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Dr. Hugh Blanton ENTC 3331

ENTC 3331 RF Fundamentals. Dr. Hugh Blanton ENTC 3331. Plane-Wave Propagation. Electric & Magnetic fields that vary harmonically with time are called electromagnetic waves:. In order to simplify the mathematical treatment, treat all fields as complex numbers.

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Dr. Hugh Blanton ENTC 3331

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  1. ENTC 3331 RF Fundamentals Dr. Hugh Blanton ENTC 3331

  2. Plane-Wave Propagation

  3. Electric & Magnetic fields that vary harmonically with time are called electromagnetic waves: Dr. Blanton - ENTC 3331 - Wave Propagation 3

  4. In order to simplify the mathematical treatment, treat all fields as complex numbers. Dr. Blanton - ENTC 3331 - Wave Propagation 4

  5. The mathematical form of the Maxwell equations remains the same, however, all quantities (apart from x,y,z,t) are now complex. Dr. Blanton - ENTC 3331 - Wave Propagation 5

  6. For • It follows that Dr. Blanton - ENTC 3331 - Wave Propagation 6

  7. The Maxwell equations (in differential form) can thus be expressed as: • In a vacuum (space) • In air (atmosphere) Dr. Blanton - ENTC 3331 - Wave Propagation 7

  8. Thus, the Maxwell equations (in differential form) and in air can be expressed as: • The Maxwell equations are fundamental and of general validity which implies • It should be possible to derive a pair of equations, which describe the propagation of electromagnetic waves. Dr. Blanton - ENTC 3331 - Wave Propagation 8

  9. We expect solutions like: • How do we get from • to Dr. Blanton - ENTC 3331 - Wave Propagation 9

  10. Recall that • and apply to both sides of • but Dr. Blanton - ENTC 3331 - Wave Propagation 10

  11. 0 Dr. Blanton - ENTC 3331 - Wave Propagation 11

  12. wave number =k2 wave equation Dr. Blanton - ENTC 3331 - Wave Propagation 12

  13. The previous two equations are called wave equations because their solutions describe the propagation of electromagnetic waves wave equation Dr. Blanton - ENTC 3331 - Wave Propagation 13

  14. In one dimension: • If this describes an electromagnetic wave, it may also hold for a single photon. Dr. Blanton - ENTC 3331 - Wave Propagation 14

  15. For a photon, is significant at the current location of the photon. • The probability of finding a photon at location x is . • This implies: Schrodinger’s equation Dr. Blanton - ENTC 3331 - Wave Propagation 15

  16. strict derivation heuristic analogy Schrodinger’s Equation (Postulates of Quantum Mechanics physics of the macroscopic world Maxwell’s equations (Newtons laws) physics of the microscopic world Wave Equation particles and waves particles-wave duality Dr. Blanton - ENTC 3331 - Wave Propagation 16

  17. What are the solutions of the electromagnetic wave equations? Dr. Blanton - ENTC 3331 - Wave Propagation 17

  18. Perform the Laplacian Dr. Blanton - ENTC 3331 - Wave Propagation 18

  19. That is: Dr. Blanton - ENTC 3331 - Wave Propagation 19

  20. Consider a uniform plane wave that is characterized by electric and magnetic fields that have uniform properties at all points across an infinite plane. Dr. Blanton - ENTC 3331 - Wave Propagation 20

  21. no component in the z-direction x y “up” wave crescents z Dr. Blanton - ENTC 3331 - Wave Propagation 21

  22. Consequently, • simplifies to Dr. Blanton - ENTC 3331 - Wave Propagation 22

  23. The most general solutions of • are • where and are constants determined by boundary conditions. Dr. Blanton - ENTC 3331 - Wave Propagation 23

  24. For mathematical simplification rotate the Cartesian coordinate system about the z-axis until • The plane wave is • The first term represents a wave with amplitude traveling in the +z-direction, and • the second term represents a wave with amplitude traveling in the –z direction. Dr. Blanton - ENTC 3331 - Wave Propagation 24

  25. Let us assume that consists of a wave traveling in the +z-direction only Dr. Blanton - ENTC 3331 - Wave Propagation 25

  26. Magnetic field, ? • We must fulfill the Maxwell equation: • But Dr. Blanton - ENTC 3331 - Wave Propagation 26

  27. Dr. Blanton - ENTC 3331 - Wave Propagation 27

  28. Dr. Blanton - ENTC 3331 - Wave Propagation 28

  29. Dr. Blanton - ENTC 3331 - Wave Propagation 29

  30. Dr. Blanton - ENTC 3331 - Wave Propagation 30

  31. Recall Dr. Blanton - ENTC 3331 - Wave Propagation 31

  32. x z y • This is possible if • Electric and magnetic field vectors are perpendicular! Dr. Blanton - ENTC 3331 - Wave Propagation 32

  33. Transversal electromagnetic wave (TEM) Dr. Blanton - ENTC 3331 - Wave Propagation 33

  34. Electromagnetic Plane Wave in Air • The electric field of a 1-MHz electromagnetic plane wave points in the x-direction. • The peak value of is 1.2p (mV/m) and for t = 0, z = 50 m. • Obtain the expression for and . Dr. Blanton - ENTC 3331 - Wave Propagation 34

  35. The field is maximum when the argument of the cosine function equals zero or multiples of 2p. • At t = 0 and z =50 m Dr. Blanton - ENTC 3331 - Wave Propagation 35

  36. Dr. Blanton - ENTC 3331 - Wave Propagation 36

  37. Dr. Blanton - ENTC 3331 - Wave Propagation 37

  38. PLANE WAVE PROPAGATION POLARIZATION

  39. y x Wave Polarization • Wave polarization describes the shape and locus of tip of the vector at a given point in space as a function of time. • The direction of wave propagation is in the z-direction. Dr. Blanton - ENTC 3331 - Wave Propagation 39

  40. Wave Polarization • The locus of , may have three different polarization states depending on conditions: • Linear • Circular • Elliptical Dr. Blanton - ENTC 3331 - Wave Propagation 40

  41. Polarization • A uniform plane wave traveling in the +z direction may have x- and y- components. • where Dr. Blanton - ENTC 3331 - Wave Propagation 41

  42. Polarization • and are the complex amplitudes of and , respectively. • Note that • the wave is traveling in the positive z-direction, and • the two amplitudes and are in general complex quantities. Dr. Blanton - ENTC 3331 - Wave Propagation 42

  43. Polarization • The phase of a wave is defined relative to a reference condition, such as z = 0 and t = 0 or any other combination of z and t. • We will choose the phase of as our reference, and will denote the phase of relative to that of , as d. • Thus, d is the phase-difference between the y-component of and its x-component. where axand ayare the magnitudesof Ex0and Ey0 Dr. Blanton - ENTC 3331 - Wave Propagation 43

  44. Polarization • The total electric field phasor is • and the corresponding instantaneous field is: Dr. Blanton - ENTC 3331 - Wave Propagation 44

  45. Intensity and Inclination Angle • The intensity of is given by: • The inclination angleψ Dr. Blanton - ENTC 3331 - Wave Propagation 45

  46. Linear Polarization • A wave is said to be linearly polarized if Ex(z,t) and Ey(z,t) are in phase (i.e., d = 0) or out of phase (d = p). • At z = 0 and d =0 or p, Dr. Blanton - ENTC 3331 - Wave Propagation 46

  47. Linear Polarization (out of phase) • For the out of phase case: • w t = 0 and • That is, extends from the origin to the point (ax ,ay) in the fourth quadrant. Dr. Blanton - ENTC 3331 - Wave Propagation 47

  48. Linear Polarization (out of phase) • For the in phase case: • w t = 0 and • That is, extends from the origin to the point (ax ,ay) in the first quadrant. y x Dr. Blanton - ENTC 3331 - Wave Propagation 48

  49. The inclination is: • If ay = 0, y = 0 or 180, the wave becomes x-polarized, and if ax = 0, y = 90  or -90 , and the wave becomes y-polarized. Dr. Blanton - ENTC 3331 - Wave Propagation 49

  50. Linear Polarization • For a +z-propagating wave, there are two possible directions of . • Direction of is called polarization • There are two independent solution for the wave equation Dr. Blanton - ENTC 3331 - Wave Propagation 50

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