ELEG 648 Plane waves

1. ELEG 648Plane waves Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Email: mirotzni@ece.udel.edu

2. SUMMARY Time Domain Frequency Domain

3. Wave Equation Time Dependent Homogenous Wave Equation (E-Field) Vector Identity

4. Source Free Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field)

5. Source Free Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field) Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field) Lossless

6. Source Free Source Free Wave Equation: Time Harmonic Time Domain Frequency Domain Lossless Lossless “Helmholtz Equation”

7. General Solution Case: Time HarmonicRectangular Coordinates Wave Number

8. Separation of Variable Solutions Assume Solution of the form:

9. Separation of Variable Solutions Assume Solution of the form:

10. Separation of Variable Solutions Assume Solution of the form: constant function of x function of y function of z

11. Separation of Variable Solutions constant function of x function of y function of z

12. Separation of Variable Solutions Solutions:

13. Separation of Variable Solutions Solutions: Traveling and standing waves Exponentially modulated traveling wave Evanescent waves or or

14. E E E E E E E H H H H H H H Wave Propagation and Polarization TEM: Transverse Electromagnetic Waves “A mode is a particular field configuration. For a given electromagnetic boundary value problem, many field configurations that satisfy the wave equation, Maxwell’s equations, and boundary conditions usually exits. A TEM mode is one whole field intensities, both E and H, at every point in space are contained in a local plane, referred to as equiphase plane, that is independent of time” Plane Waves “If the space orientation of the planes for a TEM mode are the same (equiphase planes are parallel) then the fields form a plane wave. Uniform Plane Waves “If in addition to having planar equiphases the field has equiamplitude (the amplitude of the field is the same over each plane) planar surfaces then it is called a uniform plane wave.”

15. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation Let’s begin by assuming the solution is only a function of z and has only the x component of electric field. Let’s also look at the term that represents a traveling wave moving in the +z direction

16. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation For uniform plane wave assume the solution is only a function of z and has only the x component of electric field. Lets find H

17. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation Several observations: • E and H are orthogonal to each other and to the direction of energy propagation • E and H are in phase with each other • H is smaller in amplitude than E by the term (for a uniform plane wave) Wave Impedance

18. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation How fast does the wave move?

19. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation How much power does the wave carry?

20. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation How fast does the power flow?

21. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation Relationship between phase and group velocity

22. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

23. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

24. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

25. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

26. Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

27. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

28. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

29. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

30. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

31. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

32. Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

33. Example: Skin Depth in Sea Water Skin Depth, cm Frequency, GHz

34. Example: Skin Depth in Copper Skin Depth, microns Frequency, GHz

35. Example: Skin Depth in Teflon Skin Depth, meters Frequency, GHz

36. Polarization

37. Polarization

38. Polarization

39. Polarization

40. Polarization

41. Polarization

42. Uniform Plane Waves: Propagation in Any Arbitrary Direction z H f E q y x

43. z H f E q y x Uniform Plane Waves: Propagation in Any Arbitrary Direction Since E and b are at right angles from each other. where and

44. Uniform Plane Waves: Propagation in Any Arbitrary Direction Summary and Observations: Frequency Domain Time Domain Observation 1. E, H and b vectors are pointing in orthogonal directions. Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance