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ENE 428
Microwave Engineering
Lecture 3 Polarization, Reflection
and Transmission at normal
incidence
1
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Uniform plane wave (UPW) power
transmission
from
Pavg
1
Re( E H )
2
1 Ex20 2 z j
Re
e e az
2
1 Ex20 2 z
e
cos a z
2
W/m2
2
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Polarization
• UPW is characterized by its propagation direction and
frequency.
• Its attenuation and phase are determined by medium’s
parameters.
• Polarization determines the orientation of the electric
field in a fixed spatial plane orthogonal to the direction of
the propagation.
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Linear polarization
• Consider E
in free space,
E( z, t ) E0 cos(t z )a x
• At plane z = 0, a tip of E field traces straight line
segment called “linearly polarized wave”
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4
Linear polarization
• A pair of linearly polarized wave also produces linear
polarization
E ( z , t ) Ex 0 cos(t z )a x E y 0 cos(t z )a y
At z = 0 plane
E (0, t ) Ex 0 cos(t )a x E y 0 cos(t )a y
At t = 0, both linearly polarized waves
have their maximum values.
E(0,0) Ex 0 a x Ex 0 a y
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T
E(0, ) 0
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More generalized linear polarization
• More generalized of two linearly polarized waves,
E ( z , t ) Ex 0 cos(t z x )a x E y 0 cos(t z y )a y
• Linear polarization occurs when two linearly polarized
waves are
in phase
out of phase
y x 0
y x 180 .
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Elliptically polarized wave
• Superposition of two linearly polarized waves that
y x 0 or 180
• If x = 0 and y = 45, we have
E(0, t ) Ex0 cos(t )a x Ey 0 cos(t )a y
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Circularly polarized wave
• occurs when Exo and Eyo are equal and
y x 90
• Right hand circularly polarized (RHCP) wave
y x 90
E(0, t ) Ex0 cos(t )a x Ey 0 cos(t )a y
2
• Left hand circularly polarized (LHCP) wave
y x 90
E(0, t ) Ex0 cos(t )a x Ey 0 cos(t )a y
2
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Circularly polarized wave
• Phasor forms:
from
for RHCP,
E ( z 0) Ex 0 e jx a x E y 0e
j y
ay
E ( z 0) Ex 0 (a x ja y )
for LHCP,
E ( z 0) Ex 0 (a x ja y )
Note: There are also RHEP and LHEP
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Ex1 Given
E( z, t ) 8cos(t z 30 )a x 8cos(t z 90 )a y
,determine the polarization of this wave
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Ex2 The electric field of a uniform plane wave in
free space is given byE s 100(a z ja x )e j 50 y
, determine
a) f
b) The magnetic field intensity H s
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c)
S
d) Describe the polarization of the wave
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Reflection and transmission of UPW at
normal incidence
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Incident wave
• Normal incidence – the propagation direction is normal
to the boundary
Assume the medium is lossless, let the incident electric
field to be
E1 ( z, t ) Ex10 cos(t z )a x
1
or in a phasor form E ( z ) Ex10e j1z a x
since
1
H a
E
1
1
1
then we can show that H ( z )
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Ex10
1
e j 1z a y
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Transmitted wave
• Transmitted wave
Assume the medium is lossless, let the transmitted
electric field to be
2
E ( z ) Ex20e j 2 z a x
then we can show that
2
H ( z)
Ex20
2
e j 2 z a y
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Reflected wave (1)
• From boundary conditions,
Etan1 Etan 2
H tan1 H tan 2
At z = 0, we have
Ex10 Ex20
and
Ex10
1 = 2
1
Ex20
2
are media the same?
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Reflected wave (2)
• There must be a reflected wave
1
E ( z ) Ex10e j 1z a x
and
1
H ( z)
Ex10
1
e j1z a y
This wave travels in –z direction.
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Reflection and transmission coefficients
(1)
• Boundary conditions (reflected wave is included)
Ex1 Ex 2
from
Ex1 Ex1 Ex2
therefore at z = 0
Ex10 Ex10 Ex20
(1)
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Reflection and transmission coefficients
(2)
• Boundary conditions (reflected wave is included)
H y1 H y 2
from
H y1 H y1 H y2
therefore at z = 0
Ex10
1
Ex10
1
Ex20
2
(2)
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Reflection and transmission coefficients
(3)
• Solve Eqs. (1) and (2) to get
Reflection coefficient
Ex10 2 1
e j
Ex10 2 1
Transmission coefficient
Ex20
22
1 e j1
Ex10 2 1
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Types of boundaries: perfect dielectric
and perfect conductor (1)
j2
2
0
2 j 2
From
.
Ex 20 0
Since 2 = 0
then = -1 and Ex10+= -Ex10-
Ex1 Ex1 Ex1
Ex1 Ex10e j1z Ex10e j1z
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Types of boundaries: perfect dielectric
and perfect conductor (2)
Ex1 (e j1z e j1z )Ex10
j 2Ex10 sin(1z)
This can be shown in an instantaneous form as
Ex ( z, t ) 2Ex10 sin(1z)sin t Standing wave
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Standing waves (1)
When t = m, Ex1 is 0
at
all positions.
and when z = m, Ex1 is 0
at all time.
Null positions occur at
2
1
z m
m1
z
2
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Standing waves (2)
Since
and
Ex1 H y1
Ex1 H y1
the magnetic field is
or
,
H y1
Ex10
1
H y1 ( z, t )
Hy1 is maximum when Ex1 = 0
(e j1z e j1z )
2Ex10
.
1
cos 1 z cos t
Poynting vector
S EH
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Power transmission for 2 perfect
dielectrics (1)
Then 1 and 2 are both real positive quantities and 1 = 2 =
0
2 1
real
2 1
Average incident power densities
1
P1i Re Ex1H y1
2
E
1
x1
Re Ex1 *
2
1
1 1 2
Re * Ex10
2 1
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Ex3 Let medium 1 have 1 = 100 and medium 2
have 2 = 300 , given Ex10+ = 100 V/m. Calculate
average incident, reflected, and transmitted power
densities
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Wave reflection from multiple interfaces (1)
• Wave reflection from materials that are finite in extent
such as interfaces between air, glass, and coating
• At steady state, there will be 5 total waves
1
2
3
Incident
energy
in
-l
0
z
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Wave reflection from multiple interfaces (2)
Assume lossless media, we have
23
3 2
,
3 2
then we can show that
Ex20 23 Ex20
H y20
H
y 20
1
2
Ex20
1
2
E
x 20
1
2
23 Ex20
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Wave reflection from multiple interfaces (2)
Assume lossless media, we have
23
3 2
,
3 2
then we can show that
Ex20 23 Ex20
H y20
H
y 20
1
2
Ex20
1
2
E
x 20
1
2
23 Ex20
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Wave impedance w (1)
Ex 2
Ex20e j 2 z Ex20e j 2 z
w ( z )
j 2 z
H y 2 H x 20 e
H x20e j 2 z
e j 2 z 23e j 2 z
w ( z ) 2 j 2 z
.
j 2 z
e
23e
Use Euler’s identity, we can show that
3 cos 2 z j2 sin 2 z
w ( z ) 2
2 cos 2 z j3 sin 2 z
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Wave impedance w (2)
Since from B.C.
at z = -l
Ex1 Ex1 Ex 2
H y1 H y1 H y 2
we may write
Ex10 Ex10 Ex 2
Ex10
1
Ex10
1
Ex20
w
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Input impedance in
solve to get
Ex10
in 1
in 1
Ex10
3 cos 2l j2 sin 2l
in 2
2 cos 2l j3 sin 2l
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Refractive index
n r
Under lossless conditions,
0 0
n
r
c
0 0
n
r 0
c
vp
n
v p 0
f
n
1
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