Chapter 11 Frequency Response and Filters

Chapter 11: Outline

Complex Frequency

l Complex frequency: oscillating voltages or currents with exponential amplitudes.

+=+

=

+=

tjexjemXxtj

etemX

xttemXtx

)()(Re)(

Re

)cos()(

ωσφφωσ

φωσ

ωσ js +≡ :frequencyComplex

xjemXxmXX

φφ =∠≡ :Phasor

Generalized Impedance and Admittance

IVsZ /)( ≡

IVsZ /)( ≡

VIsZsY /)(/1)( =≡

Generalized Impedance and Admittance

IVsZ /)( ≡

jω à s

Network Function

l Any response forced by a complex-frequency excitation.

=+= steXxttemXtx Re)cos()( :Input φωσ

=∠≡ xj

emXxmXXφ

φ

=+= steYyttemYty Re)cos()(:Response φωσ

=∠≡ Yj

emYYmYYφ

φ

XYsH /)( :functionNetwork ≡

Network Function (Rational)

[ ] [ ]

cases. special are admittance and Impedance

)( :functionNetwork

network,order th -nan For

ReReRe

011

1

011

1

0101

2

asasasabsbsbsb

XY

sH

xbdtdx

bdt

xdbya

dtdy

adt

yda

YsyYsyYy

eYsdt

deYeY

dtd

y

nn

nn

mm

mm

m

m

mn

n

n

stst

st

++++++

=≡

+++=+++

↔′′⇒↔′⇒↔

=

==′

−−

−−

LL

L

Network Function

frequencycomplex input theis

)())(()())((

)( 21

21

011

1

011

1

ωσ js

Xpspspszszszs

KXasasasabsbsbsb

XsHYn

mn

nn

n

mm

mm

+=

−−−−−−

=++++++

==−

−

−−

LL

Pole

Zero

Frequency Response

Frequency Response

l Frequency response is the forced response of a circuit to a sinusoid ac waveform of a particular frequency. Amplitude ratio and phase shift are typically used to characterize frequency response.

l Transfer function vs. phasor analysis:

)()( ωjHsH →

)()( ωjHsH →σ

jω

Frequency Response

XjHY )( ω=

mXmYjHa /)()( =≡ ωω :Amplitude ratio

xyjH φφωωθ −=∠≡ )()( :Phase shift

[ ][ ]tj

ym

tjxm

eYtYty

eXtXtxω

ω

φω

φω

Re)cos()(

Re)cos()(

=+=

=+=

Functions of frequency

Superposition

l Superposition for waveforms at different frequencies:

LL

++++++=++++=

))(cos()())(cos()()()cos()cos()(

2222211111

222111

φωθωωφωθωωφωφω

tXatXatytXtXtx

à Phasor analysis at different frequencies

Example 11.1: A Frequency-Selective Network

)4.82300cos(32.1)6.2620cos(94.8)(

300,20

40tan)()(

1600

40)()(

4040

)()(

00

1

2

−+−=

=

−=∠=

+==

+=⇒

+==

−

tttv

jH

jHa

jjH

RsLR

VV

sH

out

in

out

ωω

ωωθ

ωωω

ωω

0.2H

8Ω

ttvin 300cos1020cos10 +=

Frequency Response Curves

l Plots of amplitude ratio and phase shift vs. frequency. They can be obtained by analytical method or graphical method.

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

0

2121

21

21

90)()(

,0,

)(

:)(frequency high At very

)(

)(

×−+∠=

<=

=

∞→

+−∠+−∠−+−∠+−∠+∠=

−−−−

=

nmK

nmnmK

a

pjpjzjzjK

pjpjzjzj

Ka

ωθ

ω

ω

ωωωωωθ

ωωωω

ω

LL

LL

Example 11.2: An All-Pass Network

ajaj

jH

RCa

pszs

Kasas

VV

sHin

out

+−−=

=

−−

≡+−

−==

ωωω

21

)(

1 where,

21

)(

Example 11.2: An All-Pass Network

−=

−

−=∠=

==

−−−

aaajH

jHa

ωωωωωθ

ωω

111 tan2tantan)()(

pass) (all 21

)()(

Q: What does non-linear phase do?

A: Waveform Distortion

Non-Linear Phase

ωωθτωτωωωθ

ωωωθ

ω

ωθωωθωωω

ktXtX

tXtX

tXtXtytXtXtx

=+++=

+++=

+++=+=

)( ifonly ,)(cos()(cos(

))(

(cos())(

(cos(

))(cos())(cos()()cos()cos()(

2211

2

222

1

111

222111

2211

Example 11.3: Frequency-Response Calculations (Analytic Method)

50020)25(20

)( 2 +++

=ss

ssH

( )

500,500

20tan180

25tan

500,500

20tan

25tan)(

400500

62520)(

20)500()25(20

)(

22

101

22

11

222

2

2

>−

+±=

<−

−=

+−

+=

+−+

=

−−

−−

ωω

ωω

ωω

ωωωθ

ωω

ωω

ωωω

ω

a

jj

jH

Example 11.3 (Graphical Method)

)()()()(

20)(

2010,25

20

211

21

1

21

1

pjpjzj

pjpjzj

a

jppzK

−∠−−∠−−∠=

−−−

=

±−=−=

=

ωωωωθ

ωωω

ω

Example 11.3 (Cont.)

even

odd

)()()()(

)()(

ωθωθωω

ωω

−=−−==−

aajHjH

Table 11.2

Filters

Filtersl Filters are frequency-selective networks that pass certain

frequencies but suppress/reject the others.l Four common categories: lowpass, highpass, bandpass and

notch.l A positive gain constant K is assumed.l Ideal lowpass filter, ideal highpass filter, cutoff frequency,

passband and stop band.

First-Order Lowpass Filter

gain)frequency (low positive is

tan)(

)/(1)(

)/(1)(

)(

1

2

K

Ka

jK

jK

jH

sK

sH

colp

co

lp

coco

colp

co

colp

ωω

ωθ

ωωω

ωωωωω

ω

ωω

−−=

+=

+=

+=⇒

+=

First-Order Lowpass Filter

0 dB

-3 dB

-3 dB cutoff frequency

Half power point

First-Order Highpass Filter

gainfrequency high :

tan)(

)/(1)(

)/(1)(

)(

1

2

K

Ka

jK

jKj

jH

sKs

sH

cohp

co

hp

cocohp

cohp

ωω

ωθ

ωωω

ωωωωω

ω

ω

−−=

+=

−=

+=⇒

+=

First-Order Highpass Filter

radian frequency ω vs. cyclical frequency f

ω=2πfω/ ω co=f/fco

First-Order Filter Networks

RCs

RC

sCR

sCVV

RLRC

K

in

out

co

1

1

1

1filter lowpass RC: exampleFor

/ ,

1

1

+=

+=

==

=

ττ

ω

Example 11.4: Parallel Filter Network

RCK

RCj

RCj

jjH

RCs

sII

sH

co

in

c

1,1

11

11

)(

1)(

==

−=

+=

+==

ω

ωω

ωω

Example 11.5: Design of a Lowpass Filter

Design a low pass filter with fco around 4 KHz

co

co

Ls

s

Ls

s

s

out

sK

CGGsCG

GGsCG

VV

sHω

ω+

=++

=++

==/)(

/)(

Example 11.5: (Cont.)

Lseqeqco

co

Ls

s

coLs

co

co

co

Ls

s

Ls

s

s

out

RRRCR

FC

GGG

K

KHzfCC

GGsK

CGGsCG

GGsCG

VV

sH

===

≈=

=+

=

⋅===+

=

+=

++=

++==

, where,1

1401

effect) (loading 8.0

42240

1/)(

/)(

ττ

ω

µω

ππω

ωω


%)3%10(

)76cos(095.0)37cos(2.3)(

7619.0)(,3764.0)(

162,32 ,cos5.0cos5)(Let

02

01

02

01

2121

→

−+−=

−∠=−∠=

⋅=⋅=+=

tttv

jHjH

kktttv

out

s

ωω

ωω

πωπωωω

Bandpass and Notch Filters

Bandpass and Notch Filters

l Ideal bandpass filter, ideal notch filter (band-reject filter), lower cutoff frequency, upper cutoff frequency and bandwidth.

bandwidth

lower cutoff higher cutoff

Quality Factor

l Second order bandpass filter and quality factor.

)2/1(cos

180,180

41

12

,

)2/1 (i.e., :dunderdampewhen

t)coefficien damping :( 2/

)/()/(

)(

1

02

01

021

200

21

0

0

200

20

Q

pp

pp

Qj

Qpp

Q

Q

sQssQK

sHbp

−=

+=∠−=∠

==

−±−=

><=

++=

ψ

ψψ

ω

ωω

ωαααω

ωωω

Quality Factor

−α

ωd

Quality Factor

−−=

−+

=

−+

=

−

ωω

ωω

ωθ

ωω

ωω

ω

ωω

ωω

ω

0

0

1

2

0

0

2

0

0

tan)(

1

)(

1)(

Q

Q

Ka

jQ

KjH

bp

bp

bp

midband gain

(-3 dB) Bandwidth

mean) (geometric

241

1,

filtering) bandpass(for 2/1

2/)()(

1at and

20

0

020

0

0

ωωω

ωωω

ωωωω

ωω

ωω

ωω

ωω

=⋅

=−=

±+=

>

==

±=

−

lu

lu

lu

ubplbp

lu

QB

QQ

Q

Kaa

Q

symmetric)tely (approxima

21

2,

10 :high

narrowband

00

0

0

0

BQ

BQQ

B

lu ±=±≈

≥=

⇒<<

ωω

ωωω

ωω

Second-Order Notch Filter

QB

jK

Qj

K

Qj

K

jjzz

QsQ

s

ssKsHno

0

220

00220

220

022

021

0

20

02

20

2

1111)(

)( :bandwidth 3dB

,

2 ,

)2()(

ωωω

ωωωωωω

ωω

ωββωβ

ωβ

ωωωβ

=

⋅±=

−+

=+−

−−

±−≈−±−=

<<++

++=

z1p1

Second-Order Notch Filter

Table 11.3

Resonant Circuitsl Resonant circuits for bandpass and notch filters.

LC

RL

RCRQ

CL

RCRRL

Q

Q

par

ser

==≡

==≡

=

00

0

0

0

:network RLC parallel aFor

11 :network RLC series aFor

2

ωω

ωω

αω

Resonant Circuits

====

++

+=

++=

par

ser

in

no

in

bp

Q

QQ

LCK

LCs

LRs

LCs

VV

LCs

LR

s

sLR

V

V

,1

,1,0

1

1

1

0

2

2

2

ωβ

Winding Resistance (refer to 6.4)

( )↑↓

=

=<<

h notch widt ,

/

/ , if

00

0

par

parpar

wparw

Q

LRRCQ

CRLRLR

ωω

ω

Example 11.6: Design of a BandpassFilter (Parallel)

Ω=

Ω==

Ω==

==

==

→===

Ω==±

kR

kCRLR

kLQRR

nFL

C

QQ

kB

Q

RCRmHLHzkHz

wpar

par

par

w

13.8

2.13/

03.5

3.631

40

frequency)center ( 4050020

and find ,2.1,1Given 25020 :bandpass Require

0

20

00

ω

ω

ωω

Bode Plots

Bode Plotsl Amplitude ratio and frequency are converted to a

logarithmic scale.l Factored functions and decibels:

L)()()( 21 sHsKHsH =

L)()()()( 21 ωωωω aaKjHa ==

L

L

+++=

+++=≡

)(2)(1

21 )(log20)(log20log20)(log20)(

ωω

ωωωω

gg

aaKag

dBK

L+++∠=∠= )()()()( 21 ωθωθωωθ KjHdB gain00 or ±1800

Amplitude vs. dB Gain

-20-6-303620Gain in dB

1/101/22-1/2121/2210Amplitude ratio

First-Order Factors:Ramp FunctionHighpass FunctionLowpass Function

Ramp Function

0

0

90),( ,log20);(

90);();(

==

∠==⇒≡

WW

Wg

WWj

WjHWs

WsH

rr

rr

ωθω

ω

ωωω

Highpass Function

WdBg

WjHW

if

WW

g

WjWjH

WjW

if

WjWj

WjHWs

sWsH

hphp

hp

hphp

hp

hphp

10,0,0

1);(

,1

1.0,90,log20

);(

1)/(1 ,1

)/(1)/(

);();(

0

0

>=≈

≈⇒

>>

<=≈

≈⇒

≈+<<

+≡⇒

+≡

ωθ

ω

ω

ωθω

ωω

ωω

ωω

ω

Highpass Function

0

0

45,3

45707.01

);(

frequency)eak (corner/br at

=−=

∠=+

=

=

hphp

hp

dBg

jj

WjH

W

θ

ω

ω

Table 11.5: Correction Terms

Lowpass Function

frequency)(break ,45,3

10,90,log20

1.0,0,0

)/(11

);();(

0

0

0

WdBg

WW

g

WdBg

WjWjH

WsW

WsH

lplp

lplp

lplp

lplp

=−=−≈

>−=−≈

<≈≈

+≡⇒

+≡

ωθ

ωθω

ωθ

ωω

Lowpass Function

)()(

)()(

)()(

)()(

at 3

1

max

max

ωθωθωω

θωω

ω

x

x

xmx

mx

mx

dB

m

gmg

majHjH

sHsHif

WdBgg

KgKif

×=×=

∠==

=

=−==≠

Example 11.8: An Illustrative Bode Plot

0

22

2

2

180,20

)200;(2)(

)200;(2)(

)200;(1.020010

110

)200()(

±=∠−=

−∠=

−=

−=

+−=

+−= −

−

KdBK

K

gKg

sHs

ss

ssH

dB

hp

hpdB

hp

ωθωθ

ωω

First-Order Bode Plots

l Products of first-order factors: Bode plots of any transfer functions consisting entirely of first-order factors and powers of first-order factors can be constructed using the additive property of gain and phase. The important elements include: break frequencies, asymptotic gain and phase using straight line approximations and constants . and KKdB ∠

Example 11.9: Frequency Response of a Bandpass Amplifier

dBK

sHsHss

sss

ssH

dB 34

)()(50400

400100400

000,20)400)(100(

000,20)( 21

=

=++

=++

=

Hhp(s;100) Hlp(s;400)

0-1-3-2-3-10Sum (dB)

0-1-3-1000∆g2

000-1-3-10∆g1

40008004002001005010

Table 11.6


sradBul

/47575550550,75=−=

≈≈ ωω

Second-Order Bode Plots

Quadratic Factorsl Quadratic factors for complex-conjugate poles.

2,5.0,/49

16log10

,log20

10),/log(40

1.0,0

,)/(

,1),;(

)/()/(11

),;(

)/(),;(

02

0

00

0

02

0

00

02

00

200

2

20

0

=+

=∆

===>−≈

<≈>>−≈

<<≈+−

≡

++≡

ωω

ωωωωωω

ωωωωωω

ωωωωωωωω

ωω

ωωω

ω

Qg

QQ

dBg

QjH

QjQjH

sQsQsH

q

dB

q

q

q

q

Quadratic Factors

Quadratic Factors

dampedunder :1damping critical:1

21

:ratio Damping

0

<=

==

ζζ

ωα

ζQ

Example 11.10: Bode Plot of a Narrowband Filter

200,50,4.275.1log10

14

100,145log20

)5,100;()100;(14)(

)5,100;()100;(2.01020

1010010

10020)(

function ramp for the 100

1.021

,520

,100,1020

20)(

42

4

4

0

0042

===∆−=

===

++−=

=++

×=

==

=====++

=

ω

ω

ωωω

ω

ζω

ω

dBg

dBK

g

ggdBg

sHsHss

ssH

WQ

Qsss

sH

q

dB

q

qr

qr

Chapter 11: Problem Set

l 2,6,19,22,30,35,52,59,64

Chapter 11 Frequency Response and Filters

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Transcript of Chapter 11 Frequency Response and Filters