Degradation modeling of mid-power white-light LEDs by using Wiener process

Jianlin Huang; Dušan S Golubović; Sau Koh; Daoguo Yang; Xiupeng Li; Xuejun Fan; G.Q. Zhang

doi:10.1364/OE.23.00A966

1. Introduction

A rapid growth of the LED-based lighting applications has revolutionized the lighting market from backlight, flash, and automotive lamps, to general lighting including indoor and outdoor illuminations. Due to the fast evolution of LED technologies, LED packages with a lifetime exceeding 36,000 hours are widespread in the market [1–3]. As highly reliable products, LED devices may not suffer from mortal deterioration even if accelerated life tests were applied. This implies that catastrophic failures of LED devices seldom occur during the operational life in the field. Therefore, reliability tests based on traditional time-to-failure accelerated life test (ALT) and corresponding statistics method may need to be revisited and improved for LED devices.

Though failures were seldom observed during life tests, previous studies revealed that LED devices could suffer from gradual performance degradation, such as lumen decay, and color shift [4–6]. The gradual degradation mechanisms could be attributed to deterioration of the Ohmic contacts and semiconductor chips [7], silver-coating lead frame damage, encapsulant yellowing [8, 9], silicone carbonization [10, 11], as well as phosphor thermal quenching [12]. Multiple failure modes and failure mechanisms pose challenges in describing the LED degradation by using a physical model. As a consequence, the way in which a full set the degradation data is used becomes important for accurate lifetime prediction of LED devices.

In fact, using degradation data for lifetime prediction was studied by statisticians a few decades ago. As one of the pioneers, Nelson discussed a special situation in which the degradation measurement was destructive [13]. After that, Lu and Meeker [14] developed a statistical method which was so-called “general degradation path model”, in which degradation measures were used to estimate the time-to-failure distribution for a broad range of degradation models. As one special case of the general degradation path model, the IES standard TM-21-11 [15] proposed a detailed methodology, which makes use lumen degradation data to predict the lifetime of LED devices. According to the standard, an LED light source would be considered as failure as long as the lumen output drops beyond 70% of its initial value. However, the color shift, as an equally important performance characteristic of the optical degradation of LED devices, was not included in the standard.

A drawback of the IES standard TM-21-11 is that it uses regression-based methods to analyze the average lumen degradation data and cannot capture the dynamic and random characteristic of the optical degradation process [16]. The random nature of degradation processes reflects a close connection between the LED device’s performances and its lifetime. Therefore, taking the random effects into consideration allows engineers to better understand the optical degradation distribution and perform more accurate lifetime prediction. Motivated by these advantages, a few researchers proposed to use the Wiener process to deal with the LED device’s optical degradation [17–19]. The proposed Wiener process model could be easily used to describe the dynamic and random characteristics of the degradation process. However, the proposed models are only applicable for degradation processes in which the degradation data can be linearized, thus limiting their scope if degradation processes could not follow a linear behavior.

In this paper, the optical degradation data of mid-power white-light LEDs (MP LEDs) has been analyzed using a modified Wiener process, which can account for non-linear degradation mechanisms. Firstly, the modified Wiener process with a time-dependent drift parameter has been presented and an improved likelihood function has been proposed. Secondly, the Frank copula function has been employed to characterize both lumen maintenance and color shift. The joint distribution of lumen maintenance and color shift has been obtained through this function. Finally, cumulative failure distribution of MP LEDs at a specified usage condition has been predicted by performing accelerated degradation tests (ADT). The mean time to failure (MTTF) has been compared to lifetimes calculated by the IES standard TM-21-11, and the cumulative failure distributions, with combined lumen maintenance and color shift has also been presented.

2. Theory and methodology of the Wiener process

2.1 Modeling of univariate degradation by the Wiener process

The modified Wiener process ${X (t), t \geq 0}$ was defined as following expression [20, 21]:

M_{1} : X (t) = X (0) + \int_{0}^{t} η (t; θ) d t + σ_{B} B (t),

where

η (t)

is the drift rate,

σ_{B}

is the diffusion coefficient, and

B (t)

is the standard Brownian motion. Generally, the Wiener process is characterized by three basic properties as below:

(P1) The increment $Δ X (t) = X (t + Δ t) - X (t)$ is independent of $X (t),$ which means that if $0 \leq s_{1} < t_{1} < s_{2} < t_{2},$ then $X (t_{1}) - X (s_{1})$ and $X (t_{2}) - X (s_{2})$ are independent random variables, and the similar condition holds for $n$ increments.
(P2) $Δ X (t) ~ N (\int_{s}^{s + t} η (t; θ) d t, σ_{B}^{2} t) .$ Where $N (\int_{s}^{s + t} η (t; θ) d t, σ_{B}^{2} t)$ denotes the normal distribution with expected value $\int_{s}^{s + t} η (τ; θ) d τ,$ and variance $σ_{B}^{2} t .$
(P3) According to (P2), if $s = 0,$ $X (t) ~ N (\int_{0}^{t} η (t; θ) d t, σ_{B}^{2} t) .$

For this model, Si et al [22] obtained a closed form of the analytical approximation to the distribution of the first hitting time of a threshold level $C_{0}$ by a time-space transformation. Under a mild assumption, the authors gave a probability distribution function (PDF) as follows:

\begin{array}{l} f_{T} (t) = \frac{1}{\sqrt{2 π t}} (\frac{S_{B} (t)}{t} + \frac{1}{σ_{B}} η (t; θ)) \exp (- \frac{S_{B}^{2} (t)}{2 t}) \\ S_{B} (t) = \frac{1}{σ_{B}} (C_{0} - \int_{0}^{t} η (τ; θ) d τ) . \end{array}

As a special case, if $\int_{0}^{t} η (t; θ) d t = η t,$ $X (t)$ becomes the Wiener process with a constant drift, which is denoted as $M_{0},$ i.e.,

M_{0} : X (t) = X (0) + η t + σ_{B} B (t) .

In this case, the first hitting time to a critical threshold $C_{0},$ follows an inverse Gaussian distribution $I G (t | μ, λ)$ with a PDF given by [23]:

f_{T} (t) = \sqrt{\frac{λ}{2 π t^{3}}} \exp {- \frac{λ}{2 μ^{2}} \frac{{(t - μ)}^{2}}{t}},

where

t > 0,

μ = C_{0} / η > 0,

λ = C_{0}^{2} / σ_{B}^{2} .

The parameter estimation of both the models $M_{0}$ and $M_{1}$ was performed according to the aforementioned properties of the Wiener process. Assume that $X_{i j k}$ was the $j^{t h}$ degradation readout of the $i^{t h}$ unit of the samples under the $k^{t h}$ stress level. For any $0 \leq t_{j} < t_{j + 1},$ $j = 1, \dots, m_{k} - 1,$ we have independent but not identical random variables:

\begin{array}{l} Δ X_{i j k} = X_{i (j + 1) k} - X_{i j k} ~ N (\int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t, σ_{B}^{2} (t {}_{j + 1}- t_{j})) \\ k = 1, \dots, n; j = 1, \dots, m_{k} - 1; i = 1, \dots, l_{k} . \end{array}

According to Eq. (5), the likelihood function was given by Liao et al [18]:

L (θ) = \prod_{k = 1}^{n} \prod_{i = 1}^{l_{k}} \prod_{j = 1}^{m_{k} - 1} \frac{1}{\sqrt{2 π σ_{B}^{2} (t_{j + 1} - t_{j})}} \exp (- \frac{(x_{i (j + 1) k} - x_{i j k}) - \int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t}{2 σ_{B}^{2} (t_{j + 1} - t_{j})}) .

The likelihood function in Eq. (6) considers only the independent increment properties of the Wiener process. This method does not make a complete use of the information, resulting in an insufficiently accurate estimation. In order to obtain a sufficient estimation accuracy, we have proposed an estimation method which combines both property (P2) and (P3) into the likelihood function. By using this method, our likelihood function indicates a much faster convergence and higher estimation accuracy compared to Eq. (6). First, for any $0 < t_{p} < \dots < t_{q} < \dots < t_{m_{k}},$ $p = 2, \dots, m_{k} - 1, q = 3, \dots, m_{k}; p < q,$ we have two dependent random variables:

\begin{array}{l} X_{i p k} (t_{p}) ~ N (\int_{0}^{t_{p}} η_{k} (t; θ) d t, σ_{B}^{2} t_{p}) \\ X_{i q k} (t_{q}) ~ N (\int_{0}^{t_{q}} η_{k} (t; θ) d t, σ_{B}^{2} t_{q}) . \end{array}

For simplicity, we assume the interdependency of $X (t_{p})$ and $X (t_{q})$ to be negligible. With this assumption, the likelihood function reads

\begin{array}{l} L (θ) = \prod_{k = 1}^{n} \prod_{i = 1}^{l_{k}} \prod_{p = 1}^{m_{k} - 1} \frac{1}{2 π σ_{B}^{2} \sqrt{t_{p + 1} (t_{p + 1} - t_{p})}} \exp (- \frac{{(x_{i (p + 1) k} - \int_{0}^{t_{p + 1}} η_{k} (t; θ) d t)}^{2}}{2 σ_{B}^{2} t_{p + 1}}) \\ \cdot \exp (- \frac{{((x_{i (p + 1) k} - x_{i p k}) - \int_{t_{p}}^{t_{p + 1}} η_{k} (t; θ) d t)}^{2}}{2 σ_{B}^{2} (t_{p + 1} - t_{p})}) . \end{array}

We would not try to provide a mathematical proof of the feasibility of this likelihood function, but only give a comparison of the results estimated by Eq. (6) and Eq. (8), which will be presented in section 4.2.

2.2 Modeling of bivariate degradation by the Wiener process and copulas

According to Sklar’s theorem [24], if $H (x_{1}, x_{2})$ is a joint distribution with margins $G_{1} (x_{1})$ and $G_{2} (x_{2}),$ then there exists a copula $C$ such that for all $(x_{1}, x_{2}),$ in the defined range,

H (x_{1}, x_{2}) = C (G_{1} (x_{1}), G_{2} (x_{2})) .

To solve Eq. (9), the most widely used copula is the Frank copula, which is a symmetric copula (for bivariate data) given by

C_{α} (u, v) = - \frac{1}{α} \ln {1 + \frac{[\exp (- α u) - 1] [\exp (- α v) - 1]}{\exp (- α) - 1}},

with the corresponding density as

c_{α} (u, v) = \frac{α [1 - \exp (- α)] \exp [- α (u + v)]}{{1 - \exp (- α) - [1 - \exp (- α u)] [1 - \exp (- α v)]}^{2}} .

Assume there exist two performance characteristics (PCs) governed by the Wiener process at each stress level and, $X_{i j k}$ and $Y_{i j k}$ were the $j^{t h}$ degradation readouts of the two PCs (lumen degradation and color shift in this paper) of the $i^{t h}$ unit of the samples under the $k^{t h}$ stress level, then according to expression Eq. (5), we have

\begin{array}{l} Δ X_{i j k} = X_{i (j + 1) k} - X_{i j k} ~ N (\int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t, σ_{X}^{2} (t {}_{j + 1}- t_{j})) \\ Δ Y_{i j k} = Y_{i (j + 1) k} - Y_{i j k} ~ N (\int_{t_{j}}^{t_{j + 1}} β_{k} (t; θ) d t, σ_{Y}^{2} (t {}_{j + 1}- t_{j})) \\ k = 1, \dots, n; j = 1, \dots, m_{k} - 1; i = 1, \dots, l_{k} . \end{array}

Consider the joint distribution of $Δ X_{i j k}$ and $Δ Y_{i j k}$ as

H (Δ X_{i j k}, Δ Y_{i j k}) = C (Φ (\frac{Δ x_{i j k} - \int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t}{σ_{X} \sqrt{t_{j + 1} - t_{j}}}), Φ (\frac{Δ y_{i j k} - \int_{t_{j}}^{t_{j + 1}} β_{k} (t; θ) d t}{σ_{Y} \sqrt{t_{j + 1} - t_{j}}})) .

Similar to Eq. (8), the likelihood function is then given by:

\begin{array}{l} L (θ) = \prod_{k = 1}^{n} \prod_{i = 1}^{l_{k}} \prod_{j = 1}^{m_{k} - 1} [c (Φ (\frac{Δ x_{i j k} - \int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t}{σ_{X} \sqrt{t_{j + 1} - t_{j}}}), Φ (\frac{Δ y_{i j k} - \int_{t_{j}}^{t_{j + 1}} β_{k} (t; θ) d t}{σ_{Y} \sqrt{t_{j + 1} - t_{j}}})) \\ \cdot φ (\frac{Δ x_{i j k} - \int_{t_{j}}^{t_{j + 1}} η_{k} (t; θ) d t}{σ_{X} \sqrt{t_{j + 1} - t_{j}}}) \cdot φ (\frac{Δ y_{i j k} - \int_{t_{j}}^{t_{j + 1}} β_{k} (t; θ) d t}{σ_{Y} \sqrt{t_{j + 1} - t_{j}}}) \\ \cdot φ (\frac{x_{i j k} - \int_{0}^{t_{j + 1}} η_{k} (t; θ) d t}{σ_{X} \sqrt{t_{j + 1}}}) \cdot φ (\frac{y_{i j k} - \int_{0}^{t_{j + 1}} β_{k} (t; θ) d t}{σ_{Y} \sqrt{t_{j + 1}}})] . \end{array}

Let $C {}_{1}$ and $C_{2}$ be the threshold values of the two PCs. The reliability function under the use stress $S_{u}$ is

R (t) = P {\sup_{s \leq t} X (s | S_{u}) \leq C_{1}, \sup_{s \leq t} Y (s | S_{u}) \leq C_{2}} .

When $σ_{X}$ and $σ_{Y}$ is small, Eq. (14) can be approximated by [25]:

\begin{array}{l} R (t | S_{u}) = P {X (t / S_{u}) \leq C_{1}, Y (t / S_{u}) \leq C_{2}} \\ = C (R_{X} (t | S_{u}), R_{Y} (t | S_{u})), \end{array}

where

R_{X} (t | S_{u}) = P {X (t | S_{u}) \leq C_{1}},

and

R_{Y} (t | S_{u}) = P {Y (t | S_{u}) \leq C_{2}} .

Finally, the culmulative distribution function (CDF) of the first hitting time of the threshold levels $C_{1}$ and $C_{2}$ is obtained as

F_{T} (t | S_{u}) = 1 - R (t | S_{u}) .

3. Experiments

The LED device used for the ageing tests was one kind of commercially available mid-power white-light LED packages with a correlated color temperature (CCT) of 2700 K. As indicated in Fig. 1, the mid-power LED packages are composed of four components: blue die, lead-frames, phosphor-dispensing silicone and package house. More specifically, one 0.6 mm² big blue LED chip was mounted onto the Silver lead-frame by non-conductive transparent die attach. Then, gold wires were bonded to connect the blue LED chip to the lead-frame and provide electro-optical functionality of the device. After wire bonding, silicone mixed with phosphors was dispensed into the package housing in order to enable a conversions of the blue light emitted by the LED chip (with the emission wavelength in the range of 450 nm-460 nm) to green and red light. Finally, the specified CCT was achieved by carefully tuning the amount down-converted green and red light.

Fig. 1 The structure of the aged LED packages.

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At first, the LED packages were assembled onto a metal-core printed circuit (MCPCB) with the dimension of 13 cm × 13 cm. Then the MCPCB was attached on a heat sink by using thermal glue and fastened by 5 screws. After that, the test samples were placed into 3 climate chambers respectively depending on the test conditions. The stress conditions are listed in Table 1.

Table 1. Stress conditions for LED packages

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The samples were taken out of the climate chambers for optical measurement at a series of predefined time points and then put back to the chamber for further ageing. After finishing all ageing tests, all collected lumen data were normalized to a value of 100% at 0-hour for each individual sample tested, and the color shift was calculated according to the CIE 1976 color space. These data were then used for lifetime estimation based on the aforementioned Wiener process models.

4. Results and discussion

4.1 Data analysis

The lumen maintenance and color shift were presented in Fig. 2 and Fig. 3. Apparently it was found that the lumen maintenance decreases exponentially, and the color shift increases linearly after a sharp jump. In this paper, we employed the exponential degradation model to describe the lumen maintenance, as it has widely been used and verified by researchers [15, 26–28]. The reads as:

Φ (t; S_{k}) = Φ_{k} \cdot \exp (- α_{k} t),

where

Φ (t; S_{k})

is the normalized optical outputs of LED devices aged by the

k^{t h}

stress condition at time

t,

Φ_{k}

is the projected initial constant, and

α_{k}

is the decay rate constant which is related to ageing stress levels.

Fig. 2 Lumen maintenance of the stressed LEDs.

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Fig. 3 Color shift of the stressed LEDs.

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Though many results were reported on the modeling of color shift, there is no specific physic model for color shift of LED devices [29–31]. From the statistical viewpoint, Fan et al [32] proposed a nonlinear dual-exponential model to describe the chromaticity state shift process. The dual-exponential model, however, appears to be too complex to extrapolate the relationship between degradation data and stress level. On the other hand, Koh et al [33] demonstrated that a linear model could be applicable for the color shift of LED devices by analyzing both the experiment and simulation data. As shown in Fig. 3, the color shift expressed by $Δ u' v',$ is nonlinear if we include the color shift calculated from the first readout, because there is a sharp jump at this point. This phenomenon is most likely associated with early degradation during ageing tests. The early degradation been extensively observed in mid-power LED packages. However, the rest of the readout points show very good linear trend. In light of this, it is reasonable to describe color shift as linear degradation. Therefore, by excluding the early degradation from the measured color shift, the model is as follows:

Δ u' v' (t; T_{k}) = A_{k} + B_{k} t,

where

Δ u' v' (t; S_{k})

is the color shift of LED devices aged at the

k^{t h}

stress condition at time

t,

A_{k}

is the projected initial, and

B_{k}

is the decay rate constant which is related to ageing stress levels.

After determining the degradation models, the unknown parameters were estimated by means of the modified Wiener process, as will be presented in coming paragraphs.

4.2 Parameter estimation

According to the analysis above, $M_{1}$ and $M_{0}$ has been applied to describe the degradation behavior of lumen maintenance and color shift, respectively.

Let $X (0) = Φ_{k},$ and $\int_{0}^{t} η (t; θ) d t = Φ_{k} \exp (- α t),$ then substituting them into Eq. (1), the Wiener process of lumen degradation was expressed as:

Lumen maintenance : Φ (t; S_{k}) = Φ_{k} \cdot \exp (- α_{k} t) + σ_{Y} B_{Y} (t) .

Similarly to the lumen maintenance, let $X (0) = A_{k},$ and $\int_{0}^{t} η (t; θ) d t = B_{k} t,$ then substituting them into Eq. (1), the Wiener process of color shift degradation was expressed as:

Color shift : Δ u' v' (t; T_{k}) = A_{k} + B_{k} t + σ_{X} B_{X} (t) .

The parameter $α_{k}$ and $B_{k}$ are assumed to be dependent of the stress levels, such as temperature and humidity. Here, the Hallberg-Peck’s model is used to describe the effect of temperature and humidity combined on LED degradation as follows [34]:

\begin{array}{l} α_{k} = α_{0} \exp [(\frac{- E_{a_l u m e n}}{k_{b}}) (\frac{1}{T_{k}} - \frac{1}{T_{0}})] {(\frac{R H_{k}}{R H_{0}})}^{n_l u m e n} \\ B_{k} = B_{0} \exp [(\frac{- E_{a_c o l o r}}{k_{b}}) (\frac{1}{T_{k}} - \frac{1}{T_{0}})] {(\frac{R H_{k}}{R H_{0}})}^{n_c o l o r}, \end{array}

where

R H_{k}

and

T_{k}

are the relative humidity and temperature at the stress condition, respectively,

R H_{0}

and

T_{0}

is the relative humidity and temperature at the use condition,

E_{a_l u m e n}

and

E_{a_c o l o r}

are activation energy,

n_{_l u m e n}

and

n_{_c o l o r}

are humidity factor,

k_{b}

is the Boltzmann constant. In this paper,

k = 1, 2, 3,

correspond to WHTOL 85°C/90%RH, WHTOL 95°C/45%RH and WHTOL 95°C/95%RH, respectively .

For simplicity, we assumed the diffusion coefficient $σ_{X}$ and $σ_{Y}$ to be independent of the stress levels. Then, the unknown parameters $Φ_{k},$ $A_{k},$ $E_{a_l u m e n},$ $E_{a_c o l o r},$ $n_{_l u m e n},$ $n_{_c o l o r},$ $σ_{X}$ and $σ_{Y}$ have been estimated according to the likelihood function Eq. (8). The parameters including the copula constant $α$ were also estimated according to the likelihood function Eq. (13). Bayesian Markov chain Monte Carlo simulation (MCMC) has been performed because the model in such a situation is complicated and analytically intractable. More details about the algorithm can be obtained in reference [35]. After 150,000 iterations, all parameters converged. Similar results were obtained from both methods, as it could be seen in Table 2 and Table 3, respectively.

Table 2. Parameter estimation results without copula

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Table 3. Parameter estimation results with copula

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Figure 4 and 5 show the fitted models of LEDs stressed under WHTOL 85°C/90%RH. For comparison, the scatter plot of the measured data, as well as the degradation models fitted by both the likelihood function Eq. (6) and Eq. (8), were plotted together. Obviously the model fitted by Eq. (6) seriously deviates from the measured data while the one fitted by Eq. (8) matches the data very well. The results demonstrate that significant improvements of the estimation accuracy have been obtained by using likelihood function in Eq. (8). The same results, with equally improved accuracy, have also been obtained in WHTOL 95°C/45%RH and WHTOL 95°C/95%RH, but presented here.

Fig. 4 Fitted lumen degradation model by likelihood function Eq. (6) and Eq. (8).

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Fig. 5 Fitted color shift model by likelihood function Eq. (6) and Eq. (8).

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Figure 6 shows the lifetime plots which are fitted by both the modified Wiener process and the Wiener process (logarithm transformation was performed to linearize the raw data prior to MCMC simulation) for LEDs stressed under WHTOL 85°C/90%RH. It is found that both models show very similar results, indicating the feasibility of the modified Wiener process. It should be noticed that, although both the Wiener process and the modified Wiener process show very closed results in describing the exponential lumen degradation of MP LED packages, the Wiener process needs data to be linearization prior to parameter estimation, whereas the modified Wiener process does not need this type of data transformation. This advantage extends the applications of the modified Wiener process to the non-linear degradation behavior, for which the raw data cannot be linearized by any type of data transformation.

Fig. 6 Lumen degradation models fitted by using both the modified Wiener process and the Wiener process.

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4.3 Lifetime prediction at usage condition

The use condition is assumed to be $R H_{0} = 45 %$ , and $T_{0} = 85$ °C. Referring to the IES standard TM-21-11, the projected initial $Φ_{0}$ and $A_{0}$ in Eq. (19) and Eq. (20) is calculated as

Φ_{0} = \sqrt{Φ_{1} Φ_{2}}, A_{0} = (A_{1} + A_{2}) / 2 .

The rest of the unknown parameters in Eq. (19) and Eq. (20) could be found in Table 2 and Table 3. Given the critical thresholds of the lumen maintenance and color shift as $C_{1} = 0.7,$ $C_{2} = 0.007$ [15, 36], the lifetime of the LED devices is then predicted according to Eq. (2) and Eq. (4). When only lumen maintenance is considered, the mean time to failure (MTTF) is calculated to be 16,569 hours, which is much closer to that calculated by the IES standard TM-21-11, which yields the lifetime of 17,031 hours. Figure 7 shows lifetime predicted corresponding to different combinations of both lumen decay and color shift. Apparently the color shift first reaches the critical threshold $C_{1}$ before the lumen maintenance decreases to 70% of its initial value. It is also interesting to note that the dependency of lumen maintenance and color shift is not strong, because the cumulative failure functions with and without copula $α$ are almost the same, indicating that the reliability of MP LED devices can be modeled by simply considering the lumen maintenance and color shift as two independent variables.

Fig. 7 The cumulative failure function of LED devices by different methods.

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5. Conclusion

In this paper, the lifetime prediction has been performed by using a modified Wiener process which could be applied to describe the dynamic, random and non-linear degradation of MP LED devices. Under the assumption that the dependency between random variables is negligible, a simple, but efficient likelihood function with an improved accuracy of parameter estimation is proposed. On the other hand, the Frank copula function has been applied to describe the dependency between the lumen maintenance and color shift. However, the cumulative failure function demonstrates that the dependency between them is not significant. Therefore, these two performance characteristics could be simply considered as independent variables, thus simplifying the degradation modeling of the LED devices. Due to complexity of the failure distribution, we could not obtain the interval estimation in this study. However, the Wiener process had provided favorable results, making it an attractive and promising method in degradation modeling of LED devices.

Acknowledgment

The authors Jianlin Huang thank the colleagues of Philips Light Test Center for the optical measurement and testing maintenance. Thanks also to Dr. Haibo Fan, Jinfeng Li and Longgang Ding for his supports on the experiments. The author Daoguo Yang would like to thank the National Science Foundation of China for the support (Grant No. 51366003).

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Parameter	Mean	Std	MC error	2.5%	Median	97.5%
$Φ_{1}$	0.9898	0.001116	7.4E-06	0.9876	0.9898	0.9919
$Φ_{2}$	0.9959	0.001013	8.61E-06	0.9939	0.9959	0.9979
$Φ_{3}$	0.9957	0.00124	9.05E-06	0.9932	0.9957	0.9981
$E_{a_l u m e n}$	0.5469	0.04147	0.000443	0.4658	0.5469	0.6289
$n_{_l u m e n}$	1.107	0.06169	0.000814	0.9877	1.106	1.229
$α_{0}$	2.15E-05	1.09E-06	1.57E-08	1.94E-05	2.15E-05	2.37E-05
$σ_{Y}$	0.000345	9.16E-06	2.32E-08	0.000328	0.000345	0.000364
$A_{1}$	0.000839	8.32E-05	5.38E-07	0.000676	0.000839	0.001002
$A_{2}$	0.000576	7.63E-05	7.87E-07	0.000426	0.000576	0.000726
$A_{3}$	0.001055	9.31E-05	7.87E-07	0.000873	0.001055	0.001238
$E_{a_c o l o r}$	0.8235	0.08751	0.000987	0.6544	0.8226	0.9976
$n_{_c o l o r}$	1.519	0.1448	0.002329	1.247	1.515	1.817
$B_{0}$	4.99E-07	5.93E-08	9.74E-10	3.9E-07	4.97E-07	6.22E-07
$σ_{X}$	2.63E-05	7.01E-07	1.77E-09	2.5E-05	2.63E-05	2.77E-05

Parameter	Mean	Std	MC error	2.5%	Median	97.5%
$Φ_{1}$	0.9898	0.001116	7.68E-06	0.9876	0.9898	0.9919
$Φ_{2}$	0.9959	0.001018	9.83E-06	0.9939	0.9959	0.9979
$Φ_{3}$	0.9956	0.001246	9.79E-06	0.9932	0.9956	0.9981
$E_{a_l u m e n}$	0.5467	0.04143	0.000461	0.4663	0.5465	0.6286
$n_{_l u m e n}$	1.105	0.06263	0.000921	0.9844	1.104	1.229
$α_{0}$	2.15E-05	1.1E-06	1.72E-08	1.94E-05	2.15E-05	2.37E-05
$σ_{Y}$	0.000345	9.24E-06	2.53E-08	0.000327	0.000345	0.000364
$A_{1}$	0.000844	8.33E-05	6.22E-07	0.000681	0.000844	0.001008
$A_{2}$	0.000579	7.68E-05	8.79E-07	0.000428	0.000579	0.00073
$A_{3}$	0.001052	9.25E-05	6.54E-07	0.00087	0.001052	0.001234
$E_{a_c o l o r}$	0.8321	0.08831	0.000973	0.6635	0.8305	1.01
$n_{_c o l o r}$	1.528	0.1465	0.002395	1.257	1.523	1.834
$B_{0}$	4.93E-07	5.94E-08	1.01E-09	3.82E-07	4.91E-07	6.15E-07
$σ_{X}$	2.63E-05	7.03E-07	1.85E-09	2.49E-05	2.62E-05	2.77E-05
$α$	−0.1442	0.3203	0.000908	−0.7737	−0.1435	0.4806

Parameter	Mean	Std	MC error	2.5%	Median	97.5%
$Φ_{1}$	0.9898	0.001116	7.4E-06	0.9876	0.9898	0.9919
$Φ_{2}$	0.9959	0.001013	8.61E-06	0.9939	0.9959	0.9979
$Φ_{3}$	0.9957	0.00124	9.05E-06	0.9932	0.9957	0.9981
$E_{a_l u m e n}$	0.5469	0.04147	0.000443	0.4658	0.5469	0.6289
$n_{_l u m e n}$	1.107	0.06169	0.000814	0.9877	1.106	1.229
$α_{0}$	2.15E-05	1.09E-06	1.57E-08	1.94E-05	2.15E-05	2.37E-05
$σ_{Y}$	0.000345	9.16E-06	2.32E-08	0.000328	0.000345	0.000364
$A_{1}$	0.000839	8.32E-05	5.38E-07	0.000676	0.000839	0.001002
$A_{2}$	0.000576	7.63E-05	7.87E-07	0.000426	0.000576	0.000726
$A_{3}$	0.001055	9.31E-05	7.87E-07	0.000873	0.001055	0.001238
$E_{a_c o l o r}$	0.8235	0.08751	0.000987	0.6544	0.8226	0.9976
$n_{_c o l o r}$	1.519	0.1448	0.002329	1.247	1.515	1.817
$B_{0}$	4.99E-07	5.93E-08	9.74E-10	3.9E-07	4.97E-07	6.22E-07
$σ_{X}$	2.63E-05	7.01E-07	1.77E-09	2.5E-05	2.63E-05	2.77E-05

Parameter	Mean	Std	MC error	2.5%	Median	97.5%
$Φ_{1}$	0.9898	0.001116	7.68E-06	0.9876	0.9898	0.9919
$Φ_{2}$	0.9959	0.001018	9.83E-06	0.9939	0.9959	0.9979
$Φ_{3}$	0.9956	0.001246	9.79E-06	0.9932	0.9956	0.9981
$E_{a_l u m e n}$	0.5467	0.04143	0.000461	0.4663	0.5465	0.6286
$n_{_l u m e n}$	1.105	0.06263	0.000921	0.9844	1.104	1.229
$α_{0}$	2.15E-05	1.1E-06	1.72E-08	1.94E-05	2.15E-05	2.37E-05
$σ_{Y}$	0.000345	9.24E-06	2.53E-08	0.000327	0.000345	0.000364
$A_{1}$	0.000844	8.33E-05	6.22E-07	0.000681	0.000844	0.001008
$A_{2}$	0.000579	7.68E-05	8.79E-07	0.000428	0.000579	0.00073
$A_{3}$	0.001052	9.25E-05	6.54E-07	0.00087	0.001052	0.001234
$E_{a_c o l o r}$	0.8321	0.08831	0.000973	0.6635	0.8305	1.01
$n_{_c o l o r}$	1.528	0.1465	0.002395	1.257	1.523	1.834
$B_{0}$	4.93E-07	5.94E-08	1.01E-09	3.82E-07	4.91E-07	6.15E-07
$σ_{X}$	2.63E-05	7.03E-07	1.85E-09	2.49E-05	2.62E-05	2.77E-05
$α$	−0.1442	0.3203	0.000908	−0.7737	−0.1435	0.4806

Degradation modeling of mid-power white-light LEDs by using Wiener process

Abstract

1. Introduction

2. Theory and methodology of the Wiener process

2.1 Modeling of univariate degradation by the Wiener process

2.2 Modeling of bivariate degradation by the Wiener process and copulas

3. Experiments

4. Results and discussion

4.1 Data analysis

4.2 Parameter estimation

4.3 Lifetime prediction at usage condition

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (3)

Equations (23)

Optics Express

Chamber	Temperature (Unit: °C)	Relative Humidity (Unit: 1)	Current (Unit: mA)	Sample Size (Unit: pieces)
WHTOL 85°C/90%RH	85	90%	160	20
WHTOL 95°C/45%RH	95	45%	160	20
WHTOL 95°C/95%RH	95	95%	160	20