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7/28/2019 3. Transfer Functionz\ \z
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Transfer Functions, Block Diagram
and Signal Flow Graph
Transfer Function
A general n-th order LTIV differential equation (DE),
where c(t) is the output, r(t) is the input and as , bs are the
coefficients of the DE that represent the system. Taking Laplace,
If we assume all initial condition are zero,
The transfer function of the system is
Notice that the system output could be obtained using
( ) ( ) ( )sRsGsC = (2.54)
The transfer function can be represented as a following block
diagram.
LTIV : Linear Time Invariant
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The roots of numerator are called zeros and roots of denominator are
called poles.
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Block Diagrams
Basic components of a block diagram for a LTIV system
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Cascade or series subsystems,
Parallel Subsystems,
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Feedback Form
a. Feedback control system;
b. simplified model;
c. equivalent transfer function
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Moving blocks to create familiar forms,
Moving block to the left
past a summing junction
Moving block to the right
past a summing junction
Moving block to the left
past a pickoff point
Moving block to the right
past a pickoff point
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Example 1
Reduce the following block diagram to form a single transfer
function.
Solution,
G Simplify by using
feedback loop formula
H
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In class Exercise
Reduce the block diagram shows below to a single transfer function.
.
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Signal Flow Graphs
SFG may be viewed as a simplified form of block diagram. SFG
consists of arrows (represent systems) and nodes (represent signals).
Signal-flow graph components:
a. system;
b. signal;
c. interconnection of systems and signals
Converting common block diagrams to SFG
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Converting a block diagram to SFG
Signal-flow graph development:
a. signal nodes;b. signal-flow graph;
c. simplified signal-flow graph
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Mason Gain Formula
The transfer function of a given system represented by a SFG is:
( )( )
( )
==
k
kkT
sR
sCsG
where
k = no. of paths
kT = the kth forward-path gain
= 1 - loop gains + non-touching loop gains 2 at a time -
non-touching loop gains 3 at a time + non-touching
loop gains 4 at a time -
Example 1
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Example 2
Use Masons Gain formula to obtain the transfer function of the
system represented by the following SFG.
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Poles and Zeros
Consider a transfer function
Singular point of )(sF approaching to infinity is when 0))....()(( 21 =+++ npspsps . The roots are called poles
nppp ....,,, 21 .
Singular point of )(sF approaching to zero is when 0))....()(( 21 =+++ mzszszs . The roots are called zeros
mzzz ....,,, 21 .
Example:
sss
sssF
206
655)(
23
2
++
++=
)3166.33)(3166.33(
)2)(3(5)(
jsjss
sssF
+++
++=
j
Satah-s3.3166
-3 -2
-3.3166
In general transfer function can be written as
whereKdc gainType: Highest factored s of the denominator, n
Order: Highest order s of the denominator, n+jRank: Difference between the numerator and denominator, n+j-i
Example:
234
2
2062
655)(
sss
sssF
++
++=
))....()((
))....()(()(
21
21
n
m
pspsps
zszszsKsF
+++
+++=
( )( ) (( )( ) (...11
1...11)(
21
21
+++
+++=
sTsTsTs
sTsTsTKsF
bjbbn
aiaa
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)2062(
655)(
22
2
++
++=
sss
sssF
Type 2,Order4
Rank 4-2=2.
Take home Quiz(Due date:11/10/2012 before 3pm)
Use (i) the Masons rule and (ii) block reduction method, to find the transfer function of the figure below.
Compare your answer.
(a)