#### Transcript Lesson 1 and 2

```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
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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
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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.
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Propagating EM wave
 Characteristics
 Amplitude
 Phase


Angular frequency
Propagation constant
 Direction of propagation
 Polarization
 Example
 E(t,z) = Eo cos (ωt – βz) ax
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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
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Partial reflection
 This happens when there is a change in medium
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Standing EM wave
 Characteristics
 Amplitude
 Angular frequency
 Phase
 Polarization
 No net propagation
 Example
 E(t,z) = A cos (ωt ) cos( βz) ax
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Complex notation
 Recall Euler’s formula
exp(jφ) = cos (φ) + j sin (φ)
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Exercise
 Calculate the magnitude of
exp(jφ) = cos (φ)+ j sin (φ)
 Determine the complex conjugate of exp(j φ)
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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) }
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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)
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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)
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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
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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
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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 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)
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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.
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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):
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Cartesian coordinates
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Principal planes
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Unit vectors
 ax = x = i
 ay = y = j
 az = z = k
 u = A/|A|
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Handedness of coordinate system
Left handed
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Right handed
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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
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Cylindrical coordinate system
Φ = tan-1 y/x
ρ2 = x2 + y2
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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:
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Spherical coordinate system
Φ = tan-1 y/x
θ = tan-1 z/[x2 + y2]1/2
r2 = x2 + y2 + z2
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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:
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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 θ
where
 |a| and |b| denote the length of a and b
 θ is the angle between them.
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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.
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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 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.
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Cross product
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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
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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
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