COMS7300

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COMS7300
Electromagnetic Modelling of Microwave
Structures
A/Prof. Nick Shuley Room 78-535 Email
shuley_at_itee.uq.edu.au phone [phone]
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Subject Outline

• Lectures
• Assignments (computer based)
• Demonstrations
• Discussion of material
• Extended reading

For more information see the course profile
document
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Resources
Field solvers HFSS (available but
complicated) FEKO (we are really only going to
use this) Others PCaad3, IE3D, CST Microwave
Studio, Mephisto Reference list See the
extensive list of references in the course
profile.
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Subject Philosophy

• We emphasise the fundamentals.
• Difficult concepts first..
• Thinkers rather than workers.
• A mind once it is expanded
• does not regain its original dimensions

Jin Au Kong
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COMS7100
Lecture 1
Introduction to basic numerical methods in
electromagnetics
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Assumptions

• No active devices will be considered. (this
  would lead to solution of non-linear equations)
• Basics of numerical methods.
• Assumed equivalent of ELEC3100
• Assumed knowledge of vector calculus
• Solution of a linear system of equations and
  other simple numerical tasks e.g. integration,
  rootfinding, eigenvalues, eigenvectors.
• Some knowledge of a programming language (C,
  C, Fortran77 (or 90), Matlab etc)

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Effort
Electromagnetic problems are mostly described by
three methods Differential Equations ? Finite
difference (FD, FDTD) Integral Equations ? Method
of Moments (MoM) Minimization of a functional ?
Finite Element (FEM)
Theoretical effort
less
more
Computational effort
more
less
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Computational hierarchy
computational electromagnetics
High frequency
numerical methods
IE
DE
field based
current based
TD
FD
TD
FD
MoM
FDTD TLM
PO/PTD
GO/GTD
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A bit of History (according to Taflove)

• Maxwell Eqns formulated 1870.
• Military defence applications 1940s.
• Electromagnetic pulse (EMP) threat (1960s)
• High speed electronics
• Fibre
• Wireless

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CEM in action
Computed surface currents on prototype military
aircraft at 100Mhz The plane wave is incident
from left to right at nose on incidence. The
currents re-radiate back to the source radar (and
so can be detected)
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EMP
Microwave pulse penetrating a missile radome
containing a horn antenna. Wave is from right to
left at 15 from boresight.
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What happens?
Energy guided in the wall, complicated wave
interactions visible in the radome structure
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High speed electronics
Circuit theory is actually a subset of
electromagnetic field theory. It is worth noting
that
Kirchoffs current voltage laws fail in
most contemporary high speed circuits. These must
be analysed using electromagnetic field theory.
Signal power flows are not confined to the
intended metal wires or circuit paths.
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2 types of circuits
Microwave circuits, typically frequencies gt3GHz.
e.g. couplers, transmission lines, couplers,
matching networks, individual transistors.
Circuits are based on electromagnetic wave
phenomena
Digital circuits with clock rates below 2GHz.
Densely packed multiple planes of metal traces,
via pines interconnecting layers. dc feeds and
ground returns. These circuits are not based
on electromagnetic wave interaction effects.
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What can happen?

• Signals can distort as they propagate.
• Coupling between circuit path to another.
• Radiation effects can create interference to
  other circuits and systems.

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Reality
Illustration of coupling cross talk of a
high-speed logic pulse entering and leaving a
microchip embedded within a conventional dual-in
line integrated circuit package.
note how the fields associated with the pulse
are not confined to the metal circuit but spread
out and couple to adjacent circuit paths.
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In any EM problem there are three basic steps

• Pre-processing, to derive coefficients in
  algebraic equations
• Solution of the algebraic equations
• Interpretation of the results

Different methods involve different amounts of
theoretical effort and different computational
efficiencies
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Simple Problem
2D potential problem ?(x,y) ? within the
rectangle subject to the given boundary
conditions
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First Method
Finite Difference Mesh
We compute the potentials at the mesh points
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Second method
Triangular Finite Elements
We divide up the geometry into finite elements
(triangles). The field in each element is
approximated by a simple algebraic expression and
the values of the field are found by a set
of linear equations.
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Third method
More sophisticated mathematics. The potential is
expressed as an integral of charge multiplied by
a Greens function over the centre conductor.
This is an integral equation for the unknown
charge we convert the integral equation to a
matrix equation and once the charge is known the
potential can be computed everywhere.
Method of moments
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Any other methods?
Many! FDTD (Finite Difference Time Domain) TLM
(Transmission Line Method) IETD (Integral
Equation Time Domain) SDM (Spectral Domain
Method) . . Etc Etc
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A few examples
We now take a brief look (a really brief look) at
a the first and second of these techniques for
the example just quoted.
We say something about the FDM and the FEM and
illustrate the main approach to solving problems
using these methods. They are vastly different.
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Brief look at Finite Differences
Lets take a 1D example We want a solution to
k 0
k 1
k 2
k 3
k 4
x 1
x 0
Using a 3 point central difference formula
centred on k 1 gives
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Continued
similarly, we could derive 2 more equations for
the other 2 internal mesh points. We then make
use of the boundary conditions to set f(0) 0
f(4) 1
This will give us three equations with three
unknowns
now, set the boundary conditions and solve.
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Extension to 2D and 3D
The extension to 2D partial derivatives is easy,
for example
in a mesh of points Laplaces equation would take
the form
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What about the B/C?
For Dirichlet boundary conditions, the value of F
is specified on the boundary. We therefore
satisfy the equation at all internal points only.
For Neumann, the boundary condition is on the
derivative of F. This makes it a little difficult.
unknown!
exterior
interior
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Iteration
We could have solved the set of linear equations
resulting from all the nodes in the geometry
directly. However, even for small values of h, so
many unknowns result that the direct solution is
not often feasible.
We try another approach. Instead of
we rearrange to write
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Iteration (ctd)
We start with some initial values of the unknown
function F0(i,j) Then we cycle through all (i,j)
pairs to obtain an updated form.
The process then can be continued until the
solution has converged to suitable accuracy. The
convergence criteria can be a little hard to
detect from one N to another. A safer
procedure is to double the number of mesh points
and examine the changes from one to the next.
This iterative technique is known as relaxation
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Over relaxation
We convert the generalized formula
to a slight variation
R is the relaxation parameter. If R is set to 1,
regular relaxation results, Values of Rgt1 speed
convergence, but values of Rgt2 can cause
numerical instability.
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FEM
Instead of PDE s with boundary conditions,
corresponding functionals are set up and
variational expressions are applied to small
areas or volumes. The small elements are usually,
polygons (triangles) for 2D or tetrahedrals for
3D.
consider Laplace eqn in 2D
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Minimize the Functional
The solution of the PDE b/c is equivalent to
minimizing the functional
The integral is carried out as a collection of
contributions from all the small polygon
(triangle) areas. In each polygon, we approximate
? as a polynomial in x, and y.
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Apply it to an element
this formula is applied at the vertices of the
triangle
We then proceed in such a way so as to minimalize
the functional, element by element.
We will consider this in more detail when we
treat FEM
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Mathematical techniques
Electromagnetic analysis requires the resolution
of Maxwells equations plus boundary conditions.
Even for a simple structure such as microstrip we
can have very complex developments.
scan in Gardiol fig 1.5
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Towards classification
Microstrip is a good example (its used
everywhere!) e.g. TL lines, resonators, antennas,
discontinuities, filters, junctions, couplers,
First What exactly do we want to do?
What numerical method should we use? What are the
options available? What output do I get out of
it? Treat it as a circuit or an antenna? Does the
model take into account all the physical
phenomena? How long is it going to take?
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Quasi-TEM or Full Wave?
DC is of little interest. But the operation of
microstrip at low frequencies is predominately
determined by the capacitances and inductances
and these are best determined by
quasi-static approximations.
Quasi-TEM use notions of effective dielectric
constant, single mode impedance, axial current,
dispersionless propogation.
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Open or Closed?
Circuits are at some point usually enclosed
within a metal box, similarly, antennas are
designed to operate in free space. One would
expect that techniques for boxed in structures
would be used to analyse circuits and those
techniques for open structures for antennas.
This is not what happens in practice!
e.g. Antennas analysed by FDTD require b/c to be
specified. common assumptions are infinite
dielectrics and ground planes, zero thickness of
strips.
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Other considerations
2D or 3D? The infinite uniform structure is a 2D
structure (longitudinal transverse directions
only).
Circuit or antenna? Static and quasi-static
techniques are applicable to analysis of
circuits, except for spurious effects radiation
surface waves.
Time domain or frequency domain? TD is
intuitively easier , but losses are a problem.
Non linear components are best handled in the TD.
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Want to read more?

• Why Study Electromagnetics The first unit in an
  undergraduate Electromagnetics Course. Follow
  Prof Alan Tafloves Web page from
• http//www.ece.northwestern.edu/faculty-framesset.
  html
• Richard C. Booton,Computational Methods for
  Electrom,agnetics and Microwaves, Wiley
  Interscience