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Fundamental Concepts (2 sessions) . Review of Electromagnetic Theory. Maxwell’s Equations: Constitutive Relations:. is magnetic conductive current density (in volts/square meter). Boundary Conditions: Constitutive parameters are σ, ε , μ .

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slide2

Review of Electromagnetic Theory

  • Maxwell’s Equations:
  • Constitutive Relations:

is magnetic conductive current density (in volts/square meter)

  • Boundary Conditions:
  • Constitutive parameters are σ, ε,μ.
  • Linear Medium: σ, ε,μ are independent of E and H.
  • Homogeneous Medium: σ, ε,μ are not functions of space variables or.
  • IsotropicMedium: σ, ε,μ are independent of direction (scalars).

is magnetic resistivity

slide3

Review of Electromagnetic Theory

  • When a medium is source-free: J = 0, ρv = 0
  • In practice, only two of Maxwell’s equations are used:
  • Since other two are implied.
  • Also, in practice, it is sufficient to make tangential components of fields satisfy necessary boundary conditions.
  • Since normal components implicitly satisfy their corresponding boundary conditions.
  • Wave Equations:
  • Altogether there are six scalar equations for Ex, Ey, Ez, Hx, Hy, Hz the form of:
  • Time-varying Potentials:
slide4

Review of Electromagnetic Theory

  • Time-Harmonic Fields:
  • In sinusoidal steady state:
  • Source-free wave equation in phasor representation:
  • General wave equation in phasor representation:
    • Special Case 1: Poisson’s equation for static case (ω = 0):
    • Special Case 2: Laplace’s equation for static case and source-free:
slide5

Fundamental Concepts

  • Classification of EM Problems:
  • ThisClassification help to answer the question of “What method is best for solving a problem”.
  • Three independent items define a problem uniquely:
  • (1)the solution region (problem domain) of the problemas R:
  • (2) the nature of the equation describing the problem,
  • (3) the associated boundary conditions as S.
  • Classification of Solution Regions:
  • There are two classifications:
    • Solution region R is interior (inner, closed, or bounded)
    • Solution region R is exterior (outer, open, or unbounded)
  • If part or all of S is at infinity, R is exterior otherwise R is interior.
  • For example, wave propagation in a waveguide is an interior problem.
  • For example, wave propagation in free space (scattering of EM waves by raindrops, and radiation from a dipole antenna) are exterior problems.
  • Solution region R could be linear,homogeneousandisotropic.

Ris the solution region

Sis the boundary condition

slide6

Fundamental Concepts

  • Classification of Differential Equations:
  • EM problems are classified in terms of equations describing them.
  • Equations could be differential or integral or both defined as:
  • For example:
  • Another example:
  • A second-order partial differential equation (PDE):
  • or simply:
  • PDE operator:
slide7

Fundamental Concepts

  • In non-linear PDEs, coefficients are function of quantity
  • Any linear second-order PDE can be classified as elliptic, hyperbolic, or parabolic:
  • An elliptic PDE usually models an interior problem such as:
  • A Hyperbolic PDE usually models an exterior problem as:
  • A ParabolicPDE usually models an exterior problem such as diffusion (or heat) equation:

Laplace’s equation:

Poisson’s equations:

Elliptic problem

parabolic, or hyperbolic problem

slide8

Fundamental Concepts

  • Nondeterministic Problems:
  • Previous problems are deterministic, since quantity of interest can be determined directly.
  • Another type of problem where quantity is found indirectly is called nondeterministicor eigenvalue.
  • StandardEigen problem is of the form of:
  • A more general version is generalized Eigen-problem having the form of:
  • Only some particular values of λcalled eigenvaluesare permissible.
  • Eigen-problems are usually encountered in vibration and waveguide problems.
  • In these problems eigenvalues λcorrespond to physical quantities such as resonance and cutoff frequencies.

Where source term has been replaced by λ

Where M, like L, is a linear operator

slide9

Fundamental Concepts

  • Classification of Boundary Conditions:
  • Usually boundary conditions are of the Dirichlet and Neumann types.
  • Dirichlet boundary condition:
  • A good example is the charged metal plate.
    • Because all points on a metal are at same potential, a metal plate can readily be modeled by a region of points with some fixed voltage.
  • Neumannboundary condition:
  • Mixed boundary condition:
  • These conditions are called homogeneous boundary conditions.
  • General ones are inhomogeneous:
  • Dirichlet:
  • Neumann:
  • Mixed:

i.e., the normal derivative of vanishes on S

h(r) is a known function

slide10

Fundamental Concepts

  • Some Important Theorems:
  • Superposition Principle:
  • If each member of a set of functions φn , n=1,2,…,Nis a solution to PDE:
  • Then a linear combination of them also satisfies the PDE as:
  • Uniqueness Theorem:
  • This guarantees that solution a PDE with some prescribed boundary conditions is only one possible.
  • If a set of fields (E,H) is found which satisfies simultaneously Maxwell’s equations and prescribed boundary conditions, this set is unique.
  • Therefore, a field is uniquely specified by sources (ρv,J) within medium plus tangential components of E or H over boundary.
  • To prove uniqueness theorem, suppose there exist two solutions:
slide11

Fundamental Concepts

  • Uniqueness Theorem (cont.):
  • We denote the difference of the two fields as:
  • These must satisfy the source-free Maxwell\'s equations:
  • Dotting both sides with ΔEgives:
  • Integrating over volume:
  • Therefore ΔE and ΔH satisfy the Poynting theorem just as E1,2and H1,2
  • Only Etand Htcontribute to surface integral on the left side.
  • Therefore, if E1tand E2tor H1tand H2t are equal over S, ΔEtand ΔHtvanish on S.
  • Consequently, surface integral is identically zero, and hence right side must vanish also.
  • It follows that ΔE=0 due to second integral on right side and hence also ΔH=0 throughout the volume.
  • Thus E1=E2 and H1=H2, confirming that the solution is unique.

Using:

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