1 EEE 431 Computational Methods in Electrodynamics Lecture 3 By Dr. Rasime Uyguroglu

2 Energy and Power We would like to derive equations governing EM energy and power. Starting with Maxwell’s equation’s:

3 Energy and Power (Cont.) Apply H. to the first equation and E. to the second:

4 Energy and Power (Cont.) Subtracting: Since,

5 Energy and Power (Cont.) Integration over the volume of interest:

6 Energy and Power (Cont.) Applying the divergence theorem:

7 Energy and Power (Cont.) Explanation of different terms: Poynting Vector in The power flowing out of the surface S (W).

8 Energy and Power (Cont.) Dissipated Power (W) Supplied Power (W)

9 Energy and Power Magnetic power (W) Magnetic Energy.

10 Energy and Power (Cont.) Electric power (W) electric energy.

11 Energy and Power (Cont.) Conservation of EM Energy

12 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.

13 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.

14 2)Classification of differential Equations Most EM problems can be written as: L: Operator (integral, differential, integrodifferential) : Excitation or source : Unknown function.

15 Classification of Differential Equations (Cont.) Example: Poisson’s Equation in differential form.

16 Classification of Differential Equations (Cont.): In integral form, the Poisson’s equation is of the form:

17 Classification of Differential Equations (Cont.): EM problems satisfy second order partial differential equations (PDE). i.e. Wave equation, Laplace’s equation.

18 Classification of Differential Equations (Cont.): In general, a two dimensional second order PDE: If PDE is homogeneous. If PDE is inhomogeneous.

19 Classification of Differential Equations (Cont.): A PDE in general can have both: 1) Initial values (Transient Equations) 2) Boundary Values (Steady state equations)

20 Classification of Differential Equations (Cont.): The L operator is now:

21 Classification of Differential Equations (Cont.): Examples: Elliptic PDE, Poisson’s and Laplace’s Equations:

22 Classification of Differential Equations (Cont.): For both cases a=c=1,b=0. An elliptic PDE usually models the closed region problems.

23 Classification of Differential Equations (Cont.): Hyperbolic PDE’s, the Wave Equation in one dimension: Propagation Problems (Open region problems)

24 Parabolic PDE, Heat Equation in one dimension. Open region problem. Classification of Differential Equations (Cont.):

25 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.

26 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.

27 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.