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al
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Filters Tutori
Giacomino BollatiTPA R&D
12 May 2004
Department of Electronics-Univ. of Pavia STMicroelectronics
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iacomino Bollati, 12 May 2004 GmC Filters TutorialOutline
Transfer function: biquad cascade
MATLAB filter design
GmC implementation
MATLAB GmC biquad design
GmC zeros realization
Transistor level filter design:- biquad design- gm control design
Filter characterization
Second order effects
Department of Electronics-Univ. of Pavia STMicroelectronics
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cade
F onstellations able to fit the require-m
T stellation.
U king it from a standard sets (Butter-wF sets give the pole values:N ,o
It scade of first and second order (bi-q
T age that the filter is not a single bigs
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transfer function: biquad cas
or a given mask filter it is possible to find several pole-zero cents.
he first step to design a filter is to choose the pole-zero con
sually, it is possible to choose the pole-zero constellation taorth, Bessel, etc.).or each order of the filter (order = number of poles N) these/2 couples of complex conjugated pole for even order filtersdd order filters have also a real pole.
is possible to satisfy each kind of filter mask through the cauad) structures.
his approach compared to the ladder filters has the advanttructure but the cascade of “simple” blocks.
Department of Electronics-Univ. of Pavia STMicroelectronics
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S
TS an m=n=0) and the transfer functionc
T t of second order structures:
A n and to choose the sequence of theb
s z2–( )s p2–( )
--------------------
)
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
MATLAB filter design
econd order transfer function:
he above function has 2 poles and 2 zeros.tandard filters (Butterworth, Bessel, etc.) don’t have zeros (than be rewritten in terms ofωo and Q:
he filter transfer function can be represented as the produc
MATLAB model is useful to choose the best transfer functioiquad cell in the cascade.
G s( ) k1 m s n s
2⋅+⋅+
1 a s⋅ b s2⋅+ +
---------------------------------------• ks z1–( ) ⋅s p1–( ) ⋅
------------------------•= =
G s( ) k
1 sωo Q⋅---------------- s
2
ωo2
----------+ +
------------------------------------------=
H s( ) G1 s( ) G2 s( )• • Gn s(••=
Department of Electronics-Univ. of Pavia STMicroelectronics
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B
T
t
gm4+
4----
----
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Implementation 1/5iquad:
ransfer function:
+ - + - + - +gm3
C1 C2
Vin
Vou
gm1 gm2
VoutVin
------------
gm1gm4-----------
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2 C1 C2⋅
gm2 gm⋅----------------------⋅+ +
----------------------------------------------------------------------------------=
Department of Electronics-Univ. of Pavia STMicroelectronics
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F parametersk, ωo and Q in terms ofth
infinite solutions.
gm1gm4-----------
1m4------- s
2 C1 C2⋅
gm2 gm4⋅--------------------------⋅+
--------------------------------------------------
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Implementation 2/5
rom the above equation it is possible to describe the biquadegm and C parameters:
Design problem: 3 equations, 6 variables ->
G s( ) k
1 sωc ωo⋅( ) Q⋅
-------------------------------- s2
ωc ωo⋅( )2--------------------------+ +
---------------------------------------------------------------------------
1 sgm3 C⋅
gm2 g⋅-------------------⋅+
------------------------------------= =
Qgm2 gm4⋅
gm32
--------------------------C2
C1------⋅=
ωo1
ωc------- gm2 gm4⋅
C1 C2⋅--------------------------⋅=
kgm1gm4-----------=
Department of Electronics-Univ. of Pavia STMicroelectronics
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G12
S der filters having wide ranges for theω
F aving:thaa
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Implementation 3/5
ood design “rules of thumb”:- minimize the range of capacitor value used,- minimize the range of gm values used.
evere constraints on the mask filter forces the use of high oro and Q values increasing the ranges of gm and C.
or a given mask it is important to find the transfer function he lower order,nd/or the minimum range ofωo and Q,nd/or the minimum absolute values of the highestωo and Q.
Department of Electronics-Univ. of Pavia STMicroelectronics
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G an iterative procedure.
G s, for example:CT 4/Q, while from the expression forkthT rding to 2 main constraints:1 (noise filter proportional to KT/C)2 be, hence, the higher the power con-sC fy the noise requirements.O the minimum required for noise. Oneo
A of C values is the minimum (C1=C2)b le to reduce this range increasing theC
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Implementation 4/5
olden parameter set of gm and C should be found through
ivenωo, Q and k, it is possible to fix some starting condition
1=C2, gm2=gm4hus, from the expression for Q it is possible to find gm3=gm it is possible to find gm1=k*gm4e expression for ωo gives the ratio (gm4/C1)/ωc.he absolute values of the parameters must be chosen acco- The higher the C values are the lower the noise filter will be- The higher the C values are the higher the gm values mustumption will be. values should be as small as possible but sufficient to satisther requirements can force us to use capacitors bigger thanf these requirements can be the transfer function accuracy.
t this point a set of parameters has been found. The rangeut gm4/gm3 has a range as wide as the Q value. It is possib range.
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E Hz, k1=k2=1
BCL mA/V
BCL gm3=1.7mA/Vg
BBLCL *sqrt(C1*C2)=0.98mA/Vgg
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC Implementation 5/5
xample:ω1=1.603, Q1=0.805;ω2=1.430, Q2=0.522; fc=100M
QD1
1=C2, gm2=gm4, gm1=gm4, gm3=gm4/Q1et’s choose C1=1pF, gm4 =ω1*2*Π*fc*C 1 = 1mA/V, gm3=1.24
QD2
1=C2, gm2=gm4, gm1=gm4, gm3=gm4/Q2et’s choose C1=1pF, gm1=gm2=gm4=ω2*2*Π*fc*C 1=0.9mA/V,m3/gm4=1.89
QD1 gm range is very good.QD2 gm range can be reduced increasing C range.et’s redesign the cell.2=0.7*C1, gm2=gm4, gm1=gm4, gm3=gm4/Q2et’s choose C1=1.3pF, C2=0.91pF , gm1=gm2=gm4=ω2*2*Π*fcm3=gm4/Q2*sqrt(C2/C1)=1.57mA/Vm3/gm4=1.60
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2B
T
t
gm4+
4----
----
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC zeros realization 1/iquad:
ransfer function:
+ - + - + - +gm3
C1 C2
Vin
Vou
gm1 gm2
Ik
Vout
gm1gm4----------- Vin Ik
s C1⋅
gm2 gm4⋅--------------------------⋅–⋅
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2 C1 C2⋅
gm2 gm⋅----------------------⋅+ +
----------------------------------------------------------------------------------=
Department of Electronics-Univ. of Pavia STMicroelectronics
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2
C
C
mkgm1⋅
---------------- Vin⋅
C1 C2⋅
gm2 gm4⋅--------------------------⋅
--------------------------------
Ckgm1------------
Vin⋅
C1 C2⋅
gm2 gm4⋅--------------------------⋅
--------------------------------
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
GmC zeros realization 2/
ase 1: Ik = gmk*Vin
ase 2: Ik = s*Ck
Vout
gm1gm4----------- gmk
s C1⋅
gm2 gm4⋅--------------------------⋅–
Vin⋅
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2 C1 C2⋅
gm2 gm4⋅--------------------------⋅+ +
--------------------------------------------------------------------------------------
gm1gm4----------- 1 s C1
ggm2----------⋅ ⋅–
⋅
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2+ +
------------------------------------------------------= =
Vout
gm1gm4-----------
s2
Ck C1⋅ ⋅
gm2 gm4⋅-----------------------------–
Vin⋅
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2 C1 C2⋅
gm2 gm4⋅--------------------------⋅+ +
--------------------------------------------------------------------------------------
gm1gm4----------- 1 s
2 C1 ⋅
gm2 ⋅--------------⋅–
⋅
1 sgm3 C1⋅
gm2 gm4⋅--------------------------⋅ s
2+ +
------------------------------------------------------= =
Department of Electronics-Univ. of Pavia STMicroelectronics
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design 1/7
C
L of the MOS Vod=(VGS-VTH).F of the MOS.V but sufficient to achieve the requiredli
2 µ CoxWL----- Ibias⋅ ⋅⋅ ⋅
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level, filter design: biquad
MOS transconductor: differential pair.
inearity of the stage is proportional to the voltage overdrive or a given gm, Ibias is proportional to the voltage overdrive oltage overdrive should be chosen to be as small as possiblenearity.
M M
2*Ibias
gmIoutVin---------- µ Cox
WL----- VGS VTH–( )⋅ ⋅⋅ Ibias
2 VGS VTH–( )⋅-------------------------------------------= = = =
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design 2/7
T rent gm values.
S d Ibias1 it is possible to change theg a gm2=k*gm1:
It1 od12 1/k*Vod13 d2=Vod1:Ib
S1 all Vod in deep scaled technology.T achieve the target gm.2 different Vod, than it has differentli
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: biquadGm scaling
he 4 transconductors building a biquad have in general diffe
tarting from a reference transconductor with given W1, L1 anm value in several ways. Let’s suppose we want to achieve
is possible:- Acting on Ibias only. Ibias2=k2*Ibias1. In this case Vod2=k*V- Acting on the size only. W2/L2=k2*W1/L1. In this case Vod2=- Acting on the size and Ibias at the same time in order to keep Voias2=k*Ibias1, W2/L2=k*W1/L1.
olution 3 is the preferred for several reasons:- MOS quadratic law is only an approximation valid only for smhis makes it difficult to find the exact value of W/L or Ibias to- Using the first 2 “methods” Vod each transconductor has anearity.
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design 3/7
W2/L1W2/L1
Wp2/Lp1 Wp2/Lp1
Wn2/Ln1
gm2
than
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: biquadGm scaling
W1/L1W1/L1
Wp1/Lp1 Wp1/Lp1
Wn1/Ln1
gm1
Ibias 2*Ibias
Wp1/Lp1
Wn1/Ln1
If gm2 = k * gm1Wn2 = k * Wn1
W2 = k * W1Wp2 = k * Wp1
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design 4/7
T ommon mode value.T ch transconductor).In s connected together, so, only 2 com-m
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: biquadCommon mode
ransconductor output nodes need a loop to fix the voltage cheoretically 4 common mode loops are required (one for eaa biquad (gm1, gm4) and (gm2, gm3) have the output nodeon mode loops are necessary.
+ - + - + - +gm3
C1 C2
Vin
Vout
gm1 gm2 gm4+
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design 5/7
A g the output common mode voltageth n order to hold the proper commonm
C g the stability of the loop.
OPAMP
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: biquadCommon mode
possible common mode loop can be implemented sensinrough 2 high value resistor and adjust the PMOS current iode voltage.
ommon loop bandwidth doesn’t need to be very high helpin
VREF-
+
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rol design 6/7
T ductor is biased open loop: process,v on the values of gm and/or C.U riation can be as big as +/-50% mak-in
A values in order to improve the ac-c
A oaches are used:1 solute reference (for example an ex-te2 alue (the reference can be for exam-p
W le to the precision of the integratedcW er improve to few percent.
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: gm cont
he frequency accuracy of the filter is very poor if the transconoltage supply and temperature spreads have a direct effect nless closed loop solutions are used the frequency cutoff vag the filter useless for most of the applications.
lmost every gmC filter has a gm control loop to adjust the gmuracy of the filter.
ccording to the precision expected for the filter 2 main appr- Control the transconductor to match the gm value to an abrnal resistor).- Control the transconductor to match the gm value to the C vle a “switched capacitor resistor”).
ith the first approach the precision of the filter is comparabapacitors used in the filter (5-10%).ith the second approach the precision of the filter can furth
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rol design 7/7
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Transistor level filter design: gm cont
Vref
to the filter
I ref gmstage
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E
FS elay and noise.
TT
E y the comparison between the tran-sT e simulation through the followingsfnN x of matlab:p ficient is the opposite of MATLAB)Efn }fn *w2*wc)}
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Filter characterization
ldo simulator allows a complete analysis of the filter.
requency domain analysis:mall-signal frequency response: amplitude, phase, group d
ime domain analysisransient simulations -> linearity.
ldo allows the implementation of ideal functions making easistor level implementation and the target function.he ideal function can be described in a text file included in thyntax:scellname input_node output_node numerator, denominatorumerator and denominator are described with similar synta(x) = x^3 -2*x - 5 -> -5 -2 0 1 (note that the order of the coefxample:sbq1 in out1_ideal k1, 1 {1/(Q1*w1*wc)} {1/(w1*wc*w1*wc)sbq2 out1_ideal out2_ideal k2, 1 {1/(Q2*w2*wc)} {1/(w2*wc
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A put a signal having a dominant tonea posed tones at frequencies integerm
VV
In spect to the even ones.
A the Total Harmonic Distortion de-fi
U c.
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Linearity
single tone input applied to a real filter produces at the outt a frequency multiple of the input frequency having superimultiple of the fundamental.
in = a*sin(ωt)out = A*sin(ωt) + B*sin(2ωt) + C*sin(3ωt) + D*sin(4ωt)
differential circuit even harmonics are usually negligible re
way to quantify mathematically the linearity is to introduce ned as:
sually the dominant contributor to THD is the third harmoni
THD 20HiH1--------
2
i∑log⋅=
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O rrent sources are not infinite.T onductances in parallel to C1 and C2:
DQ
C2 G2 s⁄+ )
m4------------------------------
------------------------------
s2 C1 C2⋅
gm2 gm4⋅--------------------------⋅
---------------------------------------
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Second Order effectsFinite output impedance
utput resistance of the transconductor and of the PMOS cuhe effect of these resistances can be taken into account as c
C gain decreases to gm1/gm4*1/(1+G1*G2/(gm2*gm4))). decreases to:
G s( )
gm1gm4-----------
1 sgm3 C1 G1 s⁄+( )⋅
gm2 gm4⋅------------------------------------------------⋅ s
2 C1 G1 s⁄+( ) (⋅
gm2 g⋅---------------------------------------⋅+ +
------------------------------------------------------------------------------------------------------------------------=
G s( )
gm1gm4-----------
1G1 G2⋅
gm2 gm4⋅-------------------------- s
gm3 C1 G2 C1⋅ G1 C2⋅+ +⋅
gm2 gm4⋅--------------------------------------------------------------------------⋅+ + +
--------------------------------------------------------------------------------------------------------------------------------=
Qgm2 gm4⋅
gm3 G2 G1
C2C1-------⋅+ +
2-----------------------------------------------------------
C2C1-------⋅=
Department of Electronics-Univ. of Pavia STMicroelectronics
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T e to the nMOS Gate-Drain capaci-taP ), oxide caps (CGS, CGD) and metalc
al- Vg / Vd ) + Cmetal
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Second Order effectsParasitic capacitances
he Miller effect makes difficult the estimation of the term dunce.arasitic capacitances are constituted by junction caps (CDBaps:the main contributors are the oxide capacitances.
CGSn
CGDn
CGDpCDBp
CDBn
CGATE = CGSn + CGDn * ( 1 - Vd / Vg ) + CmetCDRAIN = CGDp + CDBp + CDBn + CDGn * ( 1
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tion
F ansconductors and/or the pMOS cur-rU cause cascode transistors can be sizedw act, cascode transistors don’t affecttrM ce is not multiplied for Miller effectm ode.
H tors (fc is proportional to gm/C).P total capacitance (30%-40%).D percentage of parasitic cap on eachn e nodes and 5% on other nodes.P n, metal): it is better to have on eachn % oxide, 5% junction, 2% metal).T pacitances (CGS and CGD) of thetr using gate oxide capacitance (poly-n ps as part of the load capacitors be-c ame of the load cap.
iacomino Bollati, 12 May 2004 GmC Filters Tutorial
Second Order effects reduc
inite output resistance can be increased by cascoding the trent generators.sually cascodes improves also the parasitic capacitances beith L shorter than the transconductor and pMOS driver (in fansconductance precision and offset).oreover, in cascoded transconductors gate-drain capacitanaking easier the parasitic capacitance estimation of each n
igh speed filters have small load capacitors and big transisarasitic capacitance can be a significant percentage of the esign procedure should add the constraint of having similarode: it is better to have 30% on each node than 30% on somarasitic caps are due to different contributions (oxide, junctioode similar percentage for each kind of capacitance (ex.: 20he dominant parasitic capacitance is due to the oxide caansconductor devices. If the load capacitance is made bywell capacitance) it is possible to consider oxide parasitic caause their dependence on process and temperature is the s
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