1. Fields and Waves I Lecture 20 K. A. Connor Y. Maréchal
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
Welcome to Fields and Waves I
Before I start, can those of you with pagers and cell phones please turn them off? Thanks.

2. J. Darryl Michael – GE Global Research Center, Niskayuna, NY
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 Ulaby’s textbook.
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3. Linear property 2D wave http://people.rit.edu/andpph/exhibit-3.html
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4. Overview Time Harmonic Fields EM Wave Equation Energy & Power
Maxwell’s Equations in Phasor Form
Complex Permittivity
EM Wave Equation
Uniform Plane Waves
Traveling Waves
TEM Waves
Energy & Power
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5. Full Maxwell’s Equations
Added term in curl H equation for time varying electric field that gives a magnetic field.
For high frequencies
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6. Maxwell’s equations give a wave equation.
Fully coupled fields
Maxwell’s equations give a wave equation.
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7. 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
instantaneous field
vector phasor
Valid for the other fields and their sources
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8. Maxwell’s Equations in Phasor Domain
Time domain
Phasor domain
vector phasor +
Try to symmetrize these 2 terms
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9. Homogenous wave equations
Complex Permittivity
complex permittivity
Homogenous wave equation (charge free)
Combining
and
propagation constant
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10. Plane Wave Propagation in Lossless Media
There are three constitutive parameters of the medium: σ, ε, μ
For lossless medium
Wave number
Homogenous wave
equation
for a lossless media
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11. Some typical waves
Ulaby
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12. Plane wave approximation
At large distances from physical antennas and ground, the waves can be approximated as uniform plane waves z x y
Ulaby
Uniform properties of the magnetic and electric field across x-y
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13. Maxwell’s 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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14. For a traveling direction in the +z direction only
Traveling plane waves
The Electric Field in phasor form (only x component)
General solution of the differential equation
amplitudes (constant)
For a traveling direction in the +z direction only
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15. E and H field for a plane wave
for a lossless medium
E polarized in x
traveling in +z direction
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16. Transverse Electromagnetic Wave
Spatial variation of and at t=0
Ulaby
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17. 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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18. 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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19. Example 1 – EM Waves
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20. Example 1 – EM Waves
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21. 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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22. 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
[Ω]
Intrinsic impedance
Phase velocity
[m/s]
Wavelength
[m]
If the medium is vacuum :
up=3x108 [m/s], η0 =377 [Ω]
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23. Typical values
Typical values of f, b, l for X-rays, visible light, microwaves, and FM radio in free space
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25. 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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26. Example 2 – EM Waves in Losseless Media
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27. Introduction to Electromagnetic Waves
Power and Energy

28. Electromagnetic Power Density
Poynting Vector , is defined
[W/unit area]
is along the propagation direction of the wave
Ulaby
Total power
[m/s]
[W] OR
[W]
Average power density of the wave
[W/m2]
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29. Plane wave in a Lossless Medium
[W/m2]
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30. 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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31. Example 3 – Energy & Power
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