3. transfer functionz\ \z

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)

3. transfer functionz\ \z

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