International Journal of Antennas and Propagation

Volume 2019, Article ID 1687497, 9 pages

https://doi.org/10.1155/2019/1687497

## Mutual Impedance Properties in a Lossy Array Antenna

Correspondence should be addressed to Peter L. Tokarsky; au.vokrahk.nair@yksrakot.p

Received 11 August 2019; Accepted 5 October 2019; Published 11 November 2019

Academic Editor: Hervé Aubert

Copyright © 2019 Peter L. Tokarsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The impedance matrix of an arbitrary multiport array antenna with Ohmic losses was studied. It was assumed that the partial current distributions in the array, corresponding to the alternate exciting one of its terminals while the other ones are open-circuited, are known. Consideration of the power balance in the lossy array antenna has allowed ascertaining that its impedance matrix is a sum of the radiation resistance matrix and loss resistance matrix, which are in general case complex Hermitian matrices and only in some particular cases can be real. The theoretical statements obtained are confirmed by two numerical examples, where analysis of two lossy dipole arrays was performed. In the first example, the dipoles were located above the imperfect ground, which served as a source of losses, and in the second example, the same dipoles were located in free space, and the embedded parasitic element was the source of losses. The results of the analysis showed that the asymmetric placement of energy absorbers in the array antennas leads to the appearance of imaginary parts in the matrices of radiation and loss resistances, which allow one to correctly predict the behavior of the array radiation efficiency during beam scanning.

#### 1. Introduction

The concept of mutual impedances of the circuit theory, applied to antenna problems, has been used successfully for many years to evaluate the effect of element interaction on the array antenna parameters. At first, the induced EMF method was used to calculate the mutual impedances [1–3], and then some other methods, in particular, the Poynting vector method [4–6], the variational method [7], and the integral equation method [8–10] and its modifications [11, 12]. It was generally assumed that the antennas under investigation were lossless, or the losses in them were considered very small and had no effect on the current distribution in the antennas [13, 14]. Typically, such losses were simulated by a lumped resistor connected to the antenna terminals [15], which was added to their self-impedance. In [16–20], it was shown that embedding the distributed and lumped losses into the antenna could significantly change the current distribution and, consequently, controls their parameters. In particular, in [20], using the integral equation method, it was demonstrated that by choosing the resistive loading distribution along two parallel dipoles, one can achieve a significant decrease in the mutual coupling between them.

The first attempt to study the effect of losses distributed in an array on interaction of its elements was made in [21], where it was found that loss resistances not only can be part of the self-impedance of the elements as well part of mutual impedances between them. This property of the loss resistance was confirmed by an analysis of the mutual coupling between two lossy dipoles [22], as well as by investigation of the array of vertical [23] and horizontal [24, 25] dipoles located above the imperfect ground. In recent years, in connection with the construction of new ground-based radio telescopes using large aperture arrays, as well as the reflector antennas with phased array feeds [26], explorations of the effect of losses on the different array parameters have been intensively continued [27–29]. Despite this, the mutual coupling properties in the lossy array antenna have not yet been fully studied. This article aims to partially fill this gap.

The purpose of this work is to study in detail the properties of mutual impedances between array elements with Ohmic losses and to demonstrate their use for analysis and optimal designing of the lossy array antenna. In Section 2, the basic relationships that determine the impedance matrix structure for the lossy array are derived. In Section 3, it is shown how the mutual loss resistances affect the power balance in the phased array during beam scanning, and in Section 4, two examples of numerical analysis and optimization of dipole arrays in free space and over the imperfect ground are given.

#### 2. Theory

Consider an array antenna consisting of a set of conducting bodies located inside a bounded domain in free space. Assume that the medium inside is isotropic and linear and is characterized by the dielectric permittivity , magnetic permeability , and conductivity , which are scalar functions of the coordinates.

Let us suppose that the array antenna operates in the transmission mode and is fed by the harmonic oscillators (with the time dependence ) by means of *N* regular TEM-transmission lines. We will take the reference planes in these lines as *N* array ports. Since the currents and voltages can be uniquely determined at these ports, the terminal equation of the array antenna can be written in the formwhere and are the vectors of complex amplitudes of the currents and voltages at the array ports and is an array antenna impedance matrix.

Assume also that we know the *N* partial current distributions fields and radiated by them, which are found as results of solving *N* electromagnetic boundary-value problems, when all array ports are excited one after another by current , while the other ports are open-circuited (, , ), where and are position vectors of a source point and observation point, respectively. Note that these currents should be determined by taking into account the environment of the array, which plays an essential role in the radiation process, and to which belong parasitic elements, ground, screens, a dielectric substrate, and so on.

The domain of function coupled with the *n*th array port, in fact, is an *n*th embedded element in the array with all other elements open circuited.

The current can be presented as follows:where is the dimensionless vector function characterizing the distribution of the electric current in the domain , is the cross section of the current at the *n*th terminal, and is the domain occupied by all embedded elements.

When all array ports are excited simultaneously, the total current density in the domain and the total fields and they produced are defined as the sum:

Grounding on these assumptions, let us determine the impedance at the array antenna ports using the Poynting theorem. To do this, we surround the entire array with a spherical surface located in the far zone and consider the region bounded from the outside by surface , and from the inside with surface , which is formed by the reference planes in the transmission lines selected as the array ports ().

External sources of currents and fields are absent in *V*, so the power balance equation here has the formwhere is the outward-directed unit normal to surface and “” denotes complex conjugation.

The first term in (5) determines the lost power , scattered in volume in the form of heat. Using relations (1)–(3), we express this power through the currents at the array ports:where the subscript t denotes transposition and is a loss resistance matrix of the array antenna, in which elements are determined as

From (6) and (7), is a Hermitian non-negative definite matrix. Its diagonal elements are real non-negative numbers equal to the powers of the thermal losses of the array when its ports are alternately excited by the unit current that allows to call them as self-loss resistances. Each off-diagonal element is a mutual loss resistance being a measure of nonorthogonality of the two vector functions and with respect to their weighted inner product (7).

The second term in (5) determines the power radiated by the array antenna that can be represented aswhere is a radiation resistance matrix, and are the far-field vectors related as , and is the intrinsic free space impedance.

Matrix can be determined if we substitute (4) in (8) and take into account that the embedded element patterns can be represented [30].where is the effective length, *λ* is the wavelength, are the observation point coordinates, and is the embedded element normalized pattern, given by [31]:

is the unit vector in direction.

As a result, we find the radiation resistance matrix elements:

Before, the same equation was derived in [4] to determine the mutual resistance between two arbitrary lossless antennas.

Comparing (11) and (7), it can easily be seen that , as well as , is the Hermitian matrix, which is positive-definite since for any radiating systems. Its diagonal elements are real positive numbers, which are the self-radiation resistances of the array antenna, and the off-diagonal elements are mutual radiation resistances and determine the measure of the nonorthogonality of the embedded element normalized patterns and .

The third term in (5), taken with the opposite sign, describes the power entering the array antenna from external sources through the ports. It can also be represented by the sum:where is the array antenna impedance matrix (1).

The fourth term describes the reactive power produced by the array in domain . Taking (3) and (4) into account, this power can be represented as the sum:where is an array antenna reactance matrix [4, 5].

Now, substituting (6), (8), (12), and (13) in (5) and assuming , we can derive the following equation:which can be divided into two equationswhere is the array antenna input power.

From (14) follows the equationand from (15) and (16), the following equations are obtained:

Since the matrices and are Hermitian, each of them can be divided into a symmetric real part and an antisymmetric imaginary part, namely,

Using the reciprocity principle, it is not difficult to prove thatand is a real symmetric matrix.

From (20),that allows us to write (19) as

#### 3. Discussion

Let us consider what influence the self and mutual losses resistances exert on a power balance in a lossy array antenna with phase steering on the example of a uniformly spaced *N*-element linear array. The Joule losses on it can be due to the finite conductivity of the material, the presence of embedded resistors, or the presence of closely located absorbing bodies or surfaces (array construction, aircraft body, passive scatterers, imperfect ground, etc.). We suppose that its matrices are known from the boundary problem solution, taking into account the loss distribution. In addition, the array ports are excited by currents of the uniform amplitude with a progressive phase shift Δ, i.e., . Then, the input power isand the radiated and lost powers arewhere the “plus” sign before pertains to the radiation power and the “minus” sign to the loss power. From (24), both powers and consist of three terms. The first of these, is determined only by the self-resistances and does not depend on Δ. The second and the third , being terms in (24), are determined by the real and imaginary parts of mutual resistances and are the even and the odd functions of Δ, respectively. When is changed, the input power remains unchanged and is redistributed between the radiation power and the loss power , herewith when one of them increases by , the other decreases by the same amount. This effect occurs in all multiport radiating systems with nonuniform loss distribution.

#### 4. Numerical Examples

##### 4.1. Dipole Array over Lossy Half-Space

Consider an array of two closely spaced identical symmetrical dipoles located in the immediate vicinity of the surface of a plane imperfect ground with parameters and (average ground). The *λ*/2-long dipoles are made of a thin, perfectly conducting wire with a diameter of and are oriented along the *Y*-axis of the Cartesian coordinates. Consider the two variants for arranging the dipoles above the ground, one-level array (Figure 1(a)) and two-level array (Figure 1(b)). In both arrays, dipole 1 is located at the height , and the distance between the dipoles is . The first array is named usually as a broadside array (BSA) and the second one as an end-fire array (EFA).