ene 428 microwave engineering
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1RSRS
ENE 428Microwave
Engineering
Lecture 4 Reflection and Transmission at Oblique Incidence, Transmission Lines
2RS
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,,,,,,,,,,,,,,
4RS
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,,,,,,,,,,,,,,
5RS
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.
Fields do extend into the 2nd medium where they decay exponentially with z. However, the transmitted electric and magnetic fields are 90o out of phase, so no power is trans-mitted.
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
10RS
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
iTEE0
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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
iTEE0
12RS
Average power conservation for TE polarization
It should be noted that in terms of power conservation, we only consider power directed normal to the boundary. For TE polarization, we have
tzave
rzave
izave PPP ,,,
t
t
r
r
i
i EEE
cos2
)(cos
2
)(cos
2
)(
2
20
1
20
1
20
13RS
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.
14RS
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
,,,,,,,,,,,,,,
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
,,,,,,,,,,,,,,
yzxj
rrae
EH rr ˆ)cossin(
1
0 1
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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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Total transmission for TM polarization
For TM polarization, there exists an incidence angle at which all of the wave is transmitted into the 2nd medium.
This known as the Brewster’s angle, i = BA and it can be found by first setting the numerator of the reflection coeff. equal to zero; that is,
)sin1()sin1(
coscos
coscos
222
221
222
221
12
tBA
tBA
BAt
Using Snell’s law of refraction and do some algebraic manipulation,
22
21
21
22
21
22
22 )(
sin
BA
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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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When a randomly polarized wave (such as light) is incident on a material at the Brewster’s angle, the TM polarized portion is totally transmitted but at TE component is partially reflected.
This principle is employed in gas lasers, where quartz windows at each end of the laser tube are set at the Brewster’s angle to produce linearly polarized laser output.
p = parallel s = senkrecht (german) = perpendicular
19RS
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
20RS
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
22RS
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’= capacitance 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
25RS
Telegraphist’s equations
26RS
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
where
jCjGLjR
zVdz
zVd
)'')(''(
0)()( 2
2
2
27RS
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
30RS
Low loss transmission line (2)
Therefore, we get
1 ' '( ' ' )2 ' '
C LR G
L C
1 ' '1 ( )8 ' 'G R
LCC L
31RS
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
32RS
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 (lossless: Z0 =
real)
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
W.z
avg eZ
VzP 2
0
20
2)(
z
ave
ave eP
zP 2
)0(
)(
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Power transmission
For lossy case: jeZZ 00
W. cos2
)( 2
0
20 z
avg eZ
VzP
z
ave
ave eP
zP 2
)0(
)(
34RS
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
35RS
Representation of absolute power levels is the dBm scale
Power ratios on the decibel scale (2)
( ) 10log( )1m
PG dB
mW dBm
36RS
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?
37RS
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?
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