parasitic extraction luca daniel university of california, berkeley massachusetts institute of...
TRANSCRIPT
![Page 1: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/1.jpg)
Parasitic ExtractionParasitic Extraction
Luca DanielLuca DanielUniversity of California, BerkeleyUniversity of California, Berkeley
Massachusetts Institute of TechnologyMassachusetts Institute of Technology
with contributions from:with contributions from:Alessandra Nardi, University of California, BerkeleyAlessandra Nardi, University of California, Berkeley
Joel Phillips, Cadence Berkeley LabsJoel Phillips, Cadence Berkeley LabsJacob White, Massachusetts Instit. of TechnologyJacob White, Massachusetts Instit. of Technology
![Page 2: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/2.jpg)
2
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 3: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/3.jpg)
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: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/4.jpg)
4
Parasitic ExtractionParasitic Extraction
Parasitic Extraction
thousands of wirese.g. critical pathe.g. gnd/vdd grid
tens of circuitelements for gate level spice simulation
• identify some ports• produce equivalent circuit that
models response of wires at those ports
![Page 5: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/5.jpg)
5
Parasitic Extraction (the two steps)Parasitic Extraction (the two steps)
ElectromagneticAnalysis
million of elements
thin volume thin volume filamentsfilamentswith constant with constant currentcurrent
small surface small surface panelspanelswith constant with constant chargecharge
Model OrderReduction
tens of elements
![Page 6: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/6.jpg)
6
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 7: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/7.jpg)
7
Capacitive ExtractionCapacitive ExtractionExample: Intel 0.25 micron ProcessExample: Intel 0.25 micron Process
5 metal layers Ti/Al - Cu/Ti/TiN Polysilicon dielectric.Taken from “Digital Integrated Circuits”, 2nd Edition, Rabaey, Chandrakasan, Nikolic
fringing parallel
Consider only electric field (capacitive) coupling
![Page 8: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/8.jpg)
8
Capacitive ExtractionCapacitive ExtractionWhy? E.g. Analysis of Delay of Critical PathWhy? E.g. Analysis of Delay of Critical Path
![Page 9: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/9.jpg)
9
Capacitance Extraction Capacitance Extraction Why do we need it?Why do we need it?
• Example: to produce RC Example: to produce RC tree network for elmore tree network for elmore delay analysisdelay analysis
• Example: to produce RC Example: to produce RC tree network for tree network for capacitive cross-talk capacitive cross-talk analysisanalysis
R1
C1
s
R 2
C2R 4
C4
C3
R3
Ci
Ri
1
2
3
4
i
![Page 10: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/10.jpg)
10
Capacitance ExtractionCapacitance ExtractionProblem FormulationProblem Formulation
• Given a collection of N conductors (of any shape and Given a collection of N conductors (of any shape and dimension)dimension)
fringing parallel
qvC ?
Calculate the couplingcapacitance matrix C
![Page 11: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/11.jpg)
11
Capacitance ExtractionCapacitance ExtractionSolution ProcedureSolution Procedure
• For i = 1 to N,For i = 1 to N,– 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?q
?q ?q ?q
?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 12: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/12.jpg)
12
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 13: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/13.jpg)
13
Picture Thanks to Coventor
Inductance and Resistance ExtractionInductance and Resistance ExtractionExample: IC packageExample: IC package
package
IC
wirebonding
lead frames
![Page 14: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/14.jpg)
14
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 15: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/15.jpg)
15
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 16: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/16.jpg)
16
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 at and above 100MHze.g. on PCB at and above 100MHz
– e.g. on packages at above 1GHze.g. on packages at above 1GHz
– e.g. on-chip at and above 10GHze.g. on-chip at and above 10GHz
– note. clock at 3GHz has significant harmonics at 10GHz!!note. clock at 3GHz has significant harmonics at 10GHz!!
![Page 17: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/17.jpg)
17
Inductance and Resistance ExtractionInductance and Resistance ExtractionProblem FormulationProblem Formulation
• Given a collection of Given a collection of interconnected N wires of any interconnected N wires of any shape and dimensionshape and dimension
• Identify the M input portsIdentify the M input portsPicture byPicture byM. ChouM. Chou
viLjR ??
• Calculate the MxM resistance and the Calculate the MxM resistance and the inductance matrices for the ports, inductance matrices for the ports,
• that is the real and immaginary part of that is the real and immaginary part of the impedance matrixthe impedance matrix
![Page 18: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/18.jpg)
18
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 each pair of input For each pair of input terminals,terminals,
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
and solve and solve magneto quasit-magneto quasit-static problemstatic problem (MQS)(MQS) to to calculate all terminal calculate all terminal currentscurrents
– that is one column of the that is one column of the admittance matrix [R+jwL]admittance matrix [R+jwL]-1-1
![Page 19: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/19.jpg)
19
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 20: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/20.jpg)
20
Combined RLC ExtractionCombined RLC ExtractionExample: current distributions on powergridExample: current distributions on powergrid
input terminals
![Page 21: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/21.jpg)
21
Combined RLC Extraction Combined RLC Extraction Example: analysis of resonances on powergridExample: analysis of resonances on powergrid
* 3 proximity templates per cross-section- 20 non-uniform thin filaments per cross-section
![Page 22: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/22.jpg)
22
Combined RLC ExtractionCombined RLC ExtractionExtraction Example: analysis of substrate couplingExtraction Example: analysis of substrate coupling
![Page 23: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/23.jpg)
23
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 24: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/24.jpg)
24
Combined RLC ExtractionCombined RLC ExtractionProblem FormulationProblem Formulation
• Given a collection of Given a collection of interconnected N wires of any interconnected N wires of any shape and dimensionshape and dimension
• Identify the M input portsIdentify the M input portsPicture byPicture byM. ChouM. Chou
viZ ?
• Calculate the MxM IMPEDANCE matrix Calculate the MxM IMPEDANCE matrix for the ports, for the ports,
• that is the real and immaginary part of that is the real and immaginary part of the impedance matrixthe impedance matrix
![Page 25: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/25.jpg)
25
Combined RLC ExtractionCombined RLC ExtractionSolution ProcedureSolution Procedure
• Same as RL extraction.Same as RL extraction.• Typically calculate admittance matrixTypically calculate admittance matrix• For each pair of input terminals,For each pair of input terminals,
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 and apply a unit voltage source and solve solve electro-magneto quasit-static electro-magneto quasit-static problem (EMQS)problem (EMQS) to calculate all to calculate all terminal currentsterminal currents
– that is one column of the admittance that is one column of the admittance matrix [R+jwL]matrix [R+jwL]-1-1
![Page 26: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/26.jpg)
26
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 27: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/27.jpg)
27
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 28: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/28.jpg)
28
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 29: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/29.jpg)
29
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’s IC: • clocks ~ 3GHz• harmonics ~ 30GHz • wavelengths ~ 1cm • dimensions ~ 1cm
Today’s PCB:• clocks ~ 300MHz• harmonics ~ 3GHz• wavelengths ~ 10cm• dimensions ~ 10cm
d
d
this gives resonances on PCB today,this gives resonances on PCB today,hence it might on IC tomorrow!hence it might on IC tomorrow!
![Page 30: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/30.jpg)
30
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 31: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/31.jpg)
31
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 analysisHence need FULL-BOARD analysis
1I
12 II
![Page 32: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/32.jpg)
32
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 extraction plus wave term)
![Page 33: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/33.jpg)
33
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 34: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/34.jpg)
34
Example: the most intuitive FDTD in one dimension Example: the most intuitive FDTD in one dimension
using Forward Euler (easier to explain)using Forward Euler (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 35: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/35.jpg)
35
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+1
![Page 36: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/36.jpg)
36
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 techniquescan exploit new techniques
animationsanimations for model order reductionfor model order reduction
![Page 37: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/37.jpg)
37
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 38: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/38.jpg)
38
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, MoM, PEECPEEC
MethodsMethods Finite ElementFinite Element Method of MomentsMethod of Moments
MethodMethod
![Page 39: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/39.jpg)
39
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 40: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/40.jpg)
40
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 41: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/41.jpg)
41
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 42: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/42.jpg)
42
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
![Page 43: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/43.jpg)
43
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
![Page 44: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/44.jpg)
44
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 45: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/45.jpg)
45
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 46: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/46.jpg)
46
Basis FunctionsBasis Functions
• Basis for vector spaceBasis for vector space
N
iii
NN
121 if basisa is ,,,
)()()( function if basisa is ),(),(1
21 xxxxxi
ii
• Basis functions for functional vector spaceBasis functions for functional vector space
• ExamplesExamples– exponentialsexponentials
– cos sincos sin
xje
)sin(),cos(,1 xx
)(x )(x
x x
– pieacewise constantpieacewise constant – pieacewise linearpieacewise linear
![Page 47: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/47.jpg)
47
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. Capacitance Extraction Electrostatic problemE.g. Capacitance Extraction Electrostatic problem
![Page 48: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/48.jpg)
48
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 49: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/49.jpg)
49
1
,n
i iiapprox
surface
R x x G x x x dS
iWe will pick the ' to make ( ) small. s R x
General Approach: Pick a set of test functions
0 for all .i x R x dS i
1, , , and force to be orthogonal to the setn R x
Residual Definition and MinimizationResidual Definition and Minimization
![Page 50: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/50.jpg)
50
1We will generate different methods by chosing the , , ,n
(basis = test)Galerkin Method: i ix x
Weighted Residual Method: 1 if ( ) 0 (averages)i ix x
C (point-matching)ollocation:ii tx x x
1
, 0n
i i i j jjapprox
surface
x R x dS x x dS x G x x x dS dS
Residual Minimization using Test FunctionsResidual Minimization using Test Functions
Note: Weighted Residual = Galerkin when using piecewise constant basis functions
![Page 51: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/51.jpg)
51
1
, 0i i i i
n
t t t t j jjapprox
surface
x x R x dS R x x G x x x dS
(point-matchingCollocati )on: i ix x
1
,
,i i
n
t j t jj approx
surface
i j
x G x x x dS
A
11,1 1, 1
,1 ,n
tn
n n n n t
xA A
A A x
CollocationCollocation
![Page 52: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/52.jpg)
52
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)(Centroid Collocation)
![Page 53: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/53.jpg)
53
1
,
,i i
n
t j tj line j
i j
x G x x dS
A
Centroid Collocation Centroid Collocation generates a nonsymmetric matrixgenerates a nonsymmetric matrix
1,21 22 12,1 ')',(')',( AdSxxGdSxxGApanelpanel
2x1x
![Page 54: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/54.jpg)
54
Basis Function Approach
Calculating Matrix Elements
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
![Page 55: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/55.jpg)
55
Basis Function Approach
Calculating “Self-Term”
Panel 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
![Page 56: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/56.jpg)
56
Basis Function Approach
Calculating “Self-Term” Tricks of the trade
Panel 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
![Page 57: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/57.jpg)
57
Basis Function Approach
Calculating “Self-Term” Other Tricks of the trade
Panel i
icx Collocationpoint
Integrand is singular
,
1
i
i i
panel i cx xA dS
x
yz
2) Curve panels can be handled with projection
1) If panel is a flat polygon, analytical formulas exist
![Page 58: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/58.jpg)
58
(test=basis)Galerkin: i ix x
1
, 0n
i i i j jjapprox
surface
x R x dS x x dS x G x x x dS dS
, ,If ( , ) ( , ) then =i j j iG x x G x x A A A is symmetric
1
,
,n
i j i jjapprox approx approx
surface surface surface
i ji
x x dS G x x x x dS dS
Ab
1,1 1, 1 1
,1 ,
n
n n n n n
A A b
A A b
GalerkinGalerkin
![Page 59: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/59.jpg)
59
1
,
,i i j
n
jjline line line
i i j
x dS G x x dS dS
b A
1,1 1, 1 1
,1 ,
n
n n n n n
A A b
A A b
Galerkin with Piecewise constant basesGalerkin with Piecewise constant bases
1
,
,n
i j i jj
i ji
x x dS x G x x x dS dS
Ab
source panel j
test panel i
![Page 60: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/60.jpg)
60
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 61: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/61.jpg)
61
BackgroundBackgroundKrylov 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)
guess xguess x00
REPEATREPEAT– how good was my guess? Calculate residue rhow good was my guess? Calculate residue r ii = b-Ax = b-Axii
– find next guess xfind next guess xi+1i+1 from the space x from the space x00+span{r+span{r00,Ar,Ar00,A,A22rr00,…,A,…,Aiirr00} }
which minimizes the next residue which minimizes the 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 Ax per iteration: O(N2) only need a matrix vector product Ax per iteration: O(N22))
![Page 62: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/62.jpg)
62
BackgroundBackgroundSolving the large DENSE linear system (cont)Solving the large DENSE linear system (cont)
Algorithms: Complexity: Storage:
Gaussian elimination O(n3) O(n2)
Krylov subspace iterative methods
O(n2) O(n2)
Krylov iter. + fast matrix- vector product
O(n log(n)) O(n log(n))
• Solving the dense large linear system obtained from Solving the dense large linear system obtained from the Integral Equation method:the Integral Equation method:
![Page 63: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/63.jpg)
63
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 64: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/64.jpg)
64
![Page 65: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/65.jpg)
65
![Page 66: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/66.jpg)
66
![Page 67: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/67.jpg)
67
![Page 68: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/68.jpg)
68
![Page 69: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/69.jpg)
69
O(nO(n22)) O(n log O(n log n)n)
Precorrected-FFT (e.g. capacitance extraction)Precorrected-FFT (e.g. capacitance extraction)
• Problem: solve iteratively Problem: solve iteratively Pq=Pq=ΦΦ
• At each iteration evaluate At each iteration evaluate matrix- vector products Pq:matrix- vector products Pq:
(1) Project charges into a 3D (1) Project charges into a 3D gridgrid
(2) calculate grid potentials (2) calculate grid potentials – it is a convolution use FFT. it is a convolution use FFT.
(3) Interpolate potentials from (3) Interpolate potentials from gridgrid
• but calculate close but calculate close interactions directly (pre-interactions directly (pre-correction step) correction step)
(1(1))
(2)(2)(3)(3)
Picture by J. Phillips
![Page 70: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/70.jpg)
70
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 71: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/71.jpg)
71
Frequency domain integral equation methodFrequency domain integral equation methodEMQS Mixed Potential Integral EquationEMQS Mixed Potential Integral Equation
)()(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
![Page 72: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/72.jpg)
72
EMQS Basis functions + GalerkinEMQS Basis functions + Galerkin
• Standard numerical procedure: discretize volumes and Standard numerical procedure: discretize volumes and charges into piecewise constant basis functions.charges into piecewise constant basis functions.
thin volume thin volume filamentsfilamentswith constant with constant currentcurrent
small surface small surface panelspanelswith constant with constant chargecharge
• Substitute and use Collocation or Galerkin to get branch Substitute and use Collocation or Galerkin to get branch equationsequations
c
c
c
c
c
cc VqI
PLjR
00
m
mmj
jj qvI )()()()( rrrwrJ
![Page 73: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/73.jpg)
73
Mesh AnalysisMesh Analysis[Kamon et al. Trans Packaging98][Kamon et al. Trans Packaging98]
msmT
c
cc
VIMj
PLjR
M
0
0
Imposing current conservation with Imposing current conservation with mesh analysismesh analysis
c
c
c
c
c
cc VqI
PLjR
00
Branch equationsBranch equations
![Page 74: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/74.jpg)
74
EMQS, mesh analysis, EMQS, mesh analysis, PreconditionersPreconditioners
• A Preconditioner must beA Preconditioner must be– a good approximation of a good approximation of
the inverse of the matrixthe inverse of the matrix
– easy to calculateeasy to calculate
msmT
c
cc
VIMj
PLjR
M
0
0
1
0
0diag
T
c
cc
Mj
PLjR
M
1
0
0 diag
T
c
cc
Mj
PLjR
M
Jacobi PreconditionerJacobi Preconditioner
A better choiceA better choice
M is sparse
![Page 75: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/75.jpg)
75
GMRES convergence with different preconditionersGMRES convergence with different preconditioners
![Page 76: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/76.jpg)
76
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 77: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/77.jpg)
77
Parasitic Extraction (the two steps)Parasitic Extraction (the two steps)
ElectromagneticAnalysis (Tuesday)
million of elements
thin volume thin volume filamentsfilamentswith constant with constant currentcurrent
small surface small surface panelspanelswith constant with constant chargecharge
Model OrderReduction(Today)
tens of elements
![Page 78: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/78.jpg)
78
Parasitic ExtractionParasitic Extraction
Parasitic Extraction
thousands of wirese.g. critical pathe.g. gnd/vdd grid
tens of circuitelements for gate level spice simulation
• identify some ports• produce equivalent circuit that
models response of wires at those ports
![Page 79: Parasitic Extraction Luca Daniel University of California, Berkeley Massachusetts Institute of Technology with contributions from: Alessandra Nardi, University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649e4e5503460f94b4423c/html5/thumbnails/79.jpg)
79
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.