Radar Systems Engineering Lecture 2 Review of Electromagnetism

Transcription

1 Rada Systems Engineeing Lectue 2 Review of Electomagnetism D. Robet M. O Donnell IEEE New Hampshie Section Guest Lectue IEEE New Hampshie Section Rada Systems Couse 1

2 Reasons fo Review Lectue A numbe of potential students may not have taken a 3 d yea undegaduate couse in electomagnetism Electical/Compute Engineeing Majos in the Compute Engineeing Tack Compute Science Majos Mathematics Majos Mechanical Engineeing Majos If this elatively bief eview is not sufficient, a fomal couse in advanced undegaduate couse may be equied. Rada Systems Couse 2 IEEE New Hampshie Section

3 Outline Intoduction Coulomb s Law Gauss s Law Biot - Savat Law Ampee s Law Faaday s Law Maxwell s Equations Electomagnetic Waves Rada Systems Couse 3 IEEE New Hampshie Section

4 Coulomb s Law If two electic chages, q and q2, ae sepaated by a distance,, they expeience a foce,, given by: F = q1q 4πε o 2 ˆ 2 F 1 ε = pemittivity of = o fee space 8.85 x C 2 / Chales Augustin de Coulomb ( ) 2 ( Nm ) q1 q2 Two chages of opposite sign attact; and two chages of the same sign epel each othe. The magnitude of the electic foce is popotional to the magnitude of each of the two chaes and invesely popotional to the distance between the two chages This electic foce is along the line between the two chages Rada Systems Couse 4 IEEE New Hampshie Section

5 Electic Field The electic field of a chage q1, at P a distance fom the electic chage is defined as: E() = q1ˆ 4πε o 2 q 1 P Rada Systems Couse 5 IEEE New Hampshie Section

6 Electic Field The electic field of a chage q1, at a distance fom the electic chage is defined as: E() = q1ˆ 4πε Remembe, that the foce on a chage q2 located a distance due to is give by q 1 Linea Supeposition The total electic field at a point in space is due to a numbe of point chages is the vecto sum of the electic fields of each chage q + q Electic field of a point chage o 2 F = q1q 4πε o 2 ˆ 2 q1 q2 F = q E Rada Systems Couse 6 IEEE New Hampshie Section

7 Gauss s Law Define: the Electic Flux Density : D = ε o E Then, Gauss s Law states that : D ds = Q Q Enclosed Enclosed Integating the Electic Flux Density ove a closed suface gives you the chage enclosed by the suface = Volume Chage Density ρdv Cal Feidich Gauss ( ) Using vecto calculus, Gauss s law may be cast in diffeential fom: Rada Systems Couse 7 D = ρ IEEE New Hampshie Section

8 Biot Savat Law Define: = Magnetic Field and = the Magnetic Flux Density H B The Biot-Savat law: The diffeential magnetic field geneated by a steady cuent flowing though the length d l dh is: whee is a unit vecto along the line fom the cuent element location to the measuement position of dh and R is the distance between the cuent element location and the measuement position of dh Fo an ensemble of cuent elements, the magnetic field is given by: Rada Systems Couse 8 Rˆ H = I dl x Rˆ dh = 2 4 π R I 4 π l dl x Rˆ 2 R ( A / m) Jean-Baptiste Biot ( ) Felix Savat ( ) IEEE New Hampshie Section

9 Magnetic Flux and the Absence of Magnetic Chages Magnetic Fiend of the Eath Law stating that thee ae no magnetic chages: B ds = 0 B = 0 Integating the Magnetic Flux Density ove a closed suface gives you the magnetic chage enclosed by the suface (zeo magnetic chage) This is Gauss s Law fo magnetism Law of non-existence of magnetic monopoles A numbe of physicists have seached extensively fo magnetic monopoles Find one and you will get a Nobel Pize Magnetic field lines always fom closed continuous paths, Rada Systems Couse 9 othewise magnetic souces (chages) would exist IEEE New Hampshie Section

10 Ampees Law Ampee s law (fo constant cuents): If c is a closed contou bounded by the suface, then c H ds = S J ds = I The sign convention of the closed contou is that I and obey the ight hand ule x H The line integal of aound a closed path c equals the cuent moving though that suface bounded by the closed path H = J S H Ande-Maie Ampee ( ) Rada Systems Couse 10 IEEE New Hampshie Section

11 Faaday s Law A changing magnetic field induces an electic field. c E ds = S B t ds Induced electic fields ae detemined by: B t Magnetostatic fields ae detemined by : μoj x E = B t Michael Faaday ( ) Rada Systems Couse 11 IEEE New Hampshie Section

12 Outline Intoduction Maxwell s Equations Displacement Cuent Continuity Equation Bounday Equations Electomagnetic Waves Rada Systems Couse 12 IEEE New Hampshie Section

13 Electomagnetism (Pe Maxwell) Gauss s Law D d S = ρdv D =ρ Magnetic Chages Do Not Exist Faadays s Law Ampee s Law E d s H d s B d S D = ε E = 0 B = d S t = J d S B = μ H B = 0 E H = B = t D + t J Supise! These fomulae ae inconsistent! Rada Systems Couse 13 IEEE New Hampshie Section

14 The Pe-Maxwell Equations Inconsistency Inconsistency comes about because a well known popety of vectos: ( x A) = Apply this to Faaday s law ( xe) = B t The left side is equal to 0, because of the above noted popety of vectos The ight side is 0, because If you do the same opeation to Ampee s law..touble.. 0 = t B = 0 ( B) Rada Systems Couse 14 IEEE New Hampshie Section

15 Rada Systems Couse 15 How Displacement Cuent Came to Be ( xh) = The left side is 0; but the ight side is not, geneally 0 If one applies Gauss s law and the continuity equation: ρ J + = t The above equation become: J = ρ t = t J μ So Maxwell s Equations become consistent, if we ewite Ampee s law as: A changing electic field induces an magnetic field o 0 E ( ε ) o E = εo t xh = J + D t Displacement cuent IEEE New Hampshie Section

16 Review - Electomagnetism James Clek Maxwell Plane Wave Solution No Souces Vacuum Non-Conducting Medium E,t B,t ( ) = E e o j( k jwt ) j( k jwt ) ( ) = B e o Integal Fom E d s H d s D = ε E D d S B d S = = = = 0 Maxwell s Equations ρdv B d S t D + J d S t B = μ H x y Diffeential Fom D = 4πρ B = 0 H B E = t λ D = + J t Electic Field Magnetic Field z Rada Systems Couse 16 IEEE New Hampshie Section

17 Bounday Equations Dn1 is the nomal component of D at the top of the pillbox Medium 1 Medium 2 D n2 nˆ D n 1 In the limit, when the side sufaces appoach 0, Gauss s law educes to: nˆ (D And fom 1 D2 ) = σ B d S = nˆ (B1 B2 ) = 0 s 0 D d S = ρdv The scala fom of these equations is D n1 D n2 B 1 Bn2 = n = σ 0 s Rada Systems Couse 17 IEEE New Hampshie Section

18 Bounday Equations (continued) Ht1 is the tangential component of H at the top of the pillbox Medium 1 nˆ H t1 In the limit, when the sides of the ectangle appoach 0, Ampee s law educes to: nˆ x (H 1 H 2 ) = Js Medium 2 H t2 And fom Faaday s law nˆ x (E E 2 ) 1 = 0 At the Suface of a Pefect Conducto nˆ nˆ x E = 0 x H = J Rada Systems Couse 18 s nˆ nˆ D = σ B = 0 s The scala fom of these equations is H E t1 t1 H E t2 t2 = = 0 J S IEEE New Hampshie Section

19 Outline Intoduction Maxwell s Equations Electomagnetic Waves How they ae geneated Fee Space Popagation Nea Field / Fa Field Polaization Popagation Waveguides Coaxial Tansmission Lines Miscellaneous Stuff Rada Systems Couse 19 IEEE New Hampshie Section

20 Radiation of Electomagnetic Waves Radiation is ceated by a time-vaying cuent, o an acceleation (o deceleation) of chage Two examples: An oscillating electic dipole Two electic chages, of opposite sign, whose sepaation oscillates accodingly: x = d0 sin ωt An oscillating magnetic dipole A loop of wie, which is diven by an oscillating cuent of the fom: I(t) = I0 sin ωt Eithe of these two methods ae examples of ways to geneate electomagnetic waves Rada Systems Couse 20 IEEE New Hampshie Section

21 Radiation fom an Oscillating Electic Dipole t = 0 0 cuent t = t = T T Maximum cuent Wave font initiates Wave font expands Field lines beak fom dipole t = t = cuent T T Wave font expands and foms closed loop Wave font continues to expand + T = Peiod of dipole oscillation Illustation of popagation and detachment of electic field lines fom the dipole Two chages in simple hamonic motion Rada Systems Couse 21 IEEE New Hampshie Section

22 MATLAB Movies fo Visualization of Antenna Radiation with Time Geneated via Finite Diffeence Time Domain (FDTD) solution We will study this method in a late lectue Two Cases: Single dipole / hamonic souce Two dipoles / hamonic souces Electic chages ae needed to ceate an electomagnetic wave, but ae not equied to sustain it Rada Systems Couse 22 IEEE New Hampshie Section

23 Dipole Radiation in Fee Space Vetical Distance (m) 1 Dipole* *diven by oscillating souce Rada Systems Couse Hoizontal Distance (m) 2 Coutesy of MIT Lincoln Laboatoy Used with Pemission IEEE New Hampshie Section

24 Two Antennas Radiating Vetical Distance (m) 1 Dipole 1* Dipole 2* *diven by oscillating souces (in phase) Rada Systems Couse Hoizontal Distance (m) Coutesy of MIT Lincoln Laboatoy Used with Pemission IEEE New Hampshie Section

25 Electomagnetic Waves Coutesy Bekeley National Laboatoy Rada Fequencies Rada Systems Couse 25 IEEE New Hampshie Section

26 Why Micowaves fo Rada Visible Tansmission Though Atmosphee Tansmission vs. Wavelength Tansmission (%) Thee Infaed Windows Heavy Atmospheic Attenuation Window Fo Radio and Micowaves No Tansmission Though Ionosphee 0 Wavelength 1 µm 1 mm 1 m 1 km Fequency 1 THz 1 GHz 1 MHz The micowave egion of the electomagnetic spectum (~3 MHZ to ~ 10 GHZ) is bounded by: One egion ( > 10 GHz) with vey heavy attenuation by the gaseous components of the atmosphee (except fo windows at 35 & 95 GHz) The othe egion (< 3 MHz), whose fequency implies antennas too lage fo most pactical applications Rada Systems Couse 26 IEEE New Hampshie Section

27 Electomagnetic Wave Popeties and Geneation / Calculation A adiated electomagnetic wave consists of electic and magnetic fields which jointly satisfy Maxwell s Equations EM wave is deived by integating souce cuents on antenna / taget Electic cuents on metal Magnetic cuents on apetues (tansvese electic fields) Souce cuents can be modeled and calculated Distibutions ae often assumed fo simple geometies Numeical techniques ae used fo moe igoous solutions (e.g. Method of Moments, Finite Diffeence-Time Domain Methods) z a/2 b Electic Cuent on Wie Dipole a y Electic Field Distibution (~ Magnetic Cuent) in Slot a/2 λ/2 3λ/2 λ/4 λ 2λ x Rada Systems Couse 27 IEEE New Hampshie Section

28 Antenna and Rada Coss Section Analyses Use Phaso Repesentation Hamonic Time Vaiation is assumed : E(x, y, z;t) = Real [ ] j t E ~ ω (x,y, z)e j e ω t Instantaneous Electic Field Phaso Calculate Phaso : Instantaneous Hamonic Field is : E ~ (x,y,z) E(x,y,z;t) = ê = ê E ~ (x,y,z) E ~ (x,y,z) j e cos ( ωt α + α) Any Time Vaiation can be Expessed as a Supeposition of Hamonic Solutions by Fouie Analysis Rada Systems Couse 28 IEEE New Hampshie Section

29 Field Regions Reactive Nea-Field Region R < 0.62 Enegy is stoed in vicinity of antenna Nea-field antenna Issues Input impedance Mutual coupling D 3 D R λ Reactive Nea-Field Region Fa-field (Faunhofe) Region R > 2 2D All powe is adiated out Radiated wave is a plane wave Fa-field EM wave popeties Equiphase Wave Fonts λ Polaization Antenna Gain (Diectivity) Antenna Patten Taget Rada Coss Section (RCS) H E ˆ Plane Wave Popagates Radially Out Radiating Nea-Field (Fesnel) Region Fa-Field (Faunhofe) Region Coutesy of MIT Lincoln Laboatoy Used with Pemission Rada Systems Couse 29 IEEE New Hampshie Section

30 Fa-Field EM Wave Popeties In the fa-field, a spheical wave can be appoximated by a plane wave Thee ae no adial field components in the fa field The electic and magnetic fields ae given by: whee E H η k ff ff (, θ, φ) (, θ, φ) μ ε = 2π o o E H = 377 Ω λ o o e ( θ, φ) e ( θ, φ) jk jk = 1 ˆ E η is the intinsic impedance of fee space is the wave popagation constant ff z x λ φ z θ θˆ ˆ φˆ Standad Spheical Coodinate System Electic Field Magnetic Field y x y Rada Systems Couse 30 IEEE New Hampshie Section

31 Polaization of Electomagnetic Wave Defined by behavio of the electic field vecto as it popagates in time as obseved along the diection of adiation Cicula used fo weathe mitigation Hoizontal used in long ange ai seach to obtain einfocement of diect adiation by gound eflection E θ ˆ Eθ Majo Axis E φ Mino Axis Rada Systems Couse 31 E φ E φ Coutesy of MIT Lincoln Laboatoy Used with Pemission Linea Vetical o Hoizontal Cicula Two components ae equal in amplitude, and sepaated in phase by 90 deg Right-hand (RHCP) is CW above Left-hand (LHCP) is CCW above Elliptical IEEE New Hampshie Section

32 Polaization Defined by behavio of the electic field vecto as it popagates in time θ Coutesy of MIT Lincoln Laboatoy Used with Pemission Electomagnetic Wave Electic Field φ Magnetic Field Vetical Linea (with espect to Eath) Hoizontal Linea (with espect to Eath) E (Fo ove-wate suveillance) Rada Systems Couse 32 E (Fo ai suveillance looking upwad) IEEE New Hampshie Section

33 Cicula Polaization (CP) Electic field components ae equal in amplitude, sepaated in phase by 90 deg Handed-ness is defined by obsevation of electic field along popagation diection Used fo discimination, polaization divesity, ain mitigation A θ Popagation Diection Into Pape θ 0 φ Right-Hand (RHCP) φ -A -A 0 A Left-Hand (LHCP) Phasos E A j E = Ae π θ = 2 φ Rada Systems Couse 33 Instantaneous Eθ () t = Acos( wt) E () t Asin( wt) φ = Coutesy of MIT Lincoln Laboatoy Used with Pemission Electic Field IEEE New Hampshie Section

34 Popagation Fee Space Plane wave, fee space solution to Maxwell s Equations: No Souces Vacuum j( k ωt) E(,t) = Eoe Non-conducting medium B,t j( k ωt) ( ) = B e Most electomagnetic waves ae geneated fom localized souces and expand into fee space as spheical wave. o In the fa field, when the distance fom the souce geat, they ae well appoximated by plane waves when they impinge upon a taget and scatte enegy back to the ada Rada Systems Couse 34 IEEE New Hampshie Section

35 Pointing Vecto Physical Significance The Poynting Vecto, S S, is defined as: E x H It is the powe density (powe pe unit aea) caied by an electomagnetic wave Since both E and ae functions of time, the aveage powe density is of geate inteest, and is given by: 1 ( ) * S = Re E x H WAV 2 Fo a plane wave in a lossless medium 1 2 S = E whee η = 2η H μ ε o o Rada Systems Couse 35 IEEE New Hampshie Section

36 Modes of Tansmission Fo Electomagnetic Waves Tansvese electomagnetic (TEM) mode Magnetic and electic field vectos ae tansvese (pependicula) to the diection of popagation, kˆ, and pependicula to each othe Examples (coaxial tansmission line and fee space tansmission, TEM tansmission lines have two paallel sufaces Tansvese electic (TE) mode Electic field, E, pependicula to kˆ No electic field in kˆ diection Tansvese electic (TM) mode Magnetic field, H, pependicula to kˆ No magnetic field in diection kˆ Used fo Rectangula Waveguides H TEM Mode E kˆ Hybid tansmission modes Rada Systems Couse 36 IEEE New Hampshie Section

37 Guided Tansmission of Micowave Electomagnetic Waves Coaxial Cable (TEM mode) Used mostly fo lowe powe and in low fequency potion of micowave potion of spectum Smalle coss section of coaxial cable moe pone to beakdown in the dielectic Dielectic losses incease with inceased fequency Waveguide (TE o TM mode) Metal waveguide used fo High powe ada tansmission Fom high powe amplifie in tansmitte to the antenna feed Rectangula waveguide is most pevalent geomety Coaxial Cable Coutesy of Tkgd 2007 Rectangula Waveguide Rada Systems Couse 37 Coutesy of Cobham Senso Systems. Used with pemission. IEEE New Hampshie Section

38 How Is the Size of Rada Tagets Chaacteized? Rada Coss Section = σ = R lim E E S I 2 2 By MIT OCW If the incident electic field that impinges upon a taget is known and the scatteed electic field is measued, then the ada coss section (effective aea) of the taget may by calculated. Rada Systems Couse 38 IEEE New Hampshie Section

39 Units- db vs. Scientific Notation The elative value of two quantities (in powe units), measued on a logaithmic scale, is often expessed in decibel s (db) Example: Signal-to-noise atio (db) = 10 log 10 Signal Powe Noise Powe Scientific Facto of: Notation db ,000, db = facto of 1-10 db = facto of 1/10-20 db = facto of 1/100 3 db = facto of 2-3 db = facto of 1/2 Rada Systems Couse 39 IEEE New Hampshie Section

40 Summay This lectue has pesented a vey bief eview of those electomagnetism topics that will be used in this ada couse It is not meant to eplace a one tem couse on advanced undegaduate electomagnetism that physics and electical engineeing students nomally take in thei 3 d yea of undegaduate studies Viewes of the couse may veify (o bush up on) thei skills in the aea by doing the suggested eview poblems in Giffith s (see efeence 1) and / o Ulaby s (see efeence 2) textbooks Rada Systems Couse 40 IEEE New Hampshie Section

41 Acknowledgements Pof. Kent Chambelin, ECE Depatment, Univesity of New Hampshie Rada Systems Couse 41 IEEE New Hampshie Section

42 Refeences 1. Giffiths, D. J., Intoduction to Electodynamics, Pentice Hall, New Jesey, Ulaby. F. T., Fundamentals of Applied Electomagnetics, Pentice Hall, New Jesey,5 th Ed., Skolnik, M., Intoduction to Rada Systems,McGaw-Hill, New Yok, 3 d Ed., Jackson, J. D., Classical Electodynamics, Wiley, New Jesey, Balanis, C. A., Advanced Engineeing Electomagnetics, Wiley, New Jesey, Poza, D. M., Micowave Engineeing, Wiley, New Yok, 3 d Ed., 2005 Rada Systems Couse 42 IEEE New Hampshie Section

43 Homewok Poblems Giffiths (Refeence 1) Poblems 7-34, 7-35, 7-38, 7-39, 9-9, 9-10, 9-11, 9-33 Ulaby (Refeence 2) Poblems 7-1, 7-2, 7-10, 7-11, 7-25, 7-26 It is impotant that pesons, who view these lectues, be knowledgeable in vecto calculus and phaso notation. This next poblem will veify that knowledge Poblem- Take Maxwell s Equations and the continuity equation, in integal fom, and, using vecto calculus theoems, tansfom these two sets of equations to the diffeential fom and then tansfom Maxwell s equations fom the diffeential fom to thei phaso fom. Rada Systems Couse 43 IEEE New Hampshie Section