7. plane wave propagation - sonoma.edu€¦ · review we have learned about wave propagation guided...
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7. PLANE WAVE PROPAGATION
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Review
We have learned about wave propagation Guided propagation Skywave [f=3-3- Mhz] Cable
Transmission Line Reflected wave Constructive Parameters Standing waves
Wave equation Time representation Phasor representation
Propagation constant
We have been assuming TEM waves:Direction of propagation is in Z
E is in r direction H is radial
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Sky Wave (Skip) Propagation
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Unbounded EM Waves
EM Waves can be unguided:1. EM Source radiates Spherical wave 2. Spherical wave Planer wave (far-field effect)3. Planer waves are uniform4. We consider TEM waves
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Planer Waves
Unbounded EM Waves: 1. Waves are traveling in dielectric (perfect dielectric lossless media)2. We use wave equations instead of transmission line equations3. We refer to intrinsic impedance rather than characteristic impedance, Zo4. Propagation constant = loss + Phase constant5. k = wave number (same as phase constant in transmission line)
We start by considering phasor form!
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Review of Maxwell’s Equations
We now express these in phasor form. HOW?
THIS IS WHAT WE HAVE LEARNED SO FAR……
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Review of Maxwell’s Equations –General Form
All the fields are in phasor form
Time derivatives are expressed differently:d/dt jw
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Maxwell’s Equations –Free Space Set We assume there are no charges in free space and
thus, = 0
Time-varying E and H cannot exist independently! If dE/dt non-zero dD/dt is non-zero Curl of H is non-
zero H is non-zero
If H is a function of time E must exist!
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Maxwell’s Equations –Free Space Set We assume there are no charges in free space and
thus, = 0
We will use these to derive the wave equation for EM waves.
Phasor Form
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Wave Equations -Assume no volume charges
Complex permittivity
Special Property:
Laplacian of E
For electrostatic Del of E is zero
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Homogeneous Wave Equations for E and H
Complex permittivity
Propagation Constant:
Similarly:Similarly:
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Homogeneous Wave Equations for E and H (Lossless case)
Complex permittivity
Propagation Constant:
Similarly:
Note: if lossless conductivity =0
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Our assumption was having a uniform plane
There is no change of field For example, in x-y plane: dE/dx = dE/dy = 0
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Uniform Plane Wave (x-y plane)
Must satisfyThere is no change in X and Y (uniform)Same thing for Ey and Ez:
Only non-zero vector componentSame thing for Ey and Ez:
Consider vector field E:
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General Form of the Solution:
For a wave travelling along +z only:
Application of yields:
Summary: This is a plane wave with
with
Uniform Plane Wave (x-y plane) -Solution
Propagating in +Z Propagating in
-Z
Note A
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Power and Impedance
Intrinsic Impedance of a lossless medium (analogy to Zo)
H (x,y,z)
E (x,y,z)
k (x,y,z)
Phasor FormRemember: HEK
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TEM Traveling Wave Solution Time-Domain Solution
Time Domain Representation
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Check the Simulator
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Directional Relation Between E and H
For Any TEM Wave
H (x,y,z)
E (x,y,z)
k (x,y,z)
Phasor FormNote: E and H may have x & y componentsHowever, they travel in Z direction and They are perpendicular to each other!
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Example
Note B
What is k? (it is a function of what? Which direction is it pointing at?)What is E?What is H?
H (x,y,z)
E (x,y,z)
k (x,y,z)
(HEK)
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Example
Notes B
Find l, k, E(z,t), H(z,t)
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Example cont.
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Check out the Simulator
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Polarization - General
Polarization is the orientation of electric field component of an electromagnetic wave relative to the Earth’s surface.
Polarization is important to get the maximum performance from the antennas There are different types of polarization (depending on existence and changes of
different electric fields) Linear
Horizontal (E field changing in parallel with respect to earth’s surface) Vertical (E field going up/down with respect to earth’s surface) Dual polarized
Circular (Ex and Ey) Similar to satellite communications TX and RX antennas must agree on direction of rotation
Elliptical Linear polarization is used in WiFi communications
Polarization can change as the signal travels away from the source!-Due to the magnetic field of Earth (results in Faraday rotation) -Due to reflection
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Polarization - General
Polarization is important to get the maximum performance from the antennas The polarization of the antennas at both ends of the path must use the same
polarization This is particularly important when the transmitted power is limited
(x,z)(y,z)
E-Field is Going up/down respect to Earth!(Vertical Polarization)
E-Field is Rotating (or Corkscrewed) as they are traveling
z
Propagating parallel to earth
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Wave Polarization
See Notes
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Wave Polarization
Plane wave propagating along +z : If:
then
Delta = Angle difference
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Polarization State
Polarization state describes the trace of E as a function of time at a fixed z
Magnitude of E Inclination Angle
Time domain representation
psi
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Linear Polarization:
E traces a line( in blue) as the wave traverses a fixed plane
orIn-phase Out-of-phase
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Circular Polarization
LHP:
RHP:
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LH Circular Polarization
Magnitude Inclination Angle
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RH Circular Polarization:
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Example
See notes
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LCD
Liquid CristalMolecular spiral
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Operation of a Single Pixel
5 micron
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LCD 2-D Array