EE 6340
University of Utah
Department of Electrical Engineering
Spring 2001
Numerical Techniques in Electromagnetics
Finite-Difference Time-Domain (FDTD) Techniques
Homework Assignment #4
Re-read Taflove & Hagness Sections 7.1 through 7.6 (pp. 285-308) on Perfectly
Matched Layer (PML) boundary conditions. This topic is very important for FDTD
because without PML, many desirable FDTD simulations cannot be done. For example,
antenna simulations often require accurate prediction of the sidelobe levels at positions
that are 20 to 30 dB down from the peak. With ordinary ABCs, however, small
reflections mixed into the radiation data prevent this type analysis, often giving errors as
large as 2 dB at positions 10 dB down from the peak. Although PML uses an outer layer
of 6 to 12 cells this is usually more than compensated by the fact that the ABC can place
much closer to the object simulated because it is angle independent. A full
implementation of PML takes a lot of code work and computer resources but we will try
one experiment with 1-D PML in problem 4-1, and we will try a PML waveguide load in
our final project.
The remaining two problems in this set target resonant frequency (eigenmode)
analysis of 1-D and 2-D cavities and the resulting field patterns (eigen functions). The 2D case applies to the cutoff frequencies of waveguides of any desired shape.
1. PLM at Normal Incidence. (Equation 7.55a, p. 306) Use your 1-D codes from
Homework #3 to set up a quadratically or cubically distributed "R(0)" loss over the
last 8 cells of a 1-D simulation. You can experiment with the final values of σ and
ρM to get the lowest reflection, but use an outer PEC boundary Ey = 0. Test the
reflection performance of the resulting PML ABCs using a larger comparison mesh
and a one-way launch of a smooth compact pulse.
2. 1-D cavity oscillations. It is possible to use Ey = 0 on the outer edges of a cavity to
simulate a PEC wall, but try this problem using a high conductivity wall material.
(a) Build a 1-D FDTD cavity of length 30 cells by "painting" in the material σ = 105
from cell 5 to 45, and then over-painting the cavity region from cell 11 to 39 with σ
= 0. Plot your coefficients Ca(i) see if you get a cavity region from 10 to 40 with
Ca(i) = 1 and Ca(10) = Ca(40) = -1 on each end. The FDTD algorithm will then keep
Ey(10) = Ey(40) ≅ 0 during time stepping and very little field penetration will occur
into the walls. For a fundamental mode cavity frequency of 1 GHz, compute the
value of σ when σ = 105 and compare it to copper where σ Cu = 5.76 x 107. (The
cavity length is one half wavelength for the fundamental mode.)
(b) Pre-load Ey with a half wave sinusoidal field over the range of 10 ≤ i ≤ 40 . Run the
FDTD algorithm over 1000 time steps while outputting the value Ey(35) and observe
the result. Verify that the output field has the correct spectral frequency for the cavity
fundamental mode (half wave). Repeat this exercise for the second harmonic.
(c) Drive the 1-D cavity with a one-way temporal added E-field pulse at the position i =
15. Use a raised cosine pulse of duration 20 time steps. Observe the simulation for
1000 time steps and analyze the output spectra from the Ey(35) position. This time,
you should see the fundamental and many of the higher harmonics. Calculate their
frequencies (NT) and compare the results to theory.
(d) Drive the cavity at the second harmonic frequency using an added sinusoidal Ey
source at the position i = 15. Use a raised cosine turn-on to limit excitation of higher
frequencies, and use the exact value of NT obtained from spectral analysis of this
harmonic in part (c). Again, observe the Ey(35) output over 1000 time steps where
you should see a linear growth pattern. Also observe that the field mode pattern
becomes clear as the field builds. Repeat this experiment with the value of NT
detuned by 10 % and see if you get what you expect. Explain what is happening in
this experiment.
(e) Here’s a new idea! Try driving your cavity by setting a low level of negative
conductivity for a “gain” media in the cavity. If you then seed the cavity region with
at least one tiny point of field noise, the signal will grow and excite the cavity mode.
3. 2-D cavity oscillations. EM cavities of this type simulate what happens at the cutoff
frequencies of waveguides in which there is no propagation in the z-direction, and the
waves simply resonate between (or within) the walls of a waveguide cross-section of
any shape. (This is equivalent to the BEM MoM waveguide solutions we did
previously.) For uniformly filled waveguides (or empty waveguides) these fields
come in two forms, TEz and TMz, which are the two cases we derived for 2-D FDTD
simulations. In this case, the field patterns observed for a given resonant mode at
cutoff do not change shape when the mode propagates. In this case the FDTD
simulation gives both the resonant frequencies (i.e., the eigen values − β 2⊥ ) and the
eigen functions Ψ ( x , y ) associated with the transverse part of the wave equation
(∇
)
+ β 2⊥ Ψ = 0 . When the 2-D cavity is partially filled with material, the situation is
more complicated. The resonant frequencies obtained are still waveguide cutoff
values for "hybrid" waveguide modes, but the field patterns change with frequency for
propagating modes.
2
⊥
(a) Find the Rectangular Waveguide Modes. Use a 90 x 40 FDTD mesh to simulate the
interior of an "X-band" waveguide that has inside dimensions which are 0.9" wide by
0.4" high. Use Etan = 0 on the boundary to simulate PEC walls. First, pre-load the
TE1,0 and the TE2,1 field patterns (Ex,Ey) as initial conditions and observe that the
FDTD simulation is working correctly. Then, excite the simulation with an initial 2D raised cosine pulse (20 cells diameter) placed at an off center waveguide position
such as (40, 30) so that many modes get excited. Run this simulation for 1000 time
steps and see if the output at (80,30) has a strong spectral content. (If it doesn't, move
the initial pulse position and try again.) Analyze the spectral output and correlate it
with the known waveguide spectrum. Choose one of the higher order modes such as
TE2,1 and drive the cavity with a temporal added field at the resonant frequency to
observe the resonant field pattern. This temporal added field will be stronger and
work better if it has a spatial distribution such as the raised cosine multiplier shown in
the attachment. Repeat this experiment for the TM case.
(b) Solve for a Circular Waveguide Mode. Use a 50 x 25 mesh with half of a painted-in
circular waveguide cavity of 40 cells diameter (open at the top as illustrated below).
Use the center position of each cell to determine the conductivity. It is beneficial to
use only half of the waveguide in this FDTD simulation because not only does it use
fewer cells, but also it removes the degeneracy of the circle so that only one
orientation of each mode exists. Placing the symmetry plane along the upper outside
mesh line also permits the symmetry plane at j = 25 to be set for either a PEC or a
PMC boundary condition.
For TEz modes, a PEC wall at j = 25 is set by Ex(i,25) = 0 and a PMC wall is obtained
by setting the (previously unused) outside tangential Hz(i,25) = -Hz(i,24) after each Hfield update. The boundary field is then found by updating Ex (i,25) using the outside
Hz(i,25) field. For TMz modes the field components change. A PEC wall at j = 25 is
set by Ez(i,25) = 0 and a PMC wall is obtained by setting the outside tangential
Hx(i,25) = -Hx(i,24) after each H-field update and then updating Ez(i,25) from the
result. Draw some Yee cell diagrams at this outer boundary and convince yourself
that these conditions are true. Also, before starting, do a mesh plot of the coefficient
Ca(i,j) to see if you have the correct geometry. Then, choose a circular waveguide
mode you want to simulate, and use FDTD to find its cutoff frequency and its field
profile.
PEC or
PMC
Boundary
Condition
Air
High
Conductivity
Material
Raised Cosine Spatial Coefficient
Function “coefij” for Application of
Added Driving Sources in a 2-D
Cavity.
Hz(i,j) = Hz(i,j) +
coefij(i,j)*H0(n)*sin(2*pi*n/NT)
(Can be shifted to any target postion on the full
(i,j) mesh.)
coefij(i,j) = spatial multiplier shown
plotted.
H0(n) = raised cosine “turn-on” vs time.
sin(2*pi*n/NT) = sinusoidal drive at NT
time steps/cycle.
Plot of
coefij(i,j