EEE 431Computational methods in Electrodynamics

Lecture 1By

Rasime Uyguroglu

Science knows no country because knowledge belongs to humanity and is the torch which illuminates the world.

Louis Pasteur

Methods Used in Solving Field Problems

Experimental methods Analytical Methods Numerical Methods

Experimental Methods

Expensive Time Consuming Sometimes hazardous Not flexible in parameter variation

Analytical Methods

Exact solutions Difficult to Solve Simple canonical problems Simple materials and Geometries

Numerical Methods

Approximate Solutions Involves analytical simplification to the

point where it is easy to apply it Complex Real-Life Problems Complex Materials and Geometries

Applications In Electromagnetics

Design of Antennas and Circuits Simulation of Electromagnetic Scattering

and Diffraction Problems Simulation of Biological Effects (SAR:

Specific Absorption Rate) Physical Understanding and Education

Most Commonly methods used in EM

Analytical Methods Separation of Variables Integral Solutions, e.g. Laplace Transforms

Most Commonly methods used in EM

Numerical Methods Finite Difference Methods Finite Difference Time Domain Method Method of Moments Finite Element Method Method of Lines Transmission Line Modeling

Numerical Methods (Cont.)

Above Numerical methods are applied to problems other than EM problems. i.e. fluid mechanics, heat transfer and acoustics.

The numerical approach has the advantage of allowing the work to be done by operators without a knowledge of high level of mathematics or physics.

Review of Electromagnetic Theory

Notations

E: Electric field intensity (V/ m) H: Magnetic field intensity (A/ m) D: Electric flux density (C/ m2 ) B: Magnetic flux density (Weber/ m2 ) J: Electric current density (A/ m2 ) Jc :Conduction electric current density (A/ m2 ) Jd :Displacement electric current density(A/m2)

:Volume charge density (C/m3)

Historical Background

Gauss’s law for electric fields:

Gauss’s law for magnetic fields:

.D

. 0B

Historical Background (cont.)

Ampere’s Law

Faraday’s law

DXH J

t

BXE

t

Electrostatic Fields

Electric field intensity is a conservative field:

Gauss’s Law:

0XE

. *D

Electrostatic Fields

Electrostatic fields satisfy:

Electric field intensity and electric flux density vectors are related as:

The permittivity is in (F/m) and it is denoted as

0 . 0XE or E dl

**D E

Electrostatic Potential

In terms of the electric potential V in volts,

Or

***E V

.V E dl

Poisson’s and Laplace’s Equation’s

Combining Equations *, ** and *** Poisson’s Equation:

When , Laplace’s Equation:

2 vV

0v

2 0V

Magnetostatic Fileds

Ampere’s Law, which is related to Biot-Savart Law:

Here J is the steady current density.

ˆ. .L s

H dl J nds

Static Magnetic Fields (Cont.)

Conservation of magnetic flux or Gauss’s Law for magnetic fields:

ˆ. 0sB nds

Differential Forms

Ampere’s Law:

Gauss’s Law:

XH J

. 0B

Static Magnetic Fields

The vector fields B and H are related to each other through the permeability in (H/m) as:

B H

Ohm’s Law

In a conducting medium with a conductivity (S/m) J is related to E as:

J E

Magnetic vector Potential

The magnetic vector potential A is related to the magnetic flux density vector as:

B XA

Vector Poisson’s and Laplace’s Equations

Poisson’s Equation:

Laplace’s Equation, when J=0:

2A J

2 0A

Time Varying Fields

In this case electric and magnetic fields exists simultaneously. Two divergence expressions remain the same but two curl equations need modifications.

Differential Forms of Maxwell’s equationsGeneralized Forms

.D

. 0B

BXE

t

DXH J

t

Integral Forms

Gauss’s law for electric fields:

Gauss’s law for magnetic fields:

ˆ. v equ

s v

D nds dv Q

ˆ. 0s

B nds

Integral Forms (Cont.)

Faraday’s Law of Induction:

Modified Ampere’s Law:

ˆ. .L s

BE dl nds

t

ˆ. ( ).L s

DH dl J nds

t

Constitutive Relations

D E

B H

J E

Two other fundamental equations

1)Lorentz Force Equation:

Where F is the force experienced by a particle with charge Q moving at a velocity u in an EM filed.

( )F Q E uXB

Two other equations (cont.)

Continuity Equation:

. vJt