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EE 542 Antennas and Propagation for Wireless Communications

EE 542 Antennas and Propagation for Wireless Communications. Array Antennas. Array Antennas. An antenna made up of an array of individual antennas Motivations to use array antennas: High gain  more directive pattern Steerability of the main beam Linear array: elements arranged on a line

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EE 542 Antennas and Propagation for Wireless Communications

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  1. EE 542Antennas and Propagation for Wireless Communications Array Antennas

  2. Array Antennas • An antenna made up of an array of individual antennas • Motivations to use array antennas: • High gain  more directive pattern • Steerability of the main beam • Linear array: elements arranged on a line • 2-D planar arrays: rectangular, square, circular,… • Conformal arrays: non-planar, conform to surface such as aircraft O. Kilic EE 542

  3. Radiation Pattern for Arrays Depends on: • The type of the individual elements • Their orientation • Their position in space • The amplitude and phase of the current feeding them • The total number of elements O. Kilic EE 542

  4. Array Factor The pattern of an array by neglecting the patterns of the individual elements; i.e. assume individual elements are isotropic O. Kilic EE 542

  5. Linear Receive Array j A + Receiver O. Kilic EE 542

  6. Case 1: Array Factor for Two Isotropic Sources with Identical Amplitude and Phase (d = l/2) P(x,y,z) z r1 r r2 q x (2) (1) d (I0,j0) (I0,j0) Isotropic sources are assumed for AF calculations. The radiated fields are uniform over a sphere surrounding the source. O. Kilic EE 542

  7. Radiation from an Isotropic Source r O. Kilic EE 542

  8. Case 1: Total E Field where O. Kilic EE 542

  9. Case 1: Far Field Approximation In the far field, r>>d or (d/r) <<1 O. Kilic EE 542

  10. Case 1: Far Field Approximation Similarly, Thus, in the far field O. Kilic EE 542

  11. Case 1: Far Field Geometry P(x,y,z) z r1 r dcosq r2 q x (2) (1) d If the observation point r is much larger than the separation d, the vectors r1, r and r2 can be assumed to be approximately parallel. The path lengths from the sources to the observation point are slightly different. O. Kilic EE 542

  12. Case 1: Total E in the Far Field The slight difference in path length can NOT be neglected for the exponential term!! O. Kilic EE 542

  13. Case 1: Total E for d=l/2 Note that d=l/2 O. Kilic EE 542

  14. Case 1: Array Factor The array factor is described as the magnitude of E at a constant distance r from the antenna (i.e. unit V) Normalized values O. Kilic EE 542

  15. Case 1: Radiation Pattern z q x (2) (1) (I0,j0) (I0,j0) Notice how the two element array is more directive than the single element; which is an isotropic source. O. Kilic EE 542

  16. Case 2: Array Factor for Two Isotropic Sources with Identical Amplitude and Opposite Phase P(x,y,z) z r1 r r2 q x (2) (1) d (I2,j2) (I1,j1) O. Kilic EE 542

  17. Case 2 – Far Field Geometry P(x,y,z) z r1 r dcosq r2 q x (2) (1) d (I2,j2) (I1,j1) O. Kilic EE 542

  18. Case 2: Total E in the Far Field O. Kilic EE 542

  19. Case 2: Radiation Pattern Note that d=l/2 z q x (2) (1) (I0,p+j0) (I0,j0) Observe how the pattern is rotated compared to Case1 by simply changing the phase of element 2 O. Kilic EE 542

  20. Case 3: Array Factor for Two Isotropic Sources with Identical Amplitudes and 90o Phase Shift Homework: Show that: O. Kilic EE 542

  21. Case 3 z q x (2) (1) (I0,p+j0) (I0,j0) O. Kilic EE 542

  22. Generalization to N Equally Spaced Elements r dcosq dcosq dcosq q d d d 0 1 2 3 N-1 O. Kilic EE 542

  23. General Case for Linear Array Total E field: Array Factor: O. Kilic EE 542

  24. Special Case (A) Equally Spaced Linear Array with Linear Phase Progression Fourier series O. Kilic EE 542

  25. Some Observations O. Kilic EE 542

  26. Special Case (B) Uniformly Excited, Equally Spaced Linear Array with Linear Phase Progression O. Kilic EE 542

  27. Observations • AF similar to the sinc function (i.e. sinx/x) with a major difference: • Sidelobes do not die off for increasing Y values because the denominator is a sine function, and does not increase beyond a value of 1. • AF is periodic with 2p. • Maximum value (=Io) occurs at Y=0, 2kp. O. Kilic EE 542

  28. N=4 Case Period: 2p p 3p/2 nulls p/2 O. Kilic EE 542

  29. More Observations For all k values except when y/2 becomes an integer multiple of p • Zeroes (Nulls) @ NY/2 = kp  Yok=2kp/N, k=0,1,2, … • This implies that as N increases there are more sidelobes (i.e. more secondary null points) in one period. • Sidelobe widths are 2p/N. • First null at Yo1=2p/N. • Within one period, N null points N-2 sidelobes (Because we discard k = N case, which corresponds to the second peak. Also 2 nulls create one sidelobe.) • This implies that as N increases, the main beam narrows. • Main lobe width is 2*2p/N, twice the width of sidelobes. • Max value ( = NIo) @ Y= =2kp, k=0,1,2, … O. Kilic EE 542

  30. Effect of Increasing N HW: Regenerate this plot. O. Kilic EE 542

  31. Construction of Polar Plot from AF(y) • The angle Y is not a physical quantity. • We are more interested in observing the AF as a function of angles in real space; i.e. q, j. • Since linear arrays are rotationally symmetric wrt j, we are concerned with q only. O. Kilic EE 542

  32. Case 1: Construction of Polar Plot N = 2, d = l/2, a = 0 (uniform phase) Using the general representation from Page 24 Compare to page 62 z r q x Io, j =0 Io, j =0 l/2 O. Kilic EE 542

  33. Normalized AF for Case1 Period = 2p 2p -2p -p p O. Kilic EE 542

  34. Normalized AF for Case1 – Polar Plot Visible range: q: [0-p]  Y: [-kd,kd] Y = kdcosq = pcosq Visually relate q to Y kd Circle of radius kd  q1 q2 Y; x Y2 p = kd -p Y1 f(Y) f(Y1) f(Y2) -p p Y2 Y1 Y O. Kilic EE 542

  35. Constructing the Polar Plot Circle of radius kd  f(Y1) Y; x f(Y2) Y2 p = kd -p Y1 f(Y) f(Y1) f(Y2) -p p Y2 Y1 Y O. Kilic EE 542

  36. Case 2 N = 2, d = l/2, a = p Note: AF(Y) same for all N=2 Value of Y different, depends on a, d O. Kilic EE 542

  37. Case 2: Polar Format Y = kdcosq + p Circle of radius kd  q1 Y; x Shifted by p Y1 p = kd f(Y) f(Y1) Y -p p 2p 0 Y1 O. Kilic EE 542

  38. Normalized AF for Case 2 – Polar Plot f(Y2) f(Y1) Circle of radius kd  Y2 Y1 Y; x 0 f(Y2) f(Y) f(Y1) p 0 -p Y1 Y2 Y O. Kilic EE 542

  39. Shift by a kd a + kd a - kd q1 Y a Y1= kdcosq1+a O. Kilic EE 542

  40. Generalize to Arbitrary N Visible Range: Shift by a O. Kilic EE 542

  41. General Rule • AF plot with respect to Y is identical for all cases with identical N. • The polar plot is determined by shifting the unit circle by a, the linear phase progression amount. • Visible range is always the 2kd range centered around that point. O. Kilic EE 542

  42. Shift and construct Observe the dependence of main beam direction on a, the phase progression. Main beam qpeak cos(qpeak) = a/kd q1 Y a + kd a - kd Y1 a f(Y) f(Y1) Y O. Kilic EE 542

  43. Shift and construct Observe the dependence of main beam direction on a, the phase progression. Main beam a + kd a - kd Y1 Y a f(Y) Y O. Kilic EE 542

  44. Array Pattern vs kd • If kd > 2p; i.e. d>l/2 multiple peaks can occur in the visible range. These are known as grating lobes, and are often undesirable. • Why?? • Will cause reduced directivity as power will be shared among all peaks • Likely to cause interference O. Kilic EE 542

  45. Grating Lobes Three main beams. Y, x kd -kd Y, x O. Kilic EE 542

  46. Pattern Multiplication • So far only isotropic elements were considered. • Actual arrays are made up of nearly identical antennas • AF still plays a major role in the pattern Normalized Array Pattern Normalized Array factor Normalized element pattern O. Kilic EE 542

  47. Validation with Dipoles • Consider the case of an ideal dipole array as below. r dcosq dcosq q d d d I0 I1 I2 I3 (N-1)d 0 O. Kilic EE 542

  48. Sum of the E fields For the center dipole, assuming Dz << l Normalized pattern O. Kilic EE 542

  49. Vector Potentials for Each Dipole O. Kilic EE 542

  50. Total Vector Potential O. Kilic EE 542

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