Use of radiant strips in the array antennas

Published: 08/25/2003
Original: Radio engineering and electronics (Moscow), 1992, №5, p.p.834...840
© 1992, V. I. Chulkov
© 2003, EDS–Soft,,   E-mail:

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 [1].

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

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 [3], 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


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 [4]. Similar expressions for the fields under the radiators look like


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 [5], 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 [4], we record the forms for the target factors:

for (4)

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 :

To solve the equation Galerkin’s method can be applied [3]. Accordingly to this method the target current can be recorded as series:

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:


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


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:


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 [3]; 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”.

It must be noted that the formulas found connect two operating modes of the AA: coefficients are defined under the whole array feed, and DD can be defined under the excitation of one radiator. Hereinafter both the array phasing angle and current angle DD will be designated as . Furthermore one cannot forget that the coefficients are the functions of the target current and therefore they implicitly depend on all accountable Floquet’s harmonics; that is why DD depends on them, too. The presence of only one zero Floquet’s harmonics regarding the DD is being represented by the fact that in the domain of visible angles under condition 0,5 (Т — array spacing) only this one harmonics is going to be the fast harmonics. When >0,5 there are two (and more) Floquet’s harmonics the phase velocities of which exceed the light speed. At the same time each of these harmonics can be used to describe the same diagram coinciding with the diagram of zero harmonics.

Let SC with the length L be in free space (=0) forming a part of AA at the distance from the surface on which complex surface impedance (Fig.1) is specified. For simplicity let’s assume that the field’s phases are counted off from this surface. Let the radiators have small electrical sizes () and besides be located close to the impedance (). Having the period AA Т=L and under the condition in order to show the principal properties of such radiator let’s substitute the real lumped array feed by the uniformly distributed feed, assuming that one electric terminal is connected to the SC at х=-T/2, and the second one — at х=Т/2 and between these electrical terminals voltage is applied. In addition the real part of IR in the visible angle domain and in the loss-free medium represents radiation resistance, whereas IR in the remaining angle domain is purely imaginary. Let’s limit ourselves to the zero Floquet’s harmonics in the representation of all fields and to the electric current uniform by amplitude. In addition the following formula:


for IR can be recorded, where , is a period area AA,

is the width of SC.

The analysis of the formula (10) shows, that at the real part of IR does not depend on the frequency and , and the purely imaginary one is negligible.

When =0 and IR in the SFB is purely active and does not depend on the frequency either, whereas in case of full radiator matching at the angle ==0 in the principle planes the following equations:

— E-plane

— H-plane

are valid.

Thus, having realized the surface impedance with the properties specified above (), a broadband and wide-angle AA made of the proccessable practically feasible small-size microstrip radiators.

The simplest solution would be to place the SC over a perfect conductive screen at altitude =0,25, ( — wave length corresponding to the middle of the SFB). Here =0, =-1, =1, and in the form (10) it is to be assumed that , , =0:

Fig.2. DD (firm lines) and RF SC module (hatched lines) in the H-planes within the AA over the screen, 1 − f=, 2 − f=1,175, 3 − f=1,35=, 4 − f=1,525, 5 − f=1,7

Fig.2 represents the computer calculated DD and the RF-module in the H-plane for SC at L=0,03, =0,015 ( — wave length in the lower SFB frequency). The array spacing Т=0,0З. Three harmonics type (7) in the current resolution and 242 terms of sum in the field resolution have been taken into account, the radiator is adjusted at the frequency f= (curve 3). In the frequency band from to 1,7 value changes in the range from 2,78 to -1,78. The radiator has in the frequency band with the overlap p=1,44 within the angle domain ?50° and is excited by a -generator, which is already included into the middle SC, with , where stands for the incident wave amplitude.

A layer of magnetodielectric on the screen can also be an impedance structure with an independent on the transverse coordinates impedance . If the thickness t of this layer meets the inequation


where , are relative dielectric and magnetic conductivities of the layer, then its surface impedance value

at (which, together with condition (11), conforms to ) meets the requirement necessary to provide effective operation of the radiant in AA. In order to avoid surface wave generation in the magnetic-dielectric layer in the visible angle domain, the AA period is subject to condition

Fig.3. DD (firm lines) and RF SC module (hatched lines) in the E-planes within the AA over the screen, 1 − f=, 2 − f=1,5, 3 − f=2, 4 − f=2,5, 5 − f=3

Fig.3 represents under the same approximations of the current, field and excitement, the calculated characteristics in the E-plane of SC at L=0,05 in the magnetodielectric layer with the thickness t==0,016 and under =10, =2 (within the frequency band of approximately up to 75 МHz such magnetic conductivity is typical for the m-metal rubber on the basis of caoutchouc SKI-3, containing by weight 90 percent of the ferrit powder 600 НН [7]). As work [7] shows this frequency band has practically no magnetic losses, and the electric ones do not exceed 0,2. The array spacing Т=0,05, the radiator matching is performed on the frequency (curve 3). Within the frequency band from up to 3 the value only changes from 1,08 up to 9,9. Here the better radiator matching in the frequency band and angle domain (р=2 and angle domain ±60°) can be observed than in the previous case (Fig.2) where, as numerical experiment has proven, the radiator’s efficiency is at least 0,92 in the working angle domain and wave-length band.

In the aggregate with a wider SFB and angle domain the use of a magnetodielectric allows to essen-tially (in the case here concerned next high order) decrease the altitude of radiator’s location over the screen. In case of even more increase of relative permeability this altitude tends to zero, and the fre-quency engagement factor is something like . From the physical point of view magnetodielectric layer under can be regarded as approach to the magnetic screen (on the layer surface ). Mirror views of electric current relative to the interface “magnetodielectric — free space” and relative to the screen will be in this case in such phases with the current itself, in which the fields of all currents over the array are summed up and provide for the working capacity of the concerned radiators within the AA.


The creation of a super-wideband AA (one octave and more) and a wide-domain AA (about 120° in the principle planes) can be considered as basically possible, if radiators of small electric sizes lo-cated over the complex impedance under in SFB instead of conventionally used resonance strip radiators are used.



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