A combined universal generating function and physics of failure Reliability Prediction Method for an LED driver

This is an open access article under the CC BY license (https://creativecommons.org/licenses/by/4.0/) reliability prediction, LED driver, reliability modelling, physics of failure, universal generating function. The accurate and effective reliability prediction of light emitting diode (LED) drivers has emerged as a key issue in LED applications. However, previous studies have mainly focused on the reliability of electrolytic capacitors or other single components while ignoring circuit topology. In this study, universal generating function (UGF) and physics of failure (PoF) are integrated to predict the reliability of LED drivers. Utilizing PoF, lifetime data for each component are obtained. A system reliability model with multi-phase is established, and system reliability can be predicted using UGF. Illustrated by a two-channel LED driver, the beneficial effects of capacitors and MOSFETs for the reliability of LED drivers is verified. This study (i) provides a universal numerical approach to predict the lifetime of LED drivers considering circuit topology, (ii) enhances the modelling and reliability evaluation of circuits, and (iii) bridges the gap between component and circuit system levels. Highlights Abstract

A multi-phase model is established according to • function processes of circuit.
A universal numerical approach is provided for • predicting the lifetime of LED drivers.
A combined universal generating function and physics of failure Reliability Prediction Method for an LED driver The base degradation rate of capacitors 1 2 , The surface area of the electrolytic capacitor 1 2 ,

N N
The windings d The average distance of the electrolytic capacitor 1 4 C − The capacity ϖ The conductivity of the electrolytic capacitor 1 2 ,

LS LS
The LED strings RH The relative humidity 1 2 ,

D D
The diodes in V The power supply

Introduction
As a new type of solid-state lighting (SSL), LED applications are rapidly increasing in many domains, including industrial/decorative illumination, liquid crystal display (LCD) backlighting, and automobile lamps [16] [22]. In addition to their wide range of applications, LEDs provide longer operating times and higher levels of reliability. However, compared with the lifetime of LED chips, LED drivers are found to have shorter lifetimes, as reported by the U.S. Department of Energy [33]. This discrepancy can contribute to up to 52% of total system failures [25] [29], as the operating time and reliability of LED systems are largely dependent on LED drivers [36].
In general, researchers have focused on two types of LED drivers: linear mode drivers and switch mode drivers [32] [39]. By studying LED driver schematics, many researchers have developed various approaches to predict the reliability of LED drivers. In terms of linear mode drivers, Song Lan et al. [17] presented a black box method to locate the weak point and study the reliability of a regulated LED driver circuit. A particle filter (PF) method [14] was implemented and combined with nonlinear least squares (NLS) and nonlinear mixedeffect estimation (NLME) to predict the remaining useful time (RUL). Lan et al. [15] [16] studied the degradation of a linear mode highpower LED driver, and hot carrier injection (HCI) has been revealed as the main degradation mechanism. In terms of switch mode drivers, Koosuke et al. [8] presented a diagnosis method for an output smoothing capacitor, and the lifecycle ageing rate was projected for the active circuit over the system life thanks to equivalent series resistance (ESR). Based on the PoF, Zhou [40] predicted the RUL by utilizing an accelerated ageing test.
However, several issues exist in current studies: (i) researchers have only focused on the electrolytic capacitors or other single types of components in LED drivers; (ii) the circuit topology and multiphase status of LED drivers have been ignored in previous analyses; and (iii) there has been no assurance that degradation laws and other particular factors will not change during acceleration tests. These drawbacks can lead to inaccurate prediction results, especially for complex LED driver systems with multiple phases.
To address these issues, this study aims to establish a reliability prediction model based on the operation profile of a circuit with multiple phases and bridge the gap between the component level and circuit system level. First, utilizing a PoF model, various parameters are discretized based on the failure mechanisms and the distribution model. Then, the failure probability curve can be fitted in terms of the correlation between the time and time to failure (TTF). Third, a specific two-channel switch mode LED driver is illustrated to establish the reliability model. Finally, system reliability prediction results are estimated using a UGF. The main contributions are as follows: (1) The proposed approach can provide a way to bridge the gap between the component level and circuit system level using a combined UGF and PoF approach. (2) This universal multiphase modelling approach, based on the function process of the circuit, can predict the lifetime of an LED driver.
The remainder of this paper is organized as follows. Some basic concepts, including the UGF and TTF, are introduced in Section 2. Based on the framework of the novel reliability prediction approach, the detailed modelling, analysis, and prediction of two-channel LED drivers are proposed in Section 3. The experimental procedure and re- Generally, the reliability prediction of a circuit follows competition rules [12]. The process regards the circuit as a cascaded structure and takes the minimum lifetime of components as the lifetime. However, a competition model is clearly inaccurate in reliability prediction, especially for circuit systems with high complexity such as series-parallel and backup structures. To bridge the gap between the component level and circuit system level, the UGF approach is utilized in this paper.
The UGF approach [2] [26] [34] is typically used to describe the relationship of reliability between components and a system. It was first introduced by Ushakov [34] and generally applied in reliability assessment with multi-state components [26]. Assume that there are n independent components 1 ,, n x x  , i k is the possible state of component i x , and ij p is the probability of component i x at state j . The probability distribution of each variable is discretely represented by a u-function ( ( ) i t x u e ) and then replaces t e as z , as shown in (1). In an evaluation of system reliability, the u-function combines the reliability and performance to describe the component state distribution: If the component is a binary unit, state 0 is the failure state, and state 1 is the success state. The u-function of component i x can be denoted as follows: According to a series or parallel structure, the u-function of the structure can be described as follows [2] [18]: The reliability of the system can be described as shown in (4), in which z represents the no-fault status:

Time-to-failure distribution for multiple phases
An LED driver circuit has the following characteristics: (i) the circuit system can be divided into i phases based on the direction of the current and the operation of the circuit; (ii) the components have various operation times; (iii) the TTF of components obeys various distributions; and (iv) i phases form a cycle, and the multiple cycles are serially structured [23] [35].
In accordance with the terminology introduced in [6] [7] [28], if the random variable A is in state ( , and n is the time granularity: Similarly, if A is said to be in state ( 1) n + , then A has survived the mission time T : 3. Reliability Prediction Approach for LED Driver

Framework of reliability prediction approach
In practice, the normal degenerative process of a specific LED driver system can take years to display a conspicuous change. To address this restriction, a novel reliability prediction approach based on the PoF is proposed, which is shown in Fig. 1. Integrated with the failure mechanisms and topological structure of the system, this ap-proach provides a universal numerical and simulation approach for predicting the lifetime of LED drivers. With the analysis of failure mechanisms, the major factors of degradation of components are determined (Step 1-Step 3). By utilizing the PoF method, the lifetime data for each component can be obtained by random sampling simulations (Step 4-Step 5). Then, the operation process of the circuit can be analysed and divided into multiple phases (Step 6). A reliability model can be established for the LED driver at each phase (Step 7). Finally, the reliability of the circuit system can be estimated with the lifetime curve above as the input data according to the topological structure of the multiphase circuit utilizing a UGF (Step 8).

Topology of LED driver
It is well known that the balance of current is a vital function for LED drivers [32]. A switch mode LED driver has the dominant advantage in balancing the current with full bridge and half-bridge topologies. This mode features a simple structure and precise current balance. Thus, switch mode LED drivers have received increasing attention [19] [37]. Compared with the traditional topology of a switch mode LED driver, a two-channel non-isolated LED driver has the following advantages [3] [4] [9]: (i) owing to a DC-blocking capacitor, LED currents can be balanced automatically; (ii) owing to the combination of the DC-blocking capacitor and a coupled inductor, a high-output voltage gain can be achieved; (iii) the voltage spike from the leakage inductance of the coupled inductor can be recycled to the capacitor 2 C so that low-voltage MOSFETs can be used; (iv) the proposed two-channel LED driver can be extended to a multi-channel LED driver to satisfy industry requirements.
In terms of the switch mode LED driver [10] [40], a twochannel non-isolated LED driver is regarded as an example for this study, whose topology is shown in Fig. 2. It contains two MOSFETs 1 Q and 2 Q , a coupled inductor composed of a primary winding 1 N and a secondary winding 2 N , two energy-electrolytic capacitors 1 C and 2 C , two output diodes 1 D and 2 D , two output capacitors 3 C and 4 C , and two LED strings 1 LS and 2 LS . Moreover, the coupled inductor can be modelled as an ideal transformer with a primary winding 1 N and a secondary winding 2 N connected in series with a leakage inductor 2 lk L , a magnetizing inductor m L connected in parallel with 1 N , and a leakage inductor 1 lk L . In addition, the current flows and the anode and cathode labels are shown in Fig. 2 [3].

PoF models
For different types of LED drivers, the topology of the circuit and component constitution are disparate, and the functional components are associated with different failure modes and mechanisms [38]. Previous reports [8] [14] [15] have discussed the failure mechanisms of amplifiers, diodes, resistors and capacitors in linear mode LED drivers. The reliability of a switch mode LED driver was analyzed in [30], and detailed degradation mechanisms for electrolytic capacitors, MOSFETs, inductors, diodes and sampling resistors were given. The degradation process of a two-channel LED driver is extremely complicated and is usually caused by the gradual deterioration of components over a long period of time due to electrical stress, temperature effects, material characteristics and other circumstances. Generally, the degradation of various components can be related to specific performance parameters, and the various mechanisms can have competitive relations. Hence, according to previous research [5] [13] [20] [24] [30] [31], we focus attention on the greatest effect of each component, and the mechanism models are shown in Table 1.
To obtain the parameters of the mechanism models, the least square method is applied to fit the experimental data under various conditions. Table 1 proposes the mean values and the standard deviations, and the details of experimental data analysis, test procedures, and the fitting analysis are referred to [21].
Taking the diode as an example, we first sample the parameter based on the distribution, and the TTF data of the component can be obtained according to the number of samples. Then, we organize all the TTF data from small to large and divide the data into several groups utilizing an appropriate class. By calculating the frequency of each group, the cumulative probability can be obtained, and finally, the fitted curve between the TTF and failure probability is achieved.
In Fig. 3 (a), the x-axis represents time, and the y-axis represents the failure probability, namely, the unreliability of the component diode. The relationship between time and failure probability of the diode is shown in Fig. 3 (a). Moreover, as the various parameters of distributions change, the life curve can be different, as shown in Fig. 3  (b). As shown in Fig. 3, the failure probability curve of the diode is proposed using the simulation method. As time goes on, the failure probability gradually increases and tends toward 1. Utilizing a temperature parameter that obeys the normal distribu-

Modelling rules for LED driver circuit
The failure probability of the components and their corresponding lifetime curves are mentioned above. To bridge the gap between the component level and circuit system level, modelling rules and procedures are proposed [11] [27]: Define the circuit system. In this step, the scope of the system (i) and reliability index are defined. Furthermore, its function and topology, including the components, are determined.
Determine the boundary conditions. The input, output and in-(ii) terface of the system are defined. Certain inductance and energy storage components can also be the input signals when they are at high potential.
Establish success criteria. Clearly define the success state and (iii) the output being focused on.
Analyze the circle process. Dismantle the operational process (iv) of the circuit based on the direction of the current.

Operation process of the two-channel LED driver
According to the topology structure, the operation process of the LED driver can be divided into six phases [3] [27], which are shown in Fig. 5. Switches 1 Q and 2 Q are turned on at different times, and their capacities are discharged or charged according to the electric potential. In addition, components in red represent the "OFF" state, and the "OFF" components will not deteriorate during the phase.
Model (a)-phase 1: As shown in Fig. 2 and Fig. 5 (a), 1 Q is turned ON, and 2 Q is OFF. During this state, the voltage is imposed on m L . Meanwhile, 1 C is discharged, so 1 C i or 1 lk i is increasing in the opposite direction. Moreover, 1 D is reverse biased, but 2 D is forward biased. Hence, the LED string 1 S L is supplied by 3 C , whereas the LED string 2 S L is supplied by lk i in the opposite direction. This state ends when 1 Q is turned OFF [3]. Model (b)-phase 2: as shown in Fig. 2 and Fig. 5 (b) Fig. 2 and Fig. 5 (c), but 1 Q still remains turned OFF. In addition, the behavior of the two LED strings is the reaches zero, and this state ends. Model (d)-phase 4: as shown in Fig. 2 and Fig. 5 (d), the switch 1 Q is still OFF, and the switch 2 Q is still ON. During this state, the capacitor 2 C is discharged, and the current 2 C i begins to increase in the opposite direction. In addition, the behavior of the two LED strings is the same as that in model (b). This state ends when 2 N i drops to zero. Model (e)-phase 5: as shown in Fig. 2 and Fig. 5 (e), the switch 1 Q is still OFF, and the switch 2 Q is still ON. The direction of the current is opposite model (d)-phase 4. In addition, the behavior of the two LED strings is the same as that in model (b). This state ends when 2 Q is turned OFF. Model (f)-phase 6: as shown in Fig. 2 and Fig. 5 (f), switch 1 Q is still OFF, and switch 2 Q turns OFF. The direction of the current is the same as model (d)-phase 4. This state ends when 1 Q is turned ON, and the next cycle is repeated.
Additionally, with the time-varying of current, the oscillations of electrical waveforms will create oscillations on the components, especially for the coupled inductors. Nonetheless, the problem oscillation is not the focus of our research, and thus, the effects on reliability are ignored in this study. This will be discussed in our future study.

System reliability evaluation utilizing UGF
We take model (a) as an example: the power supply in V and cou- We assume that all the components are binary, and a logic gate of k-out-of-N is used to describe the output of the system. The structure function of the circuit can be described as follows: (min(max(  (14) Other phases in Fig. 5 can be calculated using the same process above. Focusing on analyzing the impact of component failure on the LED driver, some assumptions are made to make the analysis clear and easy. As the input of the LED driver, in V has a minimal failure probability, and it can be neglected. Likewise, 1 s L and 2 s L , the light-emitting units that are the output of the system, are also ignored, namely, 1 In addition, according to the failure logic of the system, the system is working properly only if all phases are in the normal state. Therefore, the various phases are regarded as a series-mode frame to evaluate the reliability of the system in this study.
According to the approach above, the reliability prediction curve for the two-channel LED driver is displayed in Fig. 7 (a). The simulation results clearly show that the reliability of the circuit gradually degrades as a function of time due to the ageing process. It also appears that the failure rate increases sharply before 18000 h and then begins to flatten gradually in Fig. 7 (b).

Experiments and Validation
To validate the proposed method, an experiment was designed to predict the reliability of the LED driver. The proposed type of LED drivers was selected, and 60 samples were made by Everlight Elec-tronics, Ltd., which were used for testing. The real product is shown in Fig. 8 (a). In Fig. 8 (a), the two electrolytic capacitors are the components 1 C and 2 C . The output capacitors 3 C and 4 C are parallel with light emitting diodes and thus external to the board. Owing to the necessity of the diodes to glow normally, as shown in Fig. 2, the output capacitors 3 C and 4 C should be considered in the system reliability model.
According to the sample, if the nominal constant current flowing through the LED is 350 mA, the corresponding nominal forward voltage is approximately 3.5 V. The loads of the proposed drivers were kept constant by using five 3-W white LEDs for each string. In addition, the variations in the LED forward voltage are 10% ± of the nominal value, and the output power directly reflects the degradation of the LED driver, which can be conveniently measured. The main specifications of the LED driver are shown in Table 2. Fig. 8 (b) shows the experimental setup. A local oven was used to contain the LED driver and to provide local ambient temperature control. The input voltage is controlled by the programmable power source, and the output voltage can be measured periodically by the voltage meter, thus, the state of the LED driver can be obtained. This procedure was repeated until all of the samples were in the failure state.
During the validation test, a total of 60 samples were aged individually at an ambient temperature of 328 K. With a constant input voltage, the working hours of each sample were recorded by the acceleration test until the LED driver was in a state of failure. Based on empirical data, the failure criterion is defined as within 10% of the rated current increase of the LED channel [30]. Fig. 9 shows the predicted reliability of the LED driver versus the experimental results. Due to the time limit and lab constraints, the samples are not sufficient for extensive experiments, and groups of various ambient temperatures were not considered in the experiment. Nevertheless, the prediction results are in good agreement with the test results. The maximum error between the test and simulation is less than 10%.

Relevant analysis
One of the most important assumptions in modelling is that the components remain independent in the operation condition. Fig. 10 shows the reliability simulation results of the circuit system. The red line describes the degradation curve without considering the relation among the components, and the blue line displays the degradation curve with the dependence included. The results show that the reliability without considering the dependence on degrades more signifi-cantly. When the time is approximately 19500 h, the reliability that considers the relation is 0.2459, which is nearly double the value of the scenario that does not. As shown in Fig. 10, the assessment of the circuit system reliability could be more accurate with the dynamic relation between components. Fig. 11 displays the structure importance of various components. The attribute of each component to the output of the circuit system can be described quantitatively. Fig. 11 clearly shows that the higher the attribute of a component is, the greater the importance for the output of the circuit system. In Fig. 11, 7 p ( 3 C ), 9 p ( 1 Q ), and 12 p ( 4 C ) are vital components according to the entire circuit system. These components are the key objects in the structure of the circuit system. Fig. 11. The structure importance of various components

Sensitivity analysis of components
A sensitivity analysis of various components is proposed in this section. As shown in Fig. 12, 12 p ( 4 C ) is the most sensitive among the vital components mentioned above. This means that a slight improvement acting on the component can result in a vital effect being obtained on the LED driver. Although 7 p ( 3 C ) and 9 p ( 1 Q ) are vital components in the analysis of structural importance, the two components work in the first phase of the operation process. Compared with 12 p (), the improvement of the system reliability is nearly the same, as shown in Fig. 12.    Fig. 13 shows the reliability sensitivity of the system to the temperature of components within a given ambient temperature. Owing to the time-varying thermal fields, the temperature of the components is unknown and uncertain. The system reliability is simulated in this study under various component temperatures (308 K, 328 K and 348 K). As shown in Fig. 13, the simulation of system reliability under 328 K is proposed, and the data with 308 K and 348 K are proposed in Fig.  13. Based on the various temperatures of the components, the failure probabilities of the system are 0.942 (348 K), 0.868 (328 K) and 0.753 (308 K) at 19500 h.

Conclusion
This paper proposed a graphical modelled aggregation approach to estimate the reliability and predict the lifetime of two-channel LED drivers. With the basis of the PoF of all the components in the circuit system, the simulation curves of the lifetimes for components could be obtained. Bridging the gap between the component level and circuit system level, the circuit was mapped into a reliability model accord-ing to the operation process of the circuit and based on the modelling rules. Integrated with the UGF, the reliability and lifetime prediction of LED drivers can be obtained using a formalized algorithm. This study offers the following contributions: (i) the multiphase modelling of the circuit and the quantitative calculation are enhanced by the reliability model and the UGF; (ii) the gap is bridged between the component level and the circuit system level; and (iii) a universal numerical and emulation approach is provided for predicting the lifetime of LED drivers. The following conclusions are based on the above algorithm: (i) The reliability of the circuit gradually degrades as a function of time, and the failure rate increases sharply before 18000 h and then begins to flatten gradually.
(ii) The reliability with independent components will degrade more significantly, at nearly double the reliability of the dependent components at approximately 19500 h.
(iii) The capacitors and MOSFETs are identified as vital components of the circuit structure. Component is the most sensitive component.
The approach proposed in this study provides a universal approach, and it is applicable for all types of LED drivers that have the characteristics of multiphase and complex topological structures. However, a few open issues remain, such as the accuracy and precision of the fitting interval and the multiple states of components, which are to be studied in future work. For example, the failure analysis should confirm whether the failure modes used in the prediction approach are the same as the observed failure modes. The interconnection and multi-physical field coupling among various failure mechanisms are the main points of further research. Additionally, the effect of the operation process on practical circuits should be considered, such as the various internal and ambient temperatures, and oscillations on components remain challenging issues in the reliability prediction of LED drivers. Furthermore, the approach should also be extended to fault diagnosis by using the authentic data of the components.