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Mathematical modeling in design of antenna systems


A.B. Khashimov. Reviewers: V.V. Rodionov, V.M. Chernov



Publication date



The monograph


Paper book

700 Rub
Mathematical modeling has an important role in design of antenna systems. It is so due to ever increasing capabilities and complexity of antenna systems – first of all for the phased antenna arrays, where issues of radiators interference should be considered not only for the band given, but also for the scanning sector with its polarization characteristics. That leads to mathematical modeling based on the strict electrodynamics equations and taking into account the system itself for real radiators in antenna systems. In this case designer can run detailed analysis of electromagnetic radiation from antenna systems and synthesize corrective functions for amplitude-phase distribution in order to compensate the influence from antenna system as well as from metal and dielectric objects nearby.

In mathematical modeling of complicated antenna systems close attention should be paid to the validity of the results obtained. Therefore all main results of this monograph have been carefully tested on the problems that provide the strict analytical solutions. It is worth mentioning that such solutions are usually presented by infinite Fourier series of special functions which in turn have intrinsic computation accuracy. Thus, the difference between analytical and computed results while testing mathematical models often stems not only from quantization of real electrodynamics problem, but also from approximation of finite series, especially in asymptotic estimation for special functions. Therefore high correspondence between computed results and analytical solutions validates mathematical models and their applicability to antenna systems design. When appropriate boundary-value problems have no strict analytical solutions, mathematical models can be proved by direct experimental research of the designed antenna system. There are some specifics in antenna measurements, but general coincidence between measured and calculated results can serve as a criterion of mathematical model’s adequacy. Numerous modifications of antenna systems based on mathematical models developed by the author stood the test of experimental confirmation of their characteristics.

The monograph may be useful for antenna systems designers and students of radio technical specialties as a study guide for disciplines “Electrodynamics and radio-wave propagation”, “Antennas and VHF equipment”. Basic theoretical provisions for development of mathematical models are accompanied by MATLAB scripts and results of numerical research for different practical problems. Therefore results of this monograph can also be applied for studying programming in MATLAB, which is used widely in engineer design.

In Chapter 1 the issues of the asymptotic correspondence between planar and 3-D problems in antenna design are covered. It is shown that for many practical problems this correspondence allows significant simplifying of mathematical models for complex antenna systems installed near different dissipating objects. Formulating mathematical models in terms of integral equations and using efficient numeric methods to find their solutions provides a wide variety of results for estimation of parameters of antenna systems designed. MATLAB scripts can also be applied in educational activities for students of radio technical specialties.

Techniques for development of mathematical models for oscillator antennas with vertical polarization, installed nearby prolonged cylinder-like objects (metal, dielectric or mixed) are considered in Chapter 2. It is shown that asymptotic correspondence between planar and 3-D electrodynamics problems laid out in Chapter 1, allows computing the field of these antennas in the far zone using efficient mathematical models in the scalar form. Legitimacy of the transition from vector problems to considerably less complex scalar problems was proved by computation experiments. An example of practical application of developed mathematical model in design of antenna system is given, and the results of modeling are confirmed by experimental research in the active zone.

In Chapter 3 methods of mathematical models formation for slot antennas are covered. These methods have an important practical application and are based on the strict electrodynamics approach and asymptotic correspondence between planar and 3-D problems. The asymptotic approximation received for narrow slot antenna is later used for radiation field computation in relation to the solution of an appropriate diffraction problem. Comparison of modeling results to the known analytical solutions in the form of infinite series of elliptical cylinder functions also validates theoretical provisions that constitute the foundation of mathematical models of antenna systems.

Chapter 4 is devoted to forming mathematical models for dipole and slot radiators with reflectors of complicated shape. By means of the system of integral equations it is possible to determine the influence of reflector dissipation field on the radiators’ incitement characteristics and on the polarization of the radiation field. It allows to correct the matching of radiators in the band and to optimize the radiation field in the range of angles. The mathematical model developed comes down to the system of linear equations comfortable for creating, processing and storage due to its block structure. Several examples of some important practical problems are considered that illustrate the efficiency and wide applicability of proposed mathematical models.

In Chapter 5 applied mathematical models for dielectric dissipating objects are considered, based on the asymptotic correspondence between planar and 3-D problems of antenna equipment. A universal method of creating mathematical models for such objects was developed using functional matrix operators. These models are of great practical importance for antenna measurement problems in the near zone and probing. Solutions for some practical problems that can be applied to antenna systems design are also provided.

In the end of the monograph there is bibliography that cannot be regarded as complete.
Author would like to acknowledge Dr Tech. Sc. Prof. Voytovich N.I., colleagues Vorobyov M.S., Kudrin L.P., Klygach D.S. and Mr Salikhov R.R., chief designer of SPU “Radio Technical Systems” for their assistance in preparing this monograph.

The monograph was written as a part of the complex project in South-Urals State University (National Research University) “Creating of high-tech production of instrument modules and antennas for two-frequency radio beam complex of meter-band instrument landing system for category III (ICAO) civil aerodromes, including aerodromes with challenging terrain and high snow cover”.
All computations were fulfilled in the supercomputer centre in the South-Urals State University (National Research University),

Chapter 1. Correspondence between planar and 3-D electrodynamics problems. General principles of scalarization of vector problems
1.1. Planar boundary-value problems
1.2. Dissipation of arbitrary falling plane wave on the perfectly conducting infinitely extended cylinder
1.3. Dissipation field of E-polarized plane wave
1.4. Dissipation field of H-polarized plane wave
1.5. Formulation of planar dissipation problems for electromagnetic fields in terms of integral equations
1.6. Numeric solution of integral equations

Chapter 2. Mathematical models for oscillator antennas situated nearby extended cylinder-like objects
2.1. Main requirements for mathematical model of antenna system situated nearby
cylinder-like dissipating objects
2.2. Strict electrodynamics equations for construction of mathematical model
2.3. Asymptotic estimation for the surface integral S
2.4. Mathematical model for 3-D problem
2.5. Argumentation for transition from 3-D problem to equivalent planar problem
2.6. Experimental confirmation of validity of transition to planar problems

Chapter 3. Mathematical models for slot antennas situated in elongated screens
3.1. Main requirements for mathematical model of the slot antenna situated in elongated ideally conducting screen
3.2. Strict electrodynamics equations for construction of mathematical model for slot antenna
3.3. Value of the surface Sρ integral in V area
3.4. Value of the surface Spl integral in V area
3.5. Value of the space integral
3.6. Narrow slot
3.7. Numeric investigation of slot radiators

Chapter 4. Mathematical models for antenna systems with reflectors of arbitrary form
4.1. Mathematical model for oscillator antenna with finite reflector of arbitrary form
4.2. Design of mathematical model for antenna system of distance measurement equipment
4.3. Mathematical model for VHF antenna system with given broadcasting zone
4.4. Mathematical model for onboard slot antenna

Chapter 5. Mathematical models for planar dielectric objects in antenna equipment problems
5.1. Problem statemen
5.2. Mathematical model for homogeneous cylindrical objects
5.3. Mathematical model for multilayered dielectric objects
5.4. Examples of mathematical model construction for complicated dielectric objects
5.5. Mathematical models for non-homogeneous dielectric objects