Cavity-Backed Microstrip Patch Antenna Array

Another antenna array used to validate the oblique ABC is a cavity-backed microstrip patch antenna array consisting of 20 20× circular microstrip patch antenna elements backed by cylindrical cavities. Its 2 2× sub-array is shown in Fig. 7.9 [64],[65],[66]. For each antenna element, the circular patch has a radius a=0.156λ, placed in a circular aperture with a radius b=0.205λ. The cylindrical cavity has a radius c=0.27065λ and a depth h=0.08λ , and is filled with a lossless dielectric substrate with a relative permittivity εr =2.5 . Each circular patch is fed by a coaxial cable, whose inner conductor is attached to the radiating patch and the outer conductor to the ground plane. The cable has an inner radius r_in =0.004λ and an outer radius r_out =0.015λ . The distance from the feeding point on the circular patch to the center of the patch is s=0.078λ.

Similar to the previous example, the 20 20× cavity-backed patches are placed on an infinitely large ground plane which coincides with the xy-plane. The separations in the x- and y-directions are 0.5λ. For the mesh truncation of the upper half space, we employ either a hemispherical surface with a base radius of 8.75λ or a rectangular surface placed 1λ away from both the top and the side of array. Therefore, the size of the rectangular box is 12λ×12λ×1λ . The simulation is carried out at 300 MHz using (1) the conventional ABC with the hemispherical truncation surface, (2) the conventional ABC with the rectangular truncation surface, and (3) the oblique ABC with rectangular truncation surface for the main beam ( , )θ φ_{s s} steered to o o

(60 , 90 ) .

The near-zone field distributions in the yz-plane are plotted in Fig. 7.10. For such a large oblique angle, Case 3 yields a visually much better result than does Case 2, which is evident by comparing Figs. 7.10(c) and 7.10(d) to Fig. 7.10(a). Fig. 7.11 shows the radiation patterns for the three cases, and it can be seen that the result of the oblique ABC agrees much better with the reference solution than does the conventional ABC. To check the convergence rate, we plot the convergence history of Cases 2 and 3 in Fig. 7.12, which shows that the FETI-DP method converges slightly faster for this example when equipped with the oblique ABC. This simulation involves 24,251,396 unknowns, 1,516,160 dual unknowns, and 10,580 corner unknowns for the setups with the rectangular truncation surface. It takes 7.8 minutes to finish the simulation of one frequency on one computing node of the CISCO Pacini cluster. To obtain the result of the reference case, the simulation using the hybrid conformal/nonconformal domain decomposition solver involves 27,653,946 unknowns and it takes 35.9 minutes to finish one frequency on the same computer cluster.

Finally, we note that the sizes of the two phased-array antennas are relatively small for practical applications. For a large size, the performance of the oblique ABC will be even more impressive because (1) the radiated wave will be more similar to a plane wave so that the oblique ABC will be more effective and (2) the reduction of the computational domain size (from a hemispherical to a rectangular surface) will be more significant.

7.3 Summary

An oblique ABC is presented in this chapter to minimize the reflection error of the ABC truncation in the FETI-DP analysis of phased array antennas. Because of the reduced computational domain, the oblique ABC is more efficient than the conventional ABC for the analysis of large finite phased-array antennas with large scan angles. The numerical results show that the oblique ABC does not slow down the convergence of the iterative solution of the global FETI-DP interface problem. Therefore, the oblique ABC provides a simple and effective mesh truncation for the finite element analysis of large finite phased-array antennas.

7.4 Figures

Conventional ABC (Perpendicular/E Pol) Conventional ABC (Parallel/H Pol) Oblique ABC (Perpendicular/E Pol) Oblique ABC (Parallel/H Pol)

Figure 7.1: Comparison of the reflection coefficients of the conventional ABC and the oblique ABC (tuned to θ_s =60o and φ_s =0o) for the perpendicular (E) and parallel (H) polarizations. F1 F2 F3 x z F6 F5 30o (a) (b)

Figure 7.2: Computational domain designed to investigate the numerical reflection by the conventional and oblique ABCs. (a) Top view. (b) Front view.

(a) (b)

Figure 7.3: Field distributions computed using the conventional ABC. (a) Re(E_y) for the perpendicular polarization. (b) Re(E_x) for the parallel polarization.

(a) (b)

Figure 7.4: Field distributions computed using the oblique ABC. (a) Re(E_y) for the perpendicular polarization. (b) Re(E_x) for the parallel polarization.

(a)

(b)

(c)

(d)

Figure 7.5: Snapshot of Re( )E for the 20 20× Vivaldi antenna array in the xz-plane at 3.0 GHz for broadside radiation. (a) Computed using the conventional ABC with a hemispherical truncation surface. (b) Same as (a), but plotted in a limited region for the purpose of comparison. (c) Computed using the conventional ABC with a rectangular truncation surface. (d) Computed using the oblique ABC with a rectangular truncation surface.

(a)

(b)

(c)

(d)

Figure 7.6: Snapshot of Re( )E for the 20 20× Vivaldi antenna array in the xz-plane at 3.0 GHz with steering angle ( ,θ φ_{s s})=(60 , 0 )o o . (a) Computed using the conventional ABC with a hemispherical truncation surface. (b) Same as (a), but plotted in a limited region for the purpose of comparison. (c) Computed using the conventional ABC with a rectangular truncation surface. (d) Computed using the oblique ABC with a rectangular truncation surface.

−90 −60 −30 0 30 60 90 −10 0 10 20 30 40 50 60 θ (degress) Amplitude Gain (dB) Reference Conventional ABC Oblique ABC (a) −90 −60 −30 0 30 60 90 −10 0 10 20 30 40 50 60 θ (degress) Amplitude Gain (dB) Reference Conventional ABC Oblique ABC (b)

Figure 7.7: Co-polarized radiation patterns for the 20 20× Vivaldi antenna array in the

xz-plane at 3.0 GHz. (a) For the main beam steered to ( ,θ φ_{s s})=(0 , 0 )o o . (b) For the main beam steered to ( ,θ φ_{s s})=(60 , 0 )o o .

0 10 20 30 40 50 60 70 80 10−4 10−3 10−2 10−1 100 101 Number of Iterations Residue Conventional ABC, (θ_s,φ_s)=(0o,0o) Oblique ABC, (θ_s,φ_s)=(0o,0o) Conventional ABC, (θ_s,φ_s)=(60o,0o) Oblique ABC, (θ_s,φ_s)=(60o,0o)

Figure 7.8: Convergence history of the iterative solution of the global interface problem for the simulation the 20 20× Vivaldi antenna array.

r ε

Figure 7.9: Schematic view of a 2 2× sub-array in the 20 20× microstrip patch antenna array.

(a)

(b)

(c)

(d)

Figure 7.10: Snapshot of Re( )E for the 20 20× cavity-backed microstrip antenna array

in the yz-plane at 300 MHz with steering angle ( , )θ φ_{s s} =(60 , 90 )o o . (a) Computed using the conventional ABC with a hemispherical truncation surface. (b) Same as (a), but plotted in a limited region for the purpose of comparison. (c) Computed using the conventional ABC with a rectangular truncation surface. (d) Computed using the oblique ABC with a rectangular truncation surface.

−900 −60 −30 0 30 60 90 10 20 30 40 50 60 θ (degrees) Amplitude Gain (dB) Reference Conventional ABC Oblique ABC

Figure 7.11: Co-polarized radiation patterns for the 20 20× cavity-backed microstrip patch antenna array in the yz-plane at 300 MHz, with the main beam steered to

o o ( , )θ φ_{s s} =(60 , 90 ). 0 10 20 30 40 50 60 70 10−4 10−3 10−2 10−1 100 Number of Iterations Residue Conventional ABC, (θ_s,φ_s)=(60o,90o) Oblique ABC, (θ_s,φ_s)=(60o,90o)

Figure 7.12: Convergence history of the iterative solution of the global interface problem for the simulation of the 20 20× cavity-backed microstrip patch antenna array.