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1 FEDERAL AGENCY FOR EDUCATION State educational institution of higher vocational education"NORTH-WESTERN STATE CORPORATE TECHNICAL UNIVERSITY" Department of Radio Engineering ELECTRODYNAMICS AND DISTRIBUTION OF RADIO WAVES EDUCATIONAL AND METHODOLOGICAL COMPLEX Institute of Radio Electronics Specialty training of a diploma specialist: radio engineering Direction of radio engineering Bachelor's degree Publishing house 009

2 Approved by the Editorial and Publishing Council of the University UDC Electrodynamics and radio wave propagation: educational and methodological complex / comp. L. Ya. Rhodes, D.A. Chistyakov. SPb .: Publishing house of SZTU, p. The educational and methodological complex (TMC) was developed in accordance with the requirements of state educational standards of higher professional education. In the CMM, questions of the theory are considered electromagnetic field, the main methods for solving applied problems of electrodynamics in relation to the propagation of electromagnetic waves in guiding systems and radio waves on natural paths. The EMC is intended for students of the specialty studying the discipline "Electrodynamics and radio wave propagation", and bachelors of engineering and technology in the direction studying the same discipline. Considered at a meeting of the Department of Radio Engineering, approved by the Methodical Commission of the Institute of Radio Electronics, Reviewers: Department of Radio Engineering, SZTU (Head of the Department GI Khudyakov, Doctor of Technical Sciences, Prof.); V.S. Kalashnikov, Dr. sciences, prof., ch. scientific. sotr. VNIIRA. Compiled by L.Ya. Rhodes, Cand. tech. sciences, associate professor; YES. Chistyakov, Cand. tech. Sciences, Assoc. North-West State Correspondence Technical University, 008 Rhodes L.Ya., Chistyakov D.A., 008

3 1. Information about the discipline 1.1. Foreword Electrodynamics and radio wave propagation (ED and RWP) are related to the disciplines of the general professional cycle. Its volume according to the state educational standard (SES) is 170 hours. It includes two interrelated parts: part 1 - electrodynamics proper (theoretical electrodynamics) and part - radio wave propagation (applied electrodynamics). This discipline is basic for modern radio engineering. The purpose of the discipline is to acquire theoretical knowledge and skills of solving problems in the field of the theory of the electromagnetic field, the peculiarities of the interaction of electromagnetic waves with various physical media, the propagation of radio waves along guiding systems and on natural routes. The tasks of studying the discipline are mastering the basic provisions of electrodynamics and the characteristics of the propagation of radio waves. As a result of studying the discipline, the student must acquire knowledge of the discipline, formed at several levels: optical phenomena, on the vector nature of electromagnetic and optical fields, on the ranges of radio waves used in technology, the main features of the propagation of radio waves on natural paths. Know: Maxwell's equations in integral and differential forms, the physical meaning of all terms included in these equations; mechanisms of the influence of the Earth and the Earth's atmosphere on the propagation of radio waves of various ranges. 3

4 Be able to: transform Maxwell's equations into equations of electro- and magnetostatics, stationary electric and magnetic fields, into wave equations for vectors of an electromagnetic field, vector and scalar potentials; formulate a problem (select a model) for calculating the parameters of a specific radio link. Get skills: solving problems of electrodynamics by methods: separation of variables, lagging potentials, scalar and vector Kirchhoff integrals; selection of the type, size and calculation of the parameters of guiding systems (transmission lines of electromagnetic energy); calculation of radiation characteristics of elementary radiators and real antennas; choosing a model and determining the nature and degree of influence of the path of propagation of radio waves on the characteristics of a particular radio engineering system. Studying the discipline "Electrodynamics and radio wave propagation" requires mastering a number of previous disciplines. These include: mathematics (series, differential and integral calculus, vector field theory, solution differential equations); physics (electricity and magnetism, electrodynamics); informatics (algorithmic methods, numerical solution methods). In turn, the course ED and RRV underlies all the disciplines that determine the professional training of a specialist in the field of radio engineering: the basics of circuit theory, radio circuits and signals, microwave devices and antennas, devices for receiving and processing signals, devices for generating and shaping signals, radio engineering systems etc. The content, volume and procedure for studying the materials of the course "Electrodynamics and propagation of radio waves" in accordance with the requirements of the State Educational Standard are set out in the "Work Program" presented in the "Information Resources" section. There is also a "Thematic Plan" containing information on the types of reporting by topic. 4

5 1 .. The content of the discipline and types educational work The content of the discipline In accordance with the State Educational Standard, the following didactic units should be studied in the course "Electrodynamics and Radio Wave Propagation": integral and differential equations of electromagnetism; complete system of Maxwell's equations, boundary conditions; the energy of the electromagnetic field; Umov-Poynting theorem; boundary value problems of electrodynamics; analytical and numerical methods for solving boundary problems; electromagnetic waves in various environments; electrodynamic potentials; electromagnetic waves in guiding systems; electromagnetic oscillations in cavity resonators; excitation of electromagnetic fields by specified sources; radiation of electromagnetic waves into free space; retarded potential theorem; propagation of electromagnetic waves near the surface of the Earth; tropospheric propagation of radio waves; propagation of radio waves in rough terrain and in the presence of obstacles; models and methods for calculating radio paths Scope of discipline and types of educational work Total hours Type of educational work Form of study Full-time Part-time Extramural Total labor intensity of the discipline (CTD) 170 Work under the guidance of a teacher (RpRP) Including classroom sessions: Lectures Practical lessons(LR) Laboratory work (LR) The number of hours of work with the use of DOT Independent student work

6 Intermediate control, number Control work - Test Type of final control (exam), number List of types of student's academic work, current monitoring of progress and intermediate attestation - two control works (for part-time and part-time forms of study); -tests (training on topics, milestones in discipline sections, questions for self-examination, etc.); - one credit (for laboratory work, part 1 - electrodynamics); -two exams .. Working training materials. 1. Working programm(170 hours) Part 1 - electrodynamics 1.1. Section 1. Integral and differential equations of electromagnetism Basic concepts and definitions (4 hours) [1], with Basic concepts and definitions, materiality of an electromagnetic field, vectors of an electromagnetic field, classification of media in electrodynamics. Maxwell's equations - fundamental equations of electrodynamics (1 hour) [1], with Maxwell's equations in integral and differential forms and their physical meaning. Electric current continuity equation. Third-party electric and magnetic currents and charges. Complete system of EMF equations in symmetric and asymmetric forms. Maxwell's equations at harmonic - 6

7 the dependence of electromagnetic processes on time. Complex dielectric constant of media. The principle of permutation duality of Maxwell's equations. Energy characteristics of EMF (6 hours) [1], s Energy balance in EMF: localization, movement and transformation of energy. Energy characteristics with harmonic dependence of electromagnetic processes on time. Electromagnetic waves - a form of existence of EMF (6 hours) [1], with Wave equations for vectors of EMF. Electrodynamic potentials. Wave equations for electrodynamic potentials. Wave equations in complex form. Particular types of EMF equations (4 hours) [3], with Electrostatic field: system of charges, dipole, capacitance, conductors and dielectrics in an electrostatic field. Stationary field: current system, magnetic dipole, inductance. Quasi-stationary field: from Maxwell's equations to the theory of circuits..1 .. Section. Boundary problems of electrodynamics Basic methods of solving problems of electrodynamics (8 hours) [1], p. 1-7 Internal and external problems of electrodynamics. Boundary conditions and radiation condition. Uniqueness of solutions to problems of electrodynamics. Superposition principle of solutions, reciprocity theorem, equivalence theorem. Rigorous methods of solution: lagging potentials, separation of variables, Kirchhoff. Approximate solution methods: geometric and wave optics, edge waves, geometric theory of diffraction, modeling. 7

8 Plane electromagnetic waves (EMW) (10 hours) [1], p. 7-4 General properties wave processes. Plane Homogeneous Electromagnetic Waves in a Homogeneous Unlimited Isotropic Medium. Waves in a dielectric, semiconductor and conductor. Spherical EME in limitless homogeneous media. EME emission (1 hour) [1], s Types of elementary emitters. Radiation of the system of given currents. Elementary electric emitter: components of EMF vectors, directional function, power and radiation resistance. Elementary magnetic emitter. Huygens' element. Plane EME in an inhomogeneous medium (10 hours) [3], with Electromagnetic waves and optical rays. Boundary conditions for vectors of the electromagnetic field. Reflection and refraction of electromagnetic waves at a plane interface between media. Snell's laws and Fresnel's formulas. Concepts of Brewster angles, total internal reflection, surface effect Section 3. EMW in guiding systems. Electromagnetic vibrations in cavity resonators. Guided EMW and guiding systems. Waveguides (16 hours) [1], s General information about guiding systems and guided waves. Hollow metal waveguides: rectangular, round. The structure of the electromagnetic field, the main types of waves, phase and group velocities, wavelength in the waveguide, characteristic impedance, attenuation of the electromagnet - 8

9 nit waves, excitation and coupling of waveguides, selection of waveguide sizes for operation on a given wave type. Coaxial and two-wire transmission lines (4 hours) [3], p. 4-9 Features of T waves and basic parameters of T waves in coaxial and two-wire transmission lines. Phase constant, phase velocity, group velocity, line wavelength, characteristic impedance. Coaxial line single-mode operation range. Resonant resonators (8 hours) [3], with a segment of the guiding structure as a resonator. General theory cavity resonators based on rectangular, cylindrical and coaxial waveguides. Natural frequency and Q-factor of resonators. Excitation of resonators. Part of the propagation of radio waves 1.4. Section 4. Propagation of EME near the Earth's surface. Impact of obstacles. Basic concepts and definitions (4 hours), p. 4-7 Basic concepts and definitions in the theory of RRV. The role and place of radio wave propagation issues in the training of radio engineers. The history of the development of the theory of RRV. Classification of radio waves by frequency ranges and modes of propagation on natural paths. Propagation of radio waves in free space(10 hours), s Electromagnetic field of isotropic and directional emitters in free space. Ideal radio communication equations for radiators 9

10 different types. Huygens-Fresnel principle. Fresnel zones in free space. Significant and minimal areas of space in the propagation of radio waves. Transmission loss in free space propagation of radio waves. Influence of the Earth's surface on the propagation of radio waves (18 hours), s Electrical parameters of the earth's surface. Statement and general solution of the problem of radio wave diffraction around a homogeneous spherical Earth's surface. Analysis of the general solution to the problem: the influence of the electrical parameters of the Earth's surface and the distance between the corresponding points on the magnitude and behavior of the attenuation factor in space. Line-of-sight distance and line-of-sight attenuation multiplier calculation. Interference formulas. The limits of applicability of the interference formulas. Calculation of the attenuation factor in the shade and penumbra zones. Reflection of radio waves from the Earth's surface, a significant and minimal area of ​​a reflective surface. Taking into account the influence of the curvature of the Earth's surface when reflecting radio waves. Influence of inhomogeneity of electrical parameters of the Earth's surface on the propagation of radio waves along it. Influence of unevenness of the Earth's surface on the propagation of radio waves. Rayleigh criterion. General information about the propagation of radio waves near statistically uneven surfaces Section 5. Influence of the Earth's atmosphere on the propagation of radio waves. Influence of the Earth's troposphere on the propagation of radio waves (10 hours), s Composition and structure of the Earth's atmosphere. Electromagnetic parameters of the troposphere, stratosphere and ionosphere. Refraction of radio waves in the troposphere and ionosphere. Equation of the trajectory of the wave and the radius of curvature of the ray. Types of radio wave refraction in the troposphere. Equivalent radius of the Earth. Formation process and parameters of tropospheric waveguides. ten

11 Influence of the Earth's ionosphere on the propagation of radio waves (8 hours), s Trajectory of radio waves in the ionosphere. Reflection of radio waves from the ionosphere. Critical and maximum frequency. Phase and group velocities of propagation of radio waves in the ionosphere. Influence of the Earth's magnetic field on the propagation of radio waves in the ionosphere. Scattering and absorption of radio waves in the troposphere and ionosphere. Methods for experimental investigation of the troposphere and ionosphere Section 6. Models and methods for calculating radio paths. Radio links for various purposes... The ranges of frequencies used (8 hours), from the Lines of radio broadcasting, television, radio communication, radar, radio navigation, radio control and telemetry. The purpose of the radio lines, the ranges of the frequencies used and the features of the propagation of radio waves of these ranges along the radio line. Methods for calculating various radio lines, with Methods for calculating radio lines for various purposes and different ranges of radio waves. eleven

12 .. Thematic plan of the discipline ... 1. Thematic plan of the discipline for full-time students p / p Name of sections and topics Number of hours in full-time education Types of classes (hours) lectures PZ (S) LR audit. DOT audit. DOT audit. DOT Independent work Tests Types of control Control works Abstracts LR Course work TOTAL Section 1. Integral and differential equations of electromagnetism 1.1 Basic concepts and definitions 3 1. Maxwell's equations fundamental equations of electrodynamics Energy characteristics of an electromagnetic field (EMF) Electromagnetic waves form of existence of EMF Particular types of equations of EMF 7 Section. Boundary problems of electrodynamics 8.1 Basic methods of solving problems of electrodynamics 9. Plane electromagnetic waves (EMW) in a homogeneous medium 10.3 Spherical EMW in unlimited media. EME emission Plane EME in an inhomogeneous medium 1 Section 3. EME in guiding systems. Electromagnetic vibrations in cavity resonators Guided EMW and guiding systems. Waveguides Coaxial and two-wire transmission lines Resonant resonators Section 4. Propagation of 4 EMW near the Earth's surface. Impact of obstacles Basic concepts and definitions

13 18 4. Propagation of radio waves in free space Influence of the Earth's surface on the propagation of radio waves 0 Section 5. Influence of the Earth's atmosphere on the propagation of radio waves Influence of the Earth's troposphere on the propagation of radio waves 5. Influence of the Earth's ionosphere on the propagation of radio waves 3 Section 6. Models and methods for calculating radio paths Radio lines of various destination. Ranges of frequencies used 5 6. Methods for calculating various radio lines Thematic plan of the discipline for students of full-time and part-time education n / p Name of sections and topics Number of hours in daytime Forms of classes (hours) Lectures PZ LR Auditorium. DOT Auditorium DOT Auditorium DOT Samost. work Tests Types of control Contra. work PZ LR Course. work Total Section 1. Integral and differential equations of electromagnetism Basic concepts and definitions Maxwell's equations - fundamental equations of electrodynamics Energy characteristics of an electromagnetic field (EMF) Electromagnetic waves - a form of existence of EMF Particular types of EMF equations 4 7 Section. Boundary problems of electrodynamics Basic methods of solving problems of electrodynamics Plane electromagnetic waves (EMW) in a homogeneous medium Spherical EMW in unlimited homogeneous media. EME emission Plane EME in an inhomogeneous medium

14 1 Section 3. EMV in guiding systems. Electromagnetic vibrations in cavity resonators Guided electromagnetic waves and guiding systems. Waveguides Coaxial and two-wire transmission lines Resonant resonators Section 4. Propagation of electromagnetic waves near the Earth's surface. Influence of obstacles Basic concepts and definitions Propagation of radio waves in free space Influence of the Earth's surface on the propagation of radio waves Section 5. Influence of the Earth's atmosphere on the propagation of radio waves Influence of the Earth's troposphere on the propagation of radio waves Influence of the Earth's ionosphere on the propagation of radio waves Section 6. Models and methods of calculating radio paths Radio links for various purposes. Ranges of frequencies used Methods for calculating various radio lines Thematic plan of the discipline for students of extramural education p / p Name of sections and topics Number of hours in full-time education Types of classes (hours) lectures PZ (S) LR audit. DOT audit. DOT audit. DOT Independent work Tests Types of control Control work Abstracts LR Coursework TOTAL Section 1. Integral and differential equations of electromagnetism 1.1 Basic concepts and definitions 3 1. Maxwell's equations fundamental equations of electrodynamics Energy characteristics of an electromagnetic field (EMF)

15 5 1.4 Electromagnetic waves form of existence of EMF Particular types of equations of EMF Section. Boundary problems of electrodynamics Basic methods of solving problems of electrodynamics 9. Plane electromagnetic waves (EMW) in a homogeneous medium Spherical EMW in unlimited media. EME emission Plane EME in an inhomogeneous medium Section 3. EME in guiding 3 systems. Electromagnetic oscillations in cavity resonators Guided EMW and guiding systems. Waveguides Coaxial and two-wire transmission lines Resonant resonators Section 4. Propagation of 4 EMW near the Earth's surface. Influence of obstacles Basic concepts and definitions Propagation of radio waves in free space Influence of the Earth's surface on the propagation of radio waves Section 5. Influence of the Earth's atmosphere on the propagation of radio waves Influence of the Earth's troposphere on the propagation of radio waves 5. Influence of the Earth's ionosphere on the propagation of radio waves 3 Section 6. Models and methods of calculating radio paths Radio lines for various purposes. Frequency bands used 5 6. Methods for calculating various radio links

16 .3. Structural and logical diagram of the discipline Electrodynamics and radio wave propagation Section 1 Integral and differential equations - Section Boundary problems electro - Section 3 Electromagnetic waves in guides Section 4 Propagation of electromagnetic waves near Section 5 Influence of the Earth's atmosphere on propagation Section 6 Models and methods of calculating ra - Basic concepts and definition- Maxwell's equations- fundamental Basic methods for solving problems of electro- Guided electromagnetic waves and Basic concepts and definition- Influence of the Earth's troposphere on the propagation of radio lines for various purposes. Range - Energy characteristics of electric - Plane electromagnetic waves - Spherical electromagnetic waves in borderless - Coaxial and two-wire transmission lines Propagation of radio waves in the free space Influence of the Earth's ionosphere on propagation Methods for calculating various ra- Electromagnetic waves form su- Plane electromagnetic waves - Volume resonators Influence of the Earth's surface on propagation Propagation of radio waves in space Particular types of equations of electromagnet-

17 .4. Time schedule for the study of the discipline (for students involved with the use of DOT) Title of the section (topics) Duration of the study of the section (topics) 1 Section 1. Integral and differential 7 days. equations of electrodynamics Section. Boundary problems of electrodynamics 9 days. 3 Section 3. Electromagnetic waves in guiding systems. Electromagnetic vibrations in cavity resonators 7 days. 4 Section 4. Propagation of electromagnetic 7 days. waves near the surface of the Earth 5 Section 5. Influence of the Earth's atmosphere on the propagation of 4 days. radio waves 6 Section 6. Models and methods of calculating radio paths 4 days. 7 Test work 1 days. 8 Test work d. TOTAL. 5. Practical block 5.1. Practical training Practical training ( Full-time training) 4 days. Number and name of the topic Tema.3 Spherical EMW in limitless environments. EMV radiation Topic 3.1 Guided EMW and guiding systems. Waveguides Topic 4. Propagation of radio waves in free space Solving problems on the emission of EME by elementary electric and magnetic dipoles Determining the size of waveguides and characteristics of EMF in rectangular and circular waveguides Determining the parameters of radio communication lines in free (outer) space Name of topics for practical training Number of hours Topic 4.3 Influence on - Calculation of the intensity of the EMF for the

18 the surface of the Earth on the propagation of radio waves by diode lines passing near the surface of the Earth Practical lessons (correspondence and part-time forms of education). Practical classes for students of these forms of training are not provided for in the curriculum work plans ... 5 .. Laboratory work Laboratory work (full-time study) Number and name of the section (topic) Section. Boundary problems of electrodynamics Topic .. Plane electromagnetic waves Topic.4. Plane EMW in an inhomogeneous medium Section 3. EMV in guiding systems. Electromagnetic oscillations in volumetric resonators Topic 3.1. Guided EMW and guiding systems Topic 3.3. Volumetric resonators Name of laboratory work Study of electromagnetic field polarization Study of reflection and refraction of plane EMEs at a plane interface of two homogeneous dielectric media Study of the fundamental wave in a hollow rectangular metal waveguide Study of an electromagnetic field in a cylindrical volumetric resonator Number of hours

19 Section 4. Propagation of EME near the Earth's surface Topic 4 .. Propagation of radio waves in free space Topic 4.3. The influence of the Earth's surface on the propagation of radio waves Investigation of a region of space that has a significant effect on the propagation of radio waves in a homogeneous environment Investigation of the influence of the Earth's surface on the propagation of radio waves 4 4 extramural training) Number and name of the section (topic) Section. Boundary problems of electrodynamics Topic .. Plane electromagnetic waves Topic.4. Plane EMW in an inhomogeneous medium Section 3. EMV in guiding systems. Electromagnetic oscillations in volumetric resonators Topic 3.1. Guided EMW and guiding systems Topic 3.3. Volumetric resonators Name of laboratory work Study of electromagnetic field polarization Study of reflection and refraction of plane EMEs at a plane interface of two homogeneous dielectric media Study of the fundamental wave in a hollow rectangular metal waveguide Study of an electromagnetic field in a cylindrical volumetric resonator Number of hours

20 Section 4. Propagation of EMW near the Earth's surface Topic 4 .. Propagation of radio waves in free space Topic 4.3. The influence of the Earth's surface on the propagation of radio waves Investigation of a region of space that has a significant effect on the propagation of radio waves in a homogeneous environment Investigation of the influence of the Earth's surface on the propagation of radio waves 4 4 Laboratory work (correspondence course) Number and name of the section (topic) Section. Boundary problems of electrodynamics Topic .. Plane electromagnetic waves Topic.4. Plane EMW in an inhomogeneous medium Section 3. EMV in guiding systems. Electromagnetic oscillations in volumetric resonators Topic 3.1. Guided EMW and guiding systems Topic 3.3. Volumetric resonators Name of laboratory work Study of electromagnetic field polarization Study of reflection and refraction of plane EME at the plane interface of two homogeneous dielectric media Study of the fundamental wave in a hollow rectangular metal waveguide Study of the electromagnetic field in a cylindrical cavity resonator Number of hours 4

21 Section 4. Propagation of EMW near the Earth's surface Topic 4 .. Propagation of radio waves in free space Topic 4.3. Influence of the Earth's surface on the propagation of radio waves Investigation of a region of space that has a significant effect on the propagation of radio waves in a homogeneous environment Investigation of the influence of the Earth's surface on the propagation of radio waves. 6. A point-rating system for assessing knowledge when using DOT The discipline Electrodynamics and radio wave propagation, as mentioned above, consists of two parts. The study of the first part of the course (electrodynamics) is carried out in the fifth semester and ends with an exam. The first part of the course contains three sections (twelve topics), in the study of which it is necessary to complete the first test consisting of two tasks. Each topic in the pivot note ends with a checklist of self-examination questions to be considered as practice tests with open assignment... After studying each topic, it is necessary to answer the questions of the training tests of the current (intermediate) control, which contains five questions. The study of each section ends with the answer to the questions of the midterm control test, which contains ten questions. The corresponding test numbers are listed by topic. The rating points are determined as follows: - for the correct answer to the question of the midterm control test - points; - for a correctly solved problem - 0 points. With successful work with the materials of the first part of the course, the student can receive x10x3 + 0x = 100 points. Overcoming the threshold of 70 points, as well as performing a cycle of laboratory work in sections and 3 during the examination session and obtaining a 5

22 couples in laboratory work, provides admission to the exam. The second part of the course is studied in the sixth semester and ends with an exam. The second part of the course consists of three sections (seven topics), in the study of which it is necessary to complete the second test, consisting of two tasks. Each topic in the keynote summary ends with self-examination questions to be considered as open-ended practice tests. After studying each topic, it is necessary to answer the questions of the training test of current (intermediate) control, consisting of five questions. The study of each section ends with the answer to the questions of the midterm control test, which contains ten questions. The corresponding test numbers are listed by topic. Determination of rating points for the second part of the course is carried out in the same way as for the first part. With successful work with the materials of the second part of the course, the student can receive x10x3 + 0x = 100 points. Overcoming the threshold of 75 points and performing a cycle of laboratory work during the examination session ensures admission to the exam. 3. Information resources of the discipline 3.1. Bibliographic list Main: 1. Kalashnikov, V.S. Electrodynamics and propagation of radio waves (electrodynamics): letter. lectures / V.S. Kalashnikov, L. Ya. Rhodes. SPb .: Publishing house of NWTU, Rhodes, L.Ya. Electrodynamics and propagation of radio waves (propagation of radio waves): study guide. complex: study guide / L.Ya. Rhodes. - SPb .: Publishing house of SZTU, Krasyuk, N.P. Electrodynamics and radio wave propagation: textbook. manual for universities / N.P. Krasyuk, N. D. Dymovich. - M .: Higher. shk., Additional: 6

23 4. Petrov, B.M. Electrodynamics and radio wave propagation: textbook. for universities / B.M. Petrov. ed., rev. Moscow: Hotline Telecom, Krasyuk, N.P. Distribution of VHF in the heterogeneous troposphere: textbook. allowance / N.P. Krasyuk, L. Ya. Rhodes. L .: SZPI, Chistyakov, D.A. Laws and equations of electrodynamics as a consequence of Maxwell's equations: lecture notes / D.A. Chistyakov. SPb .: SZPI, Chistyakov, D.A. Fundamentals of electrodynamics in problems with solutions: letters. lectures / D.A. Chistyakov. SPb .: SZPI, Chistyakov, D.A. Maxwell's equations physical axioms of electrodynamics: letter. lectures / D.A. Chistyakov. SPb .: SZPI, In the electronic library of NWTU at the address there are sources from the bibliographic list under the numbers: 1 ;; Supporting synopsis (scenario of the educational process) Discipline Electrodynamics and radio wave propagation, as mentioned above, is a fundamental discipline and is entirely based on courses in physics and higher mathematics... In this regard, starting to study it, it is necessary to restore in memory the basic information from the second part of the course. general physics(electricity and magnetism) and the following sections of higher mathematics: equations of mathematical physics, vector analysis, field theory. The main goal of the discipline is to study Maxwell's equations, their physical meaning and the use of these equations for solving applied problems of radio physics and radio engineering. The methodology and sequence of studying the discipline correspond to the list of topics thematic plan... The material of each topic is full of mathematical relationships, the physical interpretation of which is often quite difficult, so the study of the material requires serious, thoughtful work. 7

24 3..1. Basic concepts and definitions in electrodynamics Basic concepts and definitions are set out in on pages When studying this section, it is necessary to understand the purpose of the discipline in the training of radio engineers, its place and tasks in the system of modern concepts of natural science, paying special attention to the materiality of the electromagnetic field. It is necessary to understand that the electromagnetic field in all its manifestations is fully characterized by two main and four additional vectors. The electromagnetic field exists and is considered in various environments, which are classified according to the nature of the dependence of their electromagnetic parameters on time, spatial coordinates, the magnitude and direction of the vectors of the electromagnetic field existing in the given environment. All mathematical relationships for this course are written in SI units. Questions for self-examination 1. What are the main features of the electromagnetic field, confirming its materiality ?. What is the physical meaning of the vectors characterizing the electromagnetic field? 3. What form do the material equations for the vectors of the electromagnetic field have? 4. What classifications of media are used in electrodynamics? 3 ... Maxwell's equations - fundamental equations of electrodynamics The content of this section is presented in on pages It is necessary to pay attention to the fact that Maxwell's equations are the result of a generalization of a large number of physical laws, they are fundamental dependences of macroscopic electrodynamics, allowing to obtain all the basic relations of the theory of electromagnetism.

25 foot field. It should be understood that the sources of the electromagnetic field are electrically charged particles, either moving or at rest. In practical applications, the harmonic time dependence of the quantities included in Maxwell's equations is often used; therefore, it is convenient to use the symbolic method to represent them. Questions for self-test 1. What experimental laws underlie Maxwell's equations ?. What is the physical meaning of the displacement current? 3. What is the physical meaning of Maxwell's equations in integral and differential forms? 4. What is the difference between symmetric and asymmetric forms of writing Maxwell's equations? Energy characteristics of EMF The content of this section is described in pages The electromagnetic field as a type of matter has a certain energy. The conservation law is valid for him. An analytical representation of this law is the electromagnetic energy balance equation - the Umov - Poynting theorem. Questions for self-test 1. What energy components can be included in the equation of energy balance of the electromagnetic field ?. Write down the expression for the Poynting vector in the case of harmonic in time fields Electromagnetic waves - a form of existence of EMF The content of this section is given in pages It follows from Maxwell's equations that the electromagnetic field can be

26 to live in the form of electromagnetic waves. Adequate relations describing the wave nature of the electromagnetic field are wave equations - partial differential equations of the second order, which can be obtained directly from Maxwell's equations - partial differential equations of the first order. To solve various kinds of applied problems, wave equations for field vectors and wave equations for electrodynamic potentials are usually used. With a harmonic dependence of electrodynamic processes on time, the form of recording and the solution of wave equations are greatly simplified. Questions for self-test 1. What types of wave equations are used to solve problems of electrodynamics ?. What is the meaning of the gauge ratio? 3. What is the difference between the d'Alembert and Helmholtz equations from the generalized wave equation? 4. Is there a difference between the vector potential and the Hertz vector in the case of a harmonic electromagnetic field? Particular types of EMF equations The content of this section is given in on pages The equations of stationary and static fields are obtained as special cases from the equations of electrodynamics - Maxwell's equations, provided that the sources of the electromagnetic field are either stationary (independent of time), or, in addition, are also motionless (static). Stationary and static fields are material; the law of conservation and transformation of energy is fulfilled for them, but they do not have a wave character and the equations describing their behavior do not contain a time dependence (for example, the Poisson and Laplace equations). Self-Test Questions 10

27 1. Under what conditions does the system of Maxwell's equations decompose into systems of equations of electro- and magnetostatics ?. What is the difference between stationary and static fields? 3. What determines the value of the energy of the electrostatic field? 4. Write down the second order partial differential equations for static and stationary fields. 5. What methods are used to solve the problems of electrostatics? The main methods of solving problems of electrodynamics The content of this section is set out in on pages 1 7. When mastering this section, it is necessary to study the features of the formulation and solution of internal and external problems of electrodynamics, paying special attention to the formulation of conditions for the uniqueness of the solution of electrodynamic problems for limited and unlimited volumes of space, the basic principles and theorems used in constructing solutions to practical problems. Study the rigorous and approximate methods of solution, taking into account that the results of the solution by any rigorous methods coincide, while the results of the solution of the problem, obtained by various approximate methods, differ from each other. Questions for self-examination 1. How are the internal and external problems of electrodynamics formulated ?. What is the role of the radiation condition in solving external problems? 3. How is the uniqueness theorem for solving problems of electrodynamics formulated? 4. Under what conditions is the principle of superposition of solutions valid? 5. For which environments is the reciprocity theorem satisfied and what is its essence? 6. What is the role of the equivalence theorem for external problems of electrodynamics? 7. What underlies the solution of problems by the method of delayed potentials - 11

28 cials? 8. Under what conditions can the Kirchhoff method be considered as a rigorous solution method? 9. Formulate the conditions of applicability of the methods of geometric and wave optics. 10. What is the essence of the methods of edge waves and the geometric theory of diffraction? 11. What is the essence of the method of electrodynamic modeling? Plane electromagnetic waves (EMW) The content of the section is presented in on pages 7 4. In this section, it is necessary to pay attention to the fact that the concepts of phase and amplitude wave fronts are introduced to characterize any wave process. In the general case, the phase fronts can have an arbitrary shape, but the main ones are: flat, cylindrical, and spherical. To characterize vector wave processes, in addition to the amplitude, phase and frequency of oscillations, the concept of polarization is introduced. It is necessary to study all the existing types of polarization of electromagnetic waves. Here, one should consider the solution of the Helmholtz equations for the vectors of the electromagnetic field in the form of plane waves, paying attention to the various mathematical forms of writing expressions, the mutual orientation of the vectors of the strengths of the electric and magnetic fields and the Poynting vector, as well as the relationship between them and the electromagnetic parameters of the medium. It is necessary to study the features of the propagation of a plane wave in a dielectric, semiconductor and conductor, paying attention to the specifics of the propagation of a plane wave in media with conductivity (exponential decrease in amplitude, the appearance of a phase shift and dispersion). Questions for self-test 1. What is the difference between wave processes and oscillatory processes in radio circuits? 1

29. What additional characteristic is introduced to describe vector wave processes? 3. What types of polarization are usually considered in problems of electrodynamics? 4. What are the main properties of a plane wave? 5. What is the nature of the wavenumber in different media? 6. What are the features of plane wave propagation in conductive media? 7. What is the nature of the phenomenon of dispersion during the propagation of a plane wave in a semiconducting medium? 8. What does the nonlinearity and anisotropy of the medium lead to during the propagation of a plane wave? Spherical EME in limitless homogeneous media. EME emission The content of this section is given in on pages When studying this section, it is necessary to understand the formulation of the problem of the emission of electromagnetic waves, as well as the fact that radiation is created only by electric charges moving with acceleration. It is necessary to master the purpose of introducing the concept of an elementary emitter, types of models of elementary emitters and methods for calculating their characteristics. Attention should be paid to the features of the distribution of the electromagnetic field of an elementary emitter in space, depending on the distance and angular coordinates, to learn the features of the behavior of the Poynting vector. It is also necessary to know the basic technical characteristics of the emitters, such as radiation pattern, power and radiation resistance, and directivity. Questions for self-examination 1. What is the purpose of introducing the concept of an elementary emitter? 13

thirty . How is the problem of emission of electromagnetic waves formulated? 3. What solution method is used to calculate the radiation of an elementary electric dipole? 4. Name the characteristic zones of space and the separation criteria in which it is customary to consider the radiation field. 5. Describe the energy properties of the field radiated by an elementary radiator. 6. What are the characteristics of an elementary radiator as an antenna? 7. What models are used to describe an elementary magnetic emitter? 8. Compare the emissivity of elementary electric and magnetic emitters. 9. What is the directional diagram of the Huygens element? Plane EME in an inhomogeneous medium The content of this section is presented in on pages.When studying this section, the student must understand the formulation of the problem of reflection and refraction of a plane electromagnetic wave at a plane interface between media and the physics of phenomena occurring at the interface. It is necessary to know the method of obtaining the relations for the vectors of the electromagnetic field at the interface, paying attention to the areas of use of the boundary conditions. You should also study the content and meaning of such concepts as the angle of total internal reflection, Brewster angle, surface effect. Questions for self-test 1. What is the physics of reflection and refraction of a plane wave at the interface between media ?. How is the electrodynamic problem of reflection and pre-14

31 plane wave breaking at the interface? 3. What is the point of introducing boundary conditions? 4. How is the polarization of an electromagnetic wave incident on the interface determined? 5. What is the physical meaning of the phenomenon of complete polarization? 6. What is meant by the thickness of the skin layer? 7. Draw the behavior of the module and the phase of the reflection coefficient when a plane wave is incident on the interface as a function of the angle of incidence Guided EMW and guiding systems. Waveguides The content of this section is given in on pages In this section, you should study the existing types of guiding systems, the types and main features of electromagnetic waves propagating in them, consider the solution of the wave equation for rectangular and circular waveguides. It is necessary to understand the main parameters characterizing the operation of the waveguide: critical wavelength, wavelength in the waveguide, phase and group velocities, characteristic impedance of the waveguide. It is necessary to know and be able to graphically depict the structure of the main types of oscillations in a rectangular and circular waveguide, as well as be able to choose the dimensions of the waveguide to work on a given type of oscillation. You should also have an idea of ​​the distribution of currents on the walls of the waveguide and the systems of excitation and coupling of the waveguides. Questions for self-test 1. Name the currently existing types of guiding systems .. What is the difference between electric, magnetic and transverse electromagnetic waves in transmission lines? 3. What types of waves can propagate in waveguides, coaxial and wire transmission lines? 4. Formulate the formulation of the problem of the propagation of electromagnets - 15

32 thread waves in the waveguide. 5. What boundary conditions are used to solve the wave equation in a hollow metal waveguide? 6. To what extent can the phase and group velocities of electromagnetic waves in the waveguide change? 7. What type of vibration is usually called the main one? 8. Based on what conditions is the choice of the dimensions of the waveguide cross-section? 9. Formulate the requirements for devices for exciting electromagnetic oscillations in the waveguide Coaxial and two-wire transmission lines The contents of the section are presented on pages 4 9. In this section, it is necessary to study the basic concepts related to transverse electromagnetic waves, pay attention to the features of the distribution of electromagnetic waves along the transmission line and in its cross-sections. You should also be able to write down expressions for the main parameters characterizing the data of the transmission line: wave impedance, linear capacitance and inductance, attenuation coefficient, and the amount of transferred power. Self-Test Questions 1. Formulate the basic properties of shear waves in transmission lines. Draw a picture of the lines of force of an electromagnetic wave in the cross-sectional plane of coaxial and two-wire transmission lines. 3. Write down expressions for the main parameters of the transmission lines under consideration. Resonant resonators. The content of this section is presented in on pages. When studying this section, it is necessary to understand the purpose and con- 16

33 manual features of different types of cavity resonators. To get acquainted with the method for solving the wave equation for a cavity resonator built on the basis of a rectangular waveguide, the types and structure of the simplest modes of oscillations in it, as well as methods for calculating the main parameters of the resonator. You should know the main types of oscillations in cylindrical resonant resonators, methods of determining the natural resonance frequency, Q-factor and dimensions of the resonator, methods of excitation. Self-Test Questions 1. What types of cavity resonators are used in microwave technology? What types of oscillations can exist in resonant cavities? 3. How is the Q-factor of the cavity resonator determined? 4. From what considerations are the dimensions of cavity resonators based on rectangular and circular waveguides determined? 5. What systems of excitation of resonators are used in practice? Basic concepts and definitions in the theory of RRV The content of this section is presented in on page 4. In this section, it is necessary to pay attention to the role of Russian scientists in the development of the theory and development of technology for radio broadcasting, radio communication, television, and radar systems. It should be remembered that the decimal system of dividing the frequency range of waves into sub-ranges is currently accepted all over the world. It is necessary to have an idea of ​​the features of the propagation of radio waves in these sub-bands. Questions for self-test 1. What sub-bands are divided into the entire range of radio waves ?. What are the features of the propagation of radio waves of various subbands? 17

34 Propagation of radio waves in free space The content of this section is presented in on pages In this section, you should pay attention to the energy ratios for the propagation of radio waves of non-directional and directional radiators in free space. It is necessary to be able to derive and analyze the ideal radio communication equation; using the Huygens-Fresnel principle, construct the Fresnel zones and determine the essential and minimum areas of space that affect the propagation of radio waves. It is also necessary to pay attention to the fact that even with the propagation of radio waves in free space, there is a weakening of the energy flux of the electromagnetic field with distance. You should be able to explain the physics of this phenomenon and write the mathematical expression for the transmission loss in free space. Questions for self-test 1. How to determine the energy flux density and field strength of non-directional and directional emitters in free space ?. How is the Huygens-Fresnel principle formulated? 3. How are Fresnel zones constructed in free-space RWE? 4. What are the reasons for determining the essential and minimum areas affecting the RWP in free space? 5. How to explain the process of weakening the electromagnetic field in free space? Influence of the Earth's surface on the propagation of radio waves The content of this section is presented in on pages In this section it is necessary to understand that the Earth's surface has a significant impact on the RWE. This influence is taken into account by introducing a free space field weakening factor, which is calculated based on the specific type of radio path. You need to know the electromagnetic parameters 18

35 main varieties of the earth's surface. To determine the attenuation factor, it is necessary to solve the complex problem of radio wave diffraction around the real surface of the Earth. It should be borne in mind that at present this problem, even in the most rigorous formulation, does not take into account the roughness of the Earth's surface and is solved for a smooth spherical surface. The expressions obtained, even with this formulation of the problem, are extremely complex and the calculation of the attenuation factor is possible only with the use of a computer, therefore, in engineering practice, for some radio paths, approximate solution methods are used based on interference formulas in the illuminated region and a single-term diffraction formula in the region deep shadow. Approximate methods are also used to take into account the influence of the real distribution of the Earth's parameters along the radio path and the irregularities of its surface. Attention should be paid to the following phenomena: coastal refraction (curvature of the trajectories of an electromagnetic wave); the effect of increasing the magnitude of the electromagnetic field due to obstacles; on the abrupt change in the magnitude of the electromagnetic field when crossing the border of the sections of the route with different electromagnetic parameters. Irregularities on the Earth's surface are randomly distributed, which leads to the need to apply the methods of mathematical statistics when studying the processes of radio wave propagation over such uneven surfaces. Questions for self-examination 1. How is the influence of the Earth's surface on the RRV taken into account ?. What electromagnetic parameters characterize the Earth's surface? 3. How is the problem of radio wave diffraction around the Earth's surface formulated? 4. What characteristic areas of space are customary to highlight when studying 19

Methodical instructions on the study of the disciplines "Electrodynamics and radio wave propagation" and "Electro magnetic fields and waves "for students VDBV-6-16 List of literature Main literature 1. Nikolsky V.V.,

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Article

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Zaboronkova, T.M. Fundamentals of electrodynamics and radio wave propagation:
teaching aid / T.M. Zaboronkova, E.N. Myasni-
kov. - N. Novgorod: Publishing house of FGOU VPO "VGAVT", 2009. - 133 p.

Content:
Static electric and magnetic fields,
Electrostatic field,
Constant electric current,
Stationary magnetic field,
The movement of charged particles in constant electric and magnetic fields,
Electromagnetic field, Maxwell equations,
The law of electromagnetic induction,
Displacement current, Maxwell's system of equations,
Averaged Maxwell-Lorentz equations in material media,
Boundary conditions for electric and magnetic fields,
Electromagnetic waves in free space,
Plane monochromatic electromagnetic wave,
Polarization of electromagnetic waves,
Spherical electromagnetic waves in free space,
Radiation of electromagnetic waves by an elementary vibrator,
Electromagnetic waves in homogeneous material media,
Electromagnetic waves in a homogeneous isotropic dielectric,
Electromagnetic waves in a medium with absorption,
Dispersion of dielectric constant,
Propagation of packets of electromagnetic waves group velocity,
Energy transfer by a packet of waves,
Dispersion and Resonant Absorption of Molecular Gas
Electromagnetic waves in plasma,
Ionospheric plasma parameters,
Electromagnetic waves in a homogeneous isotropic plasma,
Electromagnetic waves in a homogeneous magnetoactive plasma,
The incidence of electromagnetic waves at the interface of homogeneous media,
Reflection and refraction of waves from a flat interface between two media,
Reflection from a perfectly conductive surface,
Reflection from an imperfect conductor
Propagation of electromagnetic waves in a smoothly inhomogeneous medium,
Smoothly inhomogeneous medium, geometric optics approximation,
Refraction of radio waves in the Earth's atmosphere,
Reflection of radio waves from a layer of inhomogeneous plasma. ,
Features of the reflection of radio waves from the ionosphere when taking into account the magnetic field,
Interference and diffraction of electromagnetic waves,
Interference of plane monochromatic waves,
Huygens-Fresnel-Kirchhoff principle,
Fraunhofer diffraction,
Fresnel diffraction,
Diffraction of radio waves by random inhomogeneities of the electron density,
Propagation of radio waves in the Earth's atmosphere,
Ideal radio path, radio wave ranges,
Influence of the underlying surface on the propagation of radio waves,
Influence of the troposphere on the propagation of radio waves,
Propagation of radio waves in the ionosphere.

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1.1 Electromagnetic field

An electromagnetic field consists of an electric field that is interdependent with a magnetic field. The electric field represents the vector of electric induction, functionally dependent on the vector of the electric field strength ... Magnetic field represent the vector of magnetic induction
, functionally dependent on the magnetic field strength .

The vectors of the electromagnetic field in the general case represent a non-stationary electromagnetic vector field, which is a function of coordinates and time:

- electrical induction;

- magnetic induction.

The stationary electromagnetic vector field is a function of coordinates and does not depend on time:

- electric field strength;

- magnetic field strength;

- electrical induction;

- magnetic induction.

The speed of propagation of electromagnetic waves in a vacuum is equal to the speed of light

c = 3 · 10 8 m / s.

where λ is the wavelength, m;

T - period, s.

Frequency , Hz

c = λf

Circular frequency, s -1

ω = 2πf.

The longer the electromagnetic wavelength, the lower the frequency. Electromagnetic waves begin at a lower frequency, then radio waves of the ultra-long, long wave ranges begin, then medium waves with a higher frequency, short, ultra-short waves with an even higher frequency. Radio waves are followed by infrared radiation with a shorter wavelength, but higher frequency than radio waves. Visible light begins with red waves. The names of the flowers begin with letters in the order of the saying: "Every hunter wants to know where the pheasant is sitting." Visible light ends in purple waves. This is followed by: ultraviolet, x-ray, gamma radiation and cosmic radiation.

The theory of the electromagnetic field is based on vector calculus and vector fields, the most important provisions of which will be discussed below.

1.2 Scalar and vector fields

1.2.1 Potential (irrotational) and vortex vector fields

Potential (vortex) field linesstart at the source and end at the drain. The lines of the vortex (solenoidal) field have no sources, are always closed, continuous( see picture[ 4 ] ) .

R Figure - Potential (irrotational) and vortex fields

Circulation vector potential field along a closed loopL is zero

Flow the vortex field vector through the closed surface Sis equal to zero

The electrostatic field can only be potential (irrotational), the magnetic field is only vortex.

1.2.2 Scalar field gradient, Hamilton operator

The gradient (drop) of the scalar field φ is a vector showing in which direction φ increases most rapidly, equal in magnitude to the derivative in this direction

Conditional vector or Hamilton operator

The gradient of a scalar field φ, written using the Hamilton operator (the "nabla" operator)

The surface of the φ level contains the same values ​​φ = const of the scalar field; therefore, the gradient of the scalar field φ is perpendicular to the surface of the φ level and is directed towards increasing φ (see figure [4]).

Figure - Scalar field gradient

1.2.3 Divergence (divergence)

A vector field is given at the point (x; y; z)

where
- unit vectors (unit vectors) in the directions of the coordinate axes x, y, z, respectively.

For a vector field at the point (x; y; z), the divergence (divergence) at the point P is equal to the limit of the vector flux through the surface S, limiting scope V divided by V as V tends to zero

Divergence values ​​in points P vector fields (see figure [4]).

Figure - Divergence values

With a divergence greater than zero

inside the region V the sources of the vector field are found.

With negative divergence

inside region V are the sinks of the vector field.

With a divergence equal to zero

with the silt lines of the field cut across the area V or closed (vortex field).

1.2.4 Rotor (vortex)

The rotor (vortex) allows you to estimate the degree of rotation at some point ( x; y; z ) of the vector field

where are unit vectors (unit vectors) in the directions of the coordinate axes x, y, z, respectively.

For a vector field at the point (x; y; z), the projection of the rotor onto the normal direction to the surface, equal to the circulation limit of the vector around the contour C, divided by the areaΔ S of the surface bounded by the contour C, when tending Δ S to zero

The direction of the normal is related to the direction of traversing the contour C by the right-hand screw rule.

Rotor (vortex) of a vector field using the Hamilton operator

Vector projections
on the coordinate axis

If at point P rotor is zero

,

then there is no rotation at this point and the vector field is potential.

1.3 Types of charge distribution

Bulk charge density, C / m 3

The charge concentrated in the volume V, C

Surface charge density, C / m 2

The charge concentrated on the surface S, C

Lines charge density, C / m

Thread charge , Cl

The charge of point charges is equal to the sum of N charges of a finite value

1.4 Electric field

Electrical displacement vector (electrical induction) is equal to the electrical constant ε 0, multiplied by the parenthesis, in which the unit is added to the electrical susceptibility χ e, multiplied by the vector of the electric field strength

Electric constant

The vector of electrical displacement (electrical induction) in a substance

where ε is the absolute electrical permeability.

Electric induction vector in vacuum

.

1.5 Magnetic field

Vector of magnetic induction is equal to the magnetic constant μ 0, multiplied by the parenthesis, in which one is added to the magnetic susceptibility χ m, multiplied by the vector of the magnetic field

Magnetic constant

Vector of magnetic induction in matter

where μ - absolute magnetic permeability.

Vector of magnetic induction in vacuum

1.6 Ohm's law in differential form

Ohm's law for a section of a chain

U = IR

Current density

Let us express

We will integrate over and we obtain the dependence of the current on the current density

Ohm's law in differential form allows you to determine the current density, A / m 2

where σ is the specific conductivity of the medium, S / m.

2 Maxwell's equations

The system of Maxwell's equations in differential form describes alternating electromagnetic fields

The vectors in Maxwell's equations represent a non-stationary electromagnetic vector field that is a function of coordinates x, y, z and time t.

2.1 Special cases of electromagnetic phenomena

In special cases, Maxwell's equations can be simplified.

2.1.1 Stationary electromagnetic field

A stationary electromagnetic field is created direct currents and is described by time-independent vector functions of coordinates:

Electric field strength;

Electric induction;

Magnetic field strength;

Magnetic induction.

The vector functions do not depend on time, so the partial derivatives with respect to time in Maxwell's equations are equal to zero:

The system of Maxwell's equations in differential form takes the form that describes a stationary electromagnetic field:

2.1.2 Static electric or magnetic fields

Static fields do not change over time and do not have moving charges, therefore, currents

.

Maxwell's system of equations is divided into two independent systems of equations. The first system characterizes the electrostatic field and is called the system of differential equations of electrostatics

The second system of equations describes the magnetostatic field created by permanent fixed magnets

This system of equations can be used to describe magnetic fields created by direct currents, but in areas where the current density is zero and which are not coupled to the current (do not cover streamlines).

2.1.3 Maxwell's equations in complex form

If the vectors of the electromagnetic field change in time according to harmonic laws, then the system of Maxwell's equations can be represented in a complex form that does not contain time for complex vectors

or complex amplitudes

2.1.4 Wave equations

From Maxwell's equations in complex form, expressing separately the equations for complex vectors and we get wave Helmholtz equations for vectors

and complex amplitudes

where - wave number, for vacuum

.

3 Plane electromagnetic waves

At large distances from the source, the spherical wave element can be approximately assumed to be flat. Plane waves cannot be created by sources; they were invented to significantly simplify the theory of electromagnetic waves in individual cases.

The vectors of the electric and magnetic fields of the plane wave are in phase and oscillate along mutually perpendicular directions in the plane perpendicular to the direction of wave propagation. Such waves are transverse (see figure).

Figure - Instantaneous picture of the distribution of the strength of the electric and magnetic fields along the direction of propagation of a plane wave. In time, the picture of the field moves in space with a phase velocity v f along the z axis

The wave front is a locus of field points with the same phase: for a plane wave (see figure), one of these surfaces is the plane z = z 0, perpendicular to the direction of wave propagation. The field parameters do not change when moving within the wave front.

The front of a plane wave is a plane perpendicular to the direction of wave propagation. The field parameters do not change when moving within this plane, so the partial derivatives in the x and y directions are equal to zero:

In the news Helmholtz equationsfor a plane wave become one-dimensional for vectors

and complex amplitudes

Solving differential equations for vectors

where , - unit vectors in the direction of vectors of electric and magnetic intensities, respectively;

A, B, C, D - coefficients.

Real parts of vectors

Let us analyze the first term in the first equation. In the figure, we show the position of the maximum of the electric field at times t (point A) and t + Δ t.

Figure - Position of the maximums of the electric field

During Δ tthe position of the maximum has moved toΔ z,we can write equality

A cos (ωt - kz) = A cos (ωt + ωΔt - kz - k Δz),

in which the arguments are equal

ω t - kz = ωt + ωΔt - kz - k Δz

0 = ωΔt - kΔz

ωΔt = kΔz.

From this we obtain the phase velocity v f - wave front propagation speed

For vacuum

therefore, the phase velocity in vacuum

Substitute the values ​​of the constants

therefore, in a vacuum, the speed of propagation of the wave front is equal to the speed of light.

Phase velocity in some environment

The phase speed is independent of frequency.

Amplitudes of two points at a wavelength distance λ with phases differing by 2π are equal, therefore, the equality

cos (ωt - kz) = cos (ωt - k (z + λ) + 2π),

in which the arguments are equal

ωt - kz = ωt - k (z + λ) + 2π,

ωt - kz = ωt - kz - kλ + 2π.

Reduce ω t - kz

0 = − k λ + 2π,

k λ = 2 π.

Hence the wavelength

For any environment

,

therefore the wavelength

In a vacuum, the wavelength

Wavelength in other media

Characteristic impedance of vacuum

For dry air, the same characteristic impedance is assumed.

4 Propagation of radio waves

All electromagnetic waves, including radio waves, propagate in a vacuum at a speed of 3 · 10 8 m / s.

4.1 Propagation of radio waves in free space

The propagation of radio waves in the atmosphere, along the earth's surface, in the earth's crust, in the outer space of our galaxy and beyond will be taken as free propagation of radio waves, which we will consider.

4.1.1 Classification of radio waves by bands

Radio waves have a frequency range from thousands of hertz to thousands of gigahertz: 3 · 10 3 - 3 · 10 12 Hz. Long waves have a lower frequency than shorter waves, which have a higher frequency.

The use of radio waves is possible due to the transmitting device, the natural propagation medium of the radio waves and the receiving device, all of which form a radio link.

The Earth's atmosphere and surface are absorbing media, electrically inhomogeneous, having a conductivity not constant in time and space, dielectric constant, depending on the frequency of propagating radio waves.

Therefore, radio waves were divided into frequency ranges with approximately the same propagation conditions of radio waves within these frequency ranges. The frequency bands are adopted by the International Radio Consultative Committee (CCIR) in accordance with the Radio Regulations.

For radio communication, waves of the optical range are also used: infrared, visible and ultraviolet.

The power of electromagnetic waves depends on the frequency to the 4th power

P ~ ω 4.

Waves with a higher frequency, but with a shorter wavelength, can be more powerful.

Antennas with a narrow radiation pattern are much larger than the wavelength, for high frequencies it is easier to make such high-performance antennas.

The higher the carrier frequency, the more independent modulated channels can be transmitted by such radio waves.

4.2 Positions from antenna theory

The space around the antenna is divided into three areas with different field structures and calculation formulas: near, intermediate and far. In real communication lines, there is usually a far region (Fraunhofer zone) at distances from the antenna

where L - maximum size of the radiating area of ​​the antenna, m;

λ - wavelength, m

Characteristic (wave) resistance of a free medium

Poynting vector (Umov - Poynting vector), W / m 2

where P - power, W;

r - distance from antenna to observation point, m

where D - antenna directivity (directivity).

Average value of the Poynting vector in the far field

From the ratio

we express the amplitude of the magnetic field strength

Substitute

Let us equate the Poynting vectors

Reduce

The amplitude of the electric field in the far zone of the antenna in free space

The field strength in other directions is determined using the antenna pattern F (θ, α), in which the angles θ and α in the spherical coordinate system (r, θ, α) define the direction to the observation point:

5 Propagation of radio waves of various bands

5.1 Propagation of very long and long waves

Ultra-long waves (VLW) have a wavelength of more than 10,000 m and a frequency of less than 30 kHz. Long waves (LW) have a wavelength of 1000 to 10,000 m and a frequency of 300-30 kHz.

SDV and DV have a long wavelength, therefore they bend around the earth's surface well. The conduction currents of these radio waves are significantly higher than the displacement currents for all types of the earth's surface, therefore, there is a slight absorption of energy during the propagation of the surface wave. Therefore, VLF and VL can spread over distances of up to 3 thousand km.

LF and LW are weakly absorbed in the ionosphere. The lower the frequency of the radio wave, the lower the electron concentration of the ionosphere is required to turn the radio wave towards the Earth. Therefore, the rotation of the VLF and LW occurs in the lower boundary of the ionosphere (during the day in the D layer and at night in the E layer) at an altitude of 80-100 km. The troposphere has practically no effect on the spread of ADV and DV. Around the Earth, VLF and DV propagate, reflecting from the ionosphere and from the earth's surface in a spherical layer of 80-100 km between the lower boundary of the ionosphere and the earth's surface.

Communication lines at SDV and DV have a high stability of the electric field strength. During the day and year, the signal magnitude changes little, and also does not undergo random changes. Therefore, SDV and DV are widely used in navigation systems.

The limited frequency range (3-300 kHz) of VLF and LW does not allow placing even one television channel, which requires an 8 MHz band.

Long wavelength VLF and LW dictates the use of bulky antennas.

Despite the shortcomings, SDV and DV are used in radio navigation, radio broadcasting, radiotelephone and telegraph communications, including with underwater objects, since these and optical waves are weakly absorbed in sea water.

5.2 Propagation of medium waves

Medium waves (MW) have a wavelength of 100 to 1,000 m and a frequency of 300 kHz to 3 MHz (0.3 to 3 MHz). Terrestrial and ionospheric SWs, which are used mainly in radio broadcasting, can propagate.

Terrestrial SW radio lines are limited to no more than 1000 km in length due to significant absorption of SW by the earth's surface.

Ionospheric SW can be reflected from layer E ionosphere. Through the lowest layer D ionosphere, appearing only during the day, SW pass through and are strongly absorbed in it,practically eliminating communication during the day. Therefore, SW absorption in the ionosphere decreases significantly at night.and at distances greater than 1000 km from the transmitter, communicationis recovering.

Due to the interference of ionospheric waves with each other or (and at night) with earth waves, random signal fading (fading) occurs. Anti-fading antennas have a maximum radiation pattern pressed to the ground to combat fadingand cross modulation on SV.

5.3 Short wave propagation

Short waves (HF) have a wavelength from 10 to 100 m (10 times shorter than medium waves), a frequency from 3 to 30 MHz (10 times more than the SW frequency). HF are used primarily for radio broadcasting.

HFs are strongly absorbed by the earth's surface and poorly bend around the Earth's surface, therefore, terrestrial HFs propagate only over several tens of kilometers.

HFs experience absorption and pass in the lowest layers of the ionosphere D and E, but reflected from the layer F.

Calculation of HF communication lines consists in drawing up a schedule of operating frequencies depending on the time of day (wave schedule).

5.4 Features of propagation of ultrashort waves

Ultra-short waves (VHF) have a wavelength of less than 10 m and a frequency of more than 30 MHz. In terms of frequency, VHF borders on HF from the bottom, and infrared waves from above. The ionosphere for VHF is transparent, therefore VHF lines are used mainly within the line of sight.

VHF has a large frequency range capable of transmitting significant amounts of information. On meter and decimeter waves, 297 television channels can be placed. Only 3 television channels will be located in the entire short-wave range, and none in the entire MW range.

The development of mobile and satellite communications, the Internet and other above-mentioned reasons force radio engineering to switch to higher frequencies, therefore VHF is becoming more and more important.

5.4.1 Line-of-sight propagation of ultrashort waves

Line-of-sight VHF communication lines:

VHF and television broadcasting;

Radar stations (radar);

Radio relay communication lines (RRL);

Communication with space objects;

Mobile connection.

5.4.2 VHF propagation over the horizon

Far propagation of VHF beyond the horizon line occurs in the following ways:

Due to scattering on tropospheric irregularities;

Superrefraction in the troposphere;

Scattering by ionospheric irregularities;

Due to the reflection from the layers of the ionosphere F 2 and E S;

- due to reflection from meteor trails;

Due to the strengthening of the obstacle (see figure)

Figure - Propagation of radio waves when amplified by an obstacle

List of symbols, symbols, units and terms

D, B - vectors of electric and magnetic induction

E, H - vectors of electric and magnetic field strengths

I (r, t) - electric current

j (r, t) is the electric current density vector

P - the power of the electromagnetic field

M is the magnetization vector

P is the vector of electric polarization

q - electric charge

ε, μ - absolute permittivity and permeability

ε 0, μ 0 - dielectric and magnetic constants

ε r, μ r - relative permittivity and permeability

P - Poynting vector (Umov - Poynting vector)

ρ, ξ, τ - density of volumetric, surface and linear charge

σ - specific conductivity of the medium

ϕ - scalar electrostatic potential

χ e, χ m - electrical and magnetic susceptibility

W is the energy of the electromagnetic field

W e, W m - the energy of the electric and magnetic field

w is the energy density of the electromagnetic field

w e, w m are the energy densities of the electric and magnetic fields

k - wavenumber

SDV - superlong waves

DV - long waves

CB - medium waves

KV - short waves

VHF - ultrashort waves

Radar - radar station

RRL - radio relay line

D - antenna directivity (directivity)

G - antenna gain

F (θ, α) - antenna pattern

R 0 - the radius of the Earth (6371 km)

Z 0 - wave impedance of free space

List of sources used

1.Electrodynamics and radio wave propagation: textbook. allowance / L.A. Bokov, V.A. Zamotrinsky, A.E. Mandel. - Tomsk: Tomsk. state un-t of control systems. and radio electronics, 2013 .-- 410 p.

2.Morozov A.V. Electrodynamics and radio wave propagation: a textbook for higher education. military studies. institutions / Morozov A.V., Nyrtsov A.N., Shmakov N.P. - M.: Radiotekhnika, 2007 .-- 408 p.

3. Yamanov D.N. Fundamentals of electrodynamics and radio wave propagation. Part I. Fundamentals of electrodynamics: Lecture texts. - M .: MGTU GA, 2002 .-- 80 p.

4.Panko V.S. Lectures on the course "Electrodynamics and Radio Wave Propagation".

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