Download EEE 431 Computational methods in Electrodynamics

EEE 431
Computational methods in
Electrodynamics
Lecture 1
By
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:
.D  

Gauss’s law for magnetic fields:
.B  0
Historical Background (cont.)

Ampere’s Law

Faraday’s law
D
XH  J 
t
B
XE  
t
Electrostatic Fields

Electric field intensity is a conservative field:
XE  0

Gauss’s Law:
.D   *
Electrostatic Fields

Electrostatic fields satisfy:
XE  0 or

 E.dl  0
Electric field intensity and electric flux density
vectors are related as:
D   E **

The permittivity is in (F/m) and it is denoted as

Electrostatic Potential

In terms of the electric potential V in
volts,
E  V ***

Or
V    E .dl
Poisson’s and Laplace’s
Equation’s

Combining Equations *, ** and ***
Poisson’s Equation:
v
V 

2

When
v  0, Laplace’s Equation:
 V 0
2
Magnetostatic Fileds

Ampere’s Law, which is related to BiotSavart Law:
ˆ
H
.
dl

J
.
nds


L

s
Here J is the steady current density.
Static Magnetic Fields (Cont.)

Conservation of magnetic flux or
Gauss’s Law for magnetic fields:

s
ˆ 0
B. nds
Differential Forms

Ampere’s Law:
XH  J

Gauss’s Law:
.B  0
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:
 A   J
2

Laplace’s Equation, when J=0:
 A0
2
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 equations
Generalized Forms
B
XE  
t
.D  
D
XH  J 
t
.B  0
Integral Forms

Gauss’s law for electric fields:
ˆ    dv  Q
 D.nds
v
s
equ
v
Gauss’s law for magnetic fields:

s
ˆ 0
B.nds
Integral Forms (Cont.)

Faraday’s Law of Induction:
B
ˆ
E
.
dl


.
nds
L
s t

Modified Ampere’s Law:
D
ˆ
H
.
dl

(
J

).
nds
L
s
t
Constitutive Relations
D E
B  H
J E
Two other fundamental
equations

1)Lorentz Force Equation:
F  Q( E  uXB )

Where F is the force experienced by a
particle with charge Q moving at a
velocity u in an EM filed.
Two other equations (cont.)

Continuity Equation:
v
.J  
t