A Methodology for Analog Circuit Macromodeling A Methodology for Analog Circuit Macromodeling Rohan Batra, Peng Li and Larry Pileggi Department of Electrical and Computer Engineering Carnegie Mellon University Yu-Tsun Chien Industrial Technology Research Institute Hsinchu, Taiwan
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A Methodology for Analog Circuit Macromodeling
A Methodology for Analog Circuit Macromodeling
Rohan Batra, Peng Li and Larry PileggiDepartment of Electrical and Computer Engineering
Carnegie Mellon University
Yu-Tsun ChienIndustrial Technology Research Institute
Hsinchu, Taiwan
2
MotivationMotivationCompact sub-block macromodels are the key to whole-system verification
Back-annotation of such models facilitates system-level verification
These “reduced-order” macromodels capture the nonlinear effects
IIP3, THD, gain compression,…Dynamic range, spectral regrowth, etc…
Ana
log
Des
ign
Modeling Gap
DesignSpecifications
DesignSpecifications
System-LevelDesign
System-LevelDesign
Circuit-LevelDesign/Synthesis
Circuit-LevelDesign/Synthesis
LayoutLayout
VerificationVerification
Compact MacromodelsCompact
Macromodels
3
AgendaAgendaIntroduction
MotivationPrevious Work
Nonlinear macromodeling approachBackgroundExtraction of Volterra ParametersOverall Macromodeling Flow
Experimental resultsConclusions
4
Previous WorkPrevious Work
Reduced order modeling of time-varying systems[Roychowdhury TCAS 1999] [Phillips CICC 2000]
PWL/PWP and model order reduction[Rewienski, White ICCAD 2001] [Dong, Roychowdhury DAC 2003]
NORM : compact model order reduction of weakly nonlinear systems
[Li, Pileggi DAC 2003] Hybrid approach to nonlinear macromodel generation
Moment matching these nonlinear transfer fcts via projection for MOR
10
Reduced Order ModelingReduced Order ModelingUsing a projection-based reduction – multipoint NORM
VGVG T11
~ = VCVC T11
~ = bVb T=~ ( )VVGVG T ⊗= 22~ ( )VVCVC T ⊗= 22
~
H2
f1
f2
Fully captures interactions between transfer functions of different orders
11
Extraction of Volterra ParametersExtraction of Volterra Parameters
Vdd
M1
M2
M3 M5
M4
M8 M7
M6 M9
M10 M17 M18
M19
M20M11 M12
M13M14
M15 M16
VoutVin- Vin+
Spice models like BSIM3 include physical effects + numerical parameters which increase model complexity
Infeasible to determine the coefficients by computing higher derivatives of device model equations
12
Simulation SetupSimulation Setup
SimulationEngine
Hspice, SpectreRF ….
Numerical fitting
Commercial simulators like Hspice, SpectreRF can be used to characterize the model parameters for each transistor
For each transistor, perturb the bias voltages to generate data-points for numerical fitting
13
NonlinearitiesNonlinearitiesSecond and third order fitting of drain current using least squares
Second and third order fitting of the charge is carried out by fitting the capacitances. For instance,
Differentiating w.r.t. drain
In order to fit Qg, need to fit Cgd, Cgs and Cgg
14
Least-squares fittingLeast-squares fittingY matrix contains the voltage powers and cross terms p contains the corresponding coefficientsR contains the residue (Ids –Ids0)
]...........[
,
..........
::::::
..........
..........
32
3222
22222
3111
21111
sssdggsd
sngndndngnsndn
sgddgsd
sgddgsd
gggggp
vvvvvvv
vvvvvvv
vvvvvvv
Y
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
[ ] 021 ...... dsdsnnT
n IIIIIIR −==
15
Least-squares fittingLeast-squares fittingMinimize the error for each data-point, in matrix form:
The least-mean-square algorithm estimates the coefficients in p by minimizing the sum of squares errors:
Tne ]......[ 21 εεε=
eRYp =−
)()( RYpRYpeeF TT −−==
)(.)( 1 RYYYp TT −=
16
Improving the fit …Improving the fit …
Weighted-least squares approach
Fitting rangeLarge enough to encompass nonlinearitiesShould not cover effects outside signal swing range*
* [“Distortion in RF power amplifiers”, Vuolevi and Rahkonen, Artech House, 2003]
An OpampAn OpampModeled as a time-invariant system, linearized at the DC bias point to fit second and third-order coefficients for each transistorSecond-order nonlinearities are much higher than third-order nonlinearities for single-ended output
Vdd
M1
M2
M3 M5
M4
M8 M7
M6 M9
M10 M17 M18
M19
M20M11 M12
M13M14
M15 M16
VoutVin- Vin+
19
An OpampAn Opamp
Perform transient analysis in Hspice followed by fouriertransform to compare with model resultsFirst-order (small-signal) results
Max error between full model generated using small-signal parameters and reduced-order model is 0.07%
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 107
2
4
6
8
10
12
14
16
18
20
22
Frequency (Hz)
Orig
inal
H1
Mag
nitu
de
20
An OpampAn OpampSecond-order distortion as a function of frequency
Max error between transient simulation and full model : 5.1%Max error between full model and reduced model : 2.9%Max error between transient simulation and reduced model : 5.3%
106 107 10810-3
10-2
10-1
100
Frequency (Hz)
Seco
nd O
rder
Dis
torti
on(N
orm
aliz
ed)
Hspice SimulationFull ModelReduced Order Model
21
A Double-Balanced MixerA Double-Balanced MixerCharacterized using time-varying Volterra series w.r.t. RF input based on 1350 time-sampled circuit variablesEach nonlinearity is modeled as a third-order polynomial about the time varying operating point due to large-signal LO
+ Vout -
Vlo
Vrf
…
)()()()( 3,3
2,2,1 tvatvatvati ttt ++=
TPeriodic Time-varying Op
22
A Double-Balanced MixerA Double-Balanced Mixer
Third-order results Single tone RF input frequency varied from 300MHz to 1200MHzThird-order harmonic of the RF input down-converted w.r.t LO Max error between transient simulation and full model : 8%
020406080
100120140160180200
300 800 900 1100 1200
Frequency (Mhz)
HspiceSimulationOur model
X 10-4 LO frequency = 1 GHz
23
A Double-Balanced MixerA Double-Balanced Mixer
Reduced order model14 circuit variables as compared to 1350 in the full modelThird-order transfer function: 300Mhz ≤ f1, f2 ≤ 1.2Ghz and fLO = 1Ghz
2
4
6
8
10
12
x 108
24
68
1012
x 108
0
0.02
0.04
0.06
RelativeError
Frequency (Hz) Frequency (Hz)
24
68
1012
x 108
2
4
6
8
10
12
x 108
200
300
400
500
600
1/V 2
Frequency (Hz) Frequency (Hz)
24
ConclusionsConclusions
Reduced-order models for weakly nonlinear analog circuits can be generated from transistor-level netlists
The accuracy is comparable to transistor-level simulation using commercial simulators
Explore the adoption of these compact reduced-order models in behavioral languages like Verilog-A