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.

By using the relations given in [4], we can found the dispersion matrix of the union of two multipoles – attenuator and radiator block. If we specify the amplitude of the wave at the input of the attenuator, we can get the amplitude of the reflected wave and amplitudes of the harmonics in the Floquet channel, using which we can easily calculate the DN of the block of the phased array [1].

In [5], relations are given, which allow finding of the amplitudes of incident and reflected waves in the lines connecting the multipoles and . If we know these amplitudes, in case of the attenuator with absorbing elements, we can find power absorbed in each of the absorbing elements.

We shall note that when implementing the numeric algorithm, the method detailed above turns out to be significantly more efficient compared to the direct solution of the boundary-value problem for the block containing *M*x*N* radiators by considering an infinite phased array with the period equal to the block. Indeed, in case of waveguide radiators [1] for example, in the latter case we shall get the system of *M*x*N* operator equations, which, using the method of moments, can be reduced to the system containing linear algebraic equations. Using the latter method, we will have to invert matrix sized times. Since, when inverting an *n*-order matrix using the Gauss method, the number of operations grows as , the gain in machine time will be times.

Fig.3

Based on the mathematical model and the program for calculating characteristics of an infinite plane phased array built of round waveguides, there was developed a program for calculating characteristics of a block phased array. All calculations were performed for an array with square grid of radiators. Distances between waveguides were assumed equal to 0,7, and radius of the waveguide - to 0,34. All mentioned results correspond to the *E*-plane of waveguide radiators. Fig.3 shows the direction diagrams of the block depending on its size: *1* — for the 1x1 block; *2* — for the 2x2 block and *3* — for the 5x5 radiator block. For calculations, the input-matched attenuator was used, which provided uniform distribution inside the block [4]. For all diagrams, there are dips related to the periodicity of radiator layout and periodicity of blocks. It is known [1] that for a common infinite array, full cutoff is characteristic at the moment of appearance of difractional maximums in the area of real angles. In case of a block array, the zero is only the dip related to the periodicity of radiators ( = 25°), and the rest of the dips are nonzero. Indeed, DN of the block in an infinite block array is determined by the zero Floquet harmonics corresponding to the partial excitation with indices *k*, *l* = 0 in (4); this is why the DN of the block suffers the zero dip the same as in the common array. If waves of other partial excitation suffer full reflection, redistribution of values of amplitude coefficients takes place in the formula (4) because of interaction of waveguide radiators through the attenuator and additional dips in the DN of the block appear, which have a nonzero depth. The latter is also related to the fact that a partial excitation corresponds only to a portion of the power for a single block. Because of this, maximums of the reflection coefficient on block input do not reach the unit value unlike the case with a common infinite array. This fact is illustrated by fig. 4, a, which shows dependence of the reflection coefficients at the attenuator input for a 2x2 radiator block, the same dependence for the 5x5 radiator block is shown at fig. 4, b (curve *1*).

Fig.4

Especially interesting is the case, when during scanning not only blocks but also block radiators are phased, i.e. when the block radiators are phased in the phasing direction of the entire array. To calculate characteristics of such an array, an input-matched attenuator was used, which has phase inverters installed in its arms. The field value in the main maximum of this array is the same as in the common array with elementwise phasing. However, because of the fact that waves reflected from the aperture are re-reflected from the attenuator, passing through the phase inverters two times, side lobes appear, unlike the case with the array of semiinfinite waveguides. Besides, there is also redistribution of energy between the waveguides in effect, because of the interconnection through the attenuator. The total power of these side lobes, as the computer calculations have shown, can reach 4-6% of the incident power (in a single-beam scanning sector). Because of the fact that part of the reflected energy is re-reflected into the side lobes, such array is better matched at the attenuator input. Dependence of the reflection coefficient module on the scanning angle at the attenuator input for such array (size of the block is 5x5 reflectors) is shown at fig. 4, b (curve *2*).

It would be interesting to research characteristics of a phased array with an attenuator containing an absorbing element to provide decoupling of the attenuator outputs. When using such attenuator, a portion of power is dispersed in the absorbing element. Therefore, especially interesting are the energy characteristics of such a phased array. Fig.4, b shows dependence of power dispersed in the attenuator on the input power for an array with the 1x2 radiator block. As the plot shows, the loss of power in the attenuator is especially high at the moments of appearance of diffraction maximums in the area of real angles, when the phase difference between the reflected waves in different waveguides is at maximum.

Thus, the explained method allows for building an efficient algorithm of calculating characteristics of block phased arrays intended for scanning in a limited sector of angles and normal phased arrays with elementwise phasing considering interaction of radiators in the outside space and power wiring circuits.

The given numerical results show the possibility of considering the influence of the attenuator when designing phased antenna arrays.