Week 1 Wave Concepts Coordinate Systems and Vector Products

International System of Units (SI) Lengthmeterm Masskilogramkg Timeseconds CurrentAmpereA TemperatureKelvinK Newton = kg m/s 2 Coulomb = A s Volt = (Newton /Coulomb) m Dr. Benjamin C. Flores2

Standard prefixes (SI) Dr. Benjamin C. Flores3

Exercise The speed of light in free space is c = x 10 5 km/s. Calculate the distance traveled by a photon in 1 ns. Dr. Benjamin C. Flores4

Propagating EM wave Characteristics Amplitude Phase Angular frequency Propagation constant Direction of propagation Polarization Example E(t,z) = E o cos (ωt – βz) a x Dr. Benjamin C. Flores5

Forward and backward waves Sign Convention - βz propagation in +z direction + βzpropagation in –z direction Which is it? a) forward traveling b) backward traveling Dr. Benjamin C. Flores6

Partial reflection This happens when there is a change in medium Dr. Benjamin C. Flores7

Standing EM wave Characteristics Amplitude Angular frequency Phase Polarization No net propagation Example E(t,z) = A cos (ωt ) cos( βz) a x Dr. Benjamin C. Flores8

Complex notation Recall Euler’s formula exp(jφ) = cos (φ) + j sin (φ) Dr. Benjamin C. Flores9

Exercise Calculate the magnitude of exp(jφ) = cos (φ)+ j sin (φ) Determine the complex conjugate of exp(j φ) Dr. Benjamin C. Flores10

Traveling wave complex notation Let φ = ωt – βz Complex field E c (t, z) = A exp [j(ωt – βz)] a x = A cos(ωt – βz) a x + j A sin(ωt – βz) a x E(z,t) = Real { E c (t, z) } Dr. Benjamin C. Flores11

Standing wave complex notation E = A exp[ j(ωt – βz) + A exp[ j(ωt + βz) = A exp(jωt) [exp(–jβz) + exp(+jβz)] = 2A exp(jωt) cos(βz) E = 2A[cos(ωt) + j sin (ωt) ] cos(βz) Re { E } = 2A cos(ωt) cos(βz) Im { E } = 2A sin(ωt) cos(βz) Dr. Benjamin C. Flores12

Exercise Show that E(t) = A exp(jωt) sin(βz) can be written as the sum of two complex traveling waves. Hint: Recall that j2 sin(φ) = exp (j φ) – exp(– j φ) Dr. Benjamin C. Flores13

Transmission line/coaxial cable Voltage wave V = V o cos (ωt – βz) Current wave I = I o cos (ωt – βz) Characteristic Impedance Z C = V o / I o Typical values: 50, 75 ohms Dr. Benjamin C. Flores14

RADAR Radio detection and ranging Dr. Benjamin C. Flores15

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 E c (t, τ) = A exp [jω( t – τ )] a x Dr. Benjamin C. Flores16

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. Flores17

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. Flores18

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. Flores19

Cartesian coordinates Dr. Benjamin C. Flores20

Principal planes Dr. Benjamin C. Flores21

Unit vectors a x = x = i a y = y = j a z = z = k u = A/|A| Dr. Benjamin C. Flores22

Handedness of coordinate system Left handed Right handed Dr. Benjamin C. Flores23

Are you smarter than a 5th 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. Flores24

Cylindrical coordinate system Dr. Benjamin C. Flores25 Φ = tan -1 y/x ρ 2 = x 2 + y 2

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. Flores26

Spherical coordinate system Dr. Benjamin C. Flores27 Φ = tan -1 y/x θ = tan -1 z/[x 2 + y 2 ] 1/2 r 2 = x 2 + y 2 + z 2

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. Flores28

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. Flores29

Exercise Let a = 2x + 5y + z and b = 3x – 4y + 2z. Find the dot product of these two vectors. Determine the angle between the two vectors. Dr. Benjamin C. Flores30

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. Flores31

Cross product Dr. Benjamin C. Flores32

Exercise Consider the two vectors a= 3x + 5y + 7z and b = 2x – 2y – 2z Determine the cross product c = a x b Find the unit vector n of c Dr. Benjamin C. Flores33

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. Flores34