1. Antenna Array Dr Jaikaran Singh Reference Book : Antenna by John D. Kraus (TMH)

2. Contents Point Source Power Pattern Arrays of 2 Isotropic Point Source Principle of Pattern Multiplication Linear Arrays : BSA & EFA EFA with increased Directivity BSA with Non Uniform amplitude distribution Binomial Array Problems

3. Point Source Assume a source containing within a volume circle of radius b and we trying to find far field E at outer circle of radius R. Near field may be ignored. The far field will be distributed uniformly on outer circle uniformly. If source is rotated across that axis, min effect will be seen on far field

4. Contents If center of antenna is displaced from O to O’ for some distance d. The min effect of displacement will be shown on field pattern at Q The phase pattern differs and depend on d If d=0, the phase shift across the circle is minimum If d is large, the phase shift become large

5. Power Pattern Assume a transmitting antenna is in free space and represented by point source located at the origin of coordinate system. The radiated energy streams from the source in radial line

6. Power Pattern

7. Arrays of Two Isotropic Point Source

8. Case 1: Same Amplitude & Phase

13. Case 2: Same Amplitude but opposite phase

16. Case 3: Same Amplitude & in phase Quadrature

19. Case 4: Equal Amplitude & any phase

20. Case 5: Unequal Amplitude & any phase

22. Example Discuss the radiation pattern of a linear array of two isotropic sources spaced a distance d apart and have a phase difference  for the following cases d=0 o i) d= 0.2 ii) d= 0.5 iii) d= 1.5 d= l/3 i)  = 0 o ii)  = 60 o iii)  =120 o

23. d=0.2

24. d=0.5

25. d=1.5

26.  =0 o

27.  =60 o

28.  =120 o

30. 30 Implementation Example: 2-element array Isotropic radiators

31. Pattern Multiplication Principle Since the total radiated field for an array is the summation of the fields from each element where A n is the amplitude,  n is the relative phase, E e is the radiated field of the antenna element, and AF is called array factor. Thus the radiation pattern of an array is the product of the pattern of individual element antenna with the (isotropic source) array pattern.

33. Broadside Array

34. End Fire Array

37. Example element AF Total array AF Total array

38. Binomial Array The binomial array was investigated and proposed by J. S. Stone to synthesize patterns without side lobes. Let us first consider a 2–element array with equal current amplitudes and spacing, the array factor is given by For a broadside array (β = 0) with element spacing d less than one-half wavelength, the array factor has no sidelobes. This can be proved in the following way: where ψ = kd cos θ. The first null of this array factor can be obtained as: As long as the d < λ/2, the first null does not exist. If d = λ/2, then null will be at θ = 0 and 180. Thus, in the “visible” range of θ, all secondary lobes are eliminated. An array formed by taking the product of two arrays of this type gives:

39. This array factor, being the square of an array factor with no sidelobes, will also has no sidelobes. Mathematically, the array factor above represents a 3-element equally-spaced array driven by current amplitudes with ratios of 1:2:1. In a similar fashion, equivalent arrays with more elements may be formed Similarly for N-element the array factor can be expressed as If d ≤ λ/2, the above AF does not have side lobes regardless of the number of elements N. The excitation amplitude distribution can be obtained easily by the expansion of the binome in (6.50). Making use of Pascal’s triangle, this can be given by:

40. The relative excitation amplitudes at each element of an (N+1) element array can be determined from this traiangle. An array with a binomial distribution of the excitation amplitudes is called a binomial array. The excitation distribution as given by the binomial expansion gives the relative values of the amplitudes. It is immediately seen that there is too wide variation of the amplitude, which is a disadvantage of the binomial arrays. The overall efficiency of such an antenna would be low. Besides, the binomial array has a relatively wide beam. Its HPBW is the largest as compared to the uniform or the Dolph–Chebyshev array. An approximate closed-form expression for the HPBW of a binomial array with d = λ/2 is where L = (N-1)d is the array length. The directivity of a broadside binomial array with spacing d = λ/2 can be calculated as:

41. The array factor of a 10 element broadside binomial array (N = 10) is shown below.