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S ignals & S ystem s Chapter 4 Transfer Function Representation NC212 Signals and Systems : 2 / 2554
33

Chapter4 tf

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Page 1: Chapter4 tf

Signals & Systems

Chapter 4

Transfer Function Representation

INC212 Signals and Systems : 2 / 2554

Page 2: Chapter4 tf

Chapter 4 Transfer Function Representation

Overview Laplace transform of the differential equation Laplace transform of the convolution

integral Frequency response function Transfer functions

INC212 Signals and Systems : 2 / 2554

Page 3: Chapter4 tf

Transform of the I/O Differential Equation First-Order Case

)()()(

)(

)()(

)()0(

)(

sXsHsYas

bsH

sXas

bsY

sXas

b

as

ysY

H(s) Transfer Function (TF) of the system

)()0()()(

)()()0()(

)()()(

sbXysYas

sbXsaYyssY

tbxtaydt

tdy

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 4: Chapter4 tf

Transform of the I/O Differential Equation Example: First-Order Case

)(1

1

1

)0()(

)()0(

)(

sXRCs

RC

RCs

ysY

sXas

b

as

ysY

)(1

)(1)(

txRC

tyRCdt

tdy

0,1)0()(

1

11

1

)0()(

1

1

1

)0()(

)1()1(

teeyty

RCssRCs

ysY

sRCs

RC

RCs

ysY

tRCtRC

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 5: Chapter4 tf

Transform of the I/O Differential Equation Second-Order Case

012

01

012

01 )(;)()(asas

bsbsHsX

asas

bsbsY

)(

)0()0()0()(

)()()()0()()0()0()(

)()(

)()()(

012

01

012

1

01012

01012

2

sXasas

bsb

asas

yaysysY

sXbssXbsYayssYaysysYs

txbdt

tdxbtya

dt

tdya

dt

tyd

If initial condition = 0 :

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 6: Chapter4 tf

Transform of the I/O Differential Equation Example: Second-Order Case

x(t) = u(t) so that X(s) = 1/s; initial cond. = 0

86

2)(

)(2)(8)(

6)(

2

2

2

sssH

txtydt

tdy

dt

tyd

0,25.05.025.0)(

4

25.0

2

5.025.0)(

1

86

2)()()(

42

2

teety

ssssY

ssssXsHsY

tt

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 7: Chapter4 tf

Transform of the I/O Differential Equation Example: Second-Order Case

x(t) = u(t) with the initial condition

0,75.15.225.0)(

4

75.1

2

5.225.0)(

)86(

28

1

86

2

86

8)(

42

2

2

22

teety

ssssY

sss

sssssss

ssY

tt

2)0(

1)0(

y

y

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 8: Chapter4 tf

Transform of the I/O Differential Equation Nth-Order Case

011

1

01

011

1

01

)(

)()()(

)()(

asasas

bsbsbsH

sXasasas

bsbsbsX

sA

sBsY

NN

N

MM

NN

N

MM

)0()0()0()(

)(;)(

)()(

)(

)(

)()(;

)()()(

1

011

1011

1

0

1

0

yaysysC

asasassAbsbsbsbsB

sXsA

sB

sA

sCsY

dt

txdb

dt

tyda

dt

tyd

NN

NMM

MM

M

ii

i

i

N

ii

i

iN

N

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 9: Chapter4 tf

Transform of the I/O Convolution Integral

)(

)()(

)()(

)()()(

0,)()()()()(0

sX

sYsH

sHth

sXsHsY

tdtxhtxthtyt

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 10: Chapter4 tf

Transform of the I/O Convolution Integral Example: Determining the TF when

)(

)()(

1

1)(

4)2(

2

1

32)(

0,2cos32)(

2

2

sX

sYsH

ssX

s

s

sssY

tteety tt

sss

ssss

sssssss

ss

s

ss

ss

sssH

84

162]4)2[(

)2)(1(]4)2][(3)1(2[4)2(

)2)(1(3

)1(21

14)2(

21

32

)(

23

2

2

2

2

2

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

)()( tuetx t

Page 11: Chapter4 tf

Transform of the I/O Convolution Integral Finite-Dimensional Systems

M

ii

i

i

N

ii

i

iN

N

MM

MM

NN

N

NN

N

MM

MM

dt

txdb

dt

tyda

dt

tyd

sXbsbsbsbsYasasas

sX

sY

asasas

bsbsbsbsH

0

1

0

011

1011

1

011

1

011

1

)()()(

)()()()(

)(

)()(

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554

Page 12: Chapter4 tf

Transform of the I/O Convolution Integral Poles and zeros of a Systems

)())((

)())(()(

)(

21

21

011

1

011

1

N

MM

NN

N

MM

MM

pspsps

zszszsbsH

asasas

bsbsbsbsH

zi : “zeros of H (s)” or “zeros of system”pi : “poles of H (s)” or “poles of system”N : “number of poles of system” or “order N of system”

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 13: Chapter4 tf

Transform of the I/O Convolution Integral Example: Third-Order System

jpjpp

jzjz

jsjss

jsjssH

sss

sssH

1,1,4

3and3

)1)(1)(4(

)3)(3(2)(

8106

20122)(

321

11

23

2

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 14: Chapter4 tf

Frequency Response Functions LTI system is characterized by its impulse

response, h(t), in time domain.

H() is the CTFT of its impulse response, h(t).

It is also characterized by its frequency response or transfer function, H(), in frequency domain.

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 15: Chapter4 tf

Frequency Response Functions LTI systems

h(t)x(t) y(t) H()X() Y()

h(t) is Impulse Response H() is Frequency Response function

dtxhtxthty )()()()()( )()()( XHY

)()()( XHY

)()()( XHY

dtth )(

Assume that the system is stable:

Amplitude :

Phase :

Time domain Frequency domain

F

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 16: Chapter4 tf

Defining equations for R, L and C

)(

)(

)(

)()(

I

V

V

VH

in

out

RZ

RI

V

R

)(

)(

)(

LjZ

LjI

V

L

)(

)(

)(

CjZ

CjI

V

C

1)(

1

)(

)(

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 17: Chapter4 tf

Practical Passive Filters The RC Lowpass filter

ZR()

ZC()

+

_

+

_

Vin() Vout()

1

1)(

)()(

RCj

V

VH

in

out

)(1

1

)(1

1

)()()(

)()(

in

in

in

RC

Cout

VRCj

VRCj

Cj

VZZ

ZV

Voltage divider

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 18: Chapter4 tf

+

_

+

_

vin(t) vout(t)

ZR()

+

_

+

_

Vin() Vout()

ZL()

Practical Passive Filters The RL Lowpass filter

?)( H)(?)( inout VV Voltage divider

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 19: Chapter4 tf

ZR()+

_

+

_

Vin() Vout()ZL()ZC()

+

_

+

_

vin(t) vout(t)

Practical Passive Filters The RLC Bandpass filter

LCRCjj

RCjV

VH

in

out

1)()(

)(

)()(

2

)(?)( inout VV

Voltage divider

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 20: Chapter4 tf

ZR()+

_

+

_

Vin() Vout()ZC()

Z1()

Z2()+ V1() -

ZR()+

_

+

_

Vin() Vout()ZC()

Z1()

Z2()+ V1() -

Transfer Function Series and Parallel Connection

)()()(

)()(

)()()(

)()(

21

12

21

21

in

in

IZZ

ZI

IZZ

ZI

)()()(

)()(

)()()(

)()(

21

2

21

11

inout VZZ

ZV

VZZ

ZV

ZL()ZC()Z1() Z2()

I1() I2()Iin()

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 21: Chapter4 tf

Transfer Function Interconnections of Integrators

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 22: Chapter4 tf

Transfer Function Parallel Interconnection

H1()

H2()

X()

X()

X()

Y1()

Y2()

Y ()

)()()(

)())()((

)()()()()(

)()()(

)()()(

)()()(

21

21

21

22

11

21

HHH

XHH

XHXHY

XHY

XHY

YYY

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 23: Chapter4 tf

Transfer Function Series Connection

H1() H2()Y1()X() Y2() Y ()

)()()()()(

)()()()()(

)()()(

)()()(

2112

122

122

11

HHHHH

XHHYY

YHY

XHY

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 24: Chapter4 tf

H1()X()

H2()

Y ()Y1()

Y ()Y2()

X1 ()

Transfer Function Feedback Connection : Negative

feedback

)()(1

)()(

)()()(1

)()(

)]()()()[()(

)()()(

)()()(

)()()(

21

1

21

1

21

2

21

11

HH

HH

XHH

HY

YHXHY

YHX

YXX

XHY

)()(1

)()(

21

1

HH

HH

Positive feedback

Negative feedback

Negative feedbackPositive feedback

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 25: Chapter4 tf

Direct Construction of the TF RLC Circuits

)0()()(

)()(

LisLsIsVdt

tdiLtv

)0(1

)(1

)(

)(1

)0()(

)(1)(

vs

sICs

sV

sIC

vssV

tiCdt

tdv

)()(

)()(

sRIsV

tRitv

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 26: Chapter4 tf

Direct Construction of the TF Series and Parallel Connection

)()()(

)()(

)()()(

)()(

21

12

21

21

sIsZsZ

sZsI

sIsZsZ

sZsI

)()()(

)()(

)()()(

)()(

21

22

21

11

sVsZsZ

sZsV

sVsZsZ

sZsV

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 27: Chapter4 tf

Direct Construction of the TF Example: Series RLC Circuit

)1()(

1)(

)()1()(

1

)()1(

1)(

2

2

LCsLRs

LCsH

sXLCsLRs

LC

sXCsRLs

CssVc

)1()(

)()(

)()1()(

)(

)()1(

)(

2

2

LCsLRs

sLRsH

sXLCsLRs

sLR

sXCsRLs

RsVR

Output = VR(s)Output = VC(s)

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 28: Chapter4 tf

Direct Construction of the TF Interconnections of Integrators

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 29: Chapter4 tf

TF of Block Diagrams Parallel Interconnection

)()()(

)())()((

)()()()()(

)()()(

)()()(

)()()(

21

21

21

22

11

21

sHsHsH

sXsHsH

sXsHsXsHsY

sXsHsY

sXsHsY

sYsYsY

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 30: Chapter4 tf

TF of Block Diagrams Series Connection

)()()()()(

)()()()()(

)()()(

)()()(

2112

122

122

11

sHsHsHsHsH

sXsHsHsYsY

sYsHsY

sXsHsY

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 31: Chapter4 tf

TF of Block Diagrams Feedback Connection

)()(1

)()(

)()()(1

)()(

)]()()()[()(

)()()(

)()()(

)()()(

21

1

21

1

21

2

21

11

sHsH

sHsH

sXsHsH

sHsY

sYsHsXsHsY

sYsHsX

sYsXsX

sXsHsY

)()(1

)()(

21

1

sHsH

sHsH

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Negative feedbackPositive feedback

Page 32: Chapter4 tf

Direct Construction of the TF Example:

127

178

)4)(3(

178)(

)()4)(3(

178

)()()4)(3(

5)()()(

)()4)(3(

5

)(14

1

3

1

)]()([3

1)(

)()(3)()(

)(4

1)(

)()(4)(

2

22

2

2

12

212

1

11

ss

ss

ss

sssH

sXss

ss

sXsXss

ssXsQsY

sXss

s

sXss

sXsQs

sQ

sXsQsQssQ

sXs

sQ

sXsQssQ

INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation

Page 33: Chapter4 tf

Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554