block diagram representation -...
TRANSCRIPT
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Block Diagram Representation
This lecture we will concentrate on
● Representing system components with block diagrams
● Analyze and design transient response for systems consisting of multiple subsystems
● Reduce a block diagram of multiple systems to a single block representing the transfer function of overall system
The goal of block diagram representation is to obtain a simplified rule between the inputs and the output of the system.
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Basics
Transfer function
Adder (summer)
Take of point
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Example :
● Armature Controlled DC Motor
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Example (cont.)
Rearranging the overall system we have
In a simplified form we have
feedback form
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TF of a closed loop system
System having the following form is called closed-loop
: Forward Transfer function
: Feedback Transfer function
Note that
or
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Closed-loop Transfer Function
That is
The overall transfer function of the system
Note that
is the characteristic equation
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Block Diagram Simplification Rules
1- Cascade
2-Parallel
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Block Diagram Simplification Rules
3-Remove a block from the path
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Block Diagram Simplification Rules
4- Moving a Summing Junction
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Block Diagram Simplification Rules
5. Moving a take of point
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How about Multiple Inputs
For Linear systems you can apply Superposition !!!
Meaning when we have multiple inputs, we can tread them independently and sum the outputs
Procedure :
– Step 1 : Set all inputs except one to zero
– Step 2 : Calculate the response for the non-zero input
– Step 3 : Repeat steps 1 and 2 for all input
– Step 4 : Add all responses to obtain the overall
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Example
Find for
Solution :
Set U=0 Set R=0
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Complicated Examples
Example 1 :
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Step 1
Solution :
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Step 2
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And Finally
The solution
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Another Example
Example 2 :
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Step 1
Solution :
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Step 2
where
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And Finally
Combine all
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Examples from the Book
Example 3 : find the overall transfer function
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All Steps together :)
collapse summingjunctions
form equivalentcascaded systemin the forward pathand equivalentparallel system in thefeedback path;
form equivalentfeedback system andmultiply by cascadedG1(s)
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Alternative Approach
Signal Flow Diagrams
– Applied only to linear systems
– Equations must be in algebraic form
– represented by combination of nodes and braches
node : represents variables
branch : dependency of variables
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Signal Flow Diagrams
A path is a continous unidirectional successions of braches along which no node is passed more than once
As
or
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Signal Flow Diagrams (definitions)
An input node is a node with only outgoing branches
like
An output node is a node with only incomming braches as
A forward path is a path from an input node to an output node
A path gain is the product of the branch gains encountered
A loop gain is the product of the branch gains of the loop
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Construction of Signal Flow Diagrams
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Simplification of Signal Flow Graphs
Mason's Formulla
where
: the determinant of the graph
or
non- touching2-loops
non- touching3-loops
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Simplification of Signal Flow Graphs
Mason's Formulla
where
: the determinant of the graph
or
non- touching2-loops
non- touching3-loops
Path gain of kth forward path
The value of for the part of graph not touching kth forward path
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Example ( a warm up)
Simplifiy the following signal flow graph
Solution : Start with finding the forward paths
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Example (cont)
Find loops
Calculate and s
re calculate when is removed
The overall transfer function is then
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Another Example :
Simplifiy the Block Diagram
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Example
Solution:
First form the signal flow graph
Find the forward paths
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Example (cont.)
Find loops
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Example (still cont.)
Evaluate and s
For recalculate with removed
Similarly for recalculate with removed
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Example
Find overall transfer function
which is