LECTURE 15: LINEAR ARRAYS PART III

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1 LECTURE 5: LINEAR ARRAYS PART III (N-element linear arrays with uniform spacing and non-uniform amplitude: Binomial array; Dolph Tschebyscheff array. Directivity and design.). Advantages of Linear Arrays with Nonuniform Amplitude Distribution The most often met BSAs, classified according to the type of their excitation amplitudes, are: a) the uniform BSA relatively high directivity, but the side-lobe levels are high; b) Dolph Tschebyscheff (or Chebyshev) BSA for a given number of elements, maximum directivity is next after that of the uniform BSA; sidelobe levels are the lowest in comparison with the other two types of arrays for a given directivity; c) binomial BSA does not have good directivity but has very low side-lobe levels (when d = λ /, there are no side lobes at all).. Array Factor of Linear Arrays with Nonuniform Amplitude Distribution Let us consider a linear array with an even number (M) of elements, located symmetrically along the z-axis, with excitation, which is also symmetrical with respect to z =. For a broadside array ( β = ), 3 M j kd cosθ j kd cosθ j kd cosθ AF e = ae + ae + + am e + 3 M j kd cosθ j kd cosθ j kd cosθ + ae + ae + + am e, M e n AF = an cos kd cosθ (5.). (5.) If the linear array consists of an odd number (M+) of elements, located symmetrically along the z-axis, the array factor is AF o = a cos cos cos + ae jkd θ + a j kd 3e θ a jmkd M e θ + + (5.3) + ae jkd cosθ + ae jkdcosθ a e jmkd cosθ, 3 M + Russian spelling is Чебышёв. Nikolova 6

2 M + n ( ). (5.4) AF o = a cos n kd cosθ EVEN- AND ODD-NUMBER ARRAYS Fig. 6.7, p. 9, Balanis Nikolova 6

3 The normalized AF derived from (5.) and (5.4) can be written in the form M AF e = a n u n cos [( ) ], for N M + AF o a n u n cos[ ( ) ], for N M = = M, (5.5) = +, (5.6) π d where u = kd cosθ = cosθ. λ Examples of AFs of arrays of nonuniform amplitude distribution a) uniform amplitude distribution (N = 5, d = λ /, max. at θ = 9 ) pp , Stutzman Nikolova 6 3

4 b) triangular (::3::) amplitude distribution (N = 5, d = λ /, max. at θ = 9 ) pp , Stutzman Nikolova 6 4

5 c) binomial (:4:6:4:) amplitude distribution (N = 5, d = λ /, max. at θ = 9 ) pp , Stutzman Nikolova 6 5

6 d) Dolph-Tschebyschev (:.6:.94:.6:) amplitude distribution (N = 5, d = λ /, max. at θ = 9 ) pp , Stutzman Nikolova 6 6

7 e) Dolph-Tschebyschev (:.4:3.4:.4:) amplitude distribution (N = 5, d = λ /, max. at θ = 9 ) pp , Stutzman Notice that as the current amplitude is tapered more gradually toward the edges of the array, the side lobes tend to decrease and the beamwidth tends to increase. Nikolova 6 7

8 3. Binomial Broadside Array The binomial BSA was investigated and proposed by J. S. Stone to synthesize patterns without side lobes. First, consider a element array (along the z-axis). z x d The elements of the array are identical and their excitations are the same. The array factor is of the form j AF = + Z, where ( kd cos Z ej ψ θ+ β = = e ). (5.7) If the spacing is d λ / and β = (broad-side maximum), the array pattern AF has no side lobes at all. This is proven as follows. AF ( cos ψ) sinψ ( cos ψ) 4cos ( ψ / ) y = + + = + = (5.8) where ψ = kd cosθ. The first null of the array factor is obtained from (5.8) as π d cos π n, n, arccos λ θ =± θ =± λ d. (5.9) As long as d < λ /, the first null does not exist. If d = λ /, then θ n, =, 8. Thus, in the visible range of θ, all secondary lobes are eliminated. Second, consider a element array whose elements are identical and the same as the array given above. The distance between the two arrays is again d. US Patents #,643,33, #,75,433. Nikolova 6 8

9 z d y This new array has an AF of the form x AF Z Z Z Z = ( + )( + ) = + +. (5.) Since ( + Z) has no side lobes, ( + Z) does not have side lobes either. Continuing the process for an N-element array produces AF Z = ( + ) N. (5.) If d λ /, 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 (5.). Making use of Pascal s triangle, the relative excitation amplitudes at each element of an (N+)-element array can be determined. 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 BAs. The overall efficiency of such an antenna would be low. Besides, the BA has relatively wide beam. Its HPBW is the largest as compared to the uniform BSA or the Dolph Chebyshev array. Nikolova 6 9

10 An approximate closed-form expression for the HPBW of a BA with d = λ / is HPBW = =, (5.) N L λ L λ where L= ( N ) d is the array s length. The AFs of -element broadside binomial arrays (N = ) are given below. as The directivity of a broadside BA with spacing d = λ / can be calculated D = π cos cosθ π ( N ) D dθ, (5.3) (N ) (N 4)... =, (5.4) (N 3) (N 5)... D.77 N =.77 + L λ. (5.5) Nikolova 6

11 Fig. 6.8, p.93, Balanis Nikolova 6

12 4. Dolph Chebyshev Array (DCA) Dolph proposed (in 946) a method to design arrays with any desired sidelobe levels and any HPBWs. This method is based on the approximation of the pattern of the array by a Chebyshev polynomial of order m, high enough to meet the requirement for the side-lobe levels. A DCA with no side lobes (sidelobe level of db) reduces to the binomial design. 4.. Chebyshev polynomials The Chebyshev polynomial of order m is defined by ( ) m cosh( m arccosh z ), z, Tm ( z) = cos( m arccos( z) ), z, (5.6) cosh ( m arccosh( z) ), z. A Chebyshev polynomial T m (z) of any order m can be derived via a recursion formula, provided T m (z) and T m (z) are known: T ( z) = zt ( z) T ( z). (5.7) Explicitly, from (5.6) we see that m=, T ( z) = m=, T ( z) = z. Then, (5.7) produces: m=, T ( z) = z m= T z = z z 3, 3 3( ) 4 3 m m m m= T z = z z + 4, 4 4( ) 8 8 m= T z = z z + z 5, 5 3 5( ) 6 5, etc. If z, then the Chebyshev polynomials are related to the cosine functions, see (5.6). We can always expand the function cos(mx) as a polynomial of cos(x) of order m, e.g., for m =, cos x= cos x. (5.8) The expansion of cos(mx) can be done by observing that ( ejx ) m = ejmx and by making use of Euler s formula as Nikolova 6

13 (cos x + j sin x) m = cos( mx) + j sin( mx). (5.9) The left side of the equation is then expanded and its real and imaginary parts are equated to those on the right. Similar relations hold for the hyperbolic cosine function cosh. Comparing the trigonometric relation in (5.8) with the expression for T( z ) above (see the expanded Chebybshev polynomials after (5.7)), we see that the Chebyshev argument z is related to the cosine argument x by For example, (5.8) can be written as: z = cos x or x= arccos z. (5.) [ z ] cos(arccos z) = cos(arccos ), = =. (5.) cos(arccos z) z T ( z) Properties of the Chebyshev polynomials: ) All polynomials of any order m pass through the point (,). ) Within the range z, the polynomials have values within [,]. 3) All nulls occur within z. 4) The maxima and minima in the z [,] range have values + and, respectively. 5) The higher the order of the polynomial, the steeper the slope for z >. Nikolova 6 3

14 Fig. 6.9, pp. 96, Balanis Nikolova 6 4

15 4.. Chebyshev array design The main goal is to approximate the desired AF with a Chebyshev polynomial such that the side-lobe level meets the requirements, and the main beam width is as small as possible. An array of N elements has an AF approximated with a Chebyshev polynomial of order m, which is m= N. (5.) In general, for a given side-lobe level, the higher the order m of the polynomial, the narrower the beamwidth. However, for m >, the difference is not substantial see the slopes of Tm ( z ) in the previous figure. The AF of an N-element array (5.5) or (5.6) is shaped by a Chebyshev polynomial by requiring that T N ( z) M [ ] a cos (n ) u, M = N /, even n = M + an [ n u] M = N cos ( ), ( ) /, odd Here, u = ( πd / λ)cosθ. Let the side-lobe level be R (5.3) = Emax E = AF (voltage ratio). (5.4) sl sl Then, we require that the maximum of TN is fixed at an argument z z > ), where ( T ( z ) R max N =. (5.5) Equation (5.5) corresponds to AF( u) = AF max ( u). Obviously, z must satisfy the condition: z >, (5.6) where TN >. The maxima of TN ( z) for z are equal to unity and they correspond to the side lobes of the AF. Thus, AF( u ) has side-lobe levels equal to R. The AF is a polynomial of cosu, and the TN ( z) is a polynomial of z where z is limited to the range Nikolova 6 5

16 z z. (5.7) Since cosu, (5.8) the relation between z and cosu must be normalized as cos u = z/ z. (5.9) Design of a DCA of N elements general procedure: ) Expand the AF as given by (5.5) or (5.6) by replacing each cos( mu ) term ( m=,,..., M ) with the power series of cosu. ) Determine z such that TN ( z ) = R (voltage ratio). 3) Substitute cos u = z/ z in the AF found in step. 4) Equate the AF found in Step 3 to TN ( z) and determine the coefficients for each power of z. Example: Design a DCA (broadside) of N= elements with a major-to-minor lobe ratio of R = 6 db. Find the excitation coefficients and form the AF. Solution: The order of the Chebyshev polynomial is m= N = 9. The AF for an evennumber array is: 5 π d AF M = a n u u = λ ncos [( ) ], cosθ, 5 M =. Step : Write AF explicitly: AF = a cosu+ a cos3u+ a cos5u+ a cos 7u+ a cos9u Expand the cos( mu ) terms as powers of cosu : cos3u= 4cos 3 u 3cosu, cos5u = 6cos 5 u cos 3 u + 5cosu, cos7u= 64cos7 u cos5 u + 56cos3 u 7cosu, cos9u = 56cos9 u 576cos 7 u + 43cos5 u cos 3 u + 9cosu. Nikolova 6 6

17 Note that the above expansions can be readily obtained from the recursive Chebyshev relation (5.7), z z, and the substitution z = cosu. For example, translates into m= T z = z z 3, 3 3( ) cos(3 ) 4cos 3cos u = u u. Step : Determine z : 6 R = 6 db R = T9( z) =, cosh[ 9arccosh( z ) ] =, 9arccosh( z ) = arccosh = 3.69, arccosh( z ) =.4, z = cosh.4 z =.855. Step 3: Express the AF from Step in terms of cos u = z/ z and make equal to the Chebyshev polynomial: z AF = ( a 3a + 5a3 7a4 + 9a5 ) z z3 + ( 4a a3+ 56a4 a5) z3 z5 + ( 6a3 a4+ 43a5) z5 z7 + ( 64a4 576a5) z7 z9 + ( 56a5 ) = z9 =9z z3 + 43z5 576z7 + 56z9 T ( z) 9 Nikolova 6 7

18 Step 4: Find the coefficients by matching the power terms: 56a 9 5 = 56z a5 =.86 64a 576a = 576z7 a = a3 a4 + a5 = z5 a3 = a a3 a4 a5 z7 a = 9 = = = 5.73 a 3a 5a 7a 9a 9z a Normalize coefficients with respect to edge element (N=5): a = ; a =.357; a =.974; a =.496; a = AF =.789cos( u) +.496cos(3 u) +.974cos(5 u) cos(7 u) + cos(9 u) where π d u = cosθ. λ Fig. 6.b, p. 98, Balanis Nikolova 6 8

19 Fig. 6., p. 3, Balanis Nikolova 6 9

20 4.3. Maximum affordable d for Dolph-Chebyshev arrays This restriction arises from the requirement for a single major lobe see also equation (5.7), z z : π d z, cos u = z/ z, u = cosθ, λ π d z = z cos cosθ. (5.3) λ For a given array, when θ varies from to 8, the argument z assumes values π d from z( θ = ) = z cos (5.3) λ through z( θ = 9 ) = z (5.3) d to z( 8 ) z cos π θ= = = z( θ= ). (5.33) λ The extreme value of z to the left on the abscissa corresponds to the end-fire directions of the AF. This value must not go beyond z =. Otherwise, endfire lobes of levels higher than (higher than R ) will appear. Therefore, the inequality (5.3) must hold for θ = or 8 : Let z πd πd cos cos. (5.34) λ λ z ( ) γ = arccos z. (5.35) Remember that z > ; thus γ is a real-valued angle. Then, or π d < π γ = π arccos λ z π dmax d ( z ) λ λ π max = π arccos = arccos z γ (5.36) (5.37) Nikolova 6

21 π d max λ cosγ γ z cos( πd / λ) ILLUSTRATION OF EQUATION (5.34) AND THE REQUIREMENT IN (5.36) For the case of the previous example, d < arccos = =.873, λ π.855 π dmax =.873λ. 5. Directivity of Non-uniform Arrays It is difficult to derive closed form expressions for the directivity of nonuniform arrays. Here, we derive expressions in the form of series in the most general case of a linear array when the excitation coefficients are known. The non-normalized array factor is where N = jβn jkzncosθ n, (5.38) AF a e e a n is the amplitude of the excitation of the n-th element; β n is the phase angle of the excitation of the n-th element; z is the z-coordinate of the n-th element. n The maximum AF is Nikolova 6

22 The normalized AF is AF n AF max max AF = = AF N = a. (5.39) N The beam solid angle of a linear array is where From π we obtain Ω = π ae n ( ) n jβ n N e a n jkz n cosθ Ω = π AF θ sinθdθ, A N N A m p N m= p= an ( sin kz ( ) )cos m z jk z p m z p θ e sinθdθ =. kz ( z) π n π. (5.4) j( βm βp) jk ( zm z p)cosθ aae e sinθdθ, (5.4) m p N N 4π ( ) sin kz ( m z) j βm β p p Ω A = aae m p. (5.4) N m= p= kz ( m zp) an D = 4 π / Ω A, D = N N m= p= N an. (5.43) ( ) sin kz ( m zp) j βm β p aae m p kz ( z) m p Nikolova 6

23 For equispaced linear ( z n D = N N m= p= = nd ) arrays, (5.43) reduces to aae m p j N a n ( β ) sin [( m p) kd ] m β p ( m p) kd. (5.44) For equispaced broadside arrays, where βm = βp for any (m,p), (5.44) reduces to D = N N m= p= a a m N p an sin ( ) ( m p) kd For equispaced broadside uniform arrays, D = N N m= p= N sin ( ) [ m p kd ] [ m p kd ] ( m p) kd. (5.45). (5.46) When the spacing d is a multiple of λ /, equation (5.45) reduces to D N N an λ =, d =, λ,... (5.47) ( a ) n Example: Calculate the directivity of the Dolph Chebyshev array designed in the previous example if d = λ /. The -element DCA has the following amplitude distribution: a = ; a =.357; a =.974; a =.496; a = Nikolova 6 3

24 We make use of (5.47): D an ( 9.65) = = = 8.99 (9.5 db)..797 ( a ) Output from ARRAYS.m: D = n 6. Half-power Beamwidth of a BS DCA For large DCAs with side lobes in the range from db to 6 db, the HPBW HPBW DCA can be found from the HPBW of a uniform array HPBW UA by introducing a beam-broadening factor f given by so that f R π R, (5.48) = cosh (arccosh ) HPBW = f HPBW. (5.49) DCA In (5.48), R denotes the side-lobe level (the voltage ratio). UA Nikolova 6 4

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