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Unformatted text preview: Biquad Realization Boris Murmann
Stanford University
murmann@stanford.edu
Copyright © 2009 by Boris Murmann B. Murmann EE315A ― HO #4 1 Architectural Options for HighOrder Filters
• Cascades of (active) first and secondorder sections (biquads) • Ladder filters (passive or emulated using active components) • Specialized architectures, typically emphasizing low complexity
– Watch out for sensitivity issues B. Murmann EE315A ― HO #4 2 Building Block Options [Kuhn, IEEE TCAS II, 10/2003] B. Murmann EE315A ― HO #4 3 Example
• An interesting filter
that combines three
different approaches
– Passive LC
– Active RC
– Switched capacitor [Schreier, JSSC 12/2002] B. Murmann EE315A ― HO #4 4 The Challenge
• Way too many options available
y
y p • Deciding on which implementation is “best” may only be
possible once several options have been thoroughly compared
– I terms of both firstorder properties and secondorder
In t
f b th fi t d
ti
d
d d
nonidealities, which aren’t always easy to understand • The following discussion starts from the most basic ideas, and
derives some of the most popular solutions used in practice • For now, we will focus on the realization of biquads, and cover
ladder based
ladderbased implementations next
– The treatment of biquads will help us understand why we
may ultimately want go for a ladder implementation EE315A ― HO #4 B. Murmann 5 Passive LC Lowpass Filter H( s ) = 1
sC 1
+ R + sL
sC ωP = B. Murmann = 1
2 1 + sRC + s LC 1
LC QP = R EE315A ― HO #4 1 =
1+ s
s2
+ 2
ωPQP ωP L
C 6 Onchip Capacitors
MetalInsulatorMetal (MIM) Vertical Parallel Plate (VPP) [Ng, Trans. Electron Dev. 7/2005] [Aparicio, JSSC 3/2002] • Typically 12 fF/μm2 (1020 fF/μm2 for advanced structures)
– For 1 fF/μm2, a 10 pF capacitor occupies ~100μm x 100μm • Both MIM and VPP capacitors have good electrical properties
– Mostly worry about parasitic caps
– Series and parallel resistances are often not a concern B. Murmann EE315A ― HO #4 7 Onchip Inductors ~1mm
[Mohan, JSSC 10/1999] [Bevilacqua, ISSCC 2004] • Many nonidealities, hard to model, low “Q”
y
,
,
Q • Area inefficient, typically achieve L < 10nH • Sometimes bondwires are used as an alternative, L ~ 1nH/mm B. Murmann EE315A ― HO #4 8 Inductor Quality Factor Y= In general ⇒ Q= 1
Rs + j ωL ⇒ QL = Y= • X( ω)
R 1
R + jX( ω) ωL
Rs Onchip inductors typically achieve QL < 510 at their usable
frequencies EE315A ― HO #4 B. Murmann 9 LC Lowpass Example
• Assuming that we (very generously) use C=100pF, L=10nH
ωP = 1
= 2π ⋅ 160MHz
LC • Integrated p
g
passive LC filters become p
practical for
f > 200500MHz • For our LC lowpass, if we assume R=Rs (i.e. we only use the
parasitic resistor of the inductor no explicitly added resistance)
inductor,
QL = • ωL
Rs QP = ω
1 L QL L
=
= QL P
R C ωL C
ω This means that at ω=ωP, the magnitude peaking that we can
get is limited to the QL of the inductor (~510); not all that great B. Murmann EE315A ― HO #4 10 Summary
• Onchip capacitors are g
p p
great, even though they’re usually not as
,
g
y
y
large as we would like them to be • Onchip inductors tend to be avoided whenever possible, and
are typically not useful in a filter with poles at frequencies below
< 200500 MHz • The solution to this problem is to “simulate” the inductors using
active components
ti
t EE315A ― HO #4 B. Murmann 11 Gyrators
• Gyrators are “active inductors” T. L. Deliyannis, J. K. Fidler and Y. Sun,
ContinuousTime Active Filter Design
g
http://www.engnetbase.com/ej
ournals/books/book_summary/
summary.asp?id=475
(Section 3.5) • The above circuit is not all that useful for our lowpass filter; we
f f
f
need a “floating” inductor
− Don’t want the inductance to be ground referenced B. Murmann EE315A ― HO #4 12 Floating Gyrator
http://www.engnetbase.com/ej
ournals/books/book_summary/
summary.asp?id=475
(Section 6.4) • Floating gyrators are pretty complex
– There must be a better way to solve this problem
problem… B. Murmann EE315A ― HO #4 13 Integrators
• A circuit that we can build without much sweat is an active
integrator, e.g. using an opamp
– Many more options exist (more later) • Assuming the availability of an ideal opamp, we have vout (t ) = − 1 v in (t )
dt
C∫ R Vout ( s ) = − B. Murmann EE315A ― HO #4 1
Vin ( s )
sRC 14 StateSpace Realization
State variables
(integrator outputs)
L vc (t ) = 1
ic (t )dt
C∫ iL (t ) = 1
v L (t )dt
L∫ Vc ( s ) = 1
Ic ( s )
sC IL ( s ) = 1
V (s)
sL L L
in C C out Vc =
IL = B. Murmann 1
1
Ic =
IL = Vout
sC
sC
1
1
VL =
(Vin − ILR − Vout )
sL
sL
EE315A ― HO #4 15 Block Diagram • Looks promising, but the problem with this realization is that the
first integrator takes a voltage at the input and produces a
current at the output
– We need the opposite if we want to realize the circuit with an
RC integrator B. Murmann EE315A ― HO #4 16 Modified (Equivalent) Block Diagrams EE315A ― HO #4 B. Murmann 17 Implementation A1 A2 H( s ) = • A3 Vout ( s )
1
=−
Vout ( s )
1 + sRC + s 2LC One remaining issue is that the transfer function is inverted
– We could fix that (if needed) using a fourth opamp
– Or by pushing A2 toward the input, and utilizing both its
inverting and noninverting input
• The latter trick is used in the socalled KHN biquad B. Murmann EE315A ― HO #4 18 KHN Biquad W.J. Kerwin, L.P. Huelsman, R.W Newcomb, "StateVariable Synthesis for Insensitive Integrated Circuit Transfer
,
,
,
y
g
Functions," IEEE JSSC, vol.2, no.3, pp. 8792, Sep. 1967.
http://www.engnetbase.com/ejournals/books/book_summary/summary.asp?id=475 (Section 4.9) EE315A ― HO #4 B. Murmann 19 Highpass and Bandpass Output
• An interesting feature of some biquads (including the HKN) is
that they provide additional highpass and bandpass outputs for
“free”
τ s 2 τ2 HHP ( s ) =
1+ B. Murmann s
s2
+ 2
ωPQP ωP τ −s τ HBP ( s ) = 1+ s
s2
+ 2
ωPQP ωP EE315A ― HO #4 1 HLP ( s ) =
1+ s
s2
+ 2
ωPQP ωP 20 The General HKN Biquad
HP LP
BP
GENERAL HGENERAL ( s ) = B. Murmann b2s 2 + b1s + b0 s
s2
+ 2
1+
ωPQP ωP Implements arbitrary
poles and zeros EE315A ― HO #4 21 TowThomas Biquad P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational
amplifiers,” P
lifi
” Proc. IEEE vol. 61 pp. 662 3 M 1973
IEEE, l 61,
6623, May 1973. • General biquad using only three opamps B. Murmann EE315A ― HO #4 22 Transfer Functions
( b a − b ) s + ( b2a0 − b0 )
Vo1
= −k 2 2 1 2 1
Vin
s + a1s + a0
Vo 2 b2s 2 + b1s + b0
= 2
Vin
s + a1s + a0
Vo 3
1 ( b0 − b2a0 ) s + ( a1b0 − a0b1 )
=−
Vin
s 2 + a1s + a0
k1 a0
• Vo2/Vin implements a general biquad section with arbitrary poles
and zeros • Vo1/Vin and Vo3/Vin realize the same poles but are limited to at
most one finite zero
EE315A ― HO #4 B. Murmann 23 Design Equations
b0 =
b1 =
b2 =
a0 =
a1 =
k1 =
k2 = B. Murmann R8
R3R5R7C1C2
1 ⎛ R8 R1R8 ⎞
−
⎜
⎟
R1C1 ⎝ R6 R4R7 ⎠
R8
R6
R8
R2R3R7C1C2
1
R1C1
R2R8C2
R3R7C1
R7
R8 For given ai , bi , ki , C1, C2 and R8 :
R1 =
R2 = 1
a1C1
k1
a0 C2 R3 = 1
k1k 2 1
a0 C1 R4 = 1
1
1
k 2 a1b2 − b1 C1 R5 = k1 a0
b0C2 R6 = R8
b2 ωP = R8
R2R3R7C1C2 QP = ωP R1C1 R7 = k 2R8
EE315A ― HO #4 24 SallenKey LPF H (s ) = R.P. Sallen and E. L. Key “A Practical Method of Designing RC Active
Filters.” IRE Trans. Circuit Theory, Vol. CT2, pp. 7485, 1955 ωP = G
s
s2
1+
+ 2
ωPQP ωP
1
R1C1R2C2 • Single gain element
• Poles only, no zeros
• Sensitive to parasitic capacitances
• Versions exist for HP, BP, …
– http://en.wikipedia.org/wiki/Sallen_Key_filter QP = ωP 1
1
1− G
+
+
R1C1 R2C1 R2C2 EE315A ― HO #4 B. Murmann 25 TowThomas or SallenKey?
• Suppose we now wanted to realize a biquad that has p
pp
q
poles only
y • Should we use a TowThomas or SallenKey realization? • Clearly, from the perspective of complexity, we would probably
want to go for SallenKey • Unfortunately, the SallenKey realization comes with
disadvantages in terms of sensitivity to component variations • Let’s take a closer look… B. Murmann EE315A ― HO #4 26 Sensitivity
• The sensitivity of any variable y to any parameter x is defined as
(See e.g. Gray & Meyer, Section 4.2)
Δy x ∂y
⎛ Δy / y ⎞ x
y
Sx = lim ⎜
=
⎟ = Δlim0
Δx → 0 ⎝ Δx / x ⎠
y x → Δx y ∂x • In order to relate fractional changes in y to fractional changes in
x we can then write
Δy
y Δx
≅ Sx
y
x • Example
y
Sx = 10 • Δx
= 2%
x ⇒ Δy
≅ 20%
y Common sense: Sensitivity is a first order approximation,
accurate only for “small” errors
small B. Murmann EE315A ― HO #4 27 Parameter Variations (1)
• Discrete resistors and capacitors • Come in many different shapes and sizes and accuracies
– E metal fil resistors, ~0.1% accurate, 5
E.g.
t l film
i t
0 1%
t 5ppm/°C
/°C
– E.g. C0G dielectric capacitors, 2% accurate, very small
temperature dependence B. Murmann EE315A ― HO #4 28 Parameter Variations (2)
• Integrated resistors and capacitors
Poly resistor • MIM Capacitor Important to distinguish between
– Global process variations
– Devicetodevice mismatch
EE315A ― HO #4 B. Murmann 29 Global Process Variations
Wafer made yesterday
All NMOS are “slow”
All PMOS are “
“nominal”
i l”
All R are nominal
All C are “fast” Parameter Wafer made today
All NMOS are “fast”
All PMOS are “f ”
“fast”
All R are nominal
All C are “slow” “Slow” “Nominal” “Fast” 0.2V 0.3V 0.4V μCox (NMOS)
C 240 μA/V2
A/V 300 μA/V2
A/V 360 μA/V2
A/V μCox (PMOS) 80 μA/V2 100 μA/V2 120 μA/V2 Rpoly 60Ω/□ 50Ω/□ 40Ω/□ Rnwell 1.4 kΩ/□ 1 kΩ/□ 0.6 kΩ/□ Vt CMIM B. Murmann 1.15 fF/μm2 1 EE315A ― HO #4 fF/μm2 0.85 fF/μm2 30 DevicetoDevice Mismatch
• Upon closer inspection, device parameters not only vary from
lottolot or wafertowafer, but there are also differences
between closely spaced, nominally id ti l d i
b t
l
l
d
i ll identical devices on th
the
same chip
– These differences are called mismatch
Vt 1 − Vt 2 = ΔVt
M1 M2 C1
R1 B. Murmann C2
R2 W⎞ ⎛
W⎞
⎛
⎜ μCox
⎟ − ⎜ μCox
⎟ = Δβ
L ⎠1 ⎝
L ⎠2
⎝ C1 − C2 = ΔC
R1 − R2 = ΔR EE315A ― HO #4 31 Statistical Model
• Experiments over the past decades have shown that devicetodevice mismatch (ΔVt, ΔC, …) for properly laid out devices is
typically “random” and welldescribed by a Gaussian distribution
– With zero mean and a standard deviation that depends on
the process and the size of the device • Empirically, the standard deviation of the mismatch between two
closely spaced devices is modeled using the following
expression σΔX = AX
WL where W·L represents the area of the device, and X is the
device parameter under consideration B. Murmann EE315A ― HO #4 32 Parameters for a Typical 0.18μm Process Parameter Value AVt 5 mVμm
μ AΔβ/β 2 %μm AΔC/C (MIM capacitor) 1 %μm AΔR/R (Poly resistor) 3 %μm EE315A ― HO #4 B. Murmann 33 Example
• Example: MIMcapacitor with W=10μm, L=10μm (~100200 fF) σΔC / C = 1%
= 0.1%
100 3σΔC / C = 0.3% http://en.wikipedia.org/wiki/Image:Standard_deviation_diagram.svg B. Murmann EE315A ― HO #4 34 Sensitivity to Global Variations
TowThomas ωP = R8
1
∝
R2R3R7C1C2
RC QP = ωP R1C1 ∝ 1 SallenKey ωP = QP = 1
R1C1R2C2 ∝ 1
RC ωP 1
1
1− G
+
+
R1C1 R2C1 R2C2 • QP is independent of global variations in both realizations
– Assuming all R and C use the same device structure
structure,
respectively • ∝1 ωP varies with the RC product of the components
EE315A ― HO #4 B. Murmann 35 Sensitivity to Mismatch (SallenKey) ωP = QP = ω
ω
ω
ω
SR1P = SR2P = SC1P = SC2P = − 1
R1C1R2C2 ωP 1
1
1− G
+
+
R1C1 R2C1 R2C2 Q
Q
SR1P = −SR2P = − R2C2
1
+ QP
2
R1C1 Q
Q
SC1P = −SC2P = − ⎛ R1C2
R2C2 ⎞
1
+ QP ⎜
+
⎟
⎜ RC
2
R1C1 ⎟
2 1
⎝
⎠ Q
SG P = QPG • 1
2 R2C1
R1C2 Sensitivity depends on QP and “component spread” i.e. the
ratios of the resistors and capacitors, respectively B. Murmann EE315A ― HO #4 36 Example (1)
• Want to design a SallenKey filter with QP=10
g
y • Choice 1: All R and C are the same ⇒ G = 3 (1/QP) = 2.9
– Very nice from the perspective of component spread, but
bad for
b d f sensitivity, e.g.
iti it
Q
Q
SR1P = −SR2P = − • 1
+ QP = 9.5
2 Choice 2: Reduce sensitivity by accepting large component spread
– Can show that G=1 is a good choice
• See e.g. http://www.maximic.com/appnotes.cfm/an_pk/738/ EE315A ― HO #4 B. Murmann 37 Example (2)
• For G=1, we have
QP = ωP
1
1
+
R1C1 R2C1 and it follows that
Q
Q
SR1P = −SR2P = − • R2
1
+
=0
2 R1 + R2 R1 = R2 Unfortunately,
Unfortunately in this case
C1
2
= 4QP = 400
C2 • for for QP = 10 Bottom line: The SallenKey realization suffers from a strong
tradeoff between sensitivity and component spread B. Murmann EE315A ― HO #4 38 Case Studies MAXIM APPLICATION NOTE 738
Minimizing ComponentVariation Sensitivity in Single Op Amp Filters
http://www.maximic.com/appnotes.cfm/an_pk/738/ EE315A ― HO #4 B. Murmann 39 Sensitivity to Mismatch (TowThomas) ωP = R8
R2R3R7C1C2 ω
ω
ω
ω
ωP
ω
SRP = SR P = SR P = −SR P = SC1 = SC P = −
2
3
3
8
2 1
2 Q
SR1P = 1 QP = ωP R1C1 = R1 • R8C1
R2R3R7C2 Q
Q
Q
Q
Q
Q
SR2P = SR3P = SR7P = −SR8P = −SC1P = SC2P = − Constant sensitivities, independent of Q and component spread
– Much nicer! B. Murmann EE315A ― HO #4 40 1
2 Conclusions
• Biquads can be realized in numerous different ways • Implementation and component sizing have a big impact on
p
p
g
g p
sensitivity to variations
– Of course, we must avoid highsensitivity circuits in practice • No theory f fi di a l
N th
for finding lowsensitivity architecture
iti it
hit t
– Use proven circuits & check! • TowThomas biquad
q
– Arbitrary poles and zeros, three amplifiers
– Wellbehaved sensitivities • SallenKey biquad
– Only poles, one amplifier
– Sensitivities trade off with component spread
• Typically use G=1 and use this circuit only for “low Q” poles B. Murmann EE315A ― HO #4 41 ...
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This note was uploaded on 08/13/2009 for the course EE 315 taught by Professor Borismurmann during the Spring '09 term at Stanford.
 Spring '09
 BorisMurmann

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