EE3321 ELECTROMAGENTIC FIELD THEORY

EE3321 ELECTROMAGENTIC FIELD THEORYWeek 1Wave Concepts Coordinate Systems and Vector Products

International System of Units (SI)LengthmetermMasskilogramkgTimesecondsCurrentAmpereATemperatureKelvinK

Newton = kg m/s2Coulomb = A s Volt = (Newton /Coulomb) m

Dr. Benjamin C. Flores2Standard prefixes (SI)decahectokilomegagigaterapetaexazettayottaSymboldahkMGTPEZYFactor10010110210310610910121015101810211024decicentimillimicronanopicofemtoattozeptoyoctoSymboldcmnpfazyFactor10010110210310610910121015101810211024Dr. Benjamin C. Flores3ExerciseThe 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. Flores4Propagating EM waveCharacteristicsAmplitudePhaseAngular frequencyPropagation constantDirection of propagationPolarization

ExampleE(t,z) = Eo cos (t z) ax

Dr. Benjamin C. Flores5Forward and backward wavesSign Convention

- z propagation in +z direction+ zpropagation in z direction

Which is it? a) forward travelingb) backward traveling

Dr. Benjamin C. Flores6Partial reflectionThis happens when there is a change in medium

Dr. Benjamin C. Flores7Standing EM waveCharacteristicsAmplitudeAngular frequencyPhasePolarizationNo net propagation

ExampleE(t,z) = A cos (t ) cos( z) ax

Dr. Benjamin C. Flores8Complex notationRecall Eulers formulaexp(j) = cos () + j sin ()

Dr. Benjamin C. Flores9ExerciseCalculate the magnitude of exp(j) = cos ()+ j sin ()

Determine the complex conjugate of exp(j )

Dr. Benjamin C. Flores10Traveling wave complex notationLet = t z

Complex fieldEc(t, z) = A exp [j(t z)] ax = A cos(t z) ax + j A sin(t z) ax

E(z,t) = Real { Ec(t, z) }

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Standing wave complex notationE = 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. Flores12ExerciseShow that E(t) = A exp(jt) sin(z)can be written as the sum of two complex traveling waves. Hint: Recall thatj2 sin() = exp (j ) exp( j )

Dr. Benjamin C. Flores13Transmission line/coaxial cableVoltage waveV = Vo cos (t z)Current waveI = Io cos (t z)

Characteristic ImpedanceZC = Vo / IoTypical values: 50, 75 ohms

Dr. Benjamin C. Flores14RADARRadio detection and ranging

Dr. Benjamin C. Flores15Time delayLet 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 Ec(t, ) = A exp [j( t )] ax

Dr. Benjamin C. Flores16Definition of coordinate systemA 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. Flores17Definition of vectorA vector (sometimes called a geometric or spatial vector) is a geometric object that has a magnitude, direction and sense.

Dr. Benjamin C. Flores18Direction of a vectorA 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. Flores19Cartesian coordinates

Dr. Benjamin C. Flores20Principal planes

Dr. Benjamin C. Flores21Unit vectorsax = x = iay = y = jaz = z = k

u = A/|A|

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Handedness of coordinate system

Left handed Right handedDr. Benjamin C. Flores23Are you smarter than a 5th grader?Euclidean geometry studies the relationships among distances and angles in flat planes and flat space.truefalseAnalytic geometry uses the principles of algebra.truefalse

Dr. Benjamin C. Flores24Cylindrical coordinate system

Dr. Benjamin C. Flores25 = tan-1 y/x2 = x2 + y2Vectors in cylindrical coordinatesAny 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. Flores26Spherical coordinate system

Dr. Benjamin C. Flores27 = tan-1 y/x = tan-1 z/[x2 + y2]1/2r2 = x2 + y2 + z2Vectors in spherical coordinatesAny 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. Flores28Dot productThe 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. Flores29ExerciseLet 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. Flores30Cross productThe 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. Flores31Cross product

Dr. Benjamin C. Flores32ExerciseConsider the two vectorsa= 3x + 5y + 7z and b = 2x 2y 2zDetermine the cross product c = a x bFind the unit vector n of c

Dr. Benjamin C. Flores33HomeworkRead all of Chapter 1, sections 1-1, 1-2, 1-3, 1-4, 1-5, 1-6Read Chapter 3, sections 3-1, 3-2, 3-3Solve end-of-chapter problems 3.1, 3.3, 3.5 , 3.7, 3.19, 3.21, 3.25, 3.29Dr. Benjamin C. Flores34