EEE 431 Computational Methods in Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 3
• By
• Dr. Rasime Uyguroglu

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Energy and Power

• We would like to derive equations governing EM
  energy and power.
• Starting with Maxwells equations

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Energy and Power (Cont.)

• Apply H. to the first equation and E. to the
  second

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Energy and Power (Cont.)

• Subtracting
• Since,

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Energy and Power (Cont.)

• Integration over the volume of interest

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Energy and Power (Cont.)

• Applying the divergence theorem

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Energy and Power (Cont.)

• Explanation of different terms
• Poynting Vector in
• The power flowing out of the surface S (W).

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Energy and Power (Cont.)

• Dissipated Power (W)
• Supplied Power (W)

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Energy and Power

• Magnetic power (W)
• Magnetic Energy.

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Energy and Power (Cont.)

• Electric power (W)
• electric energy.

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Energy and Power (Cont.)

• Conservation of EM Energy

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Classification of EM Problems

• 1) The solution region of the problem,
• 2) The nature of the equation describing the
  problem,
• 3) The associated boundary conditions.

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1) Classification of Solution Regions

• Closed region, bounded, or open region,
  unbounded. i.e Wave propagation in a waveguide
  is a closed region problem where radiation from a
  dipole antenna is an open region problem.
• A problem also is classified in terms of the
  electrical, constitutive properties. We shall be
  concerned with simple materials here.

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2)Classification of differential Equations

• Most EM problems can be written as
• L Operator (integral, differential,
  integrodifferential)
• Excitation or source
• Unknown function.

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Classification of Differential Equations (Cont.)

• Example Poissons Equation in differential form .

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Classification of Differential Equations (Cont.)

• In integral form, the Poissons equation is of
  the form

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Classification of Differential Equations (Cont.)

• EM problems satisfy second order partial
  differential equations (PDE).
• i.e. Wave equation, Laplaces equation.

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Classification of Differential Equations (Cont.)

• In general, a two dimensional second order PDE
• If PDE is homogeneous.
• If PDE is inhomogeneous.

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Classification of Differential Equations (Cont.)

• A PDE in general can have both
• 1) Initial values (Transient Equations)
• 2) Boundary Values (Steady state equations)

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Classification of Differential Equations (Cont.)

• The L operator is now

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Classification of Differential Equations (Cont.)

• Examples
• Elliptic PDE, Poissons and Laplaces Equations

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Classification of Differential Equations (Cont.)

• For both cases ac1,b0.
• An elliptic PDE usually models the closed region
  problems.

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Classification of Differential Equations (Cont.)

• Hyperbolic PDEs, the Wave Equation in one
  dimension
• Propagation Problems (Open region problems)

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Classification of Differential Equations (Cont.)

• Parabolic PDE, Heat Equation in one dimension.
• Open region problem.

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Classification of Differential Equations (Cont.)

• The type of problem represented by
• Such problems are called deterministic.
• Nondeterministic (eigenvalue) problem is
  represented by
• Eigenproblems Waveguide problems, where
  eigenvalues corresponds to cutoff frequencies.

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3) Classification of Boundary Conditions

• What is the problem?
• Find which satisfies
  within a solution region R.
• must satisfy certain conditions on
  Surface S, the boundary of R.
• These boundary conditions are Dirichlet and
  Neumann types.

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Classification of Boundary Conditions (Cont.)

• 1) Dirichlet B.C.
• vanishes on S.
• 2) Neumann B.C.
• i.e. the normal derivative of vanishes on
  S.
• Mixed B.C. exits.