1. Introduction

Switch-mode power supplies (SMPS) are an integral part of modern electronic systems, known for their efficiency and providing stable power supply across a variety of output levels using high-speed switching of transistors to regulate voltage and current. This switching action introduces electromagnetic interference (EMI) that is becoming more significant with high frequency operation [1]. This has created the need for input EMI filters to protect both the power line and the switch mode power supply. This EMI arises conductive, radiated, and near-field coupling of the created by resonant peaks in the frequency response, as well as switching transients caused by the closed-loop duty cycle corrections. In Figure 1, an LC-type filter is shown in-series with a buck-boost SMPS.

EMI-filtered closed loop SMPS with constant power and controlled output voltage or current are susceptible to a condition called negative input resistance, where increases in input voltage lead to decreases in input current. This creates oscillations that can destabilize the system, dampened out by the input filter capacitors’ equivalent series resistance (ESR). Aging or variability in electrical parameters can cause a cross-over of the SMPS input impedance and EMI-filter output impedance suddenly under certain operating conditions [2]. It has been found that many SMPS devices currently being utilized in industry, due to their nonlinear behavior, would fail to meet EMC standards without extensive up-stream filtering [3]. The Middlebrook criteria dictate that for a cascaded system, like that of an EMI-filtered SMPS, stability is maintained if the output of an upstream subsystem is less than the input impedance of the downstream subsystem, preventing oscillations and preserving electrical integrity. Typically, this cross-over occurs near the resonance frequency of the LC components of the filter, where the output impedance peaks, but at higher frequencies, with increased effects of the higher dv/dt and di/dt values, predicting the cross-over frequency range that leads to increased EMI using model-based becomes intractable.

 The primary research objectives are to develop and validate parametric models for SMPS that integrate harmonic analysis and consider the degradation of both input and output filter capacitors. These models aim to precisely simulate the effects of aging and environmental variation on capacitor behavior, thereby enhancing our understanding of how such factors influence degradation. Utilizing this data, the study seeks to improve prognostic capabilities for SMPS by predicting potential failure points and frequency vulnerabilities. The effectiveness of these prognostic models will be assessed through a comparative analysis with existing predictive maintenance strategies. This analysis aims to demonstrate early detection and preemptive management of failures by identifying frequency ranges with the largest variation tied to the system degradation.

1.1. Literature Review

1.1.1. Methods of Assessing and Modeling Degradation and Reliability of EMI filtered SMPS

There has been a large amount of research into the aging effects on SMPS components and their effects on the overall frequency response of the converter. It was found that the degradation trajectory of the rise and fall time of a MOSFET switch followed random variation as a result of induced thermal aging [1, 2]. The main sources of electromagnetic emissions were the power MOSFETs, and leakage inductance from the main transformer [3, 4]. The output diodes and output inductor can also be considered as emission sources; however, they provide a more secondary contribution. Sensitivities created from passives parasitics in the printed circuit board and sub-components that require targeted identification to systematically mitigate potential causes of noise.

Several studies have found the output filter capacitor was identified as a main source of failure in SMPS that causes the increased noise and critically impacts the performance of the converter, resulting in increased stress on the peripheral components [5], [6], [7], [8].

The studies have been conducted that provide in-depth insights into the degradation mechanisms of aluminum electrolytic capacitors under varying conditions, particularly highlighting their impact on the reliability of SMPS. A diagram of a typical Aluminum electrolytic capacitor can be found in Figure 2.

In [8], the author focused on the effects of thermal overstress noting that a reduction in electrolyte volume from evaporation directly decreases capacitance and increases ESR due to a shorter liquid path length. It showed that thermal overstress from storage beyond room temperature conditions significantly compromised their longevity and performance.

In [9], the author explained that models were created to address that over half of SMPS failures are attributed to output smoothing electrolytic capacitors and proposes new models to incorporate temperature variations to forecast degradation, influenced by time, core temperature, and operation frequency.

SMPS with EMI filters are increasingly analyzed with advanced mathematical modeling techniques and system optimization methods are often employed to understand system behavior comprehensively, leading to more effective design decisions. These traditional approaches can be extended from a holistic system perspective to optimize the overall system architecture [10]. These methods often require extensive modeling of each system component without necessarily focusing on reducing the experimental or simulation effort to only the most influential factors affecting system stability and performance.

In [11], the author proposed a genetic algorithm-based method for designing front-end rectifier inductors, taking into account the effects of DC link capacitors. However, the computational intensity of genetic algorithms poses limitations. These algorithms require significant computational resources and time, particularly as system complexity increases, which can limit their practicality in iterative design processes.

In continuous conduction mode, the switch remains closed long enough for the inductor current to never fall to zero, making the system’s behavior more predictable and easier to model using linear approximations [12]. In contrast, discontinuous conduction mode is characterized by the inductor current reaching zero within each switching cycle, which introduces non-linearities that are critical for understanding the SMPS performance under light load conditions [12], [13]. The integration of these modes into a unified model involves crafting state space analyses that capture the essential dynamics of the SMPS, including its susceptibility to noise and interference as influenced by the duty cycle variations.

These state space models elucidate the intricate output to input relationships within an EMI-filtered SMPS device, highlighting how variations in the duty cycle affect the overall stability and efficiency of the power supply. The transfer functions derived from these models are instrumental in predicting the system’s behavior in response to external disturbances and internal parameter changes.

To further refine these models and achieve a more generalized representation, generalized state space averaging (GSSA) methods are employed. GSSA is a technique that averages the various state equations over one switching period to produce a smoothed model, effectively managing the rapid switching characteristics of SMPS. This method simplifies the complex dynamics of switching power supplies into manageable forms, allowing for the integration of non-linear and dynamic aspects of the system into a coherent framework. By averaging the states, GSSA reduces the computational complexity and enhances the model’s ability to predict long-term behavior under a broad range of operating conditions.

1.1.2. Time and Frequency Stability Analysis Methods

Non-invasive online condition monitoring techniques for capacitors often utilize the spectral content of voltage and current, employing various approaches estimating ESR and capacitance with minimal error. While comprehensive, this analysis requires sweeping through a broad frequency range, which can be time-consuming and may not pinpoint the most critical frequencies causing instability in complex systems.

Additionally, AC analysis methods provide a robust framework for assessing the global stability of closed-loop systems by exploring the system’s behavior across a broad frequency spectrum. This analysis is critical as it helps identify resonance peaks, phase shifts, and gain changes, offering valuable insights into how the system responds to different frequency inputs and highlighting potential instabilities. Understanding gain and phase margins is pivotal in ensuring global system stability. Gain margin refers to the amount by which the gain of a system can increase before the system becomes unstable, while phase margin is the amount of additional phase lag required to bring the system to the brink of instability.

 In [14], the author used frequency and time-domain methods to assess conducted emissions, focusing on the signal characteristics of SMPS emissions, which include a mix of medium- and high-frequency components and significant spectrum leakage. The study found that there is a need to tune and optimize these analytical methods to achieve reliable results. The Prony method tracks changes in amplitude at specific frequencies using the least squares approach, but this method requires a complete understanding of the frequencies associated with degradation for a global population of units. It has been shown that by understanding the specific EMI profile to be mitigated, selective trade-offs can be made to reduce the filter footprint, weight, and cost while maintaining performance [3].

1.1.3. Prognostic and Health Management Techniques

Accurately anticipating and addressing these shifts requires analyzing the impact of component degradation on the system’s performance. The prognostic and health management (PHM) system optimizes maintenance processes based on diagnostic and prognostic outcomes to prevent failures and enhance lifecycle management [8][15]. For SMPS, capacitance was identified as a suitable failure precursor for system failure [8].

Manufacturers often recommend a methodical approach for estimating the useful life of aluminum electrolytic capacitors using datasheet specifications. The rated ripple current, I_{AC, R} at the capacitor’s maximum specified temperature is identified and the actual operating ripple current I_AC is used to calculate their quotient. This ratio and the ambient temperature are used to estimate the capacitor’s remaining life by interpolating a given life expectancy graph and accounting for frequency variations from the standard test condition frequency, usually 100 Hz. [16].

A data-driven fault detection algorithm was introduced specifically designed for identifying failures in multilayer ceramic capacitors [17]. The algorithm utilizes regression analysis, residual detection, and prediction analysis to enhance the accuracy and reliability of fault detection. A key component of their methodology is the use of Mahalanobis distance for anomaly detection in the test data.

In [18], the author used accelerated life testing for aluminum electrolytic capacitors to evaluate how conditions such as electrolyte leakage can affect capacitance, quality factor, and dissipation factor. The study utilized statistical time-domain feature extraction and correlation-based feature selection to accurately monitor capacitor health and predict failures.

In [19], the author proposed a method that leverages noninvasive condition monitoring via time-frequency analysis of conducted EMI to evaluate the health of DC-link capacitors in three-phase inverters. This method involves a combined EMI filter and measurement board placed on the DC bus, which not only filters conducted EMI to comply with MIL-STD-461 G but also facilitates EMI measurements for condition monitoring. A continuous wavelet transform is used to create characteristic switching images, which are then used to train support vector machine (SVM) models to classify the health of DC-link capacitors into one of five stages with high accuracy. This approach uses broad-spectrum analysis, which may include frequencies that are not always relevant to condition monitoring.

A PHM system was presented that is designed to preempt failures and enhance lifecycle management for insulated gate bipolar transistors [15]. This process is organized into three main stages: Observation, Analysis, and Action. The Observation stage involves monitoring and data processing, the Analysis stage includes health evaluation and future state forecasting, and the Action stage focuses on maintenance implementation based on assessments. To overcome challenges in detecting subtle degradation signals, Principal Component Analysis (PCA) is used for feature engineering to reveal hidden trends. These trends are then fed into a Deep Neural Network for classification, enhancing the system’s ability to detect and predict failures accurately.

A PHM framework was proposed to combine traditional model-based and data-driven approaches, utilizing extensive sensor data for remaining useful life predictions [11]. This approach is designed to analyze subtle time-series patterns in large datasets by treating sensor data as continuous random processes. These functional relationships can encapsulate a significant amount of variation information across different equipment in a compact, resamplable form. This capability to adapt to time-varying data makes the approach particularly useful for the EMI-filtered SMPS usage case.

Using multivariate functional relationships as predictors for state of health can alleviate big data concerns and can be further improved by sparsity-induced optimization methods, which learn multiple classification tasks while simultaneously performing feature selection. A method of multi-task feature learning, used for analyzing brain imaging data with varied functional data sets collectively, was developed to enhance predictability and accuracy [20]. This approach can be adapted to other domains, including the health monitoring of EMI-filtered SMPS, to improve the accuracy and efficiency of remaining useful life predictions.

1.1.4. Identifying and Addressing Research Gaps

Despite extensive research on degradation and failure mechanisms in SMPS, gaps remain in predicting system instabilities caused by frequency vulnerabilities. Traditional studies, which focus on component aging and electromagnetic emissions, often employ complex models that overlook critical frequency regions impacting system stability. Moreover, these models lack the interpretability necessary to identify how specific component degradations influence overall performance in frequency-sensitive scenarios.

To address these deficiencies, we propose an approach that utilizes multivariate functional analysis and multitask least absolute shrinkage and selection operator (LASSO) regression. This method transforms complex, high-dimensional datasets into manageable, infinite-dimensional functionals conducive to resampling. It significantly enhances model accuracy and robustness, allows for the learning of multiple classification problems simultaneously, and focuses specifically on identifying critical frequency vulnerabilities in SMPS.

By isolating specific EMI components and narrowing the analysis to essential frequencies, this approach not only reduces data complexity but also improves measurement precision and system stability. This targeted analysis streamlines research, boosts efficiency, and provides clearer insights into EMI behaviors, substantially enhancing system vulnerability assessments and predictive maintenance strategies.

1.2. Fuzzy Multi-Task Functional Fusion Predictors

This study aims to develop and validate an innovative prognostic framework that leverages discrete event simulation (DES) of the EMI-filtered closed-loop SMPS and degradation models of the EMI-filter input filter capacitor and SMPS output filter capacitor. The goal is to accurately predict the degradation and remaining useful life of aluminum electrolytic capacitors in electromagnetic interference filters of switch-mode power supplies, thereby significantly enhancing system reliability and performance.

The developed SMPS system prognostic approach, termed Fuzzy Multi-Task Functional Fusion Predictor (FMT-FFP), is an advanced predictive model that combines fuzzy logic with multitask LASSO regression and B-spline convolutional-integral-based cross-correlations, integrated through functional PCA for robust and precise state-of-health forecasting in complex systems, while preserving interpretability of the input features. For the given case study, the developed approach focuses on identifying the impact of critical frequency regions associated with the degradation trajectories of aluminum electrolytic filter capacitors within an EMI-filtered SMPS system.

Generalized state-space averaging models of the LC-filtered buck-boost SMPS are developed early on in the prototyping design phase to express the k-th order harmonic content and derive the output voltage to input voltage gain frequency response, the frequency response for the output voltage to duty cycle control transfer function, and the frequency response for the EMI filter output and SMPS input impedance.

Once frequency responses for each test are collected, they are extended using convolutional integrals for each combination. Functional Principal Component Analysis (FPCA) is then used to perform dimensionality reduction and feature engineering to restrict the functional data to the principal variational modes. Multitask LASSO regression is employed to make probabilistic State of Health (SoH) estimations while identifying a sparse set of features associated with frequency-response criteria and ranges. An outline of the prognostic approach can be seen in Figure 3 below.

This study introduces a refined modeling approach tailored to the unique operational characteristics of SMPS with EMI filters. The key contributions of our research are threefold. First, a generalized state-space averaging model was developed to effectively capture the dynamic interactions within the SMPS and the input EMI filter, facilitating a deeper understanding of system behavior across varying operational states and filter capacitor degradation. This model is particularly adept at identifying critical frequency regions that directly impact the stability and efficiency of SMPS. Finally, the application of these advanced modeling techniques has led to the establishment of an efficient testing scheme. This scheme enhances traditional maintenance strategies by providing precise diagnostic capabilities that can preemptively address potential failures, thereby significantly improving system reliability and operational lifespan. Through these contributions, our research offers substantial advancements in the predictive maintenance and reliability assessment of EMI-filtered SMPS, ensuring more stable and efficient power supply systems.

The remainder of the report outlines the prognostic approach. Section 2 lays the theoretical groundwork, detailing the derivation of the generalized state-space model for the buck-boost SMPS, the discrete event simulation methodology used to create the training data, and how the probabilistic SoH estimations are converted into remaining useful life estimates. This section also elucidates the technique for identifying frequency vulnerabilities using the results of the Multitask LASSO Regression. Section 3 presents a buck-boost SMPS test fixture, featuring a microcontroller with proportional integral derivative (PID) control and voltage-based current sensors for the input filter and SMPS inductor currents. These sensors use a high-impedance, time-constant matching network for accurate current measurement and are used to collect impedance information. Section 4 presents the research analysis of the results and discusses any wider implications. Finally, in Section 5, a summary of the research findings and conclusions is provided.

2. Theory

In this research, the intricacies of EMI filter and SMPS In this research, the intricacies of EMI filter and SMPS design are addressed utilizing the generalized state-space averaging (GSSA) model. This modeling approach decomposes the state signals using Fourier analysis to account for the harmonic content evolution related to the effects of the switching frequency harmonics and the degradation effects in aluminum electrolytic capacitors within the SMPS and input filters. The analysis employs a buck-boost DC-DC SMPS topology with a general input EMI LC-type low-pass filter, as illustrated in Figure 1 in Section 1.

Parasitic resistances and losses, including the input filter inductor’s parasitic resistance, the equivalent series resistance (ESR) of input and output capacitors, and the copper losses in the SMPS inductor, are assumed. Additionally, inputs associated with the input voltage, forward voltage drops across the switching transistor and diode, and load current perturbations are considered.

This comprehensive approach allows for a detailed understanding of how various parasitic elements and operational parameters influence the performance and stability of the SMPS, thereby providing a robust foundation for predictive maintenance and reliability assessment.

2.1. State Space Averaging

The development of a linearized model for a SMPS with an EMI filter employs Kirchhoff’s Voltage Law across circuit loops, focusing on continuous conduction mode (CCM) operation. In CCM, the system oscillates between states of active switching transistor and conducting diode. By averaging these states, the model smooths out non-linear transitions due to the switch’s high slew rate, simplifying the analysis. By performing the nodal analysis and solving for the state time derivatives, state space representation of the EMI filtered SMPS can be found below in (1).

where x, u, y are the state, input, and output vectors, respectively, consisting of the input and SMPS inductor current,  and , and input and output filter capacitor voltages,  and , and input voltage, , switch and diode forward bias voltages, and , and load current perturbations, .

2.2. Steady State

The steady state behavior is derived by setting the state derivative to zero and solving for the steady state response as shown in (2).

2.3. Linearization

Linearization is used to create a small-signal model that accounts for duty cycle perturbations and their interactions with state variables:

2.4. Proportional integral derivative Control Laws

The proportional integral derivative (PID) controller logic for the output voltage tracking objective can be below in (3).

The PID controller is formulated to manage output voltage deviations as follows in (4).

where  are the respective proportional, integral, and derivative gains, the voltage reference input,  , and the output voltage, . This differential form leverages linear relationships of state and input vectors to adjust the duty cycle dynamically.

2.5. Generalized State Space Averaging

Generalized state space averaging is applied to express the state as a sum of sinusoidal functions over one switching period, enhancing the model’s ability to capture the dynamic interactions within the SMPS and input EMI filter:

where  is the angular frequency in terms of switching period, T. From the nodal analysis of the LC filtered buck boost circuit depicted in Figure 1, the original state space had four-states associated with each inductor current and capacitor voltage.The x-coefficients are found from the real and imaginary component for each Fourier coefficient, <x>_k , represents the amplitude of the k-th harmonic frequency component.

Using product rule and the chain rule, the expression for the time derivative of the Fourier coefficient is seen below in (5).

The k-th order convolution coefficient is found as the sum of the product of the Fourier coefficients of the two signals noting the ordering of the average subscript combinations.

The negative k-th average harmonic relates the complex conjugate of the signal, seen below in (6).

2.6. Extracting Transfer Function Information

The use of analyzing AC frequency response features has long been used in insuring global stability of closed loop dynamic systems [2]. From linear systems theory, the complete family of output-to-input transfer function relationships can be found for a linear time-invariant multi-input multi-output dynamic system in state space representation can be found below in (7).

The output voltage to input gain,   and control, are used to provide information concerning the different gain and phase margins which are measures of system’s stability. The EMI filter output impedance,   and SMPS input impedance, are also used to evaluate the system’s dynamic stability relying on the Middlebrook criterion as a method to assess compatibility between the SMPS and the LC-input filter, taking into account the effects of negative incremental impedance on the constant power controller voltage buck-boost SMPS [2].

2.7. Aluminum Electrolytic Degradation Models

The longevity of aluminum electrolytic capacitors is chiefly compromised by electrolyte evaporation, a consequence of elevated operating temperatures and heat from ripple currents [8][9]. A thermal model, as shown in Figure 4, simplifies the system by considering the hotspot (T_HS) and case temperature (T_C) to be approximately equal. This model integrates ripple current, capacitor ESR, and case-to-ambient thermal resistance, offering a streamlined approach to evaluating capacitor thermal behavior.

Models have been developed to account for the environmental and temporal degradation-based effects on an aluminum liquid electrolytic capacitor’s capacitance and ESR [9]. These models are contained in (8) and (9) below.Models have been developed to account for the environmental and temporal degradation-based effects on an aluminum liquid electrolytic capacitor’s capacitance and ESR [9]. These models are contained in (8) and (9) below.

2.8. Discrete Event Simulation

A discrete event simulation (DES) was performed to model the degradation trajectories of aluminum electrolytic capacitors’ capacitance and ESR. The simulation initialized circuit attributes and parasitic elements with a 10% variation to mimic real-world deviations. Throughout the simulation, the duty cycle was dynamically adjusted based on PID control logic, targeting a 15V output from a 12V input overlaid with 10% white noise to represent input perturbations. An overview of the DES is depicted in the directed graph in Figure 5.

The lifecycle of three sample SMPS was simulated by analyzing startup transient responses in 2-microsecond increments over a span of 600 increments, achieving a steady state. At this steady state, the ripple current through each capacitor was calculated for every time response. This data was then applied to incrementally update the temporal degradation. The degradation cycle was advanced every 200 hours, continuing up to a maximum of 3000 hours or until a failure condition occurred. Failure was defined as an event where the output experienced an overshoot or ripple voltage exceeding 2 volts.

2.9. State of Health Prediction Methods

The following sections cover the derivation of the different prognostic approaches. The three methods focus on dimensionality reduction routines and extracting tacit information from time-series data.

2.9.1. Method 1: (PCA-DNN)

The first method involves reducing the high dimensional data using principal component analysis for feature engineering and a deep neural network for classification of the state of health (SoH) into probabilistic fuzzy estimations.

2.9.1.1. Feature Engineering with Principal Component Analysis

The features from the AC analysis create a high dimensional data set that captures the underlying behavior but are too complex for efficient analysis. Principal component analysis (PCA) is a statistical technique that reduces the dimension of the data while preserving most of the variation by identifying correlations within the data using a covariance matrix, seen below.

Eigen analysis is used to create synthetic variables that are linear combinations of the original features.

These variables can be truncated to only include the most significant details of the variation in the original data.

2.9.1.2. Neural Networks

The reduced-dimension PCA output data serves as an input to the first layer of a deep neural network (DNN). A neural network is composed of layers of neurons that take a linear combination of inputs, x, assigns a respective weight to each input, w, and a bias, b, and applies an activation function, .

A common activation function used for the hidden neurons is the ReLu function which adds non-linearities into the model to help learn complex patterns.

To create the binned state-of-health probabilities in the output, a SoftMax function is used which normalized exponentials to create a probability distribution.

The PCA/DNN pipeline for the AC analysis feature data can be seen in Figure 6.

2.9.2. Method 2: CWT-SVM

Continuous Wavelet Transforms (CWTs) are useful in analyzing time signals that exhibit non-stationary behavior. Such as state of health classification [19].  The time and frequency analysis capability of the CWTs make them ideal at analyzing state-of health effects from harmonics associated with the interactions between the output of the LC-type input EMI-filter and the input of the closed-loop SMPS.

2.9.2.1. Continuous Wavelet Transform

The CWT formula used in this report can be found below in (14).

where,  is the wavelet function, s adjusts the wavelet’s width and  translates the wavelet thereby affecting time resolution.

2.9.2.2. Support Vector Machine

The CWT is utilized to analyze the time-frequency characteristics of the system’s output voltage. This analysis generates a high-dimensional image array, capturing intricate time-scale variations within the signal. These image arrays serve as the input for a Support Vector Machine (SVM), a powerful machine learning technique used for classification tasks. SVM operates by identifying optimal boundary support vectors among the data points that represent different classes. It constructs a hyperplane that maximizes the margin, which is the distance between the nearest data point of each class and the hyperplane itself. This maximization is crucial as it contributes to the robustness of the classification against new data. The objective function of SVM is formulated to minimize:

where w_d​ is the normal to the hyperplane. SVM uses constraints to ensure that all data points correctly classify by maintaining a distance from the hyperplane:

where w_d​ is the normal to the hyperplane. SVM uses constraints to ensure that all data points correctly classify by maintaining a distance from the hyperplane:

To assist in creating separability in the data, a radial basis function is used, that measures the similarity between the data points.

Using the kernel trick changes the constraints to the following form:

where  are Lagrange multipliers, which are optimized during training.

A diagram of the CWT-SVM methodology can be found in Figure 7, illustrating how time series data from the output voltage is processed into CWT image arrays. These arrays are then used as inputs to the kernel SVM, which classifies the system’s state based on learned patterns from the training phase.

2.9.3. Method 3: Fuzzy Multi-Task Functional Fusion Predictors

The fuzzy multi-task fusion predictor uses b-spline functional curves to form versatile low dimensional representations of the AC frequency features created from the transfer function representations. The method creates fusion between the signals by extending the features using convolution integral-based cross-correlations and multi-task learning with LASSO regression to identify a sparse subset of test conditions conducive to increasing the performance of the prognostic.

2.9.3.1. B-Spline Resampling

B-spline interpolation resamples time series data to a standard temporal scale, utilizing piecewise polynomials defined over specific intervals as can be found below in (18).

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