Transfer Function and Their Response

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Transfer Function andTheir Response

Student: Nguyen Tien D

ID: 20100908

Class: Mechatronic - K

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Content

Introduction of transfer function(TF)

Block Diagram(BD)

MATLAB representation of Transfer Func

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The Laplace Transform

Definition

L f t( )( )

0

tf t( ) e s t

d = F(s)

Here the complex frequency is s j w

The Laplace Transf orm exists w hen

0

tf t( ) e s t

d this means that the integral converges

The Laplace Transform

the Laplace transfo rmconverts integral and differential

equations into algebraic equations

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Transfer functions (TF) are frequently used to characterize

relationships or systems that can be described by Linear differential equations.

Transfer Function (TF): The transfer function (TF) of a LTI diffsystem is defined as the ratio of the Laplace transform (L(response function) to the Laplace transform (LT) of the inputunder the assumption that all initial conditions are zero.

Introduction of transfer function(TF)4


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Assume that our model is linear differential equation have the f

where D denotes the differential operator d/dt . A(D) and B(D) are polyno

Di= , the I th derivative, u(t) is the model input and y(t) its output.

write


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Transfer Function (TF): G(s)

() =()

()=

()

()


G(s)U(s) Y(s)

InputOutput

Transfer Function

Block diagram representation of a transfer function

U(s)

Laplace transform of the output variableY(s) Laplace transform of the input variableG(s) transfer function

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7 Impulse Response Function


Now, consider the output (response) of the system to a unit-impulse

when all the I.Cs are zero. Since

the LT of the output of the system is

is called the impulse response function of the system. The impulse-r

thus the response of a linear system to a unit impulse input when the

The LT of g(t ) gives the TF.


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8Block Diagrams

Block Diagrams of Dynamic Systems

A Block Diagram (BD) of a dynamic system is a graphic representafunctions performed by each component of the system and of th

within the system


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Summing Point

Block Diagrams

Figure shows a circle with a cross, the symbol that stands for a summ(+) or (-) sign at each arrowhead indicates whether the associaadded or subtracted.

It is important that the quantities being added or subtracted have thand the same units.

Summing point


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10Block Diagrams

Block Diagram of a closed-loop system

The output C(s) is fed back to the summing point, where it is coinput R(s). The closed loop nature of the system is indicatedfigure. The output C(s) is obtained by multiplying the TF G(s) the block, E(s).

Block Diagram of a closed loop-system with feedback elem


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11MATLAB representation of Transfer Functions

The transfer function of a system is represented by two arrays ofFor example, consider a system defined by

This system is represented as two arrays, each containing thethe polynomials in descending powers of s as follows

MATLAB will automatically respond with the display


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12MATLAB representation of Transfer Functions

Partial-Fraction Expansion with MATLAB

MATLAB allows us to obtain the partial-fraction expansion of theratio of two polynomials

The command

The partial fraction expansion of B(s) A(s) is given by


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MATLAB representation of Transfer Functions

Example 1:


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Example 1:


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Example 2: G(s)=

G(s)=


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Transfer Function and Their Response

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