Environmental Engineering Reference
In-Depth Information
dT
s
1
0
q
-1
V
+
u
N
0
V
-
(k+1) T
s
kT
s
Time
Figure 12.2
Signals
q
and
u
N
Assume that the switching frequency is
f
s
and the switching period is
T
s
=
1
/
f
s
. Define
1 when
S
+
is on and
S
−
is off
,
q
=
−
1 when
S
−
is on and
S
+
is off
.
Then, the corresponding voltage
u
N
, together with its given
q
, are shown in Figure 12.2.
Denote the average of
q
over a switching period by
p
. Then
p
∈
[
−
1
,
1] and the duty cycle
for
S
+
is
1
+
p
d
=
.
2
The average of
u
N
over a switching period (assuming that
V
+
and
V
−
are constant within a
switching period) is
p
2
V
DC
+
dV
+
+
(1
−
d
)
V
−
=
V
a
v
e
.
That is, approximately,
p
2
V
DC
+
u
N
=
V
a
v
e
.
(12.7)
Combining (12.3), (12.6) and (12.7), the block diagram of the neutral leg is shown in Fig-
ure 12.3, where
V
0
plays the role of an external disturbance and
p
is the control variable.
Strictly speaking, even this simplified model is non-linear, because
p
is multiplied with the
variable
V
DC
. However, since the variations of
V
DC
are relatively small, it is assumed that
the
V
DC
appearing in the first block in Figure 12.3 is equal to its nominal value
V
nom
DC
.Ifthe
variations of
V
DC
are expected to be large, then they can be compensated for by measuring
V
DC
and multiplying the output
p
of the controller with the correction factor
V
nom
DC
/
V
DC
.
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