A high gain metal-only reflectarray antenna composed of multiple rectangular grooves is analyzed. An overlapping T-block method based on mode-matching technique is used to rigorously present the scattering formulations. A reflectarray with 30 cm diameter and 5,961 rectangular grooves was fabricated and measured, achieving 42.3 dB gain at 75 GHz. The measured results agree favorably with simulations in terms of radiation patterns and gain.
Metallic rectangular-grooves based 2D reflectarray antenna excited by an open...Yong Heui Cho
This document describes a 2D reflectarray antenna composed of multiple metallic rectangular grooves excited by an open-ended parallel-plate waveguide. The scattering solutions for the antenna are obtained using an overlapping T-block method and Stratton-Chu formula. Design formulas for phase matching are proposed to achieve high directivity and beam tilting. Numerical computations of scattering characteristics like gain and focus offset are performed. A prototype antenna was fabricated for testing and measurement.
Fourier-transform analysis of a ridge waveguide and a rectangular coaxial lineYong Heui Cho
This document presents a new technique for analyzing ridge waveguides and rectangular coaxial lines using Fourier transforms. It transforms the structures into equivalent periodic transmission lines using image theory. It represents the fields using Fourier transforms and derives dispersion relations in rigorous yet simple series forms. Numerical results show fast convergence that agrees with other methods. The technique is applicable to other shielded transmission line structures.
Efficient mode-matching analysis of 2-D scattering by periodic array of circu...Yong Heui Cho
This document proposes a new mode-matching method for analyzing periodic arrays of circular cylinders with different materials. It uses a common-area concept and infinitesimal perfect magnetic conductor wires to match boundary conditions between rectangular and cylindrical coordinate systems. This allows derivation of scattering equations for the arrays. The solutions are compared to other results to validate the method. It can be used to study resonance characteristics of nanostructures and scattering behavior at different frequencies.
1) The document provides formulas for the fundamental bending frequencies of various beam configurations including cantilever beams, simply supported beams, free-free beams, fixed-fixed beams, and beams with end masses.
2) Appendices A through K derive the formulas for different beam configurations using principles of mechanics and vibration analysis. Approximate and exact solutions are provided depending on assumptions about the beam mass.
3) Examples are given for common structures like steel pipes and rocket vehicles to demonstrate application of the frequency formulas for engineering design and analysis.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
Surface crack detection using flanged parallel-plate waveguideYong Heui Cho
This document presents a method for detecting surface cracks on metal plates using a flanged parallel-plate waveguide. The method involves using Fourier transforms and mode-matching techniques to obtain simultaneous equations representing the crack signal characteristics. Parameters like crack position, width, depth, and material can be determined from the crack signal. Simulation results show good agreement with previous experiments and indicate the method can accurately detect rectangular and V-shaped cracks under different conditions.
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
Metallic rectangular-grooves based 2D reflectarray antenna excited by an open...Yong Heui Cho
This document describes a 2D reflectarray antenna composed of multiple metallic rectangular grooves excited by an open-ended parallel-plate waveguide. The scattering solutions for the antenna are obtained using an overlapping T-block method and Stratton-Chu formula. Design formulas for phase matching are proposed to achieve high directivity and beam tilting. Numerical computations of scattering characteristics like gain and focus offset are performed. A prototype antenna was fabricated for testing and measurement.
Fourier-transform analysis of a ridge waveguide and a rectangular coaxial lineYong Heui Cho
This document presents a new technique for analyzing ridge waveguides and rectangular coaxial lines using Fourier transforms. It transforms the structures into equivalent periodic transmission lines using image theory. It represents the fields using Fourier transforms and derives dispersion relations in rigorous yet simple series forms. Numerical results show fast convergence that agrees with other methods. The technique is applicable to other shielded transmission line structures.
Efficient mode-matching analysis of 2-D scattering by periodic array of circu...Yong Heui Cho
This document proposes a new mode-matching method for analyzing periodic arrays of circular cylinders with different materials. It uses a common-area concept and infinitesimal perfect magnetic conductor wires to match boundary conditions between rectangular and cylindrical coordinate systems. This allows derivation of scattering equations for the arrays. The solutions are compared to other results to validate the method. It can be used to study resonance characteristics of nanostructures and scattering behavior at different frequencies.
1) The document provides formulas for the fundamental bending frequencies of various beam configurations including cantilever beams, simply supported beams, free-free beams, fixed-fixed beams, and beams with end masses.
2) Appendices A through K derive the formulas for different beam configurations using principles of mechanics and vibration analysis. Approximate and exact solutions are provided depending on assumptions about the beam mass.
3) Examples are given for common structures like steel pipes and rocket vehicles to demonstrate application of the frequency formulas for engineering design and analysis.
1. A particle moving perpendicular to a magnetic field will follow a circular path. The radius of the path is determined by the particle's mass, charge, speed, and the magnetic field strength.
2. A velocity selector uses uniform, perpendicular electric and magnetic fields. Particles pass through undeflected if their speed equals the ratio of the field strengths.
3. A mass spectrometer accelerates ions and uses a magnetic field to cause circular orbits. Heavier ions have smaller orbit radii allowing separation based on mass.
Surface crack detection using flanged parallel-plate waveguideYong Heui Cho
This document presents a method for detecting surface cracks on metal plates using a flanged parallel-plate waveguide. The method involves using Fourier transforms and mode-matching techniques to obtain simultaneous equations representing the crack signal characteristics. Parameters like crack position, width, depth, and material can be determined from the crack signal. Simulation results show good agreement with previous experiments and indicate the method can accurately detect rectangular and V-shaped cracks under different conditions.
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
This document contains solutions to multiple physics problems involving electromagnetic waves. Problem 1 involves calculating the conductivity and penetration depth of graphite at different frequencies. Problem 2 involves propagating an electromagnetic wave in seawater and calculating various parameters like attenuation constant and phase velocity. It provides the solutions and steps for parts a, b, and c of this problem. Problem 3 involves analyzing the behavior of electromagnetic waves on a finite transmission line terminated by a load impedance and derives relevant equations.
This document provides the solution to a theoretical question regarding the thermal vibration of surface atoms.
(1) It calculates the wavelength of an incident electron beam and considers the interference between atomic rows on a surface. Two possible solutions for the phase angle are obtained.
(2) An expression is derived for the total energy of vibration of surface atoms in the direction of the surface normal as a function of time and atomic mass.
(3) Combining the energy expression with data about the material properties allows calculating the frequency and amplitude of vibration of the surface atoms. The calculated frequency is found to match results from a graph in the question.
1) A point charge moving through an electric field is shown to follow a parabolic trajectory given by z = -1.5×1010t2 m. At t = 3 μs, its position is found to be P(0.90, 0, -0.135) m.
2) A point charge moving through a uniform magnetic field follows a circular path. The equations of motion are derived and solved, giving the position, velocity, and kinetic energy at t = 3 μs.
3) Forces on current loops and filaments in various magnetic field configurations are calculated.
The document discusses deformation spectra for single-degree-of-freedom (SDF) linear systems subjected to base excitation. It presents the equations of motion for an SDF system with a moving base and defines terms like relative displacement and pseudo-acceleration. Graphs of deformation spectra are shown for half-cycle acceleration and velocity pulses. Key aspects of the spectra under different inputs are described, including asymptotic behavior and sensitivity to displacement, velocity, and acceleration portions of the input.
1) Gauss's law for magnetism states that the magnetic flux through a closed surface is always zero, since there is no magnetic monopole. Gauss's law for electricity relates the electric flux through a closed surface to the net electric charge enclosed.
2) Applying the right-hand rule, the direction of the current in a solenoid that produces a magnetic field pointing away from you is clockwise.
3) Of the gases listed, H2, CO2, and N2 are diamagnetic with magnetic susceptibility χm < 0, while O2 is paramagnetic with χm > 0.
This document proposes applying ΣΔ quantization to fusion frames for efficient analog-to-digital and digital-to-analog conversion in wireless sensor networks. It develops 1st-order and high-order ΣΔ quantization algorithms for fusion frame projections that use a minimal number of bits per subspace. It proves the canonical left inverse minimizes operator norms and introduces Sobolev left inverses, which minimize reconstruction error. Numerical experiments show the reconstruction error from high-order ΣΔ quantization decays as O(N-r).
Nonlinear Structural Dynamics: The Fundamentals TutorialVanderbiltLASIR
This presentation from Dr. Douglas Adams, Chairman of Civil & Environmental Engineering at Vanderbilt University, and Director of the Laboratory for Systems Integrity and Reliability (LASIR), introduces the fundamental concepts of nonlinear structure dynamics.
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
Smith chart:A graphical representation.amitmeghanani
The document discusses the Smith chart, which is a graphical tool used to solve transmission line problems. Some key points:
- The Smith chart was developed in 1939 and allows tedious transmission line calculations to be done graphically.
- It provides a mapping between the normalized impedance plane and the reflection coefficient plane. Circles of constant resistance and reactance are plotted, along with the reflection coefficient.
- Parameters like impedance, admittance, reflection coefficient, VSWR can all be plotted and derived from locations on the chart.
- Examples are given of using the Smith chart to determine input impedance, reflection coefficient, and stub matching of transmission lines with various termination impedances.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
1) The masses involved are 1350 kg and 5400 kg. Initial velocities are 0 m/s for both masses.
2) Using conservation of momentum and given relative velocities, the final velocities are calculated to be 10.67 km/h for the 1350 kg mass and 4.27 km/h for the 5400 kg mass.
3) The problem involves calculating final velocities using conservation of linear momentum and given relative velocities.
This document contains 26 multi-part physics problems related to electric fields, current density, and capacitance. It provides the questions and worked out solutions. Some examples include calculating the current crossing a given surface, the magnitude of the current density at a point, and the average current over a surface. Other problems involve determining capacitance values for different capacitor geometries, finding electric field and potential values given a current or charge distribution, and calculating polarization and energy storage within dielectric materials.
11903 Electromagnetic Waves And Transmission Linesguestd436758
This document contains information about an electromagnetic waves and transmission lines exam for a fourth semester engineering course. It includes 8 questions covering topics like Maxwell's equations, electric and magnetic fields, wave propagation, transmission lines, and waveguides. Students were instructed to answer any 5 of the 8 questions in the exam, which would be graded out of 80 total marks. The questions involve both theoretical concepts and calculations.
Monopole antenna radiation into a parallel plate waveguideYong Heui Cho
This document presents a rigorous solution for analyzing the radiation of a coaxially-fed monopole antenna into a parallel-plate waveguide. Fourier transform representations are used to model the scattered fields. Boundary conditions are enforced to obtain equations for the modal coefficients. A rapidly convergent series solution for the reflection coefficient is obtained and shown to agree well with other results. Input impedance and reflected power characteristics are calculated and presented.
The document contains 10 problems involving electromagnetic induction and Maxwell's equations. Problem 10.1 involves calculating the voltage and current in a circuit with a changing magnetic flux. Problem 10.2 replaces a voltmeter with a resistor and calculates the resulting current. Problem 10.3 calculates the emf induced in closed paths with changing magnetic fluxes.
This document discusses different types of electromagnetic waves that can propagate in waveguides, including transverse electric (TE), transverse magnetic (TM), and transverse electromagnetic (TEM) modes. It focuses on analyzing TE and TM modes in rectangular waveguides. Only certain discrete modes are allowed to propagate depending on the waveguide dimensions and frequency. Both TE and TM modes have cutoff frequencies below which propagation does not occur. In an X-band rectangular waveguide, only the TE10 mode can propagate from 6.56 to 13.12 GHz, which is called the dominant mode.
The Smith chart is a graphical tool used to analyze high frequency circuits. It represents all possible complex impedances in terms of the reflection coefficient. Circles of constant resistance and arcs of constant reactance intersect on the chart to indicate impedance values. The chart allows users to determine impedances, reflection coefficients, voltage standing wave ratios and other transmission line parameters through graphical techniques. It remains a popular tool decades after its original conception due to providing a clever way to visualize complex impedance functions.
Analysis of a ridge waveguide using overlapping T-blocksYong Heui Cho
This document analyzes the dispersion relation of a ridge waveguide using an overlapping T-block approach. It presents:
1) A method to represent the electromagnetic fields within a T-block in simple series based on the Green's function and mode-matching technique.
2) How to obtain rigorous yet simple dispersion equations for symmetric and asymmetric ridge waveguides by superimposing the fields of overlapping T-blocks.
3) A closed-form dominant-mode approximation that is shown to be accurate for practical applications such as coupler, filter, and polarizer designs.
The document provides information about decomposing electromagnetic fields in waveguides into longitudinal and transverse components. It introduces the key concepts of cutoff frequency, cutoff wavelength, propagation constant, transverse impedances, and relates them through important equations. Several types of waveguide modes (TEM, TE, TM, hybrid) are also defined based on which field components are nonzero.
This document contains solutions to multiple physics problems involving electromagnetic waves. Problem 1 involves calculating the conductivity and penetration depth of graphite at different frequencies. Problem 2 involves propagating an electromagnetic wave in seawater and calculating various parameters like attenuation constant and phase velocity. It provides the solutions and steps for parts a, b, and c of this problem. Problem 3 involves analyzing the behavior of electromagnetic waves on a finite transmission line terminated by a load impedance and derives relevant equations.
This document provides the solution to a theoretical question regarding the thermal vibration of surface atoms.
(1) It calculates the wavelength of an incident electron beam and considers the interference between atomic rows on a surface. Two possible solutions for the phase angle are obtained.
(2) An expression is derived for the total energy of vibration of surface atoms in the direction of the surface normal as a function of time and atomic mass.
(3) Combining the energy expression with data about the material properties allows calculating the frequency and amplitude of vibration of the surface atoms. The calculated frequency is found to match results from a graph in the question.
1) A point charge moving through an electric field is shown to follow a parabolic trajectory given by z = -1.5×1010t2 m. At t = 3 μs, its position is found to be P(0.90, 0, -0.135) m.
2) A point charge moving through a uniform magnetic field follows a circular path. The equations of motion are derived and solved, giving the position, velocity, and kinetic energy at t = 3 μs.
3) Forces on current loops and filaments in various magnetic field configurations are calculated.
The document discusses deformation spectra for single-degree-of-freedom (SDF) linear systems subjected to base excitation. It presents the equations of motion for an SDF system with a moving base and defines terms like relative displacement and pseudo-acceleration. Graphs of deformation spectra are shown for half-cycle acceleration and velocity pulses. Key aspects of the spectra under different inputs are described, including asymptotic behavior and sensitivity to displacement, velocity, and acceleration portions of the input.
1) Gauss's law for magnetism states that the magnetic flux through a closed surface is always zero, since there is no magnetic monopole. Gauss's law for electricity relates the electric flux through a closed surface to the net electric charge enclosed.
2) Applying the right-hand rule, the direction of the current in a solenoid that produces a magnetic field pointing away from you is clockwise.
3) Of the gases listed, H2, CO2, and N2 are diamagnetic with magnetic susceptibility χm < 0, while O2 is paramagnetic with χm > 0.
This document proposes applying ΣΔ quantization to fusion frames for efficient analog-to-digital and digital-to-analog conversion in wireless sensor networks. It develops 1st-order and high-order ΣΔ quantization algorithms for fusion frame projections that use a minimal number of bits per subspace. It proves the canonical left inverse minimizes operator norms and introduces Sobolev left inverses, which minimize reconstruction error. Numerical experiments show the reconstruction error from high-order ΣΔ quantization decays as O(N-r).
Nonlinear Structural Dynamics: The Fundamentals TutorialVanderbiltLASIR
This presentation from Dr. Douglas Adams, Chairman of Civil & Environmental Engineering at Vanderbilt University, and Director of the Laboratory for Systems Integrity and Reliability (LASIR), introduces the fundamental concepts of nonlinear structure dynamics.
This document provides information about the dynamics of machinery course for several mechanical engineering students. It includes the learning objectives, symbols and definitions, response of a damped system under harmonic motion, an example problem, and key concepts about magnification factor, phase angle, and total response of a system. The example calculates the total response of a single-degree-of-freedom system subjected to an external harmonic force and free vibration.
Smith chart:A graphical representation.amitmeghanani
The document discusses the Smith chart, which is a graphical tool used to solve transmission line problems. Some key points:
- The Smith chart was developed in 1939 and allows tedious transmission line calculations to be done graphically.
- It provides a mapping between the normalized impedance plane and the reflection coefficient plane. Circles of constant resistance and reactance are plotted, along with the reflection coefficient.
- Parameters like impedance, admittance, reflection coefficient, VSWR can all be plotted and derived from locations on the chart.
- Examples are given of using the Smith chart to determine input impedance, reflection coefficient, and stub matching of transmission lines with various termination impedances.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document presents a third order shear deformation theory to analyze flexure of thick cantilever beams. The theory uses a sinusoidal function in the displacement field to account for transverse shear deformation effects through the beam thickness. Governing equations and boundary conditions are derived using the principle of virtual work. Numerical examples of a cantilever beam with a cosine load distribution are presented and displacement, stress results are obtained in non-dimensional form. The results are discussed and compared to other beam theories to demonstrate the efficiency of the third order shear deformation theory.
1) The masses involved are 1350 kg and 5400 kg. Initial velocities are 0 m/s for both masses.
2) Using conservation of momentum and given relative velocities, the final velocities are calculated to be 10.67 km/h for the 1350 kg mass and 4.27 km/h for the 5400 kg mass.
3) The problem involves calculating final velocities using conservation of linear momentum and given relative velocities.
This document contains 26 multi-part physics problems related to electric fields, current density, and capacitance. It provides the questions and worked out solutions. Some examples include calculating the current crossing a given surface, the magnitude of the current density at a point, and the average current over a surface. Other problems involve determining capacitance values for different capacitor geometries, finding electric field and potential values given a current or charge distribution, and calculating polarization and energy storage within dielectric materials.
11903 Electromagnetic Waves And Transmission Linesguestd436758
This document contains information about an electromagnetic waves and transmission lines exam for a fourth semester engineering course. It includes 8 questions covering topics like Maxwell's equations, electric and magnetic fields, wave propagation, transmission lines, and waveguides. Students were instructed to answer any 5 of the 8 questions in the exam, which would be graded out of 80 total marks. The questions involve both theoretical concepts and calculations.
Monopole antenna radiation into a parallel plate waveguideYong Heui Cho
This document presents a rigorous solution for analyzing the radiation of a coaxially-fed monopole antenna into a parallel-plate waveguide. Fourier transform representations are used to model the scattered fields. Boundary conditions are enforced to obtain equations for the modal coefficients. A rapidly convergent series solution for the reflection coefficient is obtained and shown to agree well with other results. Input impedance and reflected power characteristics are calculated and presented.
The document contains 10 problems involving electromagnetic induction and Maxwell's equations. Problem 10.1 involves calculating the voltage and current in a circuit with a changing magnetic flux. Problem 10.2 replaces a voltmeter with a resistor and calculates the resulting current. Problem 10.3 calculates the emf induced in closed paths with changing magnetic fluxes.
This document discusses different types of electromagnetic waves that can propagate in waveguides, including transverse electric (TE), transverse magnetic (TM), and transverse electromagnetic (TEM) modes. It focuses on analyzing TE and TM modes in rectangular waveguides. Only certain discrete modes are allowed to propagate depending on the waveguide dimensions and frequency. Both TE and TM modes have cutoff frequencies below which propagation does not occur. In an X-band rectangular waveguide, only the TE10 mode can propagate from 6.56 to 13.12 GHz, which is called the dominant mode.
The Smith chart is a graphical tool used to analyze high frequency circuits. It represents all possible complex impedances in terms of the reflection coefficient. Circles of constant resistance and arcs of constant reactance intersect on the chart to indicate impedance values. The chart allows users to determine impedances, reflection coefficients, voltage standing wave ratios and other transmission line parameters through graphical techniques. It remains a popular tool decades after its original conception due to providing a clever way to visualize complex impedance functions.
Analysis of a ridge waveguide using overlapping T-blocksYong Heui Cho
This document analyzes the dispersion relation of a ridge waveguide using an overlapping T-block approach. It presents:
1) A method to represent the electromagnetic fields within a T-block in simple series based on the Green's function and mode-matching technique.
2) How to obtain rigorous yet simple dispersion equations for symmetric and asymmetric ridge waveguides by superimposing the fields of overlapping T-blocks.
3) A closed-form dominant-mode approximation that is shown to be accurate for practical applications such as coupler, filter, and polarizer designs.
The document provides information about decomposing electromagnetic fields in waveguides into longitudinal and transverse components. It introduces the key concepts of cutoff frequency, cutoff wavelength, propagation constant, transverse impedances, and relates them through important equations. Several types of waveguide modes (TEM, TE, TM, hybrid) are also defined based on which field components are nonzero.
Transverse magnetic plane-wave scattering equations for infinite and semi-inf...Yong Heui Cho
The document proposes plane-wave scattering equations for infinite and semi-infinite rectangular grooves in a conducting plane. For infinite grooves, the equations are derived using the overlapping T-block method and Floquet theorem, representing the magnetic fields as infinite summations. For semi-infinite grooves, large numbers of grooves are approximated using infinite groove solutions near the center and edge Green's functions, yielding efficient but approximate scattering equations. Numerical results agree with mode-matching solutions and converge rapidly.
Fourier-transform analysis of a unilateral fin line and its derivativesYong Heui Cho
This document presents a Fourier-transform analysis of a unilateral n-line and its derivatives. The key points are:
1) A unilateral n-line is transformed into an equivalent problem of multiple suspended substrate microstrip lines using the image theorem and Fourier transform.
2) New rigorous dispersion relations are derived in the form of rapidly convergent series solutions, providing an accurate yet efficient method for numerical computation.
3) The dispersion relations for derivatives of the unilateral n-line including suspended substrate microstrip lines, microstrip lines, slot lines and coplanar waveguides are also presented.
Iterative Green's function analysis of an H-plane T-junction in a parallel-pl...Yong Heui Cho
This document analyzes scattering solutions for an H-plane T-junction in a parallel-plate waveguide using an iterative Green's function method. Simple yet rigorous scattering relations are derived for reflection and transmission powers. Numerical results show good agreement with other methods and indicate the solutions converge rapidly using only a few modes. The iterative Green's function approach provides an efficient analysis of waveguide junction scattering problems.
This document discusses transmission line modes, beginning with TEM, TE, and TM waves. It then focuses on the TEM mode, deriving the electric and magnetic fields for a TEM wave. Next, it examines the TEM mode in more detail for a coaxial cable, finding the electric and magnetic fields and characteristic impedance. It concludes by briefly discussing surface waves on a grounded dielectric slab.
Torsional vibrations and buckling of thin WALLED BEAMSSRINIVASULU N V
The document discusses the torsional vibrations and buckling of thin-walled beams on elastic foundation using a dynamic stiffness matrix method. It develops analytical equations to model the behavior of clamped-simply supported beams under an axial load and resting on an elastic foundation. Numerical results are presented for natural frequencies and buckling loads for different values of warping and foundation parameters. The dynamic stiffness matrix approach can accurately analyze beams with non-uniform cross-sections and complex boundary conditions.
Computation of electromagnetic_fields_scattered_from_dielectric_objects_of_un...Alexander Litvinenko
Tools for electromagnetic scattering from objects with uncertain shapes are needed in various applications.
We develop numerical methods for predicting radar and scattering cross sections (RCS and SCS) of complex targets.
To reduce cost of Monte Carlo (MC) we offer modified multilevel MC (CMLMC) method.
Effect of the Thickness of High Tc Superconducting Rectangular Microstrip Pat...IJECEIAES
In recent years, a great interest has been observed in the development and use of new materials in microwave technology. Particularly, a special interest has been observed in the use of superconducting materials in microwave integrated circuits, this is due to their main characteristics. In this paper, the complex resonant frequency problem of a superconductor patch over Ground Plane with Rectangular Aperture is formulated in terms of an integral equation, the kernel of which is the dyadic Green‟s function. Galerkin‟s procedure is used in the resolution of the electric field integral equation. The surface impedance of the superconductor film is modeled using the two fluids model of Gorter and Casimir. Numerical results concerning the effect of the thickness of the superconductor patch on the characteristics of the antenna are presented.
Dispersion equation for groove nonradiative dielectric waveguideYong Heui Cho
This document presents a rigorous dispersion equation for a groove nonradiative dielectric (GNRD) waveguide. The equation is derived using Fourier transforms, mode matching, and an image theorem to transform the original waveguide into an equivalent structure of infinite waveguides. A closed-form dispersion relation is obtained in the form of a rapidly convergent series. Numerical results are presented that validate the dispersion equation and show good agreement with a simplified dominant-mode approximation at low frequencies. The dispersion equation provides an efficient way to model GNRD waveguides for applications in millimeter-wave antenna and circuit design.
Periodic material-based vibration isolation for satellitesIJERA Editor
This document presents research on using periodic materials to isolate satellite vibrations during launch. It begins with background on the severe vibrations satellites experience during launch and different vibration isolation methods. It then introduces the concept of using periodic composites, which exhibit frequency band gaps where vibrations are isolated. The document provides theoretical analysis to determine these band gaps based on the material properties and dimensions of the periodic isolator. It also describes preliminary experimental testing that showed a periodic foundation reduced frame vibrations by 60% compared to a non-periodic foundation. Finally, it presents a parametric study exploring how the band gaps are affected by the periodic constant and rubber filling fraction, and describes a finite element model of a satellite with a designed periodic isolator setup
Method of Moment analysis of a printed Archimedian Spiral antenna Piyush Kashyap
A single arm Archimedean spiral printed on a grounded dielectric substrate is analyzed using the method of moments. Piecewise sinusoidal subdomain basis and test functions are used over curved segments that exactly follow the spiral curvature. Results for the input impedance obtained using the curved segmentation approach on MATLAB are compared with those obtained after simulating the model on FEKO. A comparison with published results shows that the curved segment model requires fewer segments and is therefore significantly more computationally efficient than the linear segmentation model.
Overlapping T-block analysis of axially grooved rectangular waveguideYong Heui Cho
This document presents an analysis of the dispersion relations of an axially grooved rectangular waveguide using an overlapping T-block approach. The waveguide structure is divided into three overlapping asymmetric T-blocks. Field representations are derived for each T-block and combined to determine the total field. By enforcing field continuity conditions, the TE and TM dispersion relations are obtained. Numerical results for the cutoff frequencies are presented and show good agreement with other methods.
Circular Polarized Carbon-NanotubePatch Antenna embedded in Superstrates Anis...IOSRJEEE
The current distribution on surface of the Patch Antenna is radiating in the presence of the Superstrates layered anisotropic medium. The electric and magnetic field in the upper half-space are formulated using Dyadic Green Function in terms of Sommerfeld-Weyl-type integrals. The stationary phase method is used to obtain far-field expressions. Carbon-Nanotube Composite (CNT) as the radiating element for Star Patch Antenna is used. .Return loss, Axial Ratio ( <3 dB Circular Polarized), Gain and Directivity parameters of Carbon-Nanotube Star Patch Antenna are compared for different anisotropic materials
I am Grey Nolan. Currently associated with matlabassignmentexperts.com as an assignment helper. After completing my master's from the University of British Columbia, I was in search for an opportunity that expands my area of knowledge hence I decided to help students with their Signals and Systems assignments. I have written several assignments till date to help students overcome numerous difficulties they face in Signals and Systems Assignments.
Computation of electromagnetic fields scattered from dielectric objects of un...Alexander Litvinenko
This document describes using the Continuation Multi-Level Monte Carlo (CMLMC) method to compute electromagnetic fields scattered from dielectric objects of uncertain shapes. CMLMC optimally balances statistical and discretization errors using fewer samples on fine meshes and more on coarse meshes. The method is tested by computing scattering cross sections for randomly perturbed spheres under plane wave excitation and comparing results to the unperturbed sphere. Computational costs and errors are analyzed to demonstrate the efficiency of CMLMC for this scattering problem with uncertain geometry.
This document discusses electromagnetic transmission lines and the Smith chart. It introduces equivalent electrical circuit models for coaxial cables, microstrip lines, and twin lead transmission lines using distributed inductors and capacitors. The telegrapher's equations are derived from Kirchhoff's laws. For sinusoidal waves on the transmission lines, phasor analysis is used. Key concepts covered include characteristic impedance, propagation velocity, wavelength, and modeling forward and backward traveling waves.
This document analyzes a circular cylindrical dipole antenna using Fourier transforms and mode matching. It presents:
1) A rigorous solution for the input impedance and current distribution of a finite circular cylindrical dipole antenna in rapidly convergent series.
2) Numerical results showing good agreement between the presented solution and experimental data for input admittance and current distribution.
3) Analysis of the effect of cylinder radius on the angular radiation pattern, which varies less than 1dB for most angles as radius increases.
Similar to High gain metal only reflectarray antenna composed of multiple rectangular grooves (20)
The document discusses Android's Sensor Manager and how it works with sensors. It covers the SensorManager class, which allows apps to access sensor data, and the SensorEventListener interface that apps must implement to receive sensor updates. It also lists some of the different types of sensors available on Android devices like accelerometers, gyroscopes, and light sensors.
This document discusses BroadcastReceivers in Android. A BroadcastReceiver is an intent-based publish-subscribe system that allows apps to receive system events like SMS messages. BroadcastReceivers can receive and react to system broadcasts, broadcasts from other apps, and initiate broadcasts to other apps. They are registered either dynamically in code or statically in the AndroidManifest.xml file. Broadcasts are sent using the sendBroadcast or sendOrderedBroadcast methods and an Intent. Ordered broadcasts are executed in a defined order while normal broadcasts run asynchronously. The BroadcastReceiver object is only valid during the onReceive method call.
1. The document discusses the Android application lifecycle and how activities can transition between different states like onCreate, onStart, onResume, onPause, onStop, and onDestroy.
2. It also covers the activity lifecycle methods and how they relate to different states, as well as how to save and restore activity instance states.
3. Additionally, it provides comparisons between the Android and Windows lifecycles and messaging systems, and introduces concepts like handlers, loopers, threads, and the context class in Android.
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High gain metal only reflectarray antenna composed of multiple rectangular grooves
1. 1
High gain metal-only reflectarray antenna composed
of multiple rectangular grooves
Yong Heui Cho , Woo Jin Byuny, Myung Sun Songy
Department of Electrical and Computer Engineering, University of Massachusetts Amherst, on sabbatical leave
from School of Information and Communication Engineering, Mokwon University, Korea
yMillimeter-Wave Research Team, Electronics and Telecommunications Research Institute, Korea
Abstract—Using an overlapping T-block method based on
mode-matching technique and virtual current cancellation, the
scattering formulations for a metal-only reflectarray antenna
composed of multiple rectangular grooves are rigorously pre-
sented in fast convergent integrals. By matching the normal
boundary conditions at boundaries, we get the simultaneous
equations of the TE and TM modal coefficients for plane-wave
incidence and Hertzian dipole excitation. A metallic-rectangular-
grooves based reflectarray fed by a pyramidal horn antenna
was fabricated and measured with near-field scanning, thus
resulting in 42.3 [dB] antenna gain at f = 75 [GHz]. Our circular
reflectarray has 30 [cm] diameter and 5,961 rectangular grooves
on its metal surface. The simultaneous equations for a Hertzian
dipole feed are solved to approximately obtain the radiation
characteristics of a fabricated reflectarray. The measured results
are compared with those of the overlapping T-block method and
the FDTD simulation and they show favorable agreements in
terms of radiation patterns and antenna gain.
I. INTRODUCTION
In the millimeter-wave frequency bands, there have been
a lot of interesting applications in the fields of broadband
radio links for backhaul networking of cellular base stations,
Gbps-class Wireless Personal Area Network (WPAN), and
millimeter-wave imaging to detect concealed weapons and
non-metal objects [1]-[4]. Especially, 70 and 80 GHz commu-
nication systems within 1-mile distance will play an important
role in the next-generation wireless networks, because the cell
site connectivity will require more than 1 Gbps data rates.
For these millimeter-wave bands, a well-designed antenna
for narrow beamwidth and high antenna gain is essential to
compensation for severe free-space path loss and to prevention
against signal interference. Although a high gain antenna
can be manufactured by the concept of a parabolic reflector
antenna, a reflectarray antenna has been extensively studied
in [5]-[8], due to the fact that the reflectarray technology
has several advantages such as low profile, low cost, easy
manufacturing, low feeding loss, and simple controllability of
mainbeam and polarization.
Th reflectarray antenna in [5] consists of rectangular waveg-
uide arrays and a feed waveguide to reflect electromagnetic
waves efficiently. The phases of reflected waves are controlled
based on the variation of a surface impedance. In view of
a transmission line theory, the variation of waveguide depth
results in that of surface impedance at the end of each waveg-
uide. This indicates that reradiated waves can be designed
in the predefined way and then the high-gain antenna with
11,µεInfinite flange
PEC
a2
b2
x
y
22,µε
d
iθ
Incidence
z
iφ
φ
θ, x
y
Fig. 1. Geometry of a metallic rectangular groove in a perfectly conducting
plane
very low loss can be easily implemented. Modern reflectarrays
[6]-[8] have the same operational principle of the original
reflectarray [5]. In addition, the metal-only high gain antennas
in [8], [9] were proposed for the millimeter-wave bands, where
the loss characteristics are very important to maintain good
communication links.
In this work, we propose a novel analytic approach suitable
for a metal-only metallic-rectangular-grooves based reflectar-
ray. A two-dimensional (2D) metal-only reflectarray antenna
has already been analyzed and fabricated in [8]. In order
to analyze the problem of a three-dimensional (3D) metal-
only metallic-rectangular-grooves based reflectarray, we will
introduce an overlapping T-block method [8], [10] based on
mode-matching technique, virtual current cancellation, and
superposition principle, thereby obtaining analytic scattering
equations in rapidly convergent and numerically efficient inte-
grals. In view of mode-matching and Green’s function, we can
represent the discrete modal expansions for closed regions and
the continuous wavenumber spectra for open regions. Using
virtual current cancellation by means of virtual PEC covers
related to closed regions, the vector potential formulations for
open regions are easily evaluated. To apply the superposition
principle, we divide multiple rectangular grooves into several
simple T-blocks and a source block [8], [10].
II. FIELD REPRESENTATIONS FOR SINGLE GROOVE
Let’s consider the TE plane-wave incidence (Ez = 0) with
incident angles, i and i, shown in Fig. 1. The time conven-
tion e i!t is suppressed throughout. The incident magnetic
2. 2
field is
Hi(x;y;z) = 1
2
eiki r^i ; (1)
where the incident angles, i and i, are defined as i =
and i = + in terms of the - and -axes,
ki = k2 (sin i cos i^x + sin i sin i^y cos i^z) (2)
^i = cos i cos i^x cos i sin i^y sin i^z ; (3)
r = x^x + y^y + z^z, k2 = !p 2 2, and 2 =
p
2= 2. The
incident electric field for the TE plane-wave is also formulated
as
Ei (x;y;z) = ( sin i^x + cos i^y)eiki r : (4)
In order to match boundary conditions efficiently, we define
the reflected electromagnetic waves from a perfectly conduct-
ing plane at z = 0 as
Hr(x;y;z) = 1
2
eikr r^r (5)
Er(x;y;z) = (sin i^x cos i^y)eikr r ; (6)
where
kr = k2 (sin i cos i^x + sin i sin i^y + cos i^z) (7)
^r = cos i cos i^x + cos i sin i^y sin i^z : (8)
Similar to the TE plane-wave, we obtain the TM incident and
reflected plane-waves (Hz = 0) as, respectively,
Ei(x;y;z) = eiki r^i (9)
Er(x;y;z) = eikr r^r : (10)
Considering the TE (ui) and TM (vi) polarizations, the inci-
dent and reflected electric fields are represented as
Ei(x;y;z) =
h
(ui sin i + vi cos i cos i)^x
+ (ui cos i vi cos i sin i)^y
vi sin i^z
i
eiki r (11)
Er(x;y;z) =
h
(ui sin i + vi cos i cos i)^x
(ui cos i vi cos i sin i)^y
vi sin i^z
i
eikr r ; (12)
where ui and vi are polarization constants for the TE and TM
modes, respectively.
Since all electromagnetic fields can be formulated with
corresponding electric and magnetic vector potentials, we
introduce the electric vector potentials for regions (I) (z < 0)
and (II) (z 0) illustrated in Fig. 1 as
FI
z (x;y;z) = 1
1X
m=0
1X
n=0
qmn cosam(x + a)cosbn(y + b)
sin mn(z + d)uxy(a;b) (13)
FII
z (x;y;z) = 2
1X
m=0
1X
n=0
smn
h
Hmn(x;y;z)
+ RH
mn(x;y;z)
i
; (14)
b2
PEC
surface
a2
22 ,µε
d
Virtual
PEC
cover
11,µε
Radiation
boundary
xn ˆˆ =
x
yz
yn ˆˆ −=
Fig. 2. Artificial geometry for virtual current cancellation
where m+n 6= 0, qmn and smn are the unknown TE modal co-
efficients for regions (I) and (II), respectively, am = m =(2a),
bn = n =(2b), mn =
p
k2
1 a2m b2n, k1 = !p 1 1,
uxy(a;b) = u(x+a) u(x a)] u(y+b) u(y b)], and u( )
is a unit-step function. Enforcing the Hz field continuity at
z = 0 yields
smn = qmn sin( mnd) ; (15)
where we presume that RH
mn(x;y;0) = 0,
Hmn(x;y;z) = ei mnz cosam(x + a)cosbn(y + b)
uxy(a;b) ; (16)
and mn =
pk2
2 a2m b2n. Note that Hmn(x;y;z) in (16)
is formulated with virtual PEC covers placed at ( a x
a;y = b;z > 0) and (x = a; b y b;z > 0)
shown in Fig. 2. Even though the virtual PEC covers in Fig.
2 are absent from the original geometry illustrated in Fig. 1,
the PEC covers are artificially inserted to accommodate the
field formulations which will be described in (17). In order
to make the fields Hmn(x;y;z) + RH
mn(x;y;z) continuous
for z > 0, we define a scattered component RH
mn(x;y;z)
which is implicitly related to Hmn(x;y;z). Adding the vir-
tual PEC covers inevitably generates the unwanted surface
current densities on ( a x a;y = b;z > 0) and
(x = a; b y b;z > 0). By means of the Green’s
function, RH
mn(x;y;z) is utilized to remove a current density
J(r) (= ( @2
@z2 + k2
2)Hmn(r)^z ^n) produced by the Hz-field
discontinuities. Then,
@2
@z2 + k2
2 RH
mn(x;y;z)
= i! r A(r)
z component
=
Z @2
@z02 + k2
2 Hmn(r0) @
@n
h
Gxx
A (r;r0)
i
dr0 ; (17)
where ^n is an outward normal unit vector denoted in Fig.
2 and Gxx
A (r;r0) indicates the x-directional Green function
excited by the x-directed source in terms of a magnetic vector
3. 3
potential A(r). Inserting (16) into (17) yields
RH
mn(x;y;z)
= a2
m + b2
n
2 2
Z 1
0
sin( z)
(k2
2
2)( 2 2mn)
nZ 1
1
( 1)nfH(y;b; ) fH(y; b; )
Gm(a; )ei x d
+
Z 1
1
( 1)mfH(x;a; ) fH(x; a; )
Gn(b; )ei y d
o
d ; (18)
where k2
2 = 2 + 2 + 2,
fH(y;y0; ) = sgn(y y0)ei jy y0
j (19)
Gm(a; ) = i e i a ( 1)mei a]
2 a2m
; (20)
and sgn( ) = 2u( ) 1. It should be noted that our previous
assumption such as RH
mn(x;y;0) = 0 is right because of
sin( z) = 0
z=0
. To avoid pole singularities at = nm
and = k2 on the real axis, we deform the integral path
as = k2v(v i) for v 0 [10]. By using this substitution,
a radiation integral (18) can be transformed to that without
singularities as
RH
mn(x;y;z)
= k2(a2
m + b2
n) Z 1
0
(2v i) sin( z)
(k2
2
2)( 2 2mn)h
( 1)nQm(x; a;y;b;k2
2
2)
Qm(x; a;y; b;k2
2
2)
+ ( 1)mQn(y; b;x;a;k2
2
2)
Qn(y; b;x; a;k2
2
2)
i
dv ; (21)
where a precise and efficient evaluation of Qm( ) is presented
in Appendix A. For numerical computation, a path-deforming
variable v in (21) can be empirically truncated to
RVmax
0 ( ) dv
as
Vmax = max 10
vt
;1 ; (22)
where max(x;y) is the maximum value of x and y,
vt = 1
2
0
@
s
jxj
2 2
+ 1+
s
jyj
2 2
+ 1
1
A ; (23)
and 2 = 2 =k2. Applying the Green’s second integral
identity, we reduce (16) and (18) as
@2
@z2 + k2
2 Hmn(x;y;z) + RH
mn(x;y;z)
=
Z a
a
Z b
b
@2
@z02 + k2
2 Hmn(r0)
@
@z0Gxx
A (r;r0) dy0dx0
z0=0
: (24)
The integral (24) is numerically efficient for jxj a or jyj
b where the Green function Gxx
A (r;r0) in (24) does not have
any singularity in the region of jx0j a and jy0j b . Using
(24), we obtain the asymptotic Fz-potential in region (II) as
FII
z (r; ; )
eik2r
i2 r
cos
k2 sin2 2
1X
m=0
1X
n=0
qmn(a2
m + b2
n)sin( mnd)
Gm(a; k2 sin cos )Gn(b; k2 sin sin ) ; (25)
where r =
p
x2 + y2 + z2, = cos 1(z=r), and =
tan 1(y=x) shown in Fig. 1.
In the next step, the magnetic vector potentials for Fig. 1
are formulated as
AI
z(x;y;z) = 1
1X
m=0
1X
n=0
pmn sinam(x + a)sinbn(y + b)
cos mn(z + d)uxy(a;b) (26)
AII
z (x;y;z) = 2
1X
m=0
1X
n=0
h
rmnEmn(x;y;z)
+ RE
mn(x;y;z)
i
; (27)
where m n 6= 0, pmn and rmn are the unknown TM modal
coefficients for regions (I) and (II), respectively. Applying the
@Ez=@z e= field continuity at z = 0, we get
rmn = 2
1
pmn mn sin( mnd) ; (28)
where e (= 1EI
x;y ^n) is an equivalent electric charge
density which may be produced by the field discontinuities at
boundaries, we assume that @=@zRE
mn(x;y;z)jz=0 = 0, and
Emn(x;y;z) = ei mnz
i mn
sinam(x + a)sinbn(y + b)
uxy(a;b) : (29)
Similar to the Green’s function relation already described in
(17), we get the formula for RE
mn(x;y;z) as
RE
mn(x;y;z)
=
Z
J(r0)Gzz
A (r;r0) dr0
z component
= rmnREE
mn(x;y;z) ismn
! 2
REH
mn(x;y;z) ; (30)
where J(r) = Hx;y(r) ^n. To remove the Hx-field discon-
tinuities at ( a x a, y = b, z 0) and the Hy-field
discontinuities at (x = a, b y b, z 0), REE
mn(r) and
REH
mn(r) are given by, respectively,
REE
mn(x;y;z) =
Z @
@n0
h
Em(r0)
i
Gzz
A (r;r0) dr0 (31)
@2
@z2 + k2
2 REH
mn(x;y;z)
=
Z @2
@z02 + k2
2 Hmn(r0) @2
@z@tGxx
A (r;r0) dr0
+
Z @2
@z0@t0Hmn(r0) @2
@z2 + k2
2 Gzz
A (r;r0) dr0 ; (32)
4. 4
where ^n ^z = ^t illustrated in Fig. 1. The expression for
REE
mn(x;y;z) is obtained from (29) and (31), and written by
REE
mn(x;y;z)
= k2
Z 1
0
(2v i)cos( z)
2 2mnn
bn ( 1)nPm(x; a;y;b;k2
2
2)
Pm(x; a;y; b;k2
2
2)
+ am ( 1)mPn(y; b;x;a;k2
2
2)
Pn(y; b;x; a;k2
2
2)
o
dv ; (33)
where = k2v(v i) and Pm( ) is formulated in Appendix
A. Similarly, integrating (32) with (16), we get
REH
mn(x;y;z)
= k2
Z 1
0
(2v i)cos( z)
2 2mnn
( 1)nTmn(x; a;y;b;k2; )
Tmn(x; a;y; b;k2; )
( 1)mTnm(y; b;x;a;k2; )
+ Tnm(y; b;x; a;k2; )
o
dv ; (34)
where = k2v(v i) and Tmn( ) is shown in Appendix A.
Note that @=@zREE;EH
mn (x;y;z)jz=0 = 0 are satisfied due to
@=@zcos( z) = 0
z=0
.
Using the Green’s second integral identity, we simplify (29)
and (31) as
Emn(x;y;z) + REE
mn(x;y;z)
=
Z a
a
Z b
b
@
@z0
h
Emn(r0)
i
Gzz
A (r;r0) dy0dx0
z0=0
: (35)
Similarly, REH
mn(x;y;z) is reduced to
@2
@x@y
@2
@z2 + k2
2 REH
mn(x;y;z)
= k2
2
Z a
a
Z b
b
Hmn(r0)
b2
n
@2
@x2 a2
m
@2
@y2 Gzz
A (r;r0) dy0dx0
z0=0
: (36)
In the far-field, the Az-potential in region (II) is asymptotically
represented as
AII
z (r; ; )
eik2r
2 r
(
2 2
1
1X
m=0
1X
n=0
pmn mn sin( mnd)
Fm(a; k2 sin cos )Fn(b; k2 sin sin )
+ 1
i!sin2
1X
m=0
1X
n=0
qmn sin( mnd)(b2
n cot a2
m tan )
Gm(a; k2 sin cos )Gn(b; k2 sin sin )
)
; (37)
),( )1()1(
ST
),( )2()2(
ST ),( )3()3(
ST ...
)1()2(
TT − )2()3(
TT −
),( )1()1( ++ XX PP
ST ),( )2()2( ++ XX PP
ST ),( )3()3( ++ XX PP
ST
),( )12()12( ++ XX PP
ST ),( )22()22( ++ XX PP
ST ),( )32()32( ++ XX PP
ST
...
...
...
...
...
)1()1(
SS XP
−+
x
yz
Fig. 3. Geometry of multiple rectangular grooves in a perfectly conducting
plane (PX: the number of grooves in the x-axis and (T(p);S(p)): a
translation point of the pth groove in the x-y plane)
where
Fm(a; ) = am ( 1)mei a e i a]
2 a2m
: (38)
III. FIELD MATCHING FOR MULTIPLE GROOVES
Due to large number of metallic rectangular grooves illus-
trated in Fig. 3, scattering formulations and analyses are a
little complicated. These difficulties can be easily overcome
with field superposition principle [10], [11]. In the field
superposition, electromagnetic fields in open region (z 0)
related to those of each metallic rectangular groove (jxj a,
jyj b, and z < 0) for Fig. 1 are additively superimposed to
produce the total electromagnetic fields for multiple grooves
shown in Fig. 3. Based on the superposition principle, the total
electric and magnetic vector potentials are represented as
Ftot
z (x;y;z) =
PTX
p=1
T(p)
H (x T(p);y S(p);z) (39)
Atot
z (x;y;z) =
PTX
p=1
T(p)
E (x T(p);y S(p);z) ; (40)
where PT is the total number of grooves, T(p) and S(p) of
the pth groove are translation positions for the x- and y-axes,
respectively, and
T(p)
H (x;y;z) = FI(p)
z (x;y;z) + FII(p)
z (x;y;z) (41)
T(p)
E (x;y;z) = AI(p)
z (x;y;z) + AII(p)
z (x;y;z) : (42)
By matching the @Hz=@z m= and Ez fields continuities
at z = 0, we can obtain simultaneous scattering equations for
arbitrarily polarized plane-wave incidence in (11). Multiplying
the @Hz=@z m= continuity at z = 0 with cosa(r)
l (x
x0 + a(r))cosb(r)
k (y y0 + b(r)) (l = 0;1; , k = 0;1; ,
l+k 6= 0, r = 1;2; ;PT) for the rth groove and integrating
over x0 a(r) x x0 +a(r) and y0 b(r) y y0 +b(r)
yields
PTX
p=1
1X
m=0
1X
n=0
2
(p)
1
p(p)
mn
(p)
mn sin( (p)
mnd(p))IEE
mn;lk(x0;y0)
+
h
(a(r)
l )2 + (b(r)
k )2
i
i! 2
PTX
p=1
1X
m=0
1X
n=0
q(p)
mn
5. 5
(
2
(p)
1
(p)
mn cos( (p)
mnd(p))a(p)b(p)
m n ml nk pr
sin( (p)
mnd(p))
h
i (p)
mna(p)b(p)
m n ml nk pr
+ IH
mn;lk(x0;y0) +
IEH
mn;lk(x0;y0)
(a(r)
l )2 + (b(r)
k )2
i)
= 2i
! 2
Gl(a(r);k2 sin i cos i)Gk(b(r);k2 sin i sin i)
ei 0
n
ui
h
(a(r)
l )2 + (b(r)
k )2
i
cot i
+ vi
sin i
h
(b(r)
k )2 cot i (a(r)
l )2 tan i
io
; (43)
where m (= 2HII
x;y ^n) is an equivalent magnetic charge
density generated by the field discontinuities between regions
(I) and (II) placed at z = 0, ( )(p) is a parameter for the
pth groove, a(r)
l = l =(2a(r)), b(r)
k = k =(2b(r)), 0 =
k2 sin i(x0 cos i + y0 sin i), m = m0 + 1, ml is the
Kronecker delta, and IH
mn;lk( ), IEE
mn;lk( ), IEH
mn;lk( ) are defined
in Appendix B. Note that (x0 = T(r);y0 = S(r)) is a center
point of the rth groove for field matching and m is an
inevitable term which must be included in normal magnetic
field matching.
Similarly, multiplying the Ez continuity with sina(r)
l (x
x0+a(r))sinb(r)
k (y y0+b(r)) (l = 1;2; , k = 1;2; , r =
1;2; ;PT) for the rth groove and integrating with respect
to x and y gives
PTX
p=1
1X
m=0
1X
n=0
p(p)
mn
(h
(a(p)
m )2 + (b(p)
n )2
i
cos( (p)
mnd(p))a(p)b(p)
ml nk pr
+ 2
(p)
1
(p)
mn sin( (p)
mnd(p))
h(a(p)
m )2 + (b(p)
n )2
i (p)
mn
a(p)b(p)
ml nk pr + JEE
mn;lk(x0;y0)
i)
+ i
! 2
PTX
p=1
1X
m=0
1X
n=0
q(p)
mn sin( (p)
mnd(p))JEH
mn;lk(x0;y0)
= 2i! 2vi sin iFl(a(r);k2 sin i cos i)
Fk(b(r);k2 sin i sin i)ei 0 ; (44)
where JEE
mn;lk( ), JEH
mn;lk( ) are defined in Appendix B.
IV. NUMERICAL COMPUTATIONS AND MEASUREMENT
Plane-wave scattering from a rectangular metallic groove
in a perfectly conducting infinite plane is extensively studied
with numerical [12], [13] and analytic [14] techniques. In
order to validate our formulations, (43) and (44), we compared
our numerical results for a wide groove (2a = 2:5 0 and
2b = 0:25 0 in Fig. 1) with other simulations [12], [14].
Fig. 4 illustrates the normalized backscattered co-polarization
radar cross section (RCS) for an incident angle ( i) of a plane-
wave illustrated in Fig. 1. All simulated results in Fig. 4 are
strongly consistent for i 70 . In addition, Fig. 4 indicates
the convergence behavior of our simultaneous equations, (43)
0 15 30 45 60 75 90
−40
−30
−20
−10
0
Incident angle, θ
i
[Degree]
Backscatteredco−poleRCS,σ/λ
0
2
[dB]
M = 4
M = 8
M = 12
M = 16
M = 20
[12] FEM
[14] Fourier transform
Fig. 4. Behaviors of the backscattered co-polarization RCS ( = 2
0) versus
a plane-wave incident angle ( i) with PT = 1, N = 2, 2a = 2:5 0,
2b = d = 0:25 0, i = 0, ui = 0, vi = 1, 1 = 2 = 0, 1 = 2 = 0
Fig. 5. Fabricated 3D metal-only reflectarray composed of rectangular
grooves with a whole diameter (D0) = 30 [cm] and the total number of
grooves (PT ) = 5,961
and (44), where M and N denote the truncation numbers of
modal coefficients with respect to m and n in (13) and (26) for
numerical computation, respectively. As the number of modes
for m increases, the backscattered RCS converges very fast
for any i. A lower-mode solution (M = 4 and N = 2) is
very good approximation for i 35 .
Fig. 5 shows a fabricated three dimensional (3D) metal-only
reflectarray with prime focus composed of multiple rectangular
grooves. A thick circular metal plate with 30 [cm] diameter
(D0) and 1 [cm] thickness contains 5,961 rectangular metallic
grooves. A pyramidal horn antenna used as a feed has 7 [mm]
5 [mm] aperture size and 12 [mm] waveguide transition,
and an input waveguide for a feed is WR-12 (3.1 [mm] 1.5
[mm]). For simple design, we assume that each rectangular
groove illustrated in Figs. 3 and 5 has the same aperture size
of a(p) = ag, b(p) = bg and the same separations of T(p+1)
6. 6
T(p) = T, S(p+PX) S(p) = S for all p in (39) and (40),
where PX is the number of grooves in the x-axis. A depth for
the pth groove d(p) is individually calculated with a formula
based on the phase matching condition [8] as
d(p) = g
2 2
"
d0 + f0
q
(T(p))2 + (S(p))2 + (f0 d0)2
#
; (45)
where f = 78:5 [GHz], g = 2 = , =
pk2
2 =(2a(p))]2,
f0 and d0 are a focus and the maximum depth of a paraboloid,
respectively, which is effectively formed by a metal-only flat
reflectarray in Fig. 5. Because of the periodicity of reflected
phase, the depth in (45) can be limited to half the guided
wavelength [8] ( 2.38 [mm] for 78.5 [GHz]). Considering
the phase center of a pyramidal horn antenna denoted as f
9 mm], the feed in Fig. 5 is placed at (xi = 0;yi = 0;zi+f )
where is the focus of a paraboloid, f0 = zi + d0 [8].
Fig. 6 presents the H- and E-plane radiation patterns of a
metal-only reflectarray in Fig. 5 for 78.5 [GHz] and RA (Rel-
ative Aperture) = f0=D0 = 0.75. In our computations, we used
the simultaneous equations, (43) and (44), with Hertzian dipole
excitation polarized in the y-axis. This means that the right
hand sides of (43) and (44) should be modified for a Hertzian
dipole. Our formulations based on the overlapping T-block
method are compared with planar near-field measurement
[8], [16] and numerical simulation [15]. We obtained planar
near-field measurement results with a WR-10 (2.54 [mm]
1.27 [mm]) OERW (Open-Ended Rectangular Waveguide)
probe and the parameters such as distance between probe and
reflectarray = 30.8 [cm], sampling step = 1.71 [mm], and
scan range = 52.497 [cm]. In Fig. 6, the radiation behaviors
of our method and FDTD simulation agree very well for all
observation angle. The GEMS parallel FDTD simulation [15]
for Figs. 5 and 6 requires the parameters such as the number
of cells = 2 109, total memory = 66 [GB], the number of
parallel processors = 54, and simulation time = 18.7 hours.
The FDTD simulation is performed for the geometry shown in
Fig. 5 without three metallic struts to support a pyramidal horn
feed. In contrast to the FDTD simulation, our calculation time
with CPU 2 [GHz] and RAM 2 [GB] is 4.4 minutes. Fig. 6 also
shows the discrepancy between simulations and measurement
results in the side-lobe region. This noticeable difference is
caused by our simple feed modeling such as Hertzian dipole
excitation and finite measurement scan area (52.497 [cm]
52.497 [cm]) [16].
Fig. 7 shows characteristics of co- and cross-polarization
excited gain patterns for 78.5 [GHz]. The co- and cross-
polarized excitations were simulated with Hertzian dipoles
polarized in the y- and x-axes, respectively, when all pa-
rameters of a metal-only reflectarray were fixed. In case of
cross-polarized excitation, radiation patterns have the null
point at = 0 and their side-lobe levels are approximately
15 [dB] higher than those of co-polarized excitation. Fig. 8
indicates a succinct comparison of gain behaviors in terms of
our method, FDTD simulation, and near-field measurement.
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
Observation angle, θ [Degree]
H−plane(φ=0
°
)gainpatterns[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
(a) H-plane ( = 0 )
−90 −60 −30 0 30 60 90
−10
0
10
20
30
40
Observation angle, θ [Degree]
E−plane(φ=90
°
)gainpatterns[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
(b) E-plane ( = 90 )
Fig. 6. Characteristics of the H- and E-plane antenna gain patterns versus
an observation angle ( ) with f = 78.5 [GHz], PT = 5961, M = 2, N = 1,
2ag = 3:2 mm], 2bg = 2:7 mm], T 2ag = S 2bg = 0:5 mm],
xi = yi = 0, zi = 193:845 mm], 1 = 2 = 0, 1 = 2 = 0,
RA = 0:75, d0 = 25 mm], d(p)
obtained from (45)
Although the discrepancy among simulated and measured
results is maximally 3 [dB], overall tendency of gain behaviors
is not significantly different among results. It should be noted
that antenna gain of our method is higher than others, due
to the fact that our computation is based on Hertzian dipole
excitation and thus cannot include the feed characteristics. The
measured aperture efficiencies for 75, 77, 78.5 [GHz] are 30.2,
27.2, 23.3 [%], respectively.
V. CONCLUSIONS
Rigorous and analytic solutions for scattering from mul-
tiple rectangular grooves in a perfectly conducting plane are
obtained with the overlapping T-block method based on super-
position principle and Green’s function relation. The simulta-
neous scattering equations for Hertzian dipole excitation can
be utilized to predict radiation characteristics of a metal-only
7. 7
−90 −60 −30 0 30 60 90
0
10
20
30
40
Observation angle, θ [Degree]
Co−andcross−polegainpatterns[dB]
Crosspole, φ = 0°
Crosspole, φ = 90°
Copole, φ = 0°
Copole, φ = 90
°
Fig. 7. Behaviors of the co- and cross-polarization excited gain patterns
versus an observation angle ( ) (The parameters are selected from those in
Fig. 6)
73 75 77 79 81
36
38
40
42
44
46
Frequency [GHz]
Co−poleantennagain[dB]
Overlapping T−blocks
GEMS (parallel FDTD)
Measurement
Fig. 8. Co-polarization excited antenna gain variations versus a frequency
(The parameters are chosen from those in Fig. 6)
reflectarray fed by a pyramidal horn antenna. Our simulations
were compared with commercial FDTD computation and near-
field measurement and all results show favorable agreements.
The mode-matching and Green’s function approach for a
metal-only reflectarray with rectangular grooves can be ex-
tended to that with non-rectangular grooves by using suitable
modal expansions. In further work, we will investigate the gen-
eral feed modeling in near-field region and the corresponding
phase matching condition for a metal-only reflectarray.
APPENDIX A: INTEGRALS FOR Qm( ), Pm( ), AND Tmn( )
The definitions of Qm( ), Pm( ), and Tmn( ) are written by
Qm(x; a;y;y0;k2)
Original path
[ ]ηRe
[ ]ηIm
2k
Branch cut
Deformed path
k−
k
Fig. 9. Deformed integral path to remove singularities
= 1
2
Z 1
1
fH(y;y0; )Gm(a; )ei x d (46)
Pm(x; a;y;y0;k2)
= 1
2
Z 1
1
ei jy y0
j
i Fm(a; )ei x d (47)
Tmn(x; a;y;y0;k; )
=
2(a2
m + b2
n)
k2 2 Sm(x; a;y;y0;k2 2)
+ 2
mnamPm(x; a;y;y0;k2 2) ; (48)
where k2 = 2 + 2, am = m =(2a), bn = n =(2b), mn =pk2
2 a2m b2n, and
Sm(x; a;y;y0;k2)
= 1
2
Z 1
1
ei jy y0
j
Gm(a; )ei x d : (49)
To evaluate Qm( ) efficiently, we utilize the residue cal-
culus. Thus, (46) can be transformed in terms of pole and
branch-cut contributions as
Qm(x; a;y;y0;k2)
= sgn(y y0)ei mjy y0
j cosam(x + a)ux(a)
+ 1 Z 1
0
sin (y y0)]
2 a2m
h
sgn(x + a)ei jx+aj
( 1)msgn(x a)ei jx aj
i
d ; (50)
where ux(a) = u(x + a) u(x a) and k2 = a2
m + 2
m.
However, the integral (50) has rapidly oscillating behaviors
when jy y0j 1. These oscillatory characteristics can be
removed with proper path deforming. As such, we propose a
novel integral path shown in Fig. 9 which always Im ] 0
for any u as
=
8
<
:
k2u(u+ i) (u < 0)
k2u2 (0 u 1)
k2 u(u+ i) i] (u > 1)
: (51)
Based on the path parameter (51) and Fig. 9, we modify (50)
8. 8
to a fast convergent integral without singularities as
Qm(x; a;y;y0;k2)
= sgn(y y0)
(
ei mjy y0
j cosam(x + a)ux(a)
i
2
Z 1
1
d
du ei jy y0
j
2 a2m
h
sgn(x + a)ei jx+aj
( 1)msgn(x a)ei jx aj
i
du
)
: (52)
Since the integrand in (52) has complex exponential functions
and the complex numbers, m, , and in (52) have positive
imaginary parts, the integral (52) converges exponentially. This
means that the double integral (18) with Qm( ) also converges
very rapidly. For Gaussian quadrature technique, the integral
(52) can be empirically truncated to
RUmax+1
Umax ( ) du as
Umax = max
"s
ut ut + 4 2
jyj+ jy0j + v2;1
#
; (53)
where v is defined in (21) as k =
p
k2
2
2 =
k2
p1 + v2(1 v2) + 2v3i and
ut = jyj+ jy0j
2(jxj+ jaj) : (54)
When 0 u 1 and jy y0j 1, the integral (52) still has
unwanted numerical oscillations with respect to u. To avoid
numerical oscillations of integrand in (52), we analytically
reduce (52) to a finite integral as
Qm(x; a;y;y0;k2)
= ik(y y0)
2
Z a
a
H(1)
1 kRxy(x0;y0)]
Rxy(x0;y0)
cosam(x0 + a) dx0 ; (55)
where H(1)
m ( ) is the mth order Hankel function of the first
kind and Rxy(x0;y0) =
p(x x0)2 + (y y0)2. Note that
(55) is very efficient for numerical computation when jx
x0j 1 or jy y0j 1. Similar to the evaluation of Q( ), we
obtain the following integrals as
Pm(x; a;y;y0;k2)
= ei mjy y0
j
i m
sinam(x + a)ux(a)
+ iam
2
Z 1
1
d
duei jy y0
j
( 2 a2m)
h
ei jx+aj ( 1)mei jx aj
i
du
(56)
Sm(x; a;y;y0;k2)
= iamei mjy y0
j
m
sinam(x + a)ux(a)
i
2
Z 1
1
d
du ei jy y0
j
2 a2m
h
ei jx+aj ( 1)mei jx aj
i
du :
(57)
For large argument approximation (jx x0j 1 or jy y0j
1), (56) and (57) are also formulated as
Pm(x; a;y;y0;k2)
= i
2
Z a
a
H(1)
0 kRxy(x0;y0)]sinam(x0 + a) dx0 (58)
Sm(x; a;y;y0;k2)
= ik
2
Z a
a
H(1)
1 kRxy(x0;y0)] (x x0)
Rxy(x0;y0)
cosam(x0 + a) dx0 : (59)
APPENDIX B: MATCHING INTEGRALS
The matching integrals for simultaneous modal equations,
(43) and (44), are defined as
IH
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@
@zRH
mn(x;y;z)
z=0
cosa(r)
l (x x0 + a(r))cosb(r)
k (y y0 + b(r)) dydx
(60)
IEE
mn;lk(x0;y0)
= a(r)
l
Z x0+a(r)
x0 a(r)
REE
mn(x;y;0)sina(r)
l (x x0 + a(r))
cosb(r)
k (y y0 + b(r)) dx
y=y0+b(r)
y=y0 b(r)
b(r)
k
Z y0+b(r)
y0 b(r)
REE
mn(x;y;0)cosa(r)
l (x x0 + a(r))
sinb(r)
k (y y0 + b(r)) dy
x=x0+a(r)
x=x0 a(r)
(61)
IEH
mn;lk(x0;y0)
= a(r)
l
Z x0+a(r)
x0 a(r)
REH
mn(x;y;0)sina(r)
l (x x0 + a(r))
cosb(r)
k (y y0 + b(r)) dx
y=y0+b(r)
y=y0 b(r)
b(r)
k
Z y0+b(r)
y0 b(r)
REH
mn(x;y;0)cosa(r)
l (x x0 + a(r))
sinb(r)
k (y y0 + b(r)) dy
x=x0+a(r)
x=x0 a(r)
(62)
JEE
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@2
@z2 + k2
2 REE
mn(x;y;z)
z=0
sina(r)
l (x x0 + a(r))sinb(r)
k (y y0 + b(r)) dydx
(63)
JEH
mn;lk(x0;y0)
=
Z x0+a(r)
x0 a(r)
Z y0+b(r)
y0 b(r)
@2
@z2 + k2
2 REH
mn(x;y;z)
z=0
9. 9
sina(r)
l (x x0 + a(r))sinb(r)
k (y y0 + b(r)) dydx :
(64)
Since the integrands from (60) to (64) are composed of
simple elementary functions, we can easily evaluate the above
integrals in closed form. When 0 =
px2
0 + y2
0 1, (60)
through (64) are approximately formulated as
IH
mn;lk(x0;y0)
(a(p)
m )2 + (b(p)
n )2 (1 ik2 0)
2 k2
2
3
0
Gm(a(p); k2 cos 0)Gl(a(r);k2 cos 0)
Gn(b(p); k2 sin 0)Gk(b(r);k2 sin 0)eik2 0 (65)
IEE
mn;lk(x0;y0)
eik2 0
2 0
Fm(a(p); k2 cos 0)Fn(b(p); k2 sin 0)
(
a(r)
l Fl(a(r);k2 cos 0)
h
e ik2 sin 0b(r)
( 1)keik2 sin 0b(r)i
b(r)
k Fk(b(r);k2 sin 0)
h
e ik2 cos 0a(r)
( 1)leik2 cos 0a(r)i)
(66)
IEH
mn;lk(x0;y0)
eik2 0
2 0
(b(p)
n )2 cot 0 (a(p)
m )2 tan 0
Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)(
a(r)
l Fl(a(r);k2 cos 0)
h
( 1)keik2 sin 0b(r)
e ik2 sin 0b(r)i
b(r)
k Fk(b(r);k2 sin 0)
h
( 1)leik2 cos 0a(r)
e ik2 cos 0a(r)i)
(67)
JEE
mn;lk(x0;y0)
k2
2eik2 0
2 0
Fm(a(p); k2 cos 0)Fl(a(r);k2 cos 0)
Fn(b(p); k2 sin 0)Fk(b(r);k2 sin 0) (68)
JEH
mn;lk(x0;y0)
k2
2eik2 0
2 0
(b(p)
n )2 cot 0 (a(p)
m )2 tan 0
Gm(a(p); k2 cos 0)Gn(b(p); k2 sin 0)
Fl(a(r);k2 cos 0)Fk(b(r);k2 sin 0) ; (69)
where 0 = tan 1(y0=x0) and ( )(p);(r) are the parameters
for the pth and rth grooves. It should be noted that the
formulations in (65) through (69) are very useful to obtain
modal matrixes of the simultaneous scattering equations, (43)
and (44) for 0 1. This is because the simplified integrals
in (65) through (69) are in closed form without double infinite
integrals, whereas the original integrals, (60) through (64),
still have infinite integrals. By using simplified integrals, (65)
through (69), we can compute the modal matrixes for a very
large metal-only reflectarray very efficiently.
ACKNOWLEDGEMENT
This work was supported by the IT R&D program of
MKE/KCC/KCA (2008-F-013-04, Development of Spectrum
Engineering and Millimeterwave Utilizing Technology).
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