EEE 498/598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 11
• Electromagnetic Power Flow Reflection And
  Transmission Of Normally and Obliquely Incident
  Plane Waves Useful Theorems

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Lecture 11 Objectives

• To study electromagnetic power flow reflection
  and transmission of normally and obliquely
  incident plane waves and some useful theorems.

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Flow of Electromagnetic Power

• Electromagnetic waves transport throughout space
  the energy and momentum arising from a set of
  charges and currents (the sources).
• If the electromagnetic waves interact with
  another set of charges and currents in a
  receiver, information (energy) can be delivered
  from the sources to another location in space.
• The energy and momentum exchange between waves
  and charges and currents is described by the
  Lorentz force equation.

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Derivation of Poyntings Theorem

• Poyntings theorem concerns the conservation of
  energy for a given volume in space.
• Poyntings theorem is a consequence of Maxwells
  equations.

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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• Time-Domain Maxwells curl equations in
  differential form

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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• Recall a vector identity
• Furthermore,

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Derivation of Poyntings Theorem in the Time
Domain (Contd)
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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• Integrating over a volume V bounded by a closed
  surface S, we have

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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• Using the divergence theorem, we obtain the
  general form of Poyntings theorem

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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• For simple, lossless media, we have
• Note that

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Derivation of Poyntings Theorem in the Time
Domain (Contd)

• Hence, we have the form of Poyntings theorem
  valid in simple, lossless media

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Derivation of Poyntings Theorem in the Frequency
Domain (Contd)

• Time-Harmonic Maxwells curl equations in
  differential form for a simple medium

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Derivation of Poyntings Theorem in the Frequency
Domain (Contd)

• Poyntings theorem for a simple medium

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Physical Interpretation of the Terms in
Poyntings Theorem

• The terms
• represent the instantaneous power dissipated in
  the electric and magnetic conductivity losses,
  respectively, in volume V.

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• The terms
• represent the instantaneous power dissipated in
  the polarization and magnetization losses,
  respectively, in volume V.

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• Recall that the electric energy density is given
  by
• Recall that the magnetic energy density is given
  by

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• Hence, the terms
• represent the total electromagnetic energy
  stored in the volume V.

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• The term
• represents the flow of instantaneous power out
  of the volume V through the surface S.

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• The term
• represents the total electromagnetic energy
  generated by the sources in the volume V.

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Physical Interpretation of the Terms in
Poyntings Theorem (Contd)

• In words the Poynting vector can be stated as
  The sum of the power generated by the sources,
  the imaginary power (representing the time-rate
  of increase) of the stored electric and magnetic
  energies, the power leaving, and the power
  dissipated in the enclosed volume is equal to
  zero.

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Poynting Vector in the Time Domain

• We define a new vector called the (instantaneous)
  Poynting vector as
• The Poynting vector has the same direction as the
  direction of propagation.
• The Poynting vector at a point is equivalent to
  the power density of the wave at that point.

• The Poynting vector has units of W/m2.

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Time-Average Poynting Vector

• The time-average Poynting vector can be computed
  from the instantaneous Poynting vector as

period of the wave
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Time-Average Poynting Vector (Contd)

• The time-average Poynting vector can also be
  computed as

phasors
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Time-Average Poynting Vector for a Uniform Plane
Wave

• Consider a uniform plane wave traveling in the
  z-direction in a lossy medium

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Time-Average Poynting Vector for a Uniform Plane
Wave (Contd)

• The time-average Poynting vector is

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Time-Average Poynting Vector for a Uniform Plane
Wave (Contd)

• For a lossless medium, we have

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Reflection and Transmission of Waves at Planar
Interfaces
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Normal Incidence on a Lossless Dielectric

• Consider both medium 1 and medium 2 to be
  lossless dielectrics.
• Let us place the boundary between the two media
  in the z 0 plane, and consider an incident
  plane wave which is traveling in the
  z-direction.
• No loss of generality is suffered if we assume
  that the electric field of the incident wave is
  in the x-direction.

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Normal Incidence on a Lossless Dielectric (Contd)
x
medium 2
medium 1
z
z 0
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Normal Incidence on a Lossless Dielectric (Contd)

• Incident wave

known
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Normal Incidence on a Lossless Dielectric (Contd)

• Reflected wave

unknown
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Normal Incidence on a Lossless Dielectric (Contd)

• Transmitted wave

unknown
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Normal Incidence on a Lossless Dielectric (Contd)

• The total electric and magnetic fields in medium
  1 are

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Normal Incidence on a Lossless Dielectric (Contd)

• The total electric and magnetic fields in medium
  2 are

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Normal Incidence on a Lossless Dielectric (Contd)

• To determine the unknowns Er0 and Et0, we must
  enforce the BCs at z 0

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Normal Incidence on a Lossless Dielectric (Contd)

• From the BCs we have

or
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Reflection and Transmission Coefficients

• Define the reflection coefficient as
• Define the transmission coefficient as

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Reflection and Transmission Coefficients (Contd)

• Note also that
• The definitions of the reflection and
  transmission coefficients do generalize to the
  case of lossy media.
• For lossless media, G and t are real.
• For lossy media, G and t are complex.

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Traveling Waves and Standing Waves

• The total field in medium 1 is partially a
  traveling wave and partially a standing wave.
• The total field in medium 2 is a pure traveling
  wave.

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Traveling Waves and Standing Waves (Contd)

• The total electric field in medium 1 is given by

standing wave
traveling wave
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Traveling Waves and Standing Waves Example
x
medium 2
medium 1
z
z 0
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Traveling Waves and Standing Waves Example
(Contd)
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Standing Wave Ratio

• The standing wave ratio is defined as
• In this example, we have

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Time-Average Poynting Vectors
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Time-Average Poynting Vectors (Contd)
We note that
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Time-Average Poynting Vectors (Contd)

• Hence,

Power is conserved at the interface.
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Oblique Incidence at a Dielectric Interface
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Oblique Incidence at a Dielectric Interface
Parallel Polarization (TM to z)
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Oblique Incidence at a Dielectric Interface
Parallel Polarization (TM to z)
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Oblique Incidence at a Dielectric Interface
Perpendicular Polarization (TE to z)
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Oblique Incidence at a Dielectric Interface
Perpenidcular Polarization (TM to z)
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Brewster Angle

• The Brewster angle is a special angle of
  incidence for which G0.
• For dielectric media, a Brewster angle can occur
  only for parallel polarization.

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Critical Angle

• The critical angle is the largest angle of
  incidence for which k2 is real (i.e., a
  propagating wave exists in the second medium).
• For dielectric media, a critical angle can exist
  only if e1gte2.

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Some Useful Theorems

• The reciprocity theorem
• Image theory
• The uniqueness theorem

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Image Theory for Current Elements above a
Infinite, Flat, Perfect Electric Conductor
electric
magnetic
actual sources
images
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Image Theory for Current Elements above a
Infinite, Flat, Perfect Magnetic Conductor
electric
magnetic
actual sources
h
h
images