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COMM 704:COMM 704:Communication Systemsy
Lecture 4: Analog Filters (continued)Lecture 4: Analog Filters (continued)
Dr Mohamed Abd El GhanyDr. Mohamed Abd El Ghany, Department of Electronics and Electrical Engineering
Active Filter: Low-Pass FilterActive Filter: Low Pass Filter
Sallen Key LPFy
a
b
RRK +=1
CCRRK
V
2121221112
2
2121
1)111()(
CCRRS
CRK
CRCRS
CCRRVVsH
i
o
+−+++==
2Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Low-Pass FilterActive Filter: Low Pass Filter
Sallen Key LPFy
G
Using the standard notation:
22 )()(
oo
LPF
wSQwS
GsH++
=
11
CCRR
2121
1CCRR
wo =
221112
2121
111CRk
CRCR
CCRRQ
−++=
3Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Low-Pass FilterActive Filter: Low Pass Filter
Sallen Key LPFy
If no restriction is imposed on the gain K, we have five element values to design LPF circuit. Hence, we are free to make some arbitrary choices as follows:
Design 1: equal element values In this design, we letC1=C2=1F and R1=R2=R
Design 2: equal capacitance and equal feedback resistancesIn this design we chooseC1 C2 1F and R1 R2 R In this design, we chooseC1=C2=1F and Ra=Rb=R
Design 3: moderate-sensitivity Design 4: minimum sensitivityes g 3 ode ate se s t tydesignIn this design, we chooseC1=√3 Q and C2=1F and K=4/3
Design 4: minimum sensitivityIn this design, we chooseK=1 and R1=R2=1Ω
4Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Low-Pass FilterActive Filter: Low Pass Filter
Definition of sensitivityy
The sensitivity of some performance measure y with respect to a network element value x is defined by:
dxdy
yxS y
x =
If y is a function of several variable [y=f(x1,x2,….,xn)], then the sensitivity of y with respect to xi is
d
i
iyx dx
dyyxS
i= Notes:
2121 yx
yx
yyx SSS +=
2121 / yx
yx
yyx SSS −=
5Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
xxx
Active Filter: Low-Pass FilterActive Filter: Low Pass Filter
Example:
Design a fourth-order butterworth low-pass filter using the cascade of two Sallen-key biquads using Design 1 and 2 respectively. Then let wo =2Пx1000 rad/sec and use 0.1 µF capacitors.Th li d t k f ti f th t l filt
1765367.0)( 2
11 ++
=SS
GsH
The normalized network function of the two low-pass filters are:
1765367.0 ++ SS
1847759.1)( 2
22 ++
=SS
GsH
6Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: High-Pass FilterActive Filter: High Pass Filter
First order HPF (using Inverting op-amp configuration) ( g g p p g )
RsH 1)( 2−=
SCR 1
1+
S
cwSSKsH+
−=)(
1
2
RRK =
CRwc
1
1=
7Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: High-Pass FilterActive Filter: High Pass Filter
First order HPF (using Non-Inverting amplifier) ( g g p )
2
3 )1()(
SRR
sH+
+=
11
1)(
CRS
sH+
+=
S
cwSSKsH+
=)(
2
31RRK +=
11
1CR
wc =
8Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: High-Pass FilterActive Filter: High Pass Filter
Sallen Key HPFy
a
b
RRK +=1
2
)( KSVsH o ==
2121112212
2 1)111()(
CCRRS
CRK
CRCRSV
sHi +
−+++
==
9Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: High-Pass FilterActive Filter: High Pass Filter
Sallen Key HPFy
2GS
Using the standard notation:
22 )()(
oo
HPF
wSQwS
GSsH++
=
11
CCRR
2121
1CCRR
wo =
111222
2121
111CRk
CRCR
CCRRQ
−++=
10Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Band-Pass FilterActive Filter: Band Pass Filter
Sallen Key BPFy
a
b
RRK +=1
SK
21321
21
12132311
2
11
)1111()(
CCRRRRRS
CRK
CRCRCRS
SCR
VVsH
i
o
++−++++==
11Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Band-Pass FilterActive Filter: Band Pass Filter
Sallen Key BPFy
)( o SwG
Using the standard notation:
22 )(
)()(
oo
o
BPF
wSQwS
SQ
GsH
++=
RR + 21 RR +
21321
21
CCRRRRRwo
+=
12132311
21321
1111CRk
CRCRCR
CCRRRQ
−+++
=
12Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Band-rejection FilterActive Filter: Band rejection Filter
Sallen Key Band-Rejection Filtery j
a
b
RRK +=1
2 )1( SK
222
222
1)24(
)()(
CRS
RCKS
CRSK
VVsH
i
o
+−
+
+==
13Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Band-rejection FilterActive Filter: Band rejection Filter
Sallen Key Band-Rejection Filtery j
22 )(
Using the standard notation:
22
22
)(
)()(o
o
oBRF
wSQwS
wSKsH++
+=
1 1
RCwo
1= K
Q24
1−
=
14Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Multiple-Feedback Biquads (MFB)
• The MFB topology is commonly used in filters that have high Q p gy y gand require a high gain
MFBMFB
Multiple Feedback Low-pass
Multiple Feedback High pass
Multiple Feedback Band pass
Multiple Feedback
band-rejectionLow-pass biquad
High-pass biquad
Band-pass biquad
band-rejection biquad
15Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Multiple-Feedback Biquads (MFB)
Multiple Feedback Low-pass Biquad
21323211
2
2131
1)111(1
1
)(
CCRRS
RRRCS
CCRRVVsH
i
o
++++−==
21323211 CCRRRRRC
22 )()(
oo
LPF
wSQwS
GsH++
=
2132
1CCRR
wo =
3
2
2
3
1
322
1 1
RR
RR
RRRC
CQ++
=
16Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Multiple-Feedback Biquads (MFB)
Multiple Feedback High-pass Biquad
3212
2
2
1
1)(SCCCS
SCC
VVsH
i
o
++++−==
3221322 CCRRS
CCRSi ++
3221
1CCRR
wo =3
322
321
CCRCCC
Qwo ++
=
17Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Multiple-Feedback Biquads (MFB)
Multiple Feedback bandpass Biquad
2
21
]11[11
1
)(
RR
SCR
VVsH
i
o
+−==
11
212
31
2212
2 )11(CCRRRS
CRCRS +++
212
31
11
CCRRRw o
+=
11w
2212
11CRCRQ
w o +=
12
/1/)(
CCRRjwH o +
=
18Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
12 /1 CC+
Active Filter: Multiple-Feedback Biquads (MFB)Multiple Feedback band-rejection Biquad
22
2
222
1)24(
)/1()(
CRS
RCKS
CRSKVVsH
i
o
+−
+
+−==
w 1=
CRRC
RCw o =
KQ
241−
=K24
a
b
RRK += 1
19Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Active Filter: Multiple-Amplifier Biquads
The step from single-amplifier to multiple-amplifier biquadratic filter sections id l b fit Th t i t t b fit f llprovides several benefits. The most important benefits are as follows:
-Reduced passive element spread, i.e., the ratio between the largest and smallest values of resistors and/or capacitors can be reduced compared to the single-amplifier case-Multiple-amplifier biquads often provide lower sensitivities to both passive and active
tcomponent-Most of the filter parameters can be tuned independently.-Designing multiple-amplifier filters, often the values of capacitors can be chosen freely. The filter parameters are then determine by resistors, which is less costly than by
itcapacitors.Note:• On the other hand, these benefits must be paid for by increased space requirements and increased power dissipation. However, today there are low-cost and low-power i t t d i it il bl ith t f hi Th f iintegrated-circuit op-amps available with up to four op-amps on one chip. Therefore, size and power dissipation are often no longer the main problem.• There are two very famous multiple amplifier biquads;
1. Tow-Thomas Filter (TT), 2. Kerwin- Huelsman- Newcomb Filter (KHN)
20Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Multiple-Amplifier Biquads: Tow-Thomas Filter
By the direct analysis three outputsBy the direct analysis, three outputs are available;• inverting lowpass output (-VLP)• noninverting lowpass output (VLP)• inverting bandpass output (V )
R2
• inverting bandpass output (VBP)-VLP
VLPVBP
1 S1
1w = w o 1= 21 / CCRQ
213211
2
14
1)1(CCRR
SCR
S
SCR
VV
in
BP
++−=
213211
2
2142
1)1(CCRR
SCR
S
CCRRVV
in
LP
++=
2132 CCRRw o =
RCQ=
32
211 RR
RQ =
4
3)0(RRH LP =
4
1)(RRjwH oBP =
21Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
4
Multiple-Amplifier Biquads: Tow-Thomas FilterThe Tow-Thomas Filter has the
VLPVBP
following advantages:Its Q can be adjusted by varying a single resistance –> R1 without di t bi -VLP
VLPdisturbing wo
The gain can be varied by varying a single resistance -> R4 without disturbing w and Qdisturbing wo and QThe lowpass output can be either inverting or noninvertingThe sensitivity of the Tow-Thomas ybiquad are generally comparable to those of the KHN biquad
22Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
Multiple-Amplifier Biquads: Kerwin- Huesman- Newcomb FilterBy the direct analysis three outputsBy the direct analysis, three outputs are available;• noninverting High-pass output (VHP)• inverting band-pass output (-VBP)• noninverting low pass output (V )• noninverting low-pass output (VLP)
VLPVBPVHP
66532
212143
65
5
4
1)()]1)()([(
)1)()((
RSRRRS
CCRRRRRR
RR
VV
in
LP
++
+
++
=
Vin
2121511435
)()])()([(CCRRR
SCRRRR
Sin ++
+
66532
1143
65
5
4
1)()]1)()([(
)1)()((
RSRRRS
SCRRR
RRRR
VV
in
BP
++
+
++
−=
VV 1=
Notes:
2121511435
)()])()([(CCRRRCRRRR +
66532
2
43
65
5
4
1)()]1)()([(
))((
CCRRRRS
CRRRRR
RRS
SRRRR
RR
VV
in
HP
+++
+
++
=
BPLP VCSR
V22
−=
HPBP VCSR
V11
1−=
23Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011
2121511435 CCRRRCRRRR +
Notes:Notes:The general transfer function of the second order active filter in the s-plane has the following form:
22
322
1
)()(
o wSwS
KSKSKsH++
++=
)( owSQ
S ++
K1 = K2 = 0 Low-Pass
K1 = K3 = 0 Band-Pass
K2 = K3 = 0 High-Pass
K1 = 1 K2 = 0 and K3 = band-reject2wK1 = 1, K2 = 0, and K3 = band-reject
K1 = 1, K2 = , and K3 = All-Pass
ow
Qwo− 2
ow
24Dr. Mohamed Abd el Ghany Department of Electronics and Electrical Engineering
COMM 704:Communication Systems Winter 2011