1. 1 ECE 480 Wireless Systems Lecture 3 Propagation and Modulation of RF Waves

2. 2 Circular Polarization Magnitudes of the x – and y – components of are equal Phase difference is Left – Hand Circular (LHC) Polarization Right – Hand Circular (RHC) Polarization

3. 3 Left – Hand Circular (LHC) Polarization

4. 4 Convert to polar form

5. 5 Linear polarization = f (z, t)  f (z, t) Circular polarization  f (z, t) = f (z, t)

6. 6 Back to LHC Polarization Consider the LHC wave at z = 0 Inclination angle decreases with time

7. 7 Right – Hand Circular (RHC) Polarization

8. 8 The direction of polarization is defined in terms of the rotation of as a function of time in a fixed plane orthogonal to the direction of propagation

9. 9 Example RHC Polarized Wave An RHC polarized plane wave with electric field modulus of 3 mV/m is traveling in the + y direction in a dielectric medium with f = 100 MHz Obtain expressions for

10. 10 Solution The wave is traveling in the + y direction. Therefore, the field components are in the x and z directions. direction of propagation

11. 11 Assign a phase angle of 0 o to the z component of (arbitrary) The x component of will have a phase shift Both components have a magnitude of a = 3

12. 12

13. 13 Converting back to the time domain

14. 14 Elliptical Polarization Most general case The tip of traces an ellipse in the x – y plane Can be left – handed or right - handed Major axis: Minor axis: Rotation Angle Ellipticity Angle

15. 15 Rotation angle The shape and rotation are defined by the ellipticity angle axial ratio R = 1 Circular R =  Linear

16. 16 Circular Linear

17. 17 Polarization States for Various Combinations of  and 

18. 18 Positive values of corresponding to sin  > 0 define left – handed rotation Negative values of corresponding to sin  < 0 define right – handed rotation How is the type of polarization determined? a x and a y are, by definition, > 0 Two possible values of in this range

19. 19 Example: Polarization State Determine the polarization of a plane wave with an electric field given by Solution Convert the sin term to a cos term by subtracting 90 o Convert the – cos term to a + cos term by adding 180 o

20. 20 Convert to phasor form

21. 21 There are two possible solutions for  since the tan function is positive in both the first and third quadrants Which is correct?

22. 22 By a similar analysis,  The wave is elliptically polarized and the rotation is left - handed

23. 23 Plane – Wave Propagation in Lossy Media  can be written as  = attenuation constant  = phase constant

24. 24 Equate the real and imaginary parts Solve for  and 

25. 25 For a uniform plane wave traveling in the + z direction with an electric field the wave equation becomes The solution is

26. 26 The magnitude of is Decreases exponentially with e -  z also decreases exponentially with e -  z Define: Skin Depth,  s Distance that a wave must travel before it is attenuated by

27. 27 In a perfect dielectric In a perfect conductor

28. 28 Expressions are valid for any linear, isotropic, homogeneous medium Low – Loss Dielectric Quasi – Conductor (Semiconductor) Good Conductor

29. 29 Low – Loss Dielectric Consider For Divide into real and imaginary parts  - Same as for lossless medium

30. 30 Same as for the lossless case

31. 31 Good Conductor

32. 32 Semiconductors – Must use exact solution

33. 33 Example – Plane Wave in Seawater A uniform plane wave is traveling downward in the + z direction in seawater, with the x – y plane denoting the sea surface and z = 0 denoting a point just below the surface. The constitutive parameters of seawater are: The magnetic field intensity at z = 0 is given by a.Determine expressions for b.The depth at which the amplitude of E is 1% of its value at z = 0

34. 34 Solution a. The general expressions for the phasor fields are  Seawater is a good conductor at 1 KHz

35. 35 The general expression for E x0 is

36. 36 at z = 0: Compare with original expression

37. 37 Note that they are no longer in phase. The electric field always leads the magnetic field by 45 o. b. Set the amplitude to 0.01

38. 38 Electromagnetic Power Density Define: Poynting Vector Direction of S is in the direction of propagation, k Power through a surface, A unit vector normal to the surface

39. 39 Plane Wave in a Lossless Medium Consider a plane wave traveling in the + z direction Want to find the power density vector, S

40. 40 Time – Domain Approach

41. 41 Time average of

42. 42 Phasor – Domain Approach is valid for any media

43. 43 Plane Wave in a Lossy Medium

44. 44 Note that the average power decays with

45. 45 Homework The electric field of a plane wave is given by Identify the polarization state, determine the polarization angles ( ,  ), and sketch the locus of E (0, t) for each of the following cases

46. 46 Homework In a medium characterized by Determine the phase angle by which the magnetic field leads the electric field

47. 47 Radiation and Antennas An antenna may be considered as a transducer that converts a guided EM wave to a transmitted wave or an incident wave to a guided EM wave Antenna dimensions are generally referred to in wavelength units

48. 48

49. 49 Reciprocity Antenna radiation pattern: The directional function that characterizes the distribution pattern radiated by an antenna Isotropic antenna: A hypothetical antenna that radiates equally in all directions Used as a reference radiator to compare antennas Reciprocal antennas: Antennas that have the same radiation patterns for transmission as for reception

50. 50 Two aspects of antenna performance 1. Radiation Properties Direction of the radiation pattern Polarization state of the radiated wave in the TX mode (Antenna Polarization) In the RX mode, the antenna can extract only that component of the wave whose E – field is parallel to that of the antennas polarization direction 2. Antenna Impedance Pertains to the impedance match between the antenna and the generator

51. 51 Radiation Sources Two basic types 1. Current sources Dipole and loop antennas 2. Aperture fields Horn antennas

52. 52 Far – Field Region The far – field region is at a distance R where the wave may be considered to be a plane wave D = Maximum effective size of the antenna = Wavelength of the signal

53. 53 Example: Far – Field Distance of an Antenna A parabolic reflector antenna is 18" in diameter operates at 12.4 GHz. Find the operating wavelength and the far – field distance of this antenna. Solution

54. 54 Antenna Arrays Can control the phase and magnitude of each antenna individually Can steer the direction of the beam electronically

55. 55 Retarded Potentials Consider a charge distribution as shown The electric potential V (R) at a point in space specified by the position vector R is given by = position vector of an elemental volume = elemental volume = charge density inside the volume = distance between the volume and the point

56. 56 If the charge density is time – varying, the obvious solution is Problem: Does not account for reaction time Any change in the charge distribution will require a finite amount of time to change the potential

57. 57 Retarded Vector Potential Retarded Scalar Potential Delay Time Valid under both static and dynamic conditions

58. 58 Time – Harmonic Potentials In a linear system, the parameters all have the same functional dependence on time Consider a sinusoidal time – varying charge distribution = phasor representation of

59. 59

60. 60 In terms of A

61. 61 The Short (Hertzian) Dipole Approach: Develop the radiation properties of a differential antenna and use that model to predict other configurations Characteristics of a Short Dipole Current is uniform over the length

62. 62 At the point Q s = cross – sectional area of dipole limits of integration Assume R' ~ R

63. 63 Spherical propagation factor Considers both the magnitude and phase change wrt R Change to spherical coordinates

64. 64

65. 65

66. 66

67. 67 Electric Field lines

68. 68 Far – Field Approximation Independent of  Proportional to sin 

69. 69 Power Density For the short dipole:

70. 70 Define: Normalized Radiation Intensity Radiation is maximum when(azimuth plane)

71. 71 No energy is radiated by the dipole along the direction of the dipole axis and maximum radiation (F = 1) occurs in the broadside direction.