Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India

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Fundamentals of Electromagneticsfor Teaching and
LearningA Two-Week Intensive Course for Faculty
inElectrical-, Electronics-, Communication-, and
Computer- Related Engineering Departments in
Engineering Colleges in India

• by
• Nannapaneni Narayana Rao
• Edward C. Jordan Professor Emeritus
• of Electrical and Computer Engineering
• University of Illinois at Urbana-Champaign, USA
• Distinguished Amrita Professor of Engineering
• Amrita Vishwa Vidyapeetham, India

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Program for Hyderabad Area and Andhra Pradesh
FacultySponsored by IEEE Hyderabad Section, IETE
Hyderabad Center, and Vasavi College of
EngineeringIETE Conference Hall, Osmania
University CampusHyderabad, Andhra PradeshJune
3 June 11, 2009 Workshop for Master Trainer
Faculty Sponsored byIUCEE (Indo-US Coalition for
Engineering Education)Infosys Campus, Mysore,
KarnatakaJune 22 July 3, 2009
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• Module 4
• Wave Propagation
• in Free Space
• 4.1 Uniform Plane Waves in Time Domain
• 4.2 Sinusoidally Time-Varying Uniform Plane Waves
• 4.3 Polarization
• 4.4 Poynting Vector and Energy Storage

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Instructional Objectives

• 23. Write the expression for a traveling wave
  function for a
• set of specified characteristics of the
  wave
• 24. Obtain the electric and magnetic fields due
  to an infinite
• plane current sheet of an arbitrarily
  time-varying uniform
• current density, at a location away from it
  as a function of
• time, and at an instant of time as a
  function of distance, in
• free space
• 25. Find the parameters, frequency, wavelength,
  direction of
• propagation of the wave, and the associated
  magnetic (or
• electric) field, for a specified sinusoidal
  uniform plane
• wave electric (or magnetic) field in free
  space
• 26. Write expressions for the electric and
  magnetic fields of a
• uniform plane wave propagating away from an
  infinite
• plane sheet of a specified sinusoidal
  current density, in
• free space

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Instructional Objectives (Continued)

• 27. Obtain the expressions for the fields due to
  an array of
• infinite plane sheets of specified spacings
  and sinusoidal
• current densities, in free space
• 28. Write the expressions for the fields of a
  uniform plane
• wave in free space, having a specified set
  of
• characteristics, including polarization
• 29. Express linear polarization and circular
  polarization as
• superpositions of clockwise and
  counterclockwise circular
• polarizations
• 30. Find the power flow and the electric and
  magnetic stored
• energies associated with electric and
  magnetic fields

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4.1 Uniform Plane Wavesin Time Domain(EEE, Sec.
3.4 FEME, Secs. 4.1, 4.2, 4.4, 4.5)
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Infinite Plane Current Sheet Source
Example
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For a current distribution having only an
x-component of current density that varies only
with z,
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4-8
The only relevant equations are
Thus,
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4-9
In the free space on either side of the sheet, Jx
0
Combining, we get
Wave Equation
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Solution to the Wave Equation
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Where
velocity of light
represents a traveling wave
propagating in the z-direction.
represents a traveling
wave propagating in the z-direction.
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4-14
E4.1 Examples of Traveling Waves
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4-16
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Thus, the general solution is
For the particular case of the infinite plane
current sheet in the z 0 plane, there can only
be a () wave for z gt 0 and a (-) wave for z lt 0.
Therefore,
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Applying Faradays law in integral form to the
rectangular closed path abcda in the limit that
the sides bc and da?0,
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Therefore,
Now, applying Amperes circuital law in integral
form to the rectangular closed path efgha in the
limit that the sides fg and he?0,
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Thus, the solution is
Uniform plane waves propagating away from the
sheet to either side with velocity vp c.
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In practice, there are no uniform plane waves.
However,many practical situations can be studied
based on uniformplane waves. For example, at
large distances from physicalantennas and
ground, the waves can be approximated asuniform
plane waves.
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4-22
x
z
y
z 0
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E4.2
x
z gt 0
z lt 0
z
y
? z
z 0
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Review Questions

• 4.1. Outline the procedure for obtaining from the
  two
• Maxwells equations the particular
  differential equations
• for the special case of J Jx(z, t)ax.
• 4.2. State the wave equation for the case of E
  Ex(z, t)ax.
• Describe the procedure for its solution.
• 4.3. What is a uniform plane wave? Why is the
  study of
• uniform plane waves important?
• 4.4. Discuss by means of an example how a
  function
• f(t z/vp) represents a traveling wave
  propagating in the
• positive z-direction with velocity vp.
• 4.5. Discuss by means of an example how a
  function
• g(t z/vp) represents a traveling wave
  propagating in the
• negative z-direction with velocity vp.

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Review Questions (Continued)

• 4.6. What is the significance of the intrinsic
  impedance of
• free space? What is its value?
• 4.7. Summarize the procedure for obtaining the
  solution for
• the electromagnetic field due to the
  infinite plane sheet of
• uniform time-varying current density.
• 4.8. State and discuss the solution for the
  electromagnetic
• field due to the infinite plane sheet of
  current density
• Js(t) Js(t)ax for z 0.

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Problem S4.1. Writing expressions for traveling
wave functions for specified time and distance
variations
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Problem S4.2. Plotting field variations for a
specified infinite plane-sheet current source
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Problem S4.3. Source and more field variations
from a given field variation of a uniform plane
wave
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4.2 Sinusoidally Time-Varying Uniform Plane
Waves (EEE, Sec. 3.5 FEME, Secs. 4.1, 4.2, 4.4,
4.5)
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Sinusoidal function of time
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Sinusoidal Traveling Waves
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4-36
The solution for the electromagnetic field is
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Three-dimensional depiction of wave propagation
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4-38
Parameters and Properties
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4-39
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4-40
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4-42
E4.3
Then
Direction of propagation is z.
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4-43
E4.4 Array of Two Infinite Plane Current Sheets
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4-45
For both sheets,
No radiation to one side of the array. Endfire
radiation pattern.
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Depiction of superposition of the two waves
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Review Questions

• 4.9. Why is it important to give special
  consideration for
• sinusoidal functions of time and hence
  sinusoidal
• waves?
• 4.10. Discuss the quantities ?, ß, and vp
  associated with
• sinusoidally time-varying uniform plane
  waves.
• 4.11. Define wavelength. What is the
  relationship among
• wavelength, frequency, and phase
  velocity? What is the
• wavelength in free space for a frequency
  of 15 MHz?
• 4.12. How is the direction of propagation of a
  uniform plane
• wave related to the directions of its
  fields?
• 4.13. What is the direction of the magnetic field
  of a uniform
• plane wave having its electric field in
  the positive z-
• direction and propagating in the
  positive x-direction?

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Review Questions (Continued)

• 4.14. Discuss the principle of antenna array,
  with the aid of
• an example.
• 4.15. What should be the spacing and the relative
  phase angle
• of the current densities for an array of
  two infinite,
• plane, parallel current sheets of
  uniform densities, equal
• in amplitude, to confine the radiation
  to the region
• between the two sheets?

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Problem S4.4. Finding parameters and the
electric field for a specified sinusoidal uniform
plane wave magnetic field
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Problem S4.5. Apparent wavelengths of a uniform
plane wave propagating in an arbitrary direction
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Problem S4.5. Apparent wavelengths of a uniform
plane wave propagating in an arbitrary direction
(Continued)
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Problem S4.5. Apparent wavelengths of a uniform
plane wave propagating in an arbitrary direction
(Continued)
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Problem S4.6. Ratio of amplitudes of the
electric field on either side of an array of two
infinite plane current sheets
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• 4.3 Polarization
• (EEE, Sec. 3.6 FEME, Sec. 1.4, 4.5)

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Sinusoidal function of time
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• Polarization is the characteristic which
  describes how the position of the tip of the
  vector varies with time.
• Linear Polarization
• Tip of the vector
• describes a line.
• Circular Polarization
• Tip of the vector
• describes a circle.

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• Elliptical Polarization
• Tip of the vector
• describes an ellipse.
• (i) Linear Polarization
• Linearly polarized in the x direction.

Direction remains along the x axis
Magnitude varies sinusoidally with time
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Linear polarization
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Direction remains along the y axis
Magnitude varies sinusoidally with time
Linearly polarized in the y direction.
If two (or more) component linearly polarized
vectors are in phase, (or in phase opposition),
then their sum vector is also linearly
polarized. Ex
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Sum of two linearly polarized vectors in phase is
a linearly polarized vector
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• (ii) Circular Polarization
• If two component linearly polarized vectors are
• (a) equal to amplitude
• (b) differ in direction by 90
• (c) differ in phase by 90,
• then their sum vector is circularly polarized.

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Circular Polarization
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Example
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• (iii) Elliptical Polarization
• In the general case in which either of (i) or
  (ii) is not satisfied, then the sum of the two
  component linearly polarized vectors is an
  elliptically polarized vector.
• Example

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Example
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4-66

• D3.17
• F1 and F2 are equal in amplitude ( F0) and
  differ in direction by 90. The phase difference
  (say f) depends on z in the manner 2p z  (3p
  z) p z.
• (a) At (3, 4, 0), f p (0) 0.
• (b) At (3, 2, 0.5), f p (0.5) 0.5 p.

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4-67

• (c) At (2, 1, 1), f p (1) p.
• (d) At (1, 3, 0.2) f p (0.2) 0.2p.

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Clockwise and Counterclockwise Polarizations

• In the case of circular and elliptical
  polarizations for the field of a propagating
  wave, one can distinguish between clockwise (cw)
  and counterclockwise (ccw) polarizations. If the
  field vector in a constant phase plane rotates
  with time in the cw sense, as viewed along the
  direction of propagation of the wave, it is said
  to be cw- or right-circularly (or elliptically)
  polarized. If it rotates in the ccw sense, it is
  said to be ccw- or left- circularly (or
  elliptically) polarized.

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4-69
For example, consider the circularly polarized
electric field of a wave propagating in the
z-direction, given by
Then, considering the time variation of the field
vector in the z 0 plane, we note that for
and for
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Review Questions

• 4.16. A sinusoidally time-varying vector is
  expressed in
• terms of its components along the x-,
  y-, and z- axes.
• What is the polarization of each of the
  components?
• 4.17. What are the conditions for the sum of two
  linearly
• polarized sinusoidally time-varying
  vectors to be
• circularly polarized?
• 4.18. What is the polarization for the general
  case of the sum
• of two sinusoidally time-varying
  linearly polarized
• vectors having arbitrary amplitudes,
  phase angles, and
• directions?
• 4.19. Discuss clockwise and counterclockwise
  circular and
• elliptical polarizations associated with
  sinusoidally
• time-varying uniform plane waves.

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Problem S4.7. Expressing uniform plane wave
field in terms of right- and left- circularly
polarized components
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Problem S4.8. Finding the polarization
parameters for an elliptically polarized uniform
plane wave field
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• 4.4 Power Flow
• and Energy Storage
• (EEE, Sec. 3.7 FEME, Sec. 4.6)

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Consider the quantity . Then, from a
vector identity,
Substituting
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Performing volume integration on both sides, and
using the divergence theorem for the last term on
the right side, we get
where we have defined , known
as the Poynting vector. The equation is known as
the Poyntings Theorem.
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Poyntings Theorem
Source power density, (power per unit
volume), W/m3
Electric stored energy density, J/m3
Magnetic stored energy density, J/m3
Power flow out of S
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Interpretation of Poyntings Theorem
Poyntings Theorem says that the power delivered
to the volume V by the current source J0 is
accounted for by the sum of the time rates of
increase of the energies stored in the electric
and magnetic fields in the volume, plus another
term, which we must interpret as the power
carried by the electromagnetic field out of the
volume V, for conservation of energy to be
satisfied. It then follows that the Poynting
vector P has the meaning of power flow density
vector associated with the electromagnetic field.
We note that the units of E x H are volts per
meter times amperes per meter, or watts per
square meter (W/m2) and do indeed represent power
flow density.
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In the case of the infinite plane sheet of
current, note that the electric field adjacent to
and on either side of it is directed opposite to
the current density. Hence, some work has to
be done by an external agent (source) for the
current to flow, and represents
the power density (per unit volume) associated
with this work.
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Review Questions

• 4.20. What is the Poynting vector? What is the
  physical
• interpretation of the Poynting vector
  over a closed
• surface?
• 4.21. State Poyntings theorem. How is it
  derived from
• Maxwells curl equations?
• 4.22. Discuss the interpretation of Poyntings
  theorem.
• 4.23. What are the energy densities associated
  with electric and
• magnetic fields?
• 4.24. Discuss how fields far from a physical
  antenna vary
• inversely with distance from the
  antenna.

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Problem S4.9. Finding the Poynting vector and
power radiated for specified radiation fields of
an antenna
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Problem S4.10. Finding the electric field and
magnetic field energies stored in a
parallel-plate resonator
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Problem S4.10. (Continued)
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Problem S4.11. Finding the work associated with
rearranging a charge distribution
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The End