Environmental Engineering Reference
In-Depth Information
8.3 Stability Analysis
8.3.1 Without Consideration of the Sampling Effect
Assume that the load is passive and its impedance is
Z
(
s
). Furthermore, assume that the ESR
of the filter capacitor is
r
>
0. Then
⎛
⎞
1
Z
(
s
)
+
1
1
Z
c
(
s
)
v
o
,
⎝
⎠
v
o
=
=
i
(8.4)
1
sC
r
+
where
Z
(
s
)
r
1
sC
1
+
srC
sC
+
Z
c
(
s
)
=
=
1
sC
1
+
srC
sC
1
Z
(
s
)
Z
(
s
)
+
r
+
1
+
is the impedance of the augmented load after regarding the filter capacitor as part of the load
and can be represented as the feedback loop shown in Figure 8.5. Since the impedance
Z
(
s
)
of any passive network must be positive real (Smith 2002; van der Schaft 1996), the Nyquist
plot of the loop transfer function
1
+
srC
sC
1
1,0) on the
s
-plane, which means
Z
c
(
s
) is always stable for any passive
Z
(
s
). It is worth noting that the
non-zero ESR of the capacitor
r
plays an important role in guaranteeing the stability. If
r
Z
(
s
)
does not encircle the critical point (
−
,
then the parallel connection of the load and the capacitor might be oscillatory. For example, a
purely inductive load forms an oscillator with the capacitor when
r
=
0
0.
Substituting (8.4) into (8.1), then the closed-loop transfer function
G
(
s
) from
=
v
r
to
v
o
is
Z
c
(
s
)
Z
o
(
s
)
=
G
(
s
)
+
Z
c
(
s
)
+
(1
K
R
(
s
))
Z
c
(
s
)
=
(
sL
+
K
i
)
+
(1
+
K
R
(
s
))
Z
c
(
s
)
H
(
s
)
=
H
(
s
)
,
1
+
i
Z
c
(
s
)
i
o
-
1
+
srC
1
sC
Z
(
s
)
v
o
Figure 8.5
Augmented load
Z
c
(
s
)
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