In this article, based on the example of antenna array consisting of radiating strips in a two dimensional periodical antenna array located at the distance of above an impedance structure (fig. 1), boundary effects caused by finite size of the aperture of the antenna array are analyzed. As an exterior, the wave propagating in the feeders uniform-linearly with linear phase incursion.

Fig.1

To pass on from the model of infinite antenna array [1] to the model of finite antenna array we will use the following method, which allows determining characteristics of the finite antenna array in an infinite screen with impedance (fig.1) based on the solution of the boundary value problem of electrodynamics for an infinite antenna array [2].

Let’s assume that the radiators do not reveal themselves physically in the meaning of changing the characteristics of the antenna array if the currents on them are equal to zero [3]. In this case, the voltages set on their inputs form the boundary wave [4]. Let’s consider that each radiator of the antenna array consists of elements (for example, bi-polarized antenna array, multiple-frequency antenna array etc.). Then, the following method is used to analyze the resulting antenna array:

— to radiators with numbers we connect infinite resistors (idling) through virtual feeders with wave impedance *W*;

— radiators with numbers can be connected either to matched load or generators through feeders with the same wave impedance *W*.

Here *N* — is the finite set of radiator numbers which limits the fragment of the infinite periodical antenna array inside which the radiators of the finite array which are of interest are located. We will use the following notation:

— – coefficient of interdependence between the *l*−th element of the *m*−th radiators and the *s*−th element of the *n*−th radiator;

— – amplitude of the wave of fundamental type propagating in the direction of input of the *s*−th element of the *n*−th radiator;

— – amplitude of the wave of fundamental type propagating in the direction from input of the *l*−th element of the *m*−th radiator.

Then the following equation is true, written for simplicity as applicable for a one-dimensional antenna array (a line of radiators):

(1) |

We write amplitudes of the incident field as follows:

(2) |

where — are reflection coefficients of the load enabled in the *s*−th excited element of the *n*−th radiator, — the same for non−excited elements, — is the set of numbers of radiators where there is at least one excited element.

Let (i.e. there is no dependency on the radiator number). Then, substituting (2) into (1), applying Fourier transform and convolution theorem and passing on to the matrix form, it is easy to show that amplitudes of the wave of fundamental type propagating in the direction from radiator inputs satisfy the following equation of second kind:

(3) |

where the following notation is used for matrixes

*S* — is the square matrix of dispersion between radiator elements, , , — column vector , *Е* — unitary matrix, . Dimensions of all matrixes are determined by the *M* value.

The matrix operator in the equation (3) exists only if for all differential phase shifts in interval . Therefore, when (idling mode in virtual feeders) in case of antenna arrays with dielectric coating the operator exists only if there are losses in layers.

For the norm of the operator in discrete space using the Cauchy−Bunyakovsky inequality [5] the following estimate can be found:

where — are elements of the square matrix .

In case when *q* < 1 ((the operator *K* is a compression operator) to solve the equation (3) an iterative procedure can be used. In this case the following estimate is true for convergence speed [6]:

where — is vector of amplitudes on the *k*-th iteration step. As a zero approximation, the solution for the infinite antenna array can be used.

For antenna arrays with small period *Т* (, — wavelength) the convergence speed becomes unsatisfactory. In this case, the problem must be reformulated and the equation written not in relation of wave amplitudes in feeders but amplitudes of the total boundary wave [4]. If the Polozhy method [7], we get the following equation instead of (3):

where — is the matrix of the sought amplitudes of boundary wave,

— is the second iterated matrix core (), — Kronecker symbol.

After finding the sought amplitudes are determined from the equation , and the directional diagram of the finite array is determined from the formula:

where — row vector {}, and — is the directional diagram of the *s*−th element of the zero radiator in the infinite array, found using the assumption that all other elements in all radiators are loaded with matched load.

It is not hard to generalize the received result for the two-dimensional case.

Fig. 2 Directional diagram of the radiating strip of the finite array in an infinite screen in *H*−plane depending on its position in the *OX* axis (1 — = = 0; 2 — = 4, = 0; 3 — = 12, = 0).

Fig. 3 Directional diagram of the radiating strip of the finite array in an infinite screen in *H*−plane depending on its position in the *OY* axis (1 — = 0, = 4; 2 — = 0, = 12; 3 — radiator in the infinite array).

Fig. 4 Directional diagram of the radiating strip of the finite array in an infinite screen in *E*−plane depending on its position in the *OX* axis (1 — = = 0, 2 — = 4, = 0, 3 — = 12, = 0).

Fig. 5 Directional diagram of the radiating strip of the finite array in an infinite screen in *E*−plane depending on its position in the *OY* axis (1 — = 0, = 4; 2 — = 0, = 12; 3 — radiator in the infinite array).

Based on the obtained formulas (where *Г* = 0) the dependency of the direction diagram of radiating strip with length *l* = 0,14, width = 0,03 on the magnitodielectric layer with thickness *t* = 0,032 permittivity = 10, = 2 (= j1060.49, , ) on its position in the array containing 1681 radiators located in the nodes of the square grid with period *T* = 0,14 (array has size 6 x6) was calculated. The radiators are matched in the direction of normal to antenna array () provided the array is infinite. The model corresponds to the finite array in the infinite screen coated with a layer of magnitodielectric with permittivity , and thickness *t*. On the fig.2 there are directional diagrams of radiators in the *H*-plane when = 0 and = 0 (curve 1, — number of radiators in the *OX* axis, — in the *OY* axis), = 4 (curve 2), = 12 (curve 3). Since radiators are positioned symmetrically on the *OY* axis, this symmetry is retained in the diagrams. In the same plane but for other positions of the radiators the fig.3 displays directional diagram of the radiating strip for = 0 и = 4 (curve 1), = 12 (curve 2). The curve 3 displays the directional diagram of the radiator in the infinite array. Directional diagrams of radiating strips in the *E*−plane are illustrated by fig.4 and 5. All diagrams of radiators of the finite antenna array are normalized to the directional diagram of the radiator with number = = 0. It follows that:

— the directional diagram is most distorted in the *H*−plane since interdependence between radiating strips is less than in the *E*−plane;

— in the *E*−plane the directional diagram stops being distorted starting approximately from the fifth radiator from the edge of the array, in *H*−plane — from the thirteenth.

Fig. 6 Behavior of the module of the reflection coefficient of the radiating strip of the finite array in the infinite screen in *E*−plane depending on its position in the *OX* axis (1 — , 2 — , the dotted line corresponds to radiating strip in the infinite antenna array when ).

Fig. 7 Behavior of the module of the reflection coefficient of the radiating strip of the finite array in the infinite screen in *E*−plane depending on its position in the *OY* axis (1 — , 2 — , the dotted line corresponds to radiating strip in the infinite antenna array when ).

Fig. 8 Behavior of the module of the reflection coefficient of the radiating strip of the finite array in the infinite screen in *H*−plane depending on its position in the *OX* axis (1 — , 2 — , the dotted line corresponds to radiating strip in the infinite antenna array when ).

Fig. 9 Behavior of the module of the reflection coefficient of the radiating strip of the finite array in the infinite screen in *E*−plane depending on phasing angles (1 — = 0, 2 — = 4, 3 — = 12, 4 — = 20; = 0; the curve 5 corresponds to radiating strip in the infinite antenna array).

Let’s consider the characteristics of the given antenna array when exciting the entire array uniformly with linear phase incursion. The radiators like in the previous case are matched in the infinite array. The fig.6 presents curves which display the dependency of the reflection coefficient module on the number of radiator (along the radiating strip axis) when = 0 in *E*−plane for array phasing angles (curve 1) and (curve 2). The same dependencies on the number of radiator (across the radiating strip axis) when = 0 are displayed at fig.7.

Change of the reflection coefficient module for H-plane depending on the radiating strip number when phasing the array in the direction is illustrated by fig.8. The dotted line displays the reflection coefficient of the radiator in the infinite array for angle (for , ). In all cases, the module of the reflection coefficient is greater for boundary radiators and have oscillating character caused by the interference of non−excited currents and total boundary wave [4].

Analysis of the curves given on the fig.6…8 confirms the results received when analyzing the directional diagrams (fig.2…5). Behavior of the reflection coefficient of the antenna array in sector of angles is displayed at fig.9, where curve 1 corresponds to the radiator = 0, curve 2 — radiator = 4, curve 3 — = 12, curve 4 — = 20 and = 0, the dotted line corresponds to the infinite array. The charts show that in the finite array, decreasing of sector of angles where takes place. For the infinite array this sector is equal to ±60°, for radiators of the finite array with numbers = 0, 4 — ±55°, for radiator = 12 — −50°…+60°, and for boundary radiator (= 20) the sector of angles doesn’t exist .

Fig. 10 Envelopes of the maximums of directional diagrams of the fully excited array when phasing in *E*−plane (curve 1) and in *H*−plane (curve 2).

As the fig.10 shows when exciting the entire finite array, strong interdependence causes the maximums of directional diagrams to form a smooth curve in the sector of angles to ±60° preserving the average width of the diagram corresponding to the width of the directional diagram of the radiator in the infinite array.

__Conclusion__

— for multiunit radiators of finite antenna arrays, formulas received in [2] have been generalized. For calculating amplitudes of the total boundary wave, an equation of second kind is constructed, norm of the operator of which is always less than one;

— after experimental research of the diagram of the radiator of antenna array with strong connection the methods of measuring related to exciting of one element of the array and let the other elements loaded on matched loads cannot be used. For such an array, measures of maximum of diagrams must of fully excited antenna array with different phasing angles are required;

— the strongest distortions of reflection coefficient in a finite antenna array are for radiators located in the scanning plane. In this case, deviation of the reflection coefficient from the reflection coefficient of the radiator of the infinite array does not exceed 25% for radiators of the central part of the antenna array approximately starting from the distance of (1…1.15) from the edge of the array (the radiator is matched in the infinite array).