# Calculating Capacitor Reliability

## Objective

Based on vendor data and engineering estimates, calculate capacitor reliability for use in system modeling.

## Components

For a large capacitor in the system determine the expected life distribution for use in the system reliability block diagram model. The capacitors is:

DC Buss capacitor, mfg p/n E50.R16-155N10; CAP FLM 1450uF 700VDC 10%0.65Rs 40nH 116Dx165H -25+85C ROHS

From the datasheet

statistical failure rate 50 FIT*

reference service life 100000 h

at θhotspot ≤ 70 °C

*FIT is a function of temperature and voltage

## Expected Failure Mechanisms and Life Data

The most likely failure mechanism for capacitors is dielectric breakdown.

This can occur due to manufacturing defects, power line transients and over time by dielectric degradation. Assuming assembly process controls and adequate circuit protection are in place, the dominate life limiting failure mechanism is dielectric degradation and eventually dielectric breakdown.

High operating voltages relative to rated voltage and to a lesser degree high operating temperatures accelerate the dielectric degradation eventually overcoming the self healing behavior.

The ripple current contributes to the internal heating and the acceleration of the dielectric degradation.

Based on the component datasheet the following reference service life (base life) information is available:

DC Buss Capacitors

100,000 hrs @ ≤ 70°C

## Reliability Calculations

The calculation of the capacitor expected lifetime distribution based on the above information is fairly straight forward.

The basic idea uses the given single point (63.2%‘ile point along distribution), also called the service life or base life value, then we make adjustments to the base life for the voltage and temperature conditions to determine the expected operating life.

Plus, with knowledge about how capacitors fail over time, the slope, we can estimate a life distribution, since we have a point and slope.

The service life, or base life provided by the vendor, is the expected duration in hours until 63% of the population has failed while operating at or near the maximum operating conditions.

The slope, or beta value for a Weibull distribution, is best determined from testing to failure in the use environment. An estimate of beta from literature recommends at beta value of 6 to 8 for similar capacitors. [Hillman 2010] We will use a beta of 7 for our estimate.

For the film capacitor the effects of voltage and temperature adjust the provided service life value according to the following formula. [Parler 2004]

$\displaystyle L = {L _B} \times {f _1}\left( {{T _M} - {T _C}} \right) \times {f _2}(V)$

where,

LB is the base life, or the service life value, as listed on the datasheet.

TM is the maximum rated temperature for the capacitor.

TC is the actual core temperature of the capacitor under operating conditions.

f1 represents the function or adjustment to the base life value for temperature.

As expected, the formula uses the Arrhenius relationship.

$\displaystyle {f _1} = {e^{{{{E _a}} \over k}\left( {{1 \over {{T _C}}} - {1 \over {{T _M}}}} \right)}}$

where,

Ea is the activation energy, for anodic alumina Ea = 0.94

k is the Boltzmann constant, k = 8.62 e-5 eV/K

The core temperatures are calculated based on operating and thermal conditions, or roughly approximated using the case temperatures.

Similar formulas in the literature use the operating ambient temperature in place of the core temperature. And given the complexity in the calculation of core temperatures, using the case temperature, while not accurate, provides a first order approximation.

Furthermore, when the capacitor is operated close to the maximum operating temperature, f1 becomes insignificant in the adjustment of the base life value.

For the DC Buss Capacitors the maximum rated temperature is 85°C and the reference service life is listed with a core temperature of ≤ 70°C.

The measured case temperature is 81.29°C at a 45°C ambient full power condition, 480VDC. The calculated temperature rise from the case to the core, is approximately 6.5°C resulting in a core temperature of 87.8°C.

Note: the vendor mentioned that they would have calculated a core temperature of approximately 100°C (and would not share the details of the calculations). [Stoike 2010]

Therefore, the resulting calculation of f1 is 0.20. This is less than one as the expected operating hotspot temperature is higher than the reference life hotspot temperature. The actual value may be less, as datasheet suggests the life expectations are valid only below 70°C.

The second element of equation 1 above, is the f2 term related to voltage. For film capacitors, [Cooper 2005, Capsite 2010]

$\displaystyle {f _2} = {\left( {{{{E _R}} \over {{E _O}}}} \right)^7}$

where,

ER is the maximum rated voltage

EO is the operating voltage.

With the exponent at 7 this term dominates the overall impact of expected life when the operating voltage is less then the rated voltage.

For the DC Buss Capacitor, the rated voltage is 700 Vdc and the expected max operating voltage is 480 Vdc.

$\displaystyle {f _2} = {\left( {{{{E _R}} \over {{E _O}}}} \right)^7} = {\left( {{{700} \over {480}}} \right)^7} = 14.0$

For the capacitor, using equation (1) and the calculations above, we have:

f1 = 0.2

f2 = 14

Expected Life at 63.2%’ile = 2,950,000 hours

Note: This is an estimate of the characteristic life. There is a small adjustment given that the slope is not 1, yet the correction was insignificant in this case and does not change the results.

The Expected Life value is eta, η, for the Weibull function calculations.

The reliability function, R(t), for the two parameter Weibull distribution is

$\displaystyle R(t)={{e}^{-{{\left( {}^{t}\diagup{} _{\eta }\; \right)}^{\beta }}}}$

where,

t is time in operating hours (in this case)

η is the characteristic life, or the time till approximately 63.2% of units are expected to fail

β is the slope, in this case assumed to be 7.0 [Hillman]

For the DC Buss Capacitor, the expected reliability at 10 years of life, 36,500 hours, is 1.

This means that 100% (greater than 99.999%) of capacitors are expected to successfully operate over a 10 hour per day, 10 year operating life.

At 20 years, this drops to 99.994% reliable.

The expected operating temperature is at times very high and may lead to unexpected failures, yet the voltage derating significantly reduces the stress on the dielectric. Of course, other stresses (lightening strike, physical or assembly damage, etc.) may lead to capacitor failures.

## Reliability Block Diagram notes

The capacitor has the following input values for use in the reliability block diagram.

DC Buss

η = 292,000

β = 7

## References

The following references support elements of the calculations and provide background information as well.

Parler, Jr., Sam G., “Deriving Life Multipliers for Electrolytic Capacitors.” IEEE Power Electronics Society Newsletter 16, no. 1 (2004): 11-12.

Parler Jr, S G, and P E C Dubilier. “Reliability of CDE Aluminum Electrolytic Capacitors.” Cornell Dubilier Technical paper, available on the CD web site.

Hillman, Craig, CEO DFR Solutions, personal communication, beta based on DFR Solutions testing of similar capacitors and finding relatively high Weibull beta values, 7 to 8.

Stoike, Marko, Development Engineer, Electronicon, email message to author dated Dec 14th 2010.

Capsite, ” Reliability Issues.” 2009. August 16, 2010. http://my.execpc.com/~endlr/reliability.html (accessed February 17, 2011).