TC303 Antenna&Propagation Lecture 1 Introduction, Maxwell’s Equations, Fields in media, and Boundary conditions RS1

 Asst. Prof. Dr. Rardchawadee Silapunt,  Lecture: 9:00am-12:00pm Tuesday  Office hours : By appointment  Textbook: Antenna Theory: Analysis and Design, 3rd Edition, Wiley-Interscience, 2005 by C. A. Balanis RS2 Syllabus

Quiz 20% Midterm exam 40% Final exam 40% RS3 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.

 Maxwell’s equations and boundary conditions for electromagnetic fields  Uniform plane wave propagation  Electric and loop dipoles  Antenna types  Antenna arrays RS4 Course overview

RS5 Introduction

RS 6 Introduction (2)

RS7 Maxwell’s equations (1) (2) (3) (4)

 = 0,  r = 1,  r = 1  0 = 4  x10 -7 Henrys/m  0 = 8.854x Farad/m RS8 Maxwell’s equations in free space Ampère’s law Faraday’s law

RS9 Divergence theorem Stokes’ theorem

RS10 Integral forms of Maxwell’s equations

Fields are assumed to be sinusoidal or harmonic, and time dependence with steady- state conditions RS 11 Time dependence form: Phasor form:

RS12 Maxwell’s equations in phasor form (1) (2) (3) (4) Assume and e j  t time dependence,

 RS13 Fields in dielectric media (1)

RS14 Fields in dielectric media (2)  e 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

 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 RS15 Anisotropic dielectrics

An applied magnetic field causes the magnetic polarization of by aligned magnetic dipole moments where is the magnetic polarization. In the linear medium, it can be shown that Then we can write RS 16 Analogous situations for magnetic media (1)

RS17 Analogous situations for magnetic media (2) 

 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 RS18 Anisotropic magnetic material

RS19 Boundary conditions between two media H t1 H t2 E t2 E t1 B n2 B n1 D n2 D n1

 Boundary conditions at an interface between two lossless dielectric materials with no charge or current densities can be shown as RS20 Fields at a dielectric interface

 Boundary conditions at the interface between a dielectric with the perfect conductor can be shown as RS21 Fields at the interface with a perfect conductor (electric wall)

General wave equations  Consider medium free of charge (  v = 0)  Electromagnetic properties of a material are defined by  permittivity  permeability  electrical conductivity  Assume medium properties:  linear P(x) = ax  isotropic P(x) = ax, P(y) = ay, P(z) = az  homogeneous  time-invariant 22RS

Homogeneous and Isotropic medium 23 Isotropic Homogeneous Homogeneous and isotropic RS

 Consider medium free of charge  For linear, isotropic, homogeneous, and time-invariant medium, assuming no free magnetic current, (1) (2) RS24 General plane wave equations (1)

Take curl of (2), we yield From then For charge free medium RS25 General plane wave equations (2)

RS26 Helmholtz wave equations For electric field For magnetic field

 Transformation from time to frequency domain Therefore RS27 Plane waves in a general lossy medium

or where This  term is called propagation constant or we can write  =  +j  where  = attenuation constant (Np/m)  = phase constant (rad/m) RS28 Plane waves in a general lossy medium

 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 RS29 Solutions of Helmholtz equations

 Showing the forward-propagating fields without time-harmonic terms.  Conversion between instantaneous and phasor form Instantaneous field = Re(e j  t  phasor field) RS30 Solutions of Helmholtz equations in phasor forms

EX1 Convert into the phasor form RS 31

RS32 Solutions of Helmholtz equations for free space propagation (  = 0) Instantaneous form Phasor form

 RS33 Wave impedance

RS34 Propagating fields relations

Power transmission in free space RS 35 W/m 2 Time averaged power density

RS 36 Ex2 Let the electric field E = 1 V/m propagate in free space, determine the magnitude of the magnetic field H and the average power density.