Report

ENE 428 Microwave Engineering Lecture 1 Introduction, Maxwell’s equations, fields in media, and boundary conditions 1 RS Syllabus •Assoc. Prof. Dr. Rardchawadee Silapunt (Ann), [email protected] •Dr. Ekapon Siwapornsathain (Eric), [email protected], Tel: 0814389024 •Lecture: 9:00am-12:00pm Wednesday, AIT •Instructors at King Mongkut’s University of Technology Thonburi, BKK, Thailand •Textbook: Microwave Engineering by David M. Pozar (3rd edition Wiley, 2005) • Recommended additional textbook: Applied Electromagnetics by Stuart M.Wentworth (2nd edition Wiley, 2007) 2 RS 3 RS Grading Homework Quiz Midterm exam Final exam 10% 10% 40% 40% Vision Providing opportunities for intellectual growth in the context of an engineering discipline for the attainment of professional competence, and for the development of a sense of the social context of technology. 4 RS Course overview • Maxwell’s equations and boundary conditions for electromagnetic fields • Uniform plane wave propagation • Transmission lines • Matching networks • Waveguides • Two-port networks • Resonators • Antennas • Microwave communication systems 10-11/06/51 RS 5 Introduction http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52 • Microwave frequency range (300 MHz – 300 GHz) ( = 1 mm – 1 m in free space) • Microwave components are distributed components. • Lumped circuit elements approximations are invalid. • Maxwell’s equations are used to explain circuit behaviors ( H and E ) 6 RS Lumped circuit model and distributed circuit model 7 RS Introduction (2) • From Maxwell’s equations, if the electric field E is changing with time, then the magnetic field H varies spatially in a direction normal to its orientation direction • Knowledge of fields in media and boundary conditions allows useful applications of material properties to microwave components • A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation 8 RS 9 RS 10 RS Point forms of Maxwell’s equations B E H t D t M J (1) (2) D v (3) B 0 (4) 11 RS 12 RS The magnetic north can never be isolated from the south. Magnetic field lines always form closed loops. 13 RS Maxwell’s equations in free space • = 0, r = 1, r = 1 H 0 E t E 0 H t Ampère’s law Faraday’s law 0 = 4x10-7 Henrys/m 0 = 8.854x10-12 Farads/m = conductivity (1/ohm) (“constitutive parameters”) 14 RS Integral forms of Maxwell’s equations E d l t B d S (1) S H dl D d S I t (2) S D d S dV S B d S Q enc (3) V 0 (4) S Note: To convert from the point forms to the integral forms, we need to apply Stoke’s Theorem (for (1) and (2)) and Divergence theorem (for (3) and (4)), respectively. 15 RS Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions • Time dependence form: E A ( x , y , z ) co s( t ) a x • Phasor form: E s A ( x , y , z )e j ax 16 RS Maxwell’s equations in phasor form E S j B M (1) H S j D J (2) D v (3) B 0 (4) 17 RS Fields in dielectric media (1) • An applied electric field E causes the polarization of the atoms or molecules of the material to create electric D dipole moments that complements the total displacement flux, D 0 E Pe C /m 2 where P e is the electric polarization. • In the linear medium, it can be shown that Pe 0e E. • Then we can write RS D 0 (1 e ) E 0 r E E . 18 Fields in dielectric media (2) • may be complex then can be complex and can be e expressed as ' j '' • Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. • The loss of dielectric material may be considered as an equivalent conductor loss if the material has a conductivity . Loss tangent is defined as '' tan . ' RS 19 Anisotropic dielectrics • The most general linear relation of anisotropic dielectrics can be expressed in the form of a tensor which can be written in matrix form as D x xx D y yx D z zx xy yy zy xz E x Ex yz E y E y . E z zz E z 20 RS Analogous situations for magnetic media (1) • An applied magnetic field H causes the magnetic polarization of by aligned magnetic dipole moments B 0 (H P m ) Wb / m 2 where P m is the magnetic polarization or magnetization. • In the linear medium, it can be shown that Pm m H . • Then we can write B 0 (1 m ) H 0 r H H . 21 RS Analogous situations for magnetic media (2) • m may be complex then can be complex and can be expressed as ' j '' • Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. 22 RS Anisotropic magnetic material • The most general linear relation of anisotropic material can be expressed in the form of a tensor which can be written in matrix form as B x xx B y yx B z zx xy yy zy xz H x H x yz H y H y . H z zz H z 23 RS Boundary conditions between two media n Dn2 Bn2 Ht2 Et1 Ht1 Bn1 Et2 Dn1 n D 2 D1 S n B 2 n B1 E 2 E1 n M n H 2 S H1 JS 24 RS Fields at a dielectric interface • Boundary conditions at an interface between two lossless dielectric materials with no charge or current densities can be shown as n D 2 n D1 n B 2 n B1 n E1 n E 2 n H 1 n H 2. 25 RS Fields at the interface with a perfect conductor • Boundary conditions at the interface between a dielectric with the perfect conductor can be shown as nD 0 nB 0 n E M S n H 0. 26 RS General plane wave equations (1) • Consider medium free of charge • For linear, isotropic, homogeneous, and timeinvariant medium, assuming no free magnetic current, H E E H t E t (1) (2) 27 RS General plane wave equations (2) Take curl of (2), we yield E E From then ( H ) t E 2 ) E E t 2 t t t ( E A A A 2 E E 2 E t E 2 t 2 For charge free medium E 0 RS 28 Helmholtz wave equation For electric field For magnetic field E 2 E t H 2 E 2 H t t 2 H 2 t 2 29 RS Time-harmonic wave equations • Transformation from time to frequency domain t j Therefore E s j ( j ) E s 2 E s j ( j ) E s 0 2 Es Es 0 2 2 30 RS Time-harmonic wave equations or H s H 2 where 2 s 0 j ( j ) This term is called propagation constant or we can write = + j where = attenuation constant (Np/m) = phase constant (rad/m) RS 31 Solutions of Helmholtz equations • Assuming the electric field is in x-direction and the wave is propagating in z- direction • The instantaneous form of the solutions E E0 e z cos( t z ) a x E 0 e z cos( t z ) a x • Consider only the forward-propagating wave, we have E E0e z cos( t z ) a x • Use Maxwell’s equation, we get H H 0e RS z cos( t z ) a y 32 Solutions of Helmholtz equations in phasor form • Showing the forward-propagating fields without timeharmonic terms. E s E0e z H s H 0e e z j z e j z ax ay • Conversion between instantaneous and phasor form Instantaneous field = Re(ejtphasor field) 33 RS Intrinsic impedance • For any medium, Ex Hy j j • For free space Ex Hy E0 H0 0 120 0 34 RS Propagating fields relation Hs 1 a Es E s a H s where a represents a direction of propagation 35 RS