1. Chapter 1. Electromagnetic Theory Sept. 1 st, 2008

2. 2 1.1 Introduction to Microwave Engineering Microwaves: 300 MHz ~ 300 GHz (1 mm ≤ λ ≤ 1 m) 1mm ≤ λ ≤ 10mm  Millimeter waves Because of the high frequency (short wavelength), standard circuit theory generally cannot be used directly to solve microwave network problems. Microwave components: distributed elements (the phase of a voltage or current changes significantly over the physical extent of the device) Optical engineering Quasioptical

3. 3 Figure 1.1 (p. 2) The electromagnetic spectrum.

4. 4 Applications of Microwave Engineering Microwave engineering 이 필요한 이유 –Antenna gain 은 Antenna 의 electrical 크기에 비례. – 높은 주파수에서는 Bandwidth 가 더 커짐. –Microwave signal 은 Line of Sight 로 Travel. –Effective reflection area (radar cross section) 는 Target 의 electrical 크기에 비례. – 분자, 원자 및 핵공명이 Microwave 주파수에서 일어남.

5. 5 Microwave 기술의 응용 –Communication systems –Radar systems –Remote sensing –Medical systems

6. 6 A Short History of Microwave Engineering Modern electromagnetic theory 의 수립역사 1.1873 년 James Clerk Maxwell 이 이론적으로 EM wave 와, 빛은 EM energy 의 형태임을 주장 2.1885 – 1887, Oliver Heaviside 가 Maxwell 이론의 수학적 복잡성을 대부분 제거, Vector 표기 제안 및 Guided wave 와 Transmission line 의 응용 제안 3.1887-1891, Heinrich Hertz, 실험으로 증명

7. 7 Microwave 기술의 발전 1.1940 년대, 2 차대전에서 Radar 의 발명으로 큰 발전 2.MIT 에서 Radiation Lab 을 세움 –N. Marcuvitz, I.I.Rabi, J.S.Schwinger, H.A.Bethe, E.M.Purcell, C.G.Montgomery, R.H.Dicke….. –The classic 28-volume Radiation Laboratory Series of books

8. 8 Figure 1.2 (p. 4, see next slide for photograph) Original aparatus used by Hertz for his electromagnetics experiments. (1) 50 MHz transmitter spark gap and loaded dipole antenna. (2) Parallel wire grid for polarization experiments. (3) Vacuum apparatus for cathode ray experiments. (4) Hot-wire galvanometer. (5) Reiss or Knochenhauer spirals. (6) Rolled-paper galvanometer. (7) metal sphere probe. (8) Reiss spark micrometer. (9) Coaxial transmission line. (10-12) Equipment to demonstrate dielectric polarization effects. (13) Mercury induction coil interrupter. (14) Meidinger cell. (15) Vacuum bell jar. (16) High-voltage induction coil. (17) Bunsen cells. (18) Large-area conductor for charge storage. (19) Circular loop receiving antenna. (20) Eight-sided receiver detector. (21) Rotating mirror and mercury interrupter. (22) Square loop receiving antenna. (23) Equipment for refraction and dielectric constant measurement. (24) Two square loop receiving antennas. (25) Square loop receiving antenna. (26) Transmitter dipole. (27) High-voltage induction coil. (28) Coaxial line. (29) High-voltage discharger. (30) Cylindrical parabolic reflector/receiver. (31) Cylindrical parabolic reflector/transmitter. (32) Circular loop receiving antenna. (33) Planar reflector. (34, 35) Battery of accumulators. Photographed on October 1, 1913 at the Bavarian Academy of Science, Munich, Germany, with Hertz ’ s assistant, Julius Amman. Photograph and identification courtesy of J. H. Bryant, University of Michigan.

9. 9 Figure 1.2 (p. 4)

10. 10 1.2 Maxwell’s Equations Maxwell’s work was based on a large body of empirical and theoretical knowledge developed by Gauss, Ampere, Faraday, and others. The general form of time-varying Maxwell’s equations in differential form. What are the sources of electromagnetic fields?

11. 11 Faraday’s law Spatially varying field Moving loop in uniform B-field Electric generator

12. 12 Corrected Ampere’s circuital law James Clerk Maxwell conceived of displacement current as a polarization current which he used to model the magnetic field hydrodynamically and mechanically. He added this displacement current to Ampère's circuital law in his 1861 paper.

13. 13 Gauss’s law Gauss's Law is that the [net] electric flux through any closed surface is equal to the charge inside that surface divided by this constant ε.

14. 14 In free space, Since The continuity eq. can be derived by taking the Div of (1.1b)  Charge is conserved, or current is continuous since is the outflow of current at a point.

15. 15 Integral form A sinusoidal E-field in the x direction of the form At t = 0

16. 16 Figure 1.3 (p. 7) The closed contour C and surface S associated with Faraday ’ s law.

17. 17 Power and energy In phasor form

18. 18 Assuming an e jωt time dependence,

19. 19 Figure 1.4a/b (p. 9) Arbitrary volume, surface, and line currents. (a) Arbitrary electric and magnetic volume current densities. (b) Arbitrary electric and magnetic surface current densities in the z = z 0 plane.

20. 20 Figure 1.4c/d (p. 9) Arbitrary volume, surface, and line currents. (c) Arbitrary electric and magnetic line currents. (d) Infinitesimal electric and magnetic dipoles parallel to the x-axis.

21. 21 1.3 Fields in Media and Boundary Conditions For a dielectric material, an applied E  electric dipole moment  increase D where The ε″ accounts for loss in the medium (heat) due to damping of the vibrating dipole moment. The loss of a dielectric material may also be considered as an equivalent conductor loss.

22. 22 J = σE  Ohm’s law from an EM field point of view Loss tangent

23. 23 Isotropic material: P e in the same direction as E. Some materials are anisotropic (still linear): crystal structure and ionized gases For isotropic material: diagonal ε matrix.

24. 24 For magnetic materials Magnetic polarization P m, where Imaginary part of χ m or μ: loss due to damping force No magnetic conductivity  no real magnetic current Magnetic materials may be anisotropic

25. 25 If linear media are assumed (ε, μ not depending on E or H) ε, μ may be complex and may be tensors  possible phase shift between D and E, or B and H. 어떻게 하면 풀수 있을까요 ? Boundary condition

26. 26 Fields at a General Material Interface

27. 27 Figure 1.5 (p. 12) Fields, currents, and surface charge at a general interface between two media.

28. 28 Figure 1.6 (p. 12) Closed surface S for equation (1.29).

29. 29 As h  0 Similarly

30. 30 Figure 1.7 (p. 13) Closed contour C for Equation (1.33).

31. 31 If a magnetic surface current density M S exists on the surface, Similarly

32. 32 Fields at a Dielectric Interface Fields at the Interface with a Perfect Conductor (Electric Wall) (M S = 0 assumed)

33. 33 The Magnetic Wall Boundary Condition (J S = 0 assumed)

34. 34 HW1 1.Divergence Theorem 을 증명하고 물리적 의미를 설명하시오. 2.Stokes’ Theorem 을 증명하고 물리적 의미를 설명하시오. 3.Ampere 의 법칙과 Gauss 의 법칙에 대하여 설명하시오.

35. 35 1.4 The Wave Equation and Basic Plane Wave Solutions The Helmholtz Equation In a source-free, linear, isotropic, homogeneous region, Similarly Wave equation or Helmholtz equation Wavenumber or propagation constant

36. 36 Plane waves in a lossless medium ε and μ: real  k: real Consider only x component and uniform (no variation) in the x and y direction ( ) In the time domain, Phase velocity In free-space, v p = c = 2.998x10 8 m/s

37. 37 The wavelength, λ is defined as the distance between 2 successive maxima on the wave, at a fixed instant of time. Magnetic field, (1.41a)   The wave impedance for the plane (the ratio of the E and H fields) Ex 1.1

38. 38 Plane waves in a general lossy medium Define a complex propagation constant Solution?

39. 39 Associated magnetic field

40. 40 Plane waves in a good conductor Conductor current >> displacement current (σ >> ωε) The skin depth, or characteristic depth of penetration Ex 1.2 The most of the current flow in a good conductor occurs in an extremely thin region near the surface of the conductor.

41. 41 The wave impedance inside a good conductor 1.5 General plane wave solutions

42. 42 Define a wavenumber vector k a unit vector in the direction of propagation

43. 43 Also define a position vector, If

44. 44 If 2 of 3 components can be chosen independently. Magnetic field can be found from Maxwell’s eq.

45. 45 Figure 1.8 (p. 22) Orientation of the vectors for a general plane wave.

46. 46 The time-domain expression for the E-field can be found as Ex 1.3

47. 47 Circularly Polarized Plane Waves Plane waves having E-field vector pointing in a fixed direction are called linearly polarized waves. Consider the superposition of an x linearly polarized with amplitude E 1 and a y linearly polarized with E 2. - If E 1 ≠ 0, E 2 = 0  linearly polarized in x direction - If E 1 = 0, E 2 ≠ 0  linearly polarized in y direction - If E 1 ≠ 0, E 2 ≠ 0 (both real), linearly polarized at the angle

48. 48 If E 1 = E 2 = E 0, If E 1 = jE 2 = E 0,  The E-field vector changes with time or equivalently with distance along the z-axis. Pick a position (z = 0)  RHCP

49. 49 Similarly,  LHCP

50. 50 Figure 1.9 (p. 24) Electric field polarization for (a) RHCP and (b) LHCP plane waves.

51. 51 1.6 Energy and Power In the sinusoidal steady-state case, the time-average stored energy in a volume V is given by, (ε: real) Similarly, From the Maxwell’s eq.

52. 52

53. 53 Poynting’s theorem where Poynting vector

54. 54 Figure 1.10 (p. 25) A volume V, enclosed by the closed surface S, containing fields and current sources

55. 55 Power absorbed by a Good conductor To calculate attenuation and loss due to an imperfect conductor, take the interface between a lossless medium and a good conductor. A field is incident from z 0. The real average power entering the conductor volume: (E & H: fields at the interface) The contribution to the integral from S can be made 0 by proper selection of this surface.

56. 56 Figure 1.11 (p. 26) An interface between a lossless medium and a good conductor with a closed surface S 0 + S for computing the power dissipated in the conductor.

57. 57 R S : the surface resistivity of the conductor

58. 58 1.7 Plane Wave Reflection from a Media Interface Figure 1.12 (p. 27) Plane wave reflection from a lossy medium; normal incidence.

59. 59 General Medium For z < 0, In z > 0, T: Transmission coeff.

60. 60 Since E t must be continuous at z = 0,

61. 61 Lossless Medium If the region z > 0 is a lossless dielectric, σ = 0, μ & ε are real. η, Γ, T: real

62. 62 For power conservation, in z < 0 For z > 0,

63. 63 At z = 0, S - = S +, complex power flow is conserved across the interface. For z < 0, the time-average power flow through a 1-m 2 cross section, Fore z > 0,

64. 64 Good Conductor If the region z > 0 is a good conductor,  E and H will be 45 o out of phase. z < 0,

65. 65 z > 0, So, at the interface at z = 0, S - = S +, and complex power is conserved. Observe

66. 66 The time-average power flows are  power balance at z = 0. In addition,

67. 67 The electric volume current density flowing in the conducting region is given as: The average power dissipated in a 1 m 2 cross-sectional volume of the conductor

68. 68 Perfect Conductor Assume z > 0 contains a perfect conductor. The fields for z > 0 decay infinitely fast, and are identically 0 in the perfect conductor. The perfect conductor can be thought of as “shorting out” the incident E-field. For z < 0,

69. 69 At z = 0, The Poynting vector for z < 0 is  Zero real part: no real power is delivered to the perfect conductor. The volume current density

70. 70 The Surface Impedance Concept Particularly where the effect of attenuation or conductor loss is needed, the presence of an imperfect conductor must be taken into account. Consider a good conductor in z > 0. 3 ways to compute the power. 1.Joule’s law (η << η 0 ) where

71. 71 2.Poynting vector 3.Effective surface current density and the surface impedance so the total current flow per unit width in the x direction

72. 72 For large σ, Now replace the exponentially decaying volume current with a uniform volume current extending a distance of one skin depth. Let

73. 73 Find the power lost: Power loss can be accurately and simply calculated as Ex 1.4

74. 74 Figure 1.13 (p. 35) Geometry for a plane wave obliquely incident at the interface between two dielectric regions.

75. 75 Figure 1.14 (p. 38) Reflection coefficient magnitude for parallel and perpendicular polarizations of a plane wave obliquely incident on a dielectric half-space.

76. 76 1.9 Some Useful Theorems The Reciprocity Theorem

77. 77 Figure 1.15 (p. 40) Geometry for the Lorentz reciprocity theorem.

78. 78 Image Theory In many problems, a current source is located in the vicinity of a conducting ground plane. Image theory permits the removal of the ground plane by placing a virtual image source on the other side of the ground plane. Consider the surface current density A standing wave field in 0 < z < d Positively traveling wave for z > d

79. 79 Figure 1.16 (p. 42) Illustration of image theory as applied to an electric current source next to a ground plane. (a) An electric surface current density parallel to a ground plane. (b) The ground plane of (a) replaced with image current at z = – d.

80. 80 For 0 < z < d, For z > d, Boundary Condition  E x = 0 at z = 0 Since M S = 0, We have

81. 81 Total fields are Now apply the image theory to this problem. By superposition, the total fields for z > 0 can be found by combining the fields from the 2 sources individually.

82. 82 Fields due to source at z = d,

83. 83 Fields due to source at z = -d,

84. 84 Figure 1.17 (p. 44) Electric and magnetic current images. (a) An electric current parallel to a ground plane. (b) An electric current normal to a ground plane. (c) A magnetic current parallel to a ground plane. (d) A magnetic current normal to a ground plane.