Let’s consider in a spherical coordinate system with orts 
, 
, 
a field of radiation of a spherical antenna with idealized continuous distribution of radiators. Amplitude−phase distribution of the surface density of the exciting current providing for CDD equal to that of the equivalent cophasal flat aperture with collinear currents, is determined by the following expression:
 |
(1) |
where 
, 
— components of the magnetic current on an ideally conducting sphere surface, 
; 
— wave number; R — radius of the spherical antenna.
Electric field, excited by sources (1) will be found as a superposition of spherical waves of electric and magnetic kind
 |
(2) |
where 
— electric vectors of proper spherical waves of electric and magnetic kind; 
— coefficients of excitement of proper spherical waves.
Based on the orthogonality property of proper spherical waves, using the Lorenz lemma, excitement coefficients can be determined
 |
(3) |
where 
; 
— coefficients, determined by boundary conditions on the sphere surfaces; 
— norm of the mn–th wave.
Integrating (3) and substituting the result into (2), after transformations for far−field region (
) we receive the expression for DD of the antenna:
 |
(4) |
where 
.
Value of CDD in the direction of phasing of the antenna (
)
 |
(5) |
where 
— partial power, transmitted by the 1n−th wave of the electric or magnetic kind.
Using formulas (4), (5) calculations of DD and CDD were conducted with optimal from the point of view of optimal physics amplitude−phase distribution of the magnetic current with wide variation of the relative antenna size for cases of sphere excitement limited by 
, 
. The calculations have been performed with minimally possible relative error of machine number representation 
.
With the given amplitude−phase distribution of the magnetic current (1), the maximum value of CDD is reachable when exciting half of the surface of the spherical antenna (
). Other methods of exciting, which allow to get maximum CDD of non-superdirective antenna have significant oscillations of amplitude and phase, which makes it difficult to use them. For the case of exciting a half−sphere with current (1), DD were calculated using an approximate method.
Fig.1 DD of spherical antenna, kR=20: 
— strict computation; 
— approximation; 
— DD of the flat aperture.
Fig. 1 presents DD of the spherical antenna in case the half-sphere is excited computed using strict and approximate formulas. Also, for the sake of comparison, DD of the flat uniformly excited aperture is shown. It can be seen that the approximate method allows calculating of the DD of the antenna in the region of side radiation more precisely in comparison with the flat aperture diagram. The high precision provided by the approximate method can be explained by the fact that when it is used in the equivalent aperture, existence of the normal component of the current is accounted for unlike in the flat aperture. Results of the computation confirm that using of the approximate method of calculating of DD of large-sized spherical antenna is possible given the fact that the level of far-field side radiation where the approximate computation is more difficult, is small and decreases dramatically as kR grows.
Fig.2 Average level of side lobes of spherical antenna: 1 — level of side lobes of sphere; 2 — level of side lobes of half-sphere; 
= 0°; — 
= 80°; — = 90°; — = 180°.
Fig. 2 shows dependency between the level of side and rear radiation and the relative size of the spherical antenna when exciting the half−sphere and full sphere. Calculations of CDD using strict expressions (fig. 3) show that even with relatively small sizes of the antenna (
) deviation of CDD of the antenna from that of the flat aperture is less than 5%. The 
curve is asymptotic and converges to 1 as kR increases. In the area of small values of kR (when 
) converges to infinity because when the spherical radiation is infinitely small its CDD is equal to CDD of the dipole radiator length of which is equal to the radius of the sphere.
Fig.3 Dependency of between CDD and antenna size: 1 — CDD of sphere; 2 — CDD of half−sphere.
Results of calculations of CDD given for the completely excited sphere are shown at fig. 3. It can be seen that in this case, normalized value of CDD is asymptotically converging to the maximum value determined in the approximation of physical optics and equal to with the given amplitude−phase distribution of magnetic current.
Conclusion
— as a result of computations using the found strict formulas, it is shown that CDD of spherical antenna with optimal excitement when increasing the antenna radius asymptotically converges to CDD of the equivalent flat cophasal aperture with uniform amplitude−phase distribution of collinear currents.
— the results of the approximate calculation are compared with the found strict solution. It is noted that the DD computed using approximate formulas agree with the DD computed using strict formulas. It is shown that the approximation of the DD of spherical antenna by diagram of the flat aperture provides much less precision.
