filters transfer functions/bode plots frequency domain ... · enzo paterno 32 effect of poles not...
Post on 28-Mar-2020
7 Views
Preview:
TRANSCRIPT
Enzo Paterno 1
Frequency domain Analysis
Transfer Functions
Bode Plot Generation
EP
Revision
3/24/15
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 2
A transfer function is defined as the ratio of the frequency domain output
voltage to the frequency domain input voltage (i.e. Gain) with all initial
conditions equal to zero. Transfer functions are defined only for linear
systems.
Transfer functions can usually be expressed as the ratio of two polynomials
in the complex variable, s
A transfer function can be factored into the following form.
)(...))((
)(...))(()(
21
210
n
m
i pspsps
zszszsK
V
VsH
The roots of the numerator polynomial are called zeros
The roots of the denominator polynomial are called poles
FILTERS – Transfer Functions/Bode Plots
) (i.e. js
Vi (ω) G(ω)
Vo (ω)
Enzo Paterno 3
Given the transfer function. Plot the poles and zeros in
the s-plane.
)10)(4(
)14)(8()(
sss
sssH
S - plane
x x o x o 0 -4 -8 -10 -14
origin
axis
j axis
FILTERS – Transfer Functions/Bode Plots
S-plane
These zeroes and poles are real
Thus, they get plotted on the real axis
real
Imaginary
0
0
Enzo Paterno 4
FILTERS – Transfer Functions/Bode Plots
a
acbbrr
cbrar
2
4,
0
2
21
2
1.
2.
3.
1. b2 – 4ac > 0
2.
3.
b2 – 4ac = 0
b2 – 4ac < 0
)2)(2(4.3
)2)(2()2(.2
)3)(2(65.1
2
2
2
jsjss
sss
ssss
Quadratic Equation
Discriminant
real
Imaginary
0
0
1 2
3
Enzo Paterno 5
Given the transfer function. Plot the poles and zeros in
the s-plane.
)4(
)14)(8()(
2
ss
sssH
S - plane
x o
x
o 0 -8
-2j
-14
origin
axis
j axis
FILTERS – Transfer Functions/Bode Plots
S-plane real
Imaginary
0
0
x
2j
Enzo Paterno 6
FILTERS – Transfer Functions/Bode Plots
Series resonant RLC Band Pass Filter circuit
Enzo Paterno 7
ZT
sLLj
Z2
Z1
Impedance circuit diagram
sCsLRsZ
CjLjRZ
ZZZZ
T
T
T
1)(
1)(
321
Z3
sCCj
11
R
sjlet
FILTERS – Transfer Functions/Bode Plots
LjX CjX
Enzo Paterno 8
Z2 Z3
Z1
Transfer Function H(s)
RVI VO
sCsLR
R
sV
sVsH
ZZZ
Z
sV
sVG
I
O
I
O
1)(
)()(
)(
)(
321
1
sLLj sCC
j11
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 9
LCs
L
Rs
L
Rs
LCs
L
sRLC
sRCsH
LCssRC
sRC
sC
LCssRC
R
sCsLR
RsH
11)(
111)(
22
22
FILTERS – Transfer Functions/Bode Plots
22
)(
o
o
o
o
o
sQ
s
Qs
jH
o
ooLLo
QL
R
R
L
R
X
RI
XIQ
2
2
21
2
1oo
LCLCf
Quality factor
Resonant frequency
Enzo Paterno 10
LCs
L
Rs
L
Rs
sH1
)(2
FILTERS – Transfer Functions/Bode Plots
L = 100 mH, C = 10 mF, R = 11 Ω
1001
101
11.0
1001
1011000
110)(
10010
110
1000110
110)(
2
ss
s
ss
ssH
ss
s
ss
ssH
Enzo Paterno 11
Pzapplet Run: L = 100 mH, C = 10 mF, R = 11 Ω
http://home.dei.polimi.it/guariso/pzapplet/index.html
Pzapplet
(MIT)
FILTERS – Transfer Functions/Bode Plots
101
11
011.0)(
ss
ssH
1001
101
11.0)(
ss
ssH
Normalize
Enzo Paterno 12
Matlab Run: L = 100 mH, C = 10 mF, R = 11 Ω
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 13
Pspice Run: L = 100 mH, C = 10 mF, R = 11 Ω
sradsradf
sradsradf
H
L
c
c
/100/8.118)2(8.18
/10/8.8)2(4.1
FILTERS – Transfer Functions/Bode Plots
1.4 18.8
Enzo Paterno 14
Circuit Maker Run: L = 100 mH, C = 10 mF, R = 11 Ω
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 15
FILTERS
Low Pass Filter High Pass Filter
Band Pass Filter Stop Band Filter
Enzo Paterno 16
FILTERS
Low Pass Filter
High Pass Filter
Band Pass Filter
Stop Band Filter
Non Ideal Filters
22
)(
o
o
o
o
o
sQ
s
Qs
sH
22
2
)(
o
o
o
o
sQ
s
sH
22
2
)(
o
o
o sQ
s
ssH
22
22
)(
o
o
o
z
sQ
s
ssH
Enzo Paterno 17
BANDPASS
LOWPASS
HIGHPASS
BAND
REJECT
1zReal
0z
2zReal
2zComplex
FILTERS – Transfer Functions/Bode Plots
22
)(
o
o
o
o
o
sQ
s
Qs
sH
Enzo Paterno 18
BANDPASS
FILTERS – Transfer Functions/Bode Plots
MIT
PoleZero Applet
22
2
)(
o
o
o
o
sQ
s
sH
Enzo Paterno 19
LOWPASS
FILTERS – Transfer Functions/Bode Plots
MIT
PoleZero Applet
22
2
)(
o
o
o sQ
s
ssH
Enzo Paterno 20
HIGHPASS
MIT
PoleZero Applet
FILTERS – Transfer Functions/Bode Plots
22
22
)(
o
o
o
z
sQ
s
ssH
Enzo Paterno 21
BAND
REJECT
FILTERS – Transfer Functions/Bode Plots
MIT
PoleZero Applet
Enzo Paterno 22
yxy x
10log10 Such that:
Log scale spacing = log N
Linear scale
Graph paper is available in the semilog and log-log varieties.
The spacing of the log scale is determined by taking the common log
of the number. The scaling starts with 1, since log 1 = 0.
The distance between 1 and 2 is determined by
log 2 = 0.3010, (30% of the full distance of a log interval).
BODE PLOT - LOGARITHMS
Log interval
18% 12% 10% 8% 7%
Enzo Paterno 23
FILTERS
Frequency log scale
d d d d d d
Enzo Paterno 24
Semilog Graph
Enzo Paterno 25
FILTERS
bbb
ana
bab
a
baab
n
10
1
1010
1010
101010
101010
10
loglog1
log
loglog
logloglog
logloglog
01log
yxy x
10log10
Properties of logarithms
Enzo Paterno 26
DECIBELS
Power Gain:
Two levels of power can be compared using a unit of measure called the bel:
A decibel is defined as:
A decibel referenced to 1 mW is given as:
1
210
1
210 log20log10
V
V
P
PdB (decibel - dB)
1
210log
P
PB (bels)
600
101
log10mW
PdBm
Network
P1 P2
p
jj
z
jKAdB
1log20log201log20log20
Enzo Paterno 27
Given a transfer function we can generate a BODE plot.
p
jj
z
jK
p
jpj
z
jkz
pjj
zjkjwH
pss
zsksH
1
1
1
1
)(
)( Rkpz ,,
FILTERS – Transfer Functions/Bode Plots
A Bode plot is a graph of the transfer function H(jω) versus frequency, ω,
plotted with a log-frequency axis, to show the system’s frequency response.
AdB
ω
)(log20 jHAdB
Enzo Paterno 28
Effect of constant terms:
FILTERS – Transfer Functions/Bode Plots
Constant terms such as K contribute a straight horizontal line of magnitude
20 log10(K).
K10log20
p
j
z
jjK
jwH
1
1
)(
Enzo Paterno 29
Effect of a ZERO at the origin:
FILTERS – Transfer Functions/Bode Plots
A zero at the origin occurs when jω multiplies the numerator. Each
occurrence of this causes a positively sloped line passing through ω = 1
with a rise of 20 db over a decade continuously.
p
j
z
jjK
jwH
1
1
)(
10@20log20
1@0log20
10
10
dBj
dBj
db0
db20
Enzo Paterno 30
Effect of a POLE at the origin:
FILTERS – Transfer Functions/Bode Plots
A pole at the origin occurs when jω multiplies the denominator. Each
occurrence of this causes a negatively sloped line passing through ω = 1
with a drop of 20 db over a decade continuously.
p
jj
z
jK
jwH
1
1
)(10@20log20
1@0log20
10
10
dBj
dBj
db0
db20
31
Effect of ZEROS not at the origin:
FILTERS – Transfer Functions/Bode Plots
Zeros not at the origin are indicated by (1+j ω/z) terms. z in each of the
terms is called the break frequency or the critical frequency. Below the
break frequency they do not contribute to the log magnitude (0 db) and
above the break frequency, they represent a ramp function of 20 db per
decade with a positive slope continuously.
p
jj
z
jK
jwH
1
1
)(
zz
jz
z
j
dbzz
j
101010
10
log20log20;1log20
0;1log20
0.1 1 10 100 Z
20 db
dec
db0
2
2
10
10
1log20
1log20
z
zj
Enzo Paterno
Enzo Paterno 32
Effect of POLES not at the origin:
FILTERS – Transfer Functions/Bode Plots
Poles not at the origin are indicated by (1+j ω/p) terms. p in each of the
terms is called the break frequency or the critical frequency. Below the
break frequency they do not contribute to the log magnitude (0 db) and
above the break frequency, they represent a ramp function of 20 db per
decade with a negative slope continuously.
p
jj
z
jK
jwH
1
1
)(
pp
jp
p
j
dbpp
j
101010
10
log20log20;1log20
0;1log20
0.1 1 10 100
p
20 db
dec db0
2
2
10
10
1log20
1log20
p
pj
Enzo Paterno 33
BODE Plot of a transfer function
FILTERS – Transfer Functions/Bode Plots
To complete the log magnitude vs. frequency plot of a Bode diagram, we
superposition all the lines of the different terms on the same plot.
1002
1)(
ssH
50
01.0
150
1
100
1
501100
1)(
p
K
p
s
K
sssH
Example:
Enzo Paterno 34
BODE Plot of a transfer function
FILTERS – Transfer Functions/Bode Plots
501
100
1
log20s
50
01.0
1
)(p
K
p
s
KsH
300 500
Enzo Paterno 35
BODE Plot of a transfer function
FILTERS – Transfer Functions/Bode Plots
1002
1)(
ssH
2,100,0,1
)(
hgba
hsg
bsasH
Enzo Paterno 36
Let: L = 100 mH, C = 10 mF, R = 11 Ω
1001
101
11.0)(
1001100
10110
110
10010
110)(
1000110
110
1)(
22
jj
jsH
ss
s
ss
ssH
ss
s
LCs
L
Rs
L
Rs
sH22
)(
o
o
o
o
o
sQ
s
Qs
sH
BPF
FILTERS – Transfer Functions/Bode Plots
Series RLC
Enzo Paterno 37
1001
101
11.0)(
jj
jjH
100
1log2010
1log20log2011.0log20 10101010
jjjAdB
)(log20 10 jHAdB
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 38
100 101 102 103 104 105 106
dB1911.0log20 10 AdB
)(rad
40
30
20
10
0
-10
-20
-30
-40
-50
-60
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 39
100 101 102 103 104 105 106
AdB
)/( srad
40
30
20
10
0
-10
-20
-30
-40
-50
-60
1@0,20log20 10 dBslopedbj
FILTERS – Transfer Functions/Bode Plots
Enzo Paterno 40
100 101 102 103 104 105 106
AdB 40
30
20
10
0
-10
-20
-30
-40
-50
-60
10log2010,010;
101log20 1010
jdb
j
FILTERS – Transfer Functions/Bode Plots
)/( srad
Enzo Paterno 41
100 101 102 103 104 105 106
AdB 40
30
20
10
0
-10
-20
-30
-40
-50
-60
100log20100,0100;
1001log20 1010
jdb
j
FILTERS – Transfer Functions/Bode Plots
)/( srad
Enzo Paterno 42
100 101 102 103 104 105 106
AdB 40
30
20
10
0
-10
-20
-30
-40
-50
-60
100log20100,0100;
1001log20 1010
jdb
j
100
1log2010
1log20log2011.0log20 10101010
jjjAdB
FILTERS – Transfer Functions/Bode Plots
)/( srad
Enzo Paterno 43
100 101 102 103 104 105 106
AdB 40
30
20
10
0
-10
-20
-30
-40
-50
-60
FILTERS – Transfer Functions/Bode Plots
)/( srad
1000110
110)(
2
ss
ssH
Enzo Paterno 44
FILTERS – Transfer Functions/Bode Plots
H(s) = ( a0+a1s+a2s2+a3s3+a4s4+a5s5 ) / ( b0 + b1s+b2s2+b3s3+b4s4+b5s5 )
0 110 0 0 0 0 1000 110 1 0 0 0
1000110
110)(
2
ss
ssH
Enzo Paterno 45
100 101 102 103 104 105 106
AdB 40
30
20
10
0
-10
-20
-30
-40
-50
-60
2)100/1(
)10/1(100)(
ss
ssG
-20db/dec
-40 db/dec
BODE PLOT - Example
)/( srad
100log20
slog20
)10/1log(20 s
2)100/1log(20 s
5/2007 Enzo Paterno 46
R
++
VIN
C
LPF FREQUENCY RESPONSE
VOUT
22
2
22
1
1
1
1|)(|
)(
)(
RC
cX
RjH
cXR
Xc
V
VjH
JXcR
jXcVV
IN
OUT
INOUT
1 assume
1
1
2
2
2
1
RC
RC
RC
21
2
2
10
1
1
1
1
with log20
RC
RCH
dB
Enzo Paterno 47
FILTERS – ROLL OFF
1
2log20 10
H
HH
dB
Let ΔH = H2 – H1
|ΔH|dB = 20 log10 |ΔH|
|ΔH|dB = 20 log10 |H2| - 20 log10 |H1|
f1 f2
H1
H2
|H|dB
f
LPF
For a LPF, |ΔH|dB becomes:
We simplify β:
2
110log20
dBH
dBHdBHdBdB
2010@,62@ 1212
∆H
RC
1
top related