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2-D Array of a Liquid Crystal Display
1. WAVES & PHASORS
Applied EM by Ulaby, Michielssen and Ravaioli
Chapter 1 Overview
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Examples of EM Applications
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Dimensions and Units
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Fundamental Forces of Nature
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Gravitational Force
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Force exerted on mass 2 by mass 1
Gravitational field induced by mass 1
Charge: Electrical property of particles
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Units: coulomb
One coulomb: amount of charge accumulated in one second by a current of one ampere.
1 coulomb represents the charge on ~ 6.241 x 1018 electrons
The coulomb is named for a French physicist, Charles-Augustin de Coulomb (1736-1806),
who was the first to measure accurately the forces exerted between electric charges.
Charge of an electron
e = 1.602 x 10-19 C
Charge conservation
Cannot create or destroy charge, only transfer
Electrical Force
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Force exerted on charge 2 by charge 1
Electric Field In Free Space
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Permittivity of free space,
the value of the absolute
dielectric permittivity of vacuum
Linear Superposition: The total
vector electric field at a point in
space due to a system of point
charges is equal to the sum of the
elctric fields at that point due to
the individual charges.
Electric Field Inside Dielectric Medium
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In the absence of point charge, the material is electrically
neutral (each atom having positivly charged nucleus surrounded
by negative electrons
 At any point inside the material the electric field E is zero.
 When a point charge is placed inside the material, the atoms
feel forces, the center symmetry of the electron cloud is altered
with respect to the nucleus.
 One pole of the atom becomes positivly charged
 The polarized atom is called electric dipole, this distortion
process is called polarization
 The degree of the polarization depends on the distance
between the atom and the isolated point.

Electric Field Inside Dielectric Medium
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Polarization of atoms changes
electric field
New field can be accounted for by
changing the permittivity
Permittivity of the material
Another quantity used in EM
is the electric flux density D:
Magnetic Field
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Electric charges can be isolated, but magnetic poles always exist in pairs.
Magnetic field induced by a
current in a long wire
Magnetic permeability of free space
Electric and magnetic fields are
connected through the speed of light:
Static vs. Dynamic
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Static conditions: charges are stationary or moving,
but if moving, they do so at a constant velocity.
Under static conditions, electric and magnetic fields are independent,
but under dynamic conditions, they become coupled.
Material Properties
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Traveling Waves
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Waves carry energy
 Waves have velocity
 Many waves are linear: they do not affect the
passage of other waves; they can pass right through
them (When two people speak to one another, their
sound waves do not reflect from one another, but
simply pass through)

Transient waves: caused by sudden disturbance
 Continuous periodic waves: repetitive source

Types of Waves
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Longitudinal Waves:
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In a longitudinal wave the particle displacement is parallel to the
direction of wave propagation. The animation below shows a onedimensional longitudinal plane wave propagating down a tube. The
particles do not move down the tube with the wave; they simply oscillate
back and forth about their individual equilibrium positions. Pick a single
particle and watch its motion. The wave is seen as the motion of the
compressed region (i.e, it is a pressure wave), which moves from left to
right.
Transverse Waves
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
In a transverse wave the particle displacement is perpendicular
to the direction of wave propagation. The animation below shows
a one-dimensional transverse plane wave propagating from left
to right. The particles do not move along with the wave; they
simply oscillate up and down about their individual equilibrium
positions as the wave passes by. Pick a single particle and watch
its motion.
Water Waves
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
Water waves are an example of waves that involve a combination of both
longitudinal and transverse motions. As a wave travels through the waver, the
particles travel in clockwise circles. The radius of the circles decreases as the
depth into the water increases. The movie below shows a water wave travelling
from left to right in a region where the depth of the water is greater than the
wavelength of the waves. I have identified two particles in yellow to show that
each particle indeed travels in a clockwise circle as the wave passes.
Standing Waves
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Wave in space and time
A wave is a disturbance that travels from one location to another, and
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is described by a wave function that is a function of both space and
time. If the wave function was sine function then the wave would be
exressed by: Ψ ( x , t ) = A sin ( ω t ± k x )
Sinusoidal Waves in Lossless Media
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The general form of the wave equation:
y = height of water surface
x = distance
Phase velocity –
propagation velocity
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If we select a fixed height y0 (equivalent
to constant phase) and follow its progress,
then
=
Wave Frequency and Period
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Direction of Wave Travel
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Wave travelling in +x direction
Wave travelling in ‒x direction
+x direction: if coefficients of t and x have opposite signs
‒x direction: if coefficients of t and x have same sign (both positive
or both negative)
Phase Lead & Lag
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Constructive and Destructive Interference
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
Two waves (with the same amplitude, frequency, and wavelength) are
travelling in the same direction. Using the principle of superposition,
the resulting wave displacement may be written as:
y ( x , t ) = A sin ( kx - ωt ) + A sin ( kx - ωt + ϕ )
=2A cos ( ϕ/2 ) sin(kx - ωt + ϕ/2 )
The animation at left shows how two
sinusoidal waves with the same amplitude
and frequency can add either destructively
or constructively depending on their relative
phase.
When the two individual waves are exactly in
phase the result is large amplitude. When
the two gray waves become exactly out of
phase the sum wave is zero.
Two sine waves travelling in opposite directions create a
standing wave
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
Two waves (with the same amplitude, frequency, and wavelength) are
travelling in opposite directions. Using the principle of superposition, the
resulting wave amplitude may be written as:
y (x , t) = A sin( kx - ωt ) + A sin( kx + ωt )
= 2A sin( kx ) cos( ωt )


The wave amplitude as a function of position is 2Asin(kx). This amplitude does
not travel, but stands still and oscillates up and down according to cos(ωt).
Characteristic of standing waves are locations with maximum displacement
(antinodes) and locations with zero displacement (nodes).

As the movie shows, when the two waves are 180°
out-of-phase with each other they cancel,
and when they are exactly in-phase
with each other they add together.
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Wave Travel in Lossy Media
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Attenuation factor
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Example 1-1: Sound Wave in Water
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Given: sinusoidal sound wave traveling in
the positive x-direction in water
wave amplitude is 10 N/m2, and p(x, t) was
observed to be at its maximum value at t = 0
and x = 0.25 m. Also f=1 kHz, up=1.5 km/s.
Determine: p(x,t)
Solution:
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The EM Spectrum
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Tech Brief 1: LED Lighting
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
Incandescence is
the emission of
light from a hot
object due to its
temperature
Fluoresce means to emit
radiation in consequence
to incident radiation of a
shorter wavelength
When a voltage is applied in a forwardbiased direction across an LED diode,
current flows through the junction and
some of the streaming electrons are
captured by positive charges (holes).
Associated with each electron-hole
recombining act is the release of energy
in the form of a photon.
Tech Brief 1: LED Basics
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Tech Brief 1: Light Spectra
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Tech Brief 1: LED Spectra
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Two ways to generate a broad spectrum, but the phosphor-based approach is
less expensive to fabricate because it requires only one LED instead of three
Tech Brief 1: LED Lighting Cost Comparison
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Complex Numbers
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We will find it is useful to represent
sinusoids as complex numbers
j  1
z  x  jy
Rectangular coordinates
z  z   z e j
Polar coordinates
Re  z   x
Im( z )  y
Relations based
on Euler’s Identity
e j  cos  j sin
Relations for Complex Numbers
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Learn how to
perform these
with your
calculator/computer
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Phasor Domain
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1. The phasor-analysis technique transforms equations
from the time domain to the phasor domain.
2. Integro-differential equations get converted into
linear equations with no sinusoidal functions.
3. After solving for the desired variable--such as a particular voltage or
current-- in the phasor domain, conversion back to the time domain
provides the same solution that would have been obtained with
the original integro-differential equations been solved entirely in the
time domain.
Phasors:
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
A sinusoid is characterized by 3 numbers, its amplitude, its
phase, and its frequency. For example,
In a circuit there will be many signals but in the case of
phasor analysis they will all have the same frequency. For
this reason, the signals are characterized using only their
amplitude and phase.
 The combination of an amplitude and phase to describe
a signal is the phasor for that signal.
 Often it is preferable to represent a phasor using
complex numbers rather than using amplitude and phase.

Phasor Diagrams
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
A sine wave such as V(t) = Asin(ωt + δ) can be
considered to be directly related to a vector of
length A revolving in a circle with angular velocity
ω - in fact just the y component of the vector. The
phase constant δ is the starting angle at t = 0
Phasor diagram
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
When two sine waves are produced on the same
display, one wave is often said “leading” or lagging the
other. This terminology makes sense in the revolving
vector picture as shown in Figure. The blue vector is said
to be leading the red vector or conversely the red
vector is lagging the blue vector.
Phasor Diagram
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
Adding or subtracting two sine waves directly requires a great
deal of algebraic manipulation and the use of trigonometric
identities. However if we consider the sine waves as vectors, we
have a simple problem of vector addition if we ignore ω. For
example consider:
Asin(ωt + φ) = 5sin(ωt + 30°) + 4sin(ωt + 140°) ;
the corresponding vector addition is:
Ax = 5cos(30°) + 4cos(140°) = 1.26595
Ay = 5sin(30°) + 4sin(140°) = 5.07115
Thus
=> 5.23sin(ωt + 76.0°) .
Phasors: continued
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

In this case we represent the signal as a phasor (where V
is a complex number V= ):
To see that (1) and (2) are the same (remember Euler’s formula),
Phasor Domain
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Phasor counterpart of
Time and Phasor Domain
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It is much easier to deal
with exponentials in the
phasor domain than
sinusoidal relations in
the time domain
Just need to track
magnitude/phase,
knowing that everything
is at frequency w
Phasor Model of a Resistor
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Phasor Relation for Resistors
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Current through resistor
Time Domain
Frequency Domain
Time domain
i  I m cos w t   
  iR  RI m cos w t   
Phasor Domain
V  RI m 
Phasor Model of a Capacitor
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
Thus, given the phasor for the voltage across a
capacitor we can directly compute the phasor for the
current through the capacitor. Similarly, given the
phasor for the current through a capacitor we can
compute the phasor for the voltage across the
capacitor using:
Phasor Model of an Inductor
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
Thus, given the phasor for the current we can directly
compute the phasor for the voltage across the inductor.
Similarly, given the phasor for the voltage across an
inductor we can compute the phasor for the current
through the inductor using:
Phasor Relation for Inductors
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Time domain
Phasor Domain
Time Domain
Phasor Relation for Capacitors
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Time domain
Time Domain
Phasor Domain
ac Phasor Analysis: General Procedure
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Example 1-4: RL Circuit
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Cont.
Example 1-4: RL Circuit cont.
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Tech Brief 2: Photovoltaics
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Tech Brief 2: Structure of PV Cell
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Tech Brief 2: PV Cell Layers
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Tech Brief 2: PV System
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Summary
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