2. Objectives
To reduce a block diagram of multiple
subsystems to a signal block representing
the transfer function from input to output
3. Introduction
Before this we only worked with individual
subsystems represented by a block with
its input and output.
Complex systems are represented by the
interconnection of many subsystems.
In order to analyze our system, we want
to represent multiple subsystems as a
single transfer function.
4. Block diagram
A subsystems is represented as a block
with an input and output and a transfer
function.
Many systems are composed of multiple
subsystems. So, we need to add a few
more schematic elements to the block
diagram.
Summing junction
Pickoff points
6. Block diagram
Summing junction
Output signal, C(s), is the algebraic sum of the
input signals, R1(s), R2(s) and R3(s).
Pickoff point
Distributes the input signals, R(s),
undiminished, to several output points.
7. Block diagram
There are three topologies that can be
used to reduce a complicated system to a
single block.
Cascade form
Parallel form
Feedback form
8. Block diagram
Cascade form
a. cascaded subsystem
b. equivalent transfer function
Equivalent transfer function is the output
divided by the input.
9. Block diagram
Parallel form
Parallel subsystems have a common input and
output formed by the algebraic sum of the
outputs from all of the subsystems.
10. Block diagram
Feedback form
It is the same as the closed loop system that
we learn in Chapter 1.
a. closed loop system
b. closed loop, G(s)H(s) is open loop transfer
function
11. Block diagram
Moving blocks to create familiar forms
Cascade, parallel and feedback topologies are
not always apparent in a block diagram.
You will learn block moves that can be made in
order to establish familiar forms when they
almost exist. I.e. move blocks left and right
past summing junctions and pickoff points.
12. Block diagram
Block diagram
algebra for summing
junctions—
equivalent forms for
moving a block
a. to the left past a
summing junction;
b. to the right past a
summing junction
13. Block diagram
Block diagram
algebra for pickoff
points—
equivalent forms
for moving a
block
a. to the left past
a pickoff point;
b. to the right
past a pickoff
point
14. Block diagram
Block diagram reduction via familiar forms
Example:
Reduce the block diagram to a single
transfer function.
15. Block diagram
Solution:
Steps in solving
Example 5.1:
a. collapse summing
junctions;
b. form equivalent
cascaded system
in the forward path
and equivalent
parallel system in the
feedback path;
c. form equivalent
feedback system and
multiply by cascaded
G1(s)
16. Block diagram
Block diagram reduction by moving blocks
Example:
Reduce the system shown to a single
transfer function.
17. Block diagram
Solution:
First, move G2(s) to the left past the pickoff point
to create parallel subsystems, and reduce the
feedback system consisting of G3(s) and H3(s).
18. Block diagram
Second, reduce the parallel pair consisting of
1/g2(s) and unity and push G1(s) to the right past
the summing junction, creating parallel subsystems
in the feedback.
19. Block diagram
Third, collapse the summing junctions, add the two
feedback elements together, and combined the last
two cascaded blocks.
20. Block diagram
Fourth, use the feedback formula to obtain
figure below
Finally multiply the two cascaded blocks and
obtain the final result.
22. Solution
Combine the parallel blocks in the forward path. Then, push 1/s to
the left past the pickoff point.
Combine the parallel feedback paths and get 2s. Apply the
feedback formula and simplify
32. Mason’s rule
What?
A technique for reducing signal-flow graphs to
single transfer function that relate the output
of system to its input.
We must understand some components
before using Mason’s rule
Loop gain
Forward-path gain
Nontouching loops
Nontouching-loop gain
33. Mason’s rule
Loop gain
Product of branch gains found by going through a path
that starts at a node and ends at the same node,
following the direction of the signal flow, without passing
through any other node more than once.
G2(s)H1(s)
G4(s)H2(s)
G4(s)G5(s)H3(s)
G4(s)G6(s)H3(s)
34. Mason’s rule
Forward-path gain
Product of gains found by going through a path from the
input node of the signal-flow graph in the direction of
signal flow.
G1(s)G2(s)G3(s)G4(s)G5(s)G7(s)
G1(s)G2(s)G3(s)G4(s)G6(s)G7(s)
35. Mason’s rule
Nontouching loops
Loops that do not have any nodes in common.
Loop G2(s)H1(s) does not touch loops G4(s)H2(s),
G4(s)G5(s)H3(s) and G4(s)G6(s)H3(s)
36. Mason’s rule
Nontouching-loop gain
Product of gains form nontouching loops taken
two, three, four, or more at a time.
[G2(s)H1(s)][G4(s)H2(s)]
[G2(s)H1(s)][G4(s)G5(s)H3(s)]
[G2(s)H1(s)][G4(s)G6(s)H3(s)]
37. Mason’s rule
The transfer function, C(s)/R(s), of a system
represented by a signal-flow graph is
∆
∆
=
=
∑
k
k
k
T
s
R
s
C
s
G
)
(
)
(
)
(
gain
path
-
forward
kth
the
path
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of
number
=
=
k
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k
