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1 Parasitic Extraction Parasitic Extraction Step 1: Electromagnetic Field Step 1: Electromagnetic Field Solvers Solvers Luca Daniel Massachusetts Institute of Technology [email protected] http://onigo.mit.edu/~dluca/ 2009MOMiNE www.rle.mit.edu/cp

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Page 1: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

1

Parasitic ExtractionParasitic ExtractionStep 1: Electromagnetic Field SolversStep 1: Electromagnetic Field Solvers

Luca Daniel

Massachusetts Institute of Technology

[email protected]

http://onigo.mit.edu/~dluca/2009MOMiNE

www.rle.mit.edu/cpg

Page 2: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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IC Conventional Design FlowIC Conventional Design Flow

Funct. Spec

Logic Synth.

Gate-level Net.

RTL

Layout

Floorplanning

Place & Route

Front-end

Back-end

Behav. Simul.

Stat. Wire Model

Page 3: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

3

Layout parasiticsLayout parasitics

• Wires are not ideal. Wires are not ideal. Parasitics:Parasitics:– ResistanceResistance

– CapacitanceCapacitance

– InductanceInductance

• Why do we care? Why do we care? – Impact on delayImpact on delay– noisenoise– energy consumptionenergy consumption– power distributionpower distribution

Picture from “Digital Integrated Circuits”, Rabaey, Chandrakasan, Nikolic

Page 4: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

4

Conventional Design FlowConventional Design Flow

Funct. Spec

Logic Synth.

Gate-level Net.

RTL

Layout

Floorplanning

Place & Route

Front-end

Back-end

Behav. Simul.

Gate-Lev. Sim.

Stat. Wire Model

Parasitic Extrac.

Page 5: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Overview Step1 : Field SolversOverview Step1 : Field Solvers(Tuesday and Wednesday)(Tuesday and Wednesday)

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 6: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Overview Step 2: Model Order ReductionOverview Step 2: Model Order Reduction(Wednesday and Thursday)(Wednesday and Thursday)

• Problem SetupProblem Setup• Connection between circuits and State Space models Connection between circuits and State Space models • Reduction via eigenmode truncation methodReduction via eigenmode truncation method• Reduction via transfer function fittingReduction via transfer function fitting

– point matchingpoint matching– least squareleast square– quasi-convex optimization methodquasi-convex optimization method

• Reduction via Projection FrameworkReduction via Projection Framework– Truncated Balance RealizationsTruncated Balance Realizations– Krylov Subspace Moment MatchingKrylov Subspace Moment Matching

• need for orthogonalization (Arnoldi process)need for orthogonalization (Arnoldi process)• computational complexitycomputational complexity• passivity preservationpassivity preservation

• Reduction of Non-Linear SystemsReduction of Non-Linear Systems

Page 7: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Overview Step1 : Field SolversOverview Step1 : Field Solvers

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 8: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Capacitance ExtractionCapacitance ExtractionExample: Intel Process Cross-sectionExample: Intel Process Cross-section

5 metal layers Ti/Al - Cu/Ti/TiN Polysilicon dielectric.from “Digital Integrated Circuits”, 2nd Edition, Rabaey, Chandrakasan, Nikolic

fringing parallel

Consider only electric field (capacitive) coupling

Page 9: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Capacititance ExtractionCapacititance ExtractionWhy? E.g. Analysis of Delay of Critical PathWhy? E.g. Analysis of Delay of Critical Path

Page 10: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Capacitance Extraction Capacitance Extraction Why do we need it?Why do we need it?

• to produce RC tree to produce RC tree network for Elmore network for Elmore delay analysisdelay analysis

• to produce RC tree to produce RC tree network for capacitive network for capacitive cross-talk analysiscross-talk analysis

R1

C1

s

R 2

C2

R 4

C4

C3

R3

Ci

Ri

1

2

3

4

i

Page 11: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Capacitance ExtractionCapacitance ExtractionProblem FormulationProblem Formulation

• Given a collection of M conductors Given a collection of M conductors

fringing parallel

qvC ?

Calculate the couplingcapacitance matrix C

Page 12: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Capacitance ExtractionCapacitance ExtractionSolution ProcedureSolution Procedure

• For i = 1 to M,For i = 1 to M,– apply one volt to conductor i and ground all the othersapply one volt to conductor i and ground all the others

NiN

i

i

q

q

q

C

C

C

2

1

,

,2

,1

0

1

01iv?iq?1 q

?q ?q ?Nq

?q

– solve the electrostatic problem and find the resulting vector solve the electrostatic problem and find the resulting vector of charges on all conductorsof charges on all conductors

– that is the i-th column of the conductance matrix that is the i-th column of the conductance matrix

2

Page 13: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (Electrostatic)Capacitance Extraction (Electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 14: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Picture Thanks to Coventor

Inductance and Resistance ExtractionInductance and Resistance ExtractionExample: IC packageExample: IC package

packageIC

wirebondinglead

frames& pins

PCB

Page 15: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Inductance and Resistance ExtractionInductance and Resistance ExtractionWhere do we need to account for inductance?Where do we need to account for inductance?

• chip to package and package to board connections are highly chip to package and package to board connections are highly inductiveinductive

• inductance can create Ldi/dt noise on the gnd/vdd networkinductance can create Ldi/dt noise on the gnd/vdd network• inductance can limit communication bandwidthinductance can limit communication bandwidth• inductive coupling between leads or pins can introduce noiseinductive coupling between leads or pins can introduce noise

IC

on-package decouplingcapacitors

on-boarddecoupling capacitors

packagePCB

pins or solder ballsfrom package to PCB

wire bonding and lead framesor solder balls from IC to package

Page 16: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Inductance and Resistance ExtractionInductance and Resistance Extraction Why also resistance? Skin and Proximity effects Why also resistance? Skin and Proximity effects

proximity effect: opposite currents in nearby conductors attract each other

skin effect: high frequency currents crowd toward the surface of conductors

Simple ExampleSimple Example

Page 17: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Inductance and Resistance ExtractionInductance and Resistance ExtractionSkin and Proximity effects (cont.)Skin and Proximity effects (cont.)

• Why do we care?Why do we care?– Skin and proximity effects change interconnect Skin and proximity effects change interconnect

resistance and inductanceresistance and inductance

– hence they affect hence they affect performanceperformance (propagation delay) (propagation delay)

– and and noisenoise (magnetic coupling) (magnetic coupling)

• When do we care?When do we care?– frequency is high enough that wire width OR thickness frequency is high enough that wire width OR thickness

are less than two “skin-depths”are less than two “skin-depths”

– e.g. on PCB above 100MHze.g. on PCB above 100MHz

– e.g. on packages above 1GHze.g. on packages above 1GHz

– e.g. on-chip above 10GHze.g. on-chip above 10GHz

– note. clock at 3GHz has significant harmonics at 10GHz!!note. clock at 3GHz has significant harmonics at 10GHz!!

Page 18: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Inductance and Resistance ExtractionInductance and Resistance ExtractionProblem FormulationProblem Formulation

• Given a collection of Given a collection of interconnected wires with interconnected wires with M input portsM input ports

Picture byPicture byM. ChouM. Chou

viLjR ??

• Calculate the MxM resistance Calculate the MxM resistance RR and the and the inductance inductance LL matrices for the ports, matrices for the ports,

• that is the real and immaginary part of that is the real and immaginary part of the impedance matrix the impedance matrix Z=R+jwLZ=R+jwL

Page 19: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Inductance and Resistance ExtractionInductance and Resistance ExtractionSolution ProcedureSolution Procedure

• Typically instead of Typically instead of calculating impendance we calculating impendance we calculate the admittance calculate the admittance matrix.matrix.

• For i = 1 to MFor i = 1 to M

ivY

ivZ

ivLjR

1

1

MiM

i

i

i

i

i

Y

Y

Y

2

1

,

,2

,1

0

1

0

0

2

J

AjJ

JA

– apply a unit voltage source apply a unit voltage source at port i and short-circuit at port i and short-circuit the othersthe others

– solve solve magneto quasit-static magneto quasit-static problemproblem (MQS)(MQS) to calculate to calculate all port currentsall port currents

– that is the i-th column of that is the i-th column of the admittance matrix the admittance matrix

Page 20: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (Electrostatic)Capacitance Extraction (Electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 21: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC ExtractionCombined RLC ExtractionExample: current distributions on powergridExample: current distributions on powergrid

input terminals

Page 22: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC Extraction Combined RLC Extraction Example: analysis of resonances on powergridExample: analysis of resonances on powergrid

Page 23: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC ExtractionCombined RLC ExtractionExtraction Example: analysis of substrate couplingExtraction Example: analysis of substrate coupling

Page 24: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC ExtractionCombined RLC ExtractionExample: resonance of RF microinductorsExample: resonance of RF microinductors

• At frequency of operation At frequency of operation the current flows in the the current flows in the spiral and creates magnetic spiral and creates magnetic energy storage (it works as energy storage (it works as an inductor: GOOD)an inductor: GOOD)

Picture thanks to Univ. of PisaPicture thanks to Univ. of Pisa

• But for higher frequencies But for higher frequencies the impedance of the the impedance of the parasitic capacitors is lower parasitic capacitors is lower and current prefers to and current prefers to “jump” from wire to wire as “jump” from wire to wire as displacement currents (it displacement currents (it works as a capacitor: BAD)works as a capacitor: BAD)

Page 25: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC ExtractionCombined RLC ExtractionProblem FormulationProblem Formulation

• Given a collection of Given a collection of interconnected wires with interconnected wires with M portsM ports

Picture byPicture byM. ChouM. Chou

viZ ?)(

• Calculate the MxM IMPEDANCE Calculate the MxM IMPEDANCE matrix for the ports, matrix for the ports,

Page 26: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Combined RLC ExtractionCombined RLC ExtractionSolution ProcedureSolution Procedure

• Same as RL extraction.Same as RL extraction.• Typically calculate admittance matrixTypically calculate admittance matrix• For i = 1 to M:For i = 1 to M:

ivY

MiM

i

i

i

i

i

Y

Y

Y

2

1

,

,2

,1

0

1

0

jJn

J

AjJ

JA

ˆ

0

2

2– apply a unit voltage source at port i apply a unit voltage source at port i

and short-circuit the othersand short-circuit the others

– solve solve electro-magneto quasit-static electro-magneto quasit-static problem (EMQS)problem (EMQS) to calculate all port to calculate all port currentscurrents

– that is one column of the admittance that is one column of the admittance matrix matrix Y=ZY=Z-1-1

Page 27: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Setups of Parasitic Extraction ProblemsSetups of Parasitic Extraction Problems– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– RL Extraction (MQS)RL Extraction (MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers Electromagnetic solvers – classification (time vs. frequency, differential vs. integral)classification (time vs. frequency, differential vs. integral)

– integral equation solvers in detailintegral equation solvers in detail• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products

– Example: EMQS solutionExample: EMQS solution

– ConclusionsConclusions

Page 28: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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The Electromagnetic Interference (EMI)The Electromagnetic Interference (EMI)Problem descriptionProblem description

• Electronic circuits produce and are subject to Electronic circuits produce and are subject to Electromagnetic Interference (EMI).Electromagnetic Interference (EMI).– in particular when wavelengths ~ wire lengthsin particular when wavelengths ~ wire lengths

• EMI is a problem because EMI is a problem because it can severely and it can severely and randomly affect analog and digital circuit randomly affect analog and digital circuit functionality!!!functionality!!!

PCBPCB

ICIC

PCPCBB

ICIC

Page 29: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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EMI analysisEMI analysisEMI at board, package and IC levelEMI at board, package and IC level

• Traces on PCB can pick up Traces on PCB can pick up EMI EMI and transmit it to IC’s and transmit it to IC’s

• IC’s can produce high IC’s can produce high frequency frequency conducted conducted emissionsemissions that can that can radiateradiate from PCB’sfrom PCB’s

• IC’s themselves can directly IC’s themselves can directly produce produce radiated emissionsradiated emissions

– high-frequency current loops high-frequency current loops Vdd-decap-gnd on package or Vdd-decap-gnd on package or inside IC’s.inside IC’s.

– high-frequency current loops high-frequency current loops inside IC (near future)inside IC (near future)

– IC radiation amplified by heat IC radiation amplified by heat sinks!sinks!

PCBPCB

PCBPCB

ICIC

ICIC

ICIC

Page 30: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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EMI a problem for ICs design?EMI a problem for ICs design?

• So far: dimensions too small and wavelengths too largeSo far: dimensions too small and wavelengths too large• Trend: Trend: larger chip dies and higher frequencieslarger chip dies and higher frequencies

Future ICs: • clocks > 3GHz• harmonics > 30GHz • wavelengths < 1cm • dimensions > 1cm

Today’s PCB are affected by EMI:• clocks ~ 300MHz• harmonics ~ 3GHz• wavelengths ~ 10cm• dimensions ~ 10cm

d

d

this gives resonances on today’s PCB today,this gives resonances on today’s PCB today,hence it might produce resonances on future ICs!hence it might produce resonances on future ICs!

Page 31: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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EMI analysisEMI analysisSolution ProcedureSolution Procedure

• Typically, EMI analysis is a two-step process:Typically, EMI analysis is a two-step process:

1) determine accurate current distributions on conductors1) determine accurate current distributions on conductors

• 2) calculate radiated fields from the current distributions2) calculate radiated fields from the current distributions

EE

1I

2I

1I

2I

Page 32: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Need for full-board analysisNeed for full-board analysis

• Interconnect impedances depend on complicated return Interconnect impedances depend on complicated return paths.paths.

• Unbalanced currents generate most of the interferenceUnbalanced currents generate most of the interference. .

• Hence need FULL-BOARD or FULL-PACKAGE or FULL-Hence need FULL-BOARD or FULL-PACKAGE or FULL-CHIP analysisCHIP analysis

1I

12 II

Page 33: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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jJn

J

AjJ

JA

ˆ

0

22

22

Need for full-wave analysisNeed for full-wave analysis

• Circuit dementions are not negligible compared to Circuit dementions are not negligible compared to wavelengthwavelength

dt

dIi

d

ct

dt

dILv i

jij ,

coupling NOT instantaneus,speed of light creates retardation

d

Need to solve FULLWAVE equations (same as for RLC extractionplus wave term)

Page 34: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Setups of Parasitic Extraction ProblemsSetups of Parasitic Extraction Problems– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– RL Extraction (MQS)RL Extraction (MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic field solvers Electromagnetic field solvers – classification (time vs. frequency, differential vs. integral)classification (time vs. frequency, differential vs. integral)

– integral equation solvers in detailintegral equation solvers in detail• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products

– Example: EMQS solutionExample: EMQS solution

– ConclusionsConclusions

Page 35: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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The most intuitive field solver: FDTDThe most intuitive field solver: FDTDe.g. Forward Euler in 1D (easier to explain)e.g. Forward Euler in 1D (easier to explain)

t

EEH

t

HE

t

H

x

E yz

Maxwell differential equations:

In one dimension: Using forward Euler:

t

HH

x

EEn

myn

myn

mzn

mz

1

1

Page 36: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Example: 1D-FDTD with Forward Euler (cont.)Example: 1D-FDTD with Forward Euler (cont.)

1

1

1

1

1

n

my

n

mz

n

mz

n

mzn

mzn

myn

my

E

H

E

EEx

tHH

Iteration formulas:

1n

myH

n

myHn

mzE 1n

mzE

t

x

n+1

n

m m+1space

time

Page 37: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Electromagnetic Solvers Classification Electromagnetic Solvers Classification

Differential Differential Integral Integral

MethodsMethods MethodsMethods

Time-domainTime-domain FDTDFDTD PEECPEEC

MethodsMethods Finite DifferenceFinite Difference Partial ElementPartial Element

Time DomainTime Domain Equivalent Equivalent CircuitsCircuits

Frequency-domainFrequency-domain FEMFEM MoM, PEECMoM, PEEC

MethodsMethods Finite ElementFinite Element Method of MomentsMethod of Moments

MethodMethod

Page 38: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Time-domain vs. frequency domain methodsTime-domain vs. frequency domain methods

Time-domain methodsTime-domain methods Frequency-domain methodsFrequency-domain methods

can handle non-linearitiescan handle non-linearities problems with non-problems with non-linearitieslinearities

run a long simulation, excitingrun a long simulation, exciting solve for specific frequencysolve for specific frequency

all significant modes and thenall significant modes and then points of interestpoints of interest

take an FFTtake an FFT

can produce insightfulcan produce insightful can exploit new techniques can exploit new techniques

animationsanimations for model order reductionfor model order reduction

Page 39: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Differential vs. Integral methodsDifferential vs. Integral methods

Differential methodsDifferential methods Integral MethodsIntegral Methods

discretize discretize entireentire domain domain discretize discretize only “active”only “active”

regionsregions

create create hugehuge but but sparsesparse create create small small but but densedense

linear systemslinear systems linear systemslinear systems

good for inhomogeneousgood for inhomogeneous problems with problems with

materialsmaterials inhomogeneous materialsinhomogeneous materials

problems with openproblems with open good for open boundarygood for open boundary

boundary conditionsboundary conditions conditionsconditions

Page 40: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 41: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Maxwell Differential EquationsMaxwell Differential Equations

• Maxwell Differential Equations can be written in terms ofMaxwell Differential Equations can be written in terms of– the electric scalar potentialthe electric scalar potential

– and the magnetic vector potentialand the magnetic vector potential

EJ

H

E

EjH

HjE

0

jJn

J

AjJ

JA

ˆ

0

22

22

A

Page 42: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

42

AjJ

JA 22

22

Maxwell equations in integral formMaxwell equations in integral formMixed potential Integral Equation (MPIE)Mixed potential Integral Equation (MPIE)

)()(ˆ0)( srrJnrJ j current and chargecurrent and chargeconservationconservation

AB

'|'|

)'(4

)( |'|

drrr

rJrJ rr

V

jej

resistive effectresistive effect magnetic couplingmagnetic coupling

JJ

J J

J

)('|'|

)(4

1 |'|

SSSS

rr

s rdrrr

'rSS

S

je charge-voltage charge-voltage relationrelation

Page 43: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

43

Full-wave (for EMI) Full-wave (for EMI) vs. quasi-static EMQS (for RLC extraction)vs. quasi-static EMQS (for RLC extraction)

)()(ˆ0)( srrJnrJ j current and chargecurrent and chargeconservationconservation

AB

'|'|

)'(4

)( |'|

drrr

rJrJ rr

V

jej

resistive effectresistive effect magnetic couplingmagnetic coupling

JJ

J J

J

)('|'|

)(4

1 |'|

SSSS

rr

s rdrrr

'rSS

S

je charge-voltage charge-voltage relationrelation

QUASI-STATIC

QUASI-STATIC

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44

EMQS (for RLC extraction)EMQS (for RLC extraction)vs. MQS (for RL extraction)vs. MQS (for RL extraction)

0)( rJ current and chargecurrent and chargeconservationconservation

AB

'

|'|

1)'(

4

)(dr

rrrJ

rJV

j

resistive effectresistive effect magnetic couplingmagnetic coupling

JJ

J J

J

)('|'|

1)(

4

1SS

SSs rdr

rr'r

S charge-voltage charge-voltage relationrelation

MQS MAGNETO QUASI-STATIC

)()(ˆ srrJn j

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45

EMQS (for RLC extraction)EMQS (for RLC extraction)vs. electro-static (for capacitance extraction)vs. electro-static (for capacitance extraction)

)()(ˆ0)( srrJnrJ j current and chargecurrent and chargeconservationconservation

AB

'

|'|

1)'(

4

)(dr

rrrJ

rJV

j

resistive effectresistive effect magnetic couplingmagnetic coupling

JJ

J J

J

)('|'|

1)(

4

1SS

SSs rdr

rr'r

S charge-voltage charge-voltage relationrelation

ELECTRO-STATIC

ELECTRO-STATIC

Page 46: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

46

OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 47: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

47

Basis FunctionsBasis Functions

• Basis for vector spaceBasis for vector space

N

iii

NN w

121 if basis a is ,,,

)()()(function if basis a is ),(),(1

21 xwxxxxi

ii

• Basis functions for functional vector spaceBasis functions for functional vector space

• ExamplesExamples– exponentialsexponentials– cos, sincos, sin– polynomialpolynomial

xje

)sin(),cos(,1 xx

)(x )(x

x x

– pieacewise constantpieacewise constant – pieacewise linearpieacewise linear

...,,,,1 32 xxx

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48

Integral Equation:

1 if is on panel jj x x 0 otherwisej x

Discretize Surface into Panels

Panel j

1

surface

x dx x

x S

1

Basis Functions

Represent n

i ii

x x

Piecewise Constant Basis Functions. Piecewise Constant Basis Functions. E.g. Electrostatic problem for Capacitance ExtractionE.g. Electrostatic problem for Capacitance Extraction

)',( xxG Green’s Function

wi

Page 49: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

49

OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 50: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

50

1

,

,i i

n

c j cj

i j

panel j

x G x x dS

A

11,1 1, 1

,1 ,n

cn

n n n n c

xA A

A A x

Put collocation points atpanel centroids

icx Collocationpoint

Collocation on Piecewise constant basis functionsCollocation on Piecewise constant basis functions(Centroid Collocation Testing)(Centroid Collocation Testing)

1

,

,i i

n

t j t jj approx

surface

i j

x G x x x dS

A

n

jjj

surface

dSxxxGx1

')'()',()(

icxicx

icxicx

w1

wn

wj

wj

wj

Page 51: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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1

,

,i i

n

t j tj line j

i j

x G x x dS

A

Centroid Collocation Testing Centroid Collocation Testing Is the Matrix symmetric?Is the Matrix symmetric?

1,21 2 ')',( AdSxxGpanel

2x1x

icxicx

1x2x

Centroid Collocation generates a non-symmetric matrix

')',(2 12,1 dSxxGA

panel

wj

Page 52: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Panel j

icx Collocationpoint

,

1

i

i j

pa cnel j x xA dS

,

i jc centr

j

id

i

o

Panel Area

x xA

One point

quadrature Approximation

x

yz

t

4

,1 in

0.25*

i jc o

i jj p

Ar a

x xA

e

Four point

quadrature Approximation

Calculating Matrix ElementsCalculating Matrix Elements

Page 53: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Panel j=i

icx Collocationpoint

,

1

i

i i

pa cnel i x xA dS

,

0

i i

i i

c c

Panel AreaA

x x

One point quadrature

Approximation

x

yz

, is an integrable singularity1

i

i i

panel i cx xA dS

Calculating the “Self-Term”Calculating the “Self-Term”

Page 54: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

54

Panel j=i

icx Collocationpoint

,

1

i

i i

pa cnel i x xA dS

x

yz

Disk of radius R surrounding

collocation point

,

1 1

i ic c

i i

disk rest of panel

A dS dSx x x x

Disk Integral has singularity but has analytic formula

Integrate in two pieces

2

0 0

12

1

i

R

d k cis

dS rdrd Rrx x

Calculating the “Self Term” Calculating the “Self Term” Tricks of the tradeTricks of the trade

Page 55: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

55

1,1 1, 1 1

,1 ,

n

n n n n n

A A b

A A b

Galerkin Testing with Piecewise constant basesGalerkin Testing with Piecewise constant bases

source panel jtest panel i

n

jjj

surface

dSxxxGx1

')'()',()(

1

,

,i i j

n

jjline line line

i i j

x dS G x x dS dS

b A

ipanel ipanel jpanel

1

,

,n

i j i jj

i ji

x x dS x G x x x dS dS

Ab

wj

wj

wj

w1

wn

Page 56: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

56

panel 2panel 1

Galerkin Testing Galerkin Testing Is the Matrix symmetric?Is the Matrix symmetric?

Galerkin generates always a symmetric matrixif the Green function is symmetric G(x,x’)=G(x’,x)

1 22,1 ')',(

panel paneldSdSxxGA 1,22 1

')',( AdSdSxxGpanel panel

1

,

,i i j

n

jjline line line

i i j

x dS G x x dS dS

b A

ipanel ipanel jpanel

wj

Page 57: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 58: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Solution technique Computational Complexity

LU decomposition (Gaussian Elimination)

Dense: O(N3) time, O(N2) memorye.g. 10months, 80GB, for N=100,000

Iterative methods

Bottleneck: matrix-vector productO(k N2) time and memory

e.g. 7days, 80GB, for N=100,000 k=10

bxA

Computational Complexity (time and memory)Computational Complexity (time and memory)

Page 59: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Linear System SolutionLinear System SolutionKrylov subspace iterative methodsKrylov subspace iterative methods

Problem: solve Ax = b (where A is large and dense)Problem: solve Ax = b (where A is large and dense)

i=0i=0 guess xguess xii

REPEATREPEAT– how good was my guess? Calculate residue how good was my guess? Calculate residue rrii=b-Ax=b-Axii

– find next guess xfind next guess xi+1i+1 from the space from the space xx00+span{r+span{r00,Ar,Ar00,A,A22rr00,…,A,…,Aiirr00} which minimizes the } which minimizes the next residue next residue

UNTIL ||residue|| < desired accuracyUNTIL ||residue|| < desired accuracy

Advantages over gaussian elimination:Advantages over gaussian elimination:1) get a great control on accuracy (can stop and save 1) get a great control on accuracy (can stop and save

computation when desired accuracy is achieved) computation when desired accuracy is achieved) 2) only need a matrix vector product 2) only need a matrix vector product AxAxi i per iteration. Hence per iteration. Hence

computational complexity is O(Ncomputational complexity is O(N22))

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60

Suppose Ax=b converges slowly

Try PAx = Pb for some P (left pre-conditioner matrix)

If PA = I then convergence happens in one stephowever P=A-1 is VERY hard to compute

If PA ≈ I then we hope convergence happens in “few” steps

Any general idea for picking P?

:such that ~

Pick AAA

~ a)

factor!or invert easy to is ~

b) A1~

:onerPreconditi AP

e.g. if A is diagonally dominantits diagonal is: - a good approximation of A- and easy to invert

)(diag ~

AA

Linear System SolutionLinear System SolutionPreconditioners for iterative methodsPreconditioners for iterative methods

Page 61: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Integral Equation Preconditioning

1

0

0diagonerPreconditi _____

(Jacobi)oner Preconditi Diagonal ------

onerpreconditi No -.-.-.-

TM

j

PLjR

M

Page 62: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Solution technique Computational Complexity

LU decompositionDense: O(N3) time, O(N2) memory

e.g. 10months, 80GB, for N=100,000

Iterative methods + “fast” matrix-vector product

O(k N log N) time and memorye.g. 8hours, 0.3GB for N=100,000 k=10

Iterative methods

Bottleneck: matrix-vector productO(k N2) time and memory

e.g. 7days, 80GB, for N=100,000 k=10

bxA

Computational Complexity (time and memory)Computational Complexity (time and memory)

Page 63: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• example: FastMaxwell a EMQS/Fullwave solver with pFFTexample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 64: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

64

Fast-Multipole: Simplified ExampleFast-Multipole: Simplified Example

31

51

41

31

21

21

161

61

51

41

31

21

18.3

1

18.3

Example: Thin Metal Strip

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Piecewise ConstantBasis Function

Structure: Diagonals (Toeplitz)

1

1

j

jj Sx

Sxx

0

1

1,66,1 5

16 Panel Area

61

Axx

Acc

4

16,2 A

8.32 1,11,1 AA

,

i jc centr

j

id

i

o

Panel Area

x xA

centroid collocationwith one point quadrature

Page 65: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

65

Use Iterative methods: need matrix vector product

The cost of the entire matrix-vector product is O(N ·log2N)

... provided we can get the clusters cheaply.

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15 w16

N = 16

5 products4 sums≈ O(log2N)

16

2

1

w

w

w

A

16151

431

321

211 18.3 wwwwwx

5.115.55.21

8.3 1611109876543211

wwwwwwwwwwwwx

Fast-Multipole: Simplified ExampleFast-Multipole: Simplified Example

Find σ(x) on strip ↔ Find weights w1, w2, … w16 (charge on each panel)

Assume Ψ(x) = 1 given everywhere on the metal strip

Page 66: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

66

Fast-Multipole: Simplified ExampleFast-Multipole: Simplified Example

8 sums

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15 w16

w1 + w2 w3 + w4 w5 + w6 w7 + w8 w9 + w10 w11 + w12 w13 + w14 w15 + w16

w1 + w2 + w3 + w4 w5 + w6 + w7 + w8 w9 + w10 + w11 + w12 w13 + w14 + w15 + w16

w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16

4 sums

2 sums

14 sums

≈ O(N) sums

Page 67: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

67

Fast-Multipole: Simplified ExampleFast-Multipole: Simplified Example

5.105.55.32

1

1

18.3

1

1 1611109876543212

wwwwwwwwwwwwx

5.65.32

18.3

5.25.516131211

109876541

8wwww

wwwwwwww

x

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15 w16

How do we use those sums?

w1 w2 w3 w4 w5 w6 w7 w8 w9 w10 w11 w12 w13 w14 w15 w16

Each test point requires O(log2N) → total cost O(N · log2N)

Page 68: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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N

N

N N2

N

Page 69: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Fast-Multipole: Multipole representationFast-Multipole: Multipole representation

R

1 Potential

2

1 Potential

R

4

1 Potential

R

Monopole Single charge in center of cluster with sum of all chargesj = 1

DipoleTwo Chargesj = 2

QuadrapoleFour Chargesj = 4

Page 70: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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N

N

Page 71: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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N

N2

N

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What if we need to solve a different equation?What if we need to solve a different equation?

022

xx

exxG

xxj

,Can use multipole expansions but every time I change the Green function I need to come up with multipole expansions.

Helmholtz Equation (Full Wave Equation)

Fast Multipole is UNFORTUNATELYGreen Function dependent!

02 Laplace Equation (Capacitance Extraction)

xx

xxG

1,

Page 74: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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Precorrected FFT: A Simplified ExamplePrecorrected FFT: A Simplified Example

Example: Thin Metal Strip

Recognize Discrete Convolution Sum

Toeplitz

h is the unit sample response

wj Ψih[ ]

W(eiω) Ψ(eiω)H(e iω)

FFT

= W(eiω) H(e iω)

FFT –1

The convolution can be calculated efficiently using 3 FFT [O(N log2N)]

This works only if G(x,x') = G(|x – x'|) translation invariant

d

jjiji Awx

1,

51

4151

4151

41

ij

h

jiji

ji AAjixx

AreaA

ji

,,1

hwhwxd

jjiji

1

1 2 3 4 5 6 d1

1

FFT

Page 75: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

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P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

Page 77: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

77

P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

Calculate potentials on grid pointsdue to charges on grid points with FFT O(nO(n22)) O(n log O(n log

n)n)

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78

P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

Page 79: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

79

P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

Page 80: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

80

P-FFT matrix vector product

Problem: solve iteratively Aw=Ψ

At each iteration evaluate matrix- vector products Aw using the following 6 steps: Grid generation Projection Convolution Interpolation Direct computation Pre-correction

Page 81: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

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OverviewOverview

• Formulation of Parasitic Extraction Problems:Formulation of Parasitic Extraction Problems:– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– Inductance Extraction (Magneto-Quasi-Static MQS)Inductance Extraction (Magneto-Quasi-Static MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers: Electromagnetic solvers: – Classification (time vs. frequency, differential vs. integral)Classification (time vs. frequency, differential vs. integral)

– Integral equation solvers in detailsIntegral equation solvers in details• basis functionsbasis functions• equation testing (collocation vs. Galerkin)equation testing (collocation vs. Galerkin)• linear system solution (direct vs. iterative methods)linear system solution (direct vs. iterative methods)• fast matrix-vector products (eg. fastmultipole, pFFT)fast matrix-vector products (eg. fastmultipole, pFFT)• examples of integral equation solversexamples of integral equation solvers

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 82: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

82

Electromagnetic Integral Equation SolversCode Formu-

lationAnalysis Substrate/

dielectricsExtract Acceleration Public

domain

IES3

[Kapur97]

Surface EQS Green’s Function

C SVD Compression

No

FastCap

[Nabors91]

Surface EQS Green’s Function

C Fast-Multipole Yes

M.I.T.

FastHenry

[Kamon94]

Volume MQS None R, L Fast-Multipole Yes

M.I.T.

ASITIC

[Niknejad97]

Volume MQS Simplified GF

R, L none Yes

EMX

[Kapur04]

Surface Fullwave Dyadic GF Z Fast-Multipole No

EMSurf

[IBM]

Surface Fullwave None Z P-FFT Yes

FastImp

[Zhu03]

Surface Fullwave None Z P-FFT Yes

M.I.T.

FastMaxwell

[Moselhy07]

Volume EMQS +

Fullwave

Dyadic GF Z P-FFT Yes

M.I.T.

Page 83: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

83

+ + ++ + + + + + +J

J J

J

FastMaxwell: MPIE Formulation

0

ˆ

r

j

J

n J

2,S

G r r r d r

3,A

V

j G r r r d r

J

J

Current/charge conservation

Charge-voltage relation

Resistive term Magnetic coupling

AB

+ ++ + ++ + + + + + + +++ + ++

Dyadic/scalar Green’s

functions for substrate

Page 84: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

84

Piece-Wise Constant Basis FunctionsPEEC [Ruehli 74]

Volume filaments carrying constant

current

3,A

V

j G r r r d r

J

J

( ) F FR j L I V

Page 85: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

85

Discretization [Ruehli 74]

0

0 P

F

P

F

P q

R j L I V

Volume filaments carrying constant

current

Surface Panels carrying

constant charge

2,S

G r r r d r p pP q ( ) F FR j L I V

Page 86: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

86

Mesh Analysis [Kamon 97]

Imposing current conversation by writing KVL

0

0T

m m

R j L

I VPM

j

M

Mesh Matrix

A x b=

Page 87: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

87

Integral Equation Preconditioning

1

0

0diagonerPreconditi _____

(Jacobi)oner Preconditi Diagonal ------

onerpreconditi No -.-.-.-

TM

j

PLjR

M

Page 88: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

88

FastMaxwell validation against measurements on a FastMaxwell validation against measurements on a fabricated RF inductor [El Moselhy Daniel DAT07]fabricated RF inductor [El Moselhy Daniel DAT07]

• Piecewise constant basis + Galerkin • PFFT matrix vector product O(NlogN)• can handle tens of coupled RF inductors

Fabricated and measuredRF inductors to verifythe field solver

Page 89: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

89

FastMaxwell performance on large structures [El Moselhy, Daniel DATE07]

Structure Num. unknowns Memory in (MB) Time

Inductor Array 128, 033 2.5GB 3 hours

Stacked Inductor Array 139, 000 3GB 3 hours

30X30 grid 330,000 6GB 8 hours

On public domain: www.rle.mit.edu/cpg/fastmaxwell.htm

Page 90: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

90

OverviewOverview

• Setups of Parasitic Extraction ProblemsSetups of Parasitic Extraction Problems– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– RL Extraction (MQS)RL Extraction (MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers Electromagnetic solvers – classification (time vs. frequency, differential vs. integral)classification (time vs. frequency, differential vs. integral)

– integral equation solvers in detailintegral equation solvers in detail• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products

– Example: FastMaxwell a EMQS/Fullwave solver with pFFTExample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions

Page 91: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

91

Step 1. Interconnect Field Solvers

DeterministicSolvers

FASTCAP91(fastmultipole),

FRW92(Floating

Random Walk)

IES397 (SVD)

FASTHENRY94(fastmultipole) FASTIMP05

(pFFT),

EMSurf IBM(pFFT)

EQS(capacitance

electriccoupling)

MQS(inductance

magneticcoupling)

EMQS andFullwave

(full imped.EM coupling)

Substratecoupling

and losses

FASTMAXWELL (pFFT)Daniel DATE07 best paper nomin.

ASITIC97 (no acceleration),

Page 92: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

92

How do we efficiently compute the capacitances of ALL these configurations?

• The different lines correspond to different settings for lithographic dose and focus

• All patterns are topologically similar but metrically different

On/Off-chip Variations

Page 93: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

93

Step 1. Interconnect Field Solvers

DeterministicSolvers

FASTCAP91(fastmultipole),

FRW92(Floating

Random Walk)

IES397 (SVD)

FASTHENRY94(fastmultipole) FASTIMP05

(pFFT),

EMSurf IBM(pFFT)

VariationalSolvers

Adjoint SensitivityWang97,

EQS(capacitance

electriccoupling)

MQS(inductance

magneticcoupling)

EMQS andFullwave

(full imped.EM coupling)

Substratecoupling

and losses

FASTMAXWELL (pFFT)Daniel DATE07 best paper nomin.

ASITIC97 (no acceleration),

(in red our contributions)

Page 94: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

94

Variational Field Solvers

bxA

Field Solvers discretize Field Solvers discretize geometry and produce geometry and produce large linear systemlarge linear system

Field SolversField Solvers

p p

1

surface

x dx x

x S

Wcross section

,...),( hW ,...),( hW

Page 95: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

95

Finite Difference Sensitivity Analysis e.g. for capacitance extraction

bpxpA )()( Decouple random variables

(SVD or Principal Component Analysis)Lp

bxA )()(

)()( xdC T the capacitance is just the sum of some charges

dx

xdC T )()( 0

Sensitivity Matrix

dC

C

)( 0

i

iiT

id

xedxd

C

)()ˆ( 00 approximate with finite difference

Advantages: very simple to implementDisadvantages: very expensive (need to solve N+1 systems)

bedxedA iiii )ˆ()ˆ( 00

Page 96: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

96

Adjoint Sensitivity Analysis [Wang97] e.g. for capacitance extraction

bpxpA )()( Decouple random variables

(SVD or Principal Component AnalysisLp

bxA )()(

)()( xdC T the capacitance is just the sum of some charges

dx

xdC T )()( 0

0)()( 00

x

AxA

bxA )()( differentiate

)()( 01

0

xA

Ax

dxA

AxdC T )()()()( 01

00

dxA

AdxdC TT )()()()( 01

00

T

dA T )( 0solvetypically availableanalytically

Only need TWO system solves: bxA )()( 00 dA T )( 0

Page 97: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

97

Step 1. Interconnect Field Solvers

DeterministicSolvers

FASTCAP91(fastmultipole),

FRW92(Floating

Random Walk)

IES397 (SVD)

FASTHENRY94(fastmultipole) FASTIMP05

(pFFT),

EMSurf IBM(pFFT)

VariationalSolvers

Adjoint SensitivityWang97, FRW specialized

Green functionsDaniel ICCAD08

EQS(capacitance

electriccoupling)

MQS(inductance

magneticcoupling)

EMQS andFullwave

(full imped.EM coupling)

Substratecoupling

and losses

FASTMAXWELL DATE07 (pFFT)best paper nomin.

FRW Recycling Daniel ICCAD08

ASITIC97 (no acceleration),

Page 98: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

98

Capacitance Using Gauss Theorem

12)( CdnEQ

E

1 2

3

Gaussian Surface

6

54

78

• C12 can be calculated as the charge on the Gaussian surface around 1 when V1=V3=0V and V2=1V

Page 99: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

99

A Standard NON-variation aware Capacitance Method: Floating Random Walk (FRW) [Iverson92]

1 2

3

Gaussian Surface

111, drrrrGr

22211 , drrrrGr

33322 , drrrrGr

44433 , drrrrGr

55544 , drrrrGr

554521211 ,,, rrrGdrrrGdrrrGdrr

6

54

78

Page 100: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

100

The majority of the paths are not affected by dimension variations: they can be recycled! [Moselhy08]

1 2

3

Gaussian Surface

6

54

78

Page 101: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

101

1

3

2

Gaussian Surface

Full path recycle and continuation for shrinking path-termination-conductors [Moselhy08]

6

54

78

Page 102: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

102

1 2Gaussian Surface

Edge perturbation is not affecting contribution of THIS PATH to coupling capacitance C12, since it does not terminate on perturbed edge.

3

Paths can be fully recycled for shrinking path-limiting-conductors, or all small variations [Moselhy08]

6

54

78

Page 103: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

103

1 2Gaussian Surface

Perturbation affects path:

need to recompute part of it.

3

Partial path recycling and continuationfor growing conductors limiting the path [Moselhy08]

6

54

78

Page 104: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

104

Results [El Moselhy Daniel ICCAD08]

Simulation time to extract 9 configurations is less than 2x nominal time

Accuracy Speed

Page 105: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

105

100

101

102

103

104

105

10610

-4

10-3

10-2

10-1

100

Number of Similar Configurations

No

rma

lize

d R

un

Tim

e/C

on

figu

ratio

nResults [El Moselhy Daniel ICCAD08]

Average time required for a new configuration is less that 0.1% that required for nominal configuration

extrapolated6

Simulation time to extract 100,000 configurations

is less than 50x of time for nominal configuration

(2000x speed up from fastcap)

Page 106: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

106

On/Off-chip Variations

• Irregular geometries: On-chip • Rough-surfaces: On-package and on-board

[Courtesy of IBM and Cadence] [Braunisch06]

Page 107: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

107

Step 1. Interconnect Field Solvers

DeterministicSolvers

FASTCAP91(fastmultipole),

FRW92(Floating

Random Walk)

IES397 (SVD)

FASTHENRY94(fastmultipole) FASTIMP05

(pFFT),

EMSurf IBM(pFFT)

VariationalSolvers

Adjoint SensitivityWang97, FRW specialized

Green functionsDaniel ICCAD08

EQS(capacitance

electriccoupling)

MQS(inductance

magneticcoupling)

EMQS andFullwave

(full imped.EM coupling)

Substratecoupling

and losses

FASTMAXWELL (pFFT)Daniel DATE07 best paper nomin.

FRW Recycling Daniel ICCAD08

StochasticSolvers

SGM Ghanem91, FASTSIES05,

Newmann Exp.Daniel EPEP07

best papernomin.

Daniel DAC08 Speedup x1000

ASITIC97 (no acceleration),

(in red our contributions)

Page 108: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

108

1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input Impedance

PD

F

Stochastic Field Solvers

Stochastic Field Solver

Stochastic Field Solver

Geometry of interconnect structure

Distribution describing the geometrical variations

Statistics of interconnect input impedance

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Width

PD

F

widthinput impedance

PD

F PD

F

System matrix dependent on the geometry

Unknown vector of currents and charges

bpxpA

bpApx 1

C

pCppP

H

T

5.0

1

2

5.0exp

Wcross section

Page 109: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

109

Stochastic Galerkin Method FEM mechanics [Ghanem91]

expand in terms of orthogonal polynomials

write as a summation of same polynomials

Substitute in original equation and equate the

coefficients to compute the unknowns

need to decouple random variables

K

jjjxx

0

bpxpA

K

kkkAA

0

Lp

1LProblem 1: expensive multidimensional integral

Problem 2: very large linear system

Page 110: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

110

Stochastic Galerkin Method for Integrated Circuit Parasitic Extraction [Daniel DAC08]

expand in terms of orthogonal polynomials

write as a summation of same polynomials

Substitute in original equation and equate the

coefficients to compute the unknowns

need to decouple random variables

K

jjjxx

0

bpxpA

K

kkkAA

0

Lp

1LProblem 1: expensive multidimensional integral

Problem 2: very large linear system

Our solutions [DAC 08]:

• problem 1: use specialized inner product

• problem 2: use Neumann expansion

Page 111: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

111

Our Results: Accuracy Validation

• Microstrip line W=50um, L=0.5mm, H=15um

• sigma=3um, correlation length=50um

• mean: 0.0122, std (MC, SGM) = 0.001, std (New algorithm)= 0.00097

0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.0180

50

100

150

200

250

300

350

400

450

500

DC Resistance in

Prob

abilit

y D

ensi

ty F

unct

ion

New Algorithm

Monte Carlo

SFE + Thm. 1SGM +

Page 112: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

112

Results: Complexity Validation

Example TechniqueProperties for 5%

accuracyMemory Time

Long Microstrip line

DC Only

400 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

96 iid, 4753 o.p.

96 iid, 4753 o.p.

1.2 MB

1.2 MB

(72 GB)

1.2 MB

2.4 hours

0.25 hours

-

0.5 hours

Transmission Line

10 freq. points

800 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

105 iid, 5671 o.p.

105 iid, 5671 o.p.

10 MB

10 MB

(300 TB)

10 MB

16 hours

24 hours

-

7 hours

Two-turn Inductor

10 freq. points

2750 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

400 iid, 20604*

400 iid, 20604*

121 MB

121 MB

(800 PB)

121 MB

(150 hours) X 4p

(828 hours) X 4p

-

8 hours X 4p

Page 113: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

113

Conventional Design FlowConventional Design Flow

Funct. Spec

Logic Synth.

Gate-level Net.

RTL

Layout

Floorplanning

Place & Route

Front-end

Back-end

Behav. Simul.

Gate-Lev. Sim.

Stat. Wire Model

Parasitic Extrac.

Page 114: 1 Parasitic Extraction Step 1: Electromagnetic Field Solvers Luca Daniel Massachusetts Institute of Technology luca@mit.edu dluca/2009MOMiNE

114

SummarySummary

• Setups of Parasitic Extraction ProblemsSetups of Parasitic Extraction Problems– Capacitance Extraction (electrostatic)Capacitance Extraction (electrostatic)

– RL Extraction (MQS)RL Extraction (MQS)

– Combined RLC Extraction (EMQS)Combined RLC Extraction (EMQS)

– Electromagnetic Interference Analysis (fullwave)Electromagnetic Interference Analysis (fullwave)

• Electromagnetic solvers Electromagnetic solvers – classification (time vs. frequency, differential vs. integral)classification (time vs. frequency, differential vs. integral)

– integral equation solvers in detailintegral equation solvers in detail• basis functionsbasis functions• residual minimization (collocation and Galerkin)residual minimization (collocation and Galerkin)• linear system solutionlinear system solution• fast matrix-vector productsfast matrix-vector products

– Example: FastMaxwell a EMQS/Fullwave solver with pFFTExample: FastMaxwell a EMQS/Fullwave solver with pFFT

– Current Field Solvers Research DirectionsCurrent Field Solvers Research Directions