Introduction
Describe the cycle of producing an electrical
component.
Describe the effect that an electronic
component’s mechanical size has on how we
model it.
1.
2.
3.
4.
5.
a)
b)
Define specification
First design
Numerical simulation of components
Evaluation
Specs fulfilled?
Yes -> move to manufacture
No -> restart from 1-5 where
appropriate
It depends on the wavelength.
If the mechanical size is smaller than the
wavelength, we can use lumped-element models
(RLC) and the network concept (voltages and
currents).
If the mechanical size is comparable with the
wavelength, we need to consider distributed
element and the field concept. Lumped element
models aren’t sufficient.
What kind of frequency bands can we expect to
play with in electronics? What are the
corresponding wavelengths?
What is the difference between doing field
simulation and circuit simulation?
Draw the hierarchy for the numerical methods of
solving Maxwell’s equations.
Describe what ‘S-parameters’ are.
How can we get S-parameters from a time-
If the mechanical size is very large compared
with the wavelength, we need to use optical
principles (PO, GO, GTD).
kHz scale to THz scale.
Wavelength is calculated using speed of light.
E.g. for kHz, c = 3x10^8 m/s, freq = 3000, hence
wavelength = 3x10^8/3000 = 100km
For THz, the wavelength is of the order of 0.1mm
Circuit simulation:
Advantages: Very fast, libraries available
Disadvantage: Limited to available models,
parameter values limited, no multimode
propagation, coupling neglected (coupling can be
a problem if components are close to each
other)
Field simulation:
Advantages: Very flexible, can model arbitrary
things, more accurate
Disadvantage: More complex and slower
simulation
See ‘Overview’ slide, pg 17, intro lecture
Dealing with the frequency domain. Over the
range of frequency we have some curves about
reflection and transmission.
Use a Fourier transform to convert it to the
domain signal?
What is an SPDT?
What does multimode propagation mean?
What does 2.5D analysis mean?
What is a planar circuit?
What makes a calculation slow?
frequency domain.
Single Pole Double Throw. Switches a signal
between one of two outputs.
Usually we have a fundamental mode that we
introduce into a circuit. However, if we have
discontinuities in the circuit, then additional
higher modes can be generated (some reflected,
some propagated, some die out). Circuit
simulation ignores these modes, but field
simulation doesn’t.
Analysis of planar circuits consisting of layered
homogeneous media. Interconnects that
connect different layers are usually handled as
via conductor.
A circuit that can be drawn in a plane without
any component crossing over another (e.g. a
circuit that doesn’t require vias).
Number of unknowns, which depends on the
model (level of detail) and the size of the
problem.
Then there’s operations, e.g. heavy math
(matrix, eigenvalue ops).
Equations and relations
Write down the Lorentz force equation, and
state it intuitively.
Write down the quick notation for gradient
(grad), and give its expanded definition as an
operator.
Write down the quick notation for curl, and give
its expanded definition as an operator.
What does the symbol E stand for, and what are
the various possibilities for units?
What does the symbol B stand for, and what are
the various possibilities for units?
What does the symbol q stand for, and what are
its units?
What does the symbol D stand for, and what are
the possibilities for units? Describe Faraday’s
The total force F experienced by a particle of
charge q moving with a velocity v in a region of
electric and magnetic fields, E and B is given as:
F = qE + qVxB
Grad f = (df/dx1,df/dx2,df/dx3…df,dxn)
Where the d is the partial derivative and f is a
scalar function. Upside down triangle without a
dot or cross.
It’s the upside down triangle with a cross. Need
to take the determinant of the matrix. See
Wikipedia.
Electric field intensity
N/C
V/m
Magnetic flux density
(V.s)/(m^2)
(Wb)/(m^2)
Tesla T
Amount of charge (e.g. the charge of a test
particle)
Coulombs
Electric flux density (or electric displacement flux
density)
related experiment and give the formula.
What does the symbol H stand for, and what are
the possibilities for units?
What does the symbol J (or Jc) stand for, and
what are the possibilities for units?
List the equation that relates J to E. What does
the equation say intuitively?
What does σ represent, and what are its units?
What is the relationship between the vector
fields D and E?
What does ϵ represent, and what are its units?
What does P represent, what does it mean and
what are its units?
How does P fit into Maxwell’s equations?
What is the relationship between P and E?
What is the relationship between the vector
fields B and H?
What is the meaning of u (mju), and what are its
units? What is mju0 and its value?
What does the symbol Mm mean, and what does
it mean in real life? What are its units?
Units: Coulombs per square meter (C/m^2) or
(A.s/m^2)
Faraday’s experiment involved a small sphere
with +Q surrounded by a larger sphere, first
grounded then taken away from ground, results
in –Q on the larger sphere, and a flux density of
D = phi/(4piR^2)a_r
Magnetic field intensity, in ampere-turn per
meter (A/m)
Conduction current density
Amperes per square meter (A/m^2)
J = sigma E
The conduction current density is proportional to
the conductivity of the material, and is in the
same direction as the electric field, and is
proportional to the magnitude of the electric
field.
Conductivity of a material.
Units are siemens per meter (S/m)
D = epsilon E
Material permittivity.
Units of Farads per meter (F/m).
Polarization density
It is the electric dipole moment induced per unit
volume of a dialectric material (which could
either be induced by an electric field E, or could
be permanent)
Units of Coulombs per square meter (C/m^2)
D = epsilon0 E + P
Where epsilon0 is the permittivity of free space.
This basically says that the electric
(displacement) flux density would be equal to
epsilon0 E in free space, but elsewhere we have
to consider the phenomenon of polarization.
P = epsilon0 (xsie) E
Where xsie is the electric susceptibility, a
parameter that signifies the ability of the
material to get polarized.
B = mju.H
Permeability of the material, i.e. a measure of
the ability of a material to support the formation
of a magnetic field within itself.
Units: Henrys per meter (H/m) or (Vs/Am)
Mju0 is 4pi x 10^-7
It is the permeability of free space.
Magnetization vector (or magnetic dipole
moment per unit volume).
This vector is zero in free space, otherwise it
What is the relationship between Mm and H?
List the four constitutive relations.
Describe Faraday’s Law and write it in integral
and differential form. What does it mean in real
life?
What does the M stand for in Faraday’s law, and
what are its units?
Describe Ampere’s Law with the Maxwell
correction and write it in integral and differential
form. What does it mean in real life?
Describe Gauss’ Law for electric fields and write
it in integral and differential form. What does it
mean in real life?
What does the symbol p stand for in Maxwell’s
equations, and what are its units?
What does the symbol pm stand for in Maxwell’s
equations, and what are its unis?
represents the effect of magnetization.
Units of Amperes per meter (A/m)
Mm = (xsim) H
Where xsim is the magnetic susceptibility, a
parameter that signifies the ability of the
material to get magnetized.
Jc = sigma E
Jm = sigma* M
D = epsilon E
B = uH
Where sigma* is the magnetic resistivity (see
Num Techniques in Electromagnetics)
A changing magnetic flux density B, or a
magnetic current density M through a surface
induces a circulating electric field intensity on
the boundary of the surface. When a conducting
material is present along the boundary of the
surface an EMF results.
Magnetic current density. Used in the symmetric
form of Maxwell’s equations and assumes the
existence of magnetic monopoles.
Units: Volts per square meter (V/m^2)
A magnetic field can be generated by an actual
flow of charges (or conduction current density)
and also by a changing electric displacement flux
density.
The electric flux emanating from a closed surface
is equal to the charge enclosed by that surface.
Volume electric charge density, in coulombs per
cubic meter (C/m^3) or (A.s/m^3).
Volume magnetic charge density. Used in the
symmetric form of Maxwell’s equations, which
assumes the existence of magnetic monopoles.
Units are (V.s/m^3).
State the law of conservation of charge and write The current due to the flow of charges
down the corresponding equation.
emanating from a closed surface S is equal to the
time rate of decrease of charge inside the
volume V bounded by that surface.
State Lenz’s law.
An induced electromotive force (i.e. E) always
gives rise to a current whose magnetic field
opposes the original change in magnetic flux.
Write down Stokes theorem from vector
calculus.
Write down the divergence theorem from vector
calculus.
Write down all Maxwell equations in symmetric, See lecture slides.
time-harmonic integral form.
Write down all Maxwell equations in symmetric, See lecture slides.
time-harmonic derivative form.
State the continuity equation in the time
domain, and state its meaning physically.
State the continuity equation in the frequency
domain.
Write down the identity for (‘curl of the
gradient’)
Write down the identity for divergence of the
curl.
Write down the identity for divergence of the
gradient.
Write down the identity for curl of the curl.
Derive the inhomogeneous Helmholtz equation
in the time domain.
List the formula for the speed of wave
propagation (phase velocity).
List the formula for the wavelength of an EM
wave.
List the formula for the propagation (or phase)
constant of an EM wave, and describe its
meaning.
At any point in a given medium, the divergence
of the electric current density due to the flow of
charges plus the time rate of increase of the
electric volume charge density is equal to zero.
See Chapter 1 in Numerical Techniques in
Electromagnetics.
v = 1/sqrt(mju epsilon)
Lambda = v/f
Represents the change in phase per meter along
the path travelled by the wave at any instant.
Units of radians per meter.
β = 2pi/lambda = omega sqrt(mju epsilon)
Think about what the equation is saying…the
constant is equal to 2pi/lambda. That is, every
one wavelength we have 2pi radians of phase
change. So, if we had a wave with a wavelength
of 1 meter, then we’d have 2pi radians of phase
change per 1 meter. Hence the 2pi/lambda.
List the formula for calculating the wave
impedence.
List the formula for the Poynting vector, and
describe what it means.
Describe the skin effect and give the formulas for
skin depth and resistance. What happens when
the thickness of the material becomes smaller?
Write down the expanded definition of the div
operator.
What is the definition of the magnetic vector
Assuming a dielectric…
Z = sqrt (mju/epsilon)
P = ½ Re(E x H*)
Describes the time-average power density for
electromagnetic waves. The asterisk represents
the adjoint (conjugate transpose).
Skin effect is the tendency of an AC current to
distribute itself within a conductor with the
current density being largest near the surface of
the conductor, i.e. the electric current flows
mainly at the skin of the conductor at an average
depth called the skin depth.
Sd = sqrt(2/(omega mju sigma))
R=p.L/A
If the thickness becomes smaller than 2Sd, then
the boundaries overlap and the area becomes
smaller, hence the resistance R increases roughly
inversely proportionally to t. Otherwise R is
roughly constant.
Takes a vector and makes a scalar out of it.
H = curl A
potential A (German style)?
Derive the inhomogeneous Helmholtz equation
in the frequency domain based on electric
sources.
What is the definition of the electric vector
potential F?
What is the meaning of epsilon, and what are its
units? What is epsilon0 and its value?
Imagine we have a wave coming from air and
penetrating a metal. What happens to the speed
of the wave and the wavelength? What does this
mean with respect to discretization?
How can we take account of losses in Maxwell’s
equations?
What does it mean if a wave is x-polarised?
Discuss the boundary equations for the case
where a wave is normally incident on a surface
that separates 2 media. Let n be the normal
vector between the media.
What is the purpose of the Helmholtz equation?
Use the magnetic vector potential A.
Use the first two Maxwell’s equations.
Use the Lorentz condition.
Use the definition of the wave vector k.
-E = curl F
Permittivity, a measure of the resistance that is
encountered when forming an electric field in
the medium. Units are Farads per meter (F/m) or
(A.s/V.m).
Epsilon0 is the vacuum permittivity, equal to
8.85x10^-12
Mju and epsilon are higher in the metal than in
air. The speed is hence smaller, and the
wavelength is “much much smaller” for a metal
than in free space (and also smaller than for a
dielectric).
Hence we have to discretise really fine because
the wavelength is small. However…for highfrequency the current and field are confined
mainly on the surface (skin effect)…it doesn’t
penetrate on the inside. So…if we want to
discretise, yes, we need a small discretisation,
but we just do it on the surface by using the
surface impedance.
If we consider permittivity and permeability to
be complex we can take account of losses.
Say the wave is travelling in the z-direction.
Polarization describes the path taken by the tip
of the electric field intensity vector E in a plane
orthogonal (perpendicular) to the direction of
propagation. Thus, if we take a slice at z=0 and
watch E as time progresses, we’ll see the vector
bounce around in +x and –x if it’s x-polarized.
Easy.
If we have no surface current, then the magnetic
and electric field intensities are preserved
between the two regions:
n x (H1 – H2) = 0 (J if there is a surface current)
n x (E1 – E2) = 0 (-M if there is a surface current)
n dot (B1 – B2) = 0 (rho_m if there is surface
charge)
n dot (D1 – D2) = 0 (rho if there is surface
charge)
Without Helmholtz, we have many equations
with six unknowns. With it we have one equation
with two unknowns (two components of the
vector A). The trade-off is that Helmholtz is a 2nd
order DE.
How does the Helmholtz equation get even
simpler in some cases?
Once we have determined the vector potential A
by solving the Helmholtz equation, how can we
calculate the magnetic and electric field intensity
vectors?
What are the duality quantities for H, E, J, A,
epsilon, mju, rho and omega(frequency omega)?
Where are rectangular waveguides used, and
why?
When do we get TEM waves?
What is a Green’s function?
Is it possible for a rectangular waveguide to
transmit DC waves?
What can we say about the fields at the surface
of a perfect conductor?
Allows us to simplify our differential equations.
Time-independence, no charges -> Laplace
equation
Time-independence, with charges -> Poisson
equation
Time-dependence, no charges -> Wave equation
See 2.3.1 in the slides…also look it up in a book!
-E, H, MA, F, mju, epsilon, rho_m,
omega(frequency omega…same)
High-power transmission…used because there is
NO RADIATION, and it can be cooled :).
We get it always if we have two separate
conductors (e.g. coaxial cable). A rectangular
waveguide only has one conductor, so it’s either
TE or TM.
The Green function G(r) is equivalent to the
impulse response h(t) of system theory. Just as
h(t) gives the response (in time) to a temporally
impulsive source, so G(r) gives the response (in
space) to a spatially impulsive current source.
The response to a spatially distributed source is
obtained by integration, and plays the same role
in space that convolution in system theory does
in time.
No.
The electric field components that are
parallel/tangential to the conductor must be
zero.
In addition, the magnetic field components that
are normal to the conductor must also be zero.
Describe the general process for solving for all
the field components in a rectangular
waveguide.
Describe a Herzian dipole.
Thus, the electric field is normal and the
magnetic field tangential at the surface of a
perfect conductor.
It is sufficient to just find the z-component of the
vector potential (assuming the propagation
direction is z). From that, we can determine the
rest of the field components.
A Hertzian dipole is a wire of infinitesimally
small-length. At the end of the wire, two equal
and opposite charges exist.
The current in the wire leads to a magnetic field
encircling the wire.
If our source point is at r’, and our observation
point at r, then the magnetic vector potential at
point r is related to the conduction current
density J and the scalar Green’s function.
Consider a radiating Hertzian dipole. What are
the non-zero field components in spherical
coordinates at a distance r?
Consider a radiating Hertzian dipole. At a far
distance, which field components are dominant?
Consider a radiating Hertzian dipole. At a near
distance, which field components are dominant?
What does the equivalence principle state?
Give the formulas for the momentary and
average-in-time quantities of the following:
 Electric energy density
 Magnetic energy density
 Joule’s heat
 Poynting vector
The resulting magnetic field in spherical
coordinates only has one non-zero component.
The resulting electric field has two non-zero
components.
Er, Etheta, Hphi
Etheta, Hphi
Etheta, Er
If we have a surface current on a closed volume,
we can forget about what’s happening inside the
volume. We just need the tangential
components of E or H…sufficient for unique
description.
See slide 36/37 of physical basics.
Mathematics
Give an example of an inner/closed/bounded
problem in electromagnetics.
Give an example of an outer/open/unbounded
problem in electromagnetics.
Give the general format of a second-order partial
differential equation in a domain x and y, and
describe when it’s linear, and when it’s
homogeneous/inhomogeneous.
Write out Laplace’s equation in 2 dimensions.
Write out Poisson’s equation in 2 dimensions.
What kind of electromagnetics problems are
described by Laplace’s equation?
What kind of electromagnetics problems are
Transmission in a waveguide. We know that the
parallel/tangential components of the electric
field on the PEC is zero, and the magnetic field
normal component is zero. There’s also no
radiation outside the waveguide, so hence this is
an inner problem.
Radiation from an antenna.
See slide 6/34.
D^2/dx^2 (f) + D^2/dy^2 (f) = 0
D^2/dx^2 (f) + D^2/dy^2 (f) = g(x,y)
Steady state static configurations with no source
term.
Steady-state with a source term (e.g. one wall
described by Poisson’s equation?
Explain the difference between Dirichlet and
Neumann boundary conditions.
Derive the wave equation for the electric field
intensity E.
Use the series expansion technique to solve the
wave equation at a high level.
with a potential on it).
In order to have a unique solution to a DE, we
have to impose boundary conditions. If we
specify the function value on the boundary
explicitly, this is Dirichlet boundary conditions.
If we don’t specify the function value itself, but
rather its normal derivative on the boundary,
this is Neumann boundary conditions.
Start with the curlE equation and take the curl of
it. Use the vector identity to turn it into a 2nd
order PDE. Should end up with a result similar to
slide 11 CEM_3, aside from a few constants.
See slide 11 onwards, CEM_3.
CEM_4
How does the method of moments express our
unknown solution to make it easier to compute?
What are the two general choices that we have
for basis function types in the method of
moments?
What are the disadvantages of using entiredomain basis functions for the method of
moments?
What does the Galerkin condition in the method
of moments state?
Given a system Ax = b. Give the formula for
calculating each x component using Cramer’s
rule.
Given a system Ax=b with A square of dimension
n. What’s the time complexity to solve the
system using Cramer’s rule?
Given a system Ax=b with A square of dimension
n. What’s the time complexity to solve the
system using Gaussian Elimination?
What’s the advantage of using Cholesky
decomposition over Gaussian Elimination?
Describe an electric wall. What are its boundary
conditions?
It expresses the unknown solution as a sum of
known functions (expansion functions or basis
functions) with unknown coefficients.
Entire domain basis functions (orthogonal, like
sin, Chebyshev, Bessel, Lagrange etc.)
Subsectional basis functions
Loss of generality. In order to use entire-domain
basis functions, we have to know something in
advance about the field distribution, which
means a loss of generality.
Test function = basis function
Xi = det(Ai)/det(A)
Where Ai is the matrix formed by replacing the
ith column of A by the column vector b.
O(n^4)
O(n^3)
If we want the determinant of A, we can get it
simply by multiplying the main diagonal
elements of L, i.e. L11*L22*…*Lnn.
The electric wall assumption implies that the
tangential component of the E field is zero. The
normal component of E is non-zero.
The normal component of the magnetic field is
zero…it is non-zero in the tangential
components.
N cross E = 0
N cross H = Jsurface
Describe a magnetic wall. What are its boundary
conditions?
N dot D = psurface
N dot B = 0
The magnetic wall assumption implies that the
tangential component of the H field is zero. The
normal component of H is non-zero.
The normal component of the E field is zero…it is
non-zero in the tangential components.
N cross E = 0
N cross H = Jsurface
N dot D = psurface
N dot B = 0
CEM_5
When solving matrix equations, we would like
that our matrix be diagonally-dominant. Give an
example where it’s a problem when we don’t
have a diagonally-dominant matrix.
Let’s say we have an equation Ax=b. We want to
solve the equation by taking the inverse of A, i.e.
x = (A^-1)b. The dimension of A is n. What does
MATLAB actually do to solve the inverse? What’s
the time complexity?
When is the only case where we would want to
compute the inverse of a matrix?
Give the formula for an eigenvalue problem.
What are the two possibilities for general
analytical solutions to the eigenvalue problem?
Consider Ax=b
A = [1 2]
[2 1]
B = [1;1]
This doesn’t converge well.
Basically we try to solve the equation A*(A^-1) =
I
We thus have n linear systems to solve to find
the inverse! Assuming we can solve each linear
system in n^3 for GE, for example, then it will
take n^4 for a solution to the inverse problem.
If we want to use the inverse of the matrix to
solve the problem for a bunch of different b
vectors.
[A – lambdaI][x] = 0, or
Ax = lambda
[A – lambdaI][x] = 0
So, this is solved if x is zero (trivial), or
Det(A – lambdaI) = 0 (more important!)
CEM_5
What’s the basic idea of the Mode Matching
technique?
To which problems is the Mode Matching
The idea is that part of the solution is analytic,
and part is solved numerically. This results
typically in much smaller matrices.
Waveguides and optical structures.
technique mainly applied?
What are the two parameters which characterise
a transmission line, that we’re normally
interested in determining if we’re looking at e.g.
a microstrip line?
To which kind of problems is the MMT applied in
2D?
When do we have to be careful in applying a
magnetic wall in a simulation?
Draw an example of a coplanar line and a
microstrip line.
Describe a coplanar strip/line in words.
What is a dielectric?
What’s the difference between an insulator and
a dielectric?
How do we stop unsymmetric modes
propagating in a coplanar line?
When we use the MMT for solving the Helmholtz
equation for a coplanar line, are there different
propagation constants per region, or do we use a
propagation constant for the entire structure?
List the general steps for applying the MMT for
solving for the transmission line characteristic
variables (propagation constant, wave
impedence) in a coplanar line.
Propagation constant, wave impedence.
2D cross-section of a coplanar line, microstrip
line.
We have to make sure that the problem is truly
symmetric. In some cases, like a coplanar line,
unsymmetric modes might propagate, and if we
use a magnetic wall then we might lose these.
See OneNote L5.
Consists of two conducting parallel strips or
traces mounted on the same side of a dielectric
substrate or PCB.
A dielectric is an electrical insulator that can be
polarized by an applied electric field. When a
dielectric is placed in an electric field, electric
charges do not flow through the material, as in a
conductor, but only slightly shift from their
average equilibrium positions causing dielectric
polarization.
Insulator implies low electrical conduction.
Dielectric implies high polarizability.
Bring the lines to the same potential. This can be
done with very thin bond wires, which connect
the two at points of discontinuity.
Entire structure.
Partition the structure into homogeneous
sections, each one with a constant relative
permittivity. Write down the Helmholtz equation
for the potential in each region.
Do a series expansion in each region with
orthogonal basis functions that match the
boundary conditions.
Match the potential expressions at interfaces
and multiply with test functions in the MoM
style. If Galerkin is used, only one side will see
the test functions be orthogonal with the basis
functions.
Move everything over to the LHS and build up a
matrix M with unknown kz entries, such that MA
= 0, where A has the amplitudes (coefficients).
Describe the ‘relative convergence’ issue in
MMT.
Give an example of a 3D eigenvalue problem in
MMT.
Is the MMT used for antennas? Explain.
List the pros and cons of the MMT.
Solve for det(M) = 0…gives kz. Then we can find
the A values, hence the potentials, hence the
propagation constant and the wave impedence.
Consider a 3D rectangular waveguide with a
discontinuity. The idea is that we have to match
the relevant field components at the
discontinuity using MoM. For each of the two
regions in which we are matching, we are free to
choose N and M, the respective number of
modes we want to include in our approximation
for each region.
N and M should be chosen considering the
physical size of the regions and the problem. If
they’re not ‘balanced’, then unphysical solutions
may result. This is the ‘relative convergence’
issue.
Resonator, i.e. a closed waveguide without input
source.
No. MMT doesn’t consider radiation well. We
can’t break down the field behaviour into
sine/cosine functions in this case.
Pros:
 Low numerical effort, as half analytical
 Very well suited for homogeneous
transmission lines, waveguide
discontinuities and optical structures
 Finite conductor thickness/loss can be
considered
 Modelling of field inside the conductor is
possible
Cons:

What are the sources of error in the MMT?
No consideration of radiation (bad for
antennas)
 How many modes do we have to
consider (heaps for a small pin in a
waveguide)?
 What to do with the tuning screws?
Truncation due to the chosen number of modes.
Staircase approximation of round objects.
Error in evaluation of integrals.
“Relative convergence” issue.
CEM_6
Describe the flavours of the finite difference
method. Describe which equations they each
solve, which form of the equations they use, and
in which domain they operate. Give the
FDFD: Finite difference frequency domain. Based
on differential form, Helmholtz equation,
frequency domain.
abbreviations of each.
Describe the trade-off between discretization
and round-off error in choosing a mesh width in
the finite difference method.
List one way for increasing accuracy in the finite
difference method.
We’re given a 5-point 2D stencil and asked to
find the expression for the point on which the
stencil sits in terms of the other four points.
The point lies on the edge between two
materials with epsilon1 and epsilon2. Find the
expression for the field at the point.
We’re given a 5-point 2D stencil and asked to
find the expression for the point on which the
stencil sits in terms of the other four points.
The point lies on the corner between two
materials with epsilon1 and epsilon2. Find the
expression for the field at the point.
Let’s say we want to find the modes of a hollow
waveguide using the wave equation. Is this an
eigenvalue problem or a deterministic problem?
TE and TM modes use m and n subscripts to
indicate the mode. What do the m and n refer
to?
How do we get solutions for a rectangular
waveguide using the finite difference method?
FDTD: Finite difference time domain. Based on
integral form, Maxwell’s equations, time
domain.
If the mesh width is very small, then the values
of neighbours may be very similar, and taking
the difference of two similar values gives a large
error (ill-conditioned).
If the mesh width is too large, the discretization
error comes into play (this is the O(h^2) factor
for central differences, for example). A trade-off
must be reached which minimises the error.
Mesh refinement. General rule is to use a denser
mesh at the edges.
Also, the wavelength is dependent on the
frequency, and we usually need a discretization
size around 1/10 of the wavelength. Thus, we
need fine discretization for higher frequencies.
2(eps1 + eps2)*x0 = 0.5(eps1+eps2)x1 + eps1*x2
+ 0.5(eps1+eps2)x3 + eps2*x4
2(eps1 + eps2)*x0 = 0.5(eps1+eps2)x1 + eps1*x2
+ eps1*x3 + 0.5(eps1+eps2)x4
-laplace u = alpha u
Where u = the vector potential
Alpha = k^2 – B^2
Hence, eigenvalue equation.
M: number of half-wave variations of the field in
the x-direction.
N: number of half-wave variations of the field in
the y-direction.
‘Solutions’ in this case are modes, and
corresponding propagation constant.
Depending on whether or not we want TE or TM
waves, we have different boundary conditions.
For TM, we have dirichlet boundary
conditions…i.e. the vector potential has to be
zero on the boundaries. For TE, we have
Neumann boundary conditions, i.e. the vector
potential has to have a zero gradient on the
boundary.
Then we just set up the matrix based on the
finite difference stencil we’re using. We can find
the determinant of the matrix, set it to zero, and
solve for the eigenvalues. If we have N points in
our domain (mesh), then we’ll get N eigenvalues
out of it!
What are the pros and cons for the finite
difference method?
How can we use Dirichlet and Neumann
boundary conditions in finite difference method
to reduce the computational domain? What’s
the trade-off?
Each eigenvalue represents a mode, and each
eigenvector the corresponding point values for
that mode.
Pros:
 Flexible, general tool
 Low analytical effort
 Large application domain
 Good for eigenvalues
Cons:
 Large matrices in 3D
 Modelling errors
 Discretisation errors
 Roundoff error
 Truncation error of iterative solver.
Dirichlet boundary <-> electric wall
Neumann boundary <-> magnetic wall
Because we reduce the domain to half, this
means that only half of the modes are
computed.
CEM_7
Write down the expressions for dHx/dt, dHy/dt,
dHz/dt, dEx/dt, dEy/dt, dEz/dt using Maxwell’s
equations in time domain.
Draw Yee’s cell.
Describe the Leapfrog algorithm used in FDTD.
See slide 4/41 of CEM_7.
See slide 6/41 of CEM_7.
Consider Yee’s cell. We have alternating E and H
components in each direction. That is, each E
component has only H component neighbours.
At timestep n, we update the E field based on
the timestep (n-1) E field and the timestep (n0.5) H field. Thus, in timestep n, we only update
E values.
At timestep n+0.5, we update the H field based
on the timestep (n-0.5) H field and the timestep
n E field.
List the stability criterion for the FDTD algorithm
in 1D, 2D and 3D.
Thus leapfrog.
(c*deltaT)^2 <= deltaX^2
(1D)
(c*deltaT)^2 <= [1/(deltaX^2) + 1/(deltaY^2)]^-1
(2D)
Describe broadly what the difference is between
the FDTD method and the FD method.
(c*deltaT)^2 <= [1/(deltaX^2) + 1/(deltaY^2) +
1/(deltaZ^2)]^-1
(3D)
FD is in frequency domain. We take Helmholtz’s
equation, discretise in space, and we solve a big
matrix once.
FDTD is in time domain. We take Maxwell’s
equations in time domain, and we iterate in time
(leapfrog algorithm). At each timestep, we can
update each field component without having to
solve a matrix. That is, FDTD uses explicit
formulation of each field component at each
timestep.
How do we get scattering parameters from a
FDTD output?
What are the pros/cons of FDTD over other
methods?
For FDTD, we have to excite the system, usually
done with a Gaussian pulse.
FDTD output is a bunch of vectors. Each vector
represents a particular field component variation
over time. We can get the scattering parameters
by doing a Fourier Transform of this data.
Pros:
 No ‘heavy math’, i.e. no matrix
equations or eigenvalues to solve
 Doing the Fourier transform of the timedomain signal gives us the full spectrum,
which we wouldn’t get using a frequency
domain method (we choose the N in
frequency domain methods)
Cons:

Write the formulas for an FDTD Gaussian timeexcitation pulse in time and frequency domain.
What’s the best choice for a time-excitation
pulse in FDTD?
When do we truncate the Gaussian timeexcitation pulse in FDTD to zero?
Many timesteps, which are dependent
on the spatial resolution to ensure
stability. Finest detail determines
timestep
E(t) = e^(-(t-tc)/sigma)^2
E(w) = sigma*sqrt(pi)*e^(-(w)/(2/sigma))^2
Gaussian pulse. Step signals (sharp edges) lead
to unphysical behaviour and instabilities.
When it essentially gets down to machine
precision. There’s a formula for figuring this out
based on the finest detail in the structure and
the dimensionality of the problem.
Write the formula for a sine-modulated Gaussian
time-excitation pulse used in FDTD. Draw a
rough sketch of this pulse. Why might we want
to choose this instead of a standard Gaussian
pulse?
Write the formula for a single-frequency timeexcitation pulse used in FDTD. Draw a rough
sketch of this pulse. Why might we want to
choose this instead of a standard Gaussian
pulse?
We have a 3D structure that we want to solve
using FDTD. Describe the process of exciting the
structure.
Bandlimited spectrum is created. See slide 21/41
of FDTD.
Sinusoidal excitation will give only one usable
frequency point. This isn’t usually what we want.
See slide 22/41.
We need to define something like a ‘port’, which
is a 2D plane on which our input field
components (electric, magnetic) are defined.
Before we start our simulation, we have to get a
spatial distribution on that plane of the field.
This is then a 2D eigenvalue problem, which we
can solve. Otherwise, we can input an artificial
field distribution and let it run for some time
until we get the ‘natural’ field distribution.
Either of these two methods then gives us the
full-amplitude ‘imprint’ excitation for the
structure.
What happens if we truncate our input signal too
early in FDTD?
What is the tradeoff when choosing the port size
for the input signal in FDTD?
How can we avoid numerical effort in modelling
conductors in FDTD? Describe the technique.
What’s the problem?
We then choose a time excitation pulse, like a
Gaussian, and we start our simulation. The
Gaussian basically acts as a filter for the
amplitude of the imprinted spatial
excitation…starting at zero, and going up to the
nominal amplitude, then going back to zero.
We get ripples in the frequency domain, because
a rectangular pulse (i.e. the cutoff rectangle)
transforms to a sinc function (sinx/x)in the
frequency domain, and these ripples will then
overlay on our frequency plot…bad.
If we choose it too large, we can excite more
than one mode, which is bad.
If we choose it too small, the electric walls that
we use on the boundary of our port will change
the field distribution from the ‘real’ distribution,
and affect the impedence value.
Impedence boundaries. Considers the surface
impedence of the conductor, assuming that the
skin effect holds and the fields are concentrated
on the boundaries.
The problem is that the field components don’t
exist on the same grid points, so we can’t get a
reasonable approximation for the surface
impedence. To solve this we can use
interpolation.
In FDTD, what’s the problem with using the
easiest solution of electric/magnetic walls
(Dirichlet and Neumann boundary conditions) in
our domain?
What kind of box boundary conditions are the
best to use with FDTD problems?
Explain the Liao extrapolation technique for
implementing absorbing boundary conditions.
What are the problems with the Liao
extrapolation technique for doing ABCs in FDTD?
What are two ways to implement absorbing
boundary conditions in FDTD solvers?
Explain the perfectly-matched layers technique
for implementing ABCs in FDTD solvers.
List the pros and cons of FDTD.
Of course, if the conductor is very thin, the skin
effect assumption doesn’t hold any more.
If the metallic walls are close to the structure
under investigation, the field distribution is
disturbed, characteristic impedence and
propagation constant will be affected.
Resonances will shift.
If metallic box is large: box modes (resonator
modes of the metallic resonator) occur…like a
waveguide. In larger boxes, the cut-off (‘cut on’)
frequency becomes lower. This gives spikes in
the scattering parameters resulting from the box
modes, nothing to do with the structure.
ABCs…absorbing boundary conditions.
Polynomial extrapolation of field data at interior
grid points and past time steps to the desired
outer-boundary grid point at the latest time
step.
Marginally stable
Errors above 40dB
Quality dependent on angle of incident wave.
Liao extrapolation technique.
PML (perfectly matched layers)
The idea is that we add some additional
boundary points, which are lossy. The idea is
that this acts like a ‘sponge’ to soak up the
waves in this region. At the end of the lossy
material we still have a PEC which causes
reflections, but by the time it gets back the
amplitude is hopefully negligible.
Pros:
 Flexible, general 3D tool
 Lowest analytical effort
 Large application domain
 Easy implementation of lumped
elements
 Large matrices, but simple operations
 Fast for broadband structures
 Parallelization easy
Cons:



Pre- and post-processing necessary (pre:
input port definition, post: Fourier
transform to get the scattering
parameters)
Open structures need special boundaries
Rectangular mesh: staircase
List the sources of error in the FDTD method.
approximation
 Iterative process: long calculation times
 More difficult to model losses in
dispersive materials…we have to model
the entire conductor if we want it very
accurate. If we use the surface
approximation then it’s not as good.
Discretization error.
Truncation of signal (time domain)
Box modes (closed environment)
Imperfect boundaries
Modelling errors
Roundoff errors
User!
CEM_8
What’s the difference between FIT and
FD/FDTD?
Derive the FIT algebraic equations considering
the Yee-style cube. Write them next to the
corresponding Maxwell integral equation.
Let’s say we do an FIT run and we get a result. If
the result is with respect to D, how do we get E
and the path-line integral-variable of E?
If we use the time-domain algorithm in FIT,
what’s the difference between it and FDTD?
In FIT we start with Maxwell’s equations in
integral form. In FD/FDTD we start with
Maxwell’s equations in derivative form.
In FIT the quantities that are modelled are
integrals, rather than fields.
See slide 10/18 in CEM_8.
The result is the integral of D over the surface
area, so we have to divide by the surface area to
get D, then divide by epsilon to get E, then
multiply by ds to get e, and this is where we lose
exactness.
This is how we can relate D and E, but there are
approximations in between.
In FIT, the variables are the integrals over the
line, not the field values themselves.
CEM_9
In which domain, frequency or time, does the
MoL operate?
In principle, MoL is a hybrid of which two other
methods that we have studied?
List the similarities between MoL and MMT.
Frequency.
MMT and FD. It basically uses FD to discretise in
2D, and then the 3rd dimension is analytically
calculated, like with MMT.
Both are semi-analytical (half-numerical)
methods.
Both are suited for certain classes of structures.
True or false. MoL is based on finite differences.
Let’s say we want to use MoL to analyse an
infinitely long homogeneous transmission line,
e.g. a rectangular waveguide, microstrip, CPW
etc.
What is our goal? What do we want to calculate?
Which equation do we have to solve, and how?
Which discretization scheme is used in MoL, and
along how many dimensions do we discretise?
We discretise in only 1 dimension in MoL in a 3D
problem…great! What’s the catch?
When we do a 1D discretisation in MoL, do we
have the same field value on each line?
Describe the basic steps for solving a
multilayered problem using the MoL.
Both require higher effort to derive and
understand the methods, as analytical
calculation is combined with numerical
computation.
True.
Propagation constant kz
Characteristic impedence
Field distribution in the cross section
We solve the Helmholtz equation and get the kz
from an eigenvalue calculation.
Uses central finite differences, and discretization
is performed in one direction. E.g. we discretise
in the x-direction of our waveguide, and
calculate analytically in y and z.
Analytic calculation is only possible in a
homogeneous region, so we have to separate
into layers.
No…just like in FDTD with the cube, we alternate
field components.
Discretise into lines in the x-direction. Use
alternating solid/dashed lines to represent Ez
and Hz components respectively
For each layer…that is, each horizontal slide
where there is no change in the y-direction, do…
1. Set up the differential equation
(Helmholtz) and substitute in the
derivative wrt x (finite difference). This
gives us a coupled set of ODEs.
2. Do the mathematical transform of the
potential matrix to the principle axis to
get a diagonal matrix, and hence a
system of uncoupled ODEs.
3. Solve the eigenvalue problem by taking
the determinant and finding kz and the
transformed E and H fields. This is then
basically the MODES that we get! Each
mode is propagating with its own
propagation constant.
4. Propagate the modes (‘transform the
impedence in transmission line theory’)
analytically across the homogeneous
layer section in the y-direction to the
next layer interface.
5. At the interface we ‘normalise the
impedence’ by tranforming back to get
the real E and H fields, considering ALL
the modes that we have from the entire
layer.
What can we ‘leave out’ in our solution
procedure during MoL if we want to solve for the
fields in an open hollow waveguide?
When we do our geile transformation in MoL to
decouple the system into a bunch of ODEs, what
are we actually doing in a physical sense?
What’s the difference between finding modes in
the MMT and MoL? That is, is one better or
more flexible than the other?
Describe the technique used in 2D MoL to
transfer from the top and bottom layers to the
matching plane in order to find the propagation
constant.
6. Match the E and H fields with the next
layer using the continuity of the
tangential field components and repeat
:)
In an open hollow waveguide, we either have TE
or TM modes. Thus, we don’t need alternating
lines for Ez and Hz, we just need one!
We’re transforming the equation from spatial
domain into modal domain!
In MoL, we can deal with one layer at a time, and
it can be inhomogeneous in the x-direction. This
will then give us the modes of that layer in one
swoop…solving the eigenvalue problem!
Awesome!
In MMT, we had to match the modes at every
inhomogeneity.
Usually we know the values of the
field/impedence at the top and bottom of our
layered structure. We start at the top and
bottom and follow the normal MoL technique
before converging at a matching layer. At the
matching layer, we establish a relationship at the
matching plane:
Z*J = E
Now we know that the tangential E must be zero
at this metallisation, so we basically have to find
the determinant of Z where it’s zero. This gives
us the propagation constant.
(Er + 1)/2
In the 2D MoL, if we have a microstrip line with
substrate and air, what will be an approximate to
the effective epsilonRE of the fundamental mode That is, the field is partially in air and partially in
of the line?
the substrate. We can use this as a starting point
in our root-finding algorithm at the matching
plane.
What does the term ‘dielectric constant’ mean?
Synonym for relative permittivity.
How do we separate a structure for use with
2 options:
MoL in 3D?
1. Separation into longitudinally
homogeneous sections, with the cross
section allowed to be inhomogeneous.
The lines are then in the propagation
direction
2. Separation into x-axis-oriented layers
(i.e. horizontal slices of bread), with the
layers allowed to be inhomogeneous in
the x and z direction, where we use
discretization.
How does MoL work in 3D?
What’s basically the only ‘con’ of the MoL?
In the MoL tutorial, we discretised with four
types of points, filled circles and triangles, and
empty circles and triangles. What was the
purpose?
Actually, we could do it in any dimension we
want, just make sure that we look at the
problem and see which direction makes the
most sense.
Basically the same as in 2D, but we use the
Kronecker product for the matrices because
we’re doing discretisation in 2D this time.
Low flexibility…different algorithms for different
structures. Best for layered structures or long,
thin structures.
Because we need the derivatives on
neighbouring points.
CEM_10
Is an electric structure temperature agnostic?
Is the FEM usually used in time or frequency
domain?
Do we use Maxwell’s equations directly in the
variational approach of the FEM?
When we use the FEM method with the
variational technique, we try to minimise the
field energy to find the potentials at all the
points. Write down the equation for the energy
of the electrical field. How do we get from the
electric field intensity and permittivity to the
potential?
Derive the FEM equations for a single triangular
element that leads to the eventual matrix to be
solved.
We’re doing FEM and we have found our matrix
and now we want to insert the boundary
conditions for certain boundary points. How do
we go about it.
When do we have an eigenvalue problem, and
when do we have a deterministic problem?
What is the main FEM approach to use when we
have electrostatical problems with a charge
distribution? What about the case of time-
No…usually conduction properties, impedence
changes with temperature.
Frequency domain (i.e. harmonic problems).
No, we use an energy-minimisation approach.
See slide 26/64.
See slides 26-31 of CEM_10.
For the row of the matrix corresponding to that
potential point (i.e. row n for point n), we
remove all entries in that row, put a 1 in the
diagonal entry on that row, and put the known
potential value for that point on the RHS (i.e. in
the b vector).
When the vector on the right is a zero vector, we
have an eigenvalue problem (we need to find the
determinant of the matrix and set it to zero and
solve).
When the vector on the right is a nonzero vector,
we have a source term and it’s a deterministic
problem.
For the static case we can use the variational
approach with field-energy minimisation. For
time-dependent fields…don’t know yet! Finish
dependent fields?
List the FEM pipeline from problem to solution.
In which case do we have to do preprocessing of
the input in FEM?
List some of the rules for good triangular meshes
in FEM.
this!
1. Preprocessing of input
2. Proper choice of elements
3. Derivation of the governing differential
equation
4. Discretization of the 2D domain
5. Derivation of the element matrices and
vectors
6. Assembly of the global matrix system
7. Imposition of boundary conditions
8. Solution of the global matrix system
9. Convergence? If yes, do postprocessing,
otherwise repeat from 3 with finer
discretization.
When we have a deterministic problem with an
excitation. Basically the same idea as the FD
method where we calculated the field
distribution on the input port. We have to define
the ‘mask’ of the field components to imprint for
excitation.
For a triangular mesh, the shape of triangles
must be close to equilateral
Nodes must appear at source points
In regions where the solution is expected to have
large variations, the elements must be
sufficiently small
Avoid elements with very large aspect ratios, i.e.,
the ratio of the largest side to the smallest side
Number the nodes in ascending order starting
from 1. The numbering of the nodes directly
affects the bandwidth of the global matrix
There must be no overlap of elements
Neighboring elements must share a common
edge
List the pros and cons of the FEM.
An interior node (non-boundary node) must
belong to at least three elements
Pros:
 Flexible choice of elements
 For round and irregular geometries
Cons:


Elaborate mesh generation (although
adaptive mesh refinement possible)
Possible unphysical solution (spurious
What is a functional, in mathematics?
When we search for a state of a system that
minimizes the energy functional, what are we
doing exactly?
modes)
It is a map from a space of functions to output
values. That is, a functional takes a function for
its input argument, so it’s basically a function of
a function.
We look for a function that makes the functional
stationary, i.e. extrema.
CEM_11_IE
List the general steps of the integral equation
method.
Do box modes start happening for lower or
higher frequencies as the box size is increased?
What is the basic idea of the integral equation
method?
What key properties of a 2D metallisation
problem do we use in the integral equation
method to derive the equations that we use?
1. Derivation of the appropriate integral
equation (different depending on
whether we’re in static/dynamic case)
2. Conversion (discretization of the IE into a
matrix equation using basis (or
expansion) functions and weighting (or
testing) functions. This involves the
MoM.
3. Evaluation of the matrix elements
4. Solving the matrix equation and
obtaining the parameters of interest
Lower.
Reduction of a 3D problem to a 2D problem,
defined on the surface of a conductive scatterer.
That is, everything is defined on the surface of
the metallisation.
The 2D problem is described by an integral
equation, where currents and tangential fields
on the surface are connected via the dyadic
Green’s function.
So, we have a surface metallisation which has an
unknown current J. We can get the electrical
field at any position r (outside the plate) using
the integral over the Green’s function multiplied
by the current, and adding the incident field (see
slide 5).
However, we don’t know the current! So what
do we do? Well, we know that the tangential
electric field on the conducting plate has to be
zero. This then allows us to calculate the current
distribution on the plate, by satisfying the
condition that the tangential component of the
total electric field vanishes on the plate . Once
we have the current distribution on the plate, we
can calculate the electric field anywhere off the
plate using the same equation.
In the integral equation method, how many
dimensions are involved in evaluating the
integral of each matrix element?
Can the integral equation method model
dielectrics?
What’s the best method to use for modelling
radar reflections on large objects?
What determines the complexity of the
calculation of the Green’s function in the integral
equation method?
Describe the spectral domain approach in the
integral equation method.
4
No…can only model metallisations.
Integral equation method.
Whether or not the system is in an open
environment or a closed box configuration. If it’s
closed box, the Green’s function can be
expressed by sine and cosine functions, but then
we probably have the issue with box modes etc
again.
So in the integral equation method we have an
integral equation (wow) which relates the
electric field to the current via the Green’s
function.
The Green’s function is usually not really known.
So we convert everything into the spectral
domain by doing some Fourier transforms.
Is the integral equation method a local or a
global method?
Write the Coulomb integral describing the
potential of charge density in free space for the
static case.
What’s the difference in the matrices that come
out of the IE method compared to the matrices
that come out of the FEM or FDTD methods?
What determines the size of the matrix to be
solved in the IE method for a 2D conducting
surface?
What are possible sources of errors in the IE
method?
Why do we do this? By transforming into the
spectral domain, we eliminate the integral,
because it’s a convolution integral or something.
Usually a global method. The source point is
coupled to each observation point. => dense
matrix.
See slide 19/34 CEM_11
Matrices are smaller in IE than in FEM or FDTD,
but they’re dense. Also, there’s a matrix-fill stage
in IE that’s in O(n^2).
Determined by the number of elements (i.e.
small areas) on the conducting surface.





List the pros and cons of the IE method.
Pros:


Numerical calculation of the Green’s
functions.
Truncation of series of basis functions
(i.e. granularity of the discretised areas
in the domain).
Box modes in a closed environment
Modelling errors (lossy thick conductors,
vias etc)
Roundoff errors
Accurate modelling of radiation (open)
Fast-end efficient for layered structures
or structures in free space

Cons:


Adaptive frequency sweep possible.
High numerical effort for problems with
nonplanar components like dielectric
blocks
Possible high numerical effort for
calculating Green’s functions.
CEM_12
Are asymptotic methods for high or low
frequencies?
What does MLFMM stand for?
Describe the MLFMM.
What are the pros/cons of MLFMM?
High you idiot.
Multi Level Fast Multipole Method.
Multi Level Fast Multipole Method. Basically the
electromagnetic equivalent of the FMM in
molecular dynamics.
It’s an extension of the Integral Equation
method. Instead of coupling every source term
to every observer, we use a tree structure to
group source terms together, and a tree
structure to group observer points together, and
only do the interactions at high levels of the tree.
Then the interaction results are distributed down
through the tree considering distances.
Pros:
 Much faster than MoM/IE
 Applications to scattering problems,
Radar cross sections
 Parallel computing possible
 Same meshing as MoM/IE
Cons:


Give an overview of the high-frequency
approximations PO, GO, GTD and UTD with
respect to basic properties.
Inaccuracy by far field approximations
Only applicable when object >>
wavelength
Physical optics: Current-based, incoming field is
treated locally as a planar wave, good for
electrically large problems without edges
Geometrical optics (ray optics): Field (ray) based,
nature of waves not considered, good for
electrically large problems without edges
Geometric theory of diffraction: Consideration of
edges, corners, inaccurate at shadow boundaries
Uniform theory of diffraction: Shadow
boundaries
Describe physical optics in detail.
For high frequency only (wavelength >> object
size)
CURRENT-BASED!
Ignores wave effects
Incident field is treated as a planar wave. We
calculate the current on each point as the
current that would be found on a tangent plane
of similar material at that point.
Current in shadowed regions is taken as zero (no
diffraction).
Once we have the current, we then use it in the
radiation integrals to compute the scattered far
field from the target.
NOT accurate for edges (how would we define a
tangent plane?).
What do we mean by a ‘ray-based’ technique in
high-frequency methods?
Which is a simpler approach in high-frequency
methods: GO or PO?
Describe GO in detail.
PO works well for large, smooth surfaces with
low curvature.
We don’t worry about calculating the currents in
order to find the reflected fields…we just reflect
the ray and that’s it.
GO…because we need to calculate currents in
PO.
For high frequency only (wavelength >>> object
size)
Wave nature of light is not considered…we use
rays…RAY-BASED!
Each incident ray is reflected directly and
superimposed to get reflected electric fields.
Current in shadowed regions is taken as zero (no
diffraction).
What are the limitations of GO and PO? What’s
the solution?
Describe GTD in detail.
We don’t know what to do at edges (same
tangential problem as PO).
Can’t deal with edges! Solution is GTD and UTD!
Geometrical theory of diffraction. Once again,
only works if the dimensions of the body >>
wavelength. Overcomes the limitations of PO
and GO by introducing a diffraction mechanism.
GTD assigns a diffraction coefficient to a point
depending on the geometry (i.e. whether it’s an
edge, corner/tip or curvature). The coefficients
are based on known canonical geometry tests.
Draw an example of a shadow and a reflected
boundary in GTD.
What is the limitation with the geometrical
theory of diffraction? What’s the solution?
Describe the uniform theory of diffraction in
detail.
See slide 27/34.
GTD produces inaccurate results at the shadow
boundaries. UTD is a solution!
UTD approximates near EM fields as quasioptical and uses ray diffraction coefficients for
each diffracting object-source combination.
These more complex coefficients are then used
to calculate the field strength and phase for each
direction away from the diffracting point.