrad values (for different λAj and λBj combinations) per each λGj bin. For the sake of clarity, the ηrad points were marked only on the λG1 plot. It is evident that the wavelength shifts of the green LED to a higher value λGj result in a ηrad increase; however, the increases are smaller at the higher values of λGj. It is also evident that, as λAj changes from its lowest value (at λA1) to its highest value (at λA3), one observes reductions in ηrad. Again for the sake of clarity, only one set of λA1 to λA3 values are labeled on the λG1 plot. Increasing λBj leads to improvements in ηrad. The ηrad values at λB1 wavelength are lower than the ηrad values at λB7. It was interesting to observe that the difference between the ηrad(min) (A) and ηrad(max) (B) values was constant (67 lm/W) across all λGj plots.Figure 8
ηrad=f(λR1,λij). There are seven plots, one per each λGj bin, and 21 ηrad values for all possible combinations of λBj and λAj bins.


## 5. Conclusion

We have investigated the sensitivity of an optimized tetrachromatic LED spectrum to changes in the peak wavelengths,λij, of individual commercially available LEDs. The sensitivity was quantified in terms of changes in correlated colour temperature, colour-rendering indices, and luminous efficacy calculated for 882 combinations of λij. The results emphasize the fact that the influence of peak wavelength shifts in commercial LEDs on the characteristics of the LED-based light sources cannot be ignored.The correlated colour temperature increases with increases in the wavelengths of the red and green components but decreases with increases in the wavelengths of green and amber components. The changes are significant. The correlated colour temperatures deviate −21% and +36% from the original value. The maximumTc of 3976 K was calculated when the red LEDs had the highest peak wavelength value and other LEDs were at their lowest values. The Tc minimum was at minimum peak wavelengths for the red and blue LEDs and maximum peak wavelength values for the green and amber LEDs. Changes in correlated colour temperature caused by the deviation of the blue LED was one order of magnitude smaller than the changes caused by the other three LEDs. The largest Tc deviations are caused by the red LED (26%), followed by the green LED (18%) for the wavelength ranges investigated.Colour-rendering properties were also significantly affected by the variations in peak wavelengths, particularly the colour-rendering indexRb which is determined by saturated test colour samples. The results for Rb show a difference of 53 between its maximum and minimum values, ΔRb=Rb(max)-Rb(min)=53. These results make it clear that the colour rendering of strong colours can become very poor. The saturated test colour samples (red, blue, green, and yellow) had the lowest rendering index in 99.5% of all combinations. The Ra index was less sensitive to the peak wavelength shifts, decreasing from Ra(max)=92 to Ra(min)=57 (ΔRa=35). These changes in colour rendering and correlated colour temperature are significant and can be problematic in numerous applications.This study has also demonstrated that the radiant luminous efficacy,ηrad, of the optimized tetrachromatic LED spectrum changes significantly due to deviations in peak wavelength values. Its maximum value was 438 lm/W and minimum was 308 lm/W. For 67% of combinations ηrad was below its optimized value ηrad(opt)=390 lm/W. The ηrad values decrease with the increase in red and amber peak wavelengths and increase with the blue and green wavelengths. This influence is opposite to the influence caused by the same set of LED variations on the correlated colour temperature.A possible criticism of this study is that we have used the colour-rendering index, CRI, for the rendering of the white-light spectrum. While we are aware of the need for a new metric, CRI has been widely used for a number of years and is still the only officially recognized CIE method for assessing colour-rendering properties of light sources. We intend to adopt the new metric when formally adopted. In the meantime, we believe that it is appropriate to supplementRa with our additional ratings, Rb, Rc, and Rimin.This study is only focused on the tetrachromatic spectrum of the LEDs chosen from the Luxeon range, and therefore these results cannot necessarily be extrapolated to other commercially available LEDs. Nevertheless, these results make it clear that the characteristics of mixed-LED white-light sources change significantly with changes in the peak wavelengths of individual LEDs.



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*Source: 437564-2011-01-09.xml*