Measurement model of circadian action factor of phosphor-converted white LEDs based on photometric, electrical, and thermal properties

Light stimulus is confirmed to have critical non-visual biological effects on human beings, and thus evaluation of lighting quality should not be limited to traditional visual performance. The circadian action factor (CAF) has been put forward to quantify such non-visual effects of light sources. This paper focuses on studying the photometric, electrical, and thermal properties of widely used phosphor-converted white LEDs (PC-WLEDs), and establishes an indirect measurement model of CAF regarding temperature, current and excitation wavelength. During modeling, the spectral power distribution of PC-WLEDs is described as an extended Gaussian function of the double-color (blue-yellow) spectrum. In verifications, the remarkable consistency is obtained between indirect measurements and direct calculations, while the maximum relative errors do not exceed 2.6% and the mean relative errors do not exceed 0.9%. The proposed measurement model involves a series of optical tests and calibrations, which are simple for LED system designers to comply with. It offers an effective tool to quantify the PC-WLEDs’ non-visual biological effects on human beings, avoiding costly optical instruments and laborious calculations. Furthermore, it provides a potential theoretical foundation for realizing human-centric assessment, regulation and control of artificial lighting.


Introduction
Light-emitting diodes (LEDs) have successfully competed with conventional light sources and shown great potential in artificial lighting, due to their advantages of high efficiency, tunable chromaticity and environmental safety [1,2].In general lighting, phosphor-converted white LEDs (PC-WLEDs) realized by GaN chips and rare-earth phosphors become the commercial mainstream [3].Existing evaluations and optimizations of PC-WLEDs primarily focus on visual performance, such as high luminous efficiency and high color quality [4,5].As people are exposed to artificial lighting environments for longer time, biological and medical research projects carried out reported that light stimulus has critical non-visual biological effects on human beings, including work efficiency, circadian rhythm, cognitive level, etc [6][7][8].Artificial lighting, reported in [9], can alter humans' day-night cycle, interfere with melatonin secretion, and disrupt their circadian rhythm.As discussed in [10], warm-white light with a lower CCT is preferred in public areas to promote relaxation, while coolwhite light with a higher CCT is perfect for offices to enhance concentration.Besides, improper light stimulus is proved to lead to circadian disruption and increase the risk of diseases [11].
In mammals, including human beings, a special type of cells called intrinsic photosensitive retinal ganglion cells (ipRGCs) has been found to play critical roles in formation and release of melatonin, cortisol and other hormones, finally affecting their health and behaviors [12].It is observed that ipRGCs are sensitive to blue-rich light and insensitive to redrich light [13], indicating that light with inconsistent spectral compositions may have different biological effects on human beings.As people become more health conscious, they prefer natural lighting, especially indoors [14].Based on this, humancentric assessment, regulation and control of general lighting are currently attracting much attention [15].On this occasion, the question of how to estimate the non-visual biological effects of light sources on human beings becomes a crucial issue and should be addressed by sessions of the LED industry.
Based on the experimental findings of Brainhard et al [16] and Thapan et al [17], the circadian spectral sensitivity function C(λ) was proposed by Gall in 2002 [18].Subsequently, combining it with the spectral luminous efficiency function V(λ) [19], the circadian action factor (CAF) was proposed by Gall and Bieske as a quantitative indicator to assess light sources' non-visual biological effects [20].To our knowledge, no certified tools are available to directly measure the CAF of light sources.The acquisitions of CAF in existing studies are nearly calculated from its definition, which is a tedious process requiring costly optical instruments and laborious spectrumrelated integral operations [21,22].
This paper aims to develop a theoretical model to indirectly measure CAF for the widely used PC-WLEDs and to provide a convenient means to evaluate their non-visual biological effects on human beings.In modeling, interrelated properties of both the GaN chip and phosphor coating layer are taken into account, including thermal effects, electrical conditions, and phosphor excitation characteristics.Verifications are carried out with PC-WLEDs showing good agreement between indirect measurements and direct calculations, with errors within the acceptable range.This proposed model is established based on photo-electro-thermal properties, it allows CAF to be obtained with operating temperature, input current, and phosphor excitation wavelength, avoiding costly instruments and laborious calculations.Furthermore, it can be a safe theoretical basis for realizing human-centric assessment, regulation and control of artificial lighting.

Modeling with electro-thermal properties
The circadian action factor, i.e.CAF, has been put forward in the form of the spectrum-related calculation, as below [20] CAF = where P(λ) is the spectral power distribution (SPD) of the light source, C(λ) is the circadian spectral sensitivity function [18], with a peak wavelength of 450 nm [23], and V(λ) is the spectral luminous efficiency function adopted in 1924 by the International Commission on Illumination (CIE) [19], with a peak wavelength of 555 nm [24].The curves of C(λ) and V(λ) can be seen from figure 1.
Luminous flux (Φ v ) is a significant optical parameter defined considering the varying sensitivity of the human eyes to different spectral wavelengths, as below where K m is the maximum spectral luminous efficacy expressed in lumen per watt, and used as a constant of 683 lm•W −1 in spectral calculations [6].As concluded from the comparison between equations (1) and ( 2), the denominator of the CAF definition is proportional to the luminous flux Φ v .The relationship between them is determined solely by the constant coefficient K m .As P(λ) of PC-WLEDs has been confirmed to vary with operating temperature and current [25], it is reasonable to assume that specific connections exist between luminous flux Φ v and temperature, as well as between luminous flux Φ v and current.If this assumption is valid, the expression ´780nm 380nm P(λ)V(λ)dλ might also relate to temperature and current by similar connections.
The SPD of a monochromatic LED can be typically modeled as a Gaussian function [26], expressed as below where P opt is the optical power, λ peak is the peak wavelength, and σ is the rated coefficient of full-width at half-maximum (FWHM), depending on the peak wavelength and FWHM.Theoretically, the white light emitted from PC-WLEDs can be considered as a mixture of blue and yellow spectra.Hence, equation (3) can be extended to a double-color (blue-yellow) spectral model, as where P opt,b(w) , λ peak,b and σ b are respectively the optical power, peak wavelength, and rated coefficient of FWHM, of the blue spectrum; while P opt,y(w) , λ peak,y and σ y are those corresponding to the yellow spectrum.
To obtain the dependencies of P opt,b(w) and P opt,y(w) on temperature and current, a PC-WLED is tested under different temperatures at a fixed current of 120 mA and under different currents at a fixed temperature of 50 • C. The SPD and optical power, respectively, of the blue spectrum and yellow spectrum are recorded and shown in figure 2. It is observed that, the optical power of the blue spectrum is, to a large extent, linearly dependent on temperature and current, as is that of the yellow spectrum.In modeling, therefore, such interrelations are approximately described as where k 1 , k 2 , a 1 , a 2 , b 1 and b 2 are constant coefficients, of which k 1 and k 2 are related to gradients, respectively, of P opt,b(w) and P opt,y(w) changing with temperature and current.For PC-WLEDs, around the peak wavelength of the blue spectrum, the circadian spectral sensitivity function C(λ) has small gradients while the spectral luminous efficiency function V(λ) has small values; inversely, around the peak wavelength of the yellow spectrum, the C(λ) has small gradients while the V(λ) has small values.Such slight variations in peak wavelengths due to temperature or current would be considered to have little effect on the integral operations of CAF [18,19].Furthermore, the FWHMs of both blue and yellow spectra are also confirmed to present minor responses to temperature and current, and the resulting errors will be roughly negligible.Therefore, to simplify, it can be reasonably assumed that the peak wavelengths and FWHMs of the blue and yellow spectra from PC-WLEDs stay constant.As a matter of fact, similar simplifications have been generally adopted in spectral analysis and LED-related studies [27].Based on the above analysis, set the definitions as below By substituting equations ( 4)-( 6) into equation ( 1), CAF is transformed into an expression of temperature and current, as follows, From the previous analysis, C 11 , C 12 , V 11 and V 12 defined in equation ( 6) are concluded to be constants.Therefore, apparently, the numerator of equation ( 7) will linearly change with current under a fixed temperature and will linearly change with temperature under a fixed current.The denominator is found to have a similar change feature.According to that, equation ( 7) can be further simplified as where K 1 , A 1 , A 2 , B 1 and B 2 are constant coefficients that can be deduced from practical optical measurements under the fixed operating temperature T 0 and driving current I 0 .Equation ( 8) connects the calculation of CAF to temperature and current, making it a preliminary model of CAF regarding electro-thermal properties.

Modeling with phosphor excitation characteristics
Rare-earth-doped Y 3 Al 5 O 12 :Ce 3+ (i.e.YAG:Ce 3+ ) phosphors are commonly used coating materials for commercial PC-WLEDs [6][7][8][9][10].Excited by the blue spectrum, YAG:Ce 3+ phosphors will convert a portion of the blue spectrum into the yellow spectrum.From this perspective, the blue spectrum and yellow spectrum are respectively called the excitation spectrum and conversion spectrum.It has been proven that all the phosphor temperature, excitation wavelength and packaging technology greatly affect the optical properties of phosphors, as embodied heavily in the conversion spectrum [8].To further explore the excitation characteristics, a sample of YAG:Ce 3+ phosphor material is tested using the exciting spectra and thermal quenching analyzer (EX-1000), under different temperatures at a fixed excitation wavelength of 465 nm, and under different excitation wavelengths at a fixed temperature of 50 • C, as shown in figure 3.
Figures 3(a) and (b) correspond to the conversion efficiency of the YAG:Ce 3+ material varying with phosphor temperature and excitation wavelength, respectively.Figures 3(c) and (d) correspond to the ratio between the optical power of the conversion spectrum and the optical power of the excitation spectrum (i.e.P opt,p /P opt,e ) varying with phosphor temperature and excitation wavelength, respectively.
As figure 3(a) indicates, the conversion efficiency of YAG:Ce 3+ phosphors is gradually reducing with increasing temperature.The nature behind this phenomenon is that, as temperature increases, the lattice vibration within YAG:Ce 3+ phosphors is intensified and the lattice relaxation of luminescence center Ce 3+ is enhanced, leading to an increasing probability of non-radiative transition and a decreasing probability of radiative transition [28].In addition, the non-radiative transition is generally accompanied by the heat release, which will further weaken the radiative transition but instead aggravate the non-radiative transition, resulting in a more serious reduction in conversion efficiency [29].
It can be observed from figure 3(b) that there is a nonmonotonic relationship between the conversion efficiency of YAG:Ce 3+ phosphors and the excitation wavelength.In concrete terms, within the wavelength range of 435 nm-465 nm, the conversion efficiency is enhanced with the increasing excitation wavelength; within the wavelength range of 465 nm-495 nm, the conversion efficiency, on the contrary, is decreasing as the excitation wavelength continues to increase; as a consequence, the maximum value of conversion efficiency is appearing exactly at the excitation wavelength of 465 nm.This phenomenon is due to the fact that the absorption peak of YAG:Ce 3+ phosphors is located within 450 nm-470 nm (around 465 nm), thus the absorption efficiency of the blue spectrum is first improving and then reducing with increasing excitation wavelength, and thereby the highest value occurs at the excitation wavelength of about 465 nm [30].
As seen from figures 3(a) and (b), the actual phosphorus response seems to be skewed.This is because the conversion spectrum of YAG:Ce 3+ phosphors excited by the blue spectrum actually contains a large proportion of the yellow spectrum and a small fraction of the green spectrum [26].Due to the tiny composition of the green spectrum and the low resulting influence, it has not been considered in the modeling.That is why the double-color (blue-yellow) spectral model is adopted to describe the SPDs of PC-WLEDs in this study, that is, equation (4).
The curve in figure 3(c) is nearly straight, indicating that the ratio P opt,p /P opt,e is approximately linearly dependent on phosphor temperature.In contrast, the curve in figure 3(d) presents a non-linear relationship between the ratio P opt,p /P opt,e and the excitation wavelength, and is quite similar to a parabolic function.To mathematically model, at a fixed phosphor temperature T 0 , the relationship between the ratio P opt,p /P opt,e and excitation wavelength is approximately described as where f 1 , f 2 and f 3 are constant coefficients that can be extracted from phosphor excitation testing beforehand, under the fixed temperature T 0 .
For encapsulated PC-WLEDs, the YAG:Ce 3+ phosphors in the coating layer are excited by the blue spectrum emitted from the GaN chip and then convert a portion of the blue spectrum into the yellow spectrum.Within the mixed white light emitted from PC-WLEDs, define P opt,y(w) and P opt,b(w) , respectively, as the optical power of the yellow spectrum and the optical power of the blue spectrum, and the ratio between them is P opt,y(w) /P opt,b(w) .A particular relationship has been confirmed to exist between the ratio P opt,y(w) /P opt,b(w) of PC-WLEDs and the ratio P opt,p /P opt,e of YAG:Ce 3+ phosphors [31], namely where γ is the conversion coefficient dependent on the optical properties of the YAG:Ce 3+ coating layer, and is constant for encapsulated PC-WLEDs.Deduced from equation ( 10), the optical power of the yellow spectrum of PC-WLEDs can be obtained as By substituting equations ( 6) and (11) In combination with equation ( 9), the equation ( 12) of CAF is further transformed into an expression regarding excitation wavelength, that is, where K 2 , F 1 , F 2 , F 3 and F 4 are constant coefficients, which are calculated by merging coefficients C 11 , C 12 , V 11 , V 12 , f 1 , f 2 , f 3 and γ, under the stable operating condition of (I 0 , T 0 ).Based on the above analysis and simultaneously taking into account the photometric, electrical and thermal properties, the final indirect measurement model of CAF can be established by linking equations ( 8) and ( 13), as where K is a constant coefficient obtained from merging coefficients K 1 and K 2 as well as the CAF value at the operating point (I 0 ,T 0 ,λ e0 ).
As the measurement model CAF(T,I,λ e ) of equation ( 14) is established considering the interrelated properties of both the GaN chip and the YAG:Ce 3+ phosphor coating layer, it is suitable for PC-WLEDs packaged via the same production process.
The procedure of the modeling and indirect measuring of CAF can be briefly summarized in figure 4.

Verifications and discussions
The experimental setups are exhibited in figure 5, including the exciting spectral and thermal quenching analyzer (EX-1000) for YAG:Ce 3+ phosphor materials, and the Everfine HAAS-2000 spectroradiometer system with integrating sphere photometer for encapsulated PC-WLEDs.All optical measurements are performed under steady-state thermal and electrical conditions.

Electro-thermal properties of luminous flux
As discussed earlier, luminous flux Φ v is proportional to the denominator of the definition of CAF, and the only difference is the constant coefficient K m .The preliminary model CAF(T,I) in equation (8) shows that, both its numerator and denominator are linearly dependent on temperature and current.Therefore, if the luminous flux Φ v is proved to linearly vary with temperature and current, the preliminary model CAF(T,I) is verified.Briefly, for PC-WLEDs, the electrothermal properties of CAF can be indirectly verified by the electro-thermal properties of luminous flux Φ v .
To obtain the dependencies of luminous flux Φ v , respectively, on temperature and current, a PC-WLED is tested under changing temperatures and currents.The temperature is controlled from 20 • C to 80 • C with an interval of 10 • C, while the current is adjusted from 40 mA to 200 mA with an interval of 40 mA.The luminous flux Φ v versus temperature and current is displayed in figure 6, where almost straight lines indicate the nearly linear dependencies of luminous flux Φ v on temperature and current.Due to the outstanding linearity, formulate the corresponding relationships as below where c 1 , c 2 , d 1 and d 2 are constant coefficients, of which c 1 and d 1 are slopes of luminous flux Φ v , respectively, versus temperature and current.
According to equation ( 15), a two-dimensional function of luminous flux Φ v regarding temperature and current can be constructed as where e = Φ v (T 0 ,I 0 ) is the value of luminous flux at the operating point (T 0 ,I 0 ); h 1 , c 3 and d 3 are constant coefficients determined from the merging of constant coefficients c 1 , c 2 , d 1 , d 2 and e.
Referring to equation ( 16), values of the luminous flux Φ v of the tested PC-WLED are estimated and compared with actual values obtained via Everfine HAAS-2000, as displayed in figure 7. It can be observed that, the estimated values are highly consistent with their corresponding actual ones.Furthermore, according to the analysis, the maximum relative error is 3.81% and the mean relative error is 1.84%, both within an acceptable range of 5%.The combination of figure 7 and error analysis well confirms the accuracy of equation ( 16) and the validity of its establishment process.Thus, the preliminary model CAF(T,I) related to electro-thermal properties is verified.

Photo-electro-thermal properties of CAF
As the model CAF(T,I) linking thermal effects and electrical conditions is verified, verifications of the final measurement model CAF(T,I,λ e ) coupling with phosphor excitation characteristics are further carried out.In this part, several PC-WLEDs have been adopted, whose CCTs are different and within the range of 2500 K-8400 K.As is expected, all results demonstrate the high consistency between indirect measurements and their corresponding direct calculations, where maximum relative errors are not exceeding 2.6% and mean relative errors are not exceeding 0.9%.
Here, the results of two typical PC-WLEDs are displayed and discussed, representing a low CCT (2500 K-3000 K) and a high CCT (5800 K-6400 K), i.e. the so-called warm-PC-WLED and cool-PC-WLED, respectively.The geometrical dimensions and important maximum allowable parameters of these two are given in table 1, as are the parameters involved in the measurement model.It is noted that, for a PC-WLED, the peak wavelength of the GaN chip is the peak wavelength of the blue spectrum, and exactly the excitation wavelength of YAG:Ce 3+ phosphors, so λ e is equal to λ peak,b .
For the warm-PC-WLED, indirect measurement values of CAF and their corresponding direct calculation values are displayed in figure 8, along with relative errors.Those of the cool-PC-WLED are displayed in figure 9.In testing of the   Additionally, the absolute and relative errors of the results are also calculated as quantitative supplements, and the corresponding maximum and mean values are displayed in table 2. Here, the absolute error E A,i and relative error E R,i are defined as below where the subscript 'i' corresponds to the ith operating situation, CAF IM,i and CAF DC,i are the corresponding CAF values obtained from indirect measurement and direct calculation, respectively.Experimental verifications confirm that, the proposed measurement model is applicable to PC-WLEDs realized by GaN chips with YAG:Ce 3+ phosphors, in spite of possible differences in packaging structure and spectral composition.It is noted that, errors between measured and calculated ones under higher currents are slightly larger than those under lower currents.The primary causes contributing to such a phenomenon could be attributed to two aspects: (1) In general applications, the junction temperature of the GaN chip and the phosphor temperature of the coating layer may be slightly higher than the heatsink temperature.
Although the temperature-controlled heatsink is adopted in our testing, temperature differences between the controlled heatsink and GaN chip, as well as the phosphor coating layer, can only be reduced but not be eliminated completely [28].Furthermore, with the same thermal resistance and wall plug efficiency, these temperature differences become more pronounced with increasing current and fixed operating voltage.Briefly, when the PC-WLED driven by a higher current, the value of temperature T input into the model CAF(T,I,λ e ) will actually deviate from its actual value, leading to an increasing error between the measured CAF and its calculated one.(2) The higher temperature of the GaN chip and phosphors due to the higher current, will further result in variations in spectral structure.For the GaN chip, rising temperature can cause not only a decrease in optical power of the initial blue spectrum, but also a red shift of the peak wavelength that finally leads to a lower match with the excitation spectrum of YAG:Ce 3+ phosphors [32].For YAG:Ce 3+ phosphors in the coating layer, the energy difference between the ground state 4 f and excited state 5d generally reduces with rising temperature, which will contribute to degradation in conversion efficiency [33].In short, when the PC-WLED driven by a higher current, the thermal effects of the GaN chip and phosphors will have impacts on the components and structures of the mixed white light, and thus errors caused by idealization in the measurement model of CAF will also slightly increase.

Conclusion
For the PC-WLEDs, an indirect measurement model of CAF linking photo-electro-thermal properties is proposed.In modeling, the SPD of emitted white light is described as an extended double-color (blue-yellow) Gaussian function, and how thermal effects, electrical conditions, and phosphor excitation characteristics have impacts on SPD is discussed through experimental tests and theoretical deductions.Verifications are carried out with spectrophotometric analysis of PC-WLED devices, and results indicate high consistency exists between indirect measurements and direct calculations.It allows the CAF to be obtained with the operating temperature, input current, and phosphor excitation wavelength, avoiding costly optical instruments and laborious calculations.The potential contribution of this paper is to provide a theoretical basis for realizing human-centric assessment, regulation and control of artificial lighting.It is envisaged that this is the first theoretical tool to measure CAF of PC-WLEDs combining interrelated properties of both the GaN chip and phosphor coating layer.The proposed theoretical model involves a series of optical tests and calibrations beforehand, which are simple for LED system designers to comply with.It is suggested that LED manufacturers consider including the spectral modeling parameters in the datasheets, so that this theoretical tool can be used by researchers and engineers to develop new techniques for designing or controlling CAF in future LED systems.
The CAF is a quantitative indicator to describe the nonvisual biological effects of light sources, the larger the value of the CAF, the greater the non-visual effects on human beings.However, the value of CAF does not exactly correspond to a specific value of biological information.In a sense, the quantitative characteristic of CAF is relative, and it mainly contributes to the following aspects: (1) to evaluate the changing non-visual biological effects of a light source changing with operating conditions; (2) to make a contrast between different light sources in the degree of non-visual biological effects; and (3) to realize spectral structure optimization of lighting systems regarding non-visual biological effects (that is, to make an adjustment or improvement of spectral structure according to the change of CAF).
Finally, two explanations about our proposed indirect measurement model of CAF should be made here: (1) The proposed model is established based on the photometric, electrical, and thermal properties of PC-WLEDs, it therefore may work well with this kind of LED devices but not apply quite well to other LED sources.Though the PC-WLEDs have a weakness of relative poor color rendering due to the lack of red spectrum, they and their integrated LED systems at present still play important roles in general lighting in daily applications.(2) As a quantification process, either in the direct calculation or indirect measurement of CAF, there are quite a few potential influential factors of uncertainty, which may be related to instruments and devices, manual operations, data display and reading, experimental environments, etc.Since our manuscript is focusing on theoretical modeling for the indirect measurement of CAF rather than statistical analysis, and meanwhile there are almost no means to realize direct measurement of CAF for references, the issues involving measurement uncertainty analysis are not discussed here.
Despite a bit of regret, it does not prevent this paper from providing readers with valuable insights and intriguing findings, and from providing LED design engineers and manufacturers with reference or inspiration of value.Certainly, LED systems with tunable ability and adequate color rendering index as well as the measurement uncertainty analysis of CAF should be essential topics of our future research regarding spectral optimization and non-visual biological effects of artificial light sources.

Figure 2 .
Figure 2. The SPD and optical power of the tested PC-WLED: (a) SPD varies with temperature, at a fixed current of 120 mA; (b) SPD varies with current, at a fixed temperature of 50 • C; (c) optical power of blue and yellow spectra versus temperature, at a fixed current of 120 mA; (d) optical power of blue and yellow spectra versus current, at a fixed temperature of 50 • C.

Figure 3 .
Figure 3. Conversion efficiency and ratio Popt,p/Popt,e of YAG:Ce 3+ phosphors: (a) conversion efficiency versus temperature, at a fixed excitation wavelength of 465 nm; (b) conversion efficiency versus excitation wavelength, at a fixed temperature of 50 • C; (c) Popt,p/Popt,e versus temperature, at a fixed excitation wavelength of 465 nm; (d) Popt,p/Popt,e versus excitation wavelength, at a fixed temperature of 50 • C.

Figure 4 .
Figure 4.The procedure of the modeling and measuring of CAF.

Table 1 .
Information of the tested PC-WLEDs and parameters related to the measurement model of CAF.Warm-PC-WLED Cool-PC-WLED MCPCB size 20 mm * 20 mm * 2 mm 20 mm * 20 mm * -WLED, the current is adjusted from 60 mA to 200 mA with an interval of 20 mA, and the heatsink temperature is controlled from 25 • C to 85 • C with an interval of 5 • C. In testing of the cool-PC-WLED, the current is adjusted from 100 mA to 500 mA with an interval of 100 mA, while the heatsink temperature is controlled from 25 • C to 95 • C with an interval of 10 • C. The current is provided by a Rigol programmable power supply, and the heatsink temperature is controlled via the Peltier-cooled fixture CL-200.Again, all optical

Table 2 .
Errors analysis of CAF.