gmc filters tutorial - unipvmicrolab.unipv.it/castello/pdf/filtri_gmc_cast.pdf · department of...

Department of Electronics-Univ. of Pavia STMicroelectronics

G

al

iacomino Bollati, 12 May 2004 GmC Filters Tutorial

GmC Filters Tutori

Giacomino BollatiTPA R&D

12 May 2004


G
Outline

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


G

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


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.


G

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–( )

--------------------

)


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(••=


G Page 8

B

T

t

gm4+

4----

----


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⋅----------------------⋅+ +

----------------------------------------------------------------------------------=


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F parametersk, ωo and Q in terms ofth

infinite solutions.

gm1gm4-----------

1m4------- s

2 C1 C2⋅

gm2 gm4⋅--------------------------⋅+

--------------------------------------------------


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-----------=


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G12

S der filters having wide ranges for theω

F aving:thaa



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.


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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



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



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----

----


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⋅----------------------⋅+ +

----------------------------------------------------------------------------------=


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2

C

C

mkgm1⋅

---------------- Vin⋅

C1 C2⋅

gm2 gm4⋅--------------------------⋅

--------------------------------

Ckgm1------------

Vin⋅

C1 C2⋅

gm2 gm4⋅--------------------------⋅

--------------------------------


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+ +

------------------------------------------------------= =


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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⋅ ⋅⋅ ⋅


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


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


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


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


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.


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


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)}


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


G Page 24

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.


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⋅=


G Page 26

O rrent sources are not infinite.T onductances in parallel to C1 and C2:

DQ

C2 G2 s⁄+ )

m4------------------------------

------------------------------

s2 C1 C2⋅

gm2 gm4⋅--------------------------⋅

---------------------------------------


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-------⋅=


G Page 27

T e to the nMOS Gate-Drain capaci-taP ), oxide caps (CGS, CGD) and metalc

al- Vg / Vd ) + Cmetal


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


G Page 28

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.


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

gmc filters tutorial - unipvmicrolab.unipv.it/castello/pdf/filtri_gmc_cast.pdf · department of...

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