EEE 431 Computational Methods in Electrodynamics

1. EEE 431Computational 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. Classification of Differential Equations (Cont.): • Parabolic PDE, Heat Equation in one dimension. • Open region problem.

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.