Week 1 Wave Concepts Coordinate Systems and Vector
Products
Slide 2
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
Slide 3
Standard prefixes (SI) Dr. Benjamin C. Flores3
Slide 4
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
Slide 5
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
Slide 6
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
Slide 7
Partial reflection This happens when there is a change in
medium Dr. Benjamin C. Flores7
Slide 8
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
Slide 9
Complex notation Recall Eulers formula exp(j) = cos () + j sin
() Dr. Benjamin C. Flores9
Slide 10
Exercise Calculate the magnitude of exp(j) = cos ()+ j sin ()
Determine the complex conjugate of exp(j ) Dr. Benjamin C.
Flores10
Slide 11
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
Slide 12
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
Slide 13
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
Slide 14
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
Slide 15
RADAR Radio detection and ranging Dr. Benjamin C. Flores15
Slide 16
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
Slide 17
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
Slide 18
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
Slide 19
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
Slide 24
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
Slide 25
Cylindrical coordinate system Dr. Benjamin C. Flores25 = tan -1
y/x 2 = x 2 + y 2
Slide 26
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
Slide 27
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
Slide 28
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
Slide 29
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
Slide 30
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
Slide 31
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
Slide 32
Cross product Dr. Benjamin C. Flores32
Slide 33
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
Slide 34
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