rs ene 428 microwave engineering lecture 3 polarization, reflection and transmission at normal...
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RSRS
ENE 428Microwave
Engineering
Lecture 3 Polarization, Reflection and Transmission at normal incidence
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Uniform plane wave (UPW) power transmission
2
201Re
2jzx
zEe e a
from1
Re( )2
avgP E H
������������������������������������������
2
201cos
2zx
zEe a
W/m2
2
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Example 6.7: Consider an electric field incident on a copper slab such that the field in the slab is given by
xz aztetzE
)102cos(0.1),( 7
V/m
We want to find the average power density.Since copper is a good conductor, we can use
)S/m108.5)(H/m104)(Hz10( 777 f
1/m 108.47 3The intrinsic impedance is
meeoo jj 4545 17.1
Therefore, 21096-)108.47(2
2
W/m300e45cos)17.1(
)V/m1(
2
1 33
zz
zoz
ave aaem
P
At the surface (z = 0) the power density is 300 W/m2. But after only 1 skin
depth, in this case 21 m, the wave’s power density drops to e-2 (13.5%) of its surface value, or 41 W/m2 in this case.
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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.
• Specifying only the electric field direction is sufficient since magnetic field is readily found from using Maxwell’s equation
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E
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Linear polarization
• Consider in free space,
0( , ) cos( ) xE z t E t z a
��������������E��������������
• At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave”
E��������������
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• A pair of linearly polarized wave also produces linear polarization
Linear polarization
0 0( , ) cos( ) cos( )x yx yE z t E t z a E t z a
��������������
At z = 0 plane
At t = 0, both linearly polarized waveshave their maximum values.
0 0(0,0) x yx xE E a E a
��������������
(0, ) 04
�������������� TE
0 0(0, ) cos( ) cos( )x yx yE t E t a E t a
��������������
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Linear polarization• The tilt angle (tau) is the angle the line makes
with the x-axis
• The axial ratio is the ratio of the long axis of an ellipse to the short axis
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• More generalized of two linearly polarized waves,
• Linear polarization occurs when two linearly polarized waves are
• Linear polarization is a special case of elliptical polarization that has an infinite axial ratio
More generalized linear polarization
0 0( , ) cos( ) cos( )x yx x y yE z t E t z a E t z a
��������������
in phase 0y x
out of phase 180 . y x
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• Superposition of two linearly polarized waves that
• If x = 0 and y = 45, we have
Elliptically polarized wave
0 180y x or
0 0(0, ) cos( ) cos( )
4x yx yE t E t a E t a
��������������
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• occurs when Exo and Eyo are equal and
• Right hand circularly polarized (RHCP) wave
• Left hand circularly polarized (LHCP) wave
• Left and right are referred to as the handedness of wave polarization
Circularly polarized wave
90y x
0 0(0, ) cos( ) cos( )
2x yx yE t E t a E t a
��������������
0 0(0, ) cos( ) cos( )
2x yx yE t E t a E t a
��������������
90y x
90y x
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• Phasor forms:
for RHCP,
for LHCP,
Circularly polarized wave
0 0( 0) yx
jjx yx yE z E e a E e a
��������������from
0( 0) ( )x yxE z E a ja
��������������
0( 0) ( )x yxE z E a ja
��������������
Note: There are also RHEP and LHEP 11
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Ex1 Given
,determine the polarization of this wave
( , ) 8cos( 30 ) 8cos( 90 ) ��������������
x yE z t t z a t z a
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Ex2 The electric field of a uniform plane wave in free space is given by , determine
50100( ) j ys z xE a ja e
��������������
a) f
b) The magnetic field intensity sH��������������
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Assume the medium is lossless, let the incident electric field to be
or in a phasor form
since
then we can show that
1 10( , ) cos( ) xxE z t E t z a
��������������
• Normal incidence – the propagation direction is normal to the boundary
Incident wave
11
1
EH a
����������������������������
11 10( ) j z
xxE z E e a ��������������
1101
1
( ) j zxy
EH z e a
��������������
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• Transmitted wave
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Assume the medium is lossless, let the transmitted electric field to be
then we can show that
22 20( ) j z
xxE z E e a ��������������
Transmitted wave
2202
2
( )
��������������j zx
yE
H z e a
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At z = 0, we have
and
1 = 2 are media the same?
tan1 tan 2
tan1 tan 2
E E
H H
• From boundary conditions,
Reflected wave (1)
10 20x xE E
10 20
1 2
x xE E
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• There must be a reflected wave
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and
This wave travels in –z direction.
Reflected wave (2)
11 10( ) j z
xxE z E e a ��������������
1101
1
( ) j zxy
EH z e a
��������������
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• Boundary conditions (reflected wave is included)
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from
therefore at z = 0
(1)
Reflection and transmission coefficients (1)
1 2x xE E
1 1 2x x xE E E
10 10 20x x xE E E
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from
therefore at z = 0
(2)
• Boundary conditions (reflected wave is included)
Reflection and transmission coefficients (2)
1 2y yH H
1 1 2y y yH H H
10 10 20
1 1 2
x x xE E E
21
• Use Eqns. (1) and (2) to eliminate , we’ll get
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Reflection coefficient
Transmission coefficient
Reflection and transmission coefficients (3)
10 2 1
2 110
jx
x
Ee
E
120 2
2 110
21 jx
x
Ee
E
22
20xE
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Types of boundaries: perfect dielectric and perfect conductor (1)
From
.
Since 2 = 0 then = -1 and Ex10+= -Ex10
-
1 1 1x x xE E E
1 11 10 10
j z j zx x xE E e E e
22
2 2
0jj
20 0xE
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Types of boundaries: perfect dielectric and perfect conductor (2)
This can be shown in an instantaneous form as
10 1( , ) 2 sin( )sinx xE z t E z t
10 12 sin( )xj E z
Standing wave
24
101 )( 11x
zjzjx EeeE
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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
1
2z m
1
2m
z
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Standing waves (2)
Since
and ,
the magnetic field is
or .
Hy1 is maximum when Ex1 = 0
So, E and H are said to be 90o out of phase. There will be no power transmission on either side of the media
Poynting vector
1 1x yE H
1 1x yE H
1 1101
1
( )j z j zxy
EH e e
101 1
1
2( , ) cos cosx
y
EH z t z t
S E H ������������������������������������������ 26
2201
cos2
zxz
Ee a
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Power transmission for 2 perfect dielectrics (1)Then 1 and 2 are both real positive quantities and 1 = 2 = 0
Average incident power densities
11 1 1 1 *
1
1 1Re Re
2 2
xi x y x
EP E H E
2 1
2 1
real
2
10*1
1 1Re
2 xE
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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
Incident energy
in
1 2 3
-l 0 z
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Wave reflection from multiple interfaces (2)Assume lossless media, we have
then we can show that
3 223
3 2
,
20 23 20
20 202
20 20 23 202 2
1
1 1
x x
y x
y x x
E E
H E
H E E
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Wave impedance w (1)
Use Euler’s identity, we can show that
3 2 2 22
2 2 3 2
cos sin( )
cos sinw
z j zz
z j z
31
zjzj
zjzj
w
zjy
zjy
zjx
zjx
y
xw
ee
eez
eHeH
eEeE
H
Ez
22
22
22
22
23
232
2020
2020
2
2
)(
)(
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Wave impedance w (2)Since from B.C.
at z = -l
we may write
1 1 2x x xE E E
1 1 2y y yH H H
10 10 2x x xE E E
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)( lz
)( lz )( lz
(1a)
(2a)
Using eqns (1a) and (2a) to eliminate , we’ll get … 2xE
w
xxx EEE
2
1
10
1
10
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Input impedance insolve to get
10 1
110
x in
inx
E
E
3 2 2 22
2 2 3 2
cos sincos sinin
l j ll j l
33
)( lzw
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Power Transmission and Reflection
Incident energy
in
1 2 3
-l 0 z
2
1
12
in
in
in
r
P
P
22
1
1211
in
in
in
t
P
P
The power in region 2 stays constant in steady-state; power leaves that region to form the reflected and transmitted waves, but is Immediately replenished by the incident wave (from region 1)
If = 0, then there’ll be total transmission.And = 0 when in = 1 , or the input impedance is matched to that of the incident medium. So how do we achieve this?
Pin
Pr
Pt
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Half-wave matching method
Suppose 13 ml 2
ml 2
2
22ml
and
Therefore, and so
Hence, the 2nd region thickness is the multiple “half-wavelength” as measured in that medium
Using eqn: 3 2 2 22
2 2 3 2
cos sincos sinin
l j ll j l
We’ll get 3 in when 2
2ml
The general effect of a multiple half-wave is to render the 2nd region immaterial to the results on reflection and transmission. Equivalently we
have a single interface problem involving 1 and 3
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Homework
6.32: Given yo
x aztazttzE
)45sin(20)cos(10),( V/m, find the polarization and handedness.
6.38: Suppose medium 1 (z < 0) is air and medium 2 (z > 0) has r = 16. The trans-
mitted magnetic field intensity is known to be Ht = 12cos(t – β2z)ay mA/m.(a) Determine the instantaneous value of the incident electric field. (b) Find the reflected time-averaged power density
6.48: A 100-MHz TE polarized wave with amplitude 1.0 V/m is obliquely incidentfrom air (z < 0) onto a slab of lossless, nonmagnetic material with r = 25 (z > 0).
The angle of incidence is 40o. Calculate (a) the angle of transmission, (b) the reflection and transmission coefficients, and (c) the incident, reflected and transmitted fields.