IEEE Industry Applications Sociaty
Annual Meeting
New Orleans, Louisiana, October 5-9,1997
A Method for Predicting
the Expected
Life of Bus Capacitors
Michael L. Gasperi
Advanced Technology Labs
Rockwell Automation
1201 South Second St., Milwaukee, WI 53204
Email: mlgasperi @mke6.ra.rockwell.
Phone: (41 4) 382-2508
Fax: (414) 382-3500
the expected life of
Abstract - Predicting
motor drive bus capacitors is complicated by
several factors: the equivalent series resistance
of aluminum electrolytic capacitors is frequency
and temperature sensitive, the ripple current is a
composite waveform, and the capacitor expected
life is very sensitive to operating temperature.
This paper describes a method that addresses all
these factors by developing separate models for
capacitor equivalent series resistance,
motor
heat transfer conditions,
and
drive circuitry,
capacitor life.
INTRODUCTION
Voltage source inverter motor drives rectify the AC
line and use aluminum
electrolytic
capacitors
to
create a filtered DC bus.
These capacitors must
endure relatively high ripple currents which lead to
self heating. Self heating in addition to high ambient
operating temperature can make the capacitors the
weak
link
in drive
expected
life.
Capacitor
manufactures publish a rated ripple current, but it is
motor
drive
operating
nowhere
near
typical
conditions or motor drive design lifetimes.
i
Capacitor Equivalent Circuit
5?
Drive Model
Power Dissipation
-turfLife Model
Y
Expected Life
Fig. 1. Method for Predicting
Expected
Life
Fig. 1 illustrates the design method presented in this
paper. The first step is to develop a more accurate
model of equivalent series resistance ESR than a
com
simple fixed series resistance.
The next step is to
run the capacitor model in a circuit simulation to get
the expected power dissipation.
Power dissipation
becomes temperature rise by accounting for the heat
and the expected
life is
transfer
conditions,
computed from the resulting capacitor temperature.
EQUIVALENT
SERIES RESISTANCE
MODEL
The normal method for modeling capacitors utilizes a
series combination
of an ideal capacitor,
a pure
resistor, and optionally, an ideal inductor [8 and 9].
The resistor in the model is known as the equivalent
Three major sources
series resistance or ESR.
contribute
to the value of ESR in aluminum
electrolytic capacitors: the oxide dielectric produces
a frequency
sensitive
resistance,
the electrolyte
creates a temperature sensitive resistance, and the
foil, tabs, and terminals represent an additional small
source of relatively constant resistance[5].
The effect of frequency
on dielectric
resistance
appears in the ripple current multipliers provided by
capacitor manufacturers
[11 ]. The effect is caused
by energy losses in the alignment of dipoles in the
dielectric and variations in the time it takes for the
dipoles to become orientated [12]. It becomes more
significant
for higher
rated
voltage
capacitors
because their oxide layer is thicker.
For applications
with single frequency ripple current, the ESR can be
simply adjusted by using a multiplier.
Motor drives create a ripple current spectrum too
complex for simple multipliers.
Fig. 2 illustrates an
Resistance
RO
improved
capacitor
model [3].
combines the resistance of foil, tabs, and terminals
RI
accounts
for
electrolyte.
A parallel
while
combination
of Rp and Cp models the dielectric
resistance.
Equation
(1) describes
the complex
impedance
of the capacitor,
but ripple current
heating occurs only in the real part of this impedance
(2 and 3).
07803-4070-1/97/$10.00 (c) 1997 IEEE
1
T
100%
c’
&
7’2“+
90%
o
c1
I
Fig. 2. Aluminum
Electrolytic
Capacitor
,,,
,,,
,,,
Model
60% ~
Modeling with more series-parallel resistor capacitor
pairs achieves
a tighter fit to both ESR and
capacitance
behavior [1O and 14].
However, one
pair appears to be sufficient for estimating
heat
generation.
z cap
1
+R1+RO–
=
-J-+j2mf
Cz
j
—
27Cfcl
1)
H2
Where:
z cap
Complex Impedance of Capacitor
Ro
Resistance of Foil, Tabs, & Terminals
RI
Resistance
Rz
c,
Dielectric Loss Resistance (Q)
Terminal Capacitance (F)
Dielectric Loss Capacitance (F)
Frequency (Hz)
C*
f
of Electrolyte
(Q)
(Q)
‘
,,,
,,,
,,,
,,,
!l, ,
1000
Frequency (Hz)
,,, ,
,,, ,
,,, ,
,,, ,
10000
Fig. 3, Effect of Frequency on ESR
Actual at 50°C and Fit using Equation 3
Manufacturers often provide generic curves showing
the effect of temperature on ESR for their products.
These curves can be used with the ripple current
multipliers to obtain an estimate of the values for the
However,
a more
components
in the model.
accurate alternative is to experimentally measure the
ESR and capacitance over a range of temperatures
and frequencies
and use least squares to fit the
values to the data. Fig. 3 and 4 illustrate the good fit
achieved by this process and Table 1 shows the
values for an actual capacitor
180%
(2)
.,,
100
200%
ESR = Real(ZC.P)
,,, ,
,,, ,
,,, ,
~
1
I
I
50
75
(“C)
\
(j09’o
o
k-l
R2
Real(ZC.P ) =
+R1+RO
(3)
1 + (2z f)2C; R;
a 140’%
M
U 120’%0
,*
g
Increasing temperature
causes a decrease in ESR
due to the increased conductivity of electrolyte RI.
High rated voltage devices have more sensitivity to
temperature than low rated voltage devices because
they use lower conductivity electroiytes[l ]. Equation
(4) captures the temperature
sensitivity normalized
tO the base reSktiN7Ce
at room tfHTIpfMtUtE
RI base. It
is common for the elevated temperature
ESR to be
half the room temperature value.
R,
=
R,
~a,e
#,.as.-
Tcor,)/E
(4)
Where:
RI b,s.
T core
Tb,~,
RI at Base Temperature Tb.s, (Q)
Core Temperature ~K)
Base Measurement Temp =23’C=300’K
E
Temperature
100%
80%
60%
25
Temperature
Fig. 4. Effect of Temperature on ESR
Actual at 120Hz and fit using Equation 4
Table 1. Values for 400V Rated Voltage 470uF
Snap-in Capacitor
~ Series Resistor RO
0.035 Q
Series Resistor Rlba~. @ 23°C
Sensitivity
Factor
(“K-’)
100
Sensitivity
Dielectric
Factor E
Loss Resistor Rz
Series Capacitor Cl
Dielectric Loss Capacitor
07803-4070-1/97/$10.00 (c) 1997 IEEE
Cz
0.12 Q
p, OK-l
0.038 Q
470 pF
11,000 pF
MOTOR DRIVE MODEL
Ripple current heating is estimated by running the
capacitor
model in a circuit simulation
tool like
PSpice [6]. Fig. 5 shows the schematic of a simple
AC motor drive converter and inverter used to test
Two Fig. 2 style capacitor
the capacitor model.
models are used to simulate the series connection
needed to achieve a higher bus voltage than the
available
rated voltage of the capacitors.
Three
phase AC line impedance (5) has a significant effect
on the 360Hz component of the current ripple and
must be included. The PWM of the inverter coupled
with the motor impedance create the high frequency
components of ripple current.
-15
I
1
o
0.005
m
-L”
,
.
0.01
(S)
. .
0.015
0.02
Fig. 6. Ripple Current 11in Drive Simulation
:
:,:;;;;:,
;
;
,,,
,,, I
6
Recti5er
-
1
Time
7,
%
,
5
:4
.
AC
?I
Motor
~
Line
Inductance
I
-r
I-15!_l
,
-1
-12
t 11
Fig. 5. Motor Drive Simulation
100
1000
Frequency (Hz)
Model
10000
Fig. 7. Ripple Current Spectrum
i’line’% =
‘iL1”M’(2Z ’60)
PF V~~s
,00%
Where
Qd
1,
I*
Where
Zhne
L
iRMs
f6rJ
V~~s
PF
Line Impedance YO
Line Inductance (H)
RMS Rated Current (A)
Line Frequency = 60 Hz
RMS Rated Voltage (V)
Power Factor = 0.95
(6)
Q~ =1; RO +I;R1 +1; Rz
(5)
Total Power Dissipation (W)
RMS Ripple Through ROand RI (A)
RMS Current Through RZ (A)
h~
4
2
Two simulation probes are used to log the current in
Ro, RI and R2 during the run.
Fig. 6 shows the
waveform for the ripple current 11in RO and RI. The
Fig. 7 spectrum of the current shows practically
nothing below 360Hz and many components at the 4
Fig. 8 shows the
kHz PWM carrier and above.
smaller current in Rp which is primarily 360Hz and
lacks high frequency content.
At the end of the simulation run the RMS value of the
current in each resistor is used to calculate the
energy loss. Adding the resistor energy loss values
gives the estimated ripple current heating for each
capacitor (6).
so
g -2
& -4
-6
-8
-lo
o
0.005
0.01
Time (s)
0.015
0.02
Fig, 8. Dielectric Loss Current 1Pin Drive
Simulation
As capacitors age their RI ESR increases, and as
Fortunately,
they heat up their RI ESR decreases.
07803-4070-1/97/$10.00 (c) 1997 IEEE
capacitance dominates the impedance in the drive
circuit so the change in ESR has little effect on the
magnitude of the ripple current.
Fig. 9 shows a
range of values for ESR and the minimal resulting
change in ripple current.
8
0.9. Convection is modeled with (9) with the heat
transfer coefficient for natural convection modeled by
(1 O) [13].
Fig. 12 shows the heat transfer from the
two modes is nearly equal for appropriate range of
temperature
rise.
A more experimentallv , derived
equation
for screw terminal “style capacitors
is
provided in [4].
I
I
’11
—
’12
3.5
;;
,,,
,,,
,,!mlr
,,,
,,,
,,
,,,
,,,
,,,
,,
,,,
,,,
,,
,,
,,,
,,,
,
,,,
,,,
,,,
,,,
,
,,,
,,,
,,,
;
I
1
:;
0.5
;
0
:
,,,
,,,
,,,
4
0.04
RO +Rl
0.02
Fig. 9. insensitivity
0.06
(Ohm)
,,,
0.08
0.1%
of 11and IZ to ESR
Although the capacitor ESR has little effect on ripple
current, the AC line impedance
has a dramatic
effect.
Figs. 10 and 11 show the consequence
of
line impedance
on the current and the resulting
The eventual
result of line
power dissipation.
impedance on life can be calculated after the power
dissipation is related to temperature
rise through a
heat transfer model.
..1--,,
,,n
r,,
1
. . . ,,,,,
..... ‘:%:::::
,,
,,,
,,
,,,
,,, ,,
,,, :h&
,,, ,,
,,,
,,,
,,,
. .
,,,
0!”
0.1%
.
Where
T~
T case
AT
,1.
&
F
,,
. .
I
,,,
,,,
1.0%
Line Impedance
Fig. 10. Effect of Line Impedance
HEAT TRANSFER
,,
on Current
,,,
,,,
,
,’,,,
,,,
,,,
,,
,,
10.0%
on Power
(7)
(°K)
S F(T$,~, -T:
Heat dissipation from radiation
Stephan-Boltzmann
constant
)
(8)
(W)
10-8 W/m* “K4
Emissivity,
Assumed
= 0.9
Shape Factor = O to 1
QCO. = h SAT
Where
Q con
h
s
(9)
Heat dissipation from convection (W)
Heat Transfer Coefficient W/m* “C
Surface Area (m2)
MODEL
Heat transfer conditions are needed to convert the
energy loss to a temperature
rise (7).
Capacitor
banks are primarily cooled by a combination
of
natural air convection
and radiation. Equation (8)
models
radiation
where
the emissivity
of the
capacitors are assumed to be close to black body at
,,, ,
,., .
,,,
,,11.
,
Ambient Air Temperature
Case Temperature
(“K)
Temperature Rise (“C)
I
10.0%
,,,
,,
,,,
;::::::
,,,
,,,
,
AT = TCa~.- T~
= 5.669x
.,,
,,
,,,
,,
,,,
;;;;;;;
1.0%
Line Impedance
Q,,~ = os
,
,,,
Fig. 11. Effect of Line Impedance
Dissipation
Where
Qrad
G
1,,
.!.
,..
,
.
,,,
h= 1.32 (AT/2r)”4
S=27T
Where
z
Capacitor
(lo)
(rz+r2)
(11)
Height (m)
The heat transfer
equations
provide
a wav of
estimating the surface temperature of the capacitor,
07803-4070-1/97/$10.00 (c) 1997 IEEE
J
but the core temperature
is more important
in
predicting the life of the capacitor.
Equation (12)
estimates the core temperature rise by assuming the
core rise is a constant factor greater than the surface
rise. If the internal construction of the capacitor is
known, then a more accurate heat transfer model
can be built and the core temperature
more
accurately accounted.
TCO,,= T~ + a AT
(12)
Where
Relative Temperature
(x
Rise Factor = 1.5
LIFE MODEL
The primary wear-out
mechanism
for aluminum
electrolytic capacitors is vaporization
of electrolyte
and its loss through the end seal [7]. As the volume
of electrolyte
decreases
the equivalent
series
resistance
ESR of the capacitor
increases
as
modeled with (13). Eventually, the parameters drift
out of specification and the capacitor is considered
to have failed.
For applications with ripple current
the increase of ESR leads to increased
internal
heating that accelerates the wear-out process.
2,5
ESR/ESRO = (VO/V~
El Free Convection
2
Where
ESR
ESRO
v
Vo
ESRadiation
s
:1.5
.:
a
.:1
.2
n
0.5
0
O
5
10
15
20
Delta T (“C)
Fig. 12. Heat Dissipation
25
30
@ TA = 50 “C
Because the ESR drops with increasing temperature
due to increased
conductivity
of electrolyte,
an
aluminum
electrolytic
capacitor actually dissipates
less power with the same ripple current at higher
temperatures.
Fig. 13 shows the cross over point
where the actual dissipation equals the heat transfer
capability.
For this
example
the equilibrium
temperature rise is 15 “C.
Equivalent Series Resistance @ 20nC
Initial ESR (Q)
Volume of Electrolyte (Units)
Initial Volume of Electrolyte (Units)
(13)
(Q)
The life model presumes that the rate of electrolyte
loss is directly proportional to the vapor pressure of
the electrolyte [2]. A constant k in (14) is used to
characterize the quality of the end seal.
Equation
(15) relates the core temperature
to the vapor
pressure of the electrolyte that is primarily Ethylene
Glycol.
Numerical integration can be used with (13,
14, and 15) until the capacitor is at its end of life.
dV/dt = k P
Where
t
k
P
(14)
Time (h)
Leak Rate Constant (Units/mmHg/h)
Vapor Pressure of Electrolyte (mmHg)
A
-—+6
Ps
2.5
e
[1‘COW
(15)
Where
For Ethylene Glycol: A = 7060 and B =21 .63
2
0,5
0
0
5
10
15
20
Delta T (“C)
Fig. 13, Temperature
25
30
Fig. 14 shows the estimated effect of line impedance
on expected life. It was created by varying the line
inductance
in the drive simulation,
noting the
resulting values of II and 12, calculating the power
dissipation, temperature rise, and estimating the life.
It shows the significant reduction in expected life that
can accompany
applying
a motor drive to low
impedance sources.
Rise @ TA = 50 ‘C
07803-4070-1/97/$10.00 (c) 1997 IEEE
8.
1000%
9.
10.
~
. . . ....
. . . ....*.....,s,
A.
.
.
.
. . .
.
.
.
L.9.
I.LIA
.
.
. . .
.
,,.
. . . . . . . . . . . . . ff,~
,,1
1%
0.1%
. .
k.+
a.a.
.
.
..h.
.
.
.
. . .
.
.
.
. . ..4.....-
1 .
.
.
.
.
.
.
,,,
. .
.
. . .
.
.
..
k.-
.
.
.
,,,
..,1
11.
kbkw
.
,,,
.
.
. ,.f~,~
,!.1
:
1.0%
10.0%
Line Impedance
Fig. 14. Effect of Line Impedance
Life
12.
13.
on Expected
SUMMARY
14.
Prymak, J. D., “SPICE Modeling of Capacitors,”
CARTS 95, March 1995, pp. 39-46.
Schnabel, W., “Aluminum Electrolytic Capacitors
for Power
Electronics,”
CARTS-Europe
88,
October, 1988, pp. 85-92.
Scholte, J. W. A. and van Geel, W. Ch.,
“Impedances
of the Electrolytical
Rectifier,”
Philips Research Reports 8, 1953, pp. 47-72.
Electrolytic
Aluminum
United
Chemi-Con,
Capacitors, Catalog H7, Rosemont,
IL, 1995,
p265.
Young, L., Anodic Oxide Films, Academic Press,
New York, NY, 1961, p158.
Holman, J. P., F/eat Transfer, McGraw-Hill Book
Company, New York, NY, 1973, P253.
of Network Synthesis to
Tillo, J., “Application
Inductor and Capacitor Modeling,” CARTS-97,
Institute,
Inc.,
Technology
Components
Huntsville, AL, March, 1997, p76-87.
A method for estimating bus capacitor expected life
has been presented.
The method utilizes separate
models for capacitor ESR, motor drive circuitry, heat
transfer conditions, and life prediction.
The method
was used to show the significant effect of AC line
impedance on capacitor expected life.
REFERENCES
P. M., Electrolytic
Capacitors,
The
1. Deeley,
Cornell-Dubilier
Electric Corp., South Plainfield,
NJ, 1938.
Model for
M. L., “Life Prediction
2. Gasperi,
Aluminum
Electrolytic Capacitors,”
31 ‘t Annual
Meeting of the IEEE4AS, Vol. 4., October, 1996,
pp. 1347-1351.
N., Sladky, R., “A
3. Gasperi, M. L., Gollhardt,
Model for Equivalent
Series
Resistance
in
Aluminum Electrolytic Capacitors,”
CARTS 97,
Inc.,
Institute,
Components
Technology
Huntsville, AL, March, 1997, p71 -75.
and Ripple
4. Hayatee, F. G., “Heat Dissipation
Current
Rating
in Electrolytic
Capacitors,”
Electrocomponent
Science and Technology,
Vol.
2, 1975, pp. 109-114
F,
G.,
“The
Equivalent
Series
5. Hayatee,
Resistance
in
Electrolytic
Capacitors,”
Electrocomponent
Science
and
Technology,
1975, Vol. 2, pp. 67-72.
Corporation,
Circuit ArM/ysis User’s
6. MicroSim
Guide, Irvine, CA, April 1995
7. Moynihan,
John
D., Theory,
Design
and
E/ectro/ytic
Capacitors,
Application
of
Inc.,
Institute
Technology
Components
Huntsville, AL, 1982.
07803-4070-1/97/$10.00 (c) 1997 IEEE