analog filter design - unipi.itdocenti.ing.unipi.it/~a008309/mat_stud/aif/2018/lecture... · 2018....
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P. Bruschi - Analog Filter Design 1
Analog Filter Design
Part. 3: Time Continuous (TC) Filter Implementation
Sect. 3-a: Active Filters
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Motivations
• Inductors are generally difficult to miniaturizeL ~ (coil area) x (number of coils)2 x (magnetic permeability)Integrated inductors limited to a few nH (max)Stray magnetic field cause unwanted coupling
• Resistors and capacitors can be easily integrated: feasible ranges are much wider than for inductors
• Active Filters Target: Synthesis of arbitrary transfer functions using only resistors, capacitors and active elements.
P. Bruschi - Analog Filter Design 2
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Design approaches for active TC filters
• Cascade of Biquadratic (Biquad) and Bilinear cells
• State Variable Filters (MLF: Multiple Loop Feedback circuits)
• Simulation of LC filters with active RC networks
P. Bruschi - Analog Filter Design 3
System-level architectures
Circuit-level architectures
Op-amp based
OTA (Operational transconductance amplifier) – based
-
Cascade of Biquad (Bilinear) functions
P. Bruschi - Analog Filter Design 4
Biquad Transfer Function
22
2
01
2
2
0
01
2
01
2
2)(
p
p
p
z
z
z
BQ
sQ
s
bsQ
bsb
Hdsds
cscscfH
p
zBL
s
bsbHfH
010)(
b2,b0 : 0, 1
b1: 0, ±1
Bilinear Transfer Function
b1 : 0, 1
b0: 0, ±1
“Bits” b2,b1,b0 determine
which terms are present in
the numerator
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Poles vs. Biquad coefficients
P. Bruschi - Analog Filter Design 5
2 Rep
p p P
p
ss Q
s
2
* 2 * * 22 Re
p p p p p p p ps s s s s s s s s s s s s s
For a 2nd order polynomial with complex roots:
Biquads can be easily
extracted from "zpk" output
of python or Matlab filter
synthesis functions
For the zeroes, identical rules apply, with the exception of :
Zeros in the origin, s2 or s term only (b0=0)
Zeros to infinity, only constant term is present (b1,b2=0)
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Notable cases
P. Bruschi - Analog Filter Design 6
22
p
p
p
p
p
sQ
s
sQ
22
2
p
p
ps
Qs
s
22
22
p
p
p
p
sQ
s
s
22
22
p
p
p
p
p
p
sQ
s
sQ
s
22
2
p
p
p
p
sQ
s
Low pass High pass Band pass Band Stop
All pass
(phase equalizer)
All these biquads have
unity gain in their
respective pass-bands
-
Sequencing criteria for biquad cascades
P. Bruschi - Analog Filter Design 7
Requirement: Zout
-
P. Bruschi - Analog Filter Design 8
Sequencing criteria: Targets and Rules of Thumb
Pairing: couple together poles and zeroes which are closer in the s-plane
(flatter response, less component spread)
Position: Place the biquads with lower Q closer to the inputs
Keep biquads with similar frequency of maximum as far away as
possible
If possible, place LP Biquads first and HP or BP Biquads last
Gain distribution: balance the signal amplitude over the various biquads
Targets
Maximize the Dynamic Range (DR)
Minimize the transmission sensitivity (to component variations)
Minimize the pass-band attenuation
Rules
-
Biquad implementations
P. Bruschi - Analog Filter Design 9
Op-amp Based:
SAB (Single Opamp Biquad)
Finite Gain SABs – positive feedback
Finite Gain SABs – negative feedback
Infinite Gain SABs
Multiple op-amp Biquads (e.g. MFL )
OTA based (Gm-C filters)
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SABs
P. Bruschi - Analog Filter Design 10
kVV
VyVyVyI
/
0
23
3332231133
kyy
y
V
V
V
V
in
out
/3323
13
1
2
02231133 VyVyI
23
13
1
2
y
y
V
V
V
V
in
out
Finite gain Infinite gain
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Example: Sallen-Key Biquads
P. Bruschi - Analog Filter Design 11
SK General configuration SK- Low pass filter
in
out
V
V
(R.P. Sallen, E.L. Key – MIT Labs, 1955)
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Example II: SAB with infinite amplifier gainand "bridged-T" network
P. Bruschi - Analog Filter Design 12
2 1 3 4 1 43
23
2 1 3 4
Y Y Y Y Y Yiy
v Y Y Y
3 41
13
2 1 3 4
Y Yiy
v Y Y Y
13 3 4
23 2 1 3 4 1 4
y Y YH
y Y Y Y Y Y Y
Bridged –T network
For infinite (negarive) amplifier gain:
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Band-pass Delyannis-Friend Biquad
P. Bruschi - Analog Filter Design 13
1 2
2
2 1 2 1 2
( )1 1
s
s
R CH s
s sR C R R C C
1 1
2 2
3 1
4 2
1 /
1 /
Y sC
Y R
Y R
Y sC
Band-pass
Biquad
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Multiple Feedback Loop Filters
P. Bruschi - Analog Filter Design 14
Cascaded Biquads: Feedback exist only inside blocks
MFL Filters: Feedback involve all stages together
More Interaction: less sensitivity to component variations
“Follow the Leader Filter” (FLF) architecture”
Ti(s) can be:
• Integrators
• Lossy Integrators
• Biquads
coefficients
-
Integral representation of transfer functions
P. Bruschi - Analog Filter Design 15
01
1
1
01
1
1
1
2
....
....
bsbsbs
asasasa
V
Vn
n
n
n
n
n
n
1 11 1 0 2 1 1 0 1.... ....n n n n
n n ns b s b s b V a s a s a s a V
1 1 0 2 1 1 0 11 1
1 1 1 1 1 11 ,..., ,...,
n n nn n n nb b b V a a a a V
s s s s s s
2 1 1 0 2 1 1 0 11 1
1 1 1 1 1 1,..., ,...,
n n nn n n nV b b b V a a a a V
s s s s s s
divide
by sn
generic rational
transfer function
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State variable filters(MFL filters based on Integrators)
P. Bruschi - Analog Filter Design 16
1
2 1 0 2 1 2 1 2
2
1
1 1 1 0
1 1 1 1.....
....
n n n
n
n n
n
V V b V b V b Vs s s s
V K
V s b s b s b
Low pass, all poles filters
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Multi Feedback – Multi Feed Forward
P. Bruschi - Analog Filter Design 17
01
1
1
01
1
1
1
2
....
....
bsbsbs
asasasa
V
Vn
n
n
n
n
n
n
Arbitrary Functions
(poles and zeros)
'
2
1
1 1 1 0
1
....n n
n
V
V s b s b s b
-b0
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Integrators: Op-amp based solution
P. Bruschi - Analog Filter Design 18
sRCv
v
in
out 11
CRs
CR
R
R
v
v
in
out
1
11
1
1
Integrator (inverting) Lossy Integrator (inverting)
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Example: State variable Filter with Op-amp Integrators
P. Bruschi - Analog Filter Design 19
Inverting amplifiers
are required to cope
with the inverting function
of the OA based Integrators
Opamp integrators
(inverting)
Resistor values
(inverse of
summing
coefficients)
-
MLF: Example Universal 2nd order Filter
P. Bruschi - Analog Filter Design 20
Kerwin-Huelsman-Newcomb (KHN) filter(Produces LP, BP and HP outputs: Single Input – Multiple Output)
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OTA: definitions and basic circuits
P. Bruschi - Analog Filter Design 21
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 moutout
OTA-C (Gm-C) Integrator
OTA-C (Gm-C) Eq. Resistor
(0 )P out m P
P m P
I I G V
I G V
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Ota-Based summing circuits
P. Bruschi - Analog Filter Design 22
Summing amplifier
(inverting / non-inverting)
Summing Integrator
(inverting / non-inverting)
43
1
2
21
1 vvG
Gvv
sC
Gv
m
mm
out 433
2
21
3
1 vvG
Gvv
G
Gv
m
m
m
m
out
Equivalent resistor:
R=1/Gm3
-
Gm-C integrator with feed-forward input
P. Bruschi - Analog Filter Design 23
321 vvvsC
Gv mout C
Gm1
Iout
-
Gm-C integrator cascade
P. Bruschi - Analog Filter Design 24
Two signals with opposite signs
can be added at each internal
node of the cascade
-
State variable filters – alternative solution
P. Bruschi - Analog Filter Design 25
1
1 1 0
1
1 1 0
....
....
n n
out n
n n
in n
V s a s a s a
V s b s b s b
Differently from the FL filter (Follow the Leader), where all the integrator outputs are fed back to the first
integrator and fed forward to the summing node, here the output voltage is fed back to the input of each
integrator where also the input signal is fed forward. This architecture is more suitable for Gm-C
implementations, where summing several inputs would require several OTAs
Target f.d.t.
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Gm-C state variable filter: coefficients
P. Bruschi - Analog Filter Design 26
1 1 0 1 1 01 1
1 1 1 1 1 1,..., ,...,
out n out n n inn n n nV b b b V a a a a V
s s s s s s
1 1
2 1 2
0 1 2 0...
n n
n n n
n n
b
b
b
1 1 1
2 2 2
0 0 0
n n
n n n
n n n
a c
a c b
a c b
a c b
bn-1bn-2b0
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Example: First order high pass / low pass filters
P. Bruschi - Analog Filter Design 27
( )p
p
H ss
( )p
sH s
s
low pass
high pass
m
p
G
C
-
Example: State variable Gm-C biquad
P. Bruschi - Analog Filter Design 28
2 2
2 1 0
2 2
( )
p
p
p
p
p
p
B s B s BQ
H s
s sQ
01 02
01
02
p
pQ
02
in
in
in
vBv
vBv
vBv
23
12
01
}1,0{,, 210 BBB
Flexible Biquad
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State variable Flexible Biquad – OTA implementation
P. Bruschi - Analog Filter Design 29
2
2
02
1
1
01C
G
C
G mm
Function B0 B1 B2
Low pass 1 0 0
High pass 0 0 1
Band-Pass 0 1 0
Notch 1 0 1
1 2 1 2
1 2 2 1
m m m
p P
m
G G G CQ
C C G C
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Simulation of Ladder Filters with OTAs
P. Bruschi - Analog Filter Design 30
Simulation of the inductor: application of the OTA
based Gyrator
Simulation of the nodal equations by means of OTAs
(signal flow path) May require inductor simulation,
depending on the transfer function to synthesize and/or
architecture
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Gyrator
P. Bruschi - Analog Filter Design 31
0
0
1
2
G
GY
2 1 1
1 2 2
i G v
i G v
Y-parameters equivalent circuit
-
Inductance simulation by means of a gyrator and a capacitor.
P. Bruschi - Analog Filter Design 32
Generic Impedance Inversion
21GG
CLEQ
2 2
P P
V
P
v vZ
i G v
Inductor Synthesis
21
1
GG
CsZ
CsZ V
2 1 1 2
1P P
V
P P
v vZ
i G G v Z G G Z
2 2 1 pv i Z G v Z
-
OTA Based Gyrators
P. Bruschi - Analog Filter Design 33
Grounded gyrator
2 1 1
1 2 2
i G v
i G v
-
Inductor simulation with OTAs
P. Bruschi - Analog Filter Design 34
21 mm
EQGG
CL
Grounded Inductor
Floating Inductor
IA IBIOA IOB
VL+
-
VL
+
-
VC
+-
sL
VII LBA
IB
IA
Lmm
BA
CmOBBCmOAA
LmC
VsC
GGII
VGIIVGII
VGsC
V
12
22
1
;
;1
21 mm
EQGG
CL
-
P. Bruschi - Analog Filter Design 35
Inductor simulation with OTAs - Example
Initial passive filter
2
1 1
2
2 2
2
3 3
L m
L m
L m
C L G
C L G
C L G
L1
L2
L3
L1,L3: floating inductors
L2: grounded inductor
Same filter, with simulated inductors
-
Signal flow simulation of ladder (LC) networks with OTAs
P. Bruschi - Analog Filter Design 36
566
6455
5344
4233
3122
211
IZV
VVYI
IIZV
VVYI
IIZV
VVYI in
Network Equations
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Variable transformations
P. Bruschi - Analog Filter Design 37
Target: Transform current variables (I1,I3,I5)
into voltage variables
553311
111I
gVI
gVI
gV
1
1 2
2 2 1 3
3
3 2 4
4 4 3 5
5
5 4 6
6 6 5
in
YV V V
g
V gZ V V
YV V V
g
V gZ V V
YV V V
g
V gZ V
Homogeneous
equivalent
equations
-
Leap-Frog architecture
P. Bruschi - Analog Filter Design 38
423
3
3122
21
1
VVg
YV
VVgZV
VVg
YV in
4 4 3 5
5
5 4 6
6 6 5
V gZ V V
YV V V
g
V gZ V
Homogeneous equivalent equations
-
OTA implementation of the Leap-Frog structure
P. Bruschi - Analog Filter Design 39
3122
21
1
VVgZV
VVg
YV in
31222
2111
VVGZV
VVGZV
mL
inmL
2
22
1
11
m
L
m
L
G
gZZ
gG
YZ
same for all
odd indexes
same for all
even indexes
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Example: 5th Order Chebyshev Filter
P. Bruschi - Analog Filter Design 40
C1=33.9 mF
L1=17.35 mH
C2=47.7 mF
L2=17.35 mH
C3=33.9 mF
fp=10 kHz
Y1
Z2 Z4
Y3 Y5
Z6
11
1
11
m
LgGR
Z
1R
S1
1
g
1
1
1
m
LG
Z mS1k1 11 mL GZ
L
L
mm
LsC
ZGsC
g
G
gZZ
2
2
212
22
1 1
2
2 Cg
GC mL
2
1 S
10 Sm
g
G m
2339 pF
LC
-
P. Bruschi - Analog Filter Design 41
Example: 5th Order Chebyshev Filter
1
3
1
3
3
1
sLY
gG
YZ
m
L Lm
LsCgGsL
Z311
3
11 113 mL gGLC
3
1 S
10 Sm
g
G m
3
173.5 pFL
C 1
2
3
4
5
6L
1
2 6
1 k
339 pF
173.5 pF
47.7 pF
173.5 pF
Z 100 k 339 pF
1mS
G 10 S
L
L
L
L
L
m
m
R
C
C
C
C
G
m
36
2
63
2
66
6
6
1
1
1
CsGR
Gg
sCR
G
g
G
gZZ
mmmm
L
2
6m
R
gG
6 3mG C
g4 5,
L LZ Z
Same procedure 6LZ
-
Frequent choices in active ladder filters
• Inductor synthesis :
HP filters (ideal for all-grounded inductors)BP filters LP – filters with zeroes
• Leapfrog architectures:
LP all – pole filtersBP filters (resonant groups simulated by biquads)
P. Bruschi - Analog Filter Design 42
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P. Bruschi - Analog Filter Design 43
Self-Tuning of OTA FiltersIn integrated circuits, Gm's and capacitances are strongly affected by PVT variations (up to ± 30 %
variations). For these reasons, in most OTAs the Gm can be controlled by means of a voltage
applied to a proper terminal (Vtune). In this way self-tuning of the filter can be accomplished.
To understand the self-tuning
principle, consider the effect of tuning
on the phase response of a 1st order
LP filter
-
Self-Tuning of OTA Filters: master slave approach
P. Bruschi - Analog Filter Design 44
Slaves
Master
The loop varies Vtune in
such a way that wC of the
two master LP filters
equals the reference
frequency. Since the same
Vtune is fed to the slaves,
their Gm/C ratios are also
proportional to the ref.
frequency.
Reference
frequency
source
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