# EMLAB 1 Solution of Maxwell’s eqs for simple cases.

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EMLAB 1 Solution of Maxwell’s eqs for simple cases

EMLAB 2 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

EMLAB 3 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.

EMLAB 4 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

EMLAB 5 8. To find the value of C, integrate both sides of the original Helmholtz equation.

EMLAB 6 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

EMLAB 7 Source Infinitesimally small current element in free space : 3D

EMLAB 8 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.

EMLAB 9 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.

EMLAB 10 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

EMLAB 11 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.

EMLAB 12 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.

EMLAB 13 Solution in time & freq. domain

EMLAB 14 Far field approximation Electrostatic solution Biot-Savart’s law Coulomb’s law

EMLAB 15 Electric field in a phasor form

EMLAB 16 Radiation pattern of an infinitesimally small current

EMLAB 17 Example – wire antenna

EMLAB 18

EMLAB 19 Array factor :

EMLAB 20 Typical array configurations

EMLAB 21 Array antenna

EMLAB 22 Poynting’s theorem and wave power Electromagnetic wave power per unit area (Poynting vector) Average wave power per unit area

EMLAB 23 Derivation of Poynting’s theorem Thermal loss Electric energy Magnetic energy Electromagnetic wave power per unit area (Poynting vector)