ENE 428 Microwave Engineering
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Transcript of ENE 428 Microwave Engineering
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ENE 428Microwave
Engineering
Lecture 4 Reflection and Transmission at Oblique Incidence, Transmission Lines
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Plane wave propagation in general dielectrics
Assume lossless medium
The propagation directions are and The plane of incidence is defined as the plane containing both normal to the boundary and the incident wave’s propagation direction. The angle of incidence i is the angle the incident field makes with a normal to the boundary
, ,i ra a ta
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Perpendicular polarization or transverse electric (TE) polarization
is normal to the planeof incidence and tangentialto the boundary.
Only the x componentof the magnetic field is tangential.
Polarizations of UPW obliquely incident on the boundary (1)
E,,,,,,,,,,,,,,
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Parallel polarization or transverse magnetic (TM) polarization
is normal to the planeof incidence and tangentialto the boundary.
Only the x componentof the electric field is tangential.
Polarizations of UPW obliquely incident on the boundary (2)
H,,,,,,,,,,,,,,
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TE polarization
1
1
'0
'0'
1
( )
,,,,,,,,,,,,,,
,,,,,,,,,,,,,,
i j ziy
ii j z
x
E E e a
EH e a
We can write
and
1 ( sin cos )0
i ii j x zi
yE E e a ,,,,,,,,,,,,,,
1 ( sin cos )0
1
( cos sin )i i
ii j x z
x zi i
EH e a a
,,,,,,,,,,,,,,
x
zi
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Reflected and transmitted fields for TE polarization
1 ( sin cos )0
r rr j x zr
yE E e a ,,,,,,,,,,,,,,
1 ( sin cos )0
1
(cos sin )r r
rr j x z
x zr r
EH e a a
,,,,,,,,,,,,,,
Reflected fields
Transmitted fields
2 ( sin cos )0
t tt j x zt
yE E e a ,,,,,,,,,,,,,,
2 ( sin cos )0
2
( cos sin )t t
tt j x z
x zt t
EH e a a
,,,,,,,,,,,,,,
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Tangential boundary condition for the electric field
at z = 0
for this equality to hold,
Snell’s law of reflection
Snell’s law of refraction or
Snell’s laws of reflection and refraction (1)
1 21sin sinsin0 0 0
i trj x j xj xi r ty y yE e a E e a E e a
1 1 2sin sin sini r tx x x
i r
1
2
sinsin
t
i
1 1 2 2sin sinn n
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the critical angle for total reflection
If i cri, then it is total reflection and no power can be transmitted, these fields are referred as evanescent waves.
Snell’s laws of reflection and refraction (2)
1 2critical
1
( ) sini
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From the electric field’s B.C. with phases matched, we have
Tangential B.C. for the magnetic field considering matched phase and equal incident and reflected angles is
Reflection and transmission coefficients for TE polarization (1)
0 0 0. (1)i r tE E E
0 0 0
1 2
cos cos . (2)i r t
i t
E E E
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Solving Eqs. (1) and (2) gets
or
Reflection coefficient for TE polarization
2 10 0
2 1
2 1TE
2 1
cos coscos cos
cos cos.
cos cos
r ii t
i t
i t
i t
E E
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Solving Eqs. (1) and (2) gets
or
Notice that
Transmission coefficient for TE polarization
20 0
2 1
2TE
2 1
2 coscos cos
2 cos.
cos cos
t ii
i t
i
i t
E E
TE 1 TE
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Ex2 A 2 GHz TE wave is incident at 30
angle of incidence from air on to a thick slab of nonmagnetic, lossless dielectric with r = 16. Find TE and TE.
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Fields for TM polarization Incident fields
Reflected fields
Transmitted fields
1 ( sin cos )0 (cos sin )i i
i j x zix zi iE E e a a
,,,,,,,,,,,,,,
1 ( sin cos )0
1
i i
ii j x z
yE
H e a
,,,,,,,,,,,,,,
1 ( sin cos )0 (cos sin )r r
r j x zrx zr rE E e a a
,,,,,,,,,,,,,,
1 ( sin cos )0
1
r r
rr j x z
yE
H e a
,,,,,,,,,,,,,,
2 ( sin cos )0 (cos sin )t t
t j x ztx zt tE E e a a
,,,,,,,,,,,,,,
2 ( sin cos )0
2
t t
tt j x z
yE
H e a
,,,,,,,,,,,,,,
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Solving B.C.s gets
and
Notice that
Reflection and transmission coefficients for TM polarization
2TM
2 1
2 cos.
cos cosi
t i
TM
cos(1 )
cosi
TMt
2 1TM
2 1
cos coscos cos
t i
t i
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Brewster’s angle for total transmission
For lossless, non-magnetic media, we have
Total transmission for TM polarization
2 2 21 2 2 1
2 2 2 22 1 1 2
( )sini BA
1
1
2
1sin
1BA
r
r
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Ex3 A uniform plane wave is incident from air onto glass at an angle from the normal of 30. Determine the fraction of the incident power that is reflected and transmitted for a) and b). Glass has refractive index n2 = 1.45.a) TM polarization
b) TE polarization
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Transmission lines (1)
• Transmission lines or T-lines are used to guide propagation of EM waves at high frequencies.
• Examples:– Transmitter and antenna– Connections between computers in a network– Interconnects between components of a stereo system– Connection between a cable service provider and aTV set.– Connection between devices on circuit board
• Distances between devices are separated by much larger order of wavelength than those in the normal electrical
circuits causing time delay.
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Transmission lines (2)
• Properties to address:– time delay– reflections– attenuation– distortion
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Distributed-parameter model• Types of transmission lines
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Distributed-parameter model• The differential segment of the transmission
line
R’ = resistance per unit lengthL’= inductance per unit lengthC’= capacitor per unit lengthG’= conductance per unit length
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Telegraphist’s equations
• General transmission lines equations:
( , ) ( , )( , ) ' '
( , ) ( , )( , ) ' '
v z t i z ti z t R L
z ti z t v z t
v z t G Cz t
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Telegraphist’s time-harmonic wave equations
• Time-harmonic waves on transmission lines
After arranging we have
( )( ' ') ( )
( )( ' ') ( )
dV zR j L I z
dzdI z
G j C V zdz
22( )( ) 0
( ' ')( ' ') .
d V zV z
dz
R j L G j C j
where
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Traveling wave equations for the transmission line
• Instantaneous form
• Phasor form
0 0
0 0
( , ) cos( ) cos( )
( , ) cos( ) cos( )
z z
z z
v z t V e t z V e t z
i z t I e t z I e t z
0 0
0 0
( )
( )
z z
z z
V z V e V e
I z I e I e
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Lossless transmission line
• lossless when R’ = 0 and G’ = 0
0
' 'j j L C
' 'L C
1
' 'pu
L C
and
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Low loss transmission line (1)
• low loss when R’ << L’ and G’ << C’
1/ 2 1/ 2' ' ( ' ')j R j L G j C 1/ 2 1/ 2
' '' ' 1 1
' 'R G
j L Cj L j C
Expanding in binomial series gives1 x2
1 1 ......2 8x x
x for x << 1
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Low loss transmission line (2)
Therefore, we get
1 ' '( ' ' )2 ' '
C LR G
L C
1 ' '1 ( )8 ' 'G R
LCC L
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Characteristic impedance
0 00
0 0
V VZ
I I
or
For lossless line,
0
' '.
' 'R j L
ZG j C
Characteristic impedance Z0 is defined as the the ratio of the traveling voltage wave
amplitude to the traveling current wave amplitude.
0
'.'L
ZC
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Power transmitted over a specific distance is calculated.
The instantaneous power in the +z traveling wave at any point along the transmission line can be shown as
The time-averaged power can be shown as
Power transmission
22 20
0
( , ) ( , ) ( , ) cos ( ).zi
VP z t v z t i z t e t z
Z
22 20
0 00
1 1( ) ( , ) cos ( ) .
T Tz
avg i
VP z P z t dt e t z dt
T Z T
220
0
( ) zavg
VP z e
Z
W.
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A convenient way to measure power ratios
Power gain (dB)
Power loss (dB)
1 Np = 8.686 dB
Power ratios on the decibel scale (1)
( ) 10log( )out
in
PG dB
P
( ) 10log( )in
out
Pattenuation dB
P dB
dB
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Representation of absolute power levels is the dBm scale
Power ratios on the decibel scale (2)
( ) 10log( )1m
PG dB
mW dBm
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Ex1 A 12-dB amplifier is in series with a 4-dB attenuator. What is the overall gain of the circuit?
Ex2 If 1 W of power is inserted into a coaxial cable, and 1 W of power is measured 100m down the line, what is the line’s attenuation in dB/m?
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Ex3 A 20 m length of the transmission line is known to produce a 2 dB drop in the power from end to end,a) what fraction of the input power does it reach the output?
b) What fraction of the input power does it reach the midpoint of the line?
c) What is the attenuation constant?