THERE IS growing interest in predicting the future health

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1 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 1, MARCH Prognostics of Multilayer Ceramic Capacitors Via the Parameter Residuals Jianzhong Sun, Shunfeng Cheng, Student Member, IEEE, and Michael Pecht, Fellow, IEEE Abstract This paper presents a parameter residual-based method for predicting the remaining useful life (RUL) of multilayer ceramic capacitors (MLCCs) under temperature-humiditybias conditions. Three performance parameters in each MLCC were monitored: capacitance, dissipation factor, and insulation resistance. A kernel regression method was used to estimate the parameters values of interest. The residuals were generated by the difference between the estimation and the actual monitored value. Based on the features of the residual data, a linear state space model was adopted to describe the dynamics of the residuals. The future evolution of the residuals was predicted with uncertainty bounds in a Bayesian framework. The failure threshold in terms of the parameter residuals was investigated. Then, the RUL of each MLCC with uncertainty bounds was determined. By comparing the predicted results with the experimental results, it was demonstrated that the proposed prognostics approach can provide an estimation of the RUL of MLCCs. Index Terms Kernel regression, multilayer ceramic capacitors (MLCCs), parameter residual, prognostics, state space model. I. INTRODUCTION THERE IS growing interest in predicting the future health state of electronics to provide advance warning of failure and reduce risk [1]. Multilayer ceramic capacitors (MLCCs) have been widely used in electronic products to perform functions such as noise reduction (bypass), direct current blocking, filtering, timing, tuning, and energy storage. Changes in the performance of MLCCs may result in product instability and failure. Many failure mechanisms can be traced to underlying physical or chemical degradation processes. A degradation process may be linear or nonlinear. When it is possible to measure degradation, such measurements often provide valuable information about a product s health state. Defining the relationship between component failure and the amount of degradation makes it possible to use a degradation model along with monitored data to predict the failure time [2]. Compared to conventional reliability analysis approaches typically based on lifetime data, degradation analysis provides more information, Manuscript received February 6, 2011; revised May 27, 2011; accepted July 7, Date of publication July 22, 2011; date of current version March 7, This work was supported in part by the Center for Advanced Life Cycle Engineering Consortium (CALCE). The work of J. Sun was supported in part by the Chinese Scholarship Council and in part by CALCE. J. Sun is with the Nanjing University of Aeronautics and Astronautics, Nanjing , China ( sunjianzhong@nuaa.edu.cn). S. Cheng and M. Pecht are with the Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD USA ( chengsf@calce.umd.edu; pecht@calce.umd.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TDMR particularly in applications with few or no failures [3], [4]. Degradation analysis also makes it possible to assess the reliability of an individual product under actual operation conditions based on the measure of the degradation of the specific product. However, only in very few cases is it possible to measure the degradation of a product directly (e.g., tire wear [5]). The measures of product performance (e.g., temperature, resistance) are available in most situations where the degradation analysis can be carried out on the basis of performance parameters. The advantages of degradation analysis are compromised when the performance measurements are contaminated with noise or when there is no strong correlation between the actual degradation and performance measurements. Thus, in most degradation analysis, the challenge is to find variables that are highly related to the actual degradation state. For failure prognostics, another challenge is to develop an appropriate model to describe the dynamics of the specific variables against time. These challenges are investigated in this paper for prognostics of MLCCs in which the physical degradation of the components is difficult to characterize using nondestructive analysis techniques. Three electrical parameters capacitance (C), dissipation factor (DF), and insulation resistance (IR) were measured to assess the performance of the MLCCs. The performance parameters of MLCCs exhibit a variety of behavior during degradation [6], [7]. The degradation cannot be monitored directly, but failure may be determined by analyzing the performance parameters or the features extracted from the monitored performance parameters. For example, a study by J. Gu et al. [7] found that performance parameter residuals can be used to provide an advance warning of failure of MLCCs in THB tests. In that study, a simple polynomial regression model was first trained based on training data and then used to calculate the residuals. A healthy residual space was established based on the training data set. When a new residual value was out of the lower 95% bound, it was considered to be an advance warning of failure; if the residual was also outside of the lower 99.9% bound, it was defined as an estimated failure. A univariate cubic polynomial model was assumed for regression analysis based only on two of the three monitored parameters in [7]; the assumption on the model form in [7] imposed some constraints on the regression model and may not be appropriate, since no prior knowledge of the mathematical structure of the regression model is available. In this paper, a nonparametric regression method is adopted to calculate the residuals based on all three monitored performance parameters. In the experiment analysis in [7], the failure times were determined strictly according to the failure criterion /$ IEEE

2 50 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 1, MARCH 2012 TABLE I TEST MATRIX AND NUMBER OF FAILURES Sudden changes in the performance parameters of MLCCs that are very close to, but do not exceed, the failure threshold may also result in severe instability of the product, thus decreasing the reliability of a system. However, the failure criteria adopted in [6] cannot handle such a situation. This paper introduces a failure prognostics methodology for MLCCs based on the parameter residual generated by the difference between the measured capacitance and its estimation. This capacitance estimate was obtained using a nonparametric regression method based on the dissipation factor and insulation resistance. A linear state space model was designed to describe the dynamics of the residual. The soft failure times of the components were investigated, and a soft failure criterion in terms of the parameter residual was developed. Given a failure threshold in terms of the residual, we can predict the failure time of an MLCC. Fig. 1. Fig. 2. Normalized capacitance values for four training samples. Normalized dissipation factor values for four training samples. II. EXPERIMENT A total of ten MLCCs were selected for in situ monitoring and life testing in elevated temperature (85 C) and humidity (85% RH) conditions with 50 V bias voltage. During testing, three performance parameters the insulation resistance, capacitance, and dissipation factor were monitored in situ. The measurements of the electrical parameters of each capacitor were taken every 200 min, and the total THB test lasted for around 1240 h. The test matrix is listed in Table I. Fig. 3. Normalized insulation resistance values for four training samples. III. PARAMETER RESIDUAL CALCULATION The prognostics method proposed in this study was developed based on the parameter residuals. Data normalization was first performed for each parameter, and then a kernel regression method was introduced to calculate the parameter residual. A. Data Normalization Due to manufacturing variance, the performance parameters of each component may have had different initial values. Thus, some data preprocessing was applied to the raw performance data. First, the relative values of the three performance parameters were computed: ˆP i =(P i P 0 )/P 0 (1) where ˆP i is the relative value of the performance parameter P i, and P 0 is the initial value of the corresponding parameter. The performance data were then normalized, as in [7]. We selected four survived capacitors as the training samples. Their data were normal (i.e., no abnormal behavior was observed in the data). On the basis of the training data set, we cal- culated the mean and standard deviation for each performance parameter (IR, C, and DF) using μ = 1 n x i (2) n σ = n (x i μ) 2 /(n 1) (3) where μ and σ are the mean and standard deviation, and n is the number of the training samples. Then, data normalization was performed for each parameter of all the MLCCs using the following equations: NC i =(C i μ C )/σ C (4) NDF i =(DF i μ DF )/σ DF (5) NIR i =(IR i μ IR )/σ IR (6) where C i, DF i, and IR i are the relative values of the performance parameters for C, DF, and IR for the ith sample data point, respectively; and NC i, NDF i, and NIR i are the normalized C i, DF i, and IR i, respectively, as shown in Figs. 1 3.

3 SUN et al.: PROGNOSTICS OF MULTILAYER CERAMIC CAPACITORS VIA THE PARAMETER RESIDUALS 51 B. Residual Calculation After normalization, regression analysis was performed to obtain the relationships among the normalized parameters based on the training data set. Then, the parameter of interest was estimated based on the other parameters. In this paper, a nonparametric regression method was used to construct an empirical regression model. Compared with parametric methods, which are defined by sets of parameters and predefined functional relationships, the nonparametric method does not need to make any assumptions about the mathematical structure of the regression model [8]. The nonparametric method stores past data in memory and processes them when a new query is made. In statistics and empirical modeling, the process of estimating a parameter s value by calculating a weighted average of historical training sample values is known as kernel regression [9], [10]. Rather than modeling a whole input space with a parametric model, kernel regression techniques are used to construct a local model in the immediate region of a query. When a query is made, the algorithm locates training samples in its vicinity and performs a weighted regression with the corresponding responses of the located training samples. The responses are weighted with respect to their proximity to the query point. Consider the simplest inferential model (i.e., one predictor x, one response y): n ŷ(x) = {K h [d i (X i x)] Y i } n K (7) h [d i (X i x)] where X i and Y i are the predictor and response of the ith training sample, respectively; K h [d i (X i x)] is the kernel function, which generates a weight for a given distance between a training sample and a query; and ŷ(x) is an estimate of y, given x. The nonparametric model operates by comparing a query (x) to the predictor part of each training sample (X) to determine the weight. The output is the weighted average of the response of the training sample (Y ). The kernel function generates a weight for a given distance. Therefore, it should have large values for small distances and small values for large distances. Many functions can satisfy this criterion; the Gaussian kernel is a commonly used one [11]: K h (d) = 1 2πh 2 exp( d2 /2h 2 ). (8) Here, h is commonly referred to as the kernel s bandwidth and is used to control the effective distances that are deemed similar. As for the distance function, a common one is Euclidean distance: d i (X i x) = (X i x) 2. (9) To develop the nonparametric model, the training samples used were divided into sample predictor X and response Y : X 1,1 X 1,2 X 1,p X 2,1 X 2,2 X 2,p X = X n,1 X n,2 X n,p, Y = Y 1 Y 2.. Y n. Fig. 4. Process diagram for calculation of parameter residuals using kernel regression method. Here, each row, X i, is the predictor vector of the ith training sample, and Y i is the response of the ith training sample. These training samples represent the memory of the nonparametric regression model. Using this format, a query vector is represented by the 1 p vector, x =[x 1,x 2,...,x p ]. For a query predictor, the corresponding response ŷ can be obtained using the kernel regression method in three steps, as shown in Fig. 4. 1) Compute the distance between a query vector x and each predictor of memory X: d i (X i,x)= (X i,1 x 1 ) 2 +(X i,2 x 2 ) (X i,p x p ) 2. (10) This calculation is repeated for each of the n memory vectors, resulting in an n 1 vector of distances, d = [d 1,d 2,...,d n ] T. 2) Compute the weights using the Gaussian kernel function according to (8). The weight vector is obtained as: w =[w 1,w 2,...,w n ] T. (11) 3) Finally, the weight vector is combined with the response vector of the memory to make predictions using the weighted average: n / n ŷ = w i Y i w i. (12) In this paper, the normalized parameters from four training samples (MLCCs No. 1, 2, 4, and 5) were used to construct the memory for the nonparametric model. The normalized capacitance is the response of the training sample with the other two parameters as the predictor vector of the training sample. After the memory matrix was built, when new measurements arrived, they were first processed according to the data processing procedure introduced in Section III-A. The estimated capacitance was then calculated based on the other parameters according to steps (1) (3) described above. The residual was the difference between the normalized real measurements of capacitance and the estimated capacitance. Two residual sequences from MLCC No. 4 (survived) and MLCC No. 6 (failed at the 962nd h)

4 52 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 1, MARCH 2012 state space model to describe the dynamics of the degradation. We assumed that there was some estimation error for the capacitance, and the error was finally propagated into the residual error. The residual error also came from the sensor noise from the capacitance measurement. Hence, the residual error was the sum of the capacitance estimation error (which may be divided into regression model error and input error) and the error from the capacitance measured value. It is not easy to analyze all the residual error sources. Thus, in our approach, we assume that the residual error follows a Gaussian distribution with unknown variance to be estimated based on real data in a Bayesian framework. The linear state space degradation model is as follows: Fig. 5. Residual sequences for MLCC No. 4 and MLCC No. 6. MLCC No. 4 survived during the experiment, while MLCC No. 6 failed at the 962nd h according to the failure criteria. The decrease in the residual after the failure was due to the abnormal behavior of the performance parameters after failure. are shown in Fig. 5. It can be observed that as an MLCC approached the end of its lifetime, the residual increased. The residual sequences from other components also showed similar trends. IV. FAILURE PROGNOSTICS Based on the analysis of the experimental data, it was determined that the parameter residual would increase as the capacitor degrades. The parameter residual can be defined as the indirect measurement of the unobservable degradation state of the components. Then, by defining failure in terms of the parameter residuals, the failure time of the component can be predicted when it first exceeds the threshold. The failure threshold in terms of the residual can be determined based on the runto-failure data; this will be discussed in detail along with a case study in Section V. In this section, a linear growth model is introduced to describe the degradation of a component in terms of residuals and demonstrate how to predict the component s future degradation in a Bayesian framework. A. Linear Degradation Model The residuals for four failed samples (MLCCs No. 3, 6, 7, and 8) are plotted in Fig. 6. It can be observed that the parameter residual first evolved like random walking (without an evident trend), which may mean that there was little or no degradation in the early stages of its lifetime. This was followed by a rough linear growth trend with different rates as the end of life of each component approached. Many components or products with linear degradation trends in experiments have been reported in the reliability analysis literature. Suzuki, Maki, and Yokogawa [12] used linear degradation models to study the increase in resistance measurements over time. Linear degradation models were also used to study the laser degradation problem in [2], [13]. Researchers have reported using a linear model to describe the degradation of light-emitting diodes [14], semiconductor devices [15], and tire wear [5]. In this paper, a linear state space degradation model was adopted. Errors in the residual sequence due to sensor noise and data processing were also taken into account in the linear y t = μ t + v t v t N(0,V) (13) μ t = μ t 1 + β t 1 + w μ,t w μ,t N(0,W μ ) (14) β t = β t 1 + w β,t w β,t N(0,W β ) (15) where μ t is interpreted as the unobservable degradation state and β is the degradation rate. The latent degradation state was indirectly measured through the noisy parameter residual y t.to estimate the degradation rate, some dynamics were introduced into β through w β,t. This was a special case of the linear state space model with time-varying degradation rate β t for the dynamics of the degradation level μ t. Equation (13) is the observation equation, while (14) and (15) constitute the state equation. Here, we assumed that the degradation process had Gaussian noise, i.e., w μ,t N(0,W μ ) and w β,t N(0,W β ). The observations were also assumed to be contaminated with Gaussian noise (v t N(0,V)). In this model, the state vector was X t =(μ t,β t ) T, where μ t was interpreted as the level of actual degradation and β t as the degradation growth rate. The model assumed that the current level changed linearly over time and that the growth rate also evolved. It was thus more flexible than a global linear trend model and more suitable for describing the evolution of noisy residuals in our case. At the beginning, there was no evident trend. As the end of life approached, the growth rate increased to a certain level, and the component deteriorated at a constant rate. The model can be written more formally as a state space model [16]: { Yt = F t X t + v t v t N(0,V) (16) X t = G t X t 1 + w t w t N(0,W) [ μt ], G t = [ ] 1 1, F 0 1 t =[10], W = where X t = [ ] β t Wμ 0. 0 W β As the end of lifetime approached, there was a linear trend. Due to its flexibility, the linear state space model [(16)] is suitable to describe such an evolution. The state space model can adapt itself to the real data sequence with an adaptive rate parameter in the model (the degradation rate in our case). Thus, although the evolution of the residuals in Fig. 6 does not appear to be linear over the whole lifetime, the linear state space model can still track the residual evolution correctly.

5 SUN et al.: PROGNOSTICS OF MULTILAYER CERAMIC CAPACITORS VIA THE PARAMETER RESIDUALS 53 Fig. 6. Residual sequences for four failed samples. MLCC No. 8 experienced an intermittent failure at the 317th h and recovered after about 20 h. Vertical lines indicate the soft failure times. B. Degradation Estimation and Prediction Given the state space degradation model and the noisy observation of the degradation state, we obtained the optimal estimation of the actual degradation state and made further predictions regarding future degradation in a Bayesian framework. For a well-defined Gaussian linear state space model, there was a closed-form solution for state estimation and prediction. Using standard results from the multivariate Gaussian distribution, it was easily proven that the estimated and predictive state distributions were Gaussian, and it sufficed to compute their means and variances [16]. To facilitate the understanding of the following Bayesian conjugate inference and prediction, we here use the symbols closer to the traditional ones used in Bayesian inference. Let x t 1 y 1:t 1 N(m t 1,C t 1 ); then the one-step-ahead prediction is given by: where x t y 1:t 1 N(a t,r t ) (17) a t = E(x t y 1:t 1 )=G t m t 1 (18) R t =Var(x t y 1:t 1 )=G t C t 1 G T t + W t. (19) Moreover, the state estimation is given by: where x t y 1:t N(m t,c t ) (20) m t = E(x t y 1:t )=a t + R t Ft T Q 1 t e t (21) C t =Var(x t y 1:t )=R t R t Ft T Q 1 t F t R t (22) where e t = y t a t is the prediction error and Q t = F t R t F T t + V t. Equations (17) and (20) present a basic state estimation mechanism for the model in (16), where all the parameters are known. However, in practice only, rarely are the covariance matrices W t and V t completely known. In this paper, the process noise and observation noise were assumed to be Gaussian in form, but the values of the variances of the Gaussian distributions were unknown. Thus, a basic problem was estimating W t and V t jointly with the latent state. One typical method is maximum likelihood estimation. In a Bayesian framework, one alternative is the Bayesian inference, where the unknown parameters are regarded as random variables. Usually, there is no closed-form solution to these problems, and we have to turn to Monte Carlo simulation-based approaches for more complex state space models with unknown parameters, which is still an active research area. For some simple cases, such as the Gaussian linear case in this study, Bayesian inference can still be carried out in a closed form using conjugate priors. Further information can be found in [16], [17]. Suppose that at time t, we have obtained the state estimation results given by p(μ t y 1:t )=N(m t,c t ), p(β t y 1:t )= N(β,V β ), respectively, for degradation level and degradation rate. Then k-steps-ahead predictive degradation is still a Gaussian distribution, x t+k N(a t+k,r t+k ), where the mean and variance are given by (24) and (25) below. For k 1: a t+k = G t+k a t+k 1 R t+k = G t+k R t+k 1 G t+k + S (23) [ ] [ ] where a t =[m t,β] T Ct 0 W + Vβ 0, R t =, S = Substituting all known parameters, we get the mean and the variance of the predictive distribution. In this case: a t+k = m t + k β (24) R t+k = C t + k (W + V β ). (25) The variance of the predictive distribution R t+k represents the prediction uncertainty for the k-steps-ahead prediction. On the basis of the residual sequence, the degradation level and the degradation rate were estimated in a Bayesian framework. Given the estimation results, the degradation level, and the degradation rate in distribution form, degradation prediction

6 54 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 1, MARCH 2012 was carried out using (23). Once new measurements arrived, the state estimation and prediction were updated recursively. Due to the statistical nature of the Bayesian method, the degradation prediction was given in a distribution form by (23). Since the failure was defined in terms of a specified level of degradation, the failure time distribution was deduced based on the predictive distribution of the degradation state. We assumed that failure occurred when the predicted degradation level, x t, reached the predefined failure threshold x F for the first time, i.e., x t x F.LetF T (t) denote the cumulative failure time distribution (or unreliability) at time t. Then, for an upper-bounded monotonic degradation process, the cumulative distribution function (CDF) for failure time can be expressed as follows: F T (t) =Pr[T t] =Pr[x t x F ]= + x F f(x t ) dx t (26) where T is the lifetime of a component, f(x t ) is the probability density distribution of the residual at time t, and x F is the failure threshold in terms of residuals. It has to be pointed out that there is very little error between Pr[T t] and Pr[x t x F ] due to the Gaussian process noise assumption. However, in the degradation stage of interest to us, the error can be reduced so as to become negligible, and (26) can be considered valid. Here, we use ˆF T (t) to denote the estimation of F T (t): On the basis of the failure time CDF, we can calculate other reliability indices of interest, such as the remaining useful life (RUL), which may be useful for maintenance or replacement decision making. Given the cumulative failure time distribution, any quantile lifetime of the conponent can be calculated according to: ˆF T (t) =F T (t) =Pr[x t x F ]= ˆF (T α )= + + x F f(x t ) dx (27) x F f (x Tα ) dx = α (28) T α is interpreted as the α quantile of the failure time distribution (e.g., the time T 0.10 by which the component will fail with a possibility of 10%). In some applications, we may be more interested in the conditional reliability of the component or system, given the fact that the component or system has survived until the last inspection time t c. For example, the conditional failure probability is given as follows: F T (t t c )=Pr[T t T >t c ] = Pr[t c <T t] Pr[T >t c ] = F T (t) F T (t c ) 1 F T (t c ) (29) where F T (t) is given by (26). Further conditional RUL or quantile lifetime can be obtained based on (28) by substituting ˆF (T α ) with ˆF (T α t c ). Fig. 7. Evolution of normalized IR of MLCC No. 7. The IR suddenly dropped to the failure threshold at about the 856th h and was followed by an intermittent failure. Fig. 8. Evolution of NIR of MLCC No. 3. The IR suddenly dropped to the failure threshold at about the 727th h, but it was not declared a failure according to the failure criterion until the 856th h. V. R ESULTS AND DISCUSSION The proposed prognostics method was demonstrated on the failed samples. The soft failure time of the failed samples was investigated first, and then a soft failure threshold in terms of the parameter residuals was determined based on the failed samples. Failure prediction was carried out based on the residual data using the method described in Section IV. A. Failure Threshold There were hard and soft failure criteria adopted in [7]. The hard failure criteria were defined as an IR drop to a value of less than 10 7 Ohms, a change in capacitance of 10%, or a doubling of the dissipation factor maintained for five consecutive readings during the THB test. The failure time determined based on these criteria used in [7] was called the hard failure time in this study. It was observed that there was a sudden and considerable drop in the IR parameter before a hard failure for each of the failed components. For example, MLCC No. 7 (shown in Fig. 7) failed right after the sudden drop, while the others were not declared to have failed after the sudden drop (MLCCs No. 3 and No. 6, shown in Figs. 8 and 9, respectively) according the hard failure criterion. Sudden drops can result in severe instability of the product, thus decreasing the reliability of the system. This sudden drop point in time can be defined as the soft failure time

7 SUN et al.: PROGNOSTICS OF MULTILAYER CERAMIC CAPACITORS VIA THE PARAMETER RESIDUALS 55 Fig. 9. Evolution of NIR of MLCC No. 6. The IR suddenly dropped to the failure threshold at about the 839th h, but it was not declared a failure according to the failure criterion until the 962nd h. TABLE II HARD FAILURE TIMES AND SOFT FAILURE TIMES Fig. 10. Prediction of the residuals for MLCC No. 6 based on the linear growth model in a Bayesian framework. Prediction started at the 671st h (80% lifetime of MLCC No. 6). TABLE III FAILURE PROGNOSTICS RESULTS FOR MLCCS NO.3,6,AND 7 of four MLCCs. Three of them failed at different points in time (indicated by the vertical dotted lines in Fig. 6) during the test. MLCC #8 had a weak intermittent failure but soon recovered. There were no advance indications in the performance parameters or the residuals for this MLCC; thus, it was hard to predict an intermittent failure based on the performance parameters monitored. Hence, it was excluded from the testing samples. Based on the analysis, the failure threshold in this study was determined by the run-to-failure data, and the soft failure threshold with regard to the parameter residual was set at level 2. This was supported by the fact that three run-to-failure samples failed near this threshold (horizontal solid red lines in Fig. 10), and none of the healthy samples passed through this threshold until the end of the experiment. of the component. The hard and soft failure times for the failed components are compared in Table II. From Table II, it can be seen that the soft failure times were earlier than or at least equal to the hard failure time. It is difficult to determine the cause of the sudden drop due to the architectural complexity of MLCCs and insufficient knowledge of the physics-of-failure of MLCCs. It is also difficult to predict the hard failure time due to the sudden change in performance parameters near the end of life. However, it is possible to predict the soft failure time based on the soft failure threshold and the trends shown in the parameter residuals before the sudden drops. Fig. 6 shows the residual evolution trajectories B. Prognostic Results We demonstrated the proposed method on three run-tofailure components (MLCCs No. 3, 6, and 7). MLCC No. 10 failed at a very early stage of its lifetime. MLCC No. 8 experienced a weak intermittent failure at an early stage and then recovered soon. The mean times to failure with 95% confidence limits (T and T ) were predicted at different lifetime stages and are summarized in Table III. The actual times to soft failure of the components were determined based on the experimental data analysis given in Table II. Pre. Time is the time instance (%Lifetime) when the prediction was launched.

8 56 IEEE TRANSACTIONS ON DEVICE AND MATERIALS RELIABILITY, VOL. 12, NO. 1, MARCH 2012 Fig. 11. Failure time prediction launched at the 671st h (80% lifetime ) for MLCC No. 6. The mean times to failure with 95% confidence limits (T 0.025, T 0.5,andT ) are indicated by the circles in the plot. To give a comparison of the prediction results for components with different life spans, predictions were made at the same stage of the normalized lives (%Lifetime) of different components. The normalized life was computed according to the following: Current Time Normalized Life(%Lifetime) = 100%. (30) Lifetime It can be seen from the failure prognostics results listed in Table III that when the end of life approaches, the predicted mean failure time becomes closer to the actual failure time. In fact, the prediction uncertainty decreases at the same time. To illustrate the proposed method, suppose that one prediction is launched at the 671st h (in fact, at 80% lifetime) for one component, e.g., MLCC No. 6. As shown in Fig. 10, before the prediction starting point (the 671st h), the measured residuals were calculated based on the monitoring data using the kernal regression method introduced in Section III. Bayesian conjugate inference was used in this study to estimate the degradation rate as well as the variance of the noise based on the measured residuals. Then, Bayesian prediction was carried out to predict the future residuals from the starting point (the 671st h). The predicted residual evolution in a distribution form is represented by the mean (the blue dashed line) and with uncertainty limits (95% confidence limits represented by grey dotted lines) in Fig. 10. As stated previously, the predictive distribution of the residuals is a Gaussian distribution due to the linear state space model assumption. Two predictive distributions of the residuals, at the 760th h and the 890th h, are plotted in Fig. 10. Fig. 11 shows the predicted failure time distribution for MLCC No. 6. The prediction was launched at the 671st h (80% lifetime) based on all the up-to-date monitoring data. As the new measurements arrive, the predicted failure time will be updated. Generally, the prognostics results became more accurate as the end of life approached. C. Discussion Residual-based failure prognostics is dependent on the evolution pattern of the calculated residual, so it is important to establish an appropriate capacitance reconstruction model. Rather than the traditional auto-associative approach, which is a common choice for anomaly detection [8], a different reconstruction model was used for the prognostics prediction task in this paper. The capacitance reconstruction model was trained based on healthy samples (the four survived capacitors in the test). In fact, all of the components experienced normal aging in service. In our experiment, the samples that survived until the end of the test were thought of as healthy, and the others were considered unhealthy. In this way, although some aging data may have been included in the training data, the advantage was that we could also remove some of the effects from the normal aging of the components from the training data set. In practice, we could select some survived samples until some point in time (e.g., the required lifetime or test time) as the healthy ones and the others as unhealthy. On the other hand, in contrast to the traditional residual calculation method, where an autoassociative approach is often used to reconstruct the parameters of interest, in our study the capacitance was reconstructed based only on two other parameters IR and DF rather than on all the parameters, since high correlation was shown among these parameters. A more evident trend in the residuals is preferred for prognostic prediction. If the traditional auto-associative model is used, in which the measured capacitance is also used to get its estimation, the calculated residual will be masked more or less, and thus no evident trend will be available. In this situation, the measured capacitance is excluded from the input set of the reconstruction model. A linear state space model is used to describe the dynamics of the residual evolution, which is considered to be an indicator of the health state of the capacitor in this study. Instead of predicting the failure of a component directly based on the monitored performance parameters, we first predict the degradation indicator of the capacitor, and when the degradation reaches the threshold, the component is declared to have failed. It is natural since many component failures can be traced to an underlying physical or chemical degradation process. However, the degradation process typically cannot be directly measured or monitored, but may manifest itself through the features extracted from the monitored performance parameters. In Table III, we provide the 95% confidence limits, T and T 0.975, for the predicted failure time distribution. It can be seen from the tabled results that as the end of lifetime approaches, the method can give a relatively accurate and precise prediction of the failure time. T can give a conservative and safe lifetime prediction. For example, for MLCC No. 3, the failure time prediction error is considerably large, even at 95% lifetime. This can be attributed to the complex residual evolution trajectory. Even as the end of lifetime approaches, it does not show an evident linear trend. This means that the parameter residual has poor trendabilty on this sample, so the prediction result is not good enough. Selection of a more approparate prognositcs parameter or features extracted from the available performance parameters may result in a more accurate RUL prediction. Metrics for assessing the suitability of the candidate prognostics parameters or features are introduced in [18]. Soft failure criteria and times of MLCCs were investigated in this paper. In our experiment, a sudden and considerable drop in insulation resistance was first observed for each of the five failed samples before they were observed to have failed.

9 SUN et al.: PROGNOSTICS OF MULTILAYER CERAMIC CAPACITORS VIA THE PARAMETER RESIDUALS 57 In practice, such a sudden drop may have severe consequences for a highly reliable product. This drop in insulation resistance was taken as a soft failure in this study. However, this type of soft failure cannot be detected in the current widely used failure criterion. A soft failure criterion in terms of the parameter residual was presented in this paper. One advantage of our method is that the parameter residual can serve as a real-time monitor of the health state of MLCCs because it reveals the degradation trend of the component over time, which can be used for RUL prediction. VI. CONCLUSION A novel failure prognostics method for RUL prediction for MLCCs based on parameter residuals was developed and demonstrated. The parameter residuals were generated from the difference between the real measured capacitance and its estimation calculated by a nonparametric regression model based on three in situ monitored parameters. Our study showed that the parameter residuals increased as the capacitors reached their end of life. Furthermore, there was a general linear trend in the last stage of life that could be utilized for failure prognostics. The proposed prediction method can produce prognostics results with uncertainty quantified by the time to failure distribution. Other reliability indices can be deduced based on this distribution. As more monitoring data becomes available, the prognostics results can be updated in an online fashion and will become more accurate as the end of lifetime approaches. This would allow condition-based maintenance or replacement, thereby increasing product safety and availability. The resulting reductions in unplanned downtime could produce substantial savings in operational costs. ACKNOWLEDGMENT The authors would like to thank Dr. D. Das and the Center for Advance Life Cycle Engineering, University of Maryland, for their support of this research. REFERENCES [1] M. G. Pecht, Prognostics and Health Management of Electronics. New York: Wiley-Interscience, [2] W. Q. Meeker and L. A. Escobar, Statistical Method for Reliability Data. Hoboken, NJ: Wiley, [3] C. J. Lu and W. Q. Meeker, Using degradation measures to estimate a time-to-failure distribution, Technometrics, vol. 35, no. 2, pp , May [4] S. J. Bae, W. Kuo, and P. H. Kvam, Degradation models and implied lifetime distributions, Reliab. Eng. Syst. Saf.,vol.92,no.5,pp , May [5] V. Bagdonavicius, A. Bikelis, and V. Kazakevičius, Statistical analysis of linear degradation and failure time data with multiple failure modes, Lifetime Data Anal., vol. 10, no. 1, pp , Mar [6] L. Nie, M. H. Azarian, M. Keimasi, and M. 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Annu. Conf. Prognostics Health Manag. Soc., San Diego, CA, Oct Jianzhong Sun received the B.S. degree in electromechanical engineering from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2006, where he is currently working toward the Ph.D. degree in traffic information engineering and control. His research interests are data-driven-based and physics-of-failure-based prognostics methods, and aircraft system in-service reliability prediction. Shunfeng Cheng (S 08) received the B.S. and M.S. degrees in mechanical engineering from Huazhong University of Science and Technology, Wuhan, China. He is currently working toward the Ph.D. degree at the Center for Advanced Life Cycle Engineering, University of Maryland, College Park. His current research includes sensor systems for prognostics and health monitoring and datadriven-based and physics-of-failure-based prognostics methods for electrical devices. Michael Pecht (M 83 SM 90 F 92) received the M.S. degree in electrical engineering, and the M.S. and Ph.D. degrees in engineering mechanics from the University of Wisconsin-Madison, Madison. He is the Founder of the Center for Advanced Life Cycle Engineering, University of Maryland, College Park, which is funded by over 150 of the world s leading electronics companies. He is also a Chair Professor of mechanical engineering and a Professor of applied mathematics with the University of Maryland. He has written more than 20 books on electronic product development, use, and supply chain management and over 400 technical articles. He has consulted for over 100 major international electronics companies, providing expertise in strategic planning, design, test, prognostics, IP, and risk assessment of electronic products and systems. Prof. Pecht is a Professional Engineer, an ASME Fellow, an IMAPS Fellow, and an SAE Fellow. He has served as Chief Editor of the IEEE TRANSACTIONS ON RELIABILITY for 8 years and on the Advisory Board of IEEE SPECTRUM. He is Chief Editor for Microelectronics Reliability and an Associate Editor for the IEEE TRANSACTIONS ON COMPONENTS AND PACKAGING TECHNOLOGY. He was awarded the highest reliability honor, the IEEE Reliability Society s Lifetime Achievement Award in He has previously received the European Micro- and Nanoreliability Award for outstanding contributions to reliability research, the 3M Research Award for electronics packaging, and the IMAPS William D. Ashman Memorial Achievement Award for his contributions in electronics reliability analysis.