Antennas and Electromagnetic Wave Propagation

Antennas and Electromagnetic Wave Propagation Assignment

No.2 – 2019

Contents

1 Introduction 2

2 The basics behind the FDTD algorithm 2

2.1 Defining the Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Spatial Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 Time Step Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.5 Absorbing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Examples 5

3.1 example1 — Propagation in Free Space (tmax = 10 ns) . . . . . . . . . . . . . . . 5

3.2 example2 — Propagation in Free Space (tmax = 50 ns) . . . . . . . . . . . . . . . 5

3.3 example3 — Propagation in the Presence of a PEC Obstacle . . . . . . . . . . . 7

4 Assignment 10

5 Submission details 10

6 Statement of expectations 11

7 Code Listing — fdtd 1 11

∗This assignment was originally devised by Dr Michael Neve of the University of Auckland and has been

slightly modified for the purposes of the present module.

1 Introduction

The Finite-Difference Time-Domain (FDTD) method is a computational electromagnetic technique for solving for the electric and magnetic fields in arbitrary spatial domains in the time

domain. In contrast to techniques such as the Finite Element Method (FEM) and the Method of

Moments (MoM), this technique is straightforward to understand and is simple to program. A

rudimentary 2D TMz code is included in Section §7 and is used to illustrate the main features

of the method.

The aim of this assignment is to use the provided FDTD code in a series of numerical investigations, and compare quantitatively its predictions against theory, which the student is expected

to research independently after the completion of the taught part of the EMAP module. A

formal report is not required, but your assignment report needs to answer all the assignment

questions, in a self-contained manner.

2 The basics behind the FDTD algorithm

2.1 Defining the Lattice

The basic FDTD method (in Cartesian coordinates) makes use of a regular lattice of interleaved

electric and magnetic field components as originally proposed by Yee [1]. In the case of a 2D

TMz lattice1

, it is possible to derive the following from Maxwell’s equations:

i.e., the new field on the boundary is a function only of the old field on the boundary and the

field one lattice cell in from the boundary. In the case of the TMz case being considered here,

this condition need only be specified for Hx at the ±y boundaries and Hy at the ±x boundaries

— the remaining field components are established by the update equations.

3 Examples

A rudimentary FDTD code (fdtd 1) has been written in MATLAB and is included in Section

§7. Various examples using this code will be investigated in this section.

3.1 example1 — Propagation in Free Space (tmax = 10 ns)

This example is for the code included in Section §7. The source is located at (20,200), and

the total simulation time is 10 ns. The excitation waveform is plotted in Fig. 2, and the pulse

response at 10 ns is shown in Fig. 3.

The propagating wavefront is clearly seen in Fig. 3, together with evidence of a small reflection

which has arisen due to the imperfect nature of the ABC on the left-hand side of the computational domain. It is not meaningful to extract the time-harmonic response from this result, as

the solution had not reached steady state. This is now considered as example2.

3.2 example2 — Propagation in Free Space (tmax = 50 ns)

The magnitudes of the fields in Fig. 5 are noticeably smaller than those in Fig. 3, as all propagating fields have encountered the ABC on the periphery of the computational domain at least

once. However, the residual field is still of appreciable magnitude, and the only way to reduce

4 Assignment

Your report should be in three sections each providing an answer to the following questions.

The assessment criteria for each section are, (a) a clear description of what you have done: 10%,

(b) presentation of your simulation results in an appropriate form for interpretation and discussion: 20%, brief summary of relevant theory researched (including citations of key references,

but avoiding giving an unnecessary tutorial), implementation and calculation of corresponding

theoretical predictions: 30%, (c) discussion of numerical and theoretical results: 30%, and (d)

drawing conclusions: 10%.

1. Implement the Walfisch-Bertoni propagation model [6] for a typical UK scenario of cellular

telephony coverage at 900 MHz, for an area such as Selly Oak. In this scenario, the model

parameters (shown in Fig. 1 of [6]) are R = 500 m, d = 25 m, h = 10 m, H = 5 m,

and hm = 10 m. Use the theoretical model in [6] to determine the attenuation function

relative to free space in dB at a mobile receiver centred at street level. Note that the model

assumes that the buildings are knife-edge obstacles of zero thickness. [25 marks]

2. Extend the free-space simulation space of example1 so that it corresponds to the propagation scenario of #1 by selecting the correct lattice size, frequency, source location,

simulation convergence time, etc. Avoid having the source too close to the simulation

boundaries, and discuss why this is a sensible precaution. Calculate the field value at the

location which corresponds to the mobile receiver in #1, assuming the complete absence

of the ground. [25 marks]

3. Simulate the Walfisch-Bertoni propagation scenario using the FDTD code, assuming that

the buildings can be modelled as perfectly conducting knife-edge block of thickness equal

to 1 lattice interval and plot the field strength in the entire domain in logarithmic units.

to calculate the field at the receiver point, and thus the attenuation relative to free space

in dB. Compare this value to the one predicted using the Walfisch-Bertoni model in 1 and

discuss the source of any discrepancy. [25 marks]

4. Repeat #3 but now replace the knife-edges with rectangular blocks of 8.5 m thickness,

centred on the location of each knife-edge. Discuss the observed difference (if any) between