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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
Example 11.5: (Cont.)
%)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
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
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
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
Example 11.8: (Cont.)
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
Example 11.9: (Cont.)
Table 11.6
Example 11.9: (Cont.)
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
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
Example 11.10: (Cont.)
Chapter 11: Problem Set
l 2,6,19,22,30,35,52,59,64