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EMLAB 1 2. Radiation integral

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EMLAB 2 EM radiation Constant velocity Constant acceleration Periodic motion Accelerating charges radiate E and H proportional to 1/R. +q

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EMLAB 3 Basic laws of EM theory 1) Maxwell’s equations 2) Continuity equation (the relation between current density and charge density in a space) 3) Constitutive relation (explains the properties of materials) 4) Boundary conditions ( should be satisfied at the interface of two materials by E, H, D, B.)

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EMLAB 4 Continuity equation : Kirchhoff ’s current law Charges going out through dS. For steady state, charges do not accumulate at any nodes, thus ρ become constant. differential form integral form Kirchhoff ’s current law

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EMLAB 5 Boundary conditions unit vector tangential to the surface Unit vector normal to the surface Medium #1 Medium #2

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EMLAB 6 Two important vector identities 1) 2) ( ϕ : arbitrary scalar function) (A: arbitrary vector function)

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EMLAB 7 Potentials of time-varying EM theory

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EMLAB 8 Lorentz condition To find unique value of vector potential A, the divergence and the curl of A should be known. Only the curl of A is physically observed, divergence of A can be arbitrarily set. For the above choice, (3), (4) become Lorentz condition

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EMLAB 9 Fourier transform solution boundary condition : free space 1.Because the number of variables are as many as four (x, y, z, t), we apply Fourier transform to the above equations. 2.For a non-homogeneous differential equation, it is easier to substitute the source term with a delta function located at origin. (effectively it is an impulse response.)

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EMLAB 10 Solution of Maxwell’s eqs for simple cases

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EMLAB 11 Domain : infinite space Domain : interior of a rectangular cavity Domain and boundary conditions The constraints on the behavior of electric and magnetic field near the interface of two media which have different electromagnetic properties. (e.g. PEC, PMC, impedance boundary, …) Domain : interior of a circular cavity waveguide

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EMLAB 12 1-D example : Radiation due to Infinite current sheet 1.Using phasor concept in solving Helmholtz equation, x y z 2. With an infinitely large surface current on xy-plane, variations of A with coordinates x and y become zero. Then the Laplacian is reduced to derivative with respect to z. 3. If the current sheet is located at z=0, it can be represented by a delta function with an argument z. If the current flow is in the direction of x-axis, the only non-zero component of A is x-component.

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EMLAB 13 4. With a delta source, it is easier to consider first the region of z≠0. 5. Four kinds of candidate solutions can satisfy the differential equation only. Of those, exponential functions can be a propagating wave.. 6. Solutions propagating in either direction are 7. The condition that A should be continuous at z=0 forces C 1 =C 2

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EMLAB 14 8. To find the value of C, integrate both sides of the original Helmholtz equation.

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EMLAB 15 E E H H Propagating direction An infinite current sheet generates uniform plane waves whose amplitude are uniform throughout space. Plane wave 정의 E H Electric field : even symmetry Magnetic field : odd symmetry Propagating direction

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EMLAB 16 Source Infinitesimally small current element in free space : 3D

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EMLAB 17 Solution of wave equations in free space Boundary condition: Infinite free space solution. 1.As the solutions of two vector potentials are identical, scalar potential is considered first. 2.To decrease the number of independent variables (x, y, z, t), Fourier transform representation is used. 3. For convenience, a point source at origin is considered.

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EMLAB 18 Green function of free space where A suitable solution which is propagating outward from the origin is e -jkr. 1.The solution of the differential equation with the source function substituted by a delta function is called Green g, and is first sought. 2.With a delta source, consider first the region where delta function has zero value. Then, utilize delta function to find the value of integration constant. 3. With a point source in free space, the solution has a spherical symmetry. That is, g is independent of the variables , , and is a function of r only.

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EMLAB 19 Green function of free space 4. To determine the value of A, apply a volume integral operation to both sides of the differential equations. The volume is a sphere with infinitesimally small radius and its center is at the origin. 5. With a source at r’, the solution is translated such that

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EMLAB 20 6. As the original source function can be represented by an integral of a weighted delta function, the solution to the scalar potential is also an integral of a weighted Green function. 7. Taking the inverse Fourier transform, the time domain solution is obtained as follows.

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EMLAB 21 Retarded potential (Retarded potential) The distinct point from a static solution is that a time is retarded by R/c. This newly derived potential is called a retarded potential. The vector potential also contains a retarded time variable. Those A and are related to each other by Lorentz condition.

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EMLAB 22 Solution in time & freq. domain

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EMLAB 23 Far field approximation Electrostatic solution Biot-Savart’s law Coulomb’s law

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EMLAB 24 Electric field in a phasor form

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EMLAB 25 Radiation pattern of an infinitesimally small current

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EMLAB 26 Example – wire antenna Current distributions along the length of a linear wire antenna.

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EMLAB 27 Poynting’s theorem and wave power Electromagnetic wave power per unit area (Poynting vector) Average wave power per unit area

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EMLAB 28

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EMLAB 29 Array factor : Array factor z-directed array

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EMLAB 30 Array factor x-directed array Top view

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EMLAB 31 Typical array configurations

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EMLAB 32 How to change currents on elementary antennas? Magnitudes and phases of currents on elementary antennas can be changed by amplifiers and phase shifters.

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EMLAB 33 Equi-phase surface Pattern synthesis

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EMLAB 34 (1) Two element array (2) Two element array Examples

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EMLAB 35 (3) Five element array (4) Five element array (5) Five element array Beam direction

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EMLAB 36 phi=0:0.01:2*pi; %0

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EMLAB 37 N-element linear array antenna Uniform Array : Magnitudes of all currents are equal. Phases increase monotonically.

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EMLAB 38 Difference : Universal Pattern is symmetric about = Width of main lobe decrease with N Number of sidelobes = (N-2) Widths of sidelobes = (2π/N) Side lobe levels decrease with increasing N.

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EMLAB 39 Visible and invisible regions Array Factor 의 특성 Array factor has a period of 2 with respect to ψ. Of universal pattern, the range covered by a circle with radius “kd” become visible range. The rest region become invisible range 1 visible region Visible range of the linear array

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EMLAB 40 Grating Lobes Phenomenon If the visible range includes more than one peak levels of universal pattern, unwanted peaks are called grating lobes. To avoid grating lobes, the following condition should be met. 1 visible region grating lobes major lobe They have the same strength ! Example :, no grating lobe occurs

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EMLAB 41 Automotive radar antenna

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EMLAB 42 Analog beamforming : phase shifter

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EMLAB 43 LPFA/D Digital Signal Processing (Amplitude & Phase) ~ Desired signal direction LPFA/D LPFA/D LPFA/D Interference or multipath signal direction Digital Beamformer (DBF)

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ELEN 3371 Electromagnetics Fall 2008 1 Lecture 6: Maxwell’s Equations Instructor: Dr. Gleb V. Tcheslavski Contact: Office.

ELEN 3371 Electromagnetics Fall 2008 1 Lecture 6: Maxwell’s Equations Instructor: Dr. Gleb V. Tcheslavski Contact: Office.

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