1 EE 542 Antennas and Propagation for Wireless Communications Array Antennas

O. Kilic EE 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 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 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 Linear Receive Array + Receiver A 

O. Kilic EE Case 1: Array Factor for Two Isotropic Sources with Identical Amplitude and Phase (d = /2) d P(x,y,z) x z (1) (2)  r r1r1 r2r2 (I 0,  0 ) Isotropic sources are assumed for AF calculations. The radiated fields are uniform over a sphere surrounding the source.

O. Kilic EE Radiation from an Isotropic Source r

O. Kilic EE Case 1: Total E Field where

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

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

O. Kilic EE Case 1: Far Field Geometry d P(x,y,z) x z (1) (2)  r r1r1 r2r2 dcos  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 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 Case 1: Total E for d= /2 Note that d=  2

O. Kilic EE 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)  AF n 00  /2 1  0  /2 1 Normalized values

O. Kilic EE Case 1: Radiation Pattern x z (1) (2)  (I 0,  0 ) Notice how the two element array is more directive than the single element; which is an isotropic source.

O. Kilic EE Case 2: Array Factor for Two Isotropic Sources with Identical Amplitude and Opposite Phase d P(x,y,z) x z (1) (2)  r r1r1 r2r2 (I 1,  1 ) (I 2,  2 )

O. Kilic EE Case 2 – Far Field Geometry d P(x,y,z) x z (1) (2)  r r1r1 r2r2 dcos  (I 1,  1 ) (I 2,  2 )

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

O. Kilic EE Case 2: Radiation Pattern Note that d=  2 x z (1) (2)  (I 0,  0 ) (I 0,  +  0 ) Observe how the pattern is rotated compared to Case1 by simply changing the phase of element 2  AF n 01  /2 0  1  /2 0

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

O. Kilic EE Case 3  AF n 00  /2cos(  /4)  1  /2cos(  /4) x z (1) (2)  (I 0,  0 ) (I 0,  +  0 )

O. Kilic EE Generalization to N Equally Spaced Elements 0123N-1 ddd dcos   r

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

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

O. Kilic EE Some Observations

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

O. Kilic EE Observations AF similar to the sinc function (i.e. sinx/x) with a major difference: Sidelobes do not die off for increasing  values because the denominator is a sine function, and does not increase beyond a value of 1. AF is periodic with 2  Maximum value (=I o ) occurs at  k 

O. Kilic EE N=4 Case Period:  nulls  /2   /2

O. Kilic EE More Observations Zeroes N  /2 = k     k =2k  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 2  First null at   1 =2  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*2  twice the width of sidelobes  Max value ( = NI o  =2k , k=0,1,2, … For all k values except when  /2 becomes an integer multiple of 

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

O. Kilic EE Construction of Polar Plot from AF(  ) The angle  is not a physical quantity. We are more interested in observing the AF as a function of angles in real space; i.e. . Since linear arrays are rotationally symmetric wrt , we are concerned with  only.

O. Kilic EE Case 1: Construction of Polar Plot N = 2, d = /2,  = 0 (uniform phase) Using the general representation from Page 24 /2 x z I o,  =0  r Compare to page 62

O. Kilic EE Normalized AF for Case1     Period = 2 

O. Kilic EE Normalized AF for Case1 – Polar Plot Visible range:  : [0-  ]   : [-kd,kd]  = kdcos  =  cos     kd f(  )    f       x Circle of radius kd     f    kd Visually relate  to 

O. Kilic EE Constructing the Polar Plot    kd f(  )   f       x Circle of radius kd    f    f    f   

O. Kilic EE Case 2 N = 2, d = /2,  =  Note: AF(  ) same for all N=2 Value of  different, depends on , d

O. Kilic EE Case 2: Polar Format  = kdcos  +     kd f(  )    f      x 0 22 Shifted by  Circle of radius kd 

O. Kilic EE Normalized AF for Case 2 – Polar Plot   f(  )   f       x Circle of radius kd   f    f    f     

O. Kilic EE Shift by      kdcos       kd  kd kd

O. Kilic EE Generalize to Arbitrary N Shift by  Visible Range:

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

O. Kilic EE Shift and construct f(  )   f       Observe the dependence of main beam direction on , the phase progression. Main beam  + kd  - kd  peak cos(  peak ) =  /kd

O. Kilic EE Shift and construct f(  )     Observe the dependence of main beam direction on , the phase progression. Main beam  + kd  - kd

O. Kilic EE Array Pattern vs kd If kd > 2  ; i.e. d> /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 Grating Lobes Three main beams. kd , x -kd

O. Kilic EE 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 element pattern Normalized Array factor

O. Kilic EE Validation with Dipoles Consider the case of an ideal dipole array as below. I0I0 (N-1)d ddd dcos   r I1I1 I2I2 I3I3 0

O. Kilic EE Sum of the E fields For the center dipole, assuming  z << Normalized pattern

O. Kilic EE Vector Potentials for Each Dipole

O. Kilic EE Total Vector Potential

O. Kilic EE Total E field Array pattern Normalized element pattern Array factor

O. Kilic EE Directivity of Linear Arrays where

O. Kilic EE Radiation Intensity

O. Kilic EE Total Radiated Power

O. Kilic EE Directivity

O. Kilic EE Directivity for Arrays with Isotropic Elements Easier to calculate Represents an approximate solution for elements with broad patterns Uniform amplitude and equal spacing will be assumed. Using

O. Kilic EE Directivity: Isotropic Elements, Linear Phase Progression, Uniform Spacing, Uniform Amplitude

O. Kilic EE Non-Uniformly Excited Linear Arrays We have seen the effects of phase shifting on the beam direction. We can also shape the beam and control the level of sidelobes by adjusting the amplitude of the currents in an array.

O. Kilic EE Array Factor for Non-Uniform Excitation

O. Kilic EE Can we eliminate the sidelobes??? Yes! First consider the 1x2 element array as in case 1 we studied. Recall that the AF did not have any sidelobes AF = |1+e j  | = 

O. Kilic EE Binomial Series Coefficients If the amplitudes are equal to the coefficients of the binomial series, no sidelobes. Consider the array factor, which is the square of Case 1: AF = (1+Z)(1+Z)=1 + 2Z + Z 2 This corresponds to a three element array with current amplitudes in the ratio of 1:2:1 Since this array factor is simply the square of an array factor with no sidelobes  there are no sidelobes.

O. Kilic EE Dimensional Arrays The elements lie on a plane instead of a line. Many geometric shapes are possible; circle, square, rectangle, hexagon, etc. Will consider rectangular arrays

O. Kilic EE Rectangular Array Geometry dx x(m) y(n) dy  mn   r r mn z

O. Kilic EE Individual Fields

O. Kilic EE Total Field Element pattern Array factor

O. Kilic EE Array Factor

O. Kilic EE Array Factor : Linear Phase, Uniform Amplitude

O. Kilic EE Factors of Planar AF

O. Kilic EE Homework, Problem 1 Show that the Array Factor for two isotropic sources with identical amplitudes and 90 o phase shift is given by

O. Kilic EE HW Problem 2 Construct by hand after plotting the AF for N=4,  =  /2, d = /2 Hint: The AF vs  should look like this:

O. Kilic EE References Stutzman, et. al. “Antenna Theory” provides an excellent discussion on array antennas!!!