An Investigation on CCT and Ra Optimization for Trichromatic White LEDs Using a Dual-Weight-Coefficient-Based Algorithm

Spectral optimization is applied as an effective tool in designing solid-state lighting devices. Optimization speed, however, has been seldomly discussed in previous reports as regards designing an algorithm for white light-emitting diodes (WLEDs). In this study, we propose a method for trichromatic WLEDs to obtain the optimal Ra under target correlated color temperatures (CCTs). Blue-, yellow-, and red-color monochromatic spectra, produced by the GaN LED chip, YAG:Ce3+ phosphors, and CdSe/ZnSe quantum dots, respectively, are adopted to synthesize white light. To improve the effectiveness of our method, the concept of dual weight coefficients is proposed, to maintain a numerical gap between the proposed floating CCT and the target CCT. This gap can effectively guarantee that Ra and CCT ultimately move toward the targeting value simultaneously. Mechanisms of interaction between CCT, Ra, and dual-weight coefficients are investigated and discussed in detail. Particularly, a fitting curve is drawn to reveal the linear relationship between weight coefficients and target CCTs. This finding effectively maintains the accuracy and accelerates the optimization process in comparison with other methods with global searching ability. As an example, we only use 29 iterations to achieve the highest Ra of 96.1 under the target CCT of 4000 K. It is hoped that this study facilitates technology development in illumination-related areas such as residential intelligent lighting and smart planting LED systems.


Introduction
In comparison with RGB LEDs, light-conversion-material-based white light-emitting diodes (WLEDs) have played leading roles in solid-state illumination, due to their high lightconversion efficiencies in specific wavebands, stability under various junction temperatures, low cost, and feasibility in color tunability [1][2][3]. Conventionally, the color performance of WLEDs is evaluated from two aspects: the emitting color from the WLED and the releasing color of objects exposed under the WLED. Theoretically, the former is characterized by correlated color temperature (CCT), and the latter is characterized by color rendering property, in which the concept of general color rendering index (Ra) is conventionally adopted for evaluation by the International Commission on Illumination [4]. CCT expresses a warm or cold feeling when we observe the light beam, while Ra reveals the ability of a light source to express the real color of an object. Ra represents the average value of the color rendering index (CRI) of eight general colors in a WLED system. CCT and Ra are functions of monochromatic spectral power distributions (SPDs) of different colors [5]. Therefore, if we intend to adjust the circadian rhythm of humans and plantings or reduce driver's fatigue in a specific scenario, modulating the SPD of the WLED system is effective and indispensable [6]. In trichromatic WLEDs, the white-light SPD (SPD W (λ)) is conventionally generated by downconverting light-conversion materials with blue LED chips, in which light in short wavebands (such as blue light) can be effectively converted to light in long wavebands (such as red light) under the stokes effect.
In the last few decades, SPD optimization technologies have been widely investigated to facilitate illumination in areas of residential lighting [7], agriculture [8], rehabilitation therapy [9], and visible light communication [10,11]. These technologies can be mainly divided into two catalogs: the first type is to optimize the color rendering property and energy consumption by adjusting peak wavelengths, spectral bandwidth, and intensity by using Gaussian functions; the other is to optimize the color rendering property and energy consumption by adjusting the density of real light in different colors. For the first type, Guo et al. [12] conducted comprehensive numerical simulations of three-hump and four-hump SPDs in WLEDs. The changes in Ra and CCT values with the shifting peak wavelengths of full width at half maximums (FWHMs) were analyzed under different operating temperatures. The relationship between scotopic-photopic ratio and CCT was investigated as well. Wei et al. [13] proposed six-channel-based LEDs to synthesize daylight with high quality by using a genetic algorithm and Gaussian spectral model. For the second type, Zhu et al. [14] conducted a comprehensive study on illumination performances of the perovskite-based LED with four humps. Titkov et al. [15] proposed a semi-hybrid device, which combined monolithic blue-cyan LED with green-red phosphor mixture, exhibiting the highest Ra of 98.6 at CCT of 3400 K. Yuan et al. [16] manufactured a trichromatic WLED, which constitutes of blue-pump carbon dots and phosphor glass, realizing the highest Ra of 92.9 at CCT of 3610 K. Among these studies, a variety of methods are conventionally used, such as the multiple Gaussian function method [17], least-squares method [18], and iterative method of gradient descent [19]. However, these methods focus on improving the accuracy and the feasibility, as well as developing light-conversion material species with superior chromaticity; few of them discuss the improvement strategy of optimization speed for WLEDs.
With the development of the Internet of Things (IoT) and 5G technologies, the intelligent control technology of illumination lamps becomes imperative for saving energy and increasing productivity [20,21]. Therefore, improving the effectiveness of spectral optimization becomes a key issue in intelligent control. In this study, we propose a convenient method to optimize CCT and Ra values simultaneously for trichromatic WLEDs by using dual-weight coefficients. These coefficients can effectively control the variation range of CCT while searching for the optimal Ra value. Key steps to realize the proposed method is analyzed comprehensively. Compared with other conventional methods used for spectral optimization, the proposed method can greatly accelerate the calculation process while maintaining accuracy.

Monochromatic Spectra Preparation and Theory of Algorithms
As shown in Figure 1a, the blue-emissive LED chip (302 × 198 µm 2 , Hualian Co., Ltd., Xiamen, China) was selected as the excitation source, and yellow-emissive cerium-doped yttrium aluminum garment phosphors (YAG:Ce 3+ , Youyan Rare Earth Co., Ltd., Beijing, China) and red-emissive CdSe/ZnSe quantum dots (Poly OptoElectronics Co., Ltd., Jiangmen, China) were selected as light-conversion materials to fabricate white light. YAG:Ce 3+ phosphors can greatly broaden the white-light spectrum in the visible-light regime, while CdSe/ZnSe quantum dots are able to provide pure red emission with high stability and high quantum yields (QYs). Recently, QYs of YAG:Ce 3+ phosphors and CdSe/ZnSe quantum dots can reach up to 90% and~100%, respectively [22,23]. Packaging technology of the WLED has been given in [24]. To facilitate our study, spectra of monochromatic blue, yellow, and red light are referred to as SPD B (λ), SPD Y (λ), and SPD R (λ), respectively.
Here, we assumed that the peak wavelength and the FWHM of these monochromatic spectra are independent of the driven current, so SPDW(λ) can be described as a linear combination of SPDB(λ), SPDY(λ), and SPDR(λ), as described by where , , and are the proportions of the radiant power of blue, yellow, and red light, respectively. Before calculation, target CCT, test CCT, and test Ra values are defined as CCTtar, CCTtest, Ratest, respectively. CCTtest and Ratest represent current CCT and Ra values in the calculation. Figure 2a illustrates the steps for spectral optimization using the proposed method. For comparison, conventional methods I and II used for spectral optimization are illustrated in Figure 2b,c. Among these three methods, method I directly considers all the possibilities of , , and under CCTtar, while method II randomly selects values of , , and until fulfilling the cycle index, which is set as 1000 for methods I and II. Both methods I and II use the bubbling method to obtain the highest Ra within the error range of CCTtar. For the proposed method, the calculation steps are described as follows: (1) First, we initialize the procedure and load original data, such as the spectra of monochromatic light, step lengths for iteration, error ranges of Ra and CCT, and initial values of , , and ; (2) Two key problems for CCT optimization are how to adjust CCTtest and how to optimize Ra in the meantime. According to the relationship between CCT and components of different colors, we first set a floating parameter between the initial CCT and CCTtar, which is named CCTm. The relationship between CCTm and CCTtest can be expressed as CCTtest = δ1+ CCTm, where δ1 is the first weight coefficient in our algorithm. To realize CCTm, we only need to modulate the parameter of ; (3) Before realizing CCTtar, we optimize Ratest by using the bubbling method. Keeping the proportion of and unchanged, we attempt to modulate with a small step to observe the change of Ratest. If the small step helps to increase the value of Ra, we A 500 mm diameter-integrating sphere (Everfine) was utilized to measure SPD B (λ). To obtain SPD Y (λ) and SPD R (λ), we mathematically removed the superposition area of blue light of the original emission spectra produced by phosphors and quantum dots [24]. SPD B (λ), SPD Y (λ), and SPD R (λ) were normalized before optimization. From Figure 1b, we observe that the LED chip and CdSe/ZnSe quantum dots generate narrow blue and red peaks with FWHM of 54 nm and 56 nm, respectively. On the other hand, YAG:Ce 3+ phosphors produce a spectrum with FWHM of 125 nm that covers a wide range of visible light. Here, we assumed that the peak wavelength and the FWHM of these monochromatic spectra are independent of the driven current, so SPD W (λ) can be described as a linear combination of SPD B (λ), SPD Y (λ), and SPD R (λ), as described by where A B , A Y , and A R are the proportions of the radiant power of blue, yellow, and red light, respectively. Before calculation, target CCT, test CCT, and test Ra values are defined as CCT tar , CCT test , Ra test , respectively. CCT test and Ra test represent current CCT and Ra values in the calculation. Figure 2a illustrates the steps for spectral optimization using the proposed method. For comparison, conventional methods I and II used for spectral optimization are illustrated in Figure 2b,c. Among these three methods, method I directly considers all the possibilities of A B , A Y , and A R under CCT tar , while method II randomly selects values of A B , A Y , and A R until fulfilling the cycle index, which is set as 1000 for methods I and II. Both methods I and II use the bubbling method to obtain the highest Ra within the error range of CCT tar . For the proposed method, the calculation steps are described as follows: (1) First, we initialize the procedure and load original data, such as the spectra of monochromatic light, step lengths for iteration, error ranges of Ra and CCT, and initial values of A B , A Y , and A R ; (2) Two key problems for CCT optimization are how to adjust CCT test and how to optimize Ra in the meantime. According to the relationship between CCT and components of different colors, we first set a floating parameter between the initial CCT and CCT tar , which is named CCT m . The relationship between CCT m and CCT test can be expressed as CCT test = δ 1 + CCT m , where δ 1 is the first weight coefficient in our algorithm. To realize CCT m , we only need to modulate the parameter of A B ; (3) Before realizing CCT tar , we optimize Ra test by using the bubbling method. Keeping the proportion of A B and A Y unchanged, we attempt to modulate A R with a small step to observe the change of Ra test . If the small step helps to increase the value of Ra, we conduct a similar iteration until Ra test reaches the highest value; otherwise, we modulate A R in the negative direction. The relationship between CCT m and CCT tar can be expressed as CCT tar = δ 2 + CCT m , where δ 2 is the second weigh coefficient in our algorithm. (4) When the calculation result meets the required conditions, we export optimized A B , A Y , and A R values, the optimized WLED spectra, CCT test , as well as Ra test .
We first impel CCTtest to move toward CCTm and then optimize Ratest and CCTtest simultaneously, to reach the optimum Ra and CCTtar. If we directly search for CCTtar, the variation space of Ratest is very limited, due to the interaction effect of CCT and Ra. The proposal of CCTm can effectively solve this problem, rendering CCTtest reach a position near CCTtar before the optimization of Ratest. Weight coefficients of δ1 and δ2, which determine the value of CCTm and the shifting range of CCTtest, are key for the optimization result. If δ2 is too small, Ratest will not reach the highest value due to the limited shifting space; on the other hand, if δ2 is too large, Ratest can reach the highest value soon but at the expense of the error between CCTtest and CCTtar. Another problem is how to guarantee that CCTtest moves toward CCTtar instead of the reverse direction while optimizing Ratest. To solve this problem, we set the original , , and values as 0.1, 0.3, and 0.5, respectively, in which is large enough to guarantee the decreasing trend of while optimizing Ratest.  Figure 3 illustrates the variation of CCTtest, Ratest, and δ1 under different δ2 in a 3D coordinate diagram, when CCTtar is set as 8000 K. To facilitate discussion, these results are separately presented in Figure 3a,b at different view angles. In Figure 3a, CCTtest is decreasing with the increase in δ1 when δ2 equals 3000 K. The reason is that δ1 directly influences the difference between CCTtest and CCTm. If δ1 is too large, CCTtest becomes much Below is the design philosophy of the proposed algorithm. Particularly, we propose a floating CCT value named CCT m and two weight coefficients, named δ 1 and δ 2 , to control the variation range of the CCT test . There exist two main stages in the optimization process: We first impel CCT test to move toward CCT m and then optimize Ra test and CCT test simultaneously, to reach the optimum Ra and CCT tar . If we directly search for CCT tar , the variation space of Ra test is very limited, due to the interaction effect of CCT and Ra. The proposal of CCT m can effectively solve this problem, rendering CCT test reach a position near CCT tar before the optimization of Ra test .

Relationship between CCT, Ra, and Other Parameters
Weight coefficients of δ 1 and δ 2 , which determine the value of CCT m and the shifting range of CCT test , are key for the optimization result. If δ 2 is too small, Ra test will not reach the highest value due to the limited shifting space; on the other hand, if δ 2 is too large, Ra test can reach the highest value soon but at the expense of the error between CCT test and CCT tar . Another problem is how to guarantee that CCT test moves toward CCT tar instead of the reverse direction while optimizing Ra test . To solve this problem, we set the original A B , A Y , and A R values as 0.1, 0.3, and 0.5, respectively, in which A R is large enough to guarantee the decreasing trend of A R while optimizing Ra test . Figure 3 illustrates the variation of CCT test , Ra test , and δ 1 under different δ 2 in a 3D coordinate diagram, when CCT tar is set as 8000 K. To facilitate discussion, these results are separately presented in Figure 3a,b at different view angles. In Figure 3a, CCT test is decreasing with the increase in δ 1 when δ 2 equals 3000 K. The reason is that δ 1 directly influences the difference between CCT test and CCT m . If δ 1 is too large, CCT test becomes much smaller than CCT m , increasing the difficulty of reaching CCT tar while optimizing Ra test . According to the methodology of the algorithm, optimization will finally stop when we obtain the optimal Ra. By then, the final CCT test obtained may fail to reach CCT tar . curves almost lie in a similar plane with CCTtar, equal to 8000 K. However, for δ2 = 2400 K and δ2 = 3000 K, corresponding curves stretch out of this plane. In other words, their CCTtest become much smaller than CCTtar of 8000 K. Below are the explanation for this phenomenon. For δ2 = 2400 K and δ2 = 3000 K, we reserve a large variation range of CCT to support the optimization for CCTm and Ratest, causing the inaccessibility of CCTtar when the optimization process of Ra is ending. This explains the phenomenon that those points on the curve are almost far away from 8000 K when δ2 = 3000 K. When δ2 decreases from 3000 K to 2400 K, a portion of points on the curve return to the plane of 8000 K. To summarize, those points staying near the plane of 8000 K are constrained by CCTtar; those points that stretch out from the plane of 8000 K are constrained by optimization conditions of Ra. As shown in Figure 4a, Ratest increases with an increase in δ1 and δ2. If δ1 remains unchanged and δ2 decreases, it would cost more iteration steps to increase in order to improve CCTtest. Hence, even CCTtest reaches CCTm; however, the value of already becomes very large, which limits the highest Ra the WLED can realize. As lower helps the prompt enhancement of Ratest, overlarge hinders the improvement of Ratest, despite the adjustment of . On the other hand, if we keep δ2 unchanged and decrease δ1, CCTtest will reach CCTm sooner; however, Ratest cannot be fully optimized. Therefore, the value of optimal is influenced by CCTm and is finally determined by δ1 and δ2. Increasing δ1 and δ2 under high CCTtar can greatly decrease CCTm, enlarging the optimization range of Ra. In Figure 4b, CCTtest slightly decreases when we increase δ1 or δ2, indicating that CCTtest has a reverse shifting trend, compared with Ratest under different δ1 and δ2 values.

Relationship between CCT, Ra, and Other Parameters
These analyses reveal the significance of δ1 and δ2 for the optimization result. The WLED with different CCTtar values has different reactions under similar δ1 and δ2. Thus, Secondly, Ra test increases with the increase in δ 2 . Since δ 2 represents the difference between CCT m and CCT tar , when we increase δ 2 , CCT m becomes smaller. Thus, larger δ 2 provides a wider range for Ra optimization, extending the shifting area of Ra test within the permitted range of CCT test .
From Figure 3b, we observe the variation of CCT test , Ra test , and δ 1 under different δ 2 at the other view angle. When δ 2 is set as 600 K, 1200 K, and 1800 K, respectively, the curves almost lie in a similar plane with CCT tar , equal to 8000 K. However, for δ 2 = 2400 K and δ 2 = 3000 K, corresponding curves stretch out of this plane. In other words, their CCT test become much smaller than CCT tar of 8000 K. Below are the explanation for this phenomenon. For δ 2 = 2400 K and δ 2 = 3000 K, we reserve a large variation range of CCT to support the optimization for CCT m and Ra test , causing the inaccessibility of CCT tar when the optimization process of Ra is ending. This explains the phenomenon that those points on the curve are almost far away from 8000 K when δ 2 = 3000 K. When δ 2 decreases from 3000 K to 2400 K, a portion of points on the curve return to the plane of 8000 K. To summarize, those points staying near the plane of 8000 K are constrained by CCT tar ; those points that stretch out from the plane of 8000 K are constrained by optimization conditions of Ra.
As shown in Figure 4a, R atest increases with an increase in δ 1 and δ 2 . If δ 1 remains unchanged and δ 2 decreases, it would cost more iteration steps to increase A B in order to improve CCT test . Hence, even CCT test reaches CCT m ; however, the value of A B already becomes very large, which limits the highest Ra the WLED can realize. As lower A B helps the prompt enhancement of Ra test , overlarge A B hinders the improvement of Ra test , despite the adjustment of A R . On the other hand, if we keep δ 2 unchanged and decrease δ 1 , CCT test will reach CCT m sooner; however, Ra test cannot be fully optimized. Therefore, the value of optimal A B is influenced by CCT m and is finally determined by δ 1 and δ 2 . Increasing δ 1 and δ 2 under high CCT tar can greatly decrease CCT m , enlarging the optimization range of Ra. In Figure 4b, CCT test slightly decreases when we increase δ 1 or δ 2 , indicating that CCT test has a reverse shifting trend, compared with Ra test under different δ 1 and δ 2 values. balancing the relationship between δ1, δ2, and CCTtar is the next step to accelerate the optimization process.  Figure 5a illustrates the optimized spectra of the trichromatic WLED under conditions of CCTtar = 8000 K and δ1 = 200 K. When δ2 increases from 600 K to 3000 K, peaks of blue and red light slightly decrease, while Ratest increases from 66.9 to 89.7. This is because the WLED spectrum is increasingly close to the spectrum of the reference source (black body source) [14]. The shifting trend of Ratest matches well with the analysis results of Figure 3a. Figure 5b describes the shifting trend of CCTtest and Ratest in the iteration process under various CCTtar values. When CCTtar ranges from 4000 K to 8000 K, Ratest declines slowly at first, as shown in step 1; thereafter, Ratest abruptly decreases in a small step, as shown in step 2; finally, Ratest increases severely until reaching the top point before finishing the optimization steps, as shown in step 3. With the increase in CCTtar, the highest value of Ra that can be achieved decreases. A similar phenomenon has been observed in [24,25].
When CCTtar ranges from 4000 K to 8000 K, CCTtest also shows three steps to reach CCTtar; however, the shifting trend of CCTtest during the optimization process is different from that of Ratest. Ratest decreases slowly at first and then increases drastically; on the other hand, CCTtest increases slowly at first and then increases drastically. A comparison of Figure 5a with Figure 5b indicates that the increase in CCTtest in step1 sacrifices the improvement of Ratest in the initial time. In step 3, different from the optimization aim of Ratest, we only need to find a CCT value near CCTtar instead of finding the local optimal value of CCTtest. It is worth noting that only three iterations are used for optimization when CCTtar equals 3000 K, and the value of iterations increases with the increase in CCTtar. This is because the initial values of , , and are very close to optimized values of , , and under low CCTtar. As we mentioned in Figure 3, different optimization results can be obtained under different CCTtar with similar δ1 and δ2 values. To guarantee the achievement of the optimal Ratest in all cases, we should initially manage to acquire optimized values for δ1 and δ2 (δOpt1 and δOp2) under different CCTtar values. It is worth noting that there exists a strong relationship between the sum of δOpt1 and δOpt2 (∑(δOpt1, δOpt2)) and CCTtar. As shown in Figure 6, ∑(δOpt1, δOpt2) is plotted and fitted using a linear function under different CCTtar values, which ranges from 3000 K to 12,000 K. It is evident that ∑(δOpt1, δOpt2) presents a perfect linear increasing trend with the increase in CCTtar. The slope of this curve is calculated by using the linear interpolation method, and the curve can be described as These analyses reveal the significance of δ 1 and δ 2 for the optimization result. The WLED with different CCT tar values has different reactions under similar δ 1 and δ 2 . Thus, balancing the relationship between δ 1 , δ 2 , and CCT tar is the next step to accelerate the optimization process. Figure 5a illustrates the optimized spectra of the trichromatic WLED under conditions of CCT tar = 8000 K and δ 1 = 200 K. When δ 2 increases from 600 K to 3000 K, peaks of blue and red light slightly decrease, while Ra test increases from 66.9 to 89.7. This is because the WLED spectrum is increasingly close to the spectrum of the reference source (black body source) [14]. The shifting trend of Ra test matches well with the analysis results of Figure 3a. where α is calculated to be 0.63 for the proposed WLED. For WLEDs combined with different light-conversion materials, the numerical value of α should be different. Additionally, the measured data slightly deviates from the fitting curve when CCTtar equals 3000 K, which is probably because changes in δ1 and δ2 do not have visible effects on the optimization result when CCTtar is low. Once the law between α and light-conversion materials is identified, it is necessary to select the optimal δ1 and δ2 under different CCTtar values before optimization. According to Figure 4, the optimal Ratest corresponds to the largest δ1 and δ2 within the allowed range of CCTtar. Except for the linear relationship between ∑(δOpt1, δOpt2) and CCTtar, the value of δ2 should be larger than δ1, to guarantee the operation of the calculation procedure. Therefore, we had better select larger δ2 and smaller δ1 to satisfy Equation (2). This principle provides us with an effective way to accelerate the spectral optimization speed.  Figure 5b describes the shifting trend of CCT test and Ra test in the iteration process under various CCT tar values. When CCT tar ranges from 4000 K to 8000 K, Ra test declines slowly at first, as shown in step 1; thereafter, Ra test abruptly decreases in a small step, as shown in step 2; finally, Ra test increases severely until reaching the top point before finishing the optimization steps, as shown in step 3. With the increase in CCT tar , the highest value of Ra that can be achieved decreases. A similar phenomenon has been observed in [24,25].
When CCT tar ranges from 4000 K to 8000 K, CCT test also shows three steps to reach CCT tar ; however, the shifting trend of CCT test during the optimization process is different from that of Ra test . Ra test decreases slowly at first and then increases drastically; on the other hand, CCT test increases slowly at first and then increases drastically. A comparison of Figure 5a with Figure 5b indicates that the increase in CCT test in step1 sacrifices the improvement of Ra test in the initial time. In step 3, different from the optimization aim of Ra test , we only need to find a CCT value near CCT tar instead of finding the local optimal value of CCT test . It is worth noting that only three iterations are used for optimization when CCT tar equals 3000 K, and the value of iterations increases with the increase in CCT tar . This is because the initial values of A B , A Y , and A R are very close to optimized values of A B , A Y , and A R under low CCT tar .
As we mentioned in Figure 3, different optimization results can be obtained under different CCT tar with similar δ 1 and δ 2 values. To guarantee the achievement of the optimal Ra test in all cases, we should initially manage to acquire optimized values for δ 1 and δ 2 (δ Opt1 and δ Op2 ) under different CCT tar values. It is worth noting that there exists a strong relationship between the sum of δ Opt1 and δ Opt2 (∑(δ Opt1 , δ Opt2 )) and CCT tar . As shown in Figure 6, ∑(δ Opt1 , δ Opt2 ) is plotted and fitted using a linear function under different CCT tar values, which ranges from 3000 K to 12,000 K. It is evident that ∑(δ Opt1 , δ Opt2 ) presents a perfect linear increasing trend with the increase in CCT tar . The slope of this curve is calculated by using the linear interpolation method, and the curve can be described as where α is calculated to be 0.63 for the proposed WLED. For WLEDs combined with different light-conversion materials, the numerical value of α should be different. Additionally, the measured data slightly deviates from the fitting curve when CCT tar equals 3000 K, which is probably because changes in δ 1 and δ 2 do not have visible effects on the optimization result when CCT tar is low.
where α is calculated to be 0.63 for the proposed WLED. For WLEDs combined with different light-conversion materials, the numerical value of α should be different. Additionally, the measured data slightly deviates from the fitting curve when CCTtar equals 3000 K, which is probably because changes in δ1 and δ2 do not have visible effects on the optimization result when CCTtar is low. Once the law between α and light-conversion materials is identified, it is necessary to select the optimal δ1 and δ2 under different CCTtar values before optimization. According to Figure 4, the optimal Ratest corresponds to the largest δ1 and δ2 within the allowed range of CCTtar. Except for the linear relationship between ∑(δOpt1, δOpt2) and CCTtar, the value of δ2 should be larger than δ1, to guarantee the operation of the calculation procedure. Therefore, we had better select larger δ2 and smaller δ1 to satisfy Equation (2). This principle provides us with an effective way to accelerate the spectral optimization speed.

Comparison between the Proposed Method, Method I, and Method II
Tables 1-3 present calculation parameters of spectral optimization with the proposed method, method I, and method II. In Table 1, optimized CCTtest values are very close to CCTtar. Among all results, the highest Ratest reaches up to 96.1, with CCTtest of 4013 K. , , and exhibit a regular shifting trend in which increases, and reduces with Once the law between α and light-conversion materials is identified, it is necessary to select the optimal δ 1 and δ 2 under different CCT tar values before optimization. According to Figure 4, the optimal Ra test corresponds to the largest δ 1 and δ 2 within the allowed range of CCT tar . Except for the linear relationship between ∑(δ Opt1 , δ Opt2 ) and CCT tar , the value of δ 2 should be larger than δ 1 , to guarantee the operation of the calculation procedure. Therefore, we had better select larger δ 2 and smaller δ 1 to satisfy Equation (2). This principle provides us with an effective way to accelerate the spectral optimization speed.

Comparison between the Proposed Method, Method I, and Method II
Tables 1-3 present calculation parameters of spectral optimization with the proposed method, method I, and method II. In Table 1, optimized CCT test values are very close to CCT tar . Among all results, the highest Ra test reaches up to 96.1, with CCT test of 4013 K. A B , A Y , and A R exhibit a regular shifting trend in which A B increases, and A R reduces with the increase in CCT tar . The sum of δ 1 and δ 2 increases with the increasing CCT tar , which is consistent with the discussion and results in Figure 6. Optimization results of the proposed method and method I, in terms of CCT test , Ra test , A B , A Y , and A R values, are highly coincident with each other. This coincidence verifies the correctness of the proposed method. In Table 3, calculation results of method II under low CCT tar well match those of method I. Compared with the proposed method and method I, we can even obtain better optimization results of Ra test under 3000 K by using method II. However, with the increase in CCT tar , method II fails to effectively improve Ra test to obtain the optimal value. Due to the random selection rule of method II, calculation results of A B , A Y , and A R listed in Table 3 do not show a similar trend as in Tables 1 and 2. These results reveal that method II cannot effectively optimize WLED spectra under high CCTs. To apply A B , A Y , and A R in a real scenario for realizing target illumination effects, ref. [24] presented the implementation method in detail.
By using Equation (2) to find δ Opt1 and δ Opt2 under different values of CCT tar , we accelerate the optimization process. In Figure 7a, the number of iterations of these three methods under different values of CCT tar is compared. Obviously, the number of iterations of the proposed method is much less than that of the other two methods. For the proposed method, the number of iterations increases with the increase in CCT tar . The accuracy of CCT test and Ra test for these three methods can be evaluated by using the error range concept. Error ranges of CCT test and Ra test (ε C and ε R ) are calculated by |CCT tar − CCT test |/CCT tar and |100 − Ra test |/100, respectively. ε C and ε R under different values of CCT tar are given in Figure 7b,c for comparison. The ε C values of these three methods are comparable under different values of CCT tar . The ε R values of the proposed method and method I are similar, and they are relatively smaller than that of method II under high CCT tar values. These results verify the effectiveness and accuracy of the proposed method.

Conclusions
In this study, we propose an effective method to optimize the Ra of trichromatic WLEDs under different CCTs. Compared with conventional methods I and II, the proposed method exhibits superior searching ability to find the optimal Ra under target CCTs. Specifically, the highest Ra of 96.1 under 4013 K can be obtained after only 29 iterations. Three main mechanisms were investigated and analyzed for the proposed method: (1) the influence of δ1 and δ2 on the calculation results of CCTm, CCTtest, and Ratest; (2) the relationship between δ1, δ2, CCTm, , , and ; (3) the shifting rule of δOpt1 and δOpt2 under different CCTtar values. Particularly, the fitting linear curve that describes the relationship between ∑(δOpt1, δOpt2) and CCTtar can provide an effective way to greatly accelerate the optimization process under different CCTtar values. This study reveals the shifting mechanism of CCT and Ra values with dual-weight coefficients and greatly enhances the effectiveness of spectral optimization for WLEDs. Our method is hopefully applied in related areas such as residential intelligent lighting and smart planting LED systems.

Conclusions
In this study, we propose an effective method to optimize the Ra of trichromatic WLEDs under different CCTs. Compared with conventional methods I and II, the proposed method exhibits superior searching ability to find the optimal Ra under target CCTs. Specifically, the highest Ra of 96.1 under 4013 K can be obtained after only 29 iterations. Three main mechanisms were investigated and analyzed for the proposed method: (1) the influence of δ 1 and δ 2 on the calculation results of CCT m , CCT test , and Ra test ; (2) the relationship between δ 1 , δ 2 , CCT m , A B , A Y , and A R ; (3) the shifting rule of δ Opt1 and δ Opt2 under different CCT tar values. Particularly, the fitting linear curve that describes the relationship between ∑(δ Opt1 , δ Opt2 ) and CCT tar can provide an effective way to greatly accelerate the optimization process under different CCT tar values. This study reveals the shifting mechanism of CCT and Ra values with dual-weight coefficients and greatly enhances the effectiveness of spectral optimization for WLEDs. Our method is hopefully applied in related areas such as residential intelligent lighting and smart planting LED systems.