Fields and Waves I

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Fields and Waves I

• Lecture 20
• Introduction to Electromagnetic Waves
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY
• Y. Maréchal
• Power Engineering Department
• Institut National Polytechnique de Grenoble,
  France

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These Slides Were Prepared by Prof. Kenneth A.
Connor Using Original Materials Written Mostly by
the Following

• Kenneth A. Connor ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• J. Darryl Michael GE Global Research Center,
  Niskayuna, NY
• Thomas P. Crowley National Institute of
  Standards and Technology, Boulder, CO
• Sheppard J. Salon ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• Lale Ergene ITU Informatics Institute,
  Istanbul, Turkey
• Jeffrey Braunstein Chung-Ang University, Seoul,
  Korea

Materials from other sources are referenced where
they are used. Those listed as Ulaby are figures
from Ulabys textbook.
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Linear property
2D wave
http//people.rit.edu/andpph/exhibit-3.html
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Overview

• Time Harmonic Fields
• Maxwells Equations in Phasor Form
• Complex Permittivity
• EM Wave Equation
• Uniform Plane Waves
• Traveling Waves
• TEM Waves
• Energy Power

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Full Maxwells Equations
Added term in curl H equation for time varying
electric field that gives a magnetic field.
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Fully coupled fields
Maxwells equations give a wave equation.
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Time Harmonic Fields
EM wave propagation involves electric and
magnetic fields having 3 components, each
dependent on all three coordinates, in addition
to time.
e.g. Electric field
vector phasor
instantaneous field
Valid for the other fields and
their sources
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Maxwells Equations in Phasor Domain
vector phasor

Try to symmetrize these 2 terms
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Homogenous wave equations
Complex Permittivity
complex permittivity
Homogenous wave equation (charge free)

Combining
and
propagation constant
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Plane Wave Propagation in Lossless Media
There are three constitutive parameters of the
medium s, e, µ
For lossless medium
Wave number
Homogenous wave equation for a lossless media
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Some typical waves
Ulaby
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Plane wave approximation
At large distances from physical antennas and
ground, the waves can be approximated as uniform
plane waves
Ulaby
Uniform properties of the magnetic and electric
field across x-y
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Maxwells Equations in Phasor Domain
In a Source Free Region
For Plane Waves (only z dependence,
)
Note that there are now two independent field
pairs
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Traveling plane waves
The Electric Field in phasor form (only x
component)
General solution of the differential equation
0
amplitudes (constant)
For a traveling direction in the z direction only
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E and H field for a plane wave
for a lossless medium E polarized in x traveling
in z direction
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Transverse Electromagnetic Wave
http//hibp.ecse.rpi.edu/crowley/java/EMWave/emwa
ve.html
Spatial variation of and at t0
Ulaby
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Uniform Plane waves
In general, a uniform plane wave traveling in the
z direction, may have x and y components
The relationship between them
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Example 1 EM Waves
The electric field of a plane wave is given by
a. Write E in phasor form. b. Is E the solution
of a wave equation like c. Find H using the
phasor form of the ? x E equation. Assume the E
and H phasors are only a function of z. d.
Evaluate the amplitude ratio, E / H in
terms of material properties. e. If E was in
the ay direction, what direction would H be in?
f. How many independent parameters are there in
the following set?
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Example 1 EM Waves
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Example 1 EM Waves
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Transverse Electromagnetic Wave (TEM)

• A plane wave has no electric or magnetic field
  components along the direction of propagation
• Electric and magnetic fields that are
  perpendicular to each other and to the direction
  of propagation
• They are uniform in planes perpendicular to the
  direction of propagation
• At large distances from physical antennas and
  ground, the waves can be approximated as uniform
  plane waves

Ulaby
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Properties of a TEM

• Defines the connection between electric and
  magnetic fields of an EM wave
• Similar to the characteristic impedance (Z0) of
  a transmission line

O
Intrinsic impedance
Phase velocity
m/s
Wavelength
m
If the medium is vacuum up3x108 m/s, ?0
377 O
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Typical values
Typical values of f, b, l for X-rays, visible
light, microwaves, and FM radio in free space
http//www.esat.kuleuven.ac.be/sista/education/tec
hecon/
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Example 2 EM Waves in Lossless Media

• WRPI broadcasts at 91.5 MHz. The amplitude of E
  on campus is roughly 0.08 V/m. Assume a
  coordinate system in which the wave is polarized
  in the ay direction and propagating in the az
  direction.
• Assume the phase is 0 at z 0.
• What are , and for this wave?
• b. Write the electric and magnetic fields in
  phasor form.
• c. Write the electric field in time domain form.

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Example 2 EM Waves in Losseless Media
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Introduction to Electromagnetic Waves

• Power and Energy

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Electromagnetic Power Density

• Poynting Vector , is defined

W/unit area
is along the propagation direction of the wave
Ulaby
Total power
m/s
W
W
OR
Average power density of the wave
W/m2
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Plane wave in a Lossless Medium
W/m2
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Example 3 Energy Power
a. What is the average energy density of the
electric and magnetic fields for the WRPI signal
on campus? b. What is the time average Poynting
vector of the wave, Sav? Divide its magnitude by
the speed of light and compare with your answer
from part a. c. The transmitter is about 10 km
from campus. What transmitter power is required
to radiate the same power density into a sphere
of radius 10 km?
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Example 3 Energy Power