1 2 Radiation integral EMLAB 2 EM radiation

1 2. Radiation integral EMLAB

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

Basic laws of EM theory 3 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. ) EMLAB

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

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

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

Potentials of time-varying EM theory 7 EMLAB

Lorentz condition 8 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. Lorentz condition For the above choice, (3), (4) become Lorentz condition EMLAB

Fourier transform solution 9 • 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. ) EMLAB

10 Solution of Maxwell’s eqs for simple cases EMLAB

Domain and boundary conditions 11 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 rectangular cavity Domain : infinite space waveguide Domain : interior of a circular cavity EMLAB

1 -D example : Radiation due to Infinite current sheet 12 1. Using phasor concept in solving Helmholtz equation, y x 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. 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

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 EMLAB

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

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

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

Solution of wave equations in free space 17 • 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

Green function of free space 18 where 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. A suitable solution which is propagating outward from the origin is e-jkr. EMLAB

Green function of free space 19 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

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. EMLAB

Retarded potential 21 (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

Solution in time & freq. domain 22 EMLAB

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

Electric field in a phasor form 24 EMLAB

Radiation pattern of an infinitesimally small current 25 EMLAB

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

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

28 EMLAB

Array factor 29 Array factor : z-directed array EMLAB

30 x-directed array Array factor Top view EMLAB

31 Typical array configurations EMLAB

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

Pattern synthesis 33 Equi-phase surface se ha p i qu ce a f r su E EMLAB

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

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

Sample MATLAB codes 36 phi=0: 0. 01: 2*pi; %0<phi<2*pi k=2*pi; d=0. 5; % 0. 5 lambda spacing. shi=k*d*cos(phi); alpha = pi*0. 0; beta = exp(i*alpha); %Currents=[1, 2*beta, 3*beta^2, 2*beta^3, 1*beta^4]; %Current excitations Currents=[1, 1*beta^2, 1*beta^3, 1*beta^4]; %Current excitations E=freqz(Currents, 1, shi); %E for different shi values E = DB(E)+30; % 최대값에서 30 d. B 범위까지 그림. E = (E>0. ). *E; polar(phi, E); %Generating the radiation pattern EMLAB

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

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

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

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

Automotive radar antenna 41 EMLAB

Analog beamforming : phase shifter 42 EMLAB

43 Digital Beamformer (DBF) or ce en fer ter ath In on ltip recti i mu na sig ld LPF l d signa e r i s e D on directi A/D LPF A/D Digital Signal Processing (Amplitude & Phase) ~ EMLAB