1. cascade of biquadratic (biquad) and bilinear cells 2...
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Progettazione Mixed SignalPaolo Bruschi 1
Analog Active Filters: Approaches
1. Cascade of Biquadratic (Biquad) and Bilinear cells
2. State Variable Filters
3. Simulation of LC filters with active RC networks
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Progettazione Mixed SignalPaolo Bruschi 2
fc synonyms:
Corner frequency
Cut-off frequency
fCL Corner Freq – Low
fCH Corner Freq – High
Common Filter Responses
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Progettazione Mixed SignalPaolo Bruschi 3
Parameters of a real frequency response
The Low Pass filter is the
reference for other types
Frequency are general given as
angular frequencies (w)
A Transition Band (TB) is
introduced
Note: wp is generally different from wc (-3 dB frequency)
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Progettazione Mixed SignalPaolo Bruschi 4
Analog Filters: State Variable Filter
01
1
1
01
1
1
....
....
)(
)(
bsbsbs
asasasa
sv
svN
N
N
N
N
N
N
in
out
000222111
021021211 ...
bcabcabcaca
bbb
NNNNNNNN
NNNNNNN
wwwwww
It is an Integrator-based Architecture
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Progettazione Mixed SignalPaolo Bruschi 5
Biquad with State Variable Archirecture
2
002
2
000
1
2
2
)(
)(
ww
ww
sQ
s
bsQ
bsb
sV
sV
in
out Function b0 b1 b2
Low Pass 1 0 0
High Pass 0 0 1
Band Pass 0 1 0
Notch 1 0 1
02
01
0201
w
w
wwww
zp
zp
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Progettazione Mixed SignalPaolo Bruschi 6
OTA: definitions and basic circuits
Typical non-idealities:
Finite Rout
Input Capacitance
Frequency dependence of Gm
Input/Output ranges
21 vvGi mout
Ideal operation
OTA (Transconductor)
21 vvsC
G
sC
iv mout
out
OTA-C (Gm-C) Integrator
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Progettazione Mixed SignalPaolo Bruschi 7
Ota-Based summing circuits
Summing amplifier
(inverting / non-inverting)
Summing Integrator
(inverting / non-inverting)
43
1
2
21
1 vvG
Gvv
sC
Gv
m
mm
out 43
3
2
21
3
1 vvG
Gvv
G
Gv
m
m
m
m
out
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Progettazione Mixed SignalPaolo Bruschi 8
Gm-C integrator with feed-forward input
321 vvvsC
Gv m
out C
Gmw1
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Progettazione Mixed SignalPaolo Bruschi 9
Gm-C Universal Biquad
2
2
02
1
1
01C
G
C
G mm ww
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Progettazione Mixed SignalPaolo Bruschi 10
Self-Tuning of OTA Filters
Master
Slaves
Phase detector
Characteristic
Tuning law of
transconductors
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Progettazione Mixed SignalPaolo Bruschi 11
Switched Capacitors Integrator Parasitic Insensitive SC resistors
Positive SC resistorNegative SC resistor
Virtual ground
SC Integrator
Phase 1: Hold
Phase 2:
2
)1()(C
Qnvnv outout
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Progettazione Mixed SignalPaolo Bruschi 12
SC resistors: charge transfer
Phase 1
Phase 2
1
11
1
CfRVCfIVCQ
ck
EQSckEQS
Negative Resistor Positive Resistor
Q<0 Q>0
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Progettazione Mixed SignalPaolo Bruschi 13
SC Summing Integrator
1)1()1()1()(2
1
2
12
2
11 nvnvC
Cnv
C
Cnv
C
Cnvnv finfin
Fninpinoutout
ck
ck
zVC
CzV
z
z
C
CzV
z
z
C
CzV fin
Fninpinout
2
1
1
1
2
12
1
1
2
11 )(1
)(1
)(
We assume that
inputs vin-p and vin-n
are sampled at the
end of phase 2
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Progettazione Mixed SignalPaolo Bruschi 14
zVC
CzV
z
z
C
CzV
z
z
C
CzV fin
Fninpinout
2
1
1
1
2
12
1
1
2
11 )(1
)(1
)(
Discrete time integral
1
1
1)(
z
zzH IZ
wwww
w
w
ww
2221
)()(TjTjTj
Tj
Tj
TjTj
IZIF
eee
e
e
eeHjH
ww
w
2sin2
)(2
Tj
ejH
Tj
IF
TjjH
ff
TIF
ck
ww
w 1)(1
2
ckfT
1
Forward Euler Integration
Approximation for f<<fckExact response
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Progettazione Mixed SignalPaolo Bruschi 15
bilinear
Continuous Time
Euler
0.5
Integrator comparison
1
1
1
z
z
1
1
1
1
2
1
z
z
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Progettazione Mixed SignalPaolo Bruschi 16
SC-integrator equivalent block diagram
2
11
12
111
2
121
C
Ck
C
Ck
C
Cf F
F w