Part I

Use the iteration method (based on Laplace’s equation in two dimensions) to study the electric potential distribution inside a thin metal rectangular ring filled with a homogeneous material and an elliptical shape charge density in the centre. The cross-section of the rectangular ring is shown in the following figure (on the x-y plane), over which, the DC voltage V= 0 V is applied on three of the sides, while V= f(y) = 10*y+5 V is applied on x = w side. The rectangular ring is filled with a material with the dielectric constant of eT  = 45 and an elliptical charge with radii of a=20 mm and b=30mm is placed in the centre of the box at (x, y) = (w/2, h/2). The surface charge density of the ellipse varies over the surface according to the function density(x, y)= cos(0.2*x+0.2*y). The width of the rectangular plate is w = 200 mm, while the height of the rectangular ring is h =100 mm.

(The points inside the ellipse can be found by √()2  + ()2  < 1 )

(0, h)

V = 0

V = 0

(w, h)

V =f (y)

(0, 0)                                  V = 0                                (w, 0)

Task 1: Map the distribution of the electric potential V(x, y) in volt on the X-Y plane in a 1.0 mm grid resolution, and plot the potential value via a colour map along with the contour indication. The size of the computation domain must be large enough to accommodate the whole rectangle. The stopping criteria of the iteration loop can be made as to the error threshold of ∆U/U< - 100 dB.

Task 2: Based on the computation outcome from task 1, plot the electric field () in vector form

(Hint: use the ‘quiver’ function in MATLAB), and show up the intensity of the || in V/m.                  Please provide your MATLAB code with detailed comments for the above two cases (Task 1 and Task 2).

Part II

Please use the one-dimensional (1D) FDTD method to study the electromagnetic propagation problem for the through-wall radar (TWR) imaging problem.

In the EM model, you may approximate the TWR imaging scenario as a 1D layered dielectric medium, as shown below. The TWR imaging scenario has 3 layers: layer 1 (concrete wall), layer 2 (air), and layer 3 (concrete wall). Now assuming the imaging target appears in the 2nd layer, one can place a sensor outside the concrete wall to transmit and receive the electromagnetic (EM) waves. The transmitted EM waves will penetrate the concrete wall and reach the target. Most of the transmitted EM waves will be reflected and received by the sensor, while the remaining EM waves will penetrate the target and reach the concrete wall on the other side. The concrete wall on the other side will reflect the remaining EM waves and the sensor will receive them (similar processes will be repeated several times). By analysing the received signals (e.g., the matched filter), one can find the position of the target in the room.

Table 1 lists the geometry and dielectric properties of each layer, including the imaging target. The concrete wall and the imaging target can be assumed to be non-magnetic (the relative permeability is 1).

D3          D1                                                           D2                                                         D4

BC                

Concrete wall         Air          Target           Air         Concrete wall

BC

Sensor

D1: Distance between the sensor and the concrete wall (5 mm);

D2: Thickness of the room (190 mm);

D3: Distance between the boundary condition (BC) and the sensor (10 mm);

D4: Distance between the boundary condition (BC) and the concrete wall (10 mm).

Table 1 Geometry and dielectric properties of each layer

Task 1:  Simulate the sensor as an electric field source (Ey  component, hard source, single frequency (1 GHz) with amplitude of 1). Consider the Perfect Boundary Condition (BC) as shown in the above figure. Please DO NOT use any normalization for H- and E- fields in the 1D-FDTD updating equation.

Task 2: Simulate the sensor as a plane wave (Ey/Hx  mode, Gaussian pulse, the amplitude of Ey  is 1,

frequency range: 0-2.5 GHz). Consider the total field/scattered field formula for this case.                     Assume you have a Dirichlet boundary condition (BC in the above figure). Please use a normalised magnetic field in the 1D-FDTD updating equation.                                                                                   Please provide your MATLAB (or any other programming language) code with detailed comments for the above two cases (Task 1 and Task 2).