1. Prof. David R. Jackson Notes 17 Polarization of Plane Waves Polarization of Plane Waves ECE 3317 1 Spring 2013

2. Polarization The polarization of a plane wave refers to the direction of the electric field vector in the time domain. x y z We assume that the wave is traveling in the positive z direction. 2

3. Consider a plane wave with both x and y components Assume: ExEx EyEy x y Polarization (cont.) Phasor domain: 3

4. Time Domain: Depending on b/a and   three different cases arise: x y E (t)E (t) At z = 0: Polarization (cont.) 4  Linear polarization  Circular polarization  Elliptical polarization

5. Power Density: Assume lossless medium (  is real): 5 or Polarization (cont.) From Faraday’s law:

6. At z = 0: a b x y x y  or  (shown for   =  0  Linear Polarization 6 This is simply a “tilted” plane wave. + sign:  = 0 - sign:  = 

7. At z = 0: b = a AND  Circular Polarization 7

8. a x y  Circular Polarization (cont.) 8

9. a x y    RHCP LHCP IEEE convention Circular Polarization (cont.) 9 

10. There is opposite rotation in space and time. Circular Polarization (cont.) 10 Examine how the field varies in both space and time: Rotation in space vs. rotation in time Phasor domain Time domain

11. RHCP Notice that the rotation in space matches the left hand! z Circular Polarization (cont.) 11 A snapshot of the electric field vector, showing the vector at different points.

12. Circular Polarization (cont.) 12 Animation of LHCP wave http://en.wikipedia.org/wiki/Circular_polarization (Use pptx version in full-screen mode to see motion.)

13. Circular Polarization (cont.) Circular polarization is often used in wireless communications to avoid problems with signal loss due to polarization mismatch.  Misalignment of transmit and receive antennas  Reflections off of building  Propagation through the ionosphere Receive antenna The receive antenna will always receive a signal, no matter how it is rotated about the z axis. z However, for the same incident power density, an optimum linearly-polarized wave will give the maximum output signal from this linearly-polarized antenna (3 dB higher than from an incident CP wave). 13

14. Circular Polarization (cont.) Two ways in which circular polarization can be obtained: This antenna will radiate a RHCP signal in the positive z direction, and LHCP in the negative z direction. Method 1) Use two identical antenna rotated by 90 o, and fed 90 o out of phase. 14 Antenna 1 Antenna 2 x y + - + -

15. Circular Polarization (cont.) Realization with a 90 o delay line: 15 Antenna 1 Antenna 2 x y l1l1 l2l2 Z 01 Feed line Power splitter Signal

16. Circular Polarization (cont.) 16 An array of CP antennas

17. Circular Polarization (cont.) 17 The two antennas are actually two different modes of a microstrip or dielectric resonator antenna.

18. Circular Polarization (cont.) Method 2) Use an antenna that inherently radiates circular polarization. Helical antenna for WLAN communication at 2.4 GHz http://en.wikipedia.org/wiki/Helical_antenna 18

19. Circular Polarization (cont.) 19 Helical antennas on a GPS satellite

20. Circular Polarization (cont.) 20 Other Helical antennas

21. Circular Polarization (cont.) An antenna that radiates circular polarization will also receive circular polarization of the same handedness (and be blind to the opposite handedness). It does not matter how the receive antenna is rotated about the z axis. 21 Antenna 1 Antenna 2 x y l1l1 l2l2 Z0Z0 Z0Z0 Output signal Power combiner RHCP wave z RHCP Antenna

22. Circular Polarization (cont.) 22 Summary of possible scenarios 1) Transmit antenna is LP, receive antenna is LP  Simple, works good if both antennas are aligned.  The received signal is less if there is a misalignment. 2) Transmit antenna is CP, receive antenna is LP  Signal can be received no matter what the alignment is.  The received signal is 3 dB less then for two aligned LP antennas. 3) Transmit antenna is CP, receive antenna is CP  Signal can be received no matter what the alignment is.  There is never a loss of signal, no matter what the alignment is.  The system is now more complicated.

23. Includes all other cases that are not linear or circular x y  Elliptic Polarization The tip of the electric field vector stays on an ellipse. 23

24. x LHEP RHEP y  Elliptic Polarization (cont.) Rotation property: 24

25. so Elliptic Polarization (cont.) or Squaring both sides, we have Here we give a proof that the tip of the electric field vector must stay on an ellipse. 25

26. This is in the form of a quadratic expression: Elliptic Polarization (cont.) or Collecting terms, we have 26

27. Hence, this is an ellipse. so Elliptic Polarization (cont.) (determines the type of curve) 27 Discriminant:

28. x y  Elliptic Polarization (cont.) Here we give a proof of the rotation property. 28

29. Take the derivative: Hence (proof complete) Elliptic Polarization (cont.) 29

30. Rotation Rule Here we give a simple graphical method for determining the type of polarization (left-handed or right handed). 30

31. First, we review the concept of leading and lagging sinusoidal waves. B leads A Re  B A Im 31 B lags A Re  B A Im Two phasors: A and B Note: We can always assume that the phasor A is on the real axis (zero degrees phase) without loss of generality. Rotation Rule (cont.)

32. Rule: The electric field vector rotates from the leading axis to the lagging axis E y leads E x Phasor domain Re  EyEy ExEx Im 32 Now consider the case of a plane wave Rotation Rule (cont.) x y Time domain

33. Rule: The electric field vector rotates from the leading axis to the lagging axis Rotation Rule (cont.) ExEx EyEy Im Re  E y lags E x Phasor domain x y Time domain 33

34. The electric field vector rotates from the leading axis to the lagging axis. Rotation Rule (cont.) 34 The rule works in both cases, so we can call it a general rule:

35. y EzEz z x ExEx What is this wave’s polarization? Rotation Rule (cont.) Example 35

36. Example (cont.) Therefore, in time the wave rotates from the z axis to the x axis. E z leads E x EzEz ExEx Im Re Rotation Rule (cont.) 36

37. LHEP or LHCP Note: and (so this is not LHCP) Example (cont.) Rotation Rule (cont.) 37 z y EzEz x ExEx

38. LHEP Rotation Rule (cont.) Example (cont.) 38 z y EzEz x ExEx

39. x y  A B C D Axial Ratio (AR) and Tilt Angle (  ) 39 Note: In dB we have

40. Axial Ratio and Handedness Tilt Angle Note: The tilt angle  is ambiguous by the addition of  90 o. Axial Ratio (AR) and Tilt Angle (  ) Formulas 40

41. Tilt Angle: Note: The title angle is zero or 90 o if: x y Note on Title Angle 41

42. Example 42 z y EzEz x ExEx LHEP Relabel the coordinate system: Find the axial ratio and tilt angle.

43. Example (cont.) 43 LHEP y z EyEy x ExEx Renormalize: or

44. Example (cont.) 44 LHEP Hence y z EyEy x ExEx

45. Example (cont.) 45 LHEP Results

46. Example (cont.) 46 x y  Note: We are not sure which choice is correct: We can make a quick time-domain sketch to be sure.

47. Example (cont.) 47 x y 