Post on 09-Feb-2020
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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
Progettazione Mixed SignalPaolo Bruschi 2
fc synonyms:
Corner frequency
Cut-off frequency
fCL Corner Freq – Low
fCH Corner Freq – High
Common Filter Responses
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)
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
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
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
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
Progettazione Mixed SignalPaolo Bruschi 8
Gm-C integrator with feed-forward input
321 vvvsC
Gv m
out C
Gmw1
Progettazione Mixed SignalPaolo Bruschi 9
Gm-C Universal Biquad
2
2
02
1
1
01C
G
C
G mm ww
Progettazione Mixed SignalPaolo Bruschi 10
Self-Tuning of OTA Filters
Master
Slaves
Phase detector
Characteristic
Tuning law of
transconductors
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
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
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
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
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
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