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Conference Paper

NREL/CP-5400-71386

October 2018

A Review of Degradation Behavior and 

Modeling of Capacitors 

Preprint

Anunay Gupta,

1 Om Prakash Yadav,1 Douglas DeVoto,2

and Joshua Major2

1 North Dakota State University

2 National Renewable Energy Laboratory

Presented at the American Society of Mechanical Engineers (ASME) 

2018 International Technical Conference and Exhibition on Packaging and 

Integration of Electronic and Photonic Microsystems (InterPACK 2018)

San Francisco, California

August 27–30, 2018
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Conference Paper

NREL/CP-5400-71386

October 2018

A Review of Degradation Behavior and 

Modeling of Capacitors

Preprint

Anunay Gupta,

1 Om Prakash Yadav,1 Douglas DeVoto,2

and Joshua Major2

1 North Dakota State University

2 National Renewable Energy Laboratory

Suggested Citation

Anunay Gupta, Om Prakash Yadav, Douglas DeVoto, and Joshua Major. 2018. A Review of 

Degradation Behavior and Modeling of Capacitors: Preprint. Golden, CO: National Renewable 

Energy Laboratory. NREL/CP-5400-71386. https://www.nrel.gov/docs/fy19osti/71386.pdf.
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A REVIEW OF DEGRADATION BEHAVIOR AND MODELING OF CAPACITORS

Anunay Gupta

North Dakota State University

Fargo, ND, USA

Om Prakash Yadav

North Dakota State University

Fargo, ND, USA

Douglas DeVoto

National Renewable Energy Laboratory

Denver, CO, USA

Joshua Major

National Renewable Energy Laboratory

Denver, CO, USA

ABSTRACT

This paper firstly reviews the failure causes, modes and 

mechanisms of two major types of capacitors used in power 

electronic systems - metallized film capacitors and electrolytic 

capacitors. The degradation modeling related to these capacitors is 

then presented. Both physics-of-failure and data-driven 

degradation models for reliability and lifetime estimation are 

discussed. Based on the exhaustive literature review on degradation 

modeling of capacitors, we provide a critical assessment and future 

research directions.

1. INTRODUCTION

Capacitors in power electronics are used for a wide variety of 

applications, including energy storage, ripple voltage filtering, and DC 

voltage smoothing. The two major types of capacitors used in power 

electronic systems are aluminum electrolytic capacitors and metallized 

film capacitors. The state of health, or life, of these capacitors depends 

on stress factors like temperature, voltage, ripple current, chargedischarge, and humidity. Various degradation measures such as 

capacitance, equivalent series resistance, dissipation factor, and 

insulation resistance have been used to monitor the degradation state of 

capacitors. To capture the degradation behavior in a shorter time, 

several acceleration models are used to replicate the specific failure 

behavior. The acceleration models are usually based on the physics 

underlying a particular failure mechanism. Some common acceleration 

models used in capacitor lifetime prediction are the Arrhenius law for 

temperature, inverse power law for voltage and humidity stresses, and 

Eyring law for capturing the interaction of other stress factors with 

temperature stress. The major limitation of acceleration models is their

inability to track the capacitor degradation with time. That is, the 

acceleration models cannot predict the state, or health, of a capacitor at 

a particular time. Hence, there is a need for data-driven or statistical 

degradation models to ascertain the capacitor’s actual state of health at 

a given time and subsequently estimate their reliability and remaining 

useful life. Several efforts have been made to model degradation 

behavior of the capacitor considering either physics-of-failure models 

or statistical models and subsequently estimate its reliability and 

lifetime parameters. This paper reviews the behavior and the 

degradation modeling of these capacitors for reliability assessment and 

lifetime analysis.

2. FAILURE ANALYSIS

Capacitors could fail due to various factors like manufacturing and 

design defects, material wear out, operating temperature, voltage, 

current, humidity and mechanical stress. Wear-out failures signify the 

end of useful life of a product, and this section is mainly concerned with 

the wear-out failures in DC-link capacitors. Electrolyte evaporation is 

the primary wear-out mechanism in electrolytic capacitors and is caused 

by high temperatures within the capacitor core. In the case of metallized 

film capacitors, self-healing or localized dielectric breakdown due to 

overvoltage is the main wear-out mechanism. These failure mechanisms 

and their root causes, along with the other causes of capacitor failure are 

discussed in detail in the following section.

2.1. Metallized Film Capacitors

Metallized film capacitors consist of dielectric films with a metallic 

coating on the surface. The electrodes are composed of very thin layers 

(20 to 100 nm) of metal, usually aluminum or zinc, evaporated onto the 

surface of the polymer film. The most common dielectric films that are 

used in the industry are mainly polypropylene (PP) or polyethylene 

terephthalate (PET). In special applications, dielectrics such as 

polyethylene naphthalene or polyphenylene sulfide are used for hightemperature operating conditions (up to 150°C). PET film capacitors 

have higher capacitance per volume than PP film capacitors due to their

high dielectric constant and availability in thin gauges. They also have 

a better mechanical resistance and a slightly higher exploitation 

temperature than PP film capacitors, with operating temperatures up to 

125°C. The negative point is that the loss factor is ten times larger, 

which means a ten-fold increase in temperature elevation for the same 

rated power. PP film has superior electrical characteristics. PP film 

capacitors exhibit very little capacitance change over time with an 

applied voltage, making them ideal for applications where a stable level 

of capacitance is needed. The temperature and frequency dependencies 

of electrical parameters for PP film capacitors are very low. These 

capacitors may be operated up to 100°C. The dissipation factor of PP 

film capacitors is smaller than that of other film capacitors. Also, the PP

film material absorbs less moisture than PET film [1-3].

Metallized film capacitors have a unique self-healing ability. If there is 

a micro-void or defect in the dielectric film and the capacitor is 

connected to a voltage of a sufficient level (electric-field stress), a glow 

discharge occurs in the micro-void. This discharge generates heat due 

to a high current density in the breakdown region, which causes 

evaporation of the electrode layer around the void. The damaged part is 

disconnected from the capacitor and cannot degrade its performance. 
2

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This is accompanied by a slight decrease in the capacitance [4].

Metallized film capacitors have this significant advantage of selfhealing over general film capacitors or non-metallized film capacitors. 

In non-metallized film capacitors, when an overvoltage is applied and 

small localized dielectric breakdown occurs, the exposed electrodes 

make contact with each other and short, rendering the capacitor useless. 

In case of metallized films, the combination of the electrode foil

thinness and the high energy density in the fault area causes the foils to 

vaporize and the capacitor stays in operation and avoids catastrophic 

failure. This phenomenon is termed as “self-healing” [5]. The selfhealing or clearing process in metallized film capacitors depends on 

three factors, namely, the working voltage, mechanical pressure 

between the winding layers, and metallization thickness (resistivity) [6].

Accumulation of these self-healing events over time, causes a “soft” 

failure of the metallized film capacitor when the capacitance value 

decreases by a certain percentage of the initial capacitance. The 

definition of this soft failure (end-of-life criterion) varies from

manufacturer to manufacturer and between applications. With time, the 

pressure inside the capacitor casing increases due to production of 

vaporized metal by self-healing events. This can eventually cause a 

catastrophic failure either through the pressure, increasing enough to 

burst the capacitor casing, or the vaporized metal concentration 

increasing enough to become conductive and cause a flashover event [7, 

8]. Upon an increase in the working voltage, the self-healing time 

increases, and the heating of adjacent insulator layers becomes 

significant, which can be destructive for the structure of the capacitor

[9]. Li et al. [10] investigated the influence of temperature and voltage 

on the life of metallized film capacitors and the temperature dependence 

of this self-healing characteristic. It is found that a catastrophic failure 

is more likely to happen under elevated working temperature and 

voltages due to the decrease in breakdown electric field strength of the 

capacitor with a rise in temperature. During the self-healing process, 

there is an energy discharge of stored energy. This discharge of energy 

under the same breakdown voltage becomes larger under higher 

temperature conditions.

Temperature alone does not seem to play a major part in the degradation 

of metallized film capacitors. At elevated temperatures, the mechanical 

properties of dielectric films change and their volume increases due to 

thermal expansion. Hence, thermal stress occurs due to the difference 

between coefficients of the thermal expansion of the dielectric film and 

electrode metallization on the film, which can influence the structure of 

the metal film. Research work and experiments conducted in [11] point

out significant changes in the non-linearity of current-voltage 

characteristics of metallized PP film capacitors during accelerated 

thermal aging, as compared to negligible change in capacitance. These 

changes are proposed to be due to the deterioration in quality of the 

electrode metallization from temperature stresses. Apart from voltage 

and temperature, another important stress factor especially in case of 

DC link capacitors is the ripple current. By definition, a ripple current 

is an AC current through the capacitor due to incomplete suppression of 

the alternating waveform within the power supply. Root mean square 

(RMS) is a method of denoting the AC waveform as DC value that will 

produce the same heating effect, or power dissipation in the circuit. The 

ability of a capacitor to withstand a continuous RMS current depends 

on the frequency of the electrical stress and thermal power dissipation 

of the component. If the power generated by the capacitor body is 

greater than the power that can be dissipated by its surface, increased 

self-heating may occur. Self-heating increases the capacitor 

temperature, leading to a reduction of the breakdown voltage and, in the 

worst cases, even melting of the capacitor [1]. The self-heating 

temperature of a capacitor depends on equivalent series resistance 

(ESR), on the current across the capacitor, and on the thermal resistance 

between the case and the ambient temperature. The combination of high 

ripple current and voltage leads to electrochemical corrosion of the thin 

electrode metallization, gradually converting aluminum to aluminum 

oxide. The corrosion process involves the migration of oxygen and/or 

moisture of the polymer to the polymer/metallization interface. It is 

dependent on temperature, electrode thickness, stress, and frequency [2, 

12, 13]. Metallized film capacitors using PP as the dielectric are less 

affected by the ripple current than ones with a PET dielectric.

The effect of AC voltage and temperature on the degradation of 

metallized film capacitors is studied in [14] and a life model involving 

all the main stressing factors (peak voltage, RMS value, and 

temperature) is proposed, using multivariate regression. It is found that 

the peak voltage has the most significant effect on capacitor aging. 

Although thermal aging contributes to degradation, its significance is 

less than the degradation due to peak voltage and RMS voltage. The 

detachment of the “sprayed ends” or the schoopage from the capacitor 

roll is commonly found in pulsed power applications, and it normally 

results in an open-circuit failure [15]. The combined effect of electrical, 

thermal, and mechanical stresses gradually leads to the degradation of 

contact and eventually, the detachment of the sprayed ends. The high 

current peaks are mainly responsible for these effects, independent of 

the waveform energy content. As the contact is degraded, the ESR 

value, and consequently, the loss factor (tan δ) will greatly increase 

according to the number of pulses applied to the capacitor [16].

The degradation of metallized film capacitors is a concern in 

applications exposed to high humidity environments. The metallized 

electrode layer in metallized film capacitors is very thin, typically less 

than 50 nm, which is susceptible to corrosion due to the ingress of 

atmospheric moisture. The normal protective surface oxide formation 

may be comparable in thickness to that of the metal film. Corrosion 

usually progresses from the ends of the capacitor cylinder to the middle, 

related to the ingress of atmospheric moisture. Corrosion of metallized 

electrodes leads to an uneven distribution of current, which causes 

localized heating and eventually leads to catastrophic failure [17].

Regarding the effect of geometry of metallized film capacitors with the 

same specifications but different heights, experiments conducted by 

[18] show that long capacitors deteriorate faster than plate-shaped ones, 

due to higher heating under the same electrical stress.

2.2. Electrolytic Capacitors

Electrolytic capacitors are composed of an electrolyte-impregnated 

paper layer sandwiched between two highly roughened metal foils 

(usually aluminum). Grown directly onto the anode foil is a thin, highquality oxide, which acts as a dielectric possessing excellent breakdown 

voltage characteristics. One of the primary functions of electrolytic 

capacitors is the smoothing of voltage ripple and storing electrical 

energy. Electrolytic capacitors provide high capacitance values, high 

volumetric efficiency, and an excellent price over performance ratio. 

However, the electrolytic capacitor has the shortest lifespan of the 

components in power electronics. Electrolytic capacitors tend to 

degrade and fail much faster under high electrical or thermal stress 
3

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operating conditions than metallized film capacitors. The failures in 

electrolytic capacitors can be categorized into catastrophic and wear-out 

failures. The wear-out of electrolytic capacitors leads to the drift in the 

values of two important parameters, capacitance and ESR. The drift in 

these parameters ultimately leads to failure of the capacitor. Defined 

industry standards specify the end-of-life of an electrolytic capacitor 

under thermal stress if the capacitance value decreases by 10% and the 

ESR value increases by 250% or more from its initial rated value. 

Similarly, under electrical stress operations, the end-of-life is defined 

by its ESR increasing by 280% - 300% over and capacitance decreasing 

by 20% below its initial condition values. The ratings are different for 

electrical and thermal stress conditions because thermal stress 

conditions are considered for situations when the capacitor is in storage, 

and not in operation [19, 20].

The primary failure mechanism of an electrolytic capacitor is the 

evaporation of the electrolyte due to thermal overstress. A capacitance 

decrease and an ESR increase are caused by the loss of electrolyte, by 

diffusion (as vapor) through the sealing material in the wear-out failure 

period. If the electrolyte vapor pressure within the capacitor increases, 

by high temperatures for example, the diffusion rate increases. Factors 

that can increase the capacitor temperature, such as ambient temperature 

and ripple current, accelerate capacitor wear-out. Hence, estimating the 

temperature of the capacitor element (capacitor core) is of vital 

importance. The actual temperature of the capacitor element can be

greater than the ambient temperature due to the temperature rise in 

applications with ripple current. Wear-out can also be accelerated by 

high internal pressure caused by gas generation from excessive leakage 

current or attack of the cathode foil by electrolyte (oxide breakdown, 

which is mainly observed under electrical overstress conditions) [21-

26]. Table 1 summarizes the major failure causes, mechanisms and 

modes of aluminum electrolytic capacitors and metallized film 

capacitors, mainly concerned with the field aging or application phase 

of capacitors.

TABLE 1. FAILURE CAUSES, MECHANISMS AND MODES OF CAPACITORS

Root Cause Capacitor Type Major Failure Mechanism Major Failure Modes References

High Ambient 

Temperature

Electrolytic Electrolyte evaporation; increase in pressure 

inside the capacitor

Capacitance loss, ESR increase; vent opening, in 

case of excess internal pressure

21, 22, 23, 25,

26 and 52

Metallized Film Deterioration of quality of electrode 

metallization

Change in the non-linearity of current-voltage 

characteristics 7, 10 and 11

Over Voltage 

Stress

Electrolytic Degradation of oxide film, anode foil 

capacitance drop

Increased leakage current flow, capacitance loss, 

ESR increase 21, 26 and 27

Metallized Film Self-healing (“soft failure”) Capacitance loss, ESR increase 1, 5, 6, 7, 8

and 9

Excess Ripple 

Current

Electrolytic Electrolyte evaporation Capacitance loss, ESR increase 21 23 25 26

and 52

Metallized Film Electrochemical corrosion; increased core 

temperature Capacitance loss, ESR increase 2, 12, 13, 16 and 43

Continuous 

Charge/ Discharge 

Cycle (Pulsed 

Discharge)

Electrolytic

Cathode foil capacitance decrease due to 

formation of additional dielectric layer; 

electrolyte evaporation

Capacitance loss; vent opening, in case of excess 

internal pressure due to gas generated during oxide 

layer formation

21 and 27

Metallized Film Self-healing; electrochemical corrosion Capacitance loss, ESR increase; detachment of the 

sprayed ends (schoopage) under high electric stress

15, 16, 44 and 

45

Humidity Metallized Film Electrochemical corrosion; dielectric loss due to 

moisture absorption by film Capacitance loss, ESR increase 1, 17 and 37

In summary, the life of aluminum electrolytic capacitors depends on 

environmental and electrical factors. Environmental factors include 

temperature, humidity, atmospheric pressure, and vibration. Electrical 

factors include operating voltage, ripple current, and charge-discharge 

duty cycle. Among these factors, temperature (ambient temperature and 

internal heating resulting from ripple current) is the most critical to the 

life of aluminum electrolytic capacitors, whereas conditions such as 

vibration, shock, and humidity will have little effect on the actual life of 

the capacitor. Voltage within the allowed operating range has little 

effect on the actual life expectancy of a capacitor. When the applied 

voltage is above the rated voltage of the capacitor, the leakage current 

flow increases at a much faster rate, thus increasing the amount of heat 

generated [27]. Because a capacitor is essentially an electrochemical 

device, increased temperatures accelerate the chemical reaction rates 

within the capacitor (usually a 10°C rise in temperature will double the 

chemical reaction rate) [28].

3. DEGRADATION MODELING

3.1. Metallized Film Capacitors

Physics-of-Failure-Based Models. Acceleration models 

predict time-to-failure as a function of stress. Knowledge of the time to 

reach failure at an accelerated stress level, in conjunction with a 

calibrated acceleration factor, can be used to predict the equivalent timeto-failure at a different operating stress level. These models are usually 

based on the physics underlying a failure mechanism. Some common 

acceleration models used in capacitor lifetime prediction are the 

Arrhenius and Eyring models for temperature-dependent stresses, and 

the inverse power law for voltage and humidity stresses.

Arrhenius proposed a chemical kinetics model that showed the 

temperature dependence of reaction rates. Dakin proposed a lifetime 

model for electrical insulations, based on the Arrhenius model [29].

Capacitor aging as a function of the temperature follows an Arrhenius 

law, which is basically an exponential law [1, 30, 31]. The activation 

energy of a thermal process, as given in the Arrhenius law, is a stressdependent quantity [30]. The experimental observation is that 
4

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simultaneously applied stresses normally produce a degradation effect 

which is slightly lower than that derived by a simple multiplicative law. 

Therefore, an interaction term must be added. Eyring theory generalizes 

Arrhenius law to many factors besides the temperature, and captures the 

interaction of other stress factors with the temperature stress given in 

Eq. (1):

ƮEEEEEEEEEEEE(TT , SS1, SS2) = AAEETTααexp � EEaa

kkTTii

+ �BBEE + CCEE

TTii

� SS1 + �DDEE + EEEE

TTii

� SS2 � (1)

where ƮEyring is the useful life of the component, S1i and S2i are the 

supplementary considered stresses, AE, BE, CE, DE, EE and α are the 

different constants[32]. Simoni [33] and Ramu [34] proposed combined 

models for thermal and electrical stress, derived from Eyring theory, 

which take into account the effect of application of electrical stress on 

the thermal degradation process. The combined-stress life model 

considering inverse-power electrical life law is given in Eq. (2) as:

LL = LLoo �

EE

EEoo

�

−( −bb )

exp(−BB ) (2)

In Eq. (2), n, b and B are the model parameters, Lo is the life at 

temperature Θo, Eo is the value of electrical stress below which electrical 

aging can be neglected, T �= 1

ΘΘoo

− 1

ΘΘ

� is the so-called conventional 

thermal stress and Ѳ is the absolute temperature [35].

Albertini et al. [36] investigate a thermal life model for capacitors 

subjected to both constant and time-varying temperatures. The constanttemperature life models are based on the Arrhenius model and an 

Arrhenius–Miner Model is proposed for lifetime calculation under timevarying temperature stress. However, the life estimation by the 

proposed model is far from accurate when compared to the experimental 

results, and the temperatures to which capacitors are subjected in the 

experiment are much higher than the maximum recommended operating 

temperature (category temperature) of the capacitors.

Regarding voltage and humidity dependency, an inverse power law or 

an exponential law is generally used [1] [35-38]. An example of a 

degradation life model simply obtained by multiplying the aging rates 

of each stress is given in Eq. (3) as:

LL = LLoo × � RRRR

RRRRoo

�

− 3

× � VV

VVoo

�

− 1

× 2

TToo−TT

nn2 (3)

where L and L0 are the lifetime under the use condition and reference 

condition, respectively. V and V0 are the voltages at use condition and 

reference condition, respectively. T and T0 are the temperatures at use 

condition and reference condition, respectively. RH and RH0 are the 

relative humidity under the use condition and reference condition, 

respectively. The exponents for voltage, temperature and humidity 

stresses are denoted by n1, n2 and n3, respectively.

Data-Driven Models. Acceleration models are most often 

considered to predict the operating lifetime of capacitors under specific 

environmental conditions (temperature, voltage and humidity). But the 

major limitation with these models is the inability to track the capacitor

degradation with time. That is, the acceleration models cannot predict 

the state of health of the capacitor at a particular time. Hence, there is a 

need for data-driven or statistical degradation models to estimate the 

capacitor’s actual state-of health at a given time and to estimate the 

remaining useful life (RUL). In non-destructive degradation analysis, 

once the degradation information has been recorded, the measurements 

are extrapolated to the pre-defined failure level to estimate the failure 

time. The common degradation models used to perform extrapolation

of data are linear, exponential, power, and logarithmic. In destructive 

degradation analysis, theoretical population models known as lifetime 

distribution models are used to describe unit lifetimes, which are 

probability density functions defined over a range of time. The 

corresponding cumulative distribution function gives the probability 

that a randomly selected unit will fail by a specified time. The Weibull 

model is the most commonly used life distribution model, especially for 

capacitors. For a selected distribution, a degradation model is typically 

used to represent the change of the location or log-location parameter 

with time.

Sometimes there are probabilistic arguments based on the physics of the 

failure mode that tend to justify the choice of model. Other times, the 

model is used solely because of its empirical success in fitting actual 

failure data [39]. In most cases, the end-of-life criterion for metallized 

capacitors is defined as a 5% capacitance loss of the initial capacitance

[40]. As discussed before, the capacitor degradation rate increases after 

the capacitance or ESR reaches the specified end-of-life criteria. Hence, 

the estimation of the capacitor degradation indicators (e.g., capacitance, 

ESR, tan δ) is of vital importance. Soliman and Wang [41] review the 

existing condition-monitoring technologies and present the algorithms 

involved.

Makdessi et al. [42] investigate accelerated aging tests at different 

temperatures and voltage stresses for metallized film capacitors. The 

self-healing process is assumed to be a non-homogeneous phenomenon 

that follows an exponential probabilistic law. A capacitance degradation 

model is proposed where the model parameters are related to the 

capacitor at specific voltage and temperature conditions. The model 

parameters are a function of voltage and temperature and determined by 

fitting the curve to the experimental points at the applied aging test 

conditions. The results for the capacitor estimated lifetime are compared 

with the estimated lifetime given by the Eyring law (1). There is good 

agreement between the expected lifetimes from both the methods. In 

[12], a capacitance degradation model under ripple currents is proposed 

where the model parameter is a constant associated with the 

characteristic of each type of capacitor related to the gas rate diffusion 

through the oxide layer.

Hua Li et al. [43] discuss the degradation of metallized film capacitors 

stressed under a high-temperature, high-humidity environment with an 

AC voltage. Capacitance and ESR are monitored throughout the 

experiments and the electro-chemical corrosion phenomenon in two 

types of metallized films are compared. It is found that the moisture 

from the surrounding environment takes time to penetrate the capacitor.

Before this occurs, the capacitance loss is linear and mainly due to the 

pre-existing moisture in air trapped between the layers inside the 

capacitors. The capacitance then decreases faster due to the moisture 

penetrating effect. The transition time from the first stage to the latter is 

called as the ingress time of moisture. The capacitance change with time
5

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before and after the ingress of moisture is given by linear and power 

degradation models, respectively.

In [37], a set of metallized film capacitors undergo accelerated testing 

under constant temperature but varying humidity levels. The time-tofailure of the failed capacitors is fitted into a Weibull distribution curve 

and B1 life, B10 life, and mean time to failure (MTTF) is estimated 

under the required confidence intervals and the model parameter n3

(humidity derating exponent factor) for the lifetime equation (3) is 

estimated. Li et al. [44] introduce new life prediction models to capture 

the degradation of metallized film capacitors in pulsed power systems.

The capacitance loss is the accumulative result of self-healing in pulsed 

electrical shots and is divided into two parts: natural loss and sudden 

loss. The traditional life prediction models are the Weibull distribution 

model and the regression model, whose parameters are determined by 

the least squares method. The least-squares method is only appropriate 

when the capacitance loss is uniform and steady, and the lifetime 

obtained by the regression model strongly depends on the curve-fitting 

function (for instance. linear, quadratic, cubic). The Birnbaum–

Saunders distribution (also known as the fatigue life distribution) is 

introduced, where the capacitance loss resulting from a fixed number of 

shots is assumed to be an independent distribution having non-negative 

random variables. It should obey the normal distribution, whose

parameters are first calculated based on experimental data by the

method-of-moment estimation algorithm, then used to calculate the

lifetime expectancy and variance. However, this life-prediction model 

does not consider the sudden loss that occurs during lifetime testing of 

metallized film capacitors. Hence, the Poisson distribution is introduced 

to analyze the sudden capacitance loss, in which the number of random 

faults resulting in sudden drop in one shot obeys the law of Poisson 

distribution. Both the steady capacitance loss and sudden capacitance 

loss caused by random faults are assumed to be independent, identically 

distributed, and nonnegative random variables and should obey the law 

of normal distribution, whose parameters are dependent on the number 

of shots. Research conducted by Shin et al. [45] uses a similar approach 

in which the steady or the natural loss is captured by a random variable 

deterioration method and the sudden loss is captured by a gamma 

process deterioration method. The deterioration rate of the random 

variable deterioration model is obtained by using the curve-fitting

algorithm, and the shape function and scale parameter of the gamma 

process is obtained using an expectation-maximization algorithm.

3.2. Electrolytic Capacitors

Physics-of-Failure-Based Models. The Arrhenius model is 

used for the temperature lifetime dependency of electrolytic capacitors.

Gualous and Gallay [46] assumed that the capacitor life is proportional 

to the inverse reaction rate of the process and proposed a modified 

Arrhenius equation (Eq. (4)).

LL = BB ∗ ee

� EEAA

kkTToooo�

(4)

In Eq. (4), L is the expected capacitor life (in hours) for an operating

temperature Top (in Kelvin), k is the Boltzmann constant, EA is the 

activation energy, and B is a material specific parameter.

In industry, the rule of thumb related to the life of electrolytic capacitor 

is given in Eq. (5):

LL = LLoo ∗ 2�

TToo−TToooo

10 � (5)

where L is the expected life of the capacitor at the operating temperature

Top, and Lo is the expected life at the rated temperature, To. This means 

that for every 10°C increase in operating temperature, the life of the 

electrolytic capacitor reduces by half [23, 28, 47]. The temperature 

acceleration factor is approximately two over an ambient temperature 

range from 60°C to 95°C. However, according to the Arrhenius 

equation, the reciprocal of T is directly proportional to the logarithm of 

lifetime, which means that, strictly speaking, there is the temperature 

range where the theory of lifetime reducing by half for every 10°C rise 

is not applicable. Especially for capacitors whose maximum operating 

temperature is 105°C or higher, the temperature acceleration factor 

needs to be modified depending on temperature ranges of the lifetime 

estimation [28].

Dehbi and Wondrak [48] slightly modified the model given by Eq. (5) 

to account for the applied voltage. Jánó and Pitica [49] combined the 

Dehbi and Wondrak's [48] and Gualous and Gallay's [46] models and 

proposed the following (Eq. (6)):

LL = BB ∗ ee

� EEAA

kkTToooo�

∗ �

VVoo

VV � (6)

in which Vo is the maximum rated voltage, V is the operating voltage 

and n is the capacitor type parameter (0 for axial, 1 for radial). This 

suggests that only radial capacitors are affected by working voltages.

The most common monitoring methods for degradation of capacitor 

with operational time include ESR-based monitoring. Changes in 

chemical and physical properties of the capacitor electrolyte will affect 

the value of ESR [50, 51]. The conductivity of the capacitor electrolyte 

increases because of a rise in the temperature, hence the value of the 

ESR will decrease with increasing temperature, accompanied by a rise 

in the capacitance value. Gasperi [52] proposed an algorithm to estimate 

the ESR of an electrolytic capacitor. A commonly used rule-of-thumb

is that a capacitor reaches the end of life when it has lost about 40% of 

its electrolyte, that is, when the ESR increases to roughly 2.7 times of 

the initial value. Gasperi related the volume of electrolyte to the ESR of 

the capacitor as Eq. (7):

EEEE

EEEE oo

= �

VVoooo−ii

VVoooo

�

2

(7)

where, ESRo is the initial ESR, Vol is the volume of electrolyte, and Voli is the initial volume of electrolyte.

The rate of electrolyte loss is assumed to be directly proportional to the 

vapor pressure of the electrolyte. Since vapor pressure is temperature 

dependent, the core temperature of the capacitor critically affects the 

vapor pressure inside the capacitor and knowing the core temperature 

(ambient temperature + heat rise by ripple current), the vapor pressure 

of the capacitor can be estimated by an exponential relation between the 

vapor pressure of the electrolyte and the core temperature [52].
6

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Kulkarni et al. [22] presented the capacitance and ESR degradation 

models as a function of electrolyte volume loss, by analyzing the 

geometry of the capacitor (Eq. (8.1) and Eq. (8.2)).

CC(tt) = �

2εεRRεεoo

ddCC

��

VVoooo−ii − VVdd (tt)

jjee tt wwee

� (8.1)

EE EE(tt) = �

ρρEEddCCPPEE

2 �� jjee tt wweeVVoooo−ii − VVdd (tt)

� (8.2)

In these models, t is the aging time, Vdol is the dispersion volume at time 

t, ρE is electrolyte resistivity, dC is cathode oxide layer thickness, ƐR is 

relative dielectric constant, Ɛo is permittivity of free space, we is the 

volume of ethylene glycol molecules, jeo is evaporation rate, and PE is 

the correlation factor related to electrolyte spacer porosity and average 

electrolyte pathway. It is clearly seen by the models that, as the 

electrolyte volume decreases, the capacitance decreases and the ESR 

increases.

Data-Driven Models. In the research work by Kulkarni et al. [53], 

the capacitor degradation parameters are observed under nominal 

conditions. ESR drift as a function of time is determined by the 

Arrhenius model, given in Eq. (9):

1

EEEE tt

= 1

EEEE oo

(1 − kk.tt. exp( −EE

TT+273)) (9)

where t is the operating time, ESRt is the ESR value at t, T is the 

temperature in degree Celsius at which the capacitor operates, ESR0 is 

the initial ESR value when t is equal to 0, k is a constant which depends 

on the design and the construction of the capacitor and E is the

activation energy defined by Boltzmann’s constant. A third-degree 

regression model, determined to be the best fit by least squares, is 

proposed for the average capacitance degradation with time. In [54], 

capacitance loss with time under overvoltage conditions is captured by 

an exponential degradation model and the model parameters are 

estimated by a nonlinear least-squares regression algorithm.

The values of capacitance and the ESR vary both with time and the 

ambient temperature. Abdennadher et al. [55] proposed an aging 

algorithm for real-time condition monitoring of capacitors to determine 

the RUL of electrolytic capacitors used in uninterruptible power 

supplies. For a given ambient temperature of the capacitor, the total 

lifetime is given by the Arrhenius equation. Also, at that particular 

temperature, the ideal value of ESR or capacitance are obtained by 

curve-fitting equations. The actual or degraded value of capacitance or 

ESR is obtained from the experiment at a given time, which gives the 

operating time of capacitor at those values by curve-fitting equations. 

Subtracting the operating time - obtained by curve- fitting equations –

from the total lifetime obtained by Arrhenius equation yields the RUL.

Similar degradation modeling is completed in [56], where the 

relationship between temperature and the instantaneous value of ESR 

and capacitance is modeled by an inverse exponential and linear 

degradation model, respectively, and the model parameters are 

calculated with a least-square fitting method. When the ambient 

temperature and the degradation rate are constant, the evolution of ESR 

and capacitance with time are given by exponential and linear 

degradation models, respectively. Based on these statistical equations, 

a life prediction model for electrolytic capacitors operating at a constant 

temperature has been proposed to find the RUL as defined by the 

minimum residual life of either capacitance or ESR end-of-life criteria, 

whichever is shorter.

Under thermal and voltage stresses, the temperature of the capacitor 

hotspot continuously increases. More recently, Sun et al. [57] proposed 

a hybrid degradation model by combining the physics-of-failure model 

with a similar statistical model of capacitance and ESR degradation with 

time at constant capacitor temperature, where model parameters are 

assumed to follow the Arrhenius equation, as given in Eqs. (10.1) –

(10.4).

CC(tt) = CCoo. (1 + AA.tt) (10.1)

EE EE(tt) = EE EEoo. eeBB.tt (10.2)

AA(tt) = AAoo. ee

−EEaa1

(kk.TT) (10.3)

BB(tt) = BBoo. ee

−EEaa2

(kk.TT) (10.4)

Here, Co and ESRo are the initial capacitance and ESR, respectively, A

and B describe temperature-dependent degradation rates, A0 and B0 are 

base degradation rates, Ea1 and Ea2 are the activation energies, and κ is 

the Boltzmann constant. These models are then combined and 

integrated to obtain variable-temperature capacitance and ESR 

degradation models.

In [58], accelerated life tests on aluminum electrolytic capacitors under 

temperature and voltage stress are conducted to study the effect of 

applied voltage and ambient temperature on the capacitor. The 

degradation of the capacitor is studied by measuring its capacitance loss 

at random times over the test duration and a regression equation 

representing the degradation pattern is developed, which is used to 

predict the time to failure of each capacitor. Weibull analysis for the 

predicted failure times of all the capacitors at each stress level (voltage, 

temperature) is carried out and the MTTF of the population at that stress 

level is calculated. Analysis of variance (ANOVA) is also carried out in 

MINITAB software to determine which of the two stress factors has

more effect on the life of capacitor and how the interaction effect 

between these two stresses impacts the life of the capacitor. It is found 

that the applied voltage affects the mean life most, followed by ambient 

temperature and interaction effect. A regression equation that correlates 

the life of a capacitor with both stress levels is obtained that can be used 

to predict the life when the capacitor is used beyond the experimental 

stress levels. Table 2 summarizes the most significant research work

associated with the various stress types in capacitors.
7

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TABLE 2. RESEARCH WORK ASSOCIATED WITH THE VARIOUS STRESS TYPES IN CAPACITORS

Stress Type Reference Model Used Major Contribution

Metallized Film Capacitors

Temperature [36] Arrhenius model Life model proposed for time-varying thermal stress, using Miner’s cumulative 

damage theory

DC Voltage, Temperature [42] Exponential regression equation, 

Eyring law

Capacitance degradation with time; constant high temperature and voltage. 

Lifetime estimated by regression model and Eyring law

Ripple Current [12] Regression equation, based on 

underlying physics-of-failure

Electrochemical corrosion introduced as a failure mechanism in presence of 

ripple current. Regression equation proposed for capacitance loss with time 

based on theory of the oxidation kinetics.

Pulsed Discharge

[44]

Birnbaum–Saunders distribution 

(fatigue life distribution), Poisson 

distribution

Standard life prediction models (regression, Weibull) are compared with the 

newly introduced Birnbaum–Saunders distribution (to model steady 

capacitance loss) and Poisson distribution (to model sudden capacitance loss) 

for pulsed power applications.

[45]

Random variable deterioration 

method, Gamma process 

deterioration model

Steady capacitance loss is modeled by random variable deterioration method 

and sudden loss is modeled by a gamma process deterioration method.

AC Voltage, Temperature, 

Humidity [43] Regression equation Capacitance and ESR modeling according to the “ingress time” of moisture

Humidity, Temperature [37]

Weibull distribution,

combined model (Arrhenius 

model + inverse power law for 

humidity)

Time to failure is fitted into a Weibull distribution curve and B1 life, B10 life 

and MTTF are estimated under the required confidence intervals and the model 

parameter for the lifetime equation is estimated for all lifetime definitions.

Electrolytic Capacitors

Temperature

[52] Arrhenius equation

Physics-of-failure model to predict ESR change with electrolyte loss is 

proposed. Lifetime of capacitors at different temperatures is calculated using 

Arrhenius law, and the results are compared to lifetimes obtained via an 

approximation law used in industry.

[55] Regression equation,

Arrhenius equation

For a given ambient temperature of the capacitor, the total lifetime is given by 

the Arrhenius equation. The operating time of the capacitor is obtained using 

regression equations characterizing capacitance and ESR with time and 

temperature, respectively. Remaining useful life is estimated by subtracting 

total lifetime with the operating time obtained by regression equations.

[57] Regression equation,

Arrhenius equation

Hybrid degradation modeling approach is proposed in which the parameters of 

regression equations characterizing capacitance and ESR with time, follow 

Arrhenius law. Increase in output current is considered as a failure criterion. 

Time to failure can be calculated at variable-temperature operating conditions.

Temperature, Voltage [49] Combined model (Arrhenius + 

inverse power)

A new prediction method for capacitor lifetime is introduced. The lifetime 

obtained is found to be more accurate than the previous models. This prediction 

methodology suggests that only radial capacitors are actually affected by 

working voltages.

Voltage [54] Regression equation A remaining useful life prediction methodology for electrolytic capacitors 

based on Kalman filter is presented.

Nominal Conditions [53] Regression equation,

Arrhenius equation

Under nominal conditions (below rated voltage and temperature), ESR drift as 

a function of time is determined by Arrhenius model (exponential model). A 

third-degree regression model is proposed for the average capacitance 

degradation with time.

4. DISCUSSION

Although electrolytic capacitors are a popular choice for the DC-link 

applications due to high capacitance values, they generally have a low 

reliability and a short shelf life, mainly due to their sensitivity to 

temperature variations. Metallized polymer film capacitors, especially 

the capacitors with polypropylene as a dielectric, can be considered as 

a good substitute for electrolytic capacitors in fault-tolerant applications

because of their low temperature and frequency dependency, and very 

little change of capacitance with time under voltage applications due to 

their unique self-healing ability. The insulation resistance as a failure 

indicator has been largely neglected in the existing research, but it 

severely affects the capacitor degradation, contributing to phenomenon 

like thermal runaway, especially in high-DC-voltage conditions. Hence, 

both capacitance loss and decrease in insulation resistance should be 

considered to access the complete health of metallized film capacitor.

There is a need for real-time capacitor degradation models that take into 

account the operating conditions (ambient temperature, voltage, ripple 

current, humidity, frequency, time, etc.) for more accurate stress 

analysis. Also, an investigation into the interaction effect among various 

stressors on the lifetime of capacitors needs to be conducted [30]. There 

is a need for hybrid models, which combine the physics-of-failure and 

data-driven models, for capturing the capacitor degradation more 

realistically. This is particularly needed in the case of metallized film 

capacitors, as little research has been published in this area apart from 

the work of Makdessi et al. [12, 42]. Also, regression-based degradation 

modeling seems to be the most widely used among the data-driven 

approaches. But regression-based models require large sample sets to 
8

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increase their predictive accuracy. In regression analysis, many 

variables are susceptible to noise in the data or certain unnecessary 

information, therefore reducing the prediction power of regression 

analysis. Grey models may provide better performance using small 

sample size, not only in model fitting, but also in forecasting [59].

ACKNOWLEDGMENTS 

This work was authored in part by Alliance for Sustainable Energy, 

LLC, the manager and operator of the National Renewable Energy 

Laboratory for the U.S. Department of Energy (DOE) under Contract 

No. DE-AC36-08GO28308. The views expressed in the article do not 

necessarily represent the views of the DOE or the U.S. Government. 

The U.S. Government retains and the publisher, by accepting the article 

for publication, acknowledges that the U.S. Government retains a 

nonexclusive, paid-up, irrevocable, worldwide license to publish or 

reproduce the published form of this work, or allow others to do so, for 

U.S. Government purposes.

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