The final project is optional and meant to increase your final grade to A. For instance, if you have completed all the bonus exercises, your starting grade is B, and you will obtain an A as the final grade if yous successfully complete the final project
If you do not intend to submit the optional final project, please upload a pdf file stating the expected grade.
We propose below a final project to complete the course and obtain A. However, if you feel strongly about a different project you would like to complete, we accept a different topic of your choice. For instance, you are welcome to explore further and study MEEP or other CEM software of your choice for solving simple problems. If this is the case, it needs to be a project limited in scope and a maximum of 1-2 weeks for the implementation. If you want to perform a different project, you can email me with the suggested project for approval.
Proposed Project - 3D FDTD for Filter Modeling
1. Introduction and Background Information
The goal of the project is to develop a 3D Finite Difference Time Domain Solver for studying the reflection and transmission coefficients in a 3D waveguide using Matlab, Python, pyMEEP, or Julia.
Waveguides and filters are important components of many complex microwave systems. Here we consider the characteristic features for some relatively simple structures that provide a filtering functionality in waveguides. The quantities of interest are
- reflection coefficient
- transmission coefficient
as a function of frequency. This project focuses on waveguide structures with rectangular cross-sections and a finite-difference time-domain (FDTD) scheme.
1.1 Modal Representation for a Rectangular Waveguide
In an air-filled rectangular waveguide with Lx and Ly's transverse dimensions, we can decompose the electric and magnetic fields into transverse electric (TE) and transverse magnetic (TM) modes.
Each mode has its propagation constant:
where kt2 are the eigenvalues of the transverse problem for Hz (TE case) or Ez (TM case), i.e.,
For TE modes, nx and ny are nonnegative integers that satisfy nx + ny > 0. For TM modes, both nx and ny are positive integers. Numbering the modes from 1 to inf, we can express the electric field in the waveguide as a superposition of both TE and TM modes by
where Vm(z,t) is the modal amplitude, or voltage, of mode m (which can be either a TE or a TM mode) and em(x,y) is its modal field.
The modal field, em(x,y), for the TE10 mode is shown in the Figure below.
We will consider situations where the frequency range of interest and the dimensions (Lx, Ly) of the waveguide are chosen such that kz is real for the TE10 mode and imaginary for all other modes.
Thus, the only mode that propagates is the TE10 mode. All the other modes are evanescent and decay exponentially along the waveguide axis. Consequently, the TE10 mode is the only mode present at a sufficiently large distance from any source or irregularity in the waveguide that may excite higher-order modes.
1.2 Computation of Scattering Parameters
A filter can be characterized in terms of its reflection and transmission coefficient. These are often referred to as the scattering parameters or simply the S-parameters.
The Figure below shows a rectangular waveguide (without the filtering structure) truncated at two ports for computational modeling purposes.
The S-parameters can be computed given the relationship between the amplitudes of the TE10 mode at the ports:
(1) an incident wave is launched at one port
(2) the reflected wave is recorded at the same port
(3) the transmitted wave is recorded at the other port.
Let s1,in(t) be the amplitude of the incoming TE10 wave at port 1, and let s1,out (t) and s2,out(t) be the amplitudes of the outgoing TE10 waves at ports 1 and 2, respectively.
The Fourier transform of these signals gives S1,in() S1,out() and S2,out(). The relation between the amplitudes at the two ports is usually described by the so-called S-parameters:
The scattering parameter S11 is recognized as the reflection coefficient and S12 as the transmission coefficient.
1.2 Numerical Modeling
An FDTD grid discretizes the interior of the waveguide.
A wave can be launched at one of the ports and then propagated through the waveguide by means of Maxwell's equations represented by the FDTD scheme applied to the grid in between the ports.
A filtering structure can be modeled in detail by the FDTD scheme and its reflection and the transmission coefficient computed from the fields at the ports.
The particular type of boundary condition that is required at the ports is already implemented in the MATLAB program provided as a starting point for the tasks below. However, it is useful and interesting to have some understanding of the port algorithm, especially if you intend to implement it in a language different from Matlab.
The algorithm is briefly summarized as follows:
At each time step, n, we extract the transverse electric field one cell away from the port boundary. Let us denote this by Etnp,q,Nz-1. Clearly, this field can be represented as a superposition of waveguide modes that propagate along with both directions of the waveguide. Subsequently, we consider a port that does not have an incident wave for simplicity.
- With this result, we can compute the voltages VmnNz-1 of the different modes m one cell away from the boundary:
For a waveguide port without an incident field, the decomposed field only consists of waveguide modes propagating away from the interior of the computational domain. For a port with an incident field, we could easily compute the incident field at the plane one cell away from the port boundary. Then we subtract the incident field from the total field to get the field associated with modes that are propagating away from the computational domain.
- Each mode can be modeled by a 1D wave equation:
which can be discretized as
The impulse response Imn = Vmn1 for this 1D wave equation can be computed. Given this impulse response, we can use a 1D convolution to compute the voltages on the boundary that coincides with the port:
- Now, we know the modal voltages on the port boundary. The total electric field on the boundary is a linear combination of the modal fields
and this solution is explicitly written into the FDTD grid before the next update of the interior grid points that are located inside the computational domain.
1.3 Matlab Example Codes - Starting Point for Implementation
The following supporting MATLAB script and functions are set up such that they can be used directly as a basis for the solution of the assignments that follow. Here is a brief description of each MATLAB file:
- CourseProj.m Setup of the problem with the allocation of memory for variables that store the electromagnetic fields, port information, excitation pulse, etc. Given the setup of the computational problem, the routine contains a loop that should include the update expressions of the FDTD scheme in the bulk of the computational domain. Further, extraction of the scattering parameters and the addition of possible metal objects are included in this routine.
- ComputeIR.m Computes the impulse response for all the modes included in the analysis.
- ComputeTEModes.m Computes the transverse electric field for the TE modes and the corresponding cutoff frequencies associated with the FDTD discretization.
- ComputeTMModes.m Computes the transverse electric field for the TM modes and the corresponding cutoff frequencies associated with the FDTD discretization.
If you prefer to use another programming language, e.g. pyMEEP, these MATLAB files could also be used as a starting point. In that case, you will have to implement this functionality in the language of your choice.
The 3D FDTD solver is explained in Section 5.2 of the textbook and we used it in Assignment II. In that case, the code we used for studying the eigenfrequencies in a cubical cavity is presented here.
2. Project Assignment
Consider an empty 3D waveguide with the dimensions:
- Lx = 40.0 mm
- Ly = 22.5 mm
- Lz = 160.0 mm
The waveguide ports are located at z = 0 mm and z = 160.0 mm.
The waveguide is excited by a TE10 mode at z = 0 by a Gaussian-modulated sinusoidal pulse, which contains energy in the frequency interval from 4 to 7 GHz.
The port located at z = 0 is transparent to the reflected field, which essentially corresponds to the rectangular waveguide's continuing indefinitely for the region z < 0.
Similarly, the port located at z = Lz is transparent to the transmitted field, which continues to propagate in the positive z-direction as if the waveguide continued indefinitely for the region z > Lz. The MATLAB implementation (which may be used as a starting point) is set up for these particular dimensions and this excitation.
Tasks to complete in the project
a) Implement the update loops for Faraday's and Ampere's law according to the FDTD scheme in three dimensions for an empty rectangular waveguide. Make an implementation such that it is possible to change the cell size to allow for convergence studies.
b) Answer the questions:
- What is the expected reflected s1,out (t) / and transmitted s2,out (t) solution for an empty waveguide given the Gaussian excitation pulse? Test your code and see if the result is what you expected.
- What is the cutoff frequency of the TE10 mode? Which mode has the second-lowest cutoff frequency, and what frequency is that?
c) Implement a postprocessing step that transforms the time-domain scattering amplitudes s1,out(t) and s2,out(t) to their corresponding frequency-domain quantities, and provide code that evaluates the scattering parameters.
d) Write and submit a report describing your implementation. The report should include your names, expected final grade. The report should include the following sections:
- Introduction. Provide some background on the performed problem and your approach to the problem
- Methodology. Explain all the different steps in your implementation
- Results. Present and discuss the results.
- Discussion and Conclusion. Discuss the results briefly and summarize the report.
- References. Provide a list of articles and books you consulted for preparing the project.
The maximum number of pages for the final report is six, including the references.
Deadline: January 15, 2021, 17:00 PM.
Please, upload to Canvas the final report as a pdf file.