Chapter 1. Electromagnetic Theory

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

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

4. Applications of Microwave Engineering • Microwave engineering이 필요한 이유 • Antenna gain은 Antenna의 electrical 크기에 비례. • 높은 주파수에서는 Bandwidth가 더 커짐. • Microwave signal은 Line of Sight로 Travel. • Effective reflection area(radar cross section)는 Target의 electrical 크기에 비례. • 분자, 원자 및 핵공명이 Microwave 주파수에서 일어남.

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

6. A Short History of Microwave Engineering • Modern electromagnetic theory의 수립역사 • 1873년 James Clerk Maxwell이 이론적으로 EM wave와, 빛은 EM energy의 형태임을 주장 • 1885 – 1887, Oliver Heaviside가 Maxwell이론의 수학적 복잡성을 대부분 제거, Vector 표기 제안 및 Guided wave와 Transmission line의 응용 제안 • 1887-1891, Heinrich Hertz, 실험으로 증명

7. Microwave 기술의 발전 • 1940년대, 2차대전에서 Radar의 발명으로 큰 발전 • MIT에서 Radiation Lab을 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. 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. Figure 1.2 (p. 4)

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

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

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

17. Power and energy In phasor form

18. Assuming an ejωt time dependence,

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 = z0 plane.

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

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

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

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

26. Fields at a General Material Interface

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

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

29. As h 0 Similarly

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

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

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

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

34. HW1 • Divergence Theorem을 증명하고 물리적 의미를 설명하시오. • Stokes’ Theorem을 증명하고 물리적 의미를 설명하시오. • Ampere의 법칙과 Gauss의 법칙에 대하여 설명하시오.

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

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, vp = c = 2.998x108 m/s

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

39. Associated magnetic field

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

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

43. Also define a position vector, • If

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

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

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

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 E1 and a y linearly polarized with E2. - If E1≠ 0, E2 = 0  linearly polarized in x direction - If E1 = 0, E2 ≠ 0  linearly polarized in y direction - If E1 ≠ 0, E2 ≠ 0 (both real), linearly polarized at the angle

48. If E1 = E2 = E0, • If E1 = jE2 = E0,  The E-field vector changes with time or equivalently with distance along the z-axis. • Pick a position (z = 0)  RHCP

49. Similarly,  LHCP

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