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Week 1 Wave Concepts Coordinate Systems and Vector Products

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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

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Standard prefixes (SI) Dr. Benjamin C. Flores3

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Exercise The speed of light in free space is c = 2.998 x 10 5 km/s. Calculate the distance traveled by a photon in 1 ns. Dr. Benjamin C. Flores4

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

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

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

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

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Complex notation Recall Eulers formula exp(j) = cos () + j sin () Dr. Benjamin C. Flores9

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

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

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Standing wave complex notation E = A exp[ j(t z) + A exp[ j(t + z) = A exp(jt) [exp(jz) + exp(+jz)] = 2A exp(jt) cos(z) E = 2A[cos(t) + j sin (t) ] cos(z) Re { E } = 2A cos(t) cos(z) Im { E } = 2A sin(t) cos(z) Dr. Benjamin C. Flores12

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

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

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RADAR Radio detection and ranging Dr. Benjamin C. Flores15

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Time delay Let r be the range to a target in meters = t r = [ t (/)r ] Define the phase velocity as v = / Let = r/v be the time delay Then = (t ) And the field at the target is E c (t, ) = A exp [j( t )] a x Dr. Benjamin C. Flores16

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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 (radius on x-y plane), (azimuth angle), and z (height) Spherical coordinate system with coordinates r (radius or range), (azimuth angle), and (zenith or elevation angle) Dr. Benjamin C. Flores17

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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

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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

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Cartesian coordinates Dr. Benjamin C. Flores20

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Principal planes Dr. Benjamin C. Flores21

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

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

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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

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

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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

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

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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

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Dot product The dot product (or scalar product) of vectors a and b is defined as a b = |a| |b| cos where |a| and |b| denote the length of a and b is the angle between them. Dr. Benjamin C. Flores29

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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

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Cross product The cross product (or vector product) of vectors a and b is defined as a x b = |a| |b| sin n where is the measure of the smaller angle between a and b (0 180), 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

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Cross product Dr. Benjamin C. Flores32

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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

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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