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