LECTURE 15: LINEAR ARRAY THEORY - PART I

Transcription

1 LECTURE 5: LINEAR ARRAY THEORY - PART I (Linear arrays: the two-element array; the N-element array with uniform amplitue an spacing; broa - sie array; en-fire array; phase array). Introuction Usually the raiation patterns of single-element antennas are relatively wie, i.e. they have relatively low irectivity (gain). In long istance communication, antennas with very high irectivity are often require. This type of antenna is possible to construct by enlarging the imensions of the raiating element (maximum size much larger than λ ). This approach however may lea to the appearance of multiple sie lobes an technologically inconvenient shapes an imensions. Another way to increase the electrical size of an antenna is to construct it as an assembly of raiating elements in a proper electrical an geometrical configuration antenna array. Usually the array elements are ientical. This is not necessary but it is more practical, simple an convenient for esign an fabrication. The iniviual elements may be of any type (wire ipoles or loops, apertures, etc.) The total fiel of an array is a vector superposition of the fiels raiate by the iniviual elements. To provie very irective pattern, it is necessary that the partial fiels (generate by the iniviual elements) interfere constructively in the esire irection an interfere estructively in the remaining space. There are five basic methos to control the overall antenna pattern: a) the geometrical configuration of the overall array (linear, circular, spherical, rectangular, etc.) b) the relative isplacement between elements c) the excitation amplitue of iniviual elements ) the excitation phase of each element e) the relative pattern of each element

2 . Two-element array Let us represent the electric fiels in the far-zone of the array elements in the form: E E = M E ( θ, φ ) β j kr ˆ ρ n r β j kr + e = MEn( θ, φ) ˆ ρ r e (5.) (5.) P z θ r r θ r θ y Here: M, M fiel magnitues (o not inclue the /r factor); E n, En normalize fiel patterns; r, r istances to the observation point P; β phase ifference between the fee of the two array elements; ˆρ, ˆρ polarization vectors of the far-zone E fiels.

3 The far-fiel approximation of the two-element array problem: P r z θ r θ y r θ cosθ Let us assume that: ) the array elements are ientical, i.e., En( θ, φ ) = En( θ, φ ) = En( θ, φ ) (5.3) ) they are oriente in the same way in space (they have ientical polarization) i.e. ˆ ρ = ˆ ρ = ˆ ρ (5.4) 3) their excitation is of the same amplitue, i.e. M = M = M (5.5) 3

4 Then, the total fiel can be erive as: E = E = E β β jk r cosθ + j jk r+ cosθ j E = ˆ ρmen ( θ, φ) e + e r k β k M j cosθ j cos jkr E = ˆ ρ e En ( θ, φ) e + e r jkr e cos ˆ k θ + β E = ρm En ( θ, φ) cos r β + θ+ AF (5.6) (5.7) The total fiel of the array is equal to the prouct of the fiel create by a single element locate at the origin an a factor calle array factor, AF: k cosθ + β AF cos (5.8) = Using the normalize fiel pattern of a single element, (, ) an the normalize AF, AF n k cosθ + β = cos E θ φ, n (5.9) the normalize fiel pattern of the array can be foun as their prouct: f θ, φ = E θ, φ AF θ, φ (5.0) ( ) ( ) ( ) n n n The concept illustrate by (5.0) is the so-calle pattern multiplication rule vali for arrays of ientical elements. This rule hols for any array consisting of ientical elements, where the excitation magnitues, the phase shift between elements an isplacement between them are not necessarily the same. The total pattern, therefore, can be controlle via the single element E θ, φ, or via the AF. pattern, ( ) n 4

5 The AF, in general, epens on: number of elements geometrical arrangement relative excitation magnitues relative phases Example : An array of two horizontal infinitesimal ipoles locate at a istance = λ / 4 from each other. Fin the nulls of the total fiel, if the excitation magnitues are the same an the phase ifference is: a) β = 0 b) β = π / c) β = π / θ = 0 θ = 90 z λ 8 λ 8 0 y θ = 80 The element factor, E (, ) n θ φ, oes not epen on β an it prouces in all three cases one an the same null. Since E θ, φ = cosθ,itisat n ( ) θ = π / (5.) 5

6 The AF epens on β an will prouce ifferent results in these 3 cases: a) β = θ k cosθn AFn = cos = 0 π π π cos cosθn = 0 cosθn = cosθn = 4 4 The solution oes not exist. In this case the total fiel pattern has only null at θ = 90. Fig 6.3, pp 55, Balanis 6

7 π b) β = π π AFn = cos cosθn + = π π ( cosθn + ) = cosθn + = cosθn = θ = 0 4 The equation π π ( cosθ n + ) = 4 oes not have a solution. The total fiel pattern has nulls at θ = 90 an at θ = 0 : Fig 6.4, pp.56, Balanis 7

8 c) π β = π π AFn = cos cosθn = π π ( cosθn ) =± cosθn =± θ = π 4 The total fiel pattern has nulls: at θ = 90 an at θ = 80. fig 6.4b, pp.57, Balanis 8

9 Example #: Consier a -element array of two ientical ipoles (infinitesimal ones) oriente along the y-axis. Fin the angles of observation where nulls of the pattern occur as a function of the istance between the ipoles,, an the phase ifference, β. The normalize total fiel pattern is: k cosθ + β fn = cosθ cos (5.) In orer to fin the nulls, the equation k cosθ + β fn = cosθ cos = 0 (5.3) will be sole. The element factor, cosθ, prouces one null at θ = π / (5.4) The amplitue factor leas to the following solution: k cosθ + β k cosθ + β n + cos = 0 =± π λ θn = arccos ( β ± ( n+ ) π), n= 0,,... π (5.5) When there is no phase ifference between the two elements ( β = 0), the separation must satisfy: λ in orer at least one null to occur ue to (5.5). 9

10 3. N-element linear array with uniform amplitue an spacing It is assume that each succeeing element has a β progressive phase lea current excitation relative to the preceing one. An array of ientical elements with ientical magnitues an with a progressive phase is calle a uniform array. The AF can be obtaine by consiering the iniviual elements as point (isotropic) sources. If the elements are of any other pattern, the total fiel pattern can be obtaine by simply multiplying the AF by the normalize fiel pattern of the iniviual element. The AF of an N-element linear array of isotropic sources is: j( k cosθ + β ) j( k cosθ + β ) j( N )( k cosθ + β ) AF = + e + e + + e (5.6) z θ to P θ θ r θ cosθ y 0

11 Phase terms of partial fiels: 3 st n r e th e e jkr N e Equation (5.6) can be re-written as: N n= ( cosθ ) jk r jk r ( cosθ ) ( ( ) cosθ ) jk r N jn ( )( kcosθ + β ) AF = e (5.7) AF N jn ( ) ψ = e (5.8) n= where ψ = k cosθ + β. From (5.8), it is obvious that the AFs of uniform linear arrays can be controlle by the relative phase β between the elements. The AF in (5.8) can be expresse in a close form, which is more convenient for pattern analysis. jψ N n= jnψ AF e = e (5.9) jψ jnψ AF e AF = e N N N j ψ j ψ j ψ e e e jnψ e AF = = jψ ψ ψ ψ e j j j e e e N N sin ψ j ψ AF = e ψ sin (5.0)

12 Here, N shows the location of the last element with respect to the ψ j( N ) reference point in steps with length. The phase factor e is not important unless the array output signal is further combine with the output signal of another antenna. It represents the phase shift of the array s phase centre relative to the origin an it woul be ientically equal to one if the origin were to coincie with the array s centre. Neglecting the phase factor gives: N sin ψ AF = (5.) ψ sin For small values of ψ, (5.) can be reuce to: N sin ψ AF = (5.) ψ To normalize (5.) or (5.) one nees the maximum of the AF. N sin ψ AF = N (5.3) ψ N sin

13 The function: sin( Nx) f ( x) = Nsin( x) has its maximum at x = 0, π,, an the value of this maximum is f max =. Therefore, AFmax = N. 0.9 sin( Nx) f( x) = 0.8 Nsin( x) N = N = 5 0. N = The normalize AF is obtaine as: or AF n AF n ψ sin N = (5.4) N ψ N sin N sin ψ =,forsmall ψ (5.5) N ψ 3

14 Nulls of the AF To fin the nulls of the AF, equation (5.4) is set equal to zero: sin N N N ψ = 0 ψ =± nπ ( kcosθ + n β ) =± nπ λ n θn = arccos β ± π, π N n=,,3 ( n 0, N, N,3 N ) (5.6) (5.7) When n= 0, N, N,3N,theAF attains its maximum values (see the case below). The values of n etermine the orer of the nulls. For a null to exist, the argument of the arccosine must not excee unity. Maxima of the AF They are stuie in orer to etermine the maximum irectivity, the HPBW s, the irection of maximum raiation. The maxima of (5.4) occur when (see the plot in page 3): ψ = ( kcosθm + β ) =± mπ (5.8) λ θm = arccos ( β ± mπ), m= 0,, π (5.9) When (5.8) is true, AF n =, i.e. these are not maxima of minor lobes. The inex m shows the maximum s orer. It is usually esirable to have a single major lobe, i.e. m=0. This can be achieve by choosing / λ sufficiently small. Then, the argument of the arccosine function is (5.9) becomes greater than unity for m =, an equation (5.9) has a single solution: βλ θm = arccos π (5.30) 4

15 The HPBW of a major lobe It can be calculate by setting the value of AF n equal to /. For AF n in (5.5): N N ψ = ( k cos θ h + β ) =±.39 λ.78 θh = arccos β ± π N (5.3) For a symmetrical pattern aroun θ m (the angle at which maximum raiation occurs), the HPBW ca be calculate as: HPBW = θ m θ h (5.3) Maxima of minor lobes (seconary maxima) They are the maxima of AF n,where AFn. Thisisclearly seen in the plot of the array factors as a function of ψ = k cosθ + β for a uniform equally space linear array (N=3,5,0) sin( Nψ / ) f ( ψ ) = N sin( ψ / ) ψ = k cosθ + β N = 3 N = 5 N = ψ 5

16 Only approximate solutions will be given here. For the approximate AF n of equation (5.5), the seconary maxima occur approximately where the numerator attains a maximum: N sin ψ =± N s+ ( k cosθ + β ) =± π (5.33) λ s + θs = arccos β π π ± N or (5.34) π λ s + θs = arccos β ± π π N (5.35) 4. Broasie array A broasie array is an array, which has maximum raiation at θ = 90 (normal to the axis of the array). For optimal solution both, the element factor an the AF, shoul have their maxima at θ = 90. From (5.8), it follows that the maximum of the AF woul occur when ψ = k cosθ + β = 0 (5.36) Equation (5.36) is vali for the zero-th orer maximum, m = 0.If θ = π /, then: m ψ = β = 0 (5.37) The uniform linear array will have its maximum raiation at θ = 90, if all array elements have the same phase excitation. To ensure that there are no maxima in other irections (grating lobes), the separation between the elements shoul not be equal to multiples of a wavelength: nλ, n=,,3, (5.38) 6

17 If nλ, then aitional maxima, AF n =, appear. Assume that = nλ. Then, π ψ = k cosθ = nλcosθ = πn cosθ (5.39) λ If equation (5.39) is true, the maximum conition: ψ m = mπ, m= 0, ±, ± (5.40) will be fulfille not only for θ = π / but also for m θ g = arccos, m =±, ± (5.4) n If, for example, = λ ( n= ), equation (5.4) results in two aitional major lobes at: θ = arccos ± θ = 0,80 g ( ) g, If = λ ( n= ), equation (5.4) results in four aitional major lobes at: θg = arccos ±, ± θg = 0,60,0,80,,3,4 The best way to ensure the existence of only one maximum is to choose max < λ. Then, in the case of the broasie array ( β = 0), equation (5.9) prouces no solution for m. 7

18 5. Orinary en-fire array An en-fire array is an array, which has its maximum raiation along the axis of the array ( θ = 0,80). It might be require that the array raiates only in one irection either θ = 0 or θ = 80. ψ = k cosθ + β θ = 0 = k + β = 0 (for max. AF) Then: β = k, for θ = 0 (5.4) θ = 80 max ψ = k cosθ + β = k + β = 0 (for max. AF) β = k, for θmax = 80 (5.43) If the element separation is multiple of a wavelength, = nλ,then in aition to the en-fire maxima there also exist maxima in the broasie irections. As with the broasie array, in orer to avoi grating lobes, the maximum spacing between the element shoul be less than λ : max < λ (Show that an en-fire array with = λ / will have maxima for β = k :atθ = 0 an at θ = 80 ) 8

19 AF pattern of an EFA: N=0, = λ / 4 fig 6-, pp 70, Balanis 6. Phase (scanning ) arrays It was alreay shown that the zero-th orer maximum (m=0) of AF n occurs when ψ = k cosθ0 + β = 0 (5.44) The relation between the irection of the main beam θ 0 an the phase ifference β is obvious. The irection of the main beam can be controlle by the phase shift β. This is the basic principle of electronic scanning for phase arrays. The scanning must be continuous. That is why the feeing system shoul be capable of continuously varying the progressive phase β between the elements. This is accomplishe by ferrite or ioe shifters (varactors). 9

20 Example: Values of the progressive phase shift β as epenent on the irection of the main beam θ 0 for a uniform linear array with = λ /4. From equation (5.44): πλ π β = k cosθ0 = cosθ0 = cosθ0 λ 4 θ0 β The HPBW of a scanning array is obtaine using eq.(5.3), where β = k cosθ : 0 λ.78 θh = arccos β, ± π N (5.45) The total beamwith is: HPBW = θ θ (5.46) h h λ.78 HPBW = arccos k cosθ0 π N λ.78 arccos k cosθ0 + π N Since k = π / λ :.78 HPBW = arccos cosθ0 Nk.78 arccos cosθ0 + Nk (5.47) (5.48) 0

21 One can use the substitution N = ( L+ )/ to obtain: λ HPBW = arccos cosθ L + (5.49) λ arccos cosθ L+ Here, L is the length of the array. Equations (5.48) an (5.49) can be use to calculate the HPBW of a broasie array, too ( θ 0 = 90 = const ). However, it is not vali for en-fire arrays,where.78 HPBW = cos Nk (5.50)