ELEG 840: Advanced Computational Electromagnetics

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ELEG 840 Advanced Computational Electromagnetics

• Lecture 1
• Introduction and Review

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Instructor Dennis W. Prather, Professor Phone
302-831-8170 Office 217B, Evans Hall Email
dprather_at_ee.udel.edu Text Computational
Electrodynamics The finite-difference
time-domain method Authors
Allen Taflove Susan Hagness Publisher Artech
House 2nd Ed.
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Grading Policies

• Periodic Computing Assignments 50
• Midterm Project (that I design) 20
• Final Project (You can choose) 30

All source code and plotted results are required
when handing in projects. The MATLAB programming
environment is preferred for all projects and
results.
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Course Topics

• Review of Electromagnetics and analytic
  solutions.
• 1D and 2D Finite Difference Time Domain Methods
• FDTD Properties
• a) Stability
• b) Dispersion
• c) Source Conditions
• d) Absorbing Boundary Conditions
• 3D FDTD
• FDTD symmetries
• a) 2-fold, 4-fold symmetry
• b) Axial symmetry

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• Propagation Methods
• a) Stratton-Chu
• b) Near-to-Far transformation
• c) Plane Wave Spectra
• 7) Material Properties
• a) Anisotropic
• b) Dispersive
• c) Non-linear
• d) Active Materials
• 1) Semi-conductors
• 2) Quantum wells
• Parallel Computational Aspects
• a) Primarily message passing interface (MPI)

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• Applications
• a) Scattering
• b) Antennas
• c) Micro-Optics
• d) Photonic Band Gaps
• e) Semiconductor lasers
• General Comments
• All computer assignments are to be done
  individually.
• Both results AND source code must be turned in.

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Motivation For This Class
A near universal trend in advanced electronics
and photonics is for ¹smaller and ²faster
devices. 1) As things become faster then
operational wavelength becomes
smaller. a) Hence the electrical
length of the associated devices becomes
larger. b) Ramifications is that higher
frequencies/smaller wavelengths give rise to
enhanced radiation properties. - Good for
small antennas, such as cell phones. - Bad
for integrated microsystems, such as integrated
circuits.
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2) As things become smaller, their interaction
with neighboring devices
becomes much more complicated. a)
Ramifications of (2) is that the radius of the
interaction of a given device with its
surroundings become large in
comparison to device dimensions. b) The
main problem here is that taking all of these
issues into account is nearly
impossible with purely analytic techniques.
c) To solve such problems one can use
computational methods - FEM - MOM - BEM -
FDTD, which is the post popular because 1)
It is conceptually simple 2) It is simple
to code 3) It is applicable to a large
class of problems
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Before we dive into the FDTD method, it is first
necessary to review the foundation of
electromagnetics.
1) Underlying Principles a) The field of
electromagnetics (EM) is concerned with the study
of charges (electric and magnetic) in
motion and at rest. b) To a large extent,
circuit theory is a special case of EM, in that
EM principles reduce to circuit equations
when the dimension of them are small
compared to a wavelength.
c) To this end, the theoretical concepts
are described by a set of basic laws, formulated
through experiments performed during the 18th and
19th centuries. - Faradays Law
- Amperes Law - Gauss Law d)
Later, they were into a self-consistent set of
equations by Maxwell. As such,
these are known as Maxwells Equations
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Understanding Analytic Solutions
We will begin will Coulombs Law
This Equation was empirically derived in 1785 by
French Physicist Charles Coulomb. We can
re-express this force in terms of a potential,
known as the electric field (E).
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Super position is valid for electric fields.
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We can also define the electric flux
Element Surface
Normal to DA
The total flux is
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For a Spherical Surface
This is known a Gauss Law. More succinctly, we
can write
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Where D ?E
Electric field
Electric permitivity
Electric displacement vector
We can use the divergence theorem
You can see this by working an elemental area
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Substitute in
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Magnetic fields can go through a similar
deviation with the result being
However, Qm 0 because there are no magnetic
monopoles, thus
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In 1820, Jean-Baptist Biot and Felix Savart
presented the relationship between the magnetic
field and the current in a wire. This was
empirically derived as
Current loop
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For a straight wire that reduces to
Now we wish to evaluate
Where the path surrounds a straight wire.
where Ip represents the current enclosed by the
path
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Alternatively, Ip can be expressed as
Current density
Now we can write
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Using Stokes Theorem
Simplify to
This is known as Amperes Law.
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A quick explanation of Stokes Theorem
z
s
x
y
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From a Taylor series expansion
Thus
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Which is equal to
Cross section
Stokes Theorem
What if its integrated over a capacitor, where
Jp Ip 0? To explain this, Maxwell introduced
the concept of Displacement Current
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Therefore, Amperes Law can be written as
Around the same time, Michael Faraday discovered
that an electric field could be produced by a
time varying magnetic field.
s
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This was expressed as EMF, electro motive force
Lenzs Law says that the direction of the induced
EMF is such that it opposes the change in the
magnetic flux producing it. Using Stokes
Theorem
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Collectively we have
Maxwells Equations