ENE 428
Microwave 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 ai , ar , and at
 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
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2
Polarizations of UPW obliquely
incident on the boundary (1)
 Perpendicular polarization or transverse electric (TE)
polarization
 E is normal to the plane
of incidence and tangential
to the boundary.
 Only the x component
of the magnetic field is
tangential.
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Polarizations of UPW obliquely
incident on the boundary (2)
 Parallel polarization or transverse magnetic (TM)
polarization
 H is normal to the plane
of incidence and tangential
to the boundary.
 Only the x component
of the electric field is
tangential.
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TE polarization
x
i
E  E0i e  j 1z ' a y
i
z
i
H 
E0i
e  j 1z ' (a x ' )
1
We can write
i
E  E0i e  j 1 ( x sini  z cosi ) a y
and
i
H 
E0i
1
e j1 ( x sini  z cosi ) ( cos i a x  sin i a z )
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Reflected and transmitted fields for
TE polarization
Reflected fields
r
E  E0r e j 1 ( x sinr  z cosr ) a y
r
H 
E0r
1
e j1 ( x sinr  z cosr ) (cos r a x  sin r a z )
Transmitted fields
t
E  E0t e j 2 ( x sint  z cost ) a y
t
H 
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E0t
2
e j2 ( x sint  z cost ) ( cos t a x  sin t a z )
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Snell’s laws of reflection and
refraction (1)
Tangential boundary condition for the electric field
 j  x sin
 j  x sin
at z = 0 E0i e 1 i a y  E0r e j1x sinr a y  E0t e 2 t a y
for this equality to hold,
1x sin i  1x sin r  2 x sin t
Snell’s law of reflection
Snell’s law of refraction
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i   r
1 sin t or

 2 sin i
n1 sin 1  n2 sin 2
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Snell’s laws of reflection and
refraction (2)
the critical angle for total reflection
(i )critical
 2 
 sin  
 1 
1
If i  cri, then it is total reflection and no power can be
transmitted, these fields are referred as evanescent waves.
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Reflection and transmission
coefficients for TE polarization (1)
From the electric field’s B.C. with phases matched, we
have
E0i  E0r  E0t .
(1)
Tangential B.C. for the magnetic field considering
matched phase and equal incident and reflected angles
is
E0i  E0r
1
cos i 
E0t
2
cos t .
(2)
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Reflection coefficient for TE
polarization
Solving Eqs. (1) and (2) gets
 2 cos i  1 cos t i
E 
E0
 2 cos i  1 cos t
r
0
or
 TE
 2 cos i  1 cos t

.
 2 cos i  1 cos  t
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Transmission coefficient for TE
polarization
Solving Eqs. (1) and (2) gets
2 2 cos i
E 
E0i
 2 cos i  1 cos  t
t
0
or
 TE
Notice that
2 2 cos i

.
 2 cos i  1 cos  t
 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
i
E  E0i e j 1 ( x sini  z cosi ) (cos i a x  sin i a z )
i
H 
E0i
1
e j1 ( x sini  z cosi ) a y
 Reflected fields
r
E  E0r e j1 ( x sinr  z cosr ) (cos  r a x  sin  r a z )
r
H 
E0r
1
e j 1 ( x sinr  z cosr ) a y
 Transmitted fields
t
E  E0t e j 2 ( x sint  z cost ) (cos t a x  sin t a z )
t
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H 
E0t
2
e j2 ( x sint  z cost ) a y
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Reflection and transmission
coefficients for TM polarization
Solving B.C.s gets
 TM 
and
Notice that
2 cos t  1 cos i
2 cos t  1 cos i
 TM
22 cos i

.
2 cos t  1 cos i
 TM
cos i
 (1   TM )
cos t
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Total transmission for TM
polarization
Brewster’s angle for total transmission
i   BA  sin
1
 22 ( 22  12 )
 22 12  12  22
For lossless, non-magnetic media, we have
 BA  sin
1
1
1
 r1
r2
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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 length
L’= inductance per unit length
C’= capacitor per unit length
G’= conductance per unit length
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Telegraphist’s equations
• General transmission lines equations:
v( z, t )
i ( z, t )

 i ( z, t ) R ' L '
z
t
i ( z, t )
v( z , t )

 v( z, t )G ' C '
z
t
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Telegraphist’s time-harmonic wave
equations
• Time-harmonic waves on transmission lines
dV ( z )
 ( R ' j L ') I ( z )
dz
dI ( z )
 (G ' jC ')V ( z )
dz
After arranging we have
d 2V ( z )
  2V ( z )  0
dz
where   ( R ' j L ')(G ' jC ')    j  .
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Traveling wave equations for the
transmission line
• Instantaneous form
v( z, t )  V0 e z cos(t   z )  V0e z cos(t   z )
i( z, t )  I 0 e z cos(t   z )  I 0e z cos(t   z )
• Phasor form
V ( z )  V0 e z  V0e z
I ( z )  I 0 e z  I 0e z
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Lossless transmission line
• lossless when R’ = 0 and G’ = 0
 0
  j   j L ' C '
   L ' C '
and

1
up  

L 'C '
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Low loss transmission line (1)
• low loss when R’ << L’ and G’ << C’
1/ 2

    j    R ' j L ' (G ' jC ')1/ 2 


1/ 2
1/ 2

R'  
G'  
 j L ' C ' 1 
1


 
j

L
'
j

C
'
 
 

Expanding 1  x in binomial series gives
2
x x
1  x  1    ......
2 8
for x << 1
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Low loss transmission line (2)
Therefore, we get
1
C'
L'
  (R '
G'
)
2
L'
C'
R' 
 1 G'
   LC 1  (

)
 8 C '  L ' 
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Characteristic impedance
 Characteristic impedance Z0 is defined as the
the ratio of the traveling voltage wave
amplitude to the traveling current wave
amplitude.
V0
V0
Z0     
I0
I0
or
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R ' j L '
Z0 
.
G ' jC '
For lossless line,
L'
Z0 
.
C'
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Power transmission
 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
2
V
Pi  ( z, t )  v( z, t )i( z, t )  0 e2 z cos 2 (t   z ).
Z0
 The time-averaged power can be shown as
2
T
T
V
1
1



2

z
2
Pavg ( z )   Pi ( z, t )dt  0 e
cos
(t   z )dt.

T0
Z0
T0
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2
V

Pavg
( z )  0 e2 z
Z0
W.
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Power ratios on the decibel scale (1)
 A convenient way to measure power ratios
 Power gain (dB)
Pout
G (dB)  10 log(
)
Pin
dB
 Power loss (dB)
Pin
attenuation(dB)  10 log(
) dB
Pout
 1 Np = 8.686 dB
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Power ratios on the decibel scale (2)
 Representation of absolute power levels is the
dBm scale
P
G(dBm )  10log(
)
1mW
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?
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