RS ENE 428 Microwave Engineering Lecture 3 Polarization, Reflection and Transmission at normal...

RSRS

ENE 428Microwave

Engineering

Lecture 3 Polarization, Reflection and Transmission at normal incidence

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Uniform plane wave (UPW) power transmission

2

201Re

2jzx

zEe e a

from1

Re( )2

avgP E H

��

2

201cos

2zx

zEe a

W/m2

2

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Example 6.7: Consider an electric field incident on a copper slab such that the field in the slab is given by

xz aztetzE

)102cos(0.1),( 7

V/m

We want to find the average power density.Since copper is a good conductor, we can use

)S/m108.5)(H/m104)(Hz10( 777 f

1/m 108.47 3The intrinsic impedance is

meeoo jj 4545 17.1

Therefore, 21096-)108.47(2

2

W/m300e45cos)17.1(

)V/m1(

2

1 33

zz

zoz

ave aaem

P

At the surface (z = 0) the power density is 300 W/m2. But after only 1 skin

depth, in this case 21 m, the wave’s power density drops to e-2 (13.5%) of its surface value, or 41 W/m2 in this case.

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Polarization• UPW is characterized by its propagation direction

and frequency.

• Its attenuation and phase are determined by medium’s parameters.

• Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.

• Specifying only the electric field direction is sufficient since magnetic field is readily found from using Maxwell’s equation

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E

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Linear polarization

• Consider in free space,

0( , ) cos( ) xE z t E t z a

��E��

• At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave”

E��

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• A pair of linearly polarized wave also produces linear polarization

Linear polarization

0 0( , ) cos( ) cos( )x yx yE z t E t z a E t z a

��

At z = 0 plane

At t = 0, both linearly polarized waveshave their maximum values.

0 0(0,0) x yx xE E a E a

��

(0, ) 04

�� TE

0 0(0, ) cos( ) cos( )x yx yE t E t a E t a

��

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Linear polarization• The tilt angle (tau) is the angle the line makes

with the x-axis

• The axial ratio is the ratio of the long axis of an ellipse to the short axis

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• More generalized of two linearly polarized waves,

• Linear polarization occurs when two linearly polarized waves are

• Linear polarization is a special case of elliptical polarization that has an infinite axial ratio

More generalized linear polarization

0 0( , ) cos( ) cos( )x yx x y yE z t E t z a E t z a

��

in phase 0y x

out of phase 180 . y x

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• Superposition of two linearly polarized waves that

• If x = 0 and y = 45, we have

Elliptically polarized wave

0 180y x or

0 0(0, ) cos( ) cos( )

4x yx yE t E t a E t a

��

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• occurs when Exo and Eyo are equal and

• Right hand circularly polarized (RHCP) wave

• Left hand circularly polarized (LHCP) wave

• Left and right are referred to as the handedness of wave polarization

Circularly polarized wave

90y x

0 0(0, ) cos( ) cos( )


��

0 0(0, ) cos( ) cos( )


��

90y x

90y x

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• Phasor forms:

for RHCP,

for LHCP,

Circularly polarized wave

0 0( 0) yx

jjx yx yE z E e a E e a

��from

0( 0) ( )x yxE z E a ja

��

0( 0) ( )x yxE z E a ja

��

Note: There are also RHEP and LHEP 11

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Ex1 Given

,determine the polarization of this wave

( , ) 8cos( 30 ) 8cos( 90 ) ��

x yE z t t z a t z a

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Ex2 The electric field of a uniform plane wave in free space is given by , determine

50100( ) j ys z xE a ja e

��

a) f

b) The magnetic field intensity sH��

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c)

d) Describe the polarization of the wave

S��

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Reflection and transmission of UPW at normal incidence

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Assume the medium is lossless, let the incident electric field to be

or in a phasor form

since

then we can show that

1 10( , ) cos( ) xxE z t E t z a

��

• Normal incidence – the propagation direction is normal to the boundary

Incident wave

11

1

EH a

��

11 10( ) j z

xxE z E e a ��

1101

1

( ) j zxy

EH z e a

��

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• Transmitted wave

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Assume the medium is lossless, let the transmitted electric field to be


22 20( ) j z

xxE z E e a ��

Transmitted wave

2202

2

( )

��j zx

yE

H z e a

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At z = 0, we have

and

1 = 2 are media the same?

tan1 tan 2

tan1 tan 2

E E

H H

• From boundary conditions,

Reflected wave (1)

10 20x xE E

10 20

1 2

x xE E

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• There must be a reflected wave

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and

This wave travels in –z direction.

Reflected wave (2)

11 10( ) j z

xxE z E e a ��

1101

1

( ) j zxy

EH z e a

��

19

• Boundary conditions (reflected wave is included)

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from

therefore at z = 0

(1)

Reflection and transmission coefficients (1)

1 2x xE E

1 1 2x x xE E E

10 10 20x x xE E E

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from

therefore at z = 0

(2)

• Boundary conditions (reflected wave is included)


1 2y yH H

1 1 2y y yH H H

10 10 20

1 1 2

x x xE E E

21

• Use Eqns. (1) and (2) to eliminate , we’ll get

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Reflection coefficient

Transmission coefficient


10 2 1

2 110

jx

x

Ee

E

120 2

2 110

21 jx

x

Ee

E

22

20xE

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Types of boundaries: perfect dielectric and perfect conductor (1)

From

.

Since 2 = 0 then = -1 and Ex10+= -Ex10

-

1 1 1x x xE E E

1 11 10 10

j z j zx x xE E e E e

22

2 2

0jj

20 0xE

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Types of boundaries: perfect dielectric and perfect conductor (2)

This can be shown in an instantaneous form as

10 1( , ) 2 sin( )sinx xE z t E z t

10 12 sin( )xj E z

Standing wave

24

101 )( 11x

zjzjx EeeE

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Standing waves (1)

When t = m, Ex1 is 0 at all positions.and when z = m, Ex1 is 0 at all time.

Null positions occur at

1

2z m

1

2m

z

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Standing waves (2)

Since

and ,

the magnetic field is

or .

Hy1 is maximum when Ex1 = 0

So, E and H are said to be 90o out of phase. There will be no power transmission on either side of the media

Poynting vector

1 1x yE H

1 1x yE H

1 1101

1

( )j z j zxy

EH e e

101 1

1

2( , ) cos cosx

y

EH z t z t

S E H �� 26

2201

cos2

zxz

Ee a

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Power transmission for 2 perfect dielectrics (1)Then 1 and 2 are both real positive quantities and 1 = 2 = 0

Average incident power densities

11 1 1 1 *

1

1 1Re Re

2 2

xi x y x

EP E H E

2 1

2 1

real

2

10*1

1 1Re

2 xE

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Ex3 Let medium 1 have 1 = 100 and medium 2 have 2 = 300 , given Ex10

+ = 100 V/m. Calculate average incident, reflected, and transmitted power densities

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Wave reflection from multiple interfaces (1)• Wave reflection from materials that are finite in

extent such as interfaces between air, glass, and coating

• At steady state, there will be 5 total waves

Incident energy

in

1 2 3

-l 0 z

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Wave reflection from multiple interfaces (2)Assume lossless media, we have


3 223

3 2

,

20 23 20

20 202

20 20 23 202 2

1

1 1

x x

y x

y x x

E E

H E

H E E

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Wave impedance w (1)

Use Euler’s identity, we can show that

3 2 2 22

2 2 3 2

cos sin( )

cos sinw

z j zz

z j z

31

zjzj

zjzj

w

zjy

zjy

zjx

zjx

y

xw

ee

eez

eHeH

eEeE

H

Ez

22

22

22

22

23

232

2020

2020

2

2

)(

)(

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Wave impedance w (2)Since from B.C.

at z = -l

we may write

1 1 2x x xE E E

1 1 2y y yH H H

10 10 2x x xE E E

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)( lz

)( lz )( lz

(1a)

(2a)

Using eqns (1a) and (2a) to eliminate , we’ll get … 2xE

w

xxx EEE

2

1

10

1

10

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Input impedance insolve to get

10 1

110

x in

inx

E

E

3 2 2 22

2 2 3 2

cos sincos sinin

l j ll j l

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)( lzw

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Power Transmission and Reflection

Incident energy

in

1 2 3

-l 0 z

2

1

12

in

in

in

r

P

P

22

1

1211

in

in

in

t

P

P

The power in region 2 stays constant in steady-state; power leaves that region to form the reflected and transmitted waves, but is Immediately replenished by the incident wave (from region 1)

If = 0, then there’ll be total transmission.And = 0 when in = 1 , or the input impedance is matched to that of the incident medium. So how do we achieve this?

Pin

Pr

Pt

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Half-wave matching method

Suppose 13 ml 2

ml 2

2

22ml

and

Therefore, and so

Hence, the 2nd region thickness is the multiple “half-wavelength” as measured in that medium

Using eqn: 3 2 2 22

2 2 3 2

cos sincos sinin

l j ll j l

We’ll get 3 in when 2

2ml

The general effect of a multiple half-wave is to render the 2nd region immaterial to the results on reflection and transmission. Equivalently we

have a single interface problem involving 1 and 3

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Refractive index

rn

Under lossless conditions,

0 0 r

nc

0 0

0

1

rn

p

cv

n

0pv

f n

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Homework

6.32: Given yo

x aztazttzE

)45sin(20)cos(10),( V/m, find the polarization and handedness.

6.38: Suppose medium 1 (z < 0) is air and medium 2 (z > 0) has r = 16. The trans-

mitted magnetic field intensity is known to be Ht = 12cos(t – β2z)ay mA/m.(a) Determine the instantaneous value of the incident electric field. (b) Find the reflected time-averaged power density

6.48: A 100-MHz TE polarized wave with amplitude 1.0 V/m is obliquely incidentfrom air (z < 0) onto a slab of lossless, nonmagnetic material with r = 25 (z > 0).

The angle of incidence is 40o. Calculate (a) the angle of transmission, (b) the reflection and transmission coefficients, and (c) the incident, reflected and transmitted fields.

RS ENE 428 Microwave Engineering Lecture 3 Polarization, Reflection and Transmission at normal...

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