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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