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ECE 421, Spring 2006
Solutions to HW Assignment #2
Problem B31
A parallel con
f
guration is de
f
ned by two or more blocks having the same input signal and the outputs
of the blocks being added together at a summing junction. Therefore, blocks
G
1
and
G
2
are in parallel,
and blocks
G
3
and
G
4
are in parallel. The equivalent block for a set of blocks in parallel is the sum of
the transfer functions in the individual blocks, taking into account the appropriate signs at the summing
junction. Therefore, the equivalent transfer function for the blocks
G
1
and
G
2
is
G
1
+
G
2
,
and the equivalent
transfer function for the blocks
G
3
and
G
4
is
G
3
−
G
4
.
The feedback con
f
guration is de
f
ned by the output of one block going to the input of the second block,
and the output of the second block going to the input of the
f
rst block, generally after being combined with
an external input signal in a summing junction. Therefore, the blocks
G
=
G
1
+
G
2
and
H
=
G
3
−
G
4
form
a feedback con
f
guration, with
G
be
ingintheforwardpathand
H
being in the feedback back (since
R
(
s
)
is
the input signal and
C
(
s
)
is the output signal). The
f
nal closedloop transfer function is
C
(
s
)
R
(
s
)
=
T
CL
(
s
)=
G
(
s
)
1+
G
(
s
)
H
(
s
)
=
G
1
+
G
2
1+(
G
1
+
G
2
)(
G
3
−
G
4
)
(1)
Problem B32
The
f
rst step is to interchange the order of the summing junctions in the upper left corner of the
diagram. Then combine the two blocks in parallel in the feedback path,
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 Spring '08
 Cook,G

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