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## Use of radiant strips in the array antennas

Published: 08/25/2003
Original: Radio engineering and electronics (Moscow), 1992, №5, p.p.834...840
© 2003, EDS–Soft,  http://www.eldys.org,   E-mail: publications@eldys.org

In the recent years the number of works on the analysis of properties of resonant microstrip radiators in the array antennas (AA) has increased. If using such radiators the service frequency band width (SFB) is generally limited up to 35% at voltage standing wave ratio =1,5 .

This work shows that the use of microstrip radiators with small electrical sizes placed over the impedance surface in the AAs is one of the possibilities to provide simultaneously a wide SFB and scanning angle domain up to ±60° within the principal planes.  Fig.1. One period of AA made of SCs in a dielectric layer on the impedance specified, 1 − Floquet’s channel, 2 − radiator, 3 − impedance surface

Let’s make a mathematical model of a flat periodic AA made of strip conductors (SC) placed parallel to the surface for which the surface impedance , (Fig.1). The conductors may be in one or several dielectric layers. Within the above model the array is considered to be periodically completed by the radiators to form an infinite array, and the SCs are considered to be infinitely thin (which is true at thickness of the real SCs meeting the condition , where stands for skin-layer thickness and stands for wave-length), having surface impedance , . Assuming the SC width to be much less than their length and the wave-length we only consider the electric current component which coincides with the direction of the conductor’s longitudinal axis.

Let AA be excited by the primary , . The secondary (diffractional) electromagnetic field is designated as , . Then the electrodynamics boundary problem regarding AA over the impedance structure can be formulated as follows. Find the secondary electromagnetic field satisfying:

— Maxwell’s heterogeneous equations;
— boundary conditions at the radiators (1)
where = — normal vector to the SC surface;

— the condition of absence of secondary waves coming from infinity;
— the condition at the arris of each SC.

Let the primary field drive the radiators with linear phase progression at equal amplitudes. In this case Floquet theorem can be applied.

We introduce two planes parallel to the aperture AA and by analogy with [2, p.317] designate as the “reflection” factor of the i-Floquet’s harmonics from the lower plane, and as the one from the upper plane (i — is a generalized index of Floquet’s harmonics , Fig.1). These factors depend on the distance between the planes and their positions as respects to the AA aperture (phase origin). The space V between the introduced planes contains the SC and is homogeneous. The and factors allow to abstract from the unessential properties of the space located outside V and can be either specified (e.g. via surface impedance ), or defined by the solution of another electrodynamic task.

The tangent electric and magnetic fields over the radiators can be recorded as (2)

where is an amplitude of the i-Floquet’s harmonics over the radiator (Fig. 1), and the electric and magnetic fields of the subwaves relate to the Floquet’s vector harmonics as defined per . Similar expressions for the fields under the radiators look like (3)

where is an amplitude of the i-Floquet’s harmonics under the radiator.

Concerning the volume limited by the closed surface and containing the electric current the Lorentz lemma can be recorded in the form of integral , previously supposing the electric and magnetic currents = , = = =0 within this volume. Concerning electromagnetic fields , and , последовательно consecutively consider that and are defined by the relations (2), and , equal respectively  and are defined by the relations (3), and , equal respectively The «-k» index here corresponds to the flat wave propagating at the angles , ( , - are angles of propagation with the “k”-index)

Using the conditions of the field quasi-periodicity and orthogonal property of the subwaves in the formula of (34) from the work , we record the forms for the target factors: for (4) for Here the z value refers to the point of observation and — to the source, S — SC surface, — wave admittance of the i-Floquet’s harmonics. The formulas similar to (2)-(4) have been originally concluded in the works [2,6]. Now, using (2) and (3) and the boundary condition (1), the second genus integral equation can be obtained as regards to : (5)
To solve the equation Galerkin’s method can be applied . Accordingly to this method the target current can be recorded as series: (6)

where is a unitary vector directed lengthwise the SC axis, stands for a definable expansion coefficients, , — stands for an orthogonal local system of coordinates on the SC surface, and N stands for a number of accountable basis function.

Function is introduced to describe the current behavior pattern at the arris of an infinitely thin impedance body. The function’s specific mode depends on the impedance value of the radiator surface. The full orthonormal system of functions: (7)

is used as a basis . Here angles , define the phasing direction and L stands for the length of the radiant strip.

After the projection of equation (5) on the function system (7) the coefficients can be found, and the formula (6) defines the current . This allows to define all SC characteristics in the AA such as directional diagram (DD) and , polarization characteristics, reflection factor (RF) Г, input resistance (IR) . In particular DD of the (m,n)-radiator can be found by use of certain formula (8)

where — stands for the aperture area of AA, — stands for the radius-vector of a point on the AA surface, — stands for the radius-vector of a point of observation, , , , — electric and magnetic fields over the AA surface under excitation of the (m,n)-radiator and under the condition that all other radiators are loaded with the matched load: (9)

where is a “transmission” ratio of the i-Floquet’s harmonics from V into the homogeneous range over the array, and the coefficient is defined by the from (4). In the relation (9) and — mean differential phase shifts.

After having substituted (9) into (8) and performed not-complicated transformations we get simple formulas for the DD: where index “100” conforms to the zero vector H-Floquet’s harmonics; index “200” — conforms to the zero vector E-Floquet’s harmonics ; coefficients can be defined from the relation (4) assuming i =p00; — are transmission coefficients of the zero Floquet’s harmonics over the interface “magnetodielectric — free space”.

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