In chapter 4.4 of the paper [3], the asymptotic formulas for the “partial” harmonics have been obtained in regard to cylindrical periodic structures. The expressions (7) and (8) of this paper are more general since they are oriented towards periodical structures connected with bi-dimensional-convex surfaces.
Further, expressing with the help, for example, of the Lorentz lemma, the unknown coefficients 
in the form of quadratures of the current in the radiator in the unit cell and using the boundary conditions of electrodynamics on the IS and on the surface of the radiator, we can obtain the system of operator equations, solution to which allows us determine the coefficients and, therefore, all the characteristics of the convex array. This method was explored in detail in the paper [2].
On the basis of the expressions obtained in this paper, the algorithm and programs have been developed, which allows calculation of the characteristics of CWSR in CA.
The influence of the shape of the CA on the characteristics of an azimuth- and axial- oriented CWSR in CA is illustrated by the curves shown in Fig.1…6. We will give the form of cross–section by the means of the canonic ellipse equation:
 |
and examine the characteristics for the radiator located in the period with the coordinates x=y=0, z=b. The CWSR are excited in the middle by the 
–generators, have length l=0.05
( is the wavelength at lower frequency 
) and located on the layer of magneto-dielectric having thickness of 0.016 and permeability 
=2, 
=10. The width of the radiator is 0.015, 
=0.224, the geometry of the array is a square grid with period 0.05.
The pattern of the radiator of the elliptical array having a=5
, dependent on b, is shown in Fig.1, and with b=5, dependent a, is shown in Fig.2. The results of calculations correspond to the physical meaning: when the radius of the equivalent circular tangent to the point of position of the radiator in question increase, the pattern oscillations decrease until the scan angle 
. The same is valid for the radiation level in the “shadow” region at (
).
The absolute values of reflection coefficients (RC) of the azimuth strip radiators of elliptic array are shown in Fig.3 and in Fig.4 within the frequency band. The parameters of the array are: a=5, b=20 (Fig.3) and a=20, b=5 (Fig.4). The rest of the sizes are unchanged.
The behavior of absolute values of RC of axial SR located on a magneto-dielectric layer of the elliptic array, with the same parameters as in the previous case, in the frequency bound, are shown in Fig.5 and Fig.6.
Fig. 1 Directivity pattern of an azimuth wide-band radiator on the elliptic cylinder, which has a=5 (1 − b=10; 2 − b=20; 3 is a plane array)
Fig. 2 Directivity pattern of an azimuth broadband radiator on the elliptical cylinder, which has b=5 (1 − a=10; 2 − a=20; 3 is a plane array)
Fig. 3 The behavior of the reflection coefficient modulo of an azimuth broadband radiator on the elliptical cylinder (a=5, b=20) in the frequency band (1 − f=; 2 − f=1.5; 3 − f=2; 4 is the radiator in a plane array with f=)
Fig. 4 The behavior of the amplitude of the reflection coefficient of an azimuth broadband radiator on the elliptical cylinder (a=20,
b=5) in the frequency band (1 − f=; 2 − f=1.5; 3 − f=2; 4 is the radiator in a plane array where f=)
Fig. 5 The behavior of the of the reflection coefficient modulo of an axial broadband radiator on the elliptical cylinder (a=5,
b=20) in the frequency band (1 − f=; 2 − f=1.5; 3 − f=2; 4 is the radiator in a plane array with f=)
Fig. 6 The behavior of the reflection coefficient modulo of an axial broadband radiator on the elliptical cylinder (a=20, b=5) in the frequency band (1 − f=; 2 − f=1.5; 3 − f=2; 4 is the radiator in a plane array with f=)
Based on the results obtained in this paper and the numerical experiment we can conclude:
1. The mathematical models of CWSR taking into account the interaction between adjacent radiators in a CA on a convex surface of double curvature have been constructed, on the condition that the array is bi-dimensionally-periodic, infinite along the generating line and has a large slowly changing curvature radius. The CA can have multi-layer coating and the screen can have losses.
2. The principal difference between conformal and plane arrays is the behavior of the reflection coefficient in the scan angle region near ±90°. While the plane array has |Г|=1, the conformal array has |Г|<0.75 at scan angle 90° within the three-repeated frequency band. This fact should be accounted for when examining the problems related to e.g. decoupling of CA.
3. While designing the wide-band CA of CWSR we should keep in mind that curving of the antenna array aperture until the curvature radii 
does not lead to significant changes in the internal and external characteristics of radiators of the system, as compared to the plane aperture, to the angles 
. In this sector, good matching (voltage standing-wave ratio 
) in the twofold frequency band for the two main planes of scanning can be provided.
