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Professor John N. Sahalos is a professional engineer and consultant to industry and Head of the Radio Communications  Laboratory, Department of Physics, University of Thessaloniki, Greece. He is a member of the New York Academy of Science and the Technical Chamber of Greece. In 2002, he was appointed to serve a five-year term on the Board of Directors of the Hellenic Telecommunications Organization S.A. His research interests are in the areas of applied electromagnetics, antennas, high frequency methods, communications, microwaves and biomedical engineering.

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The first time that such a complete systematic analysis of the mathematical and numerical techniques related to the orthogonal methods has been given.

With the explosion of the wireless world, greater emphasis than ever before is being placed on the effective design of antennas. Orthogonal Methods for Array Synthesis outlines several procedures of orthogonal methods suitable for antenna array synthesis. The book presents a simple approach to the design of antenna arrays to enable the reader to use the classical Orthogonal Method for synthesis of linear arrays.

This theory-based book, which includes rapid, effective solutions to design problems for communications applications and broadcasting, is amply illustrated with real-world examples and case studies. Also included in the book is the ORAMA MS Windows-compatible computer tool, patented by Professor Sahalos and his team.

• Provides comprehensive coverage of the basic principles of orthogonal methods including an analytical explanation of the orthogonal method (OM) and the orthogonal perturbation method (OP)
• Gives rapid, cost-effective solutions to antenna design problems for communications applications and broadcasting
• Illustrates all theory with practical applications gleaned from the author’s extensive experience in the field of orthogonal advanced methods for antennas

Providing a complete guide to the theory and applications of the Orthogonal Methods, this book is a must-read for antenna engineers and graduate students of electrical and computer engineering and physics.

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Orthogonal Methods for Array Synthesis

John Wiley & Sons

Chapter One

1.1 Introduction

Antennas have become ubiquitous devices and occupy a salient position in wireless systems. Radio and TV as well as satellite and new generation mobile communications have experienced the largest growth among industry systems. The global wireless market continues to grow at breakneck speed and the strongest economic and social impact nowadays comes from cellular telephony, personal communications and satellite navigation systems. All of the above systems have served as motivation for engineers to incorporate elegant antennas into handy and portable systems.

Many textbooks provide in-depth resources on antennas. Especially on antenna arrays, there are digests, studies and books containing extensive data and techniques. In the references given herewith, there are some of the best-known and most highly recommended books.

A device able to receive or transmit electromagnetic energy is called an 'antenna'. As seen in [6], the antenna plays the role of a transitional structure between free space and a guiding device. An antenna consists of one or more elements. A single-element antenna is usually not enough to achieve technical needs. That happens because its performance is limited. A set of discrete elements, which constitute an antenna array, offers the solution to the transmission and/or reception of electromagnetic energy. The geometry and the type of elements characterize an antenna array. For simplicity, implementation and fabrication reasons, the elements are chosen in such a way so as to be identical and parallel. For the same reasons, uniformly spaced linear arrays are mostly encountered in practice.

In the following paragraphs, the properties of various antenna arrays will be presented.

1.2 Antenna Array Factor

The radiation characteristics of antennas have mostly to do with the far field (Fraunhofer) region. In this region, the field expression is a multiplication of two parts. One part contains the distance r dependence of the observation point (receiver location) and the other contains its spherical coordinate angles θ and φ dependence. The angular distribution of the field is independent of the distance r. For a typical antenna element (see Fig. 1.1), the far electric field is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-1)

The angular-dependent vector f_n(θ, φ) gives the directional characteristics of the nth element electric field [11]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-2)

where

J_n(r'_n) = electric current density of the nth element

r'_n = distance of a source point from the origin

r = distance of the observation point from the origin

ß = 2π/λ the free space wave number

ω = the angular frequency and

µ = the magnetic permeability of the space

The total electric field of an N element antenna array is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3)

Moreover, the total magnetic field is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-4)

where η = √µ/[member of] ([member of] is the electric permeability of the space).

For identical and identically oriented elements, the current distribution of each element is approximately the same except for a constant complex multiplier. In (1-1), f_n(θ, φ) can be expressed as

f_n(θ, φ) = I_n f(θ, φ) (1-5)

f (θ, φ) is called the 'pattern function' of the element and I_n is the complex excitation of the nth element of the array.

(1-1), (1-2) and (1-5) are combined and give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-6)

(r_n, θ_n, φ_n) are the spherical coordinates of a convenient reference point of the nth element and cos ξ_n = sin θ sin θ_n cos(φ - φ_n) + cos θ cos &theta_n.

The last term of (1-6) is expressed separately as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

AF(θ, φ) is called the 'array factor'. This factor is actually the array pattern of N isotropic point sources positioned at the reference points of the elements of the original array.

From (1-6) and (1-7), we have the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-8)

Expression (1-8) states the following pattern multiplication principle: An array consisting of identical and identically oriented elements has a pattern, which can be expressed as the product of the element pattern and the array factor.

An antenna engineer has to make an anticipatory and compatible choice of elements according to technical requirements. Once the element pattern is derived, the design effort is mainly directed at the array factor.

1.3 Elements and Array Types

Element types of antenna arrays are delineated in the literature. Dipoles, monopoles, loops, slots, microstrip patches and horns are the most common types of array elements. Recent studies and innovations have resulted in new types of elements. Some of them are the monolithic, the superconducting, the active and the electronically and functionally small antenna elements.

In parallel with the development of elements, antenna arrays have experienced a tremendous growth. Their list starts with the linear broadside and end-fire arrays, the planar, the circular, and the conformal, and goes up to the adaptive arrays. Moreover, flat plate slot arrays, digital beam forming, dichroic, slotted and fractal arrays are some of the recent types.

It was mentioned previously that antenna analysis and synthesis focuses mostly on the array factor. Consequently, in the following paragraphs we devote the analysis mainly to this factor.

1.4 Antenna Parameters and Indices

In many cases, it is necessary to characterize the performance of antennas by referring to specific parameters and indices. Most of these parameters and indices have been defined by the committees of the institutions in charge (IEEE, ETSI etc.) and will be presented in this paragraph.

It is well known that antennas have to do with applications of time varying fields. Sources with e^jωt time variation produce fields that also vary in the same way. In the literature, the above fields are called 'time-harmonic fields'. If a signal with certain bandwidth is present in an antenna, the Fourier transform can derive the time varying forms of the electromagnetic quantities. The procedure is analogous to that of solving electric circuit problems. In this book, time-harmonic fields with a proper choice of working frequency/ies are assumed.

1.4.1 Radiated Power

The time average power density, which is the average Poynting vector, can be written in the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-9)

The 1/2 in (1-9) appears because the electric and magnetic fields represent the peak values.

The radiated power P_r, through a closed surface S surrounding the antenna, can be obtained by integrating the normal component of the average Poynting vector over the entire surface. On the basis of the above definition we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-10)

The surrounding surface can be of arbitrary shape, and without losing generality a sphere is used.

1.4.2 Radiation Intensity

Radiation intensity in a given direction is the power radiated per unit solid angle in the above-mentioned direction. It is expressed in watts per unit solid angle and is related to the far field of the antenna. In a spherical coordinate system (r, θ, φ), radiation intensity is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-11)

where r denotes the distance between the antenna and the observation point and [??] is the corresponding radial unit vector.

U(θ, φ) can be expressed only by the electric or the magnetic far-zone field:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-12)

where η is the intrinsic impedance of the medium. (For the free space η = 120π Ohms).

The total power P_r radiated is obtained by taking the integral of the radiation intensity over all angles around the antenna. Thus, P_r is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-13)

Another parameter related to U(θ, φ) is the average radiation intensity, defined as the radiation intensity of an isotropic source radiating the same power as that of the actual antenna. Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-14)

where dΩ is the element of solid angle.

It is obvious that an isotropic source (antenna) is not realizable; however, it is often used as a reference element for many antennas.

1.4.3 Directivity

The directivity of an antenna in a given direction is defined as the ratio of the radiation intensity in the above-mentioned direction to the average radiation intensity. If the direction is not specified, then it is implied in the direction of maximum radiation. By using (1-12) and (1-13), directivity D is

D = U/U_ave (1-15)

Directivity can be more conveniently expressed by

D = 4π[U/P_r] (1-16)

Maximum directivity is given by

D_max = D_o = 4π[U_max/P_r (1-17)

The directivity of an isotropic source is unity. All other sources have maximum directivity greater than unity.

1.4.4 Antenna Efficiency

It is well known that only one part of the input power at the input terminals of an antenna is radiated. Various reasons, like the mismatch between the transmission line and the antenna or conduction and dielectric losses of the antenna itself, reduce the power radiated.

The total efficiency of an antenna can be expressed by

e_o = e_re_ce_d (1-18)

where e_o is the total efficiency, e_r is the mismatch efficiency = (1 - |Γ|²), e_c is the conduction efficiency and e_d is the dielectric efficiency. Γ is the voltage reflection coefficient at the input of the antenna terminals expressed by

Γ = [Z_in - Z_o]/[Z_in + Z_o] (1-19)

where Z_in is the antenna input impedance and Z_o is the characteristic impedance of the transmission line.

(1-18) is expressed in an alternative form as

e_o = e_cde_r = e_cd(1 - |Γ|²) (1-20)

e_cd is the radiation efficiency that can be determined experimentally or, if possible, numerically.

1.4.5 Gain

The performance of an antenna can also be described by the gain. The gain is related to directivity. It is an index that takes the directional properties and the efficiency of the antenna into account. The gain G(θ, φ) is defined by

G(θ, φ) = radiation intensity for the direction(θ, φ/[[1/4π](power input to the antenna)] (1-21)

When the direction is not specified, the gain is taken in the direction of maximum radiation. Expression (1-21) counts the losses of the antenna itself and can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-22)

If one takes the mismatch losses into account, then the absolute gain G_abs is introduced. G_abs is expressed by

G_abs = e_oD (1-23)

Finally, maximum absolute gain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is related to maximum directivity D_o. It is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-24)

The above indices are given in terms of decibels instead of dimensionless quantities. The relation of an index X in dB and its corresponding dimensionless value is

X[dB] = 10 log X[dimensionless] (1-25)

1.4.6 Antenna Patterns

One of the main characteristics of an antenna is its radiation pattern. It presents graphically the radiation properties and can be measured by moving a probe antenna around the antenna under test at a constant distance in the far field (see Fig. 1.2a). The response, as a function of the angular coordinates, constitutes the radiation pattern. Depending on probe type and orientation, the appropriate component of an electric or a magnetic field can be measured. If a probe is moved over the spherical surface, its terminal voltage will present the 3-D radiation pattern. A pattern taken on one plane is known as a 'plane pattern'. The pattern that contains the electric field vector is the E-plane pattern while the pattern that contains the magnetic field vector is the H-plane. The above two are referred to as the 'principal plane patterns'. As an example, Fig. 1.2b presents the 3-D radiation pattern of a uniform linear array of three collinear vertical electric dipoles with equal mutual distance of 0.75λ. Figures 1.2c and d show the E- and H-plane patterns of the array.

A plane pattern can be depicted as a polar or as a rectangular plot. The units of the patterns are either linear (Fig. 1.3a) or logarithmic (dB) (Fig. 1.3b). The lobe in the direction of maximum radiation is called the 'main lobe' or the 'main beam'. Any lobe of the pattern other than the main lobe is a 'minor lobe'. Usually, a minor lobe is also called a 'side lobe'. A side lobe in a pattern is, in general, any lobe other than that of the intended one. Since the intended lobe is usually the main lobe, it is obvious that minor and side lobes are the same. A measure of the characteristics of the pattern is given by certain quantities:

1. The half-power (or 3 dB power) beam width (HPBW) is the angular width between the angular points half-power (3 dB) below that of the main-beam maximum of the antenna (see Fig. 1.3b).

2. The first null beam width (BW_null) is defined as the angular width between the first zero crossing of either side of the main-beam maximum of the antenna (see Figs. 1.3a and b).

3. The bandwidth [increment of f] of an antenna is defined by the frequency limits at which the maximum gain is reduced to its half value (3 dB). The fractional bandwidth is given by [increment of f]/f. f is the mean operating frequency of the antenna.

4. The side lobe level (SLL) is the ratio of the pattern value of a side lobe peak to the corresponding value of the main lobe. Usually, SLL in an antenna is defined as the largest side lobe level for the whole pattern. A special case of the inverse SLL is the front to back ratio (F/B). This is the ratio of the pattern value in the main lobe maximum to the corresponding value in the direction of 180 degrees from it. If there is a minor lobe in the back direction, this is called a 'back lobe' (see Fig. 1.4a).

5. The grating lobe is any maximum equal to the maximum of the main lobe of the pattern. One or more grating lobes are formed in antenna arrays when the spacing between the elements is more than λ (see Fig. 1.4b).

Depending on its radiation pattern, an antenna is called 'broadside', 'end-fire' or 'intermediate'. A broadside antenna is an antenna for which the direction of the main lobe maximum is normal to the plane containing the antenna. If the above direction is within the plane, then the antenna is an end-fire antenna. An antenna is intermediate if it is neither broadside nor end-fire. Figures 1.5a, b and c represent the pattern of the three types of antennas. The beam of the antenna in Fig. 1.5a is known as a 'fan beam'. The single lobe of the antenna in Fig. 1.5b is called a 'pencil beam'.

It is noticed that, owing to reciprocity, the radiation pattern of an antenna in the transmitting mode is identical to that in the receiving mode. The reciprocity theorem is well known from circuit analysis. It states: If a constant current (voltage) source is placed in one branch of a reciprocal network, and a voltage (current) reaction is measured in another, then interchanging the locations between the source and the branch of measurement, we have unchanged measurement results. A network is reciprocal when it is composed of linear, bilateral elements. In antennas, the materials of the construction, the transmission line and the medium of wave propagation must be linear. Nonlinear devices (diodes, transistors) make the antenna nonreciprocal. In electromagnetics, the reciprocity theorem is represented with the help of Maxwell's equations and is called the 'Lorentz reciprocity theorem'. Application of the above theorem to open (unbounded) problems is of importance to antennas. Without missing the point of this book, it would be useful to give some more details.

(Continues...)

Excerpted from Orthogonal Methods for Array Synthesisby John N. Sahalos Copyright © 2006 by John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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