Fundamental Concepts (2 sessions)

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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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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

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:

Review of Electromagnetic Theory

• Time-Harmonic Fields:
• 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:

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

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:

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

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

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

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:

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: