computational electromagnetics for circuits simulations
TRANSCRIPT
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ComputationalComputational ElectromagneticsElectromagnetics for Circuits for Circuits SimulationsSimulations----the Challengesthe Challenges
W.C. Chew,W.C. Chew,11 M.K. Li,M.K. Li,11 and L.J. Jiangand L.J. Jiang22
11CCEML, Department of Electrical and Computer EngineeringCCEML, Department of Electrical and Computer EngineeringUniversity of Illinois, Urbana, IL 61801University of Illinois, Urbana, IL 61801--29912991
22IBM TJ Watson Research Center, Yorktown Heights, NY 10598IBM TJ Watson Research Center, Yorktown Heights, NY 10598
Future Direction in IC and Packaging Design WorkshopFuture Direction in IC and Packaging Design WorkshopEPEP 2005EPEP 2005
October 23, 2005October 23, 2005Austin, TXAustin, TX
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Acknowledgements
JunshengJunsheng Zhao, Chris Pan, Zhao, Chris Pan, Yunhui ChuYunhui Chu, I, I--Ting Chiang, Yuan Liu, Ting Chiang, Yuan Liu, ZhiguoZhiguo QianQian, Andrew , Andrew HesfordHesford, Greg Sorenson, Shinichiro , Greg Sorenson, Shinichiro OhnukiOhnuki, , HsuehHsueh--Yung Yung ChaoChao
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Outline
Solving Maxwell’s equation in the “twilight Solving Maxwell’s equation in the “twilight zone”?zone”?Solving Solving Laplace’s Laplace’s EquationEquation
–– Center of Charge MethodCenter of Charge Method–– Layered Medium Modeling Layered Medium Modeling
Solving problems in the lowSolving problems in the low--frequency regimefrequency regimeContact region modelingContact region modelingA new broadband fast solverA new broadband fast solverEquivalence Principle Algorithm (EPAL)Equivalence Principle Algorithm (EPAL)ConclusionsConclusions
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Electromagnetics at Low Frequency
Decoupling of Electric and Magnetic Fields Decoupling of Electric and Magnetic Fields at DCat DCThe Maxwell’s equations can be written at zero frequency as follows:
It is easy to see that the electric and magnetic fields are decoupled at DC and the current can be decomposed as
∇× = ∇⋅ = = ∇⋅
∇ × = ∇⋅ =→
E E J
H J H
0
00
, lim /
,
ε ρ ω
µω
i
,sol irr= +J J J
divergence-free
H
curl-free
E
( ), , 0irr sol irrO ω ω>> →J J J∼
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Another POV
Helmholtz Helmholtz system when k tends to zero:system when k tends to zero:
( )2 2
2
k s
s
φ
φ
∇ + =
∇ =Electromagnetic system when k tends to Electromagnetic system when k tends to zero:zero:
2k ii
ωµωµ
∇×∇× − =∇×∇× =
E E JE J
A singular perturbation problem!A singular perturbation problem!
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Multi-Scale Problem--Vertically
MultiMulti--scalescaleDC to tens of GHzDC to tens of GHzStable and fast solversStable and fast solvers
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Static Problems—Laplace’s Equation
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Some Popular Methods
Random walkRandom walkStatic Fast Static Fast Multipole Multipole for Homogeneous Mediafor Homogeneous Media
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Center of Charge Method (CCM) for Laplace’s Equation
First OrderFirst Order
Second OrderSecond Order
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CCM CPU Time
CCM can be as much as 10 times faster than FASTCAP type CCM can be as much as 10 times faster than FASTCAP type algorithmalgorithm
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Multiple-Reflections in the Multilayer Green’s Function
(Laplace Problem)
Outgoing to local Outgoing to local multipolemultipoletranslatortranslator
Closed form of the outgoing Closed form of the outgoing to local to local multipolemultipole translatortranslator
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Multiple-Reflection in the Multilayers (Laplace Problem)
Special box positions for the outgoing to local translatorSpecial box positions for the outgoing to local translatorThe substrate parameter is still used for the translationThe substrate parameter is still used for the translationClosed forms of the translators are still validClosed forms of the translators are still valid
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Solving Laplace’s Equation(1,204,352 unknowns)
Uses 1.41 GB of MemoryUses 1.41 GB of Memory
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Six Coupling Lines—Acceleration with Discrete Complex Image Method (DCIM)
Charge Distribution of the Six Coupling LinesCharge Distribution of the Six Coupling Lines
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Two Signal Lines—DCIM Acceleration
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Solving Low Frequency Problems
Involves full Maxwell’s equationsInvolves full Maxwell’s equationsIn In electromagneticselectromagnetics, LF is decided by , LF is decided by
size/wavelength ratiosize/wavelength ratio
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Low Frequency—Quasi-Helmholtz Decomposition
– Loop Basis: divergence free
– Star Basis: quasi-curl-free.
– Tree Basis: RWG basis with the basis along a cut removed.The cut prevents the rest of the
RWG basis (tree basis) from forming any loop.
LS or LT formulation isolates the contribution of vector potential and scalar potential.
Information of vector potential will not be lost due to machine precision.
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Acceleration Methods in Low Frequency
FFTFFTFast Fast Multipole Multipole AlgorithmAlgorithmStabilization is necessary for the fast Stabilization is necessary for the fast multipole multipole methodmethod
–– Stabilized via frequency normalization techniqueStabilized via frequency normalization technique
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Large-Scale Low-Frequency Computation with LF-FMA Acceleration
A PEC Sphere with Over One Million A PEC Sphere with Over One Million RWG UnknownsRWG Unknowns
CurrentCurrent
ChargeCharge
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Hertzian Dipole from Zero to Microwave Frequencies
Input admittance/impedance of a Input admittance/impedance of a Hertzian Hertzian dipole at very low dipole at very low frequencies and at higher frequenciesfrequencies and at higher frequencies
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Low-Frequency Fast Multipole Algorithm
Wire Spiral Inductor with Ground PlaneWire Spiral Inductor with Ground Plane
Frequency (Hz)
Input susceptance (1/Ohm)12 TurnsRadius of wire = 2.76 µmTrack spacing = 24 µmDiameter of the inductor = 680 µm
µm
µm
250µmInductance (nH)
Frequency (Hz)
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CrossCross--talk Simulation Suggested by talk Simulation Suggested by MazumderMazumder at Intelat Intel
Setup for the SimulationSetup for the Simulation
+
-1
3
2
4
Resistor: 50 (ohm)Capacitance: 50E-15 FaradRadius of the wire: 0.4 umLength of the wires: 2000 umDistant between axes of wires: 1.6 umConductivity of the wire: 2.38E7Conductivity of the plate: 2.893E7Thickness of the plate: 2.0 umHeight of the wire: 60umRising/falling time: 50 ps
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CrossCross--talk Simulationtalk Simulation
Current and Voltage at Port 1Current and Voltage at Port 1
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Example 3: a more complicated crisscross line structure with impedance load
Resistor: 50 (ohm)Side width of the wire: 2 umLength of the wires: 200 umDistant between axes of wires: 4 umHeight of the axes of upper layer wires: 11 umHeight of the axes of upper layer wires: 15 umRising/falling time: 20 ps
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Port definition
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Electric current and electric charge distribution at 10 GHz
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Electric current and electric chargedistribution at 200 GHz
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Voltage response at port 10 and port 30 when port 9 is excited
0 200 400 600-0.5
0
0.5
1
1.5
time ( ps )
voltage on port 10
0 200 400 600-4
-2
0
2
4x 10
-4
time ( ps )
voltage on port 30
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Voltage response at port 30 when all the driven ports of the higher layer are excited
0 200 400 600-0.5
0
0.5
1
1.5
time ( ps )
voltage on port 10
0 200 400 600-2
-1
0
1
2x 10
-3
time ( ps )
voltage on port 30
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Surface Integral Equation (SIE) Multi-Region Problem
ContactContact--Region Modeling and PMCHWT Formulation Region Modeling and PMCHWT Formulation
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CRM--Numerical Results
An Inductance StructureAn Inductance Structure
Input ReactanceInput Reactance
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CRM--Numerical Results
Validation of CRMValidation of CRM
Inconsistent meshInconsistent mesh1980 unknowns1980 unknowns
Consistent meshConsistent mesh1986 unknowns1986 unknowns
Inconsistent meshInconsistent mesh32172 unknowns32172 unknowns
Electric Electric ChargeCharge
Magnetic Magnetic CurrentCurrent
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CRM--Numerical Results
Inconsistent meshInconsistent mesh1980 unknowns1980 unknowns
Consistent meshConsistent mesh1986 unknowns1986 unknowns
Inconsistent meshInconsistent mesh32172 unknowns32172 unknowns
Electric Electric CurrentCurrent
Magnetic Magnetic ChargeCharge
Total CPU Total CPU TimeTime 38 min.38 min. 41 min.41 min. 858 min.858 min.
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Large-Scale Low-Frequency Computation (cont’d)
OnOn--Chip Spiral InductorChip Spiral InductorCurrent Magnitude Distribution in Current Magnitude Distribution in Logarithmic Scale at 1 GHz with Logarithmic Scale at 1 GHz with
91,485 RWG Unknowns91,485 RWG UnknownsD=95 um, w = 10 um, t=3 um, s = 5 umD=95 um, w = 10 um, t=3 um, s = 5 um
Computed on a Sun Blade 2000 (1 CPU)Computed on a Sun Blade 2000 (1 CPU)Total CPU Time: 5.25 hoursTotal CPU Time: 5.25 hours
Total Memory Usage: 599 MbTotal Memory Usage: 599 Mb
Imaginary Part of Input Impedance Imaginary Part of Input Impedance with 9,894 RWG Unknownswith 9,894 RWG Unknowns
CPU Time/Freq. Pt. : 25 minutesCPU Time/Freq. Pt. : 25 minutesTotal Memory Usage: 155 MbTotal Memory Usage: 155 Mb
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Numerical Results (cont’d)
939,897 Unknowns939,897 Unknowns
CurrentCurrent ChargeCharge
7 x 9 Array7 x 9 Array
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Mixed Form Fast Multipole Algorithm
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Full Band MF-FMA
7 x 7 fork structure7 x 7 fork structure
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Full Band MF-FMA
Incident Wave: 1 MHzIncident Wave: 1 MHz
θθ=45deg=45deg
ΦΦ=45deg=45deg
No of triangles: 487,354No of triangles: 487,354
No of unknowns: 731,031No of unknowns: 731,031
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One Example of Equivalence Principle Algorithm
1incJ1
incM
1scaM 1
scaJ
2incJ
2incM
2scaM 2
scaJ
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One Example of Equivalence Principle Algorithm (cont’d)
1incJ1
incM
1scaM 1
scaJ
2incJ
2incM
2scaM 2
scaJ
1
2
0.5
0.5
0.5
0.6
0.4
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One Example of Equivalence Principle Algorithm (cont’d)
Frequency:Frequency: 100 MHz100 MHz
Left Conductor:Left Conductor: 0.5 x 0.5 x 0.06 (m)0.5 x 0.5 x 0.06 (m) 306 Unknowns (84/λ)306 Unknowns (84/λ)
Right Conductor:Right Conductor: 0.5 x 0.6 x 0.04 (m)0.5 x 0.6 x 0.04 (m) 372 Unknowns (142/λ)372 Unknowns (142/λ)
Huygens’ Surface:Huygens’ Surface: 1.0 x 1.0 x 1.0 (m)1.0 x 1.0 x 1.0 (m) 144 Unknowns (25/λ)144 Unknowns (25/λ)
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One Example of Equivalence Principle Algorithm (cont’d)
Frequency:Frequency: 0.1 GHZ0.1 GHZ
Incident wave:Incident wave: θ=0 degreeθ=0 degree
Φ=0 degreeΦ=0 degree
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Second Example ofEquivalence Principle Algorithm (cont’d)
Frequency:Frequency: 100 MHz100 MHz
Coil Conductor:Coil Conductor: 0.5 x 0.5 x 0.06 (m)0.5 x 0.5 x 0.06 (m) 306 Unknowns (84/λ)306 Unknowns (84/λ)
Fork Conductor:Fork Conductor: 0.5 x 0.6 x 0.04 (m)0.5 x 0.6 x 0.04 (m) 372 Unknowns (142/λ)372 Unknowns (142/λ)
Huygens’ Surface:Huygens’ Surface: 1.0 x 1.0 x 1.0 (m)1.0 x 1.0 x 1.0 (m) 144 Unknowns (25/λ)144 Unknowns (25/λ)
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Second Example of Equivalence Principle Algorithm (cont’d)
Frequency:Frequency: 0.1 GHZ0.1 GHZ
Incident wave:Incident wave: θ=0 degreeθ=0 degree
Φ=0 degreeΦ=0 degree
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Antenna on Vehicle---Geometry and MeshParameters for Substrate and Radome
Radome Substrate
GHzjZmm.d
TDS
r
3325.2@ 2.4823 016.2
−===ε
mmmmd
r
74.0Probe ofDiameter 0.47.21
===ε
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Animation of Current Distribution on Car
Antenna Antenna generates a generates a circularly circularly polarized field polarized field and current and current on rooftop of on rooftop of the car.the car.2 GHz antenna 2 GHz antenna on a 4.25 m on a 4.25 m size car is size car is equivalent to equivalent to 100 GHz on a 100 GHz on a 8 cm board8 cm board
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Conclusions
Need for easy modeling of complex structures.Need for easy modeling of complex structures.Contact region modeling allows for the inclusion of Contact region modeling allows for the inclusion of materials in low frequency modeling.materials in low frequency modeling.MultiscaleMultiscale problems are the challenge problems of the problems are the challenge problems of the future.future.Circuit physics and wave physics have to be captured Circuit physics and wave physics have to be captured simultaneously in a simulation.simultaneously in a simulation.Mixed form FMA a new fast algorithm, will allow a Mixed form FMA a new fast algorithm, will allow a seamless way to simulate structures all the way from seamless way to simulate structures all the way from static to microwave.static to microwave.Equivalence principle algorithm (EPAL) allows the Equivalence principle algorithm (EPAL) allows the decoupling of circuit physics from wave physics.decoupling of circuit physics from wave physics.