EEE 498/598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 10
• Uniform Plane Wave Solutions to Maxwells
  Equations

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Lecture 10 Objectives

• To study uniform plane wave solutions to
  Maxwells equations
• In the time domain for a lossless medium.
• In the frequency domain for a lossy medium.

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Overview of Waves

• A wave is a pattern of values in space that
  appear to move as time evolves.
• A wave is a solution to a wave equation.
• Examples of waves include water waves, sound
  waves, seismic waves, and voltage and current
  waves on transmission lines.

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Overview of Waves (Contd)

• Wave phenomena result from an exchange between
  two different forms of energy such that the time
  rate of change in one form leads to a spatial
  change in the other.
• Waves possess
• no mass
• energy
• momentum
• velocity

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Time-Domain Maxwells Equations in Differential
Form
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Time-Domain Maxwells Equations in Differential
Form for a Simple Medium
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Time-Domain Maxwells Equations in Differential
Form for a Simple, Source-Free, and Lossless
Medium
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Time-Domain Maxwells Equations in Differential
Form for a Simple, Source-Free, and Lossless
Medium

• Obviously, there must be a source for the field
  somewhere.
• However, we are looking at the properties of
  waves in a region far from the source.

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Derivation of Wave Equations for Electromagnetic
Waves in a Simple, Source-Free, Lossless Medium
0
0
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Wave Equations for Electromagnetic Waves in a
Simple, Source-Free, Lossless Medium

• The wave equations are not independent.
• Usually we solve the electric field wave equation
  and determine H from E using Faradays law.

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Uniform Plane Wave Solutions in the Time Domain

• A uniform plane wave is an electromagnetic wave
  in which the electric and magnetic fields and the
  direction of propagation are mutually orthogonal,
  and their amplitudes and phases are constant over
  planes perpendicular to the direction of
  propagation.
• Let us examine a possible plane wave solution
  given by

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The wave equation for this field simplifies to
• The general solution to this wave equation is

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The functions p1(z-vpt) and p2 (zvpt) represent
  uniform waves propagating in the z and -z
  directions respectively.
• Once the electric field has been determined from
  the wave equation, the magnetic field must follow
  from Maxwells equations.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The velocity of propagation is determined solely
  by the medium
• The functions p1 and p2 are determined by the
  source and the other boundary conditions.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• Here we must have

where
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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• h is the intrinsic impedance of the medium given
  by
• Like the velocity of propagation, the intrinsic
  impedance is independent of the source and is
  determined only by the properties of the medium.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• In free space (vacuum)

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• Strictly speaking, uniform plane waves can be
  produced only by sources of infinite extent.
• However, point sources create spherical waves.
  Locally, a spherical wave looks like a plane
  wave.
• Thus, an understanding of plane waves is very
  important in the study of electromagnetics.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• Assuming that the source is sinusoidal. We have

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The electric and magnetic fields are given by

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The argument of the cosine function is the called
  the instantaneous phase of the field

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The speed with which a constant value of
  instantaneous phase travels is called the phase
  velocity. For a lossless medium, it is equal to
  and denoted by the same symbol as the velocity of
  propagation.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The distance along the direction of propagation
  over which the instantaneous phase changes by 2p
  radians for a fixed value of time is the
  wavelength.

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• The wavelength is also the distance between every
  other zero crossing of the sinusoid.

Function vs. position at a fixed time
l
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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• Relationship between wavelength and frequency in
  free space
• Relationship between wavelength and frequency in
  a material medium

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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• b is the phase constant and is given by

rad/m
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Uniform Plane Wave Solutions in the Time Domain
(Contd)

• In free space (vacuum)

free space wavenumber (rad/m)
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Time-Harmonic Analysis

• Sinusoidal steady-state (or time-harmonic)
  analysis is very useful in electrical engineering
  because an arbitrary waveform can be represented
  by a superposition of sinusoids of different
  frequencies using Fourier analysis.
• If the waveform is periodic, it can be
  represented using a Fourier series.
• If the waveform is not periodic, it can be
  represented using a Fourier transform.

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Time-Harmonic Maxwells Equations in Differential
Form for a Simple, Source-Free, Possibly Lossy
Medium
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Derivation of Helmholtz Equations for
Electromagnetic Waves in a Simple, Source-Free,
Possibly Lossy Medium
0
0
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Helmholtz Equations for Electromagnetic Waves in
a Simple, Source-Free, Possibly Lossy Medium

• The Helmholtz equations are not independent.
• Usually we solve the electric field equation and
  determine H from E using Faradays law.

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Uniform Plane Wave Solutions in the Frequency
Domain

• Assuming a plane wave solution of the form
• The Helmholtz equation simplifies to

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• The propagation constant is a complex number that
  can be written as

attenuation constant (Np/m)
phase constant (rad/m)
(m-1)
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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• a is the attenuation constant and has units of
  nepers per meter (Np/m).
• b is the phase constant and has units of radians
  per meter (rad/m).
• Note that in general for a lossy medium

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• The general solution to this wave equation is

• wave traveling in the -z-direction

• wave traveling in the z-direction

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Converting the phasor representation of E back
  into the time domain, we have

• We have assumed that C1 and C2 are real.

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• The corresponding magnetic field for the uniform
  plane wave is obtained using Faradays law

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Evaluating H we have

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• We note that the intrinsic impedance h is a
  complex number for lossy media.

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Converting the phasor representation of H back
  into the time domain, we have

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• We note that in a lossy medium, the electric
  field and the magnetic field are no longer in
  phase.
• The magnetic field lags the electric field by an
  angle of fh.

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Note that we have
• These form a right-handed coordinate system

• Uniform plane waves are a type of transverse
  electromagnetic (TEM) wave.

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Relationships between the phasor representations
  of electric and magnetic fields in uniform plane
  waves

unit vector in direction of propagation
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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Example
• Consider

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)
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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Properties of the wave determined by the source
• amplitude
• phase
• frequency

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Uniform Plane Wave Solutions in the Frequency
Domain (Contd)

• Properties of the wave determined by the medium
  are
• velocity of propagation (vp)
• intrinsic impedance (h)
• propagation constant constant (gajb)
• wavelength (l)

• also depend on frequency

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Dispersion

• For a signal (such as a pulse) comprising a band
  of frequencies, different frequency components
  propagate with different velocities causing
  distortion of the signal. This phenomenon is
  called dispersion.

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Plane Wave Propagation in Lossy Media

• Assume a wave propagating in the z-direction
• We consider two special cases
• Low-loss dielectric.
• Good (but not perfect) conductor.

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Plane Waves in a Low-Loss Dielectric

• A lossy dielectric exhibits loss due to molecular
  forces that the electric field has to overcome in
  polarizing the material.
• We shall assume that

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Plane Waves in a Low-Loss Dielectric (Contd)

• Assume that the material is a low-loss
  dielectric, i.e, the loss tangent of the material
  is small

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Plane Waves in a Low-Loss Dielectric (Contd)

• Assuming that the loss tangent is small,
  approximate expressions for a and b can be
  developed.

wavenumber
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Plane Waves in a Low-Loss Dielectric (Contd)

• The phase velocity is given by

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Plane Waves in a Low-Loss Dielectric (Contd)

• The intrinsic impedance is given by

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Plane Waves in a Low-Loss Dielectric (Contd)

• In most low-loss dielectrics, er is more or less
  independent of frequency. Hence, dispersion can
  usually be neglected.
• The approximate expression for a is used to
  accurately compute the loss per unit length.

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Plane Waves in a Good Conductor

• In a perfect conductor, the electromagnetic field
  must vanish.
• In a good conductor, the electromagnetic field
  experiences significant attenuation as it
  propagates.
• The properties of a good conductor are determined
  primarily by its conductivity.

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Plane Waves in a Good Conductor

• For a good conductor,
• Hence,

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Plane Waves in a Good Conductor (Contd)
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Plane Waves in a Good Conductor (Contd)

• The phase velocity is given by

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Plane Waves in a Good Conductor (Contd)

• The intrinsic impedance is given by

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Plane Waves in a Good Conductor (Contd)

• The skin depth of material is the depth to which
  a uniform plane wave can penetrate before it is
  attenuated by a factor of 1/e.
• We have

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Plane Waves in a Good Conductor (Contd)

• For a good conductor, we have

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Wave Equations for Time-Harmonic Fields in Simple
Medium