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TRANSCRIPT
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University of HoustonCullen College of Engineering
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ing Design/Modeling for Periodic
N St t f EMC/EMI
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pute
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in
Ji Chen
Nano Structures for EMC/EMI
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Department of Electrical and Computer Engineering
University of HoustonHouston, TX, 77204
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Outline• Introduction
• Composite Materials Design with Numerical
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Com
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in • Composite Materials Design with Numerical Mixing-Law
• FDTD design of Nano-scale FSS
• Stochastic analysis
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• Conclusions
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IntroductionElectromagnetic Compatibility /Electromagnetic Interference
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http://en.wikipedia.org/wiki/Electromagnetic_compatibility
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ing Shielding
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NASA JSC Nanotubehttp://research.jsc.nasa.gov/BiennialResearchReport/PDF/Eng-8.pdf http://www.ursi.org/Proceedings/ProcGA05/pdf/E01.3(01681).pdf
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Shielding Materials with Periodic Structurestr
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650nm P itch450nm length
100nm A rm s
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FSS
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Composite Materials with Numerical Mixing-Law
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Maxwell-Garnett equation
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q
∑
∑
=
=
+−
−
+−
+= N
k ek
ekk
N
k ek
ekk
e
f
f
1
1eeff
21
23
εεεεεεεε
εεε
L
L
L
L
+++−
−++
+++−
++−=
+−
)2(/)()(2)2(
)2(/)()2()(
212
31
3212
11
1
31
321
11
eff
eff
εεεεεεεε
εεεεεεεε
εεεε
aa
aa
fee
e
eee
e
e
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Composite Materials with Numerical Mixing-Lawtr
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Periodic Composite Material 3D Periodical Structure
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7Unit Element
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FDMFDMZmax
V
PBC
V
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Xmax
Ymax
Voltage
PBC
PBC
PBC
Voltage
Voltage
Voltage
PBC: periodic boundary condition
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PBC: periodic boundary condition
Unit Element
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Effect of Effect of IInclusion nclusion EElectrical lectrical PPropertiesropertiestr
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_
_ _ _
=1cm, 1,
3, 1r host
rx inclusion ry inclusion rz inclusion
ε
ε ε ε
=
= = =
l
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Effect of Effect of IInclusion nclusion EElectrical lectrical PPropertiesroperties
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_
_ _ _
=1cm, 1,
3 , 1r host
rx inclusion ry inclusion rz inclusion
ε
ε ε ε
=
= = =
l _
_ _ _
=1cm, 1,
30, 3r host
rx inclusion ry inclusion rz inclusion
ε
ε ε ε
=
= = =
l
30 3
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Electrical Properties Versus FrequencyElectrical Properties Versus Frequency
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Cubic InclusionCubic Inclusiontr
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Vf=0.764
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Numerical Results Versus FrequencyNumerical Results Versus Frequency
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(1 )eff incl f host fV Vε ε ε= × + × −
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Optimization for Multiphase Mixture
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volume fractions 0.1.
espr: 4.42 and 4.04
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Conductivity : 0.00715 S/m and 1.369 S/m
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FDTD Modeling
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Challenge in the Modeling of IR FSSsChallenge in the Modeling of IR FSSs
PEC assumption is not valid any more
The metal is highly frequency-dependent no
Lorentz-Drude Model (gold)
( ) ( ) ( )f bε ω ε ω ε ω+
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in dependent nowIt has both negative permittivity and conductive lossBut all of the tradition microwave designs are based on this assumption.
FDTD Modeling for Periodic Structures
( ) ( ) ( )
( ) ( )2 2
2 210
= 1
fr r r
kp i p
i i i
fj j
ε ω ε ω ε ω
ωω ω ω ω ω=
= +
⎛ ⎞⎛ ⎞Ω⎜ ⎟− +⎜ ⎟⎜ ⎟ ⎜ ⎟− Γ − + Γ⎝ ⎠ ⎝ ⎠∑
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Finite-Sized Electromagnetic Source
?PBC PBC
?
z zk0
kx
kz
“Brute Force”
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Plane Wave Incidence Finite Size Source Incidence
Plane wave incidence
a ax x
( ) ( ) ( ) ( ), , xjk aE x a H x a E x H x e−⎡ ⎤ ⎡ ⎤+ + =⎣ ⎦ ⎣ ⎦
….…. ….….
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Periodic boundary condition (PBC) can be applied for above equation in both frequency and time domains
Finite size source incidence
Assumption is no longer validIn time domain, “Brute Force” FDTD simulation is needed
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ASM-FDTD Methodz z( , , )
th unit cell
ntot x na y t
n+E
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in ….a x
….a x
….….
Spectral domain transformation of the finite size source
The field in the 0th unit cell excited by this source can be represented as
( ) ( ) ( ) ( )0 0 0 0, , , xjk nai ix
nJ x y k J x y x x na y y eδ δ
∞−
=−∞
= − − −∑% ( ) ( )/
0 0/
,2
ai i
x xa
aJ x y J k dkπ
ππ −
= ∫ %
/0 0( ) ( )
aa k dkπ
∫E E
n=0 n=1 n=2n=-2 n=-1
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The 0th unit cell spectral domain solution can be obtained by FDTD simulation when following PBC is applied
The field in nth cell is found by
0 0
/
( , , ) ( , , )2tot tot x x
a
x y t k y t dkππ −
= ∫E E
0 0( , , ) (0, , ) xjk atot tota y t y t e−=E E
0 ( , , )tot xk y tE
/0
/
( , , ) ( , , )2
x
ajk nan
tot tot x xa
ax na y t k y t e dkπ
ππ−
−
+ = ∫E E
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Numerical Examples
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dipole source 45 mm above the structure
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dipole source 45 mm above the structuredipole strip 12 mm by 3 mmperiodicity 15 mm in both directionsfield sampled 90 mm under the dipole source
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Different Model Impacts On The Final ResultDifferent Model Impacts On The Final Resulttr
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Simulated transmission intensity of an Au cross slot array on the quartz substrate (a=650nm, L=500nm, W=110nm, thickness=300nm and substrate dielectric constant is 2.1316
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Three Practical Patterns I: Standard CaseThree Practical Patterns I: Standard Case
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Simulated transmission intensity of an Au cross slot array on the quartz substrate (a=660nm, L=510nm, W=160nm, thickness=220nm and substrate dielectric constant is 2.1316
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Five Practical Patterns II: Faced Centered CaseFive Practical Patterns II: Faced Centered Casetr
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Simulated transmission intensity of an Au cross slot array on the quartz substrate (a=660nm, L=500nm, W=170nm, thickness=220nm and substrate dielectric constant is 2.1316
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Variation quantification
Composite mixtures have inherent randomness
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The homogenized EM property needs to be evaluated
Sources of randomness includes-- material electrical properties
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-- component geometry
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Variation quantification techniques
Monte-Carlo (MC) : Simple to implement, computationally expensive
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Perturbation: Limited to small fluctuation
Stochastic collocation method (SCM): Can handle large fluctuations highly efficient transforms the stochastic analysis into a
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fluctuations, highly efficient, transforms the stochastic analysis into a series of deterministic simulations
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Suppose we have N random parameters
– we use the abbreviation
Introduction to SCM
1{ }Nn nξ =1{ }Nn nξ =1{ }Nn nξ =
1{ }Nn nξ =
1 2{ , ,..., }Nξ ξ ξ ξ=r
1 2{ , ,..., }Nξ ξ ξ ξ=r
( )
( )n nρ ξ
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in – the parameters could be distributed according to a joint PDF
– each could be distributed independently according to its probability density function (PDF)
1{ }Nn nξ =
( )ρ ξr
nξ( )n nρ ξ
( )1
( )N
n nn
ρ ξ ρ ξ=
=∏r
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Realization = a output from the deterministic simulation tool for a specific choice of
( )f ξr
1{ }Nn nξ ξ ==
r
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Introduction to SCM (cont.)One may be interested in statistics of outputs
– average or expected value
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– variance
[ ] ( ) ( )E f f dξ ρ ξ ξΓ
= ∫r r r
[ ]
( )2 2
22
[ ] [ ]
( ) ( ) ( ) ( )
Var f
E f E f
f d f dξ ρ ξ ξ ξ ρ ξ ξ
= −
∫ ∫r r r r r r
( ( )) ( )G f dξ ρ ξ ξΓ∫
r r r
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– higher moments
( )( ) ( ) ( ) ( )f d f dξ ρ ξ ξ ξ ρ ξ ξΓ Γ
= −∫ ∫
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Integrals of the type
Introduction to SCM (cont.)
( ( )) ( )G f dξ ρ ξ ξΓ∫
r r r
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in cannot, in general, be evaluated exactly
Thus, these integrals are approximated using a quadrature rule
1
( ( )) ( ) ( ) (( ( )))Q
q q qq
G f d G fξ ρ ξ ξ ω ρ ξ ξΓ
=
=∑∫r r r r r
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for some choice of quadrature weights
andquadrature points
q
1{ }Qq qω =
1{ }Qq qξ =
r
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Introduction to SCM (cont.)To use such a rule, one needs to know the simulation output at each of the quadrature points
-- for this purpose, one can build a polynomial approximation
( )f ξr
{ } 1
Qq qξ
=
r
( ( ))G f ξr
%
( ( ))G f ξr
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pute
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in and then evaluate that approximation at the quadrature points
-- the simplest means of doing this is to use the set of Lagrange interpolation polynomials corresponding to the sample points( ){ }
1
sampleNs s
L ξ=
r
( ) ( ) ( )( ) ( )sampleN
s sG f G f Lξ ξ ξ= ∑r r
%
{ } 1
sampleNs sξ
=
r
1
( ( )) ( )
( ) (( ( )))Q
q q qq
G f d
G f
ξ ρ ξ ξ
ω ρ ξ ξ
Γ
=
=
∫
∑
r r r
r r
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Then we have the approximation for the mean:
( ) ( ) ( )1
( ) ( )s ss
G f G f Lξ ξ ξ=∑
[ ] ( )1 1
( ) ( ) ( ) ( )sampleN Q
s q s qqs q
E f f d f Lξ ρ ξ ξ ξ ω ρ ξ ξΓ
= =
= = ∑ ∑∫r r rr r r
( )
1
1 1
( ( )) ( )sample
q
N Q
s q s qqs q
G f Lξ ω ρ ξ ξ
=
= =
= ∑ ∑rr r
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i'ε
_
= 1.0 m,
1, 0
= =r host host
x y z μ
ε σ= =
l l l
Numerical example
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_
[ ] : 15%[ 8, [] ] 1r inclusion inclusion
E VFE Eε σ= =
Mean values:
Goal: Evaluating the global sensitivity of effective permittivity
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g g y p ydue to variation in certain mixing parameters
Varying parameters: inclusion relative permittivity , inclusion conductivity and volume fraction; all of which are assumed to be Gaussian variables.
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Convergence Test
Operating Frequency: 1e10 Hz
Variable: inclusion permittivity, has a variance around its mean value30%±
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mean Standard deviation
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The Effect of Inclusion Relative Permittivity Variation
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_ 8r inclusionε =
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This case involves a variance of 30% around the mean value
_ 8r inclusionε =
_0
hostc r host jσε ε
ωε= −
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The Effect of Inclusion Conductivity Variationtr
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σ
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This case involves a variance of 30% around the mean value 1inclusionσ =
_0
inclusionc r inclusion jσε ε
ωε= −
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The Effect of Volume Fraction Variation
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This case involves a variance of 30% around the mean value % 15%VF =
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The Effect of Volume Fraction Variationtr
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This case involves a variance of 30% around the mean value % 15%VF =
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Conclusion
• New FDTD methods for periodic structures
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in • Applications include– FSS– Meta-materials– Nano-scale devices
• Multi layer periodic structures at different
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• Multi-layer periodic structures at different periodicities