filters transfer functions/bode plots frequency domain ... · enzo paterno 32 effect of poles not...

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


) (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


S-plane

These zeroes and poles are real

Thus, they get plotted on the real axis

real

Imaginary

0

0

Enzo Paterno 4


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


S-plane real

Imaginary

0

0

x

2j

Enzo Paterno 6


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


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


Enzo Paterno 9

LCs

L

Rs

L

Rs

LCs

L

sRLC

sRCsH

LCssRC

sRC

sC

LCssRC

R

sCsLR

RsH

11)(

111)(

22

22


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


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)


101

11

011.0)(

ss

ssH

1001

101

11.0)(

ss

ssH

Normalize

http://home.dei.polimi.it/guariso/pzapplet/index.html

Enzo Paterno 12

Matlab Run: L = 100 mH, C = 10 mF, R = 11 Ω


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


1.4 18.8

Enzo Paterno 14

Circuit Maker Run: L = 100 mH, C = 10 mF, R = 11 Ω


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


22

)(

o

o

o

o

o

sQ

s

Qs

sH

Enzo Paterno 18

BANDPASS


MIT

PoleZero Applet

22

2

)(

o

o

o

o

sQ

s

sH

Enzo Paterno 19

LOWPASS


MIT

PoleZero Applet

22

2

)(

o

o

o sQ

s

ssH

Enzo Paterno 20

HIGHPASS

MIT

PoleZero Applet


22

22

)(

o

o

o

z

sQ

s

ssH

Enzo Paterno 21

BAND

REJECT


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 ,,


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:


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:


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:


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:


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:


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


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



501

100

1

log20s

50

01.0

1

)(p

K

p

s

KsH

300 500

Enzo Paterno 35



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


Series RLC

Enzo Paterno 37

1001

101

11.0)(

jj

jjH

100

1log2010

1log20log2011.0log20 10101010

jjjAdB

)(log20 10 jHAdB


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


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


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


)/( 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


)/( 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


)/( srad

Enzo Paterno 43

100 101 102 103 104 105 106

AdB 40

30

20

10

0

-10

-20

-30

-40

-50

-60


)/( srad

1000110

110)(

2

ss

ssH

Enzo Paterno 44


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

filters transfer functions/bode plots frequency domain ... · enzo paterno 32 effect of poles not...

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