high pass filters, 2nd order filters, active filters,resonances.pdf
Post on 16-Apr-2015
121 Views
Preview:
TRANSCRIPT
Lecture 20Lecture 20High Pass Filters, 2nd Order g ,
Filters, Active Filters, RResonances
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Series Resonance
fjRfLjfZs π 12)( −+=
resonanceForfC
jfjfs π
11:
2)(
2
For resonance the reactance of the inductor and the capacitor cancel:
f
LCf
CfLf
πππ
1)2(1
212 2
20
00 =→=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
LCf
π20 =
Series ResonanceQuality factor QS
Resistance resonance at inductance of ReactanceQs ≡
RLf
Resistance02π
=
LCffrom
CfLSubstitute
R
020
2 21
)()2(1
ππ==
CRfQ
LCCf
s
0
21
2)()2(
π
ππ
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
CRf02π
Series Resonance−+=
fCjfLjRfZs 2
12)(π
π
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
fRCRfLjR
2121
ππ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
⎥⎦⎢⎣ ⎠⎝
LCffff
RLf
jR
f
20
)2(12
1π
==
⎥⎦⎢⎣⎟⎠
⎜⎝
RLf
QandfSubstitute
LCfffR
s0
0
02
0
21
)2(
π
π
⎥⎤
⎢⎡
⎟⎟⎞
⎜⎜⎛
−+=ffjQRfZ
RQ
LCf s
0
0
1)(
2π
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎥⎥⎦⎢
⎢⎣
⎟⎟⎠
⎜⎜⎝
+ff
jQRfZ ss0
1)(
Series Resonance
⎟⎟⎠
⎞⎜⎜⎝
⎛−=∠⎟⎟
⎠
⎞⎜⎜⎝
⎛−+= −
ff
ffQTanZ
ff
ffQ
RZ
sss01
2021
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎟⎠
⎜⎝
⎟⎠
⎜⎝ ffffR 00
Series Resonant Band-Pass Filter
Rss /VVI
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
==
ff
ffjQ
fZs
s
s
s
0
01
)(I
⎟⎟⎞
⎜⎜⎛
−+=→
⎟⎟⎞
⎜⎜⎛
−+==
⎠⎝
ffjQffjQR
s
RsR
00 1
1
1VVV
IV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎟⎟⎠
⎜⎜⎝
−+⎟⎟⎠
⎜⎜⎝
−+ff
jQff
jQ ss00
11
Series Resonant Band-Pass Filter
1V( )ffffjQss
R
0011
−+=
VV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Series Resonant Band-Pass Filter
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Series Resonant Band-Pass Filter
fffB 0
sLH Q
fffB 0=−=
sQ
For Q >>1
B
For QS>>1
20BffH +≅
20BffL −≅
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
22
Example 6.6
HzFxHLC
f 1000)1015920)(159260(2
12
160 ===
−ππ
HHzR
LfQ
FxH
s 10100
)15926.0)(1000(22)101592.0)(15926.0(2
0 =Ω
==πππ
BB
HzHzQf
Bs
10010
10000 ===
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
HzHzHzBffHzHzHzBff LH 9505010002
10505010002 00 =−=−≈=+=+≈
Example 6.6
:resonanceAt
Ω
Ω===
1000111000)1592.0)(1000(22 0L
jjjZ
jHHzjLfjZ ππ
Ω=++=
Ω−=−=−=−
100
1000)101592.0)(1000(22 6
0
CLs
C
ZZRZ
jFxHz
jCf
jZππ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
CLs
Example 6.6
oo
001001∠=
∠== SV
I
oo 01)0010)(100(
001.0100
∠=∠==
∠===
R
Z
R
s
IV
I
oo 9010)001.0)(1000(
01)001.0)(100(
∠=∠==
∠=∠==
jZ
R
LL
R
IV
IV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
oo 9010)001.0)(1000( −∠=∠−== jZCC IV
Example 6.6
L Q VV = ssL Q VV
ssC Q VV =
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parallel Resonance
Z 1= ( ) ( )fLjfCjR
Z p ππ 2121 −+=
At resonance Z is purely resistive:At resonance ZP is purely resistive:
( ) fLfjCfj ππ 1212 000 =→=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( )LC
ffjfjπ2
000
Parallel ResonanceQuality factor QP
ResistanceQP ⎟⎟⎞
⎜⎜⎛
≡
Rresonance at inductance of Reactance
QP
=
⎟⎟⎠
⎜⎜⎝
≡
ffromLSubstitute
Lf0
112π
=
CRfQLC
ffromCf
LSubstitute
P 0
020
2
22)()2(
πππ
=
==
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
fQP 0
Parallel Resonance
( ) ( )⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=−+
=
fLRjfRCj
RfLjfCjR
Z P
221
21211
ππ
ππ
⎟⎞
⎜⎛
=⎟⎞
⎜⎛
=
⎟⎠
⎜⎝
f
R
f
RfL
11
2π
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟
⎟⎠
⎞⎜⎜⎝
⎛−+
RfLCff
fjQfLCff
fRCfj P0
200
20
0 )2(11
)2(121
πππ
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=
ff
ffjQ
R
P0
01
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎠⎝
Parallel Resonance
⎟⎟⎞
⎜⎜⎛
+==
ffjQ
RZPout01
IIV Vout for constant current, varying the frequency
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎟⎟⎠
⎜⎜⎝
−+ff
jQP0
1
Parallel Resonance
Fi d th h t I I d I f
2.159,2.159,10,010 3 ==Ω=∠= − pFCHLkR μoI
Find the phasor currents IR, IC and IL for:
10)2.159)(101(2
102
1012
16
4
0
60 =====
HHzxLfRQHzx
LCf P μπππ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Parallel Resonanceoo 010)10(010 43 ∠=∠== −Rout IV
oo
010010
)(
344 ∠=
∠=== −
Routout
R
out
VVI
oo
90100101010
2
44
∠=∠
=== −
R
outout
R
VVI
o010
9010102 3
0
∠
−∠====jLfjZ L
L π
VV
I
o901010
010
2
23 ∠=
−∠
=−
== −
jCfjZ
out
C
outC
VVI
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
2 0Cfπ
II PC Q= II PC Q=
II PL Q=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Ideal Filters
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Noise Rejectionj
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Second Order Low-Pass Filter
−− jj
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=−+
=++
=
LCffff
RLfj
fRC
fCjfLjR
fCZZZ
Zininin
CLR
Cout
00
0
2121
2
22
2
ππ
π
ππ
π VVVV
⎟⎞
⎜⎛
−=
⎟⎞
⎜⎛
−
==
⎠⎝
ffffjQ
fLffRC
j
fH S
in
out
0
0
0
00
)/(12
2)(π
πVV
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
ff
ffjQ
LCffff
RLfj S
in 0
000
0 12
121π
πV
Second Order Low Pass FilterSecond-Order Low-Pass Filter
( )ffjQV( ) ( )( )00
0
1 ffffjQffjQ
fHs
s
−+−
==in
outVV
( )( ) ( )122
0
1
90
ffffQTffffQ
ffQs
∠
−∠=
−
o
( ) ( )
( ) ( )0
0012
0021
ffQfH
ffffQTanffffQ
s
sS −∠−+ −
( ) ( )( )200
20
1 ffffQfH
S
s
−+=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Second Order Low-Pass Filter
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Second Order High-Pass Filterg
At low frequency the capacitor i i iis an open circuit
At high frequency the capacitor is a short and the inductor is open
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Second Order Band-Pass Filter
At low frequency the capacitor i i iis an open circuit
At high frequency the inductor is an open circuit
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Second Order Band-Reject Filterj
At low frequency the i icapacitor is an open
circuit
At high frequency the inductor is an open circuit
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 6.8
Design a filter that passes frequency components higher than 1 kHz and rejects components lower than 1 kHz. Chose L=50 mHChose L 50 mH
FxxLf
CLC
kHzf μπππ
507.0)1050()101()2(
1)2(1
211 3232220 ===→==
−
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
xxLfLC πππ )1050()101()2()2(2 0
Example 6.8
To avoid amplifying the signal at f0 choose Qs=1
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Example 6.8
Ω−
1314)1050)(1(222 300 HxkHzLf
RLf
Q πππΩ===→= 1.314
1))((00
Qf
RRf
Qs
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.20
Design a filter with QS=1 that passes frequency components lower than 5 kHz and rejects components higher than 5 kHz. Chose L=5 mHChose L 5 mH
FxxLf
CLC
kHzf μπππ
2026.0)105()105()2(
1)2(1
215 32322
020 ===→==
−
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
xxLfLC πππ )105()105()2()2(2 0
Exercise 6.20
Ω−
1157)105)(5(222 300 HxkHzLf
RLf
Q πππΩ===→= 1.157
1))((00
Qf
RRf
Qs
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.21
Design a filter that passes frequency components between fL=45 kHz and fH=55 kHz. Chose L=1 mH
nFxxLf
CLC
kHzf 13.10)101()1050()2(
1)2(1
2150 32322
020 ===→==
−πππ
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Exercise 6.21=−=
55010
0 kHzfQ
kHzffB LH
=
===
2
510
0
0
LfR
kHzBf
Q
π
Ω==−
83.62)101)(50(2 3 HxkHz
QR
πΩ83.62
5
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Active Filters
Ideally, an active filter circuit should:Ideally, an active filter circuit should:
1. Contain few components. Co ta ew co po e ts
2. Have a transfer function that is insensitive to . ave a t a s e u ct o t at s se s t ve to component tolerances
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3. Place modest demands on the op amp’s gain–p p gbandwidth product, output impedance, slew rate, and other specifications
4. Be easily adjusted
5. Require a small spread of component values
6. Allow a wide range of useful transfer functions to be realized
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Active First-Order Low-Pass Filter
ff
i
f
i
o
CfRjZZ
VV
fH
π21111
)( −==
f
ff
f
f
ff
R
RCfRj
RfCj
RZπ
π
21
21111
+=+=
ff
ff
ff
RZ
CfRjR
Zπ
1
21
⎟⎞
⎜⎛
+=
f
ffi
f
i
f
R
CfRjRZfH
π
1
211)(
⎥⎤
⎢⎡⎟⎟⎞
⎜⎜⎛−=
+⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
ffB
Bi
CRf
ffjR
π21
)/(1
=
⎥⎦
⎢⎣ +⎟⎟⎠
⎜⎜⎝
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ff
A low-pass filter with a dc gain of -Rf/Ri
t
( ) ( )dttvRC
tvt
o in1∫−=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
RC 0∫
Active First-Order High-Pass Filterg
i
f
i
o
ZZ
vv
fH )( −==
ff
ffi
ii
RZfH
RZCfj
RZπ
)(
21
==
=+=
ifiif
ii
i
CRfjRCRfjCfj
RZfH
πππ
222
1)(
⎟⎞
⎜⎛
+−=−=
Bf
ii
if
i
i
ii
if
ffjffj
RR
CRfjRCRfj ππ
)/(1)/(
2121
⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
+⎟⎟⎠
⎜⎜⎝−=
+−=
iiB
Bi
CRf
ffjR
π21
)/(1
=
⎥⎦
⎢⎣ +⎟⎠
⎜⎝
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
A high-pass filter with a high frequency gain of -Rf/Ri
Differentiator Circuit
d( )dt
dvRCtvoin−=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
dt
Higher Order Active Filtersg
n fHfHfHfH ∗∗∗= )()()()( 21n
B
n
i
fn
ffjRR
⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
)/(11)1(
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Butterworth Transfer FunctionButterworth filters are characterized by having a particularly flat pass-band.
( )( ) nff
HfH2
0
1=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
( ) nBff 21+
Sallen Key CircuitsSallen–Key Circuits
RCfB π2
1=
RCπ2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Sallen–Key Circuitsy
vR
v f
vv
vRKR
v
i
off
f
=
−+=− )1(
KR
RKA
v
vv
fv
o
i
=−
+==
= −
)1(1
Rv fv
i
N i ti lifi ith i f K
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Non-inverting amplifier with a gain of K
K Values for Low-Pass and High-Pass Butterworth Filters of Various Orders
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Low-Pass Active Filter Designg
Design a fourth order low pass Butterworth
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Design a fourth-order low-pass Butterworth filter having a frequency cut-off of 100 Hz
Low-Pass Active Filter Designg
= F0 1CChoose μ
==− HzxCf
RB )100)(1010(2
12
1F0.1C Choose
6ππ
μ
Ω= kHzxCfB
92.15)100)(101.0(22 ππ
From the table a fourth order filter requires K values of 1.152 and 2.235. The DC gain is (1.152)(2.235) = 2.575
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Low-Pass Active Filter Designg
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Low-pass Active Filter Designg
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Low-pass Active Filter Designg
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Review for Next Lecture!
• Magnetic fields (B flux density H magneticMagnetic fields (B flux density, H magnetic field intensity, magnetic permeability μ)
• Right hand rule• Right hand rule• Forces on charges and current carrying wires
i i i fi ldmoving in a magnetic field• Faraday’s Law• Lenz’s Law• Ampere’s Law
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Ampere s Law
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Equal Component Sallen-Key Low-Pass Active Filter
1RC
fB π21
=
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Equal Component Sallen-Key Low-Pass Active Filter
va
0=−
+−
+− −
Rvv
Zvv
Rvv aoaina
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
RZR C
Equal Component Sallen-Key Low-Pass Active Filter
0−−− −vvvvvv aoaina
1
0=++
RRZR
a
C
oaina
1)1(
=−+
=− vK
vRKR
Rv oo
ff
f
0/
=−
+−
+−
RKvv
Zvv
Rvv oa
C
oaina
0/
)( =−
+−+−
RKvv
CjvvR
vv oaoa
ina
C
ω
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
RR
Equal Component Sallen-Key Low-Pass Active Filter
va
/
0
−
=+−−
vKvZv
Rvv
C
aa
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
0/
=+ CjvR
vKva
ao ω
top related