Wave Wave IncidenceIncidence
[Chapter 10 cont, Sadiku][Chapter 10 cont, Sadiku]
Dr. Sandra Cruz-PolElectrical and Computer Engineering
Dept. UPR-Mayagüez
Ex. Light traveling in Ex. Light traveling in airair encounters the encounters the waterwater; another medium.; another medium.
Wave incidenceWave incidence
• For many applications, [such as For many applications, [such as fiber fiber optics, line power transmission]optics, line power transmission], it, it’’s s necessary to know what happens to a necessary to know what happens to a wave when it meets a different wave when it meets a different medium.medium.
– How much is How much is transmittedtransmitted? ?
– How much is How much is reflectedreflected back?back?
We will look at…We will look at…
I.I. NormalNormal incidence incidenceWave arrives at 0o from normal
– ““Standing waves”Standing waves”
II.II. ObliqueOblique incidence incidenceWave arrives at another angle
– SnellSnell’’s Law and Critical s Law and Critical angleangle
– ParallelParallel or or PerpendicularPerpendicular– BrewsterBrewster angle angle
Reflection at Normal IncidenceReflection at Normal Incidencex
z
Ei
Hiak
Incident wave
Et
Htak
Transmitted wave
Er
Hr
Reflected wave
akr
yz=0
Medium 1 Medium 2
Now in terms of equations …
Incident wave
xeEzE ziois ˆ)( 1
yeE
yeHzH zioziois ˆˆ)( 11
1
Ei
Hiak
Incident wave
Reflected wave
It’s traveling along –z axis
xeEzE zrors ˆ)( 1
yeE
yeHzH zrozrors ˆ)ˆ()( 11
1
Er
Hr
Reflected wave
akr
Transmitted wave
xeEzE ztots ˆ)( 2
yeE
yeHzH ztoztots ˆˆ)( 22
2
Et
Htak
Transmitted wave
The total fieldsThe total fields
• At medium 1 and medium 2At medium 1 and medium 2
• Tangential components must be Tangential components must be continuouscontinuous at the interfaceat the interface
)0()0()0(
)0()0()0(
tri
tri
HHH
EEE
tri
tri
HHHHH
EEEEE
21
21
Define Define
• Reflection coefficient, Reflection coefficient,
• Transmission coefficient, Transmission coefficient,
12
12
io
ro
E
E
12
22
io
to
E
E
Note:
•1+ •Both are dimensionless and may be complex
• 0≤||≤1
PE 10.8PE 10.8A 5GHz uniform plane wave Eis =10e-jz ax in free space is
incident normally on a large plane, lossless dielectric slab (z>0) having = 4o and =o.
Find: the reflected wave Ers and
the transmitted wave Ets.
Answer:
-3.33 ej1z x V/m,
6.67 e-j2z x V/m
where 2 = 21 = 200 /3
SEE http://www.acs.psu.edu/drussell/Demos/reflect/reflect.html
Case 1:Case 1:
• Medium 1: perfect dielectric, Medium 1: perfect dielectric, 11=0=0
• Medium 2: perfect conductor, Medium 2: perfect conductor, 22=∞=∞
Halla impedancias intrínseca.Halla impedancias intrínseca.
Reflección,Reflección,
TransmisiónTransmisión
y camposy campos
http://www.phy.ntnu.edu.tw/java/waveSuperposition/waveSuperposition.html
The EM field forms aThe EM field forms aStanding Wave on medium 1Standing Wave on medium 1
xtzEE io ˆsinsin2 11
1,3,5 4
2
5,
2
3,
2- @ Maxima
2,,0- @ Minima
1
1max
1
1
nnn
z
z
z
Conducting material
z
2Eio
|E1|
2
01
Standing Wave Applets Standing Wave Applets
• http://www.phy.ntnu.edu.tw/java/waveSuperposition/waveSuperposition.html
• http://www.ngsir.netfirms.com/englishhtm/StatWave.htm
• http://www.physics.smu.edu/~olness/www/03fall1320/applet/pipe-waves.html
• http://www.walter-fendt.de/ph14e/stwaverefl.htm
Case 2:Case 2:Medium 1: perfect dielectric 1=0
Medium 2: perfect dielectric 2=0
0,1,2,3 2
1
1max n
nnz
real. are and
,0
, If 12
0,1,2,3 4
)12(
2
)12( 1
1min
n
nnz
)1(
)(11
11
2
1
zjzjoi
zjzjoi
rsiss
eeE
eeE
EEE
,...3,2,,0or
...6,4,2,02
max1
max1
z
z
Standing waves due to reflectionStanding waves due to reflection)1()( 1111 2
1zjzj
oizjzj
oiri eeEeeEEEE
0,1,2,3 2
1
1max n
nnz
12
z
|E1|
Lossless Medium 2
***At every half-wavelength, everything repeats! ***
Lossless Medium 1
Eio (1+||)01
02 0
Case 3:Case 3:
Medium 1 = perfect dielectric 1=0
Medium 2 = perfect dielectric 2=0
0,1,2,3 2
1
1min n
nnz
real. are and ,0
, If 12
1,2,3 4
)12(
2
)12( 1
1max
n
nnz
Standing waves due to reflectionStanding waves due to reflection)1()( 1111 2
1zjzj
oizjzj
oiri eeEeeEEEE
0,1,2,3 4
)12( 1max
n
nz
12
z
|E1|
Lossless Medium 2
At every half-wavelength, all em properties repeat
Lossless Medium 1
Eio (1-||)
Eio (1+||)
0
Standing Wave Ratio, Standing Wave Ratio, ss
Measures the amount of reflections, the more reflections, the larger the standing wave that is formed.
The ratio of |E1|max to |E1|min
or
1
1
min1
max1
min1
max1
H
H
E
Es
1
1
s
s
Ideally
s=1 (0 dB)No
reflections
PE 10.9PE 10.9• The plane wave The plane wave EE=50 sin=50 sin ( (tt – 5 – 5xx) ) aayy V/m in a lossless medium (V/m in a lossless medium (=4=4oo, , ==oo) ) encounters a lossy medium (encounters a lossy medium (==oo, , =4=4oo, , =0.1 mhos/m) normal to the =0.1 mhos/m) normal to the x-x-axis at axis at xx=0. Find=0. Find
• • • ss• EErr
• EEtt
• http://www.walter-fendt.de/ph14e/stwaverefl.htmhttp://www.walter-fendt.de/ph14e/stwaverefl.htm
←Answers:
=0.8186 exp(j171o)
=0.23 exp(j33.56o)
=10.03
= 40.93 sin (t +5x + 171o) y
= 11.5 e -6.02x sin (t - 7.83x + 33.6o) y V/m
Ex. Antenna Ex. Antenna RadomeRadome
A 10GHz aircraft radar uses a A 10GHz aircraft radar uses a narrow-beam scanning antenna narrow-beam scanning antenna mounted on a gimbal behind a mounted on a gimbal behind a dielectric radome. dielectric radome.
• Even though the radome shape is Even though the radome shape is far from planar, it is far from planar, it is approximately planar over the approximately planar over the narrow extent of the radar beam. narrow extent of the radar beam.
• If the radome material is a If the radome material is a lossless dielectric with lossless dielectric with rr=1 and =1 and rr=9, choose its thickness =9, choose its thickness dd such such that the radome appears that the radome appears transparent to the radar beam. transparent to the radar beam.
• Mechanical integrity requires Mechanical integrity requires dd to to be greater that 2.3 cm.be greater that 2.3 cm.
Antenna with no radome
Antenna with radome
Answer:/2=.5cm, d=2.5cm
Power Flow in Medium 1Power Flow in Medium 1• The net The net average power densityaverage power density flowing in medium 1flowing in medium 1
r
avei
ave
io
zjzjiozjzjio
ave
PP
Ez
eeE
yeeEx
HEzP
2
1
2
*
1
*
*111
12
ˆ
ˆˆRe2
1
]Re[2
1)(
1111
Ei
Hiak
Incident wave
Et
Htak
Transmitted wave
E
r
H
rReflected wave
ak
r
yz=0
Medium 1 Medium 2
Power Flow in Transmitted wavePower Flow in Transmitted wave
• The net average power density The net average power density flowing in medium 2flowing in medium 2
zawhere
Ez
eE
yeEx
HEzP
k
io
zjiozjio
ave
ˆˆ
2ˆ
ˆˆRe2
1
]Re[2
1)(
2
2
2
2
**
*222
22
Ei
Hiak
Incident wave
Et
Htak
Transmitted wave
E
r
H
rReflected wave
ak
r
yz=0
Medium 1 Medium 2
Power in Lossy MediaPower in Lossy Media
*2
2
2
2
2
1Re
2ˆ)( 2
c
z
io
ave eE
zzP
wherewhere
zz
c
io
ave eeE
zzP 11 222
1
2
1 2ˆ)(
2
222
21
21 and
jc
cc
cc
We will look at…We will look at…
I.I. Normal incidenceNormal incidenceWave arrives at 90o from the
surface– Standing wavesStanding waves
II.II. Oblique incidence Oblique incidence ((losslesslossless) ) Wave arrives at an angle
– SnellSnell’’s Law and Critical angles Law and Critical angle– ParallelParallel or or PerpendicularPerpendicular– BrewsterBrewster angle angle