Fundamental Concepts (2 sessions)

1. Fundamental Concepts(2 sessions)

2. 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

3. 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:

4. 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:

5. 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

6. 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:

7. 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

8. 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

9. 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

10. 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:

11. 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: