# RS 1 ENE 428 Microwave Engineering Lecture 1 Introduction, Maxwell’s equations, fields in media, and boundary conditions.

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RS 1 ENE 428 Microwave Engineering Lecture 1 Introduction, Maxwell’s equations, fields in media, and boundary conditions

RS 2 Syllabus Asst. Prof. Dr. Rardchawadee Silapunt, rardchawadee.sil@kmutt.ac.th rardchawadee.sil@kmutt.ac.th Lecture: 9:30pm-12:20pm Tuesday, CB41004 12:30pm-3:20pm Wednesday, CB41002 Office hours : By appointment Textbook: Microwave Engineering by David M. Pozar (3 rd edition Wiley, 2005) Recommended additional textbook: Applied Electromagnetics by Stuart M.Wentworth (2 nd edition Wiley, 2007)

RS 3 Homework 10% Quiz 10% Midterm exam 40% Final exam 40% Grading 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.

10-11/06/51 RS 4 Course overview Maxwell’s equations and boundary conditions for electromagnetic fields Uniform plane wave propagation Waveguides Antennas Microwave communication systems

RS 5 Microwave frequency range (300 MHz – 300 GHz) Microwave components are distributed components. Lumped circuit elements approximations are invalid. Maxwell’s equations are used to explain circuit behaviors ( and ) Introduction http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52

RS 6 From Maxwell’s equations, if the electric field is changing with time, then the magnetic field 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 Introduction (2)

RS 7 Maxwell’s equations (1) (2) (3) (4)

RS 8 Maxwell’s equations in free space  = 0,  r = 1,  r = 1  0 = 4  x10 -7 Henrys/m  0 = 8.854x10 -12 farad/m Ampère’s law Faraday’s law

RS 9 Integral forms of Maxwell’s equations

RS 10 Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions Time dependence form: Phasor form:

RS 11 Maxwell’s equations in phasor form (1) (2) (3) (4)

RS 12 Fields in dielectric media (1) An applied electric field causes the polarization of the atoms or molecules of the material to create electric dipole moments that complements the total displacement flux, where is the electric polarization. In the linear medium, it can be shown that Then we can write

RS 13 Fields in dielectric media (2) may be complex then  can be complex and can be expressed as 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

RS 14 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

RS 15 Analogous situations for magnetic media (1) An applied magnetic field causes the magnetic polarization of by aligned magnetic dipole moments where is the electric polarization. In the linear medium, it can be shown that Then we can write

RS 16 Analogous situations for magnetic media (2) may be complex then  can be complex and can be expressed as Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments.

RS 17 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

RS 18 Boundary conditions between two media H t1 H t2 E t2 E t1 B n2 B n1 D n2 D n1

RS 19 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

RS 20 Fields at the interface with a perfect conductor Boundary conditions at the interface between a dielectric with the perfect conductor can be shown as

RS 21 General plane wave equations (1) Consider medium free of charge For linear, isotropic, homogeneous, and time- invariant medium, assuming no free magnetic current, (1) (2)

RS 22 General plane wave equations (2) Take curl of (2), we yield From then For charge free medium

RS 23 Helmholtz wave equation For electric field For magnetic field

RS 24 Time-harmonic wave equations Transformation from time to frequency domain Therefore

RS 25 Time-harmonic wave equations or where This  term is called propagation constant or we can write  =  +j  where  = attenuation constant (Np/m)  = phase constant (rad/m)

RS 26 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 Consider only the forward-propagating wave, we have Use Maxwell’s equation, we get

RS 27 Solutions of Helmholtz equations in phasor form Showing the forward-propagating fields without time- harmonic terms. Conversion between instantaneous and phasor form Instantaneous field = Re(e j  t  phasor field)