Transcript
Signals & Systems
Chapter 4
Transfer Function Representation
INC212 Signals and Systems : 2 / 2554
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
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
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
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
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
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
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
Transform of the I/O Convolution Integral
)(
)()(
)()(
)()()(
0,)()()()()(0
sX
sYsH
sHth
sXsHsY
tdtxhtxthtyt
Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554
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
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
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
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
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
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
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
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
+
_
+
_
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
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
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
Transfer Function Interconnections of Integrators
INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation
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
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
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
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
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
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
Direct Construction of the TF Interconnections of Integrators
INC212 Signals and Systems : 2 / 2554 Chapter 4 Transfer Function Representation
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
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
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
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
Chapter 4 Transfer Function RepresentationINC212 Signals and Systems : 2 / 2554
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