1. 1 EEE 498/598 Overview of Electrical Engineering Lecture 11: Electromagnetic Power Flow; Reflection And Transmission Of Normally and Obliquely Incident Plane Waves; Useful Theorems

2. Lecture 11 2 Lecture 11 Objectives To study electromagnetic power flow; reflection and transmission of normally and obliquely incident plane waves; and some useful theorems. To study electromagnetic power flow; reflection and transmission of normally and obliquely incident plane waves; and some useful theorems.

3. Lecture 11 3 Flow of Electromagnetic Power Electromagnetic waves transport throughout space the energy and momentum arising from a set of charges and currents (the sources). Electromagnetic waves transport throughout space the energy and momentum arising from a set of charges and currents (the sources). If the electromagnetic waves interact with another set of charges and currents in a receiver, information (energy) can be delivered from the sources to another location in space. If the electromagnetic waves interact with another set of charges and currents in a receiver, information (energy) can be delivered from the sources to another location in space. The energy and momentum exchange between waves and charges and currents is described by the Lorentz force equation. The energy and momentum exchange between waves and charges and currents is described by the Lorentz force equation.

4. Lecture 11 4 Derivation of Poynting’s Theorem Poynting’s theorem concerns the conservation of energy for a given volume in space. Poynting’s theorem concerns the conservation of energy for a given volume in space. Poynting’s theorem is a consequence of Maxwell’s equations. Poynting’s theorem is a consequence of Maxwell’s equations.

5. Lecture 11 5 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Time-Domain Maxwell’s curl equations in differential form Time-Domain Maxwell’s curl equations in differential form

6. Lecture 11 6 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Recall a vector identity Recall a vector identity Furthermore, Furthermore,

7. Lecture 11 7 Derivation of Poynting’s Theorem in the Time Domain (Cont’d)

8. Lecture 11 8 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Integrating over a volume V bounded by a closed surface S, we have Integrating over a volume V bounded by a closed surface S, we have

9. Lecture 11 9 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Using the divergence theorem, we obtain the general form of Poynting’s theorem Using the divergence theorem, we obtain the general form of Poynting’s theorem

10. Lecture 11 10 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) For simple, lossless media, we have For simple, lossless media, we have Note that Note that

11. Lecture 11 11 Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Hence, we have the form of Poynting’s theorem valid in simple, lossless media: Hence, we have the form of Poynting’s theorem valid in simple, lossless media:

12. Lecture 11 12 Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) Time-Harmonic Maxwell’s curl equations in differential form for a simple medium Time-Harmonic Maxwell’s curl equations in differential form for a simple medium

13. Lecture 11 13 Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) Poynting’s theorem for a simple medium Poynting’s theorem for a simple medium

14. Lecture 11 14 Physical Interpretation of the Terms in Poynting’s Theorem The terms The terms represent the instantaneous power dissipated in the electric and magnetic conductivity losses, respectively, in volume V.

15. Lecture 11 15 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) The terms The terms represent the instantaneous power dissipated in the polarization and magnetization losses, respectively, in volume V.

16. Lecture 11 16 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) Recall that the electric energy density is given by Recall that the electric energy density is given by Recall that the magnetic energy density is given by Recall that the magnetic energy density is given by

17. Lecture 11 17 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) Hence, the terms Hence, the terms represent the total electromagnetic energy stored in the volume V.

18. Lecture 11 18 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) The term The term represents the flow of instantaneous power out of the volume V through the surface S.

19. Lecture 11 19 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) The term The term represents the total electromagnetic energy generated by the sources in the volume V.

20. Lecture 11 20 Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) In words the Poynting vector can be stated as “The sum of the power generated by the sources, the imaginary power (representing the time-rate of increase) of the stored electric and magnetic energies, the power leaving, and the power dissipated in the enclosed volume is equal to zero.” In words the Poynting vector can be stated as “The sum of the power generated by the sources, the imaginary power (representing the time-rate of increase) of the stored electric and magnetic energies, the power leaving, and the power dissipated in the enclosed volume is equal to zero.”

21. Lecture 11 21 Poynting Vector in the Time Domain We define a new vector called the (instantaneous) Poynting vector as We define a new vector called the (instantaneous) Poynting vector as The Poynting vector has the same direction as the direction of propagation. The Poynting vector has the same direction as the direction of propagation. The Poynting vector at a point is equivalent to the power density of the wave at that point. The Poynting vector at a point is equivalent to the power density of the wave at that point. The Poynting vector has units of W/m 2.

22. Lecture 11 22 Time-Average Poynting Vector The time-average Poynting vector can be computed from the instantaneous Poynting vector as The time-average Poynting vector can be computed from the instantaneous Poynting vector as period of the wave

23. Lecture 11 23 Time-Average Poynting Vector (Cont’d) The time-average Poynting vector can also be computed as The time-average Poynting vector can also be computed as phasors

24. Lecture 11 24 Time-Average Poynting Vector for a Uniform Plane Wave Consider a uniform plane wave traveling in the + z -direction in a lossy medium: Consider a uniform plane wave traveling in the + z -direction in a lossy medium:

25. Lecture 11 25 Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) The time-average Poynting vector is The time-average Poynting vector is

26. Lecture 11 26 Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) For a lossless medium, we have For a lossless medium, we have

27. Lecture 11 27 Reflection and Transmission of Waves at Planar Interfaces medium 2medium 1 incident wave reflected wave transmitted wave

28. Lecture 11 28 Normal Incidence on a Lossless Dielectric Consider both medium 1 and medium 2 to be lossless dielectrics. Consider both medium 1 and medium 2 to be lossless dielectrics. Let us place the boundary between the two media in the z = 0 plane, and consider an incident plane wave which is traveling in the + z - direction. Let us place the boundary between the two media in the z = 0 plane, and consider an incident plane wave which is traveling in the + z - direction. No loss of generality is suffered if we assume that the electric field of the incident wave is in the x -direction. No loss of generality is suffered if we assume that the electric field of the incident wave is in the x -direction.

29. Lecture 11 29 Normal Incidence on a Lossless Dielectric (Cont’d) medium 2medium 1 z x z = 0

30. Lecture 11 30 Normal Incidence on a Lossless Dielectric (Cont’d) Incident wave Incident wave known

31. Lecture 11 31 Normal Incidence on a Lossless Dielectric (Cont’d) Reflected wave Reflected wave unknown

32. Lecture 11 32 Normal Incidence on a Lossless Dielectric (Cont’d) Transmitted wave Transmitted wave unknown

33. Lecture 11 33 Normal Incidence on a Lossless Dielectric (Cont’d) The total electric and magnetic fields in medium 1 are The total electric and magnetic fields in medium 1 are

34. Lecture 11 34 Normal Incidence on a Lossless Dielectric (Cont’d) The total electric and magnetic fields in medium 2 are The total electric and magnetic fields in medium 2 are

35. Lecture 11 35 Normal Incidence on a Lossless Dielectric (Cont’d) To determine the unknowns E r0 and E t0, we must enforce the BCs at z = 0 : To determine the unknowns E r0 and E t0, we must enforce the BCs at z = 0 :

36. Lecture 11 36 Normal Incidence on a Lossless Dielectric (Cont’d) From the BCs we have From the BCs we have or

37. Lecture 11 37 Reflection and Transmission Coefficients Define the reflection coefficient as Define the reflection coefficient as Define the transmission coefficient as Define the transmission coefficient as

38. Lecture 11 38 Reflection and Transmission Coefficients (Cont’d) Note also that Note also that The definitions of the reflection and transmission coefficients do generalize to the case of lossy media. The definitions of the reflection and transmission coefficients do generalize to the case of lossy media. For lossless media,  and  are real. For lossless media,  and  are real. For lossy media,  and  are complex. For lossy media,  and  are complex.

39. Lecture 11 39 Traveling Waves and Standing Waves The total field in medium 1 is partially a traveling wave and partially a standing wave. The total field in medium 1 is partially a traveling wave and partially a standing wave. The total field in medium 2 is a pure traveling wave. The total field in medium 2 is a pure traveling wave.

40. Lecture 11 40 Traveling Waves and Standing Waves (Cont’d) The total electric field in medium 1 is given by The total electric field in medium 1 is given by traveling wave standing wave

41. Lecture 11 41 Traveling Waves and Standing Waves: Example medium 2medium 1 z x z = 0

42. Lecture 11 42 Traveling Waves and Standing Waves: Example (Cont’d) -2-1.5-0.500.51 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 z/ 0 Normalized E field

43. Lecture 11 43 Standing Wave Ratio The standing wave ratio is defined as The standing wave ratio is defined as In this example, we have In this example, we have

44. Lecture 11 44 Time-Average Poynting Vectors

45. Lecture 11 45 Time-Average Poynting Vectors (Cont’d) We note that

46. Lecture 11 46 Time-Average Poynting Vectors (Cont’d) Hence, Hence, Power is conserved at the interface.

47. Lecture 11 47 Oblique Incidence at a Dielectric Interface

48. Lecture 11 48 Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z)

49. Lecture 11 49 Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z)

50. Lecture 11 50 Oblique Incidence at a Dielectric Interface: Perpendicular Polarization (TE to z)

51. Lecture 11 51 Oblique Incidence at a Dielectric Interface: Perpenidcular Polarization (TM to z)

52. Lecture 11 52 Brewster Angle The Brewster angle is a special angle of incidence for which  =0. The Brewster angle is a special angle of incidence for which  =0. For dielectric media, a Brewster angle can occur only for parallel polarization. For dielectric media, a Brewster angle can occur only for parallel polarization.

53. Lecture 11 53 Critical Angle The critical angle is the largest angle of incidence for which k 2 is real (i.e., a propagating wave exists in the second medium). The critical angle is the largest angle of incidence for which k 2 is real (i.e., a propagating wave exists in the second medium). For dielectric media, a critical angle can exist only if  1 >  2. For dielectric media, a critical angle can exist only if  1 >  2.

54. Lecture 11 54 Some Useful Theorems The reciprocity theorem The reciprocity theorem Image theory Image theory The uniqueness theorem The uniqueness theorem

55. Lecture 11 55 Image Theory for Current Elements above a Infinite, Flat, Perfect Electric Conductor actual sources images electric magnetic

56. Lecture 11 56 Image Theory for Current Elements above a Infinite, Flat, Perfect Magnetic Conductor actual sources images electric magnetic h h