Giai đoạn mảng anten P8

A simple approximate analysis method is called the “large array method,” wherein the element currents are assumed to be the Floquet infinite array are summed over the finite array to get

Direct impedance (admittance) matrix methods were described in Section 7.3.2

1974; Bailey and Bostian, 1974; Cha and Hsiao, 1974; Steyskal, 1974; Bird, 1979; Luzwick and Harrington, 1982; Clarricoats et al., 1984; Pozar, 1985, 1986; Fukao et al., 1986; Deshpande and Bailey, 1989; Silvestro, 1989; Usoff

half~wave dipoles, for example), sizeable planar arrays may be solved When the element current distribution is complicated the number of moment method

expansion functions needed for good convergence will restrict this method to small arrays

An elegant solution to the semiminfinite array, that is, an array extending to infinity on three sides, is given by Wasylkiwskyj (1973) The Weiner-Hopf

factorization procedure is extended to finite Fourier transforms, resulting in

an expression for ~~~~ ~e~ec~~~~ c~~~~c~e~~ of the semiminfinite array in terms of ache ~~~ee~~~~ ~~e~~~~e~~ for the corresponding infinite array, and an integral of

a phased sum of the infinite array ache ~rn~ed~~ces

Still another approach embeds the finite array in a matrix of identical

1995) For single-mode elements (thin dipoles, thin slots, thin patches) the procedure is simple First, the ache eZerne~~ ~~~~e~~ (SEP) is computed for the corresponding infinite array This SEP is then Fourier transformed [usually

by discrete Fourier transform (DFT)] back to the aperture Third, a periodic structure consisting of equally spaced finite arrays, with each array having the desired amplitude distribution and scan phase, is transformed Fourth, these two transforms are convolved Next the result is inverse transformed to get

273

Phased Array Antennas Robert C Hansen

ISBNs: 0-471-53076-X (Hardback); 0-471-22421-9 (Electronic)

274 FNTE ARRAYS

the SEPs over the periodic structure Last, the SEP for one finite array is extracted, and multiplied by the isolated element pattern, yielding the final finite array scan element pattern

When the type of element allows multiple modes or nonsimple currents, the changing mutual coupling over the finite array may change the mode mix or change the current distribution For these cases, each element needs a moment method (or modal) expansion, with all of the equations representing the expan- sion functions and number of elements solved simultaneously A term of the impedance matrix for one expansion function consists of a double sum (over the elements) containing an integral of the appropriate Green’s function times the expansion and test functions, times the exponential scan factor Through the Poisson sum formula this is transformed to the spectral domain An exam- ple of this formula is given in Section 7.4.1 The term is now written as a convolution product of this spectral domain result and the transform of the array amplitude distribution The simultaneous equations are solved in the

impedance for each element To reduce errors introduced by aliasing between nearby arrays, it is necessary for the blank spaces to be greater than the width

of the finite array But as the ratio of blank to array becomes larger, the convergence deteriorates This method, like all the others, has both good and bad aspects

A simple approximate analysis method is called the “large array method,” wherein the element currents are assumed to be the Floquet infinite array

are summed over the finite array to get scan impedance Amplitude tapers are readily included Impedance of the center element is given by

The scan element pattern is obtained from this scan impedance, as shown in Section 7.2.3:

(8 2) l

Here R(u, V) and I?@, u) are the scan resistance and scan reflection coefficient; giso(u, V) is the isolated element (and backscreen) power pattern Both resis- tance and pattern are normalized to broadside values

Results from the large-array method show finite array oscillations super- imposed onto infinite array results (Hansen, 1990) Figures 8.1 and 8.2 show scan element pattern for the O.% lattice without screen, for array sizes 41 x 41 and 101 x 101 As expected, the number of oscillations in the pattern is about half the number of elements along x or y, but only half the symmetric pattern is shown Somewhat surprisingly, the wide-angle (largest) oscillations appear to have an amplitude independent of the array size Because of the oscillatory effect of adding a row or column of elements to the array, it might be expected

275

I -2O~""""""""""""""'f"""""""""""""""""""""""""""'~"'

L = 0.5 A

that adding half the mutual impedances for the perimeter elements would reduce the oscillations This proves to be the case, a 1 dB reduction is obtained,

so all results shown utilize half-perimeter mutual impedances Figures 8.3 and 8.4 show SEP for the 0.7h lattice case Again oscillations are superimposed on the infinite array results and, as expected, the number of oscillations is gov- erned by the number of elements in the scan direction The 41 x 41 array of Fig 8.3 provides a coarse approximation to the infinite array results; the

101 x 101 array of Fig 8.4 is better The array size in the transverse direction

is less critical; representation of deep grating lobe blind angles requires 41 or more elements

Addition of a back screen appears to fill in the cyclic behavior of the dipole- dipole impedances, with the result that the large-array approximation gives very small oscillations Figure 8.5 shows that very good correlation with infi- nite array results occurs for 41 x 41 array elements For 21 x 21 array, the oscillations are less than 1 dB The large-array method appears useful for arrays where the element pattern is of dipole~screen type

Computational cost is reduced when a finite-by-infinite array is considered, with scan across the finite width of the array (Denison and Scharstein, 1995) Only principal plane scans are allowed here Even when moment meth- ods are applied, relatively large arrays can be simulated From this type of

Figure 8.3 Dipole array, large array approximation, 41 x 41 I dX = dY = 0.7 A,

278

simulator much can be gleaned about the behavior of edge elements in an array As has been shown many times, half-wave thin dipole arrays exhibit all the pertinent features of mutual coupling Thus complex array elements and moment method solutions are not needed at this stage By considering collinear

or parallel dipoles, both H-plane and E-plane scans can be accommodated The computational model then consists of N infinite linear arrays as sketched

in Figures 8.6 and 8.7 Each infinite array (“stick”) is composed of half-wave thin dipoles Thus mutual impedance between a dipole and the dipoles in one infinite “stick” is then computed The set of these for all the sticks is used to form the usual impedance matrix, with the solution providing the scan impe- dance of a dipole in each stick

The reference scan impedance for the infinite array is provided by the spec- tral domain formulation pioneered by Oliner and Malech (1996b) This give the scan resistance as a single term for each main lobe or grating lobe and a spectral summation of trigonometric type functions for the scan reactance For array spacings close to h/2 the formulation converges extremely rapidly

For both the N-plane and E-plane scan cases, the dipole stick impedances have been computed both in the spatial domain and in the spectral domain In the spatial domain a subroutine based on Carter’s mutual impedance between two thin wire dipoles (Hansen, 1972) is used to obtain dipole-dipole impedance

Za This involves Sine and Cosine Integrals which are calculated in double precision, to allow many terms to be summed For the collinear dipole sticks (H-plane scan), the dipole to linear array impedance is just

Figure

Figure 8.7 Finite-by-infinite dipole array with E-plane scan

As shown in Figure 8.8 the isolated dipole offset is a, the parallel shift is x0, the perpendicular spacing is yo, the array spacing is d, and dipole half-lengths are h For parallel dipole sticks (E-plane scan), the dipole to linear array impedance is

Dimensions are shown in Fig 8.9 Difficulty occurs in that the sums oscillate, with this worsening as the dipole separation decreases As the period of the

X

sum oscillations changes with dimensions, no simple compensation method

1000 terms Convergence acceleration techniques include those of Shanks (1955) and Levin (1973) Both have been used, but the Levin is superior as it does not concatenate roundoff errors The Levin transformation is a rearrange- ment of partial sums (Singh et al., 1990; Singh and Singh, 1993):

This gives the transform of m order m

0 is the binomial coefficient

Typically, 21 Levin cycles gave excel&t accuracy for all cases; because par- allel dipole mutual impedance decays asymptotically as l/r, instead of the col- linear value of 1 /r*, more Levin cycles are needed to achieve convergence for that case For both scan planes, larger element spacing gives faster convergence The spectral domain summation technique was also used; it is a simplifica- tion of the periodic moment method, developed by Ben Munk and colleagues at

written as a spectral (Floquet) summation over the wavenumber, where the wavenumber includes the element spacing For the H-plane scan array, the

281

with

P= cos kh~ - cos kh

0.01 ohm Larger element spacings require more terms

For the parallel (E-plane) case, a significant difference exists from the col- linear case: the Hankel function is now inside the generalized pattern function integral, as shown by Skinner and Munk (1992) and Skinner et al (1995), and the integral cannot be integrated in closed form The formulation, with details

of the derivation omitted, is

VW

If d, does not allow a grating lobe, the argument of J!?f) is imaginary except for

n = 0; Hf) is replaced by (2/n)Ko(arg) Numerical integration was used for each term in the series above For y > 1, a complex 32.point Gaussian was used, with a 128-point Gaussian for the closer spacings (Stroud and Secrest, 1966) Excellent convergence was achieved with 501 terms, but the result was a slow program

With proper choice of limits of both spectral and spatial summations, the plots of Scaa impedance from these two techniques lay on top of one another In general the spectral summation was slower and less satisfactory Results are given in Section 8.3.1

8.2 SCAN PERFORMANCE OF SMALL ARRAYS

dance and scan element pattern of several small arrays of dipoles Figure 8.10 shows scan resistance for the center element of an array of 7 x 9 half-wave dipoles on a h/2 square lattice, with the infinite array results, shown dashed, for comparison In Figure 8.11 the same array is over a ground screen at h/4 spacing The screen removes the high values of the ILI- and diagonal planes Small arrays exhibit behavior that oscillates around that of a large array Similar oscillations occur in the scan reactance of small arrays In general, as might be expected, larger arrays have more oscillations, but of smaller ampli- tude

Scan element patterns are of interest for small arrays although it is no longer valid to multiply the array factor by a single lean eZ~ment pattern in order to get the overall arrav rclattern Figure 8.12 shows calculated and

\

0

Chapter 3 of TR381, Lincoln Laboratory, March 1965

30* 0* 30* 60° 90*

(0)

RELATIVE POWER, db

a 7 x 9 array of dipoles for various lattice spacings, From these curves it is apparent that the xa~t element pattern limits the scan angle range As more elements are added to the array, the number of oscillations in the pattern increases and the curves become smoother Also the falloff becomes steeper Figure 8.15 and 8.16 compare center, edge center, and corner scan element patterns for a similar 9 x 11 array of dipoles The corner elements do not show the symmetric mutual coupling effects that tend to flatten the scan

array to be scanned if space is available

- ISOLATED X/2 DIPOLE 4

-em 0, = 0, = 0.5 0

* 0~ Dp 0.6 l

-80 -60 -40 -20

6 (deg)

Figure 8.14 H-plane scan element patterns of center element of 7 x 9 dipole/screen array (Courtesy Diamond, B L., “Theoretical Investigation of Mutual Coupling Effects,” Chapter 3 of TR381, Lincoln Laboratory, March 1965.)

SCAN P~R~OR~A~C~ OF SMALL ARRAYS 285

An interesting example of edge effects in a linear array is given by Gallegro (1969), for half-wave dipoles with and without a ground plane Spacing between elements is 0.586& which gives a just-visible grating lobe at

$ = 90 deg for 45 deg scan; dipole orientation is both parallel and collinear The center element is matched at 0 deg scan angle, with all elements using the same matching network All arrays have 51 elements, and the ground plane spacing is 0.2% The collinear arrays are well behaved even at grating lobe incidence; the parallel arrays are somewhat less well behaved up to grating lobe (GL) incidence, but at incidence there was a marked edge effect, especially for the array without ground plane Figures 8.17 and 8.18 show element impe- dance by number with and without ground plane for scan angles of Odeg, 40deg, and 45 deg for parallel dipoles It appears that the edge elements in the scan direction are more affected than the edge elements in the reverse direction, but the VSWR of the reverse edge elements is higher owing to the grating lobe pointing in that direction A rough tabulation of the number of edge elements where the edge element impedance is outside the center cluster is shown in Table 8.1 It must not be supposed that these results apply to planar arrays, but the trends should be useful

72 R ,fig6 120 144

Figure 8.17 Scan impedance of linear array of parallel dipoles (Courtesy Gallegro, A D., Mutual Coupling and Edge Effects in Linear Phased Arrays, MS thesis, Polytechnic

287

scan

Gallegro, A D., Mutual Coupling and Edge Effects in Linear Phased Arrays, MS thesis, Polytechnic lnstitute of Brooklyn, 1969.)

Dipoles Ground Plane Scan direction Reverse direction Parallel

8.3 FINITE-BY-INFINITE ARRAY GIBBSIAN MODEL

The finite-by-infinite array simulators described in Section 8.1.2 were used to compute xa~l impedance across the array, for several arrays from large to small Results for H-plane for arrays of 201 resonant dipoles are shown as