cullen college of engineering university of houstonweb.mst.edu/~jfan/slides/chen2.pdf ·...

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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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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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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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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

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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


ε

ε ε ε

=

= = =

l _

_ _ _

=1cm, 1,

30, 3r host


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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Five Practical Patterns II: Faced Centered CaseFive Practical Patterns II: Faced Centered Casetr

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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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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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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

cullen college of engineering university of houstonweb.mst.edu/~jfan/slides/chen2.pdf ·...

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