# Unit 4: Block Diagram Reduction Block Diagram Reduction by kxb86934

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```									                   Block Diagram Reduction                                                       Block Diagram Reduction
Signal-Flow Graphs                                                            Signal-Flow Graphs

1   Block Diagram Reduction
Unit 4: Block Diagram Reduction                                         Cascade Form
Parallel Form
Engineering 5821:                                      Feedback Form
Control Systems I
Moving Blocks

Faculty of Engineering & Applied Science                          Example
Memorial University of Newfoundland

1   Signal-Flow Graphs
February 27, 2009

ENGI 5821     Unit 4: Block Diagram Reduction                                 ENGI 5821     Unit 4: Block Diagram Reduction

Parallel Form
Block Diagram Reduction
Feedback Form
Signal-Flow Graphs
Moving Blocks
Example

First we summarize the elements of block diagrams:
Block Diagram Reduction
Subsystems are represented in block diagrams as blocks, each
representing a transfer function. In this unit we will consider how
to combine the blocks corresponding to individual subsystems so
that we can represent a whole system as a single block, and
therefore a single transfer function. Here is an example of this
reduction:

We now consider the forms in which blocks are typically connected
and how these forms can be reduced to single blocks.
Reduced Form:

ENGI 5821     Unit 4: Block Diagram Reduction
Parallel Form
Block Diagram Reduction
Feedback Form
Signal-Flow Graphs
Moving Blocks
Example

That is, a subsystem’s output remains the same no matter what
When multiple subsystems are connected such that the output of                 the output is connected to. If another subsystem connected to the
one subsystem serves as the input to the next, these subsystems                output modiﬁes that output, we say that it loads the ﬁrst system.
are said to be in cascade form.                                                Consider interconnecting the circuits (a) and (b) below:

The overall TF is not the product of the individual TF’s!
The algebraic form of the ﬁnal output clearly shows the equivalent
system TF—the product of the cascaded subsystem TF’s.

ENGI 5821     Unit 4: Block Diagram Reduction

Parallel Form
Block Diagram Reduction
Feedback Form
Signal-Flow Graphs
Moving Blocks
Example
should have a high input impedance so it does not load its source,
Parallel Form
and low output impedance so it appears as a pure voltage source                Parallel subsystems have a common input and their outputs are
to the subsystem it feeds into.                                                summed together.

If no actual gain is desired then K = 1 and the “ampliﬁer” is
referred to as a buﬀer.
The equivalent TF is the sum of parallel TF’s (with matched signs
at summing junction).
ENGI 5821     Unit 4: Block Diagram Reduction
Feedback Form

Systems with feedback typically have the following form:

We can easily establish the following two facts:

E (s) = R(s) ∓ C (s)H(s)
Noticing the cascade form within the feedforward and feedback
C (s) = E (s)G (s)
paths we can simplify:
We can now eliminate E (s) to obtain,

G (s)
Ge (s) =
1 ± G (s)H(s)

Moving Blocks

A system’s block diagram may require some modiﬁcation before    Or we may need to move blocks to the left or right of a pickoﬀ
the reductions discussed above can be applied.                  point:
We may need to move blocks either to the left or right of a
summing junction:
Parallel Form
Block Diagram Reduction
Feedback Form
Signal-Flow Graphs
Moving Blocks
Example

Example

Reduce the following system to a single TF:

We can now recognize the parallel form in the feedback path:

First we can combine the three summing junctions together...                 We now have G1 cascaded with a feedback subsystem:

ENGI 5821     Unit 4: Block Diagram Reduction

Example 2

Reduce the following more complicated block diagram:

Reduce parallel form involving 1/G2 and unity
Steps:
Push G1 to the right past the summing junction to create a
Rightmost feedback loop can be reduced                                       parallel form in the feedback path
Create parallel form by moving G2 left
Reduce parallel form on left                                      Reduce feedback form on left

Signal-Flow Graphs

We can convert the cascaded, parallel, and feedback forms into
signal-ﬂow graphs:

Signal-ﬂow graphs are an alternative to block diagrams. They
consist of branches which represent systems (a) and nodes which
represent signals (b). Multiple branches converging on a node
implies summation.

V (s) = R1 (s)G1 (s) − R2 (s)G2 (s) + R3 (s)G3 (s)
C1 (s) = V (s)G4 (s)
C2 (s) = V (s)G5 (s)
C3 (s) = V (s)G6 (s)
e.g. Convert the following block diagram to a signal-ﬂow graph:

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