In the first case, the approach detailed below allows researching the array that provides for scanning in a narrow angle sector involving several collateral principal maximums in the area of real angles, which are suppressed because of the narrow block direction diagram, considering the attenuator. In the second case, the influence of a feeder on the characteristics of phased array with elementwise phasing can be researched.
Fig.1 displays a generic scheme of a block phased through-type array. In the general case, it includes radiator blocks 1, controlled phase inverters used to phase the block radiators 2, attenuators that provide necessary amplitude-phase distribution in the block limits 3 and controlled phased inverters used to phase the blocks 4. To analyze this structure, we shall use methods of the theory of microwave circuits. We shall consider the radiator block and the attenuator as microwave multipoles characterized by generic dispersion matrices 
and 
correspondingly (fig. 2). Let us consider the dispersion matrix of the attenuator known. It can be found either analytically or experimentally. We shall dwell on the generic dispersion matrix of the radiator block.
In the supplying feeders of the radiators, we shall consider 
types of waves and renumber them in their number ascension order. Some of the harmonics will be propagating, and the rest will be supercritical. To find the dispersion matrix we shall consider an infinite plane phased array with semi-finite supplying feeders, phased by blocks. Fields of incident waves for such an array can be written as follows:
 |
(1) |
where m, n — indices of the radiator, s, t — indices of the block in the infinite array; m, n, s, t = (-∞,∞); 
— complex amplitude of the 
, harmonic, incident to the input of the radiator with indices р, q inside the block; М, N — number of radiators in the block on x and y axes correspondingly; 
, 
— phase shifts between adjacent blocks on the corresponding axes; 
— Kroneker symbol.
In (1), let us single out the periodical factor 
 |
(2) |
The function is periodic with periods М and N on axes x and y correspondingly, thus it can be factorized into a discrete Fourier series:
 |
(3) |
where 
— Fourier coefficients [3]. By substituting (3) in (2), we shall get
 |
(4) |
where 
, 
.
Thus excitation of a regular block array is represented as a sum of feeds of a common infinite phased array with differential phase shifts between the 
and 
radiators on the х and у axes correspondingly. According to the superposition principle, we can now get the solution of a boundary-value problem for an infinite regular block phased array as the sum of solutions for a common array, excited by the wave specter with the .
Now, we shall represent the radiator block as a multipole having 
pairs of input terminals and two pairs of output terminals. The output terminals of this multipole correspond to the Floquet harmonics of H and Е [1] types with zero indices for partial excitation k, l = 0, in the (4) factorization, which define DN of the phased array block (group of inputs А at fig. 2). Besides these harmonics, other Floquet harmonics can also be propagating. Therefore, the multipole in question is a lossy multipole generally. The input terminals of the multipole correspond to the harmonics, propagating in the supplying feeders of the block radiators (group of inputs В at fig.2).
Using the solutions to the boundary value problem for the infinite plane phased array [1] for all partial feeds in (4), we can determine the generic dispersion matrix . Indeed, let us consider that block excitation is determined by a single wave 
, incident to the input of the radiator with indices p, q inside the block. Then, if we determine the Fourier series coefficients for this excitation and add the complex amplitudes of waves, propagating in the feeders from radiators, we can easily find the coefficients of the dispersion matrix, which characterize the group of inputs В (fig. 2) of the multipole . If we know the coefficients 
of the Floquet harmonic amplitude of the Н and Е kind with zero indices for partial excitation k=0, l=0 in (4), we can find the transfer coefficients from the group of inputs В to the group of inputs А.
If we limit the block phased array to transmission only, the coefficients of the dispersion matrix found above are enough to determine the justification and radiation ratios. Characteristics of the receiving phased array can be found according to the reciprocity principle.
