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ELEG 840 Advanced Computational Electromagnetics

- Lecture 1
- Introduction and Review

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.

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.

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

- 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)

- 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.

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.

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

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

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).

Super position is valid for electric fields.

We can also define the electric flux

Element Surface

Normal to DA

The total flux is

For a Spherical Surface

This is known a Gauss Law. More succinctly, we

can write

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

(No Transcript)

Substitute in

Magnetic fields can go through a similar

deviation with the result being

However, Qm 0 because there are no magnetic

monopoles, thus

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

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

Alternatively, Ip can be expressed as

Current density

Now we can write

Using Stokes Theorem

Simplify to

This is known as Amperes Law.

A quick explanation of Stokes Theorem

z

s

x

y

From a Taylor series expansion

Thus

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

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

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

Collectively we have

Maxwells Equations