laplacetransform…transfer function

7/30/2019 LaplaceTransformTransfer Function

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EECE 301

Signals & Systems

Prof. Mark Fowler

Note Set #29

C-T Systems: Laplace Transform Transfer Function Reading Assignment: Section 6.5 of Kamen and Heck


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Ch. 1 Intro

C-T Signal Model

Functions on Real Line

D-T Signal Model

Functions on Integers

System Properties

LTICausal

Etc

Ch. 2 Diff EqsC-T System Model

Differential Equations

D-T Signal ModelDifference Equations

Zero-State Response

Zero-Input Response

Characteristic Eq.

Ch. 2 Convolution

C-T System Model

Convolution Integral

D-T Signal Model

Convolution Sum

Ch. 3: CT FourierSignalModels

Fourier Series

Periodic Signals

Fourier Transform (CTFT)

Non-Periodic Signals

New System Model

New Signal

Models

Ch. 5: CT FourierSystem Models

Frequency Response

Based on Fourier Transform

New System Model

Ch. 4: DT Fourier

SignalModels

DTFT

(for Hand Analysis)DFT & FFT

(for Computer Analysis)

New Signal

Model

Powerful

Analysis Tool

Ch. 6 & 8: LaplaceModels for CT

Signals & Systems

Transfer Function

New System Model

Ch. 7: Z Trans.

Models for DT

Signals & Systems

Transfer Function

New System

Model

Ch. 5: DT Fourier

System Models

Freq. Response for DT

Based on DTFT

New System Model

Course Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between

the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).


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6.5 Transfer FunctionWeve seen that the system outputs LT is:

)()(

)(

)(

)()( sX

sA

sB

sA

sCsY +=

Part due to ICs

So, if the system is in zero-state then we only get the second term:

)()(

)()( sXsA

sBsY =

)()()( sXsHsY =

Define:)(

)()(

sA

sBsH

=

System effect in zero-state case is completely set by the transfer function

Part due to Input


4/17

Note: If the system is described by a linear, constant coefficient differential

equation, we can getH(s) by inspection!! Lets see how

)()()()()( 01012

sXbssXbsYassYasYs +=++

)()(

)()()(

01012

2

txbdt

tdxbtya

dt

tdya

dt

tyd+=++

To illustrateTake the LT of a Diff. Eq. under the zero-state case:

Solve for Y(s) and identify theH(s):)()(

)(01

2

01 sXasas

bsbsY

sH44 344 21

=

++

+=

With zero ICs we have that each higher derivative

corresponds to just another power ofs.

We can then apply this idea to get the Transfer Function

)()0()0()()( )1(21)( txxsxssXstx nnnnn L&Recall:


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)()()()()( 01012

2

txbdt

tdxbtyadt

tdyadt

tyd +=++

01

2

01)(

asas

bsbsH

++

+=

So now it is possible to directly identify the TFH(s) from the Diff. Eq.:


6/17

But, we have also seen that for the zero-state case the system output is:

)(*)()()()(0

txthdtxhtyt

==

ButWe have an LT property for convolution that says:

)(*)()()()()( txthtysXsHsY ==

where: { })()( thsH L=

{ }ResponseImpulseFunctionTransfer L=

So, we have two ways to getH(s)

- Inspect the Diff. Eq. and identify the transfer functionH(s)

- Take the LT of the impulse response h(t)

This gives an easy way to get the impulse response from a Diff. Eq.:- Identify the H(s) from he Diff. Eq. and then find the ILT of that


7/17

Recall: If the ROC ofH(s) includes thejaxis, then

js

sHH=

= )()(

This is the connection between

Note that the transfer function does essentially the same thing that the frequency

response does

)(*)()()()()( txthtysXsHsY ==

)(*)()()()()( txthtyXHY ==

Recall that the LT is a generalized, more-powerful version of the FT this

result just says that we can do the same thing withH(s) that we did withH(),

but we can do it for a larger class of systems

There are some systems for which we can use either method those are the

ones for which the ROC ofH(s) includes thejaxis.


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Poles and Zeros of a system

Given a system with Transfer Function:

01

1

1

011

1

...

...)(

asasas

bsbsbsbsH

N

N

N

MM

MM

++++++++=

)(sB

)(sA

We can factorB(s) andA(s): (Recall:A(s) = characteristic polynomial)

))...()((

))...()(()(

21

21

N

MM

pspsps

zszszsbsH

=

Assume any common factors inB(s) andA(s) have been cancelled out

So we know thatH(s) is completely described by the Diff. Eq. Therefore

we should expect that we can tell a lot about a system by looking at the

structure of the transfer functionH(s) This structure is captured in the idea

of Poles and Zeros

Note: MisHizs

...,,2,10)( ===

NisHips

...,,2,1)( ===

{ } )"(ofpoles"calledare sHpi

Note: pi are the roots of the char. polynomial

{ } )"(ofzeros"calledare sHzi


9/17

Note that knowing the sets { } { }Nii

M

iipz

11&

==

tells us whatH(s) is: (up to the multiplicative scale factor bM)-bMis like a gain (i.e. amplification)

Pole-Zero Plot

This gives us a graphical view of the systems behavior

jplane-s

1

1

Pole-Zero Plot for thisH(s)

)1)(1)(4(

)3)(3(2

8106

20122)(

23

2

jsjss

jsjs

sss

sssH

++++

+++=

+++

++=

Real coefficients complex conjugate pairs

Example:


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From the Pole-Zero Plots we can Visualize the TF function on the s-plane:


11/17


12/17

As the pole moves closer to thej

axis it has a stronger effect on the

frequency responseH(). Poles

close to thejaxis will yield

sharper and taller bumps in the

frequency response.

By being able to visualize what |H(s)|will look like based on where the poles

and zeros are, an engineer gains the

ability to know what kind of transfer

function is needed to achieve a desiredfrequency response then through

accumulated knowledge of electronic

circuits (requires experience

accumulated AFTER graduation) the

engineer can devise a circuit that will

achieve the desired effect.

Can also look at a pole-zero plot and see the effects on Freq. Resp.


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Differential

Equation

Differential

Equation

Transfer

Function

Transfer

Function

FrequencyResponse

FrequencyResponse

Impulse

Response

ImpulseResponse

Fourier Transform

Laplace Transform

Inspection in Practice

Time Domain Freq Domain

Continuous-Time System Relationships

h(t)

011

1

0111)(asasas

bsbsbsbsHN

NN

MMMM

++++++++=

L

L

jssHH == |)()(

s = j

)()()()(

)()()()(

011

1

1

011

1

1

txbdt

tdxb

dt

txdb

dt

txdb

tyadt

tdya

dt

tyda

dt

tyd

M

M

MM

M

M

N

N

NN

N

++++

=++++

L

L

Char.

Modes

LT in Theory

Pole-Zero

Diagram

Pole-Zero

Diagram


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1. From the differential equation you can get:

a. Transfer function, then the impulse response, the pole-zeroplot, and if allowable you can get the frequency response

2. From the impulse response you can get:

a. Transfer function, then the Diff. Eq., the pole-zero plot, andif allowable you can get the frequency response

3. From the Transfer Function you can get:

a. Diff. Eq., the impulse response, the pole-zero plot, and ifallowable you can get the frequency response

4. From the Frequency Response you can get:

a. Transfer function, then the Diff. Eq., the pole-zero plot, andthe impulse response

5. From the Pole-Zero Plot you can get:

a. (up to a scaling factor) Transfer function, then the Diff. Eq.,the impulse response, and possible the Frequency Response

In practice you may need to start your work in any spot on this diagram


15/17

Similarly, we can get the transfer function using the s-domain impedances:

sLsZsC

sZRsZ LCR === )(1

)()(

Recall: We get the frequency response from a circuit by using frequency

dependent impedances

LjZCj

ZRZ LCR

=== )(1

)()(

and then doing circuit analysis.

In Practice we often get the transfer function from a circuit Heres an example:


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Example 6.37 (Assume zero-state)

)(tx )(ty

+

)(sY

+

)(sX

Now we treat everything here as if we have a

DC Circuit which leads to simple algebraic

manipulation rather than differential equations!

For this circuit the easiest approach is to use the Voltage Divider

)()/1(

/1

)( sXCsLsR

Cs

sY

++=

)()/1()/(

/1

2 sXLCsLRs

LC

++=

)(sH

A standard form:

a ratio of two polynomials in s

unity coefficient on the highest power in the denominator


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Once you start including linear amplifiers with gain > 1 this may not be true

If you include non-linear devices the system becomes non-linear

But, you may be able to linearize the system over a small operating range

E.g. A transistor can be used to build a (nearly) linear amplifier even though

the transistor is itself a non-linear device

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laplacetransform…transfer function

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