EEE 431 Computational methods in Electrodynamics

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EEE 431Computational methods in Electrodynamics

• Lecture 1
• By
• Rasime Uyguroglu

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

• Louis Pasteur

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Methods Used in Solving Field Problems

• Experimental methods
• Analytical Methods
• Numerical Methods

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Experimental Methods

• Expensive
• Time Consuming
• Sometimes hazardous
• Not flexible in parameter variation

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Analytical Methods

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

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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

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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

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Most Commonly methods used in EM

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

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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

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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.

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Review of Electromagnetic Theory
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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)

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Historical Background

• Gausss law for electric fields
• Gausss law for magnetic fields

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Historical Background (cont.)

• Amperes Law
• Faradays law

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Electrostatic Fields

• Electric field intensity is a conservative field
• Gausss Law

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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

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Electrostatic Potential

• In terms of the electric potential V in volts,
• Or

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Poissons and Laplaces Equations

• Combining Equations , and Poissons
  Equation
• When , Laplaces Equation

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Magnetostatic Fileds

• Amperes Law, which is related to Biot-Savart
  Law
• Here J is the steady current density.

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Static Magnetic Fields (Cont.)

• Conservation of magnetic flux or Gausss Law for
  magnetic fields

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Differential Forms

• Amperes Law
• Gausss Law

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Static Magnetic Fields

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

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Ohms Law

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

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Magnetic vector Potential

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

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Vector Poissons and Laplaces Equations

• Poissons Equation
• Laplaces Equation, when J0

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Time Varying Fields

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

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Differential Forms of Maxwells
equationsGeneralized Forms
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Integral Forms

• Gausss law for electric fields
• Gausss law for magnetic fields

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Integral Forms (Cont.)

• Faradays Law of Induction
• Modified Amperes Law

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Constitutive Relations
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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.

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Two other equations (cont.)

• Continuity Equation