sizing cmos circuits by means of the g /i methodology and a
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1MOS-AK workshop, Dec 13, 2008. P.G.A. Jespers
MOS-AK workshop13 Dec. 2008
Sizing CMOS circuits by means of the gm/ID methodology and a compact model.
P.G.A. JespersUniversité Catholique de Louvain
paul.jespers@uclouvain.be
2
sizing....
find D.C. currents and transistor sizes meeting :
• a prescribed gain-bandwidth product• minimal power consumption• minimal area• large gain• ….
short channel devices,..low-voltage, low-power circuits
BWRC, Dec 12, 2008. P.G.A. Jespers
low- voltage, low-power MOS circuitslow- voltage, low-power MOS circuits
3BWRC, Dec 12, 2008. P.G.A. Jespers
Outline
Sizing... • the Intrinsic Gain Stage (I.G.S.)
gm/ID semi-empirical methodology gm/ID compact model methodology
L-V, L-P, short channel I.G.S. • the Miller Op. Amp.
Conclusion
4BWRC, Dec 12, 2008. P.G.A. Jespers
The Intrinsic Gain Stage (I.G.S.)
IDW/L
Vin (sat) C
Vout
gm ID
.VA
log ω
A (dB)
- 20 dB/decade
gm CωT =
gm = ωT.C
gain-bandwidth sizing:
find ID and W/Lachieving ωT
5
ID
Vin (sat) C
Vout
W/L
!
gm ="TC
1) (strong inversion)
!
W
L=
n "TC( )
2
2µ # C ox
$1
ID
!
ID
power decreases, gain increases
?
!
" = µ # C ox
W
L
gm =$ ID
$VG
=2" ID
n
!
A =gm
IDVA =
2"
nIDVA
BWRC, Dec 12, 2008. P.G.A. Jespers
6
2) weak inversion)
!
ID = Io expVG
nUT
"
# $
%
& '
gm =ID
nUT
!
W
L
!
ID
ID = nUT! T C
!
Amax
= "VA
nUT
ID
Vin (sat) C
Vout
W/L
!
gm ="TC
sizing in moderate inversion?
BWRC, Dec 12, 2008. P.G.A. Jespers
7BWRC, Dec 12, 2008. P.G.A. Jespers
Outline
Sizing... • the Intrinsic Gain Stage (I.G.S.)
gm/ID semi-empirical methodology gm/ID compact model methodology
L-V, L-P, short channel I.G.S. • the Miller Op. Amp.
Conclusion
8
what does gm/ID represent ?
BWRC, Dec 12, 2008. P.G.A. Jespers
!
gm
ID
"
# $
%
& ' =
1
ID
(ID(VGS
=(
(VGSlog ID( )
0 0.5 1 1.5 2 2.5 310
-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
ID
S.I. approx(quadratic)
W.I. approx(expon)
VG
0 0.5 1 1.5 2 2.5 30
5
10
15
20
25
30
35
VG
gm/ID
9
• gm/ID does not depend on the transistor widthgm and ID are proportional to W
• gm/ID bridges a small signal and a large signal quantity
• gm/ID controls gain, power consumption ...
!
gm " ID
!
A =gm
IDVA
why gm/ID?
BWRC, Dec 12, 2008. P.G.A. Jespers
10
The gm/ID sizing methodology(semi-empirical)
BWRC, Dec 12, 2008. P.G.A. Jespers!
W VGS( ) =ID VGS( )ID ref VGS( )
W( )ref
!
gm ="T C
!
ID VGS( ) =gm
gm
ID
"
# $
%
& '
!
"
"VGSlog IDref VGS( )( )
• measurements• reconstructed data (BSIM, PSP...)• model E.K.V...
monitors the mode of operationof the MOS transistor
!
Wref
11
ID (A)
W/L
fT = 1 GHz; Co = 1 pf; L = 120 nm; VS = 0; VDS = 0.6 V; Wmax = 1 µm;
BWRC, Dec 12, 2008. P.G.A. Jespers
mobility degradation
parasitic drain junction
Example
12
A gm/ID Based Methodology for the Design of CMOS Analog Circuits andIts Application to the Synthesis of a Silicon-on-Insulator Micropower OTA
F. Silveira, D. Flandre, P.G.A. Jespers
IEEE JOURNAL OF SOLID STATE CIRCUITS, VOL. 31, NO. 9, SEPTEMBER 1996p. 1314 ...
BWRC, Dec 12, 2008. P.G.A. Jespers
First paper
13BWRC, Dec 12, 2008. P.G.A. Jespers
Outline
Sizing... • the Intrinsic Gain Stage (I.G.S.)
gm/ID semi-empirical methodology gm/ID compact model methodology
L-V, L-P, short channel I.G.S. • the Miller Op. Amp.
Conclusion
14
- An MOS transistor model for analog circuit design Ana I. Cunha, M.C. Schneider, C. G. Montoro. IEEE JSSC,vol 33,n°10,oct, 1998.
- An analytical MOS transistor model valid in all regions of operation and dedicated to Low-Voltage and Low-current applications. Chr. C.Enz, F. Krummenacher, E. A. Vittoz. Analog Integrated Circuits and Signal Processing, Kluwer Ac. Publ. 1995.
A.C.M.
E.K.V.
The ACM and EKV compact models
+ continuous model (saturation, weak to strong inversion)+ few parameters:
n subthreshold slope factorISu unary specific currentVTo threshold voltage
- uniformly doped substrate, no mobility degradation,- gradual channel approximation (1D)
BWRC, Dec 12, 2008. P.G.A. Jespers
15
The compact model (1)
normalized drain current
!
i =ID
IS
!
2nUT
2µ " C ox1 2 4 3 4
W
L
6 7 4 4 8 4 4
!
ISu
specific current
unary specific current (W = L)
drain current normalization
BWRC, Dec 12, 2008. P.G.A. Jespers
EKV ACM
!
1
2nU
T
2µ " C ox
W
L
16
The compact model (2)
BWRC, Dec 12, 2008. P.G.A. Jespers
!
q = "# Q i
2nUT# C ox
normalized mobile charge density
pinch - off voltage
gate voltage
!
VP
=VG"V
To
n
!
i = q2
+ q
norm. drain current (saturation)
!
VP "V =UT 2 q "1( ) + log q( )[ ] channel voltage
17
!
i = qF2 + qF - qR
2 + qR( )!
VP "VS =UT 2 qF "1( ) + log qF( )[ ]
!
VP "VD =UT 2 qR "1( ) + log qR( )[ ]
sat sat
VDVSVS
VD
ID IDForward IDReverse= -
VG VG
The compact model (3)
BWRC, Dec 12, 2008. P.G.A. Jespers
18
!
VP = UT 2 qF "1( ) + logqF( )
!
VG
= nVP
+VTo
example : IDu(VG) of grounded source (VS = 0 V) saturated (q => qF) transistor
!
i = qF2
+ qF
!
IDu
= i ISu
parametric method
% data UT = .026;n = 1.2;Isu = 1e-6; VTo = 0.4; % computeqF = logspace(-4,1.2,50);i = qF.^2 + qF;ID = i*Isu;VP = UT*(2*(qF-1) + log(qF));VG = n*VP + VTo; % plotsemilogy(VG,ID); grid
IDu (A)
VG (V)
W.I.
S.I.
VTo
!
qF
BWRC, Dec 12, 2008. P.G.A. Jespers
19
!
gm
ID=
1
nUT
qF
i=
1
nUT
1
qF +1
!
gm
ID=d log i( )dVG
!
d log i( ) =di
i=2qF +1
idqF
and
dVG = n dVP = nUT 2 +1
qF
"
# $
%
& ' dqF = nUT
2qF +1
qFdqF
gm/ID of the saturated transistor
BWRC, Dec 12, 2008. P.G.A. Jespers
20
% datafT = 1e8;C = 1e-12;n = 1.2;Isu = 1e-6;VTo = 0.4; % computeUT = .026;ωT = 2*pi*fT;gm = ωT*C;qF = logspace(-4,1.5,50);gmoverID = 1./(n*UT*(1+qF));ID = gm./gmoverID;IDu = Isu*(qF.^2 + qF);WsL = ID./IDu;VP = UT*(2*(qF+1) + log(qF));VG = n*VP + VTo; % plotloglog(ID,WsL,'b',ID,VG,'r');
sizing the Intrinsic Gain Stage by means of theE.K.V. model
E.K.V. param
gm/ID sizing
specs
BWRC, Dec 12, 2008. P.G.A. Jespers
21
ID (A)
VGS (V)
W / L
fT = 100 MHzC = 1 pF
strong inversion approx.
weak inversion approx.
gm/ID sizing
sizing the Intrinsic Gain Stage by means of theE.K.V. model
BWRC, Dec 12, 2008. P.G.A. Jespers
22
• The basic EKV / ACM model does not apply to short channel devices!
BWRC, Dec 12, 2008. P.G.A. Jespers
L = 100 nm
L = 500 nmVDS = 0.2 V
VDS = 1.2 V
VGS (V)
ID (A)
90 nm technologyN channelW = 10 µmVSB = 0 V
• Real ID(VGS) characteristics however look very similar.
23
p
source drain
gate silicide
n-
tox
n+
poly
• The spatial distribution of electrical fields in the substrate boils down to a 2D problem controlled mainly by L, VSB, VDS, little by VGS.• The inversion layer confines to a 1D problem controlled by VGS and L, VSB, VDS.
BWRC, Dec 12, 2008. P.G.A. Jespers
• Is it possible to model ID(VG) characteristics by means of the EKV / ACM model with parameters that are functions of L, VS and VD.?
24BWRC, Dec 12, 2008. P.G.A. Jespers
E.K.V. Identification
to be performed in the common source configuration
For L, VS and VDS
S.I.
W.I.M.I.
log(ID) n
VGS
gm/ID
1/nUT
VGS
80 to 70 %mobility
degradation
VTo
25BWRC, Dec 12, 2008. P.G.A. Jespers
!
VGS " VTo
n=VP # q
!
IDuVGS( )i
= ISuVGS( )
!
ISuo
polynomial fit
VGS (V)
!
ISu
= 2nUT
2 µo" C ox
ISuo
1 2 4 3 4 # i( )
mobility degradation factor
(µA)
Isu (specific current)
For L, VS and VDS
26
IDu (A)
VGS (V)
N-channel; L = 100 nm; VDS = 0.6 V; VSB = 0.6 V.
n slope factor
n,VTo and ISuo
+++ n,Vto, Isuo and θ(i)
‘experm.’ data
BWRC, Dec 12, 2008. P.G.A. Jespers
Verification : Iu(VGS) reconstruction
27
gm/ID (V-1)
VGS (V)
edge conductance effect ?
VDS = 0.6 VVSB = 0 VL = 100 nm
mob. degradation
BWRC, Dec 12, 2008. P.G.A. Jespers
+++++‘experimental’model model (no mob degradation)
28BWRC, Dec 12, 2008. P.G.A. Jespers
Outline
Sizing... • the Intrinsic Gain Stage (I.G.S.)
gm/ID semi-empirical methodology gm/ID compact model methodology
L-V, L-P, short channel I.G.S. • the Miller Op. Amp.
Conclusion
29BWRC, Dec 12, 2008. P.G.A. Jespers
!
gm
IDdoes not depend on W
as long as W >> Wmin
(true for most analogue circuits)
depends on L, VDS and VSB for- VT roll-off- D.I.B.L.- C.L.M.- mobility degradation- ….
30
fT = 1 GHz; Co = 1 pf; L = 120 nm; VS = 0; VDS = 0.6 V;
ID (A)
W/L
VGS (V)
stronginversionapprox.
weak inversion approx.
n, VTo and Isuono mobilty degradation: θ = 1
BWRC, Dec 12, 2008. P.G.A. Jespers
31
fT = 1 GHz; Co = 1 pf; L = 120 nm; VS = 0; VDS = 0.6 V;
ID (A)
W/L
VGS (V)
stronginversionapprox.
weak inversion approx.
n, VTo and Isuowith mobilty degradation: θ(ι)
BWRC, Dec 12, 2008. P.G.A. Jespers
32
ID (A)
W/L
VGS (V)
Add drain junction cap. to Co
fT = 1 GHz; Co = 1 pf; L = 120 nm; VS = 0; VDS = 0.6 V; Wmax = 1 µm;
partitioning
BWRC, Dec 12, 2008. P.G.A. Jespers
33
ID (A)
W/L
VGS (V)
fT = 1 GHz; Co = 1 pf; L = 120 nm; VS = 0; VDS = 0.6 V; Wmax = 1 µm;
partitioning
Comparison with semi-empirical method (+++ )
BWRC, Dec 12, 2008. P.G.A. Jespers
34
VDS (V)
L µm 0.100 0.110 0.120 0.130 0.140 0.160 0.500 1.000 4.000
D.I.B.L.
Param. dependence on VDS1) n small2) VTo
BWRC, Dec 12, 2008. P.G.A. Jespers
VSB = 0 V.
35
VGS (V)
VDS (V)
mobility degradation (vert)
channel lengthmodulation
Desaturation
!
ISu
µA( )
VSB = 0 V.
mobility (longitudinal)
BWRC, Dec 12, 2008. P.G.A. Jespers
Param. dependence on VDS1) n small2) VTo3) ISu
36BWRC, Dec 12, 2008. P.G.A. Jespers
Param. dependence on VDS1) n small2) VTo3) ISuo
C.M.L.
L µm 0.100 0.110 0.120 0.130 0.140 0.160 0.500 1.000 4.000
VSB = 0 V.
37
vertical field longitudinal field
sat.
not sat.
BWRC, Dec 12, 2008. P.G.A. Jespers
Param. dependence on VDS1) n small2) VTo3) Isuo4) θ(i)
VSB = 0 V.
38
+++++‘experimental’model model (no mob degradation)
0.5 V
0.6 V
0.7 V
VGS
X 10-5
Verification : IDu(VDS) reconstruction (S.I.)
BWRC, Dec 12, 2008. P.G.A. Jespers
39
0.00 V
0.05 V
0.10 V
VGS
X 10-9
+++++‘experimental’model model (no mob degradation)
Verification : IDu(VDS) reconstruction (W.I.)
D.I.B.L.
BWRC, Dec 12, 2008. P.G.A. Jespers
40BWRC, Dec 12, 2008. P.G.A. Jespers
Outline
Sizing... • the Intrinsic Gain Stage (I.G.S.)
gm/ID semi-empirical methodology gm/ID compact model methodology
L-V, L-P, short channel I.G.S. • the Miller Op. Amp.
Conclusion
41
!
"T = gm1
Cm
z = gm2
Cm = Z"T
"ndp = gm2
Cm
Cm2
C1+C2( )Cm + C1C2
= NDP"T
e.g. Z = 10 and NDP = 4… 5 for phase margin of approx. 60°
1) fix NDPole and Zero with respect to ωT to meet phase margin
BWRC, Dec 12, 2008. P.G.A. Jespers
Bias
VDD
VSS
Q5
Cm- vin/2 + vin /2
C1
node 3 node1
node 2
C2
Q4
Q2
Q1a Q1b
Q3a Q3b
C3
ID22 ID1
vout
42
2) Size the ‘A’ transistors.
Bias
VDD
VSS
Q5
Cm- vin/2 + vin /2
C1
node 3 node1
node 2
C2
Q4
Q2
Q1a Q1b
Q3a Q3b
C3
ID22 ID1
vout
BWRC, Dec 12, 2008. P.G.A. Jespers
!
gm2 = Zgm1
ID2 =
gm2
gm
ID
!
" # #
$
% & &
2
qF2
!
W2
L2
= ID2
IDu2
ID1 =
gm1
gm
ID
!
" # #
$
% & &
1
!
W1
L1
= ID1
IDu1
!
gm1 = "TCm
spec. initial guess
qF1
43
3) Size the ‘B’ transistors.
more constraints
BWRC, Dec 12, 2008. P.G.A. Jespers
Bias
VDD
VSS
Q5
Cm- vin/2 + vin /2
C1
node 3 node1
node 2
C2
Q4
Q2
Q1a Q1b
Q3a Q3b
C3
ID22 ID1
vout
!
IDu4
= IDu5
!
W
L
"
# $
%
& '
4
= ID2
IDu4
!
W
L
"
# $
%
& '
5
= 2 I
D1
IDu4
• choose bias so that Q4 and Q5 are in strong inversion
• zero systematic offset
VG3 = VG2
!
W
L
"
# $
%
& '
3
= ID1
IDu2
44
Choose …L1 medium (voltage gain)L2 min. sizeL3 large for min 1/f noise (beware from doublet!)L4 matching + sizeL5 matching + common mode rejection
4) Estimate C1, C2, C3 and compute Cm
• the parasitic cap. are estimated knowing W’s and L’s + techno. data
BWRC, Dec 12, 2008. P.G.A. Jespers
reiterate until Cm gets constant
Cm = 0.5NDP
Z! C1 +C2 + C1 +C2( ) + 4
Z
NDPC1C2
"
# $
%
& '
• a new Cm is extracted from inverted NDP equation
45
fT = 50 MHz;C = 1 pF;VDD = 1.2 V;
BWRC, Dec 12, 2008. P.G.A. Jespers
Q 5
Cm
C1
node 3 node1
node 2
C2
Q4
Q2
Q 1
Q 3
C3
ID22 ID1
1.2 V
0 V
Spec:
example
qF1 = 0.0316 --> 3.16qF2 = 0.10 --> 2qF4 = 2,90
Z = 10NDP = 4
L1 = 1 µmL2 = 0.5 µmL3 = 1 µmL4 = 0.5 µmL5 = 1 µm
phase margin
qF exploration space
46BWRC, Dec 12, 2008. P.G.A. Jespers
W1 W2
W3 W4
VG1VG2
47BWRC, Dec 12, 2008. P.G.A. Jespers
constant active area (µm2)
48BWRC, Dec 12, 2008. P.G.A. Jespers
constant supply (A)constant active area (µm2)
49BWRC, Dec 12, 2008. P.G.A. Jespers
constant gain (dB)constant supply (A)constant active area (µm2)
50BWRC, Dec 12, 2008. P.G.A. Jespers
constant gain (dB)constant supply (A)constant active area (µm2)
51BWRC, Dec 12, 2008. P.G.A. Jespers
0.6242CmpF
0.0576C3pF
1.0375C2pF
0.1965C1pF
39.245.1
7.691
57.830.5
2.231
47.50.5
67.31
2 x 8.1143.72 x 8.1
0.490.490.400.400.29
2.902.901.2361.0650.433qF
VGS (V)
I (µA)
gain (dB)
Q1 Q2 Q3 Q4 Q5
gain = 81 dBpower consump. = 191 µW
WL
(µm)
Q 5
Cm
C1
node 3 node1
node 2
C2
Q4
Q2
Q 1
Q 3
C3
ID22 ID1
1.2 V
0 V
The selected point
52
Conclusion
gm/ID• relates a small signal param. to a large signal quantity• does not vary with transistor widths• controls the mode of operation, power consump, gain ...
paves the way for sizing CMOS circuits• semi-empirically
(look-up tables : ID, gm, gd, ..)• by means of the E.K.V./A.C.M. model
(parameters look-up tables or fitting functions)• simple expressions of ID, gm/ID, gd/ID• qF monitors mode of operation• increased physical insight
suitable for sub-micron low-voltage low-power circuits
BWRC, Dec 12, 2008. P.G.A. Jespers
53
gm/ID sizing methodologyfor low-power/voltage
CMOS circuitsby P.G A. Jespers
to be published 2009 by Springer
paul.jespers@uclouvain.be
BWRC, Dec 12, 2008. P.G.A. Jespers
54
A list of references concerning the gm/ID methodology:
1) A gm/ID based methodology for the design of CMOS analog circuits and its applicationto the synthesis of a silicon-on-insulator micropower OTA.F. Silveira, D. Flandre and P.G.A. JespersIEEE Journal of Solid-State Circuits, vol 31, n° 9, sept 1996, p. 1314 - 1319.(the first reference)2) A CAD methodology for optimizing transistor current and sizing in analog CMOS design.D.M. Binkley, C.E. Hopper, S.D. Tucker, B.C. Moss, J.M. Rochelle and D.P. Foty.IEEE Trans. on computer-aided design of integrated circuits and systems,vol 22, n° 2, Febr. 2003.3) gm/ID-based mosfet modeling and modern analog design.D. Foty, D. Binkley, Matthias Bucher.Presented at MIXDES, Wroclaw, Poland, 20 June 2002.4) Une méthodologie de conception des amplificateurs opérationnels à faible consommationP. Jespers.FTFC’2001 records, mai-juin, Paris, p.99-1065) Automated design methodology for CMOS analog circuit blocks in complex systems.R. Ionita, A. Vladimirescu and P.G.A. Jespers.contact Prof Vladimirescu, UCBerkeley, BWRC,2208 Allston Way, Berkeley, CA 94704.6) Sizing of MOS transistors for amplifier design.R.L. Oliveira Pinto, M.C. Schneider and C.G. Montoro.ISCAS 2000.7) A behavioral model of a 1.8-V flash A/D converter based on device parameters.M. Hasan, H.H.P. Shen, D.R. Allee, M. Pennell.IEEE Trans. on computer-aided design of integrated circuits and systems, vol 19, n° 1,Jan 2000, p 69-82
BWRC, Dec 12, 2008. P.G.A. Jespers
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