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Prof. David R. Jackson

ECE 3317. Prof. David R. Jackson. Spring 2013. Notes 17 Polarization of Plane Waves . Polarization. z. y. x. The polarization of a plane wave refers to the direction of the electric field vector in the time domain. We assume that the wave is traveling in the positive z direction.

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Prof. David R. Jackson

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

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

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

  4. Polarization (cont.) y Time Domain: E(t) At z = 0: x Depending on b/aand b, three different cases arise: • Linear polarization • Circular polarization • Elliptical polarization

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

  6. Linear Polarization y x y b x a b = 0 or At z = 0: + sign: = 0 - sign:  =  (shown forb =0) This is simply a “tilted” plane wave.

  7. Circular Polarization b = a ANDb = p /2 At z = 0:

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

  9. Circular Polarization (cont.) y b =+p / 2 a LHCP f x b =-p / 2 RHCP IEEE convention b = p /2

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

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

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

  13. Circular Polarization (cont.) Receive antenna z 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 The receive antenna will always receive a signal, no matter how it is rotated about the z axis. 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).

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

  15. Circular Polarization (cont.) Realization with a 90o delay line: y Antenna 2 Antenna 1 x Z01 l1 Signal Z01 l2 Feed line Power splitter

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

  17. Circular Polarization (cont.) 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

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

  20. Circular Polarization (cont.) 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). y It does not matter how the receive antenna is rotated about the z axis. Antenna 2 RHCP Antenna Antenna 1 x Z0 l1 Output signal Z0 l2 z RHCP wave Power combiner

  22. Circular Polarization (cont.) 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. Elliptic Polarization y f x Includes all other cases that are not linear or circular The tip of the electric field vector stays on an ellipse.

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

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

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

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

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

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

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

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

  32. Rotation Rule (cont.) Im Ey b Ex Re Phasor domain Now consider the case of a plane wave EyleadsEx Rule: The electric field vector rotates from the leading axis to the lagging axis y x Time domain

  33. Rotation Rule (cont.) Im Ey lagsEx Ex b Re Ey Rule: The electric field vector rotates from the leading axis to the lagging axis Phasor domain y x Time domain

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

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

  36. Rotation Rule (cont.) Im EzleadsEx Ez Re Ex Example (cont.) Therefore, in time the wave rotates from the z axis to the x axis.

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

  38. Rotation Rule (cont.) z Ez y Ex x Example (cont.) LHEP

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

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

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

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

  43. Example (cont.) y Ey z Ex x LHEP Renormalize: or

  44. Example (cont.) y Ey z Ex x Hence LHEP

  45. Example (cont.) Results LHEP

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

  47. Example (cont.) y  x

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