Environmental Engineering Reference
In-Depth Information
An equivalent series inductance would also be added in series in the ESR
model for a more realistic complete equivalent. The equivalent series inductance of
a film capacitor (EC35
m
F, 500 V) is 35 nH. For the ESR model shown the
dielectric loss time constant,
t
d
=
R
d
C
d
is taken as 20 times the capacitor bulk
capacitance times series resistance time constant in order to model the dielectric
loss factor. For a 470
m
F ceramic dielectric capacitor the electric dissipation
factor,
D
e
, or, equivalently, the tangent of the loss angle is equal to 0.036 at 100 Hz.
Figure 4.25 illustrates the definition of the loss tangent.
X
c
D
e
= 1/
Q
e
= tan(
d
e
)
PF
= sin(
d
e
) = cos(
q
e
)
d
e
q
e
ESR
Figure 4.25 Construct for dc link capacitor dissipation factor from ESR
The capacitor loss factor is a measure of deviation from ideal capacitive
reactance caused by the presence of ESR. Equation (4.10) summarizes the defini-
tion of dissipation factor, or loss angle, in terms of the capacitor's conductivity,
permittivity and frequency:
1
Q
e
¼
tan
ð
s
we
d
e
Þ¼
tan
ð
90
D
e
¼
q
e
Þ¼
ð
4
:
10
Þ
D
e
ð
1
þ
D
e
Þ
PF
e
¼
sin
ðd
e
Þ¼
cos
ðq
e
Þ¼
p
For the ceramic capacitor example, the internal ESR is 0.122
W
at 100 Hz. If we
further assign values of 6 and 23 m
W
to the electrolyte and foil resistances, we obtain
a total package ESR = 151 m
W
. From these data the capacitor has an inherent time
10
6
=71
constant,
t
c
= ESR
s. Using the empirical relation
stated above, we assign a dielectric time constant
t
d
=20
t
c
= 1.46 ms, fromwhich the
dielectric capacitor value,
C
d
, computes to 12,000
C = 0.151
470
m
F. These data are then put in
the model shown in Figure 4.25 and the ESR solved as a function of frequency using
the empirical relation given in (4.11) [30]:
m
R
d
ð
1
þw
2
t
d
Þ
þ
R
e
e
ð
T
c
T
b
Þ
ESR
¼
þ
R
f
ð
4
:
11
Þ
se
