Electromagnetic properties of nanostructured materials · Electromagnetic properties of nanostructured materials Ian D. Flintoft, ... In Section 3 the design and testing of a coaxial

University of York 1 10 July 2015

Electromagnetic properties of nanostructured materials

Ian D Flintoft John F Dawson Iain G Will Linda Dawson

Physical Layer Research Group Department of Electronics University of York

10th July 2015

Contents1 Introduction 3

2 Electromagnetic parameters of materials 3

21 Dielectric properties 4

22 Magnetic properties 5

23 Reflection and transmission at a plane interface 6

24 Reflection and transmission of a TEM wave from a slab 10

25 Reflection and transmission of a TETM wave from a slab in a waveguide 14

26 Contributions to the sample transmission 14

27 Parameter extraction methods 17

271 Nicholson-Ross-Weir parameter extraction 17

272 Thin slab limit 24

273 The effective parameter method 26

274 Sample position and thickness independent methods 28

275 NIST iterative inversion 29

276 MATLAB implementation and experiment simulation 32

3 A coaxial measurement jig 33

31 Design of a coaxial measurement jig 33

32 Coaxial jig test sample manufacture 38

33 Coaxial jig test results 40

4 Nanoparticle film and material fabrication 47

5 Conclusions and Plans for future work 48

51 Material characterisation capability 48

52 Application of nano-particle films 48

53 Proposals and collaboration 49

6 References 49

7 Bibliography 50



Glossary

CFC Carbon Fibre Composite

CNT Carbon Nano-Tube

DC Direct Current

EMT Effective Medium Theory

GA Genetic Algorithm

NRW Nicholson-Ross-Weir

PTFE Polytetrafluoroethylene

RF Radio Frequency

SE Shielding Effectiveness

SMA Sub-miniature Version A

SNR Signal-to-Noise Ratio

SWCNT Single Walled Carbon Nano-Tube

TE Transverse Electric

TEM Transverse Electromagnetic

TM Transverse Magnetic

VNA Vector Network Analyser



1 Introduction

This report summarises the progress to date of work funded by the University of York Research

Priming Fund on ldquoElectromagnetic properties of nanostructured materialsrdquo

The inclusion of nanoscale structures within materials has the potential to provide enhanced or

customised electromagnetic properties Applications include shielding of sensitive systems from

electromagnetic radiation (eg as widely used in the electronics and aerospace industries) and the

ability to allow some radio signals to pass whilst preventing others (eg privacy and security

applications) The work has allowed the Physical Layer Research Group to develop its measurement

capability so that the electromagnetic properties of nanostructured materials can be measured and

also to begin working on materials by fabricating a range of nano-particle based films

In Section 2 a review of the electromagnetic properties of materials and the extraction of

electromagnetic parameters from various measurements is discussed The software developed for

parameter extraction is then summarised In Section 3 the design and testing of a coaxial shielding

measurement jig is described In Section 4 the potential nano-structured materials that could be

fabricated to address currently funded research application areas are identified Section 5 provides

some concluding remarks and describes future and ongoing work in this area

2 Electromagnetic parameters of materials

Functionally the electromagnetic performance of materials are often specified in terms of their

transmission and reflection coefficients which define the amplitude and phase of plane

electromagnetic waves that propagate through and are reflected from a sample respectively The

proportion of the incident electromagnetic energy that is absorbed within the material (ie neither

reflected nor transmitted) may also be an important consideration in many application areas These

properties of the material depend on both its internal micro-structure and the overall macroscopic

shape of the material For the purposes of material characterisation a planar sample is often used

since the electromagnetic scattering and transmission from an infinite plane sheet of homogeneous

material is amenable to exact theoretical analysis thereby providing a solid reference for validation

of measurement procedures and numerical simulations

If probed by sufficiently high frequency electromagnetic waves the micro-structure of the material

can be discerned in the characteristics of the scattered waves This applies at different length scales

associated with the physical structure of the material The shortest length scale of interest

corresponds to the nano-scale level in which the atomic structure can be resolved Most

engineering materials have structure at larger length scales associated with their fabrication and

application For example carbon-fibre reinforced composites (CFCs) consist of filaments of carbon

with diameters of the order micro-meters embedded in a dielectric resin The microstructure

therefore has apertures between the fibres with a similar length scale that influences the

electromagnetic properties of the composite even at low frequencies

In many applications the frequencies of interest are far below the length-scale of the microstructure

and a homogenised view of the material can be taken averaging the electromagnetic fields at the

surface of the material to give simpler effective parameters for the material properties This

approach is called Effective Medium Theory (EMT) This is essentially a generalisation of the classical

macroscopic treatment of dielectric and magnetic materials in Maxwellrsquos electromagnetic theory by



averaging over the atomic level electric and magnetic dipole moments caused by the induced charge

movements in the material when subjected to an external electromagnetic field This leads to the

definition of the classical electric permittivity and magnetic permeability of the material

In this chapter a brief review of the electromagnetic parameters is presented followed by a review

and implementation of the different methodologies that can be employed to extract material

parameters from different measurement system including the coaxial shielding jig described in

Section 3

21 Dielectric properties

Linear isotropic materials can be described by the introducing the electric flux density D that is

related to the electric field E via the complex permittivity ()Ƹߝ

۲() = ()Ƹ()۳ߝ = ε۳() + () = ε۳() + εχොக()۳()

Here ε is the permittivity of free-space P the electric polarisation vector and χොக() the electricsusceptibility The complex permittivity can be decomposed into real and imaginary parts

()Ƹߝ = minusᇱߝ jεᇱᇱ≝ Ƹ()εߝ

where the relative permittivity is

()Ƹߝ = εᇱminus jε

ᇱᇱ= 1 + χොக()ߝ

The real part characterises energy storage in material while the imaginary part characterisesloss due to the bound charge movement The loss tangent is defined ratio of energy lost percycle to energy stored per cycle and given by

tanߜக =εᇱᇱ

εᇱ

allowing the relative permittivity to be written

()Ƹߝ = εᇱ(1 minus j tanߜக)

Some materials have free ionic charges that respond to a DC electric field by producing aconduction current

۸() = ()ୈେ۳ߪ

The overall effective relative permittivity is then

()Ƹߝ = εᇱminus j൬ε

ᇱᇱ+ୈେߪε

൰

In real materials there are often partially bound charges which blur the distinction betweendielectric loss and conduction effects The AC conductivity may be written [Bake2005]

ߪ = minusᇱߪ jߪᇱᇱasymp ୈେߪ minus jߪᇱᇱ

The overall dielectric response is then characterised the a complex relative permittivity

()Ƹߝ = εᇱminus

ᇱᇱߪ

εminus jቆε

ᇱᇱ+ᇱߪ

εቇ



In order for the material to have a causal time response the complex permittivity must

satisfy the Kramers-Kronig relations

εᇱ() minus ஶߝ = minus

2

ߨන

εߠᇱᇱ(ߠ)minus ε

ᇱᇱ()

minusଶߠ ଶdߠ

ஶ

εᇱᇱ() = minus

2

ߨන

εᇱ(ߠ) minus ε

ᇱ()

minusଶߠ ଶdߠ

ஶ

These can also be written in ldquoonce subtracted formrdquo

εᇱ() minus ε

ᇱ() = minus minus

ߨPVන

εᇱᇱ(ߠ)

minusߠ) minusߠ)( )dߠ

ஶ

ஶ

εᇱᇱ() minus ε

ᇱᇱ() = minus

ߨPVන

εᇱ(ߠ)


ஶ

Various models for the frequency response of a material have been developed- some arelisted in Table 1

Model Dispersion Relation

Drude σ =

ଶ

1

ߛ + j=

ଶ

minusߛ j

ߛଶ + ଶ

Debye()ߝ = ஶߝ +

ߝ∆

1 + j

Lorentz ()ߝ = ஶߝ +ఌଶ

ఌଶ + Γఌω minus ଶ

Cole-Cole[Gabr1996] ()ߝ = ஶߝ +

ߝ∆1 + (j )ଵఈ

+ୈେߪjε

Table 1 Common dispersion models for dielectrics and conductors

22 Magnetic properties

In a completely analogous way the magnetic permeability of a material can be defined by

()Ƹߤ = minusᇱߤ jߤᇱᇱ= +ߤσlowast

j≝ ߤ()Ƹߤ

where σlowast is the magnetic conductance The relative permeability is then

()Ƹߤ = ߤᇱminus jߤ

ᇱᇱ= ߤ +σlowast

jߤ



Figure 1 Interfacial transmission of TE and TM waves at z-normal boundary

23 Reflection and transmission at a plane interface

The reflection coefficient at an infinite plane interface between two different materials can be

determined from the boundary conditions on the electromagnetic field at the surface Consider a

plane-wave propagating in the z-x plane with wave vector

ܓ ൌ ௫ܠො ௭ܢො

in an isotropic medium with complex permittivity Ƹandߝ permeability Ƹߤ The wave can be

decomposed into transverse electric (TE) and transverse magnetic (TM) components

۳(ܚǢ ) ൌ Ǣܧe୨ܓήܡܚො

۶(ܚǢ ) ൌ Ǣܧ

1

Ƹߤe୨ܓήܚ(െ௭ܠො ௫ܢො)

and

۳ Ǣܚ) ) ൌ Ǣܧ e୨ܓήܚ൬ܠොെ

௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ

௭e୨ܓήܡܚොǡ

where Ǣܧ and Ǣܧ

are the transverse field amplitudes and the dispersion relation is

௫ଶ ௭

ଶ ൌ ଶߤƸߝƸǤ

If the medium is lossless then the wave-vector is real and can be written

ܓ ൌ መܓ ൌ ොܠߠ) (ොܢߠ



where is the angle between the z-axis and the wave vector and = ߝߤradic The total amplitudes of

the TE and TM fields are then related to the transverse field amplitudes by

ܧ = ܧ

ܧ = ܧ

cosߠ

The complete transverse fields can be written

۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො

where the transverse wave impedances are defined by

ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ

Writing the total transverse field as

۳ = ܧ +ොܠ ܧ

ܡො

۶ = ܪ minusොܡ ܪ

ܠො

noting the minus sign for the x component of the TE magnetic field the propagation in the medium

is reduced to two uncoupled one-dimensional transmission line problems of the form

(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ

ାe୨௭minus ܧeା୨௭൯=

ܧା(ݖ)

ߟ൫1 minus Γ(ݖ)൯

where the e୨௫ phase factor has been suppressed This is illustrated in Figure 1 The propagation

of the forward and backward waves can be described by a propagation matrix

ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0

0 e୨(௭మ௭భ)൨ܧା(ݖଶ)

ܧ(ଶݖ)

൨

while the total transverse fields propagate according to a ldquochain matrixrdquo given by



(ଵݖ)ܧ(ଵݖ)ܪ

൨= cos ௭(ݖଶminus (ଵݖ jߟ sin ௭(ݖଶminus (ଵݖ

jߟଵ sin ௭(ݖଶminus (ଵݖ cos ௭(ݖଶminus (ଵݖ

൨(ଶݖ)ܧ(ଶݖ)ܪ

൨

The electric field reflection coefficient is

Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)

and propagates according to

Γ(ݖଵ) = Γ(ݖଵ)eଶ୨(௭మ௭భ)

while the wave impedance

(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as

(ଵݖ) = ߟ(ଶݖ) cos ௭(ݖଶminus (ଵݖ + jߟ sin ௭(ݖଶminus (ଵݖ

ߟ cos ௭(ݖଶminus (ଵݖ + j(ݖଶ) sin ௭(ݖଶminus (ଵݖ

Note that

Γ(ݖ) =(ݖ) minus ߟ(ݖ) + ߟ

At a plane interface between two semi-infinite media denoted left (L) and right (R) located at z = 0

the transverse electric and magnetic fields must be continuous Matching the phase in the x-

direction at the interface leads to Snellrsquos Law

௫ = ௫

If the left medium is lossless then

௫ = sinߠ= ඥߤƸߝƸ sinߠ = ௫

where is the angle in incidence and hence from the dispersion relationships

൫ ௫൯

ଶ+ ൫ ௭

൯ଶ

= ଶߤƸߝƸ ≝ ൫ ൯ଶ

we find that

௭ = ටଶߤƸߝƸ minus ൫ ௫

൯ଶ

= ඥଶߤƸߝƸ minus ( sinߠ)ଶ

For the left medium ௭ = cosߠ Matching the electric and magnetic fields either side of the

boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ

leads to a matching matrix condition at the interface

ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ

where the interfacial reflection and transmission coefficients for incidence from the left are given by

Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ

The inverse matching condition is

ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and

Ԧ = 1 + Ԧߩ Ԧߩ = ശߩminus ശ = 1 + ശߩ = 1 minus Ԧߩ Ԧ ശ = 1 minus ଶ(Ԧߩ)

Note that the wave impedance is continuous across the interface

=ܧ

ܪ

=ܧ

ܪ

=

The reflection coefficients on either side of the boundary are related by

Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ

The scattering matrix for the interface is given by

ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ



Figure 2 Oblique incidence on a slab in terms of transverse fields

24 Reflection and transmission of a TEM wave from a slab

We now consider the reflection and transmission from a slab of material formed by two interfaces as

shown in Figure 2 The interfacial reflection and transmission coefficients for the two interfaces are

ԦǢଵߩ =Ǣଵߟ െ ǢߟǢଵߟ Ǣߟ

ԦǢଶߩ =Ǣୠߟ െ ǢଵߟǢୠߟ Ǣଵߟ

ԦǢଵ ൌ ͳ ԦǢଵߩ

ԦǢଶ ൌ ͳ ԦǢଶߩ

where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ

௭Ǣ= ටଶߤƸߝƸെ ൫ ௫Ǣ൯ଶ

= ඥଶߤƸߝƸminus (ୟߠୟ)ଶ

If the medium either side of the slab is lossless then phase matching in the x-direction gives

௫ୟ ൌ ௫

ୠ ୟߠୟ ൌ ୠߠୠ

Specifically if the medium on either side of the slab is the same

௭Ǣ = ඥଶߤୟߝୟminus (ୟߠୟ)ଶ ൌ ඥߤୟߝୟඥ1 minus ଶ(ୟߠ)



௭ଵ = ඥଶߤƸଵߝƸଵminus (ୟsinߠୟ)ଶ = ඥߤୟߝୟටߤƸଵߝƸଵminus (sinߠୟ)ଶ = ඥߤୟߝୟට ෝଵଶminus (sinߠୟ)ଶ

where ෝଵ = ඥߤƸଵߝƸଵ fraslୟߝୟߤ is the relative refractive index of the slab relative to the material either

side

The matching and propagation matrices for the two interfaces and one layer are

ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ

Ԧଵߩ 1ቈଵܧା

ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ

Ԧଶߩ 1ቈଶܧା

ଶܧ

which can be put together to give

ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e

୨ఋభ 00 e୨ఋభ

൨ቈ1 Ԧଶߩ


ଶܧ

where

≝ଵߜ ௭ଵ(ݖଶminus (ଵݖ ≝ ௭ଵ ଵ

Changing the dependent and independent variables in the linear system leads to the scattering

matrix

ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ቈଵܧା

ଶܧ=

ଵଵଵ ଵଶଵ

ଶଵଵ ଶଶଵ൨ቈଵܧା

ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where

Γ =Ԧଵߩ + Ԧଶeଶ୨ఋభߩ

1 + Ԧଶeଶ୨ఋభߩԦଵߩ

ሬ=Ԧଵ Ԧଶe୨ఋభ


Γ = minusԦଶߩ + Ԧଵeଶ୨ఋభߩ

1 + Ԧଶeଶ୨ఋభߩԦଵߩ=

ശଶߩ + ശଵeଶ୨ఋభߩ

1 + ശଶeଶ୨ఋభߩശଵߩ

ሬ=൫1 minus Ԧଵߩ

ଶ ൯

Ԧଵ

൫1 minus Ԧଵߩଶ ൯

Ԧଶ

e୨ఋభ


ശଵശଶe୨ఋభ


An alternative arrangement of the linear equations gives the transmission scattering matrix for the

slab



ቈଵܧ

ଵܧା=

1

ሬቈminusdet ଵ Γ

minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ

ଶଵଵ ଶଶଵ൨ቈଶܧ

ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1

ଶଶଵቈ ଵଶଵ det ധଵ1 minus ଶଵଵ

ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ

ଶଵଵ ଶଶଵ൨=

1

ଶଵଵቈminusdet ଵ ଵଵଵ

minus ଶଶଵ 1

ଵ = ଵଵଵ ଵଶଵ


1


det ଵ = ΓΓ minus ሬሬ


det ധଵ = ሬ

Note that for a matched section of line

det ଵ = minusሬሬ

and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1

e୨ఋభeଶ୨ఋభ 0

0 1൨= e


൨

The complex wave vector in a lossy medium can be written in terms of propagation and attenuation

coefficients as

ଵܓ = ௫ଵܠො+ ௭ܢො= minusଵࢼ jࢻଵ = ൫ߚ௫ଵminus +ොܠ௫ଵ൯ߙ ൫ߚ௭ଵminus ොܢ௭ଵ൯ߙ

subject to

ଵܓ ∙ ଵܓ = ඥߤƸଵߝƸଵ

The spatial variation of the internal fields in the slab therefore has the form

e୨௭e୨௫ = e୨൫ఉభఈభ൯௭e୨൫ఉభఈభ൯௫ = e൫ఈభ௭ାఈభ௫൯e୨൫ఉభ௭ାఉభ௫൯

The condition for zero reflection Γ rarr 0 from the slab is

Ԧଵߩ + Ԧଶeଶ୨ఋభߩ = 0

or

eଶ୨ఋభ = eଶఈభభeଶ୨ఉభభ = minusԦଵߩ

Ԧଶߩ

For a lossless slab with a lossless medium at either side at normal incidence in a TEM wave structure

Ԧଵߩ fraslԦଶߩ is real so this condition requires either



eଶ୨ఉభభ = 1 andߩԦଶ = Ԧଵߩminus

or

eଶ୨ఉభభ = minus1 andߩԦଶ = Ԧଵߩ

The first case corresponds to the slab being a multiple of a half-wavelength (in the medium) thick

and further requires the medium to be the same on either side of the slab

௭ଵߚ2 ଵ =ߨ2

ଵߣଵ = ߟandߨ2 = ߟ

The second case corresponds to the slab being a quarter-wavelength (in the medium thick) and

imposes a matching condition on the transverse impedances

௭ଵߚ2 ଵ =ଶగ

ఒభଵ = (2 + ଵߟandߨ(1

ଶ = ߟߟ

In this lossless case zero reflection requires total transmission Γ rarr 0 rArr ሬ= 1 since there is no

absorption in the slab

Now consider illumination from the left

ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା

and the reflected and transmitted power are

ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ

ୟߟ= ℛሬሬ୧୬

ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬

where the reflectance transmittance and absorbance of the sample are respectively

ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ

≝ሬ୧୬ minus ሬ

minus ሬ୲ୟ୬ୱ

ሬ୧୬

= 1 minus ℛሬminus ሬ= 1 minus หΓหଶminusୟߟ

ୠߟหሬห

ଶ



Here we have assumed that the left and right media are lossless If the left and right media are the

same then the ratio of intrinsic impedances is unity

25 Reflection and transmission of a TETM wave from a slab in a waveguide

For transverse electric (TE) and transverse magnetic (TM) waves the formulation is essentially the

same as the oblique incidence TEM case with a redefinition of transverse impedances and dispersion

relation

௭= ට ଶminus ୡ

ଶ = ටଶߤƸߝƸminus ୡଶ

ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ

Typically TE10 mode is used for material characterisation If the medium either side of the slab is the

same and lossless we have

௭ୟ = ට ୟଶminus ୡ

ଶ = ୟඥ1 minus ( ୡ ୟfrasl )ଶ = ඥߤୟߝୟඥ1 minus (ୡ frasl )ଶ =ߨ2

ୟߣඥ1 minus ୟߣ) fraslୡߣ )ଶ

where

ୡ≝ୡ

ඥߤୟߝୟ≝ ୟ ୡ≝ ୟ

ߨ2

ୡߣ

Then

௭ଵ = ටଶߤƸଵߝƸଵminus ୡଶ = ඥߤୟߝୟටߤƸଵߝƸଵminus (ୡ frasl )ଶ =

ߨ2

ୟߣට ෝଵ

ଶminus ୟߣ) fraslୡߣ )ଶ

Also for TE waves only

ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ

ඥ1 minus (ୡ frasl )ଶ=

ୟߟ

ඥ1 minus (ୡ frasl )ଶ

௭ߟ = Ƹߤ

26 Contributions to the sample transmission

The transmission through a slab can be factorised into three components due to the initial reflection

from the front face absorption through the slab and multiple reflections

ሬ= ሬ ሬୟୠୱ

ሬ୫ ୳୪୲୧

where

ሬ = 1 minus Ԧଵߩ

ଶ =ߟ4

൫ߟ + 1൯ଶ



ሬୟୠୱ = = e୨ఋభ = e୨൫ఉభఈభ൯భ = eఈభభe୨ఉభభ

ሬ୫ ୳୪୲୧=

1

1 minus Ԧଵߩଶ ଶ

= ൫ߩԦଵ൯ଶ

ஶ

ୀ

For samples with large absorption || ≪ 1 and ሬ୫ ୳୪୲୧ is small Note that the overall reflection

coefficient likewise contains three terms The first Ԧଵߩ is the reflection from the front face of the

sample the second 1 minus ଶ accounts for the initial reflection from the back face and is small if

absorption in the sample is significant The third term 1 1 minus Ԧଵߩଶ ଶfrasl is a multiple reflection term

that is only important for thin or low loss materials

For a good conductor with no magnetic losses

ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ

ଵߤminusଵߝ ଵߪ frasl

asymp (1 + )ඨଵߤߝ

ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ

where େ୳ߪ =58 MSm Taking the magnitude in decibels [Paul1992 eqn (1131)]

หሬ ห[dB] = 10 logଵ൬ߝߨ32େ୳ߪ

൰+ 10 logଵቆଵߤ

ଵߪቇ= minus16814 + 10 logଵቆ

ଵߤ

ଵߪቇ

The absorption term can be written

ሬୟୠୱ = = eఈభభe୨ఉభభ

where

minus௭ଵߚ ௭ଵߙ = ඥߤƸଵߝƸଵ = ඥߤଵ(ߝଵminus ଵߪ frasl ) asymp (1 minus )ටଵߪଵߤ

2=1 minus

ୱଵߜ

and the skin depth is

ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence

௭ଵߚ asymp ௭ଵߙ asymp1

ୱଵߜ

and

ሬୟୠୱasymp eభ ఋ౩భfrasl e୨భ ఋ౩భfrasl



or taking the magnitude in decibels [Paul1992 eqn (1132)]

ሬୟୠୱ [dB] = minus20 logଵ(e)

ଵ

ୱଵߜ= minus20 logଵ(e)ඥߤߨߪେ୳ ଵඥߤଵߪଵ= minus13143 ଵඥߤଵߪଵ

The multiple reflection terms is

ሬ୫ ୳୪୲୧=

1


=1

1 minus ൬minusߟ 1ߟ + 1

൰ଶ

eଶభ ఋ౩భfrasl eଶ୨భ ఋ౩భfrasl

asymp1

1 minus eଶభ ఋ౩భfrasl eଶ୨భ ఋ౩భfrasl

For thick samples ( ଵ≫ (ୱଵߜ we see that ሬ୫ ୳୪୲୧rarr 1 For thin conducting samples

ሬ୫ ୳୪୲୧rarr

1

1 minus (1 minus 2 (ଵߜ=

1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ

หሬ୫ ୳୪୲୧ห[dB] = minus3263 minus 10 logଵ൫ߤଵߪଵ ଵଶ൯

Note that in this limit the product

ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ

is independent of frequency and determines the DC transmission through the sample

หሬ ሬ୫ ୳୪୲୧ห[dB] = minus20077 minus 20 logଵ൫ߪଵ ଵ൯= minus4550 minus 20 logଵ(ߪଵ ଵ)

This can also be written as

หሬ ሬ୫ ୳୪୲୧ห[dB] = minus4550 + 20 logଵ൫ ୗଵ൯

where the surface resistance of the sample is

ୗଵ =1

ଵߪ ଵ

ldquoohms per squarerdquo This follows from the fact that the resistance across the ends of a thin film of

thickness ଵ width and lengthܮ is

=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ

ୀௐሱ⎯ሮ ୗଵ

The corresponding shielding effectiveness defined here as the reciprocal of the magnitude of the

transmission coefficient

SE [dB] = 4550 minus 20 logଵ൫ ୗଵ൯

is shown in Figure 3



Figure 3 DC shielding effectiveness of a thin conductive sample as a function of its surface resistance

27 Parameter extraction methods

The complex permittivity and permeability of a material can be determined from a measurement of

its complex reflection and transmission coefficient in a TEM or TETM wave measurement cell In

this section we review these techniques and present MATLAB implementations of the most

promising ones

271 Nicholson-Ross-Weir parameter extraction

The reflection and transmission coefficient of a slab in a TEM wave and TETM waveguide structure

can both be written





where the complex phase shift in the slab is

ଵߜ = ௭ଵ ଵ

For TEM waves

௭ୟ = ඥߤୟߝୟඥ1 minus (sinߠୟ)ଶఏୀሱ⎯⎯ሮ ඥߤୟߝୟ = ୟ

௭ଵ = ඥߤୟߝୟට ෝଵଶminus (sinߠୟ)ଶ

ఏୀሱ⎯⎯ሮ ෝଵඥߤୟߝୟ = ෝଵ ୟ

while for TETM waves in a waveguide

௭ୟ = ඥߤୟߝୟඥ1 minus (ୡ frasl )ଶ =ߨ2

ୟߣඥ1 minus ୟߣ) fraslୡߣ )ଶ ≝

ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)

Surface Resistance (ohms per square)



௭ଵ = ඥߤୟߝୟට ෝଵଶminus (ୡ frasl )ଶ =

ߨ2

ୟߣට ෝଵ

ଶminus ୟߣ) fraslୡߣ )ଶ ≝ߨ2

ଵߣ

Here the guided wavelengths are

ୟߣ =ୟߣ

ඥ1 minus ୟߣ) fraslୡߣ )ଶ

ଵߣ =ୟߣ

ඥ ෝଵଶminus ୟߣ) fraslୡߣ )ଶ

In the latter case the dispersion relation includes the effects of both the complex material

parameters and the dispersion characteristics of the waves For both types of wave the transverse

impedances are given by

ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ

and the interfacial reflection coefficients at the two interfaces are

Ԧଵߩ =minusଵߟ ୟߟ

ଵߟ + ୟߟ

Ԧଶߩ =minusୠߟ ଵߟ

ୠߟ + ଵߟ

Since the medium on both sides is the same we find that

Ԧଵߩ = Ԧଶߩminus

Ԧଵ = 1 + Ԧଵߩ

Ԧଶ = 1 + Ԧଶߩ = 1 minus Ԧଵߩ

and the coefficients can be written

Γ = Γ = Ԧଵߩ

1 minus eଶ୨ఋభ

1 minus Ԧଵߩଶ eଶ୨ఋభ

= Ԧଵߩ

1 minus ଶ


ሬ= ሬ=൫1 minus Ԧଵߩ

ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯


where the transmission factor through the slab is

≝ e୨ఋభ

and the relative transverse impedance is



≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ

these can also be written

Γ = Γ =൫ߟ

ଶ minus 1൯(1 minus ଶ)

൫ߟ+ 1൯ଶminus ൫ߟminus 1൯

ଶଶ

ሬ= ሬ=ߟ4


ଶଶ

and the ratio is given by

Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2

From the definition of we can also obtain the relationships

1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ

j tanߜଵ =1 minus ଶ

1 + ଶ

j tanଵߜ2

=1 minus

1 +

The reflection and transmission parameters can thus also be written [Barr2012]

Γ =൫ߟ

ଶ minus 1൯j sinߜଵ

൫ߟ+ 1൯ଶ

j sinߜଵ + eߟ2୨ఋభ

ሬ=ߟ2

൫ߟ+ 1൯ଶ


The NRW method inverts these equations directly [Nico1970Weir1974] We start by defining

ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that

ଵ ଶ = ൫ሬ+ Γ൯൫ሬminus Γ൯= ሬଶminus Γଶ

ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ

Factorising the combinations

ଵ ଶfrasl = ሬplusmn Γ =൫ minus Ԧଵߩ

ଶ ൯plusmn ൫ߩԦଵminus Ԧଵߩଶ൯


=൫1 ∓ Ԧଵ൯൫ߩ plusmn Ԧଵ൯ߩ

൫1 minus Ԧଵ൯൫1ߩ + Ԧଵ൯ߩ

we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ

and hence inverting the first relation for and the second for Ԧଵweߩ find

=ଵminus Ԧଵߩ

1 minus Ԧଵߩ ଵ=

൫ሬ+ Γ൯minus Ԧଵߩ

1 minus +Ԧଵ൫ሬߩ Γ൯

Ԧଵߩ = minus ଶ

1 minus ଶ=

minus ൫ሬminus Γ൯

1 minus ൫ሬminus Γ൯

Further considering the product

ଵ ଶ = ሬଶminus Γଶ =൫ + Ԧଵ൯൫ߩ minus Ԧଵ൯ߩ

൫1 minus Ԧଵ൯൫1ߩ + Ԧଵ൯ߩ=

ଶminus Ԧଵߩଶ


we can construct the term

൫1 minus Ԧଵߩଶ ଶ൯൛1 plusmn ൫ሬଶminus Γଶ൯ൟ= 1 minus Ԧଵߩ

ଶ ଶ plusmn ൫ ଶminus Ԧଵߩଶ ൯= ൫1 ∓ Ԧଵߩ

ଶ ൯(1 plusmn ଶ)

Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=

1 + ൫ሬଶminus Γଶ൯

2 ሬ

Υ ≝1 minus ଵ ଶ

ଵminus ଶ=1 minus ൫ሬଶminus Γଶ൯

2Γ

we can deduce

χ =1 + ൫ሬଶminus Γଶ൯

2 ሬ=൫1 minus Ԧଵߩ

ଶ ൯(1 + ଶ)

2൫1 minus Ԧଵߩଶ ൯

=1 + ଶ

2



Υ =1 minus ൫ሬଶminus Γଶ൯

2Γ=൫1 + Ԧଵߩ

ଶ ൯(1 minus ଶ)

Ԧଵ(1ߩ2 minus ଶ)=

1 + Ԧଵߩଶ

Ԧଵߩ2

These quadratic equations can be solved to give

= χ plusmn ඥχଶminus 1 with || le 1

Ԧଵߩ = Υplusmn ඥΥଶminus 1 withหߩԦଵหle 1

where the signs are chosen to maintain a modulus less than or equal to unity Note that

Υ plusmn 1 =൫1 plusmn Ԧଵߩ

ଶ ൯ଶ

ሬሬሬሬଵߩ2ଶ

It is also possible to determine the relative transverse impedance and propagation factor directly in

terms of the scattering parameters [Ziol2003]

ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ

1 minus ଵ∙1 minus ଶ

1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ

൫Γ minus 1൯ଶminus ሬଶ

with Re le൧ߟ 0

= e୨ఋభ = cosߜଵminus j sinߜଵ =1 + ଶ

2minus1 minus ଶ

2=

1 + ሬଶminus Γଶ

2 ሬminus

2Γ

൫ߟminus 1 fraslߟ ൯ሬ

Direct inversion then proceeds from the transmission factor through the slab

e୨ఋభ = e୨భభ =

by taking the logarithm of both sides

minusj ௭ଵ ଵ = log()

allowing the complex wave vector to be obtained as

௭ଵ =ߨ2

ଵߣ=

j

ଵlog()

The complex logarithm has multiple branches corresponding to the thickness of the slab being

multiples of the wavelength in the slab ଵߣ Since ଵߣ is a-priori unknown since the material

parameters are unknown this causes an ambiguity in determining the phase of the wave number

that has to be resolved as discussed below From the dispersion relation we have


ߨ2

ୟߣට ෝଵ

ଶminus ୟߣ) fraslୡߣ )ଶ =j

ଵlog()

and hence the relative complex refractive index is determined as

ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ



For non-magnetic materials we can assume Ƹଵߤ = 1 and obtain the relative permittivity as

Ƹଵߝ =Ƹଵߝୟߝ

=ƸୟߤƸଵߤ

ෝଵଶ

ఓෝ౨భୀଵሱ⎯⎯⎯ሮ ෝଵ

ଶ

In the general case the permeability can be obtained from the relative transverse impedance (for

TEMTE waves only) using

ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving

Ƹଵߤ =ƸଵߤƸୟߤ

=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=


ඥ1 minus ୟߣ) fraslୡߣ )ଶቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

The permittivity then follows either from the relative refractive index


=ෝଵଶ

Ƹଵߤ

or by inverting the dispersion relation

ෝଵଶ = ƸଵߝƸଵߤ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ

+ ൬ୟߣୡߣ൰ଶ

to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ

This can also be written


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ= ඥ1 minus (ୡ frasl )ଶƸୟߤƸଵߤቆ1 minus Ԧଵߩ

1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+

ƸୟߤƸଵߤቀୡቁଶ

The complex wave number

௭ଵ =ߨ2

ଵߣ=

j

ଵlog()

is a multi-valued complex function Writing

= ||e୨థe୨ଶగ with minus geߨ lt ߨ

we define the principal value of the logarithm by

Log() ≝ log|| + j

so that the branches are given explicitly by



log() = Log() + j2ߨ= log|| + j( + (ߨ2

where ni ℤ and = 0 for the principal branch (this is compatible with MATLAB) Hence

௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ

The phase constant is

௭ଵߚ = Re ௭ଵ൧=ߨ2

Re ଵ൧ߣ= minus

+ ߨ2

ଵ

so the electrical length of the slab is

ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus

For the principal branch = 0 and we find that geߨminus le 0 corresponds to ଵ le Re ଵ൧ߣ 2frasl At

low enough frequency we therefore expect to be in the principal branch however at higher

frequencies gt 0 corresponding to the slab being multiple wavelengths thick

One way to resolve the branch ambiguity is to use a stepwise approach to determine the phase at

each frequency point ൛= 1 hellip ൟfrom that at the last frequency point assuming that the first

frequency in the series lies in the principal branch ଵ le Re ଵ൧ߣ 2frasl and that the interval between all

the frequency points is such that ൫ ൯minus ൫ ଵ൯lt ߨ [Luuk2011] For the first frequency we

calculate

( ଵ) = arg[( ଵ)] s t geߨminus ( ଵ) le 0

௭ଵ( ଵ) ଵ = j log|( ଵ)| minus ( ଵ)

and then for successive frequencies we calculate

൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that

௭ଵ൫ ൯ଵ = j logห ൫ ൯หminus ( ଵ) minus argቈ( )

( ଵ)

ୀଵ

(gt 1)

This is equivalent to unwrapping the phase of the principal argument of log() [Barr2012] Note

that phase unwrapping has the same requirements the lowest frequency should be in the principal

(p=0) branch and ൫ ൯minus ൫ ଵ൯lt ߨ

Another way to deal with the ambiguity is to measure the group delay ୫ through the slab

[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d

and compare it to the group delay predicted by the algorithm

௭ଵߚ =ߨ2

ଵ ୫ ୟୱ= minus

+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2

minus ୫ ୟୱ൨= roundቈminusIm[Log()]

ߨ2minus ୫ ୟୱ

We round to the nearest integer since minus2) ߨ(1 le le +2) ߨ(1 corresponds to minus2minus)

1) Re ଵ൧ߣ 2frasl le ଵ le +2minus) 1) Re ଵ൧ߣ 2frasl

For TEM waves the dispersion relations reduce to [Ghod1989Ghod1990]

௭ୟ = ඥߤୟߝୟ

௭ଵ = ඥߤୟߝୟෝଵ = ඥߤୟߝୟඥߤƸଵߝƸଵ

Once the wave number and transverse interfacial reflection coefficient have been found from the

inversion above the material parameters can be obtained via the relative impedance

ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ

௭ୟቆ1 minus Ԧଵߩ

1 + Ԧଵߩቇ

Ƹଵߤ = ෝଵߟ =௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit

For thin slabs the transmission factor can be approximated to first order using [Ziol2003]

1 minus = 1 minus e୨ఋభ భభ൧≪ଵሱ⎯⎯⎯⎯⎯⎯⎯⎯ሮ 1 minus (1 minus jߜଵ) = jߜଵ = j ௭ଵ ଵ

Hence

1 minus asymp j ௭ଵ ଵ =൫1 minus ሬminus Γ൯൫1 + Ԧଵ൯ߩ




=ߟ1 +

1 minus ∙1 minus ሬ+ Γ

1 + ሬminus Γasymp

2

j ௭ଵ ଵ∙1 minus ሬ+ Γ

1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ

൫1 minus ሬminus Γ൯൫1 + Ԧଵ൯ߩ


ߟ ௭ଵ =2

j ଵ∙1 minus ሬ+ Γ

1 + ሬminus Γ

In the first equation the interfacial reflection coefficient determined from the NRW inversion or

directly from the scattering parameters can be used to determine the wave number without any

issues related to the logarithm But it seems to give a relatively inaccurate result for ௭ଵ For TEM

and TE waves

ߟ ௭ଵ =ଵߟ ௭ଵ

ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1

ඥ1 minus (ୡ frasl )ଶ∙1 minus ሬ+ Γ

1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=


ඥ ෝଵଶminus (ୡ frasl )ଶ

so

ෝଵଶ = ƸଵߝƸଵߤ = ቆ

Ƹଵߤ

ߟቇ

ଶ

൬1 minus ቀୡቁଶ

൰+ ቀୡቁଶ

Using the equation for the relative impedance gives

Ƹଵߝ =Ƹଵߤ

ߟଶ ൬1 minus ቀ

ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ

ఠౙrarrሱ⎯⎯ሮ

Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ

Alternatively from the dispersion relation


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߤƸଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ

These equations can also be derived by considering

Vଵminus 1

Vଵ + 1=ሬ+ Γ minus 1

ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1

Vଶ + 1=ሬminus Γ minus 1

ሬminus Γ + 1= minusjߟtan

ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ

The second equation is the same as the one above and allows the permeability to be determined

The permittivity can then be determined from the dispersion relation as above or the first equation

can be used to determine

௭ଵ

ଵߟ=

௭ଵଶ

Ƹଵߤ=ୟߝୟߤƸଵߤ

൜ෝଵଶminus ቀ

ୡቁଶ

ൠ

and thus

ෝଵଶ =

ƸଵߝƸଵߤୟߝୟߤ

=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+

ୟߤƸଵߤቀୡቁଶ

For TEM waves the permittivity and permeability can both be obtained directly since

ୟߟ ௭ଵ

ଵߟ=ୟߝୟߤୟߟ

Ƹଵߤ

( ෝଵଶminus (ୡ frasl )ଶ)

ඥ1 minus (ୡ frasl )ଶ ఠౙrarrሱ⎯⎯ሮ ୟ

Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ

ඥ1 minus (ୡ frasl )ଶఠౙrarrሱ⎯⎯ሮ ୟ

Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j

ୟ ଵ∙ሬ+ Γ minus 1

ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j

ୟ ଵ∙ሬminus Γ minus 1

ሬminus Γ + 1

These equations do not require any of the NRW inversion variables and can be evaluated directly

However the permeability equation may not be as well behaved as the permittivity equation since it

contains the 1 plusmn ଵ terms [Ziol2003]

273 The effective parameter method

We define the effective relative permittivity and permeability such that

௭ଵ ≝ ௭ୟඥߤƸଵߝƸଵ

≝ߟ ඨƸଵߤƸଵߝ



This is valid for quasi-TEM structures ndash such as microstrip and stripline - as well true TEM lines We

can also reformulate waveguide using effective parameters The wave number can be written

௭ଵ = ඥߤୟߝୟට ෝଵଶminus (ୡ frasl )ଶ ≝ ௭ୟඥߤƸଵߝƸଵ

by taking

Ƹଵߤ = Ƹଵߤ

Ƹଵߝ = ൬1 minus ቀୡቁଶ

൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ

The effective parameter problem is isomorphic to TEM problem The solution is straightforward

once the wave numbers in the two regions and the relative impedance are known [Boug1997]

Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ

We can therefore determine effective parameters from the wave number and wave impedance (or

interfacial reflection coefficient) from an NRW inversion

Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ

ෝ ଵ = ƸଵߝƸଵߤ = ቆ௭ଵ

௭ୟቇ

ଶ

If the material is non-magnetic with Ƹଵߤ = 1 then

Ƹଵߝ = Ƹଵ൯ߤƸଵ൫ߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ

where isin ℝ A number of choice for n have been made in the literature

n = -1 Stuchly [Stuc1978]

n = 0 NRW [Nich1970]

n = 1 Boughriet non-iterative method [Boug1997]



The choice n = 1 is numerically more robust as it eliminates the problematic 1 minus Ԧଵߩ 1 + fraslԦଵߩ

terms however it does not help for magnetic materials

274 Sample position and thickness independent methods

The scattering parameters at the reference planes of the material are

Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯


The scattering parameters measured by a VNA including sections of assumed lossless propagation

of length ୟୠon each port are

ଵଵ ଵଶ

ଶଵ ଶଶ൨= Γeଶ୨ఋ ሬe୨(ఋାఋౘ)

ሬe୨(ఋାఋౘ) Γeଶ୨ఋౘ൨

where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ

Explicitly these give

ଵଵ = eଶ୨ఋߩԦଵ



ଵଶ = ଶଵ = e୨(ఋାఋౘ)൫1 minus Ԧଵߩ

ଶ ൯e୨ఋభ


ଶଶ = eଶ୨ఋౘߩԦଵ



Uncertainty in ୟୠcan be a significant cause of error The alleviate this we can form various

combinations of the scattering parameters that are independent of the position of the sample in

line We assume that the total length of the measurement system is

= ୟ + ଵ + ୠ

and that the length of the sample is also known accurately Hence

minus ଵ = ୟ + ୠ

is known but not the individual lengths ୟ and ୠ Invariant combinations include [Bake1990

Chal2009]



det= ଵଵ ଶଶminus ଵଶ ଶଵ = eଶ୨( భ)Ԧଵߩଶ minus eଶ୨ఋభ


= eଶ୨( భ)Ԧଵߩଶ minus ଶ


≝ minuseଶ୨( భ)ܤ

ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ

൫1 minus eଶ୨ఋభ൯ଶ

eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ

= e୨భ൫1 minus Ԧଵߩ

ଶ ൯


where

ଵଶ = ଶଵ

= e୨

is the transmission factor through the empty cell We deduce that

ܤ =ଶminus Ԧଵߩ

ଶ


rArr ଶ =ܤ + Ԧଵߩ

ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=

Ԧଵߩଶ (1 minus ଶ(ܤ

൫ܤ + Ԧଵߩଶ ൯൫1 + Ԧଵߩܤ

ଶ ൯

This can be rearranged into a quadratic equation for Ԧଵߩଶ and solved to give

Ԧଵߩଶ =

1)ܣminus + (ଶܤ + (1 minus ଶ(ܤ

ܤܣ2= plusmn

ඥminus4ܣଶܤଶ + 1)ܣ + (ଶܤ minus (1 minus ଶଶ(ܤ

ܤܣ2

We choose the sign so that หߩԦଵหle 1 The remaining ambiguity from taking the square root must be

resolved by using the techniques above

275 NIST iterative inversion

NIST have developed a robust inversion method based on the equations [Bake1990]

ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=

eଶ୨ + eଶ୨ౘ

2∙Ԧଵ(1ߩ minus ଶ)


They use an iterative solver to find the solution of the non-linear equations

ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨

ܠ)୧ܨ ω β) = Re Imfrasl ൬ଵଶ + ଶଵ

2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ

minus ቆe୨( భ)൫1 minus Ԧଵߩ

ଶ ൯


+ ߚeଶ୨ + eଶ୨ౘ



ቇቤܠ

൩

The solution is found independently at each frequency point The NRW inversion can be used to

provide an initial guess is varied according to the sample length measurement noise and material

For low loss materials the transmission is strong so we take = 0

For high loss materials the reflection is strong so we take to be large

Generally we take as a ratio of noiseuncertainty in S21 to that in S11

The solution can be achieved using a multivariate Newton-Rhapson approach [Bake1990AppE ]

ܠ = minusଵܠ (ଵܠ)۴(ଵܠ)۸

where the Jacobian matrix is

(ܠ)۸ = ൦

part ଵ fraslଵݔpart part ଵ fraslଶݔpart

part ଶ fraslଵݔpart part ଶ fraslଵݔpart⋯ part ଵ fraslݔpart

hellip part ଶ fraslݔpart⋮ ⋮

part fraslଵݔpart part fraslଵݔpart⋱ ⋮hellip part fraslݔpart

൪ተ

ܠ

or another nonlinear equation solver The Jacobian can be evaluated using finite difference

approximations in order to obtain a derivative-free problem

Note that the second term is not reference plane invariant For an invariant approach we can use

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2

instead We can generalise this to obtain the permeability by adding a second complex equation The

approach can also be restated as an optimisation problem

minܠ

ܠ) ω β)

where



ܠ) ω β) = อ൬ଵଶ + ଶଵ

2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ

[Bart2010] fitted the calculated ଵଵ ଵଶ with a polynomial approximation

ߝᇱ() =

ே

ୀ

= prod ൫ minus ൯ேୀଵஷ

prod ൫ minus ൯ேୀଵ

ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ

of the permittivitypermeability to the measured values This uses the Lagrange representation of

the polynomials in terms of functional pairs ൫ ߝᇱ()൯ with linearly spaced frequencies The

order of polynomial is increased until the prescribed error bound is met The bound is informed by

the measurement uncertainty to limit the polynomial order and avoid ldquofitting to noiserdquo

=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ

2+ (1 minus (ߚ det ൰ฬ

୫ ୟୱఠೖ

minus ൬ߚଵଶ + ଶଵ

2minus (1 minus (ߚ det ൰ฬ

ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)

୫ ୧୬ gt ୫ ୟୱrArr rarr + 1

This gives smooth results with some immunity to noise and systematic effects but does not enforce

physical constraints on the extracted permittivity

Another approach is to enforce a prescribed model for permittivitypermeability and use an

optimisation approach This can directly enforce physical constraints but it requires prior knowledge

of likely form of material parameters [Tric2009][Zhan2010] For example a one term Debye

relaxation with DC conductivity

()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε

can be fitted using the GA schema



=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)

Many variations on the objective function are possible

276 MATLAB implementation and experiment simulation

1 A number of the extraction algorithms reviewed above have been implemented in the AEG

MATLAB Materials Toolbox The functions are summarized in Table 2

Function Description

matExtractNRW Basic NRW algorithm

matExtractNRWThin NRW algorithm with thin-sheet assumption

matExtractEffective Effective parameter algorithm

matExtractNIST NIST iterative algorithm

matExtractGA GA based optimization algorithm

matTestExtract Test function implementing simulated measurement with errors

Table 2 Parameter extraction functions implemented in the AEG Materials Toolbox

In addition to the extraction functions themselves a test function that implements a simulated

measurement was also developed In outline this function carries out the following steps

1 A set of known material parameters (relative permittivity permeability and thickness are

taken as inputs

2 The scattering parameters of the planar material slab referenced to the surfaces of the

sample are determined using a multilayer transmission and reflection code

3 The reference planes are moved to the locations of the two ports of a test cell of prescribed

geometry

4 Gaussian noise is added to the scattering parameters to give a defined signal-to-noise ratio

(SNR)

5 Systematic errors are introduced into the locations of the reference plane and sample

thickness according to a specified relative error



6 Each of the extraction algorithms is then applied to the simulated measured scattering

parameters to determine the relative permittivity and permeability

The extracted parameters are finally compared to the known input parameters

The results from a simple example consisting of a 5 mm thick slab of non-magnetic material with a

frequency independent relative permittivity of 4 and conductivity of 01 Sm are shown in Figure 4

In this case the SNR was set at 50 dB and no systematic errors were introduced For this simple low

permittivity material all the algorithms are fairly reliable however the direct inversion algorithms

show significant errors at some frequencies due to the noise on the simulated measurement data

The indirect algorithms are much less susceptible to the simulated measurement noise

Another more realistic example is shown in Figure 5 In this case the input material parameters are a

parametric model for the permittivity of infiltrated fat derived from experimental data [Gabr1996]

The SNR was again set at 50 dB and a 1 systematic error in the sample location in the cell was

introduced The direct inversion algorithms are seen to be particularly sensitive to systematic errors

while the indirect methods are generally more reliable

3 A coaxial measurement jig

In order to measure the shielding effectiveness of a sheet material it is necessary to illuminate it on

one side with an electromagnetic wave and measure how much of the wave passes through the

material to the other side The main problems are

Preventing any energy passing around the edges of the sample

Ensuring that unwanted standing waves are not excited as they will affect the frequency

response of the system

The solution chosen here is to incorporate the sample sheet into a coaxial transmission line Given

the difficulty in fabricating large samples of new materials incorporating nano-particles it is also

desirable that the sample be as small as possible A small sample size helps with preventing

unwanted standing waves However the sample must be large enough to allow easy handling of the

jig and sample and for the jig to be manufactured by the Physics and Electronics mechanical

workshops Also it must be possible to attach the jig to a suitable RF cable by means of a standard

connector suitable for 20 GHz operation

31 Design of a coaxial measurement jig

Due to the difficulty of manufacturing a suitable connector adaptor an SMA female-female adaptor

was chosen as the basis for the jig to cable interface and is shown in yellow on the right hand side of

Figure 6 A sample diameter of about 40 mm was small enough to be manufactured and an inner

diameter for the central conductor of 10 mm was chosen as viable for manufacture (orange in Figure

6) which results in a 23 mm diameter for the inside of the outer conductor in order to achieve a

50 ohm characteristic impedance for the jig A 10 mm wide flange was chosen resulting in an overall

outside diameter at the flange of 43 mm The flange size is sufficient to allow for 3 mm bolts to hold

the two halves of the jig together



Figure 4 Extracted parameters for a 5mm thick slab of material with r = 4 and = 01 Sm The results for the directinversion algorithms are shown at the top and those for the indirect algorithms at the bottom



Figure 5 Extracted parameters for a 5mm thick slab of infiltrated fat using a parametric model for the complexpermittivity The results for the direct inversion algorithms are shown at the top and those for the indirect algorithms atthe bottom



Figure 6 Cross sectional CAD view of one half of jig

Figure 7 3D views of jig CAD

The conductors must be tapered down to meet the SMA adaptor and as the SMA adaptor has a

PTFE dielectric a tapered PTFE insulator can be used to locate the central conductor (grey in Figure

6) A suitable curve must be computed to ensure that a 50 ohm impedance is maintained at the

transition from PTFE to air The outer conductor of the jig (grey in Figure 6) is made in two parts

joined by a threaded section This allows the alignment of the ends of the inner and outer

conductors to be adjusted

A stand was also manufactured to hold the jig in a vertical position during operation Figure 8 shows

a view from above of a completed jig in the stand with the top half removed A removable centre pin

was also added to the centre conductor to aid alignment and to locate the centre of the reference

sample for some measurements Brass (conducting) and nylon (non-conducting) versions of the pin

were manufactured



Figure 8 Completed lower half of the jig in its stand

The fully assembled jig with a sample inserted can be seen in Figure 9 The changes in outline and

fabrication of the threads on the outer conductor can be seen

Figure 9 Completed jig in stand with connector shell attached to the top



32 Coaxial jig test sample manufacture

In order to fully characterise the performance of the new coaxial measurement jig outlined in

Section 31 it was necessary to prepare a preliminary validation sample This was used to make

semi-quantitative measurements of the jigs performance and will allow the characterisation of

future materials which incorporate nanoscale structures that modify their electromagnetic response

Such materials are thought to be potentially very useful for a variety of applications including RF

absorbent materials filters and adaptive lsquosmartrsquo materials

Our validation sample consisted of a thin continuous film of copper grown under vacuum on a 50 μm

thick supporting polyamide film substrate known commercially as Kapton HN manufactured by

DuPont Kapton was chosen for the substrate because it has high thermal and mechanical stability

which is needed to support the very thin conductive copper films Kapton is also virtually transparent

to RF in this frequency range and thickness it is electrically non-conductive A thermal evaporator

was used to grow the copper film from a tungsten crucible and the thickness was monitored during

film growth by a quartz crystal microbalance as illustrated in Figure 10(a) The substrate had several

apertures to allow it to be accurately located within the measurement jig and clamped firmly in

place these can be seen in Figure 10(b) A repeatable method of sample location within the jig is

important if consistent and reproducible measurements are to be made It is also important to

clamp the sample firmly between the two halves of the jig to avoid the introduction of gaps from

which signal loss can occur during measurements

Figure 10 (a) The thermal evaporation chamber used to grow the copper thin films atop the polyamide Kaptonsubstrate The film thickness was monitored during film growth using the quartz microbalance shown in the figure (b)shows the kapton substrate including the central locating hole and surrounding clamping holes



A representative (2 x 2) cm square test piece was grown at the same time as the validation film then

removed to our measurements laboratory where its sheet resistance was measured using a four-

point-probe Hall effectresistometer as seen in Figure 11(a) and (b) The film thickness and

associated sheet resistance of the validation sample were determined as 339 nm and 1303 Ω

respectively The RF validation measurement results are included in Section 33 below

Figure 11 (a) The four-point-probe resistometer used to measure the sheet resistances of the representative square testpiece for the validation sample (b) the sample holder card and gold contacts

Ultimately a range of film thicknesses will be prepared to enable a complete validation of the

coaxial measurement jig Increasingly thicker films cause greater attenuation to the test signals In

addition to the preliminary validation sample a range of relatively thick perforated brass calibration

samples were manufactured whose performance is relatively easy to describe using a predictive

mathematical model These are shown in Figure 12 Comparative measurements can be performed

when the perforated calibration samples are replaced with either the validation sample or future

test samples incorporating nanomaterials

Figure 12 Brass calibration samples



33 Coaxial jig test results

The transmission through samples in the measurement jig was measured using a vector network

analyser The shielding effectiveness of a sample is the reciprocal of the transmission coefficient of

the jig with the sample present

Figure 13 Coaxial shielding test jig measurement setup

In order to test the jig performance one of the perforated plate test samples was measured as

shown in Figure 14

Figure 14 Perforated plate test sample on the open coaxial jig



Preliminary tests on the original jig design showed that the mechanical stability of the threaded

sections was not adequate Figure 15 shows the results of four measurements of a perforated plate

test sample demonstrating poor measurement repeatability

Figure 15 Repeated shielding measurements of a perforated plate with 075 mm diameter holes at 25 mm pitchshowing variability due to movement of the threaded sections and connector inserts

The shielding with no sample in the jig should remain at zero decibels but as can be seen in Figure 18

there were some significant features which also exhibited a degree of variability from measurement

to measurement

Figure 16 Repeated shielding measurements of empty jig showing variability due to movement of the threadedsections and connector inserts



Locking rings (see Figure 17) were manufactured for the threaded sections and connector inserts

which resulted in improved repeatability between measurements

Figure 17 Modified jig showing locking ring and connector locking tab

Figure 18 shows the results of measurements of the modified shielding with no sample in the jig The

variability has been reduced a little though there are still some significant features which warrant

further investigation

A solid brass test sample was then measured to determine the upper limit of shielding measurement

due to the network analyser sensitivity Figure 19 shows the resulting upper limit it is repeated on

other following graphs along with the empty jig results to indicate the valid measurement range

Figure 20 shows two measurements of the shielding effectiveness of the 03 mm thick perforated

plate sample with 075 mm holes on a 25 mm pitch It can be seen that where the shielding is within

the measurement limits the shape and level of the measured result corresponds to the analytic

estimate though the absolute value matches better if the hole size is set to 085 mm The brass

sample was formed by etching and the actual hole size and shape are somewhat variable A small

number of the holes were subsequently investigated using an optical microscope Figure 21 shows

images of four of the holes The hole diameter varied between 082 mm and 089 mm The

measurement can be seen to oscillate around the analytic value This is likely to be due to

imperfections in the jig causing unwanted reflections and further investigation is required

A second perforated plate with 15 mm holes at a 5 mm pitch was also fabricated as part of the jig

test set Figure 22 shows the measured shielding effectiveness for two separate measurements

compared with an analytic estimate of the shielding of the plate Again the measurement is seen to

oscillate about the analytic solution



Figure 18 Repeated shielding measurements of empty jig showing reduced variability

Figure 19 Measurement of a 03 mm thick solid brass sample which shows the upper limit of shielding measurement

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

Empty jig 1 brass pin

Empty jig 2 no pin ferrites on cables

Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)

Solid brass sample 1 brass pin

Solid brass sample 2 no pin

Solid brass sample 3 no pin ferrites on cables



Figure 20 Shielding measurement of the perforated plate sample with 075 mm holes on a 25 mm pitch compared withan analytic estimate

Figure 21 Optical microscope images of four holes in the perforated plate sample with 075 mm holes on a 25 mmpitch Clockwise from the top left the hole diameters (verticalhorizontal) are 084088 mm 084089 mm082087 mm and 083088 mm

2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)


075mm hole 25mm pitch brass sample 1 brass pin

075mm hole 25mm pitch brass sample 2 no pin


075mm hole 25mm pitch 03mm thick plate analytic




Figure 22 Shielding measurement of the perforated plate sample with 15 mm holes on a 5 mm pitch in comparisonwith an analytic estimate

The copper coated Kapton sample described earlier is shown in Figure 23 on the open jig Figure 24

shows the measured shielding for this sample Based on the sheet resistance measurement a

shielding effectiveness of 43 dB was expected It can be seen in Figure 24 that the measured

shielding effectiveness oscillates about the expected value at frequencies between 2 and 20 GHz

Below 2 GHz the measured shielding value is approximately 40 dB which corresponds to a sheet

resistance 165 times higher than that originally measured The analytic calculations here ignore the

Kapton substrate and consider only the effect of the copper Figure 25 shows measurements of the

Kapton substrate compared with the empty jig It can be seen that with the Kapton sample present

the result is practically identical to that of the empty jig

Figure 23 339 nm copper on Kapton substrate sample on the open jig

2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)








Figure 24 Shielding measurement of 339 nm copper on Kapton substrate in comparison with analytic estimates

Figure 25 Shielding measurement of Kapton substrate compared with empty jig

2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)


339nm Cu on kapton (IGW 30115) brass pin


339nm Cu Analytic from thickness

339nm Cu Analytic from sheet resistance

339nm Cu Analytic from 165x sheet resistance

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)


Kapton sample 1 brass pin

Kapton sample 2 no pin

Kapton sample 3 no pin ferrites on cables



4 Nanoparticle film and material fabrication

Nanomaterials are becoming increasingly more interesting as fillers for other structural materials in

order to manufacture composites that have very different physical properties to those of their

individual component materials Examples include self-cleaning glass paints and super strength

composites for the sports industry eg carbon nanotubes (CNTs) used in bicycle frames We expect

that nanomaterials will be useful for modifying electromagnetic properties and are especially

interested in their use as electromagnetic shielding materials The advent of 3D printers makes it

possible to print housings for electronic circuits that require electromagnetic shielding in which

nanomaterials can be incorporated during the printing process

The wavelength of shielded electromagnetic signals must be scaled to the dimensions of the

nanostructures when designing new shielding materials that exploit conductive losses For example

microwaves have a wavelength of the order of 10 -2 m much larger than any nanostructure

Electrically isolated nanostructures with dimensions less than a half wavelength can be transparent

to some useful frequencies This can be overcome by electrically connecting nanostructures to form

continuously conducting macrostructures Clearly conventional thin film technologies can achieve

this however it may also be possible to create materials with anisotropic (directional) shielding

properties by incorporating for example co-aligned CNTs as shown in Figure 26 Materials of this

kind can polarise electromagnetic radiation so that cross polarised variable filters can be devised

In order to work towards this goal we intend to process commercially acquired CNTs in order to

attach cobalt or nickel metal nanoparticles at their ends in a controlled manner The nanoparticles

have a large magnetic moment that will cause the CNTs to align magnetically end-to-end Useful

conductive shielding can be accomplished when this forms a continuous conductive path through

planar materials ie very thin flat sheets These sheets may also have a polarising effect as mention

above when the CNTs are co-aligned We will suspend CNTs in a supportive matrix so that they are

free to align with an external magnetic field This gives the opportunity to alter the polarisation

angle of the material

Figure 26 SEM images of CNTs (a) co-aligned and (b) randomly orientated

The same CNTs may also have interesting dielectric effects when they form a discontinuous film ie

are nonconductive In this case the individual CNTs develop an electric dipole which opposes the

driving signal and therefore attenuates it to some extent Again co-aligned dielectric CNTs may lead

to some degree of polarisation when they greatly attenuate the signal in only one orientation It has

been shown that an attenuation of 24 dB is possible using CoNi in single walled CNTs (SWCNTs)

which is attributed to dielectric (electronic and interfacial polarization) and magnetic (eddy current

and natural resonance) losses [Singh2015] This new material showed a microwave shielding



effectiveness value greater than that required for commercial applications Consequently nano-

composites are promising microwave shielding materials in the Ku band We intend to magnetically

align functionalised CNTs in a nonconductive resin to further explore this effect

Figure 27 SEM image of a conductive graphene foam showing an open matrix of graphene flakes randomly orientated ina dimethyl siloxane composite [Chen2011]

In addition to our CNT work we will also develop quasi-continuous materials such as abutted

graphene flakes Recently a three dimensional graphene foam has been developed with an

exceptionally high electrical conductivity of sim10 S cmminus1 at an ultra-low graphene loading of sim05 wt

(sim022 vol) [Chen2011] This is shown in Figure 27 where the open foam matrix can be seen

Graphene has a large surface area but is only one atom thick the potential for weight savings

especially in aircraft is obvious In our work we will control the connectivity between each individual

flake and hope to thereby control the sheet resistance We will aim to achieve this by using a

rheological fluid to electrically couple the graphene flakes and more permanently by using metallic

atoms as conductive bridges When rheological fluids are used they can be manipulated by external

magnetic fields giving the opportunity to create smart adaptive materials for electromagnetic

absorbers

5 Conclusions and Plans for future work

51 Material characterisation capability

The work so far has allowed us to better understand the electromagnetic characterisation of

materials and the development of a coaxial test jig The coaxial test jig initial design was fabricated

(complete April 2015) and modified after initial tests The jig is capable of measuring the shielding

effectiveness of planar samples but suffers from some measurement artefacts which are believed to

be due to mechanical imperfections in the design It is planned to build a second version with

improved performance once further analysis of the current jig has been fabricated The MATLAB

software developed for material parameter extraction will aid future characterisation work

52 Application of nano-particle films

Due to the delay in jig manufacture and the remedial work to improve the mechanical stability the

production of test samples was delayed and is just about to commence Going forward we aim to

explore the anisotropic conductive and dielectric shielding of functionalised carbon based

nanostructures such as CNTs and graphene flakes when used in combination with magnetic metallic

atoms including Co and Ni We anticipate that the dielectric properties will dominate over

conductive effects in terms the shielding properties due to the effects of scale



53 Proposals and collaboration

Future proposals are currently under discussion within the Physical Layer Research Group and with

interested outside parties

6 References

[Bake1990] J Baker-Jarvis ldquoTransmissionreflection and short-circuit line permittivity

measurementsrdquo NIST Technical Note 1341 National Institute of Standards and Technology Boulder

CO July 1990 URL httpwwweeelnistgovadvanced_materials_publicationsBaker-

Jarvis20TN2090pdf

[Bake2005] J Baker-Jarvis M D Janezic R F Riddle R T Johnk P Kabos C Holloway R G Geyer

and C A Grosvenor ldquoMeasuring the permittivity and permeability of lossy materials Solids liquids

metals building materials and negative-index materialsrdquo NIST Technical Note 1536 National

Institute of Standards and Technology Boulder CO 2005 URL

httpwwweeelnistgovadvanced_materials_publicationsBaker-Jarvis20TN1536pdf

[Barr2012] J J Barroso and A L de Paula ldquoRetrieval of permittivity and permeability of

homogeneous materials from scattering parametersrdquo Journal of Electromagnetic Waves and

Applications vol 24 no 11-12 pp 1563-1574 2010 DOI 101163156939310792149759

[Boug1997] A-H Boughriet C Legrand and A Chapoton ldquoNoniterative stable

transmissionreflection method for low-loss material complex permittivity determinationrdquo

Microwave Theory and Techniques IEEE Transactions on vol 45 no 1 pp 52-57 Jan 1997 DOI

10110922552032

[Chal2009] K Chalapat K Sarvala J Li and G S Paraoanu ldquoWideband reference-plane invariant

method for measuring electromagnetic parameters of materialsrdquo Microwave Theory and

Techniques IEEE Transactions on vol 57 no 9 pp 2257-2267 Sept 2009 DOI

101109TMTT20092027160

[Chen2011] Z Chen W Ren L Gao B Liu S Pei and H Cheng ldquoThree-dimensional flexible and

conductive interconnected graphene networks grown by chemical vapour depositionrdquo Nature

Materials Vol 10 June 2011 DOI 101038NMAT3001

[Gabr1996] S Gabriel R W Lau and C Gabriel ldquoThe dielectric properties of biological tissues III

Parametric models for the dielectric spectrum of tissuesrdquo Physics in Medicine and Biology vol 41

pp 2271-2293 1996 DOI 1010880031-91554111003

[Ghod1989] D K Ghodgaonkar V V Varadan and VK Varadan ldquoA free-space method for

measurement of dielectric constants and loss tangents at microwave frequenciesrdquo Instrumentation

and Measurement IEEE Transactions on vol 38 no 3 pp 789-793 Jun 1989 DOI

1011091932194

[Ghod1990] D K Ghodgaonkar V V Varadan and V K Varadan ldquoFree-space measurement of

complex permittivity and complex permeability of magnetic materials at microwave frequenciesrdquo

Instrumentation and Measurement IEEE Transactions on vol 39 no 2 pp 387-394 Apr 1990 DOI

1011091952520



[Luuk2011] O Luukkonen S I Maslovski and S A Tretyakov ldquoA stepwise NicolsonndashRossndashWeir-

based material parameter extraction methodrdquo Antennas and Wireless Propagation Letters IEEE

vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897

[Nico1970] A M Nicolson and G F Ross ldquoMeasurement of the intrinsic properties of materials by

time-domain techniquesrdquo Instrumentation and Measurement IEEE Transactions on vol 19 no 4

pp 377-382 Nov 1970 DOI 10101109TIM19704313932

[Paul1992] C R Paul ldquoIntroduction to Electromagnetic Compatibilityrdquo John Wiley New York 1992

[Singh2015] B P Singh D K Saket A P Singh Santwana Pati T K Gupta V N Singh S R

Dhakate S K Dhawan R K Kotnala and R B Mathur ldquoMicrowave shielding properties of CoNi

attached to single walled carbon nanotubesrdquo J Mater Chem A May 2015 Vol 3 13203-13209

DOI 101039C5TA02381E

[Stuc1978] S Stuchly C Sibbald and J Anderson ldquoA new aperture admittance model for open-

ended waveguidesrdquo IEEE Transactions on Microwave Theory and Techniques vol 42 no 2

pp 192ndash198 Feb 1994 DOI 10110922275246

[Tric2009] S Tricarico F Bilotti and L Vegni ldquoA genetic algorithm based procedure to retrieve

effective parameters of planar metamaterial samplesrdquo Proc SPIE 7353 Metamaterials IV 73530H

12 May 2009 DOI 10111712820648

[Weir1974] W B Weir ldquoAutomatic measurement of complex dielectric constant and permeability at

microwave frequenciesrdquo Proceedings of the IEEE vol 62 no 1 pp 33- 36 Jan 1974 DOI

101109PROC19749382

[Zhan2010] J Zhang M Y Koledintseva G Antonini J L Drewniak A Orlandi and K N Rozanov

ldquoPlanar transmission line method for characterization of printed circuit board dielectricsrdquo Progress

In Electromagnetics Research (PIER) vol 102 pp 267-286 2010 URL

httpwwwjpierorgPIERpierphppaper=10012807

[Ziol2003] R W Ziolkowski ldquoDesign fabrication and testing of double negative metamaterialsrdquo

Antennas and Propagation IEEE Transactions on vol 51 no 7 pp 1516-1529 July 2003 DOI

101109TAP2003813622

7 Bibliography

[Agil85071E] Agilent 85071E ldquoMaterials measurement software Technical overviewrdquo 2012 URL

httpwwwhomeagilentcomagilentproductjspxnid=-

53690247553688326800amplc=engampcc=GB

[Agil2006] Agilent ldquoBasics of measuring the dielectric properties of materialsrdquo Agilent Technologies

Application Note 5989-2589EN 2006 URL httpcpliteratureagilentcomlitwebpdf5989-

2589ENpdf

[ASTM4935] ASTM Standard D4935-10 ldquoStandard test method for measuring the electromagnetic

shielding effectiveness of planar materialsrdquo ASTM International West Conshohocken PA June

2010 DOI 101520D4935-10



[ASTMD5568] ASTM Standard D5568-08 ldquoStandard test method for measuring relative complex

permittivity and relative magnetic permeability of solid materials at microwave frequenciesrdquo ASTM

International West Conshohocken PA May 2012 DOI 101520D5568-08

[Badi2002] M Badic and M-J Marinescu ldquoThe failure of coaxial TEM cells ASTM standards methods

in HF rangerdquo Electromagnetic Compatibility 2002 EMC 2002 IEEE International Symposium on

vol 1 pp 29-34 19-23 Aug 2002 DOI 101109ISEMC20021032442

[Bake1990b] J Baker-Jarvis E J Vanzura and W A Kissick ldquoImproved technique for determining

complex permittivity with the transmissionreflection methodrdquo Microwave Theory and Techniques

IEEE Transactions on vol 38 no 8 pp 1096-1103 Aug 1990 DOI 1011092257336

[Barr2001] J J Barroso and U C Hasar ldquoUnambiguous determination of constitutive parameters of

a metamaterial slabrdquo Microwave and Optoelectronics Conference (IMOC) 2011 SBMOIEEE MTT-S

International pp 498-502 29 Oct - 1Nov 2011 DOI 101109IMOC20116169261

[Barr2012b] J J Barroso and U C Hasar ldquoConstitutive parameters of a metamaterial slab retrieved

by the phase unwrapping methodrdquo Journal of Infrared Millimeter and Terahertz Waves vol 33

no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3

[Bart2010] P G Bartley and S B Begley ldquoA new technique for the determination of the complex

permittivity and permeability of materialsrdquo Instrumentation and Measurement Technology

Conference (I2MTC) 2010 IEEE pp 54-57 3-6 May 2010 DOI 101109IMTC20105488184

[Caij2011] Zhao Caijun Jiang Quanxing and Jing Shenhui ldquoCalibration-independent and position-

insensitive transmissionreflection method for permittivity measurement with one sample in coaxial

linerdquo Electromagnetic Compatibility IEEE Transactions on vol 53 no 3 pp 684-689 Aug 2011

DOI 101109TEMC20112156416

[Catr1992] J Catrysse M Delesie and W Steenbakkers ldquoThe influence of the test fixture on

shielding effectiveness measurementsrdquo Electromagnetic Compatibility IEEE Transactions on vol 34

no 3 pp 348-351 August 1992 DOI 10110915155853

[Hasa2009] U C Hasar and C R Westgate ldquoA broadband and stable method for unique complex

permittivity determination of low-loss materialsrdquo Microwave Theory and Techniques IEEE

Transactions on vol 57 no 2 pp 471-477 Feb 2009 DOI 101109TMTT20082011242

[Hong2003] Y K Hong C Y Lee C K Jeong D E Lee K Kim and J Joo ldquoMethod and apparatus to

measure electromagnetic interference shielding efficiency and its shielding characteristics in

broadband frequency rangesrdquo Review of Scientific Instruments vol 74 no 2 pp 1098-1102 2003

DOI 10106311532540

[Krup2006] J Krupka ldquoFrequency domain complex permittivity measurements at microwave

frequenciesrdquo Measurement Science and Technology vol 17 p R55 2006 DOI 1010880957-

0233176R01

[Popo2005a] D Popovic L McCartney C Beasley M Lazebnik M Okoniewski S C Hagness and J

H Booske ldquoPrecision open-ended coaxial probes for in vivo and ex vivo dielectric spectroscopy of



biological tissues at microwave frequenciesrdquo Microwave Theory and Techniques IEEE Transactions

on vol 53 no 5 pp 1713- 1722 May 2005 DOI 101109TMTT2005847111

[Popo2005b] D Popovic L McCartney C Beasley M Lazebnik M Okoniewski S C Hagness and J

H Booske ldquoCorrections on lsquoPrecision open-ended coaxial probes for in vivo and ex vivo dielectric

spectroscopy of biological tissues at microwave frequenciesrsquordquo Microwave Theory and Techniques

IEEE Transactions on vol 53 no 9 pp 3053 Sept 2005 DOI 101109TMTT2005854214

[RnS] Rhode amp Schwarz Application Note RAC-0607-0019 ldquoMeasurement of dielectric material

propertiesrdquo 23 March 2012 URL httpwww2rohde-schwarzcomfileRAC-0607-0019_1_5Epdf

[Sart2006] M S Sarto and A Tamburrano ldquoInnovative test method for the shielding effectiveness

measurement of conductive thin films in a wide frequency rangerdquo Electromagnetic Compatibility

IEEE Transactions on vol48 no2 pp 331- 341 May 2006 DOI 101109TEMC2006874664

[Toro2000] A Toropainen P Vainikainen and A Drossos ldquoMethod for accurate measurement of

complex permittivity of tissue equivalent liquidsrdquo Electronics Letters vol 36 no 1 pp 32-34 6 Jan

2000 DOI 101049el20000126

[Vasq2009] H Vasquez L Espinoza K Lozano H Foltz and S Yang ldquoSimple device for

electromagnetic interference shielding effectiveness measurementrdquo IEEE Electromagnetic

Compatibility Society Newsletter pp 62-68 Winter 2009 URL

httpwwwemcsorgacstrialnewsletterswinter09pp2pdf

[WikiMatSim] Wikipedia ldquoMatrix similarityrdquo Wikipediaorg URL

httpenwikipediaorgwikiMatrix_similarity [Last Modified 7 August 2012 at 2141]

[Wils1988] P F Wilson M T Ma and J W Adams ldquoTechniques for measuring the electromagnetic

shielding effectiveness of materials I Far-field source simulationrdquo Electromagnetic Compatibility

IEEE Transactions on vol 30 no 3 pp 239-250 August 1988 DOI 101109153302



Glossary

CFC Carbon Fibre Composite

CNT Carbon Nano-Tube

DC Direct Current

EMT Effective Medium Theory

GA Genetic Algorithm

NRW Nicholson-Ross-Weir

PTFE Polytetrafluoroethylene

RF Radio Frequency

SE Shielding Effectiveness

SMA Sub-miniature Version A

SNR Signal-to-Noise Ratio

SWCNT Single Walled Carbon Nano-Tube

TE Transverse Electric

TEM Transverse Electromagnetic

TM Transverse Magnetic

VNA Vector Network Analyser



1 Introduction

















































Section 3




۲() = ()Ƹ()۳ߝ = ε۳() + () = ε۳() + εχොக()۳()







tanߜக =εᇱᇱ

εᇱ




۸() = ()ୈେ۳ߪ



ᇱᇱ+ୈେߪε

൰




()Ƹߝ = εᇱminus

ᇱᇱߪ

εminus jቆε

ᇱᇱ+ᇱߪ

εቇ






2

ߨන


ᇱᇱ()

minusଶߠ ଶdߠ

ஶ

εᇱᇱ() = minus

2

ߨන

εᇱ(ߠ) minus ε

ᇱ()

minusଶߠ ଶdߠ

ஶ


εᇱ() minus ε

ᇱ() = minus minus

ߨPVන

εᇱᇱ(ߠ)


ஶ

ஶ

εᇱᇱ() minus ε

ᇱᇱ() = minus

ߨPVන

εᇱ(ߠ)


ஶ



Drude σ =

ଶ

1

ߛ + j=

ଶ

minusߛ j

ߛଶ + ଶ

Debye()ߝ = ஶߝ +

ߝ∆

1 + j




ߝ∆1 + (j )ଵఈ

+ୈେߪjε





j≝ ߤ()Ƹߤ




jߤ












۶(ܚǢ ) ൌ Ǣܧ

1


and


௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ


where Ǣܧ and Ǣܧ


௫ଶ ௭








ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












1 Introduction

















































Section 3




۲() = ()Ƹ()۳ߝ = ε۳() + () = ε۳() + εχොக()۳()







tanߜக =εᇱᇱ

εᇱ




۸() = ()ୈେ۳ߪ



ᇱᇱ+ୈେߪε

൰




()Ƹߝ = εᇱminus

ᇱᇱߪ

εminus jቆε

ᇱᇱ+ᇱߪ

εቇ






2

ߨන


ᇱᇱ()

minusଶߠ ଶdߠ

ஶ

εᇱᇱ() = minus

2

ߨන

εᇱ(ߠ) minus ε

ᇱ()

minusଶߠ ଶdߠ

ஶ


εᇱ() minus ε

ᇱ() = minus minus

ߨPVන

εᇱᇱ(ߠ)


ஶ

ஶ

εᇱᇱ() minus ε

ᇱᇱ() = minus

ߨPVන

εᇱ(ߠ)


ஶ



Drude σ =

ଶ

1

ߛ + j=

ଶ

minusߛ j

ߛଶ + ଶ

Debye()ߝ = ஶߝ +

ߝ∆

1 + j




ߝ∆1 + (j )ଵఈ

+ୈେߪjε





j≝ ߤ()Ƹߤ




jߤ












۶(ܚǢ ) ൌ Ǣܧ

1


and


௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ


where Ǣܧ and Ǣܧ


௫ଶ ௭








ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


















Section 3




۲() = ()Ƹ()۳ߝ = ε۳() + () = ε۳() + εχොக()۳()







tanߜக =εᇱᇱ

εᇱ




۸() = ()ୈେ۳ߪ



ᇱᇱ+ୈେߪε

൰




()Ƹߝ = εᇱminus

ᇱᇱߪ

εminus jቆε

ᇱᇱ+ᇱߪ

εቇ






2

ߨන


ᇱᇱ()

minusଶߠ ଶdߠ

ஶ

εᇱᇱ() = minus

2

ߨන

εᇱ(ߠ) minus ε

ᇱ()

minusଶߠ ଶdߠ

ஶ


εᇱ() minus ε

ᇱ() = minus minus

ߨPVන

εᇱᇱ(ߠ)


ஶ

ஶ

εᇱᇱ() minus ε

ᇱᇱ() = minus

ߨPVන

εᇱ(ߠ)


ஶ



Drude σ =

ଶ

1

ߛ + j=

ଶ

minusߛ j

ߛଶ + ଶ

Debye()ߝ = ஶߝ +

ߝ∆

1 + j




ߝ∆1 + (j )ଵఈ

+ୈେߪjε





j≝ ߤ()Ƹߤ




jߤ












۶(ܚǢ ) ൌ Ǣܧ

1


and


௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ


where Ǣܧ and Ǣܧ


௫ଶ ௭








ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126















2

ߨන


ᇱᇱ()

minusଶߠ ଶdߠ

ஶ

εᇱᇱ() = minus

2

ߨන

εᇱ(ߠ) minus ε

ᇱ()

minusଶߠ ଶdߠ

ஶ


εᇱ() minus ε

ᇱ() = minus minus

ߨPVන

εᇱᇱ(ߠ)


ஶ

ஶ

εᇱᇱ() minus ε

ᇱᇱ() = minus

ߨPVන

εᇱ(ߠ)


ஶ



Drude σ =

ଶ

1

ߛ + j=

ଶ

minusߛ j

ߛଶ + ଶ

Debye()ߝ = ஶߝ +

ߝ∆

1 + j




ߝ∆1 + (j )ଵఈ

+ୈେߪjε





j≝ ߤ()Ƹߤ




jߤ












۶(ܚǢ ) ൌ Ǣܧ

1


and


௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ


where Ǣܧ and Ǣܧ


௫ଶ ௭








ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126





















۶(ܚǢ ) ൌ Ǣܧ

1


and


௫

௭ො൰ܢ

۶ Ǣܚ) ) ൌ Ǣܧ

Ƹߝ


where Ǣܧ and Ǣܧ


௫ଶ ௭








ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














ܧ = ܧ

ܧ = ܧ

cosߠ


۳ = ܧ

e୨௭e୨௫ܡො

۶ = minus

ܧ

ߟ

e୨௭e୨௫ܠො

and

۳ = ܧ

e୨௭e୨௫ܠො

۶ =

ܧ

ߟ

e୨௭e୨௫ܡො


ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ


۳ = ܧ +ොܠ ܧ

ܡො


ܠො



(ݖ)ܧ = ܧା(ݖ) + ܧ

(ݖ) = ܧାe୨௭ + ܧ

eା୨௭ = ܧା(ݖ)൫1 + Γ(ݖ)൯

(ݖ)ܪ = ܪା(ݖ) + ܪ

(ݖ) =1

ߟ൫ܧ


ܧା(ݖ)




ܧା(ݖଵ)

ܧ(ଵݖ)

൨= e୨(௭మ௭భ) 0


ܧ(ଶݖ)

൨




(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












(ଵݖ)ܧ(ଵݖ)ܪ




൨


Γ(ݖ) =ܧ(ݖ)

ܧା(ݖ)




(ݖ) =(ݖ)ܧ

(ݖ)ܪ= ߟ

1 + Γ(ݖ)

1 minus Γ(ݖ)

propagates as



Note that





௫ = ௫




൫ ௫൯

ଶ+ ൫ ௭

൯ଶ


we find that


൯ଶ



boundary

ܧ = ܧ

ା + ܧ = ܧ

ା + ܧ = ܧ



ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












ܪ =

1

ߟ൫ܧ

ା minus ܧ൯=

1

ߟ൫ܧ

ା minus ܧ൯= ܪ


ቈܧା

ܧ=

1

Ԧ

1 ԦߩԦߩ 1

൨ቈܧା

ܧ


Ԧߩ =ߟ minus ߟ

ߟ + ߟ

Ԧ =ߟ2

ߟ + ߟ


ቈܧା

ܧ=

1

ശ

1 ശߩശߩ 1

൨ቈܧା

ܧ

where

ശߩ =ߟminus ߟ

ߟ + ߟ

ശ =ߟ2

ߟ + ߟ

and



=ܧ

ܪ

=ܧ

ܪ

=


Γ =ܧ

ܧା

=Ԧߩ + Γ

1 + ԦΓߩhArr Γ =

ܧ

ܧା

=ശߩ + Γ

1 + ശΓߩ


ቈܧ

ܧା=

Ԧߩ ശԦ ശߩ

൨ቈܧା

ܧ











where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126




















where

Ǣߟ =

Ƹߤ

௭Ǣ

Ǣߟ =

௭Ǣ

Ƹߝ




௫ୟ ൌ ௫








side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














side


ቈଵܧା

ଵܧ=

1

Ԧଵቈ

1 Ԧଵߩ


ଵܧ

ቈଵܧା

ଵܧ= e

୨భ(௭మ௭భ) 00 e୨భ(௭మ௭భ)

൨ቈଶܧା

ଶܧ

ቈଶܧା

ଶܧ=

1

Ԧଶቈ

1 Ԧଶߩ


ଶܧ


ቈଵܧା

ଵܧ=

1

Ԧଵ Ԧଶቈ

1 Ԧଵߩ

Ԧଵߩ 1e


൨ቈ1 Ԧଶߩ


ଶܧ

where



matrix

ቈଵܧ

ଶܧା= Γ ሬ


ଶܧ=

ଵଵଵ ଵଶଵ


ଶܧ≝ ଵቈ

ଵܧା

ଶܧ

where










ଶ ൯

Ԧଵ


Ԧଶ

e୨ఋభ





slab



ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












ቈଵܧ

ଵܧା=

1


minusΓ 1ቈଶܧ

ଶܧା=

ଵଵଵ ଵଶଵ


ଶܧା= ധ

ଵቈଶܧ

ଶܧା

ቈଶܧ

ଶܧା=

1


ቈଵܧ

ଵܧା

ധଵ =

ଵଵଵ ଵଶଵ


1


minus ଶଶଵ 1



1




det ധଵ = ሬ



and therefore

ധଵ =

1

ሬቈdet ଵ Γ

minusΓ 1=

1


0 1൨= e


൨


coefficients as


subject to






or


Ԧଶߩ






or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126













or




௭ଵߚ2 ଵ =ߨ2






ଶ = ߟߟ




ቈଵܧ

ଶܧା= Γ ሬ

ሬ Γ൨ଵܧା

0൨

so that

ଵܧ = Γܧଵ

ା

ଶܧା = ሬܧଵ

ା


ሬ =

1

ୟߟหܧଵ

หଶ

=1

ୟߟหΓห

ଶหܧଵ

ାหଶ≝ ℛሬ

หܧଵାห

ଶ


ሬ୲ୟ୬ୱ =

1

ୠߟหܧଶ

ାหଶ

=1

ୠߟหሬห

ଶหܧଵ

ାหଶ≝ ሬ

หܧଵାห

ଶ

ୟߟ= ሬሬ

୧୬


ℛሬ≝ሬ

ሬ୧୬

= หΓหଶ

≝ሬ୲ୟ୬ୱ

ሬ୧୬

=ୟߟ

ୠߟหሬห

ଶ



ሬ୧୬


ୠߟหሬห

ଶ








relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126

















relation



ߟ ≝

Ƹߤ

௭

ߟ ≝

௭

Ƹߝ






where

ୡ≝ୡ


ߨ2

ୡߣ

Then


ߨ2

ୟߣට ෝଵ



ୟߟ ≝

Ƹୟߤ

௭ୟ=

ඥߤୟ fraslୟߝ


ୟߟ


௭ߟ = Ƹߤ





ሬ୫ ୳୪୲୧

where


ଶ =ߟ4

൫ߟ + 1൯ଶ




ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126













ሬ୫ ୳୪୲୧=

1


= ൫ߩԦଵ൯ଶ

ஶ

ୀ







ߟ =1

ߟඨƸଵߤƸଵߝ

=1

ߟඨ



ଵߪ2≪ 1

and hence

ሬ = ߟ4 = 4(1 + )ඨ

ଵߤߝߨ

ଵߪ= 4(1 + )ඨ

ଵߤߝߨ

େ୳ߪଵߪ= 4(1 + )ඨ

ߝߨେ୳ߪ

ඨଵߤ

ଵߪ





ଵߤ

ଵߪቇ



where


2=1 minus

ୱଵߜ


ୱଵߜ = ඨ2

ଵߪଵߤ= ඨ

1

ߨଵߪଵߤ=

1

ඥߤߨߪେ୳

1

ඥߤଵߪଵ

Hence


ୱଵߜ

and






ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














ଵ



ሬ୫ ୳୪୲୧=

1


=1


൰ଶ


asymp1




1


1

2 ଵߜ=

ୱଵߜ

2(+ 1) ଵ=

1

ඥߤߨߪେ୳

1

2(+ 1) ଵ

1

ඥߤଵߪଵ



ሬ ሬ୫ ୳୪୲୧=

2

େ୳ߪߟ

1

ଵߪ ଵ






ୗଵ =1

ଵߪ ଵ



=ܮଵߩ

ܣ=

ܮ

ଵߪ ଵ=

1

ଵߪ ଵ

ܮ

= ୗଵ

ܮ













promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126

















promising ones



can both be written






ଵߜ = ௭ଵ ଵ

For TEM waves







ߨ2

ୟߣ

0

20

40

60

80

100

120

0001 001 01 1 10

Sh

ield

ing

Eff

ec

tiv

en

es

s(d

B)





ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126













ߨ2

ୟߣට ෝଵ


ଵߣ


ୟߣ =ୟߣ


ଵߣ =ୟߣ





ߟ =

Ƹߤ

௭

ߟ =

௭

Ƹߝ



ଵߟ + ୟߟ


ୠߟ + ଵߟ



Ԧଵ = 1 + Ԧଵߩ



Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯



≝ e୨ఋభ




≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












≝ߟଵߟ

ߟ

Noting that

Ԧଵߩ =minusߟ 1

ߟ + 1hArr ߟ =

1 + Ԧଵߩ

1 minus Ԧଵߩ

minusߟ1

ߟ=

Ԧଵߩ2ଶ

1 minus Ԧଵߩଶ


Γ = Γ =൫ߟ



ଶଶ

ሬ= ሬ=ߟ4


ଶଶ


Γ

ሬ=

Ԧଵߩ2

1 minus Ԧଵߩଶ

1 minus ଶ

2=ߟଶ minus 1

ߟ2∙1 minus ଶ

2=

1

2ቆߟminus

1

ߟቇ1 minus ଶ

2


1 + ଶ

2= cosߜଵ

1 minus ଶ

2= j sinߜଵ


1 + ଶ

j tanଵߜ2

=1 minus

1 +


Γ =൫ߟ


൫ߟ+ 1൯ଶ


ሬ=ߟ2

൫ߟ+ 1൯ଶ



ଵ≝ ሬ+ Γ

ଶ≝ ሬminus Γ



so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












so that


ଵ+ ଶ = 2 ሬ

ଵminus ଶ = 2Γ







we obtain

ଵ = + Ԧଵߩ

1 + Ԧଵߩ

ଶ = minus Ԧଵߩ

1 minus Ԧଵߩ


=ଵminus Ԧଵߩ




Ԧଵߩ = minus ଶ

1 minus ଶ=






ଶminus Ԧଵߩଶ






Defining

χ ≝1 + ଵ ଶ

ଵ + ଶ=


2 ሬ



2Γ

we can deduce



ଶ ൯(1 + ଶ)


=1 + ଶ

2




2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126













2Γ=൫1 + Ԧଵߩ



1 + Ԧଵߩଶ

Ԧଵߩ2






ଶ ൯ଶ




ߟଶ =

Υ + 1

Υ minus 1=

1 + ଵ


1 + ଶ=൫Γ + 1൯

ଶminus ሬଶ


with Re le൧ߟ 0


2minus1 minus ଶ

2=


2 ሬminus

2Γ







௭ଵ =ߨ2

ଵߣ=

j

ଵlog()






ߨ2

ୟߣට ෝଵ


ଵlog()


ෝଵଶ = ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

=1

ୟଶ൬

j

ଵlog()൰

ଶ

+ ቀୡቁଶ





=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














=ƸୟߤƸଵߤ

ෝଵଶ


ଶ



ߟ =ଵߟ

ߟ=ƸଵߤƸୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

ଵߣ

ୟߣ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

giving


=ୟߣ

ଵߣቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ=



1 + Ԧଵߩ

1 minus Ԧଵߩቇ



=ෝଵଶ

Ƹଵߤ



௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

= ቆୟߣଵߣ

ቇ

ଶ


to give


=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ=ୟߣଶ

Ƹଵߤቆ

1

ଵߣଶ +

1

ୡߣଶቇ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


1 + Ԧଵߩቇ൬

௭ଵ

ୟ൰+



௭ଵ =ߨ2

ଵߣ=

j

ଵlog()




Log() ≝ log|| + j




log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












log() = Log() + j2ߨ= log|| + j( + (ߨ2


௭ଵ =ߨ2

ଵߣ=

j

ଵlog() =

j

ଵlog|| minus

+ ߨ2

ଵ



Re ଵ൧ߣ= minus

+ ߨ2

ଵ


ଵ

Re ଵ൧ߣ=

ଵߚ௭ଵ

ߨ2= minus

+ ߨ2

ߨ2= minus

ߨ2

minus








calculate




൫ ൯= ൫ ଵ൯+ argቈ൫ ൯

൫ ଵ൯= ( ଵ) + argቈ

( )

( ଵ)

ୀଵ

(gt 1)

so that


( ଵ)

ୀଵ

(gt 1)





[Weir1974Chal2009]



୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












୫ ୟୱ= minus1

ߨ2

d୫ ୟୱ

d= ଵ

dߚ௭ଵ

d=

ଵ

ߨ2

dߚ௭ଵ

d


௭ଵߚ =ߨ2


+ ߨ2

ଵ

to give

= minusߨ2

minus ୫ ୟୱ

= roundminusߨ2


ߨ2minus ୫ ୟୱ








ߟ = ඨƸଵߤ

Ƹଵߝ=

1 + Ԧଵߩ

1 minus Ԧଵߩ

using

Ƹଵߝ =ෝଵߟ

=௭ଵ


1 + Ԧଵߩቇ


௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

272 Thin slab limit



Hence





=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












=ߟ1 +



2


1 + ሬminus Γ

can be reduced to

௭ଵ =1

j ଵ



ߟ ௭ଵ =2


1 + ሬminus Γ




and TE waves


ୟߟ=Ƹଵߤୟߟ

hence

Ƹଵߤ =ୟߟ2


1 + ሬminus Γ=

ୟߟ2j ଵ

1


1 + ሬminus Γ

Now

ߟ =ଵߟ

ୟߟ=Ƹଵߤୟߤ

௭ୟ

௭ଵ= Ƹଵߤ

௭ୟ

௭ଵ=



so


Ƹଵߤ

ߟቇ

ଶ


൰+ ቀୡቁଶ


Ƹଵߝ =Ƹଵߤ


ୡቁଶ

൰+1

Ƹଵߤቀୡቁଶ


Ƹଵߤ

ߟଶ =

ୟߟ2

j ଵ∙1 minus ଵ

1 + ଵ



=1

Ƹଵߤቊ൬

௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ


௭ଵ

ୟ൰ଶ

+ ቀୡቁଶ

ቋ


Vଵminus 1


ሬ+ Γ + 1=

minusj

ߟtan

ଵߜ2asymp minusj

ଵ

2௭ଵ

ߟ= minusj

ଵ

2

ୟߟ ௭ଵ

ଵߟ



Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












Vଶminus 1



ଵߜ2asymp minusj

ଵ

2ߟ ௭ଵ = minusj

ଵ

2

Ƹଵߤୟߟ




௭ଵ

ଵߟ=

௭ଵଶ



ୡቁଶ

ൠ

and thus

ෝଵଶ =


=Ƹଵߤ

ୟߝୟߤ

௭ଵ

ଵߟ+ ቀ

ୡቁଶ

giving

Ƹଵߝୟߝ

=1

ୟߝቆ

௭ଵ

ଵߟቇ+



ୟߟ ௭ଵ


Ƹଵߤ



Ƹଵߝୟߝ

Ƹଵߤୟߟ

=Ƹଵߤୟߟ


Ƹଵߤୟߤ

giving

Ƹଵߝୟߝasymp

2j


ሬ+ Γ + 1

Ƹଵߤୟߤ

asymp2j


ሬminus Γ + 1













by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126















by taking

Ƹଵߤ = Ƹଵߤ


൰ߝƸଵ +1

Ƹଵߤቀୡቁଶ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ

Ƹଵߤ =௭ଵ

௭ୟߟ



Ƹଵߝ =௭ଵ

௭ୟ

1

ߟ=

௭ଵ


1 + Ԧଵߩቇ

Ƹଵߤ =௭ଵ

௭ୟ=ߟ

௭ଵ

௭ୟቆ

1 + Ԧଵߩ

1 minus Ԧଵߩቇ

As usual

ƸଵߤƸଵߝ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଶ


௭ୟቇ

ଶ



= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ୟߣ

௭ଵߣቇ

ାଵ

= ቆ1 + Ԧଵߩ

1 minus Ԧଵߩቇ

ଵ

ቆ௭ଵ

௭ୟቇ

ାଵ











Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126
















Γ = Γ = Ԧଵߩ



= Ԧଵߩ

1 minus ଶ



ଶ ൯e୨ఋభ


=൫1 minus Ԧଵߩ

ଶ ൯




ଵଵ ଵଶ



where

ୟߜ = ௭ୟ ୟ

ୠߜ = ௭ୠ ୠ = ௭ୟ ୠ






ଶ ൯e୨ఋభ








= ୟ + ଵ + ୠ




Chal2009]








ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126

















ଵଶ + ଶଵ

2= e୨( భ)



ܣ =ଵଵ ଶଶ

ଵଶ ଶଵ=

Ԧଵߩଶ


ଶ


eଶ୨ఋభ=

Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ

=ଵଶ + ଶଵ

ଵଶ + ଶଵ


ଶ ൯


where

ଵଶ = ଶଵ

= e୨



ଶ



ଶ

1 + Ԧଵߩܤଶ

rArr ܣ =Ԧଵߩଶ


ଶ

(1 minus ଶ)ଶ

ଶ=



ଶ ൯


Ԧଵߩଶ =


ܤܣ2= plusmn


ܤܣ2





ଵଶ + ଶଵ

2= e୨( భ)



ଵଵ + ଶଶ

2=





ܠ)۴ ω β) =

where



=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












=ܠ ߝᇱ

ߝᇱᇱ൨

ܠ)۴ ω β) = ܠ)ܨ ω β)ܠ)୧ܨ ω β)

൨


2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

൩









(ܠ)۸ = ൦





൪ተ

ܠ




ߚଵଶ + ଶଵ

2+ (1 minus (ߚ det

or

ߚଵଶ + ଶଵ

2+ (1 minus (ߚ

| ଵଵ| + | ଶଶ|

2



minܠ

ܠ) ω β)

where




2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126













2+ ߚ

ଵଵ + ଶଶ

2൰ฬ୫ ୟୱ


ଶ ൯





ቇቤܠ

อ

ଶ


ߝᇱ() =

ே

ୀ



ߝᇱ()

ே

ୀ

ߝᇱᇱ() =

ே

ୀ



ߝᇱᇱ()

ே

ୀ





=ܠ ൦

ߝᇱ()⋮

ߝᇱᇱ()

⋮

൪

ଶேtimesଵ

൦

1⋮0⋮

൪le geܠ

maxߝ

⋮maxߝ

⋮

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)








()Ƹߝ = ஶߝ + ߝ∆

1 + j

+ୈେߪjε




=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126












=ܠ ൦

infinߝߝ∆DCߪ

൪

൦

1000

൪le geܠ ൦

maxߝmaxߝ ݔ

maxߪ

൪

ܠ) β) =1

ቤ൬ߚ

ଵଶ + ଶଵ


୫ ୟୱఠೖ



ୡୟ୪ୡఠೖ

ቤ

ଶே

ୀଵ

minܠ

ܠ) β)
















taken as inputs




geometry


(SNR)






























































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126





































































were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126






























were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126



























were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126
























were manufactured





















































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126






























































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126
























































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


































shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


















shown in Figure 14










to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


















to measurement






























2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126






































2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)



Empty jig 3 no pin


2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

120

Frequency (MHz)

SE

(dB

)








2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)




















2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126























2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)










2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














2000 4000 6000 8000 10000 12000 14000 16000 18000 200000

20

40

60

80

100

Frequency (MHz)

SE

(dB

)







2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0

2

4

6

8

10

12

14

16

18

Frequency (MHz)

SE

(dB

)

























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126






























































absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


























absorbers






















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126















6 References




Jarvis20TN2090pdf












10110922552032




101109TMTT20092027160






pp 2271-2293 1996 DOI 1010880031-91554111003




1011091932194




1011091952520





vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126














vol 10 pp 1295-1298 2011 DOI 101109LAWP20112175897



pp 377-382 Nov 1970 DOI 10101109TIM19704313932





DOI 101039C5TA02381E



pp 192ndash198 Feb 1994 DOI 10110922275246



12 May 2009 DOI 10111712820648



101109PROC19749382







101109TAP2003813622

7 Bibliography






2589ENpdf



2010 DOI 101520D4935-10

















no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126


























no 2 pp 237-244 2012 DOI 101007s10762-011-9869-3







DOI 101109TEMC20112156416



no 3 pp 348-351 August 1992 DOI 10110915155853







DOI 10106311532540



0233176R01


















2000 DOI 101049el20000126

























2000 DOI 101049el20000126










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