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ENE 428Microwave

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 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,,,,,,,,,,,,,,

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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,,,,,,,,,,,,,,

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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

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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

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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

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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

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

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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 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

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Telegraphist’s equations

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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

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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

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Low loss transmission line (2)

Therefore, we get

1 ' '( ' ' )2 ' '

C LR G

L C

1 ' '1 ( )8 ' 'G R

LCC L

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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

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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(

)(

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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

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Representation of absolute power levels is the dBm scale

Power ratios on the decibel scale (2)

( ) 10log( )1m

PG dB

mW 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?