Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Uniaxial Anisotropic Dielectrics Benjamin D. Braaten* North Dakota State

Current Distribution of a Printed Dipole with Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Arbitrary Length Embedded in Layered

Uniaxial Anisotropic DielectricsUniaxial Anisotropic DielectricsBenjamin D. Braaten*Benjamin D. Braaten*

North Dakota State University, Fargo ND, USANorth Dakota State University, Fargo ND, USA

David A. RogersDavid A. RogersNorth Dakota State University, Fargo ND, USANorth Dakota State University, Fargo ND, USA

Robert M. NelsonRobert M. NelsonUniversity of Wisconsin – Stout, Menomonie WI, USAUniversity of Wisconsin – Stout, Menomonie WI, USA

North Dakota State UniversityNorth Dakota State University

TopicsTopics

Problem DefinitionProblem Definition Spectral domain immittance Spectral domain immittance

functionsfunctions Spectral domain moment methodSpectral domain moment method Results/DiscussionResults/Discussion ConclusionConclusion


Problem DefinitionProblem DefinitionConsider:Consider:


The spectral domain immittance The spectral domain immittance functionsfunctions


Start with the following Hertz vector potentials:Start with the following Hertz vector potentials:

andand

Electric Hertz Electric Hertz potentialpotential

Magnetic HertzMagnetic Hertzpotentialpotential



Next, only the y-direction of the Hertz vector Next, only the y-direction of the Hertz vector potential is needed.potential is needed.

andand

This is because the optical axis is in the y-This is because the optical axis is in the y-direction direction

and and this component satisfies the higher order TE this component satisfies the higher order TE

and TM tangential boundary conditions.and TM tangential boundary conditions.



Now define the following expression for the magneticNow define the following expression for the magneticand electric field:and electric field:

where the Hertzian vector potentials are solutions to where the Hertzian vector potentials are solutions to the following equations:the following equations:



andand



To simplify evaluating the previous expressions, we To simplify evaluating the previous expressions, we define the following Fourier transform:define the following Fourier transform:

This results in the following relations:This results in the following relations:



This results in the following simplified expressions:This results in the following simplified expressions:

wherewhere

andand



Similarly forSimilarly for

andand



Next, consider:Next, consider:

Enforcing the B.C. with the Enforcing the B.C. with the previous expressions previous expressions results in the following results in the following simple relation:simple relation:

The Spectral Domain Moment The Spectral Domain Moment MethodMethod


Next, the x-component of the electric field in each Next, the x-component of the electric field in each region can be written in the spatial domain as:region can be written in the spatial domain as:

Since is in the spatial domain, the two-dimensional Since is in the spatial domain, the two-dimensional inverse Fourier transform inverse Fourier transform

will need to be applied towill need to be applied to



Proceeding in this manner results in the following Proceeding in this manner results in the following expression:expression:

Next, the current in terms of the basis functions:Next, the current in terms of the basis functions:



Defining weighting functions and rearranging gives:Defining weighting functions and rearranging gives:

wherewhere

andand



Notice that the following expression is in the previous Notice that the following expression is in the previous rearrangement:rearrangement:

This is the 2D FT of the basis function.This is the 2D FT of the basis function.

This can be useful if a basis function with an This can be useful if a basis function with an analytical FT is chosen.analytical FT is chosen.



Using this simplification results in the following Using this simplification results in the following expression:expression:

NoticeNotice

Integration only on a Integration only on a single planesingle plane



For this work, PWS basis functions were used:For this work, PWS basis functions were used:



The alpha-beta plane of integration:The alpha-beta plane of integration:

Numerical ResultsNumerical Results


Single LayerSingle Layer

L = .5L = .5λλ00

W = .0004W = .0004λλ00

dd11 = .1016 = .1016λλ00

1V source1V source



Single LayerSingle Layer

Notice: Notice that the imaginary part can be individually modified Notice: Notice that the imaginary part can be individually modified (compare the solid lines with the dashed lines)(compare the solid lines with the dashed lines)



Triple LayerTriple Layer

L = .25L = .25λλ00

W = .00083W = .00083λλ00

dd11 = .0026 = .0026λλ00

dd22 = .0026 = .0026λλ00

dd33 = .0026 = .0026λλ00

εε11 = 3.25 (iso.) = 3.25 (iso.)

1V source1V source



Triple LayerTriple Layer

Notice: Notice that both the real and imaginary parts of the current Notice: Notice that both the real and imaginary parts of the current change from the isotropic case when each permittivity component is change from the isotropic case when each permittivity component is modified.modified.

ConclusionConclusion


A summary on the spectral domain moment method A summary on the spectral domain moment method has been presented.has been presented.

A single printed dipole on a single anisotropic A single printed dipole on a single anisotropic substrate has been investigated.substrate has been investigated. It was shown that with certain permittivity components, the It was shown that with certain permittivity components, the

imaginary part of the current could be modified while the real imaginary part of the current could be modified while the real part of the current remained unchanged.part of the current remained unchanged.

A single printed dipole in three anisotropic layers has A single printed dipole in three anisotropic layers has been investigated.been investigated. It was shown that each component of the permittivity in the It was shown that each component of the permittivity in the

superstrate and substrate had an effect on both the real and superstrate and substrate had an effect on both the real and imaginary part of the current.imaginary part of the current.

QuestionsQuestions

Thank you for listeningThank you for listening


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Current Distribution of a Printed Dipole with Arbitrary Length Embedded in Layered Uniaxial Anisotropic Dielectrics Benjamin D. Braaten* North Dakota State