Elementary slot and microstrip patch antennas are commonly used singly or as array elements for conformal application. However, these radiators provide ideal performance only when they use plane conducting surfaces. Ideal theory can be used when the radii of curvature of the surfaces are large compared to the operating wavelength. In other cases both the impedance and radiation characteristics may be affected significantly.

The radiation patterns of slot antennas can be significantly altered by the curved mounting surface. Pathak and Kouy — oumjian (9) give a convenient extension of the geometrical theory of diffraction (GTD) for apertures in curved surfaces.

Figure 1. Patterns of a thin axial slot on a perfectly conducting cylinder. (After Ref. 9.) |

Figure 1, taken from Ref. 9, shows the patterns of an axial slot element on perfectly conducting circular cylinders of various radii; the results indicate the accuracy of the approximate theory. The effects of the cylinder radius on the patterns shown in Fig. 1 should be noticed. A similar slot on a flat ground plane would have a constant pattern from ф = 0 to 180°. Radiation patterns of slots on a variety of other generalized surfaces are discussed in Refs. 10-12. Mailloux (13) summarizes some of the results of Pathak and Kouyoumjian (9) shown in Fig. 2, which gives the radiated power pattern in the upper half plane (в < 90°) for an infinitesimal slot in a cylinder of radius a. The angular extent of the transition zone is on the order of (koa)~1/3 on each side of the shadow boundary, k0 = 2n/A0 being the propagation constant in free space. The results indicate that above the transition zone (i. e., the illuminated zone) the circumferentially polarized radiation is nearly constant but the axially polarized radiation has a cos в pattern. Compared with the field strength in the в = 0° direction, the field strengths in the в = 90° area are found to be about 0.7 and 0.4(2/k0a)1/3 for circumferential and axial polarizations, respectively. It should be noted that in the case of flat surface the field reduces to zero in the в = 90° area.

Figure 2. Approximate pattern of a thin slot on a conducting cylinder of radius a; k0 is the free-space propagation constant. (After Ref. 6.) |

Slots and slot arrays on metallic cones are found advantageous to use for missile or missilelike bodies. For efficient design of such arrays, the self — and mutual admittances must be taken into account. Theoretical and experimental investigation of slot antennas on metallic cones are discussed in Ref. 14, where the effects of scattering from a sharp tip on the mutual admittances have been investigated for pairs of circumferential and radial slots on a semi-infinite metallic cone. The base of the conical model used in the experimental study was terminated in a spherical cap to minimize scattering from the finite length of the apparatus. The two slot antennas configurations considered are shown in Fig. 3. Self — and mutual admittance expressions for pairs of slots shown in Fig. 3 have been derived by Golden, Stewart, and Pridmore-Brown (14), and the results have been confirmed by measurements. These admittance results can be immediately applied to determine the aperture voltages required for the analysis of W-element slots on cones.

In Ref. 14 the circumferential slot results illustrate interference effects between the direct coupling from slot-to-slot via the geodesic path over the conical surface and the tip back scattering. For the radial slot configuration, the results indicate negligible tip scattering effects. Golden and Stewart (15) have found that the current distribution near a slot for a sharp cone can be approximated by the distribution on an equivalent cylinder if scattering from the apex (on tip) is small. Thus, the mutual admittance between two slots can be approximately calculated by using a cylindrical model with the same local radii of curvature as the cone, provided the wave scattering from either the tip or the base region of the vehicle is negligible. The slotted cone and equivalent cylinder are shown in Fig. 4, which reveals that the cylinder has a radius equal to the radius of the circular cross section of the cone midway between the two slots antennas. For small-angle cones (в0 ~ 180°), the radial separation of the slots on the cone can be equated to the axial separation of the slots on the equivalent cylinder.

Frequency = 9.0 GHz — Theory 0 Measurements, z0 = 3.81 cm 30 60 90 120 150 180 210 Ф0, deg Figure 6. Mutual coupling for axial slots on cylinder, p0 = 5.057 cm. (After Ref. 14.) |

cient; in the case of radial slots (azimuthal electric fields) there is no radial component of the magnetic field in the far field of tip and therefore no contribution to the mutual admittance. More detailed results and discussions are given in Refs. 14 and 15.

Frequency = 9.0 GHz — Theory □ Measurements, z0=10.16 cm A Measurements, z0=7.62 cm 0 Measurements, z0=5.08 cm |

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Mutual coupling (|S12| parameter) results versus azimuthal separation for two circumferential and axial slots on a cylinder are shown in Figs. 5 and 6, respectively. The mutual coupling between two radial slots on a 12 • 2° half-angle cone is shown in Fig. 7 as a function of frequencies. Figure 8 shows the mutual coupling versus frequencies for circumferential slots on a 12 • 2° (half-angle) cone. The results illustrate the interference effects between the direct and tip scattered components. The mutual coupling between circumferential slots on an 11° (half-angle) cone is shown in Fig. 9. Using the results given in Ref. 14, it may be concluded that for the case of circumferential slots (radial electric fields) the tip scattered portion of the azimuthal magnetic field at the slot aperture can be expressed in terms of an appropriate diffraction coeffi- |

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Microstrips on Curved Surfaces Microstrip or patch is a popular low-profile, flush-mounted antenna developed in the 1970s. Detailed descriptions of the research and development of microstrip antennas can be found in Refs. 16 and 17. Such antennas generally use a metallic patch on a dielectric substrate backed by a planar ground plane, and they are excited either by a strip line or a coaxial line. The shape of the patch can be rectangular, circular, or some other shape, in general, of which the first two are the most popular. We shall mostly describe the basic rectangular patch antenna whose one dimension is A/2 at the operating wavelength in the substance and the other dimension is slightly less than the former. Ideally, such antennas produce similar E — and H-plane patterns that have maxima in the broadside direction; generally, the polarization is linear and parallel to the patch plane but they can be designed to produce circular polarization also. For conformal applications, it is necessary to take into account the effects of nonplanar surfaces on the performance of such antennas. Cylindrical-Rectangular Patch Antenna The geometry of a rectangular microstrip patch antenna mounted on a conducting cylinder is shown in Fig. 10. Reso- |

— Cylinder calculations о Cone measurements r1 = Г2 = 45.53 cm ф0 = 60.8 deg 0c = 12.2 deg |

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Figure 5. Mutual coupling for circumferential slots on cylinder, p0 5.057 cm. (After Ref. 14.)

Figure 7. Mutual coupling for radial slots versus frequency, p0 9.622 cm. (After Ref. 14.)

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r1 = r2 = 45.53 cm. r3 = 92.39 cm ~“ Cylinder calculations о Cone measurements ■■■■ Cylinder calculations |

ф0 = 60.8 deg 12.2 deg |

Figure 8. Mutual coupling for circumferential slots versus frequency, p0 = 9.622 cm. (After Ref. 14.) |

Figure 10. Geometry of a cylindrical-rectangular microstrip patch antenna. |

nant frequencies and radiation characteristics of this antenna are discussed in Refs. 18 and 19. For thin substrate satisfying h < a, Luk, Lee, and Dahele (19) give the following expression for the (transverse magnetic mode with respect to p) TMp resonant frequencies for the antenna

I mn — q,—

2V

where c is the velocity of light in free space, er is the dielectric constant of the substrate, and m, n = 0, 1, 2, . . ., but m = n Ф 0. Equation (1) indicates that if the dimensions of the patch—that is, 2(a + h)e and 2b—are fixed, the resonant frequencies of the TMp modes are not affected by the curvature of the thin substrate. However, to account for fringing fields, effective values of the dimensions are to be used in Eq. (11), as mentioned by Carver and Mink (20). Luk, Lee, and Dahele (19) discuss the E — and H-plane radiation patterns produced by the antenna using er = 1.06, er = 2.32, and different values a. It is found that the patterns are not sensitive to the thickness. For a curved patch, there is significant radiation in the lower hemisphere for the TM01 mode; the deviation from the flat patch results increases for larger value of er. Compared to

the TM10 mode, there is less radiation in the lower hemisphere for the TM01 mode. Wong and Ke (21) describe the design of this antenna for circular polarization by using the TM01 and TM10 modes excited by a single coaxial feed located on a diagonal line and the operating frequency chosen between the two lowest frequencies f01 and f10 given by Eq. (1).

Kashiwa, Onishi, and Fukai (22) describe the application of a strip-line-fed cylindrically curved rectangular patch antenna as a small, portable antenna for mobile communication. It has been found that near the resonant frequency the real part of the input impedance approaches 50 П. The radiation patterns near the broadside direction are found to be similar to those of the equivalent planar antenna; however, significant differences have been found in large off-broadside directions.

Radiation patterns of a cavity-backed microstrip patch antenna on a cylindrical body of arbitrary cross section have been investigated theoretically and experimentally by Jin, Berrie, Kipp, and Lee (23). The finite-element method has been used to characterize the microstrip patch antennas, and then the reciprocity theorem is applied in conjunction with a two-dimensional method of moments to calculate the radiated field. The method can be extended to characterize the radiation patterns of conformal microstrip patch antennas on general three-dimensional bodies.

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Г1 = 27.03 cm r2 = 25.88 cm r3 = 77.47 cm ф0 = 80 deg ec = 11 deg |

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Cone calculations Cylinder calculations Cone measurements |