Introduction - Electrical and Information Technology

ETEN10:1 Antenna Technology Mats Gustafsson Department of Electrical and Information Technology Lund University ETEN10, HT2, 2015 Outline 1 ETEN10 overview 2 Antenna background 3 Antenna parameters Mats Gustafsson, ETEN10:1(2)-2015, EIT, Lund University, Sweden ETEN10 Antenna technology: Aim The student shall I acquire fundamental knowledge of antenna theory. I acquire good ability to analyse and design antennas. I acquire good knowledge of antenna parameters such as directivity and radiation pattern. SonyEricsson T68, 2001 [7] I carry through and document a project in which an antenna is designed, fabricated and measured. VLA in New Mexico 2006 Mats Gustafsson, ETEN10:1(3)-2015, EIT, Lund University, Sweden Rome 2012. Patch antenna, cf., the lab project. Examination In order to pass the course you must: 1. do all laboratory sessions 2. write a laboratory report (covering the laboratory sessions) 3. do a written five hour examination (max 60p, grade 3 for 30-39p, 4 for 40-49p, and 5 for 50-60p). Pass on the electronic quiz offers 5 bonus points on the written exam in January 2016. Retake exams in April and August. Note, no bonus points on the retake exams. Mats Gustafsson, ETEN10:1(4)-2015, EIT, Lund University, Sweden Overview I 2 lectures per week. I 1 problem solving class (computer exercise)/week. I 3 laboratory sessions. I Electronic quiz (7 tests, 5p on the written exam in January 2016). I Written examination (max 60p, pass 30p). I http://www.eit.lth.se/course/ETEN10 I Mats Gustafsson, Anders J. Johansson, Doruk Tayli, Zachary Miers Mats Gustafsson, ETEN10:1(5)-2015, EIT, Lund University, Sweden Overview: Literature I (KM) Antennas by J.D. Kraus and R.J. Marhefka. I (S0) Electromagnetic Waves and Antennas by S.J. Orfanidis. (www.ece.rutgers.edu/~orfanidi/ewa/) I Chapter 8 and 9 in (RB) ’Fundamentals of Engineering Electromagnetics’ by R. Bansal. I Parts of chapter 7 in (BRI) ’Computational Electromagnetics’ by A. Bondeson, T. Rylander and P. Ingelström. I Download the e-books from the homepage. I Lecture notes (download from moodle). Alternative literature: Antenna Theory: Analysis and Design by Balanis, Antenna Theory and Design by Stutzman and Thiele, Foundations of Antenna Engineering by Kildal (download from http://www.kildal.se), ... Mats Gustafsson, ETEN10:1(6)-2015, EIT, Lund University, Sweden Schedule 2015 See homepage for details. Laboratory lessons preliminary for week 3, 5, and 6, see the homepage. Mats Gustafsson, ETEN10:1(7)-2015, EIT, Lund University, Sweden input impedance beam steering pattern synthesis reflection coefficient efficiency Arrays Parameters side lobes gain mobile phones Q-factor MIMO array factor directivity power transfer lens horn reflector radio, TV patch Applications radar antenna elements Antenna technology dipole DRA astronomy monopole nano, optics Yagi-Uda RFID anechoic chamber helix modes energy VNA, Sparameters GO,PO near field Measurements Analysis current to far field FEM SAR far field Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden FDTD MoM Arrays side lobes -factor MIMO Many different types, sizes, and shapes of antennas. Some common antenna types are: array factor lens horn reflector patch antenna elements Antenna technology dipole DRA monopole Yagi-Uda helix modes energy VNA, Srameters GO,PO Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden Arrays side lobes -factor MIMO array factor lens horn reflector patch antenna elements Antenna technology dipole DRA monopole Yagi-Uda helix modes energy VNA, Srameters GO,PO Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden Antenna elements collected into arrays. input impedance beam steering pattern synthesis reflection coefficient Arrays Parameters side lobes Q-factor MIMO array factor directivity lens horn reflector patch Antenna Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden antenna input impedance beam steering pattern synthesis 0o 45o reflection 0 45 coefficient o 45o o 45 o −30−20−10 90o 90o dB 90o Parameters 135o 135o 180o −30−20−10 90o dB Arrays 135o 135o 180o side lobes Q-factor MIMO array factor directivity lens horn reflector patch Antenna Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden antenna Parameters used to characterize antennas include: input impedance pattern synthesis reflection coefficient efficiency Arr Parameters gain mobile phones Q-factor MIMO arra directivity power transfer radio, TV Applica- Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden Antenna The impedance at the antenna terminals Z = V /I input impedance Radiated power normalized by the input power. efficiency Parameters Radiation intensity normalized by the accepted power. G = P/PT mobile phones power transfer Applica- gain Reflected pattern signal Γ Uin , Z−Z 0 . Often synthesis Γ = Z+Z 0 reflection |Γ | ≤ 1/3 for coefficient matched antennas. Stored energy normalized by the average dissipated MIMO Q-factorenergy Q = Arr 2ω max{We ,Wm } . Pd directivity Fractional bandwidth B ≈ 0.7/Q for arra Γ0 = 1/3. Radiation intensity normalized by the average radiated radio, TV power. D = P/Pave Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden Antenna Parameters gain Q-f Common applications for antennas include: directivity power transfer mobile phones radio, TV Applications radar te astronomy nano, optics RFID anechoic chamber VN para Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden near field Measure- Parameters gain directivity power transfer mobile phones Q-f radio, TV Applications radar te astronomy nano, optics RFID anechoic chamber VN para Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden near field Measure- Applications radar Antenna technology astronomy nano, optics Antennas are measured using: RFID anechoic chamber mod VNA, Sparameters GO,PO near field Measurements Ana FEM SAR far field Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden FDTD Applications radar Antenna technology astronomy nano, optics RFID anechoic chamber mod VNA, Sparameters GO,PO near field Measurements Ana FEM SAR far field Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden FDTD antenna elements Antenna technology dip DRA monopole Antennas are analyzed using e.g., : anechoic chamber Yagi-Uda helix modes energy VNA, Sparameters GO,PO Measurements ld Analysis current to far field FEM SAR FDTD Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden MoM antenna elements Antenna technology dip DRA modal solutions to Maxwell’s eq. monopole Yagi-Uda anechoic chamber modes Geometrical optics and physical optics. High frequency VNA, approximations forSparameters Maxwell’s eq. ld helix energy GO,PO Measurements Analysis Finite Element Method. Stored energy in fields or currents. Fourier transform to current to determine the far-field far field from a current density. FEM Method of Moments SAR FDTD Finite Differences in the Time-Domain. Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden MoM (integral solver for Maxwell’s eq.) input impedance beam steering pattern synthesis reflection coefficient efficiency Arrays Parameters side lobes gain mobile phones Q-factor MIMO array factor directivity power transfer lens horn reflector radio, TV patch Applications radar antenna elements Antenna technology dipole DRA astronomy monopole nano, optics Yagi-Uda RFID anechoic chamber helix modes energy VNA, Sparameters GO,PO near field Measurements Analysis current to far field FEM SAR far field Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden FDTD MoM input impedance beam steering First three lectures: pattern synthesis reflection coefficient efficiency I dipole antennas I antenna parameters Arrays Parameters I lobes determine side gain mobile phones Q-factor the far-field from current distributions MIMO radio, TV Method oflensmoments inhornthe computer reflector exercise patch Applications radar I array factor directivity power transfer antenna elements Antenna technology dipole DRA astronomy monopole nano, optics Yagi-Uda RFID anechoic chamber helix modes energy VNA, Sparameters GO,PO near field Measurements Analysis current to far field FEM SAR far field Mats Gustafsson, ETEN10:1(8)-2015, EIT, Lund University, Sweden FDTD MoM CST Microwave studio I I Commercial program (www.cst.com). You can run the program on your PC. I I I Also available in E:4118, E:4119, use: Note, create your own folder (use your name) and store your data on some external memory, e.g., USB. Mats Gustafsson, ETEN10:1(9)-2015, EIT, Lund University, Sweden CST Microwave studio I I Commercial program (www.cst.com). You can run the program on your PC. I I I Also available in E:4118, E:4119, use: Note, create your own folder (use your name) and store your data on some external memory, e.g., USB. Mats Gustafsson, ETEN10:1(9)-2015, EIT, Lund University, Sweden CST Microwave studio I I Commercial program (www.cst.com). You can run the program on your PC. I I I Also available in E:4118, E:4119, use: Note, create your own folder (use your name) and store your data on some external memory, e.g., USB. Mats Gustafsson, ETEN10:1(9)-2015, EIT, Lund University, Sweden First computer exercise In this first computer exercise you should 1. investigate basic properties of dipole antennas. 2. practice on antenna parameters such as reflection coefficient, impedance, radiation pattern, and directivity. 3. get familiar with the program CST. You will use it later in the course and in the laboratory part of the course. Preparation: Study the lecture notes and Ch. 15 in Orfanidis. Do the exercises in Ch. 15, see the instructions (ETEN10_E1.pdf download from moodle). Mats Gustafsson, ETEN10:1(10)-2015, EIT, Lund University, Sweden Moodle (lecture notes and electronic quiz) You log in using your STIL or lucat identity. More about moodle and the quiz next lecture. Mats Gustafsson, ETEN10:1(11)-2015, EIT, Lund University, Sweden Moodle (lecture notes and electronic quiz) You log in using your STIL or lucat identity. More about moodle and the quiz next lecture. Mats Gustafsson, ETEN10:1(11)-2015, EIT, Lund University, Sweden Moodle (lecture notes and electronic quiz) You log in using your STIL or lucat identity. More about moodle and the quiz next lecture. Mats Gustafsson, ETEN10:1(11)-2015, EIT, Lund University, Sweden Outline 1 ETEN10 overview 2 Antenna background 3 Antenna parameters Mats Gustafsson, ETEN10:1(12)-2015, EIT, Lund University, Sweden Antennas Antenna definition (IEEE) ”That part of a transmitting or receiving system that is designed to radiate or to receive electromagnetic waves.” I Transition from guided waves on transmission lines to free space waves. I EM waves on transmission lines are studied in other courses (e.g., Ch. 10) EM-waves in free space Transmission line with guided EM-waves Antenna EM waves in free space are studied in other courses (e.g., Ch. 2) See also http://en.wiktionary.org/wiki/antenna and http://en.wikipedia.org/wiki/Antenna_(radio) I Mats Gustafsson, ETEN10:1(13)-2015, EIT, Lund University, Sweden Early history I J.C. Maxwell [1] I 1800s Wire telegraph I 1864 James Clerk Maxwell (EM-theory) I 1887 Heinrich Rudolf Hertz (Experimental verification of EM waves) I Guglielmo Marconi develops ”wireless” and radio (Nobel Prize 1909) and Loomis, Popov, Tesla,... G. Marconi [5] Mats Gustafsson, ETEN10:1(14)-2015, EIT, Lund University, Sweden Early history II I Heinrich Rudolf Hertz (1857–1894) I Experimental verification of EM waves. Hertz did not realize the practical importance of his experiments. He stated that, Heinrich Rudolf Hertz [6] ”It’s of no use whatsoever[...] this is just an experiment that proves Maestro Maxwell was right - we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there.” Asked about the ramifications of his discoveries, Hertz replied, ”Nothing, I guess.” Mats Gustafsson, ETEN10:1(15)-2015, EIT, Lund University, Sweden Applications SonyEricsson T68, 2001 I Communication Radio, TV, mobile phone, Satellite I Radar (RAdio Detection And Ranging), Ground-penetrating radar (GPR), Microwave tomography I RFID (Radio-frequency identification) I Positioning, e.g., GPS I Radio astronomy I Optics, nano antennas VLA in New Mexico [4] RFID-TAG [3] Mats Gustafsson, ETEN10:1(16)-2015, EIT, Lund University, Sweden Wireless communications Base stations: I High efficiency I Directivity Terminal antennas: I Many small antennas in modern mobile phones. I MIMO (many antennas), RFID, TV... I Co-exist with camera, battery, ..., user I Must be cost efficient. Base station in Nattavaara by. Mats Gustafsson, ETEN10:1(17)-2015, EIT, Lund University, Sweden SonyEricsson T68, 2001 [7] Radio astronomy I Often large reflector antennas. I Often large arrays. I The VLA (Very Large Array) in Socorro, New Mexico has 27 antennas that together offer a resolution of approximately 0.2 arcsec ≈ 1 µrad at λ = 3 cm. The VLA and Washington DC Mats Gustafsson, ETEN10:1(18)-2015, EIT, Lund University, Sweden Outline 1 ETEN10 overview 2 Antenna background 3 Antenna parameters Mats Gustafsson, ETEN10:1(19)-2015, EIT, Lund University, Sweden Antenna characteristics What parameters are interesting? Radiated EM-waves Antenna Mats Gustafsson, ETEN10:1(20)-2015, EIT, Lund University, Sweden Antenna characteristics Circuit (transmission line) properties: J + I V - Radiated EM-waves Power flow in direction, µ,' I Input impedance Z(ω) and reflection 0 coefficient Γ = Z−Z Z+Z0 . I Matched antennas |Γ | ≈ 0 I Determine Z = V /I in the feed point I Or measure the reflection coefficient Γ . Radiation properties: z µ x I Radiated power in direction θ, φ. I Determine the currents on the antenna and integrate to obtain the far-field. I Or measure the received (or transmitted) field. Mats Gustafsson, ETEN10:1(21)-2015, EIT, Lund University, Sweden Circuit (transmission line) parameters (Ch. 10) Antenna input impedance, Z(ω) I + Z V - The impedance at the antenna terminals. Function of the (angular) frequency ω = 2πf . Reflection coefficient, Γ (ω) The reflected signal on the transmission line (characteristic impedance Z0 ) is Γ uin , where Γ = Z − Z0 Z + Z0 and Z = Z0 1+Γ 1−Γ Measure with a network analyzer. Standing wave ratio (SWR or VSWR) Voltage standing wave ratio VSWR = Vmax /Vmin = (1 + |Γ |)/(1 − |Γ |) Mats Gustafsson, ETEN10:1(22)-2015, EIT, Lund University, Sweden Antenna radiation parameters Coordinate system (Ch. 14.8) ẑ Use the spherical coordinate system {r, θ, φ} with θ x = r cos φ sin θ y = r sin φ sin θ φ ŷ z = r cos θ x̂ The radiation pattern depends on {θ, φ}. We also consider the polarization, often expressed in the directions θ̂ and φ̂ directions, see also 39 . Mats Gustafsson, ETEN10:1(23)-2015, EIT, Lund University, Sweden Antenna radiation parameters Coordinate system (Ch. 14.8) r̂ ẑ φ̂ θ Use the spherical coordinate system {r, θ, φ} with x = r cos φ sin θ y = r sin φ sin θ φ ŷ θ̂ z = r cos θ x̂ The radiation pattern depends on {θ, φ}. We also consider the polarization, often expressed in the directions θ̂ and φ̂ directions, see also 39 . Mats Gustafsson, ETEN10:1(23)-2015, EIT, Lund University, Sweden Antenna radiation parameters Coordinate system (Ch. 14.8) ẑ Use the spherical coordinate system {r, θ, φ} with θ r̂ φ̂ x = r cos φ sin θ y = r sin φ sin θ φ ŷ θ̂ z = r cos θ x̂ The radiation pattern depends on {θ, φ}. We also consider the polarization, often expressed in the directions θ̂ and φ̂ directions, see also 39 . Mats Gustafsson, ETEN10:1(23)-2015, EIT, Lund University, Sweden Latitude, longitude, and cardinal directions We can compare the spherical coordinate system that we use to describe antenna parameters with a geographic coordinate system. On earth, we want to describe Earth. Radiated EM-waves Power flow in direction, µ,' z µ I positions. I directions. This is similar to the radiation pattern from an antenna, where we want to describe I the radiation intensity far away from the antenna (cf., the position on earth). I polarization of the E-field (directions on earth). x Radiated field. Mats Gustafsson, ETEN10:1(24)-2015, EIT, Lund University, Sweden Latitude, longitude, and cardinal directions Use a set of coordinates that describe positions and directions on earth (here we neglect altitude). I Latitude, angle between −90◦ (south pole) and 90◦ (north pole). I Longitude, angle between −180◦ (west) and 180◦ (east) from the reference meridian, 0◦ at Greenwich, UK. I Directions: north (latitude) and east (longitude). I Note the change of area close to the north and south poles. latitude Latitude and longitude on a sphere. Lund at 55◦ N, 13◦ E 90 75 60 45 30 15 0 −15 −30 −45 −60 −75 −90 −180 −150 −120 −90 −60 −30 0 30 longitude 60 90 120 150 180 Latitude and longitude in a Cartesian coordinate system. Mats Gustafsson, ETEN10:1(25)-2015, EIT, Lund University, Sweden Spherical coordinates for antennas Radiated EM-waves µ=0 Power flow in direction, µ,' z µ=30 µ x Á=60 Á=330 Á=30 Á=0 The {θ, φ} coordinate system on a sphere. µ 0 15 30 45 60 75 90 105 120 135 150 165 180 0 30 60 90 120 150 180 210 240 270 300 330 360 Á The {θ, φ} coordinate system. Place the antenna in the center of a sphere (the earth). Use a spherical coordinate system (similar as the latitude and longitude) with θ = 0◦ at the north pole, θ = 180◦ at the south pole, and 0◦ ≤ φ < 360◦ . Mats Gustafsson, ETEN10:1(26)-2015, EIT, Lund University, Sweden Spherical coordinates for antennas Place the antenna in the center of a sphere (the earth). Use a spherical coordinate system (similar as the latitude and longitude) with θ = 0◦ at the north pole, θ = 180◦ at the south pole, and 0◦ ≤ φ < 360◦ . Directions: µ=0 ^ Á µ=30 ^ µ ^ Á ^ µ Á=60 I The unit vector θ̂ is in the direction of increasing θ, i.e., towards south. This is also the polarization (direction of the electric field) of an ẑ-directed dipole antenna. I The unit vector φ̂ is in the direction of increasing φ, i.e., towards east. I Note the change of area close to the poles, cf., the sin θ in many integrals. Á=330 Á=30 Á=0 Spherical coordinates, {θ, φ}, on a sphere with the directions θ̂ and φ̂. µ 0 15 30 45 60 75 90 105 120 135 150 165 180 ^ Á µ^ 0 30 ^ Á µ^ 60 90 120 150 180 210 240 270 300 330 360 Á The {θ, φ} coordinate system. Mats Gustafsson, ETEN10:1(27)-2015, EIT, Lund University, Sweden Spherical coordinate system r̂ ẑ φ̂ θ φ ŷ θ̂ x̂ ẑ θ r̂ φ x̂ Useful vector formulas   r̂ = x̂ sin θ cos φ + ŷ sin θ sin φ + ẑ cos θ θ̂ = x̂ cos θ cos φ + ŷ cos θ sin φ − ẑ sin θ   φ̂ = −x̂ sin φ + ŷ cos φ φ̂ ŷ and   x̂ = r̂ sin θ cos φ + θ̂ cos θ cos φ − φ̂ sin φ ŷ = r̂ sin θ sin φ + θ̂ cos θ sin φ + φ̂ cos φ   ẑ = r̂ cos θ − θ̂ sin θ You have them in the formula sheet, see also 43 . θ̂ Mats Gustafsson, ETEN10:1(28)-2015, EIT, Lund University, Sweden Antenna radiation parameters (Ch. 15.2) Radiation intensity P (θ, φ) Angular distribution of the radiated power around the antenna. Directivity D(θ, φ) = P (θ, φ)/Pave z V x V1 y Radiation intensity normalized by the isotropic (average) intensity Pave = Prad /(4π). The direction of maximum radiation is implied if no direction is specified, i.e., D = maxθ,φ D(θ, φ). Gain G(θ, φ) = P (θ, φ)/(PT /(4π)) Radiation intensity normalized by the power accepted by the antenna. Efficiency εr = Prad /PT Anechoic chamber at EIT, LU [2] Radiated power normalized by the input power. Mats Gustafsson, ETEN10:1(29)-2015, EIT, Lund University, Sweden Electromagnetic fields I Electric field intensity E unit V/m. I Magnetic field intensity H unit A/m. I (Electric) current density J unit A/m2 . I Poynting vector S = 21 E × H ∗ unit W/m2 . I Frequency f , angular frequency ω = 2πf , wavenumber k = ω/c0 , wavelength λ = c0 /f = 2π/k. ∇ × E = −jωB I Free space impedance η0 ≈ 377 Ω. ∇×H = J +jωD I Speed of light c0 = 299792458 m/s ≈ 3 108 m/s. J.C. Maxwell [1] Time harmonic fields, e.g., E(t) = Re{Eejωt }. PhysWorld 2004 Mats Gustafsson, ETEN10:1(30)-2015, EIT, Lund University, Sweden Current to radiation pattern I The radiated field is determined from the current density, J , on the antenna element as Z 0 F (θ, φ) = F (r̂) = J (r 0 )ejkr̂·r dV0 , ^ r r µ^ r0 J(r 0 ) where F is the radiation vector 44 and ^ E(r) r̂ = x̂ cos φ sin θ + ŷ sin φ sin θ + ẑ cos θ. The electric field far from the antenna is E(r) = −jkη0 Mats Gustafsson, ETEN10:1(31)-2015, EIT, Lund University, Sweden e−jkr F⊥(r̂) 4πr e−jkr = −jkη0 θ̂Fθ + φ̂Fφ . 4πr Current to radiation pattern II The Poynting vector (power flux) far from the antenna is Re S = P (θ, φ) 1 Re{E × H ∗ } = r̂, 2 r2 with the radiation intensity P (θ, φ) = z V x V1 y η0 k 2 |F⊥(θ, φ)|2 . 32π 2 In power flux is in the radial direction r̂, decays as r−2 , and depends on the direction {θ, φ}. Approximate current distributions are known for some simple antennas. We use this approach to analyze dipole, loop, and patch antennas. Accurate current distributions require numerical solutions, e.g., with MoM. Mats Gustafsson, ETEN10:1(32)-2015, EIT, Lund University, Sweden Near field and far field I I Reactive near field: the region close to the antenna where the reactive field dominates. Radiative near field (Fresnel region): weak reactive fields but radiation in different directions. Far-field region Radiative near field D Far field (Fraunhofer region): the Reactive region far from the antenna where the near field fields decay as 1/r, i.e., −jkr E(r) ≈ −jkη0 e4πr F⊥(r̂). Often defined as distances r > rf = 2D2 /λ and r λ, where D denotes the diameter of the antenna. We analyze the far-field (radiation pattern) in this course. I Mats Gustafsson, ETEN10:1(33)-2015, EIT, Lund University, Sweden rf Receiving antenna parameters Reciprocity Most antennas are reciprocal, i.e., they have similar properties in transmission and reception. E in ^ k Z0 Effective area (aperture) A(θ, φ) = PR /Pin Where PR is the power absorbed by the antenna in watts, and Pin is the power density incident on the antenna in watts per square meter. It is assumed that the antenna is terminated with a matched load. Gain and effective area A(θ, φ) = λ2 G(θ, φ) 4π Mats Gustafsson, ETEN10:1(34)-2015, EIT, Lund University, Sweden and G(θ, φ) = 4πA(θ, φ) λ2 Friis transmission equation Transmitting antenna PT, GT, A T r Receiving antenna PR, GR, A R The radiated power density at the distance r from the transmitting antenna is Pin = P/r2 = PT GT /(4πr2 ). The received power is hence PR = Pin AR = PT G T GT GR λ 2 G T GR A = P = PT R T 4πr2 (4πr)2 (4πr/λ)2 Here, we assume that the antennas are matched and have aligned polarizations. This will be generalized later in the course. Harald T. Friis, Proc. IRE, 1946. Mats Gustafsson, ETEN10:1(35)-2015, EIT, Lund University, Sweden Next lecture date 11-02 11-04 11-05 I I Lec. 1 Exe. Lab S.O. 15 K.&M. 2 16 2,6 1 2 R.B. topic Introduction, antenna param Computer exercise (E:4118, Dipoles, antenna parameter Computer exercise in E:4118 and E:4119, preparations: definitions and terminology e.g., input impedance, reflection coefficient, directivity, gain, radiation pattern. Read these lecture notes and Orfanidis Ch 15. Download the exercise instructions ETEN10 E1.pdf from moodle. Lecture 2: dipole antennas, short dipole, half-wave dipole, simple analytic expressions, download the lecture notes ETEN10 2.pdf from moodle. Read before next lecture. Mats Gustafsson, ETEN10:1(36)-2015, EIT, Lund University, Sweden References [1] James Clerk Maxwell. http://en.wikipedia.org/. [2] The anechoic chamber at EIT, LU, 2002. LTH-Nytt. [3] RFID TAG, 2008. http://en.wikipedia.org/. [4] Hajor. The Very Large Array at Socorro, New Mexico, US, 2004. http://en.wikipedia.org/. [5] Nobel foundation. Marconi, 1909. http://nobelprize.org/. [6] Oliver Heaviside: Sage in Solitude. Heinrich Rudolf Hertz, 1894. http://en.wikipedia.org/. [7] Sony Ericsson. T68, 2001. Mats Gustafsson, ETEN10:1(37)-2015, EIT, Lund University, Sweden Outline 4 Appendix: Additional material Vectors Potentials and far fields Mats Gustafsson, ETEN10:1(38)-2015, EIT, Lund University, Sweden Appendix: vectors We use vectors to describe positions and directions. In a Cartesian coordinate system {x̂, ŷ, ẑ}, we have r = xx̂ + yŷ + zẑ and r 0 = x0 x̂ + y 0 ŷ + z 0 ẑ with the scalar product r · r 0 = (xx̂ + yŷ + zẑ) · (x0 x̂ + y 0 ŷ + z 0 ẑ) = xx0 + yy 0 + zz 0 as x̂ · x̂ = 1, x̂ · ŷ = 0, x̂ · ẑ = 0,... The length of a vector is p √ r = |r| = r · r = x2 + y 2 + z 2 The distance between the two points r and r 0 is p p |r−r 0 | = (r − r 0 ) · (r − r 0 ) = (x − x0 )2 + (y − y 0 )2 + (z − z 0 )2 or |r − r 0 | = p p (r − r 0 ) · (r − r 0 ) = r2 − 2r · r 0 + r02 Mats Gustafsson, ETEN10:1(39)-2015, EIT, Lund University, Sweden Vector fields The electric field E(r), magnetic field H(r), and current density J (r) are vector fields, where a vector is assigned to every point in space. −y x̂ + xŷ p = − cos φx̂ + sin φŷ = φ̂ x2 + y 2 xx̂ + yŷ p = cos φx̂ + sin φŷ = r̂ x2 + y 2 2 2 y y 1 1 x x −2 −1 1 2 −2 −1 1 −1 −1 −2 −2 Mats Gustafsson, ETEN10:1(40)-2015, EIT, Lund University, Sweden 2 Matrix notation A Cartesian coordinate system {x̂, ŷ, ẑ} with vectors r = xx̂ + yŷ + zẑ can be identified with the matrix notation          0 1 0 0 x x 0          r = xx̂+yŷ+zẑ = x 0 +y 1 +z 0 = y and r = y 0  0 0 1 z z0 where we have scalar (or inner) product  0  T  0  x x x r · r 0 = y  y 0  = x y z y 0  = xx0 + yy 0 + zz 0 z0 z0 z The matrix notation is not suitable for spherical coordinate systems and we do not use it in this course. Mats Gustafsson, ETEN10:1(41)-2015, EIT, Lund University, Sweden Spherical coordinate system A vector r = xx̂ + yŷ + zẑ can be described in a spherical coordinate system {r, θ, φ} ẑ θr x = r cos φ sin θ y = r sin φ sin θ φ z = r cos θ ŷ x̂ ẑ I θ φ x̂ r I ŷ I p r = |r| = x2 + y 2 + z 2 is the length of r (the distance from the center of the coordinate system to the point r). θ is the polar angle, i.e., the angle from ẑ to r. φ is the azimuthal angle, i.e., the angle from x̂ to the projection of r in the xy-plane. Mats Gustafsson, ETEN10:1(42)-2015, EIT, Lund University, Sweden Spherical coordinate system r̂ ẑ φ̂ θ φ ŷ θ̂ x̂ A spherical coordinate system {r, θ, φ} has the unit vectors   r̂ = x̂ sin θ cos φ + ŷ sin θ sin φ + ẑ cos θ θ̂ = x̂ cos θ cos φ + ŷ cos θ sin φ − ẑ sin θ   φ̂ = −x̂ sin φ + ŷ cos φ ẑ θ that defines a local coordinate system with unit vectors r̂ φ̂ I I φ ŷ I r̂ in the direction of increasing r. θ̂ in the direction of increasing θ. φ̂ in the direction of increasing φ. x̂ θ̂ Mats Gustafsson, ETEN10:1(43)-2015, EIT, Lund University, Sweden Appendix: Potentials, far field, and radiation vector The (time harmonic) Maxwell equations are (also assuming free space D = 0 E, B = µ0 H) ∇ × E = −jωB ∇ × H = jωD + J ∇·B =0 ∇·D =ρ Continuity equation from ∇ · (∇ × H) = 0 0 = jω∇ · D + ∇ · J = jωρ + ∇ · J or jωρ = −∇ · J Mats Gustafsson, ETEN10:1(44)-2015, EIT, Lund University, Sweden The divergenceless of B (∇ · B = 0) implies the existence of a vector potential A such that B = ∇ × A. Moreover, ∇ × E = −jωB = −jω∇ × A implies that there exists a scalar potential φ such that E = −∇φ − jωA. The vector and scalar potentials are not uniquely defined and here we impose the Lorenz gauge ∇ · A = −c−2 0 jωφ. The vector potentials satisfy the Helmholtz equations ∇2 φ + k 2 φ = −ρ/0 ∇2 A + k 2 A = −µ0 J Mats Gustafsson, ETEN10:1(45)-2015, EIT, Lund University, Sweden We have the solutions (Green’s function G(r) = e−jkr /(4πr), where r = |r|) 1 φ(r) = 0 0 V 1 ρ(r 0 )e−jk|r−r | dV0 = 0 4π|r − r | 0 V J (r 0 )e−jk|r−r | dV0 = µ0 4π|r − r 0 | Z Z A(r) = µ0 0 Z ρ(r 0 )G(r − r 0 ) dV0 V Z J (r 0 )G(r − r 0 ) dV0 V giving the electric and magnetic fields (ωµ0 = kη0 , k/η0 = ω0 ) Z 0 0 0 0 E(r) = −jωµ0 J (r 0 )G(r − r 0 ) + −1 0 ρ(r )∇ G(r − r ) dV VZ 1 = η0 −jkJ (r 0 )G(r − r 0 ) − ∇0 · J (r 0 )∇0 G(r − r 0 ) dV0 jk Z V J (r 0 ) × ∇0 G(r − r 0 ) dV0 H(r) = V Mats Gustafsson, ETEN10:1(46)-2015, EIT, Lund University, Sweden The expressions for the potentials and the fields simplify in the far field. For r r0 , we can approximate the distance R = |r − r 0 | = p p (r − r 0 ) · (r − r 0 ) = r2 − 2r · r 0 + r02 p = r 1 − 2r̂ · r 0 /r + (r/r0 )2 ≈ r − r̂ · r 0 as r → ∞, where r = |r|, r̂ = r/r, and r0 = |r 0 |. The potentials are Z e−jkr 0 φ(r) ≈ ρ(r 0 )ejkr ·r̂ dV0 4π0 r V Z e−jkr µ0 e−jkr µ0 0 F (r̂) A(r) ≈ J (r 0 )ejkr ·r̂ dV0 = 4πr 4πr V where F is the radiation vector. The electric field is E(r) ≈ −jω(r̂ × A(r)) × r̂ = −jkη0 Mats Gustafsson, ETEN10:1(47)-2015, EIT, Lund University, Sweden e−jkr (r̂ × F (r̂)) × r̂ 4πr

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