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Week 1 Wave Concepts Coordinate Systems and Vector Products International System of Units (SI) Length meter kilogram second Ampere Kelvin Mass Time Current Temperature m kg s A K Newton = kg m/s2 Coulomb = A s Volt = (Newton /Coulomb) m Dr. Benjamin C. Flores 2 Standard prefixes (SI) Symbol Factor 100 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 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)] = 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. 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 j2 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 = x = i ay = y = j az = z = k u = A/|A| Dr. Benjamin C. Flores 22 Handedness of coordinate system Left handed Dr. Benjamin C. Flores Right handed 23 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. Flores 24 Cylindrical coordinate system Φ = tan-1 y/x ρ2 = x2 + y2 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/[x2 + y2]1/2 r2 = x2 + y2 + z2 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 = 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. 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= 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. 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