EE 3321 ELECTROMAGENTIC FIELD THEORY Week 1 Wave

EE 3321 ELECTROMAGENTIC FIELD THEORY Week 1 Wave Concepts Coordinate Systems and Vector Products

International System of Units (SI) �Length �Mass �Time �Current �Temperature meter kilogram second Ampere Kelvin m kg s A K �Newton = kg m/s 2 �Coulomb = A s � Volt = (Newton /Coulomb) m Dr. Benjamin C. Flores 2

Standard prefixes (SI) Symbol Factor 100 deca hecto kilo mega giga tera peta exa zetta yotta da h k M G T P E Z Y 101 102 103 106 109 1012 1015 1018 1021 1024 deci centi milli micro nano pico femto atto zepto yocto d c m µ n p f a z y 10− 1 10− 2 10− 3 10− 6 10− 9 10− 12 10− 15 10− 18 10− 21 10− 24 Symbol Factor 100 Dr. Benjamin C. Flores 3

Exercise �The speed of light in free space is c = 2. 998 x 105 km/s. Calculate the distance traveled by a photon in 1 ns. Dr. Benjamin C. Flores 4

Propagating EM wave �Characteristics �Amplitude �Phase � Angular frequency � Propagation constant �Direction of propagation �Polarization �Example �E(t, z) = Eo cos (ωt – βz) ax Dr. Benjamin C. Flores 5

Forward and backward waves �Sign Convention - βz propagation in +z direction + βz propagation in –z direction Which is it? a) forward traveling b) backward traveling Dr. Benjamin C. Flores 6

Partial reflection �This happens when there is a change in medium Dr. Benjamin C. Flores 7

Standing EM wave �Characteristics �Amplitude �Angular frequency �Phase �Polarization �No net propagation �Example �E(t, z) = A cos (ωt ) cos( βz) ax Dr. Benjamin C. Flores 8

Complex notation �Recall Euler’s formula exp(jφ) = cos (φ) + j sin (φ) Dr. Benjamin C. Flores 9

Exercise �Calculate the magnitude of exp(jφ) = cos (φ)+ j sin (φ) �Determine the complex conjugate of exp(j φ) Dr. Benjamin C. Flores 10

Traveling wave complex notation �Let φ = ωt – βz �Complex field Ec(t, z) = A exp [j(ωt – βz)] ax = A cos(ωt – βz) ax + j A sin(ωt – βz) ax E(z, t) = Real { Ec(t, z) } Dr. Benjamin C. Flores 11

Standing wave complex notation �E = A exp[ j(ωt – βz) + A exp[ j(ωt + βz) = A exp(jωt) [exp(–jβz) + exp(+jβz)] = 2 A exp(jωt) cos(βz) �E = 2 A[cos(ωt) + j sin (ωt) ] cos(βz) �Re { E } = 2 A cos(ωt) cos(βz) �Im { E } = 2 A sin(ωt) cos(βz) Dr. Benjamin C. Flores 12

Exercise Show that E(t) = A exp(jωt) sin(βz) can be written as the sum of two complex traveling waves. Hint: Recall that j 2 sin(φ) = exp (j φ) – exp(– j φ) Dr. Benjamin C. Flores 13

Transmission line/coaxial cable � Voltage wave �V = Vo cos (ωt – βz) �Current wave �I = Io cos (ωt – βz) � Characteristic Impedance �ZC = Vo / Io �Typical values: 50, 75 ohms Dr. Benjamin C. Flores 14

RADAR �Radio detection and ranging Dr. Benjamin C. Flores 15

Time delay �Let r be the range to a target in meters � φ = ωt – βr = ω[ t – (β/ω)r ] �Define the phase velocity as v = β/ω �Let τ = r/v be the time delay �Then φ = ω (t – τ) �And the field at the target is Ec(t, τ) = A exp [jω( t – τ )] ax Dr. Benjamin C. Flores 16

Definition of coordinate system �A coordinate system is a system for assigning real numbers (scalars) to each point in a 3 -dimensional Euclidean space. �Systems commonly used in this course include: �Cartesian coordinate system with coordinates x (length), y (width), and z (height) �Cylindrical coordinate system with coordinates ρ (radius on x-y plane), φ (azimuth angle), and z (height) �Spherical coordinate system with coordinates r (radius or range), Ф (azimuth angle), and θ (zenith or elevation angle) Dr. Benjamin C. Flores 17

Definition of vector �A vector (sometimes called a geometric or spatial vector) is a geometric object that has a magnitude, direction and sense. Dr. Benjamin C. Flores 18

Direction of a vector �A vector in or out of a plane (like the white board) are represented graphically as follows: �Vectors are described as a sum of scaled basis vectors (components): Dr. Benjamin C. Flores 19

Cartesian coordinates Dr. Benjamin C. Flores 20

Principal planes Dr. Benjamin C. Flores 21

Unit vectors �ax = i �ay = j �az = k �u = A/|A| Dr. Benjamin C. Flores 22

Handedness of coordinate system Left handed Right handed Dr. Benjamin C. Flores 23

Are you smarter than a 5 th grader? �Euclidean geometry studies the relationships among distances and angles in flat planes and flat space. �true �false �Analytic geometry uses the principles of algebra. �true �false Dr. Benjamin C. Flores 24

Cylindrical coordinate system Φ = tan-1 y/x ρ2 = x 2 + y 2 Dr. Benjamin C. Flores 25

Vectors in cylindrical coordinates �Any vector in Cartesian can be written in terms of the unit vectors in cylindrical coordinates: �The cylindrical unit vectors are related to the Cartesian unit vectors by: Dr. Benjamin C. Flores 26

Spherical coordinate system Φ = tan-1 y/x θ = tan-1 z/[x 2 + y 2]1/2 r 2 = x 2 + y 2 + z 2 Dr. Benjamin C. Flores 27

Vectors in spherical coordinates �Any vector field in Cartesian coordinates can be written in terms of the unit vectors in spherical coordinates: �The spherical unit vectors are related to the Cartesian unit vectors by: Dr. Benjamin C. Flores 28

Dot product �The dot product (or scalar product) of vectors a and b is defined as �a · b = |a| |b| cos θ where �|a| and |b| denote the length of a and b �θ is the angle between them. Dr. Benjamin C. Flores 29

Exercise �Let a = 2 x + 5 y + z and b = 3 x – 4 y + 2 z. �Find the dot product of these two vectors. �Determine the angle between the two vectors. Dr. Benjamin C. Flores 30

Cross product �The cross product (or vector product) of vectors a and b is defined as a x b = |a| |b| sin θ n where �θ is the measure of the smaller angle between a and b (0° ≤ θ ≤ 180°), �a and b are the magnitudes of vectors a and b, �and n is a unit vector perpendicular to the plane containing a and b. Dr. Benjamin C. Flores 31

Cross product Dr. Benjamin C. Flores 32

Exercise �Consider the two vectors a= 3 x + 5 y + 7 z and b = 2 x – 2 y – 2 z �Determine the cross product c = a x b �Find the unit vector n of c Dr. Benjamin C. Flores 33

Homework �Read all of Chapter 1, sections 1 -1, 1 -2, 1 -3, 1 -4, 1 -5, 1 -6 �Read Chapter 3, sections 3 -1, 3 -2, 3 -3 �Solve end-of-chapter problems 3. 1, 3. 3, 3. 5 , 3. 7, 3. 19, 3. 21, 3. 25, 3. 29 Dr. Benjamin C. Flores 34