April 23, 2013

1- Wave Reflection and Transmission III

- General Relation for E and H

2- Review

Transmission Line

Examples: Optical Fiber Metal Waveguide

Page 379

Examples: Coaxial Line Two-Wire Line Parallel-Plate Line

Transverse Electromagnetic (TEM)

0&0 zz HE

Non Transverse Electromagnetic Transverse Electric (TE) Transverse Magnetic (TM)

Ez = 0 H z ¹ 0

H z = 0 Ez ¹ 0

- Below 30 GHz coaxial cables are mostly used

Disadvantages of coaxial cable for high frequency power transmission:

Dielectric Losses ↑

PowerDf cableTEM-mode constraint for fabrication

Time varying electric field lines extending between the inner conductor of a coaxial cable and inside surface of the guide can excite an EM wave in the waveguide

Page 379

For transmission of high power high frequency (5-100 GHz) metal waveguides are used.

t

DJH

t

BE

B

D

0

EjJH

H

HjE

E v

~~~0

~

~~/~~

linear, isotropic, and homogeneous medium:

Time variation is a sinusoidal function:

tjezyxEetzyxE ),,(~

);,,(

HB

ED

Assumptions:

0~ v

Charge free

medium

EJ~~

jcom p lex

EjH

H

HjE

E

complex

~~0

~

~~0

~

EjJH

H

HjE

E

~~~0

~

~~0

~

complex 22

Propagation Constant

0~~

0~~

22

22

HH

EE

lossless medium

0

jcom p lex

Wavenumber

222 k0

~~0

~~

22

22

HkH

EkE

Plane Wave

Cartesian Coordinate

Propagation in z-direction 0~

0~

~

0~

~0

~

0~

~

0~

~

2

2

2

2

2

2

2

2

2

2

2

2

z

y

y

xx

z

y

y

xx

H

Hkdz

Hd

Hkdz

Hd

E

Ekdz

Ed

Ekdz

Ed

Solution:

)cos(ˆ)cos(ˆ)(~

),( yyxx

tj kztaykztaxezEetzE

)cos(ˆ)cos(ˆ)(~

),( yyxx

tj kztbykztbxezHetzH

EzH~

ˆ1~

fuk

p

1

Intrinsic Impedance

Objective: Drive expressions for E and H for the TE and TM modes

Maxwell Equations

Solution Procedure

)~

,~

(~

),~

,~

(~

),~

,~

(~

),~

,~

(~

zzyzzxzzyzzx HEHHEHHEEHEE

Wave Equations

Use of boundary condition to find a general solution

EjH

HjE~~

~~

Maxwell Equations

y

H

x

E

k

jH

x

H

y

E

k

jH

x

H

y

E

k

jE

y

H

x

E

k

jE

zz

c

y

zz

c

x

zz

c

y

zz

c

x

~~~

~~~

~~~

~~~

2

2

2

2

zyx

zyx

HzHyHxH

EzEyExE

~ˆ

~ˆ

~ˆ

~

~ˆ

~ˆ

~ˆ

~

222 kkc

Cutoff Wavenumber

...

~

,~

~

...~

,),(~),,(~

z

EEj

z

E

EeyxezyxE

k

y

xx

y

zj

xx

x

E

k

jH

y

E

k

jH

y

E

k

jE

x

E

k

jE

z

c

y

z

c

x

z

c

y

z

c

x

~~

~~

~~

~~

2

2

2

2

y

H

k

jH

x

H

k

jH

x

H

k

jE

y

H

k

jE

z

c

y

z

c

x

z

c

y

z

c

x

~~

~~

~~

~~

2

2

2

2

TE mode Ez=0 TM mode Hz=0

zjzj

zz eb

yn

a

xmEeeE

sinsin~~

00

~zH

TM mode

zj

c

y

zj

c

x

zj

c

y

zj

c

x

eb

yn

a

xmE

a

m

k

jH

eb

yn

a

xmE

b

n

k

jH

eb

yn

a

xmE

b

n

k

jE

eb

yn

a

xmE

a

m

k

jE

sincos~

cossin~

cossin~

sincos~

02

02

02

02

Each combination of the integers m and n represents a variable solution, or a mode, denoted TMnm..

Using the Boundary Conditions:

x

E

k

jH

y

E

k

jH

y

E

k

jE

x

E

k

jE

z

c

y

z

c

x

z

c

y

z

c

x

~~

~~

~~

~~

2

2

2

2

zjzj

zz eb

yn

a

xmHehH

sinsin

~~0

0~zE

TE mode

Page 385

zj

c

y

zj

c

x

zj

c

y

zj

c

x

eb

yn

a

xmH

b

n

k

jH

eb

yn

a

xmH

a

m

k

jH

eb

yn

a

xmH

a

m

k

jE

eb

yn

a

xmH

b

n

k

jE

sincos~

cossin~

cossin~

sincos~

02

02

02

02

y

H

k

jH

x

H

k

jH

x

H

k

jE

y

H

k

jE

z

c

y

z

c

x

z

c

y

z

c

x

~~

~~

~~

~~

2

2

2

2

222

2

0

d

p

b

n

a

muf

p

mnp

,...3,2,1/ pdp

TE mode: 10& pnm

TM mode: 01& pnm

Page 392

Quantization of β:

Design a cavity with f101=12.6 GHz

Assume: a=b=d, m=1, n=0, p=1, and up0=c

Hza

f2

1023 8

101

a=1.68 cm

EM Modes Applet

d

a

b

1- Wave Reflection and Transmission III

- General Relation for E and H

2- Review

Chapter 6: Example 6.2, 6.3, 6.5, 6.7 and 6.8 Chapter 7: Example 7.1 Exercise 7.3, 7.5, 7.6 and 7.9 Problem 7.26 Chapte 8:

Example 8.5 Exercise 8.7 and 8.8 Problem 8.3

Differential Form Integral Form

Gauss’s Law

Faraday’s Law

Gauss’s Law for

Magnetism

Ampere’s Law

vD

t

BE

S

QSdD

Sdt

BldE

SC

0 B

0S SdB

t

DJH

Sdt

DJldH

SC

Page 256

t

BE

Sdt

BldE

SC

tSdB

tldE

SC

Magnetic Flux

tSdB

tldEV

SCemf

Electromotive

Force

tVtVs cos)( 0

t

DJH

Sdt

DJldH

SC

PED

0

t

P

t

E

t

DJD

0Displacement

Current

Displacement Field

zjkii

i

zjkii

eE

yzE

zzH

eExzE

1

1

1

0

1

0

ˆ)(

~

ˆ)(~

ˆ)(~

Incident wave

zjktt

t

zjktt

eE

yzE

zzH

eExzE

2

2

2

0

2

0

ˆ)(

~

ˆ)(~

ˆ)(~

Transmitted wave

zjkrr

r

zjkrr

eE

yzE

zzH

eExzE

1

1

1

0

1

0

ˆ)(

~

)ˆ()(~

ˆ)(~

Reflected wave

Wave Reflection and Transmission at Normal Incident

iit

iir

EEE

EEE

00

12

20

00

12

120

2

iE0is a known quantity

Locus

)cos(ˆ)cos(ˆ)(~

),( yyxx

tj kztaykztaxezEetzE

20 0

sin2sin)2tan( 0

x

y

a

a0tan0

4/4/

22

0

0

0

Left polarization

Linear polarization

Right polarization

Find:

Ψ0 Auxiliary angle

Ellipticity angle

circularor oo 4545

0

0

0

Left circular polarization

Linear polarization

Right circular polarization

Page 275

*~~

2

1HEeSav

)/(2

~

ˆ 2

2

mWE

zSav

)/(cos2

ˆ)( 22

2

0mWe

EzzS z

c

av

90o

Page 337

view

x

z y

It is more convenient to first decompose the incident wave:

),(),(),( ||||

iiiiii HEHEHE ),(&),( ||||

rrrr HEHEdetermine Add the components

t

r

i r

.

ik

iH||

iE||

t

tk tH||

tE||

rkrH||

rE||

z

x

iE

iE||

E

E

E n

i

i r

x

k

iH

iE

t

tk

tH

tE

rk

rH

rE

z

ik

iH

iE

kHE ˆˆˆ

For normal incidence, Γ and are independent of polarization. This is not the case when .0i

t

i

it

i

i

t

it

it

i

r

E

E

E

E

cos

cos1

coscos

cos2

coscos

coscos

||

12

2

0||

0||

||

12

12

0||

0||

||

1coscos

cos2

coscos

coscos

12

2

0

0

12

12

0

0

ti

i

i

t

ti

ti

i

r

E

E

E

E

Parallel Polarization Perpendicular Polarization

The Brewster angle ΘB is defined as the incidence angle Θi at which the reflection coefficient Γ=0.

Perpendicular component Parallel component

22/1

12211

1

)/(1sin

B

1

21

|| tan

B

does not exist for nonmagnetic materials

For nonmagnetic materials, the Brewster angle exists

only for parallel polarization and depend on ε2/ε1

0.5-MHz antenna

Normally

incident plane

mVE /3000

mS

r

r

/4

1

72

d=?

Minimum Signal

Amplitude required: )/(01.0 mV

0.5-MHz antenna

Normally

incident plane

mVE /3000

mS

r

r

/4

1

72

d=?

Minimum Signal

Amplitude required: )/(01.0 mV

dit eEE 2

0

2000

Sea water is a good conductor

Use the equations for a good

conductor

dit eEE 2

0

d=7.54 m

Brewster 1

?2 Parallel polarized

plane wave

Polyester

6.21 rr

Air

)/(10 mVE

=50o

?

Parallel polarized beam

?iP

Compare the polarization states of each of the following pairs of plane waves:

Compare the polarization states of each of the following pairs of plane waves:

Find:

Ψ0 Auxiliary angle

γ Rotation angle

Χ Ellipticity angle

sin2sin)2tan( 0

x

y

a

a0tan

0

Example 7.3: Determine the polarization state of a plane wave with electric field )45sin(4ˆ)30cos(3ˆ),( oo kztykztxtzE

20 0

sin2sin)2tan( 0

x

y

a

a0tan0

4/4/

22

0

0

0

Left circular polarization

Linear polarization

Right circular polarization

Find:

Ψ0 Auxiliary angle

Ellipticity angle

Example 7.3: Determine the polarization state of a plane wave with electric field )45sin(4ˆ)30cos(3ˆ),( oo kztykztxtzE

Example 7.3: Determine the polarization state of a plane wave with electric field

)45sin(4ˆ)30cos(3ˆ),( oo kztykztxtzE

22

4/4/

auxiliary angle

)45cos(4ˆ)30cos(3ˆ),( oo kztykztxtzE

E(z, t) = x3cos(wt -kz+30o)+ y4cos(wt -kz+135o)

x

y

a

a0tan

x

y

a

a1

0 tan

20 0

o1.533

4tan 1

0

89.0105cos2.106tancos2tan)2tan( 0 oo

oo 105sin2.106sinsin2sin)2tan( 0

oo or 2.698.20 Since cosδ<0 Sign of γ the same as cosδ; see the rule on page 328

o0.34;0cos0

;0cos0

if

if

0

0

0

Left circular polarization

Linear polarization

Right circular polarization

Page 275

o0.34

Wave traveling out of the page

1r 4r

mVeyE zxji /20ˆ~ 43

90o

view

x

z y

t

rn

iiE

1r

0

4

zat

r

mVeyE zxji /20ˆ~ 43

a) The polarization of the incident wave

b) The angle of incident

)cossin(

01ˆ

~ii zxjkii eEyE

Equation (8.48a)

90o

view x

z y

t

r

E

E

E n

i

mVeyE zxji /20ˆ~ 43

c) The time-domain expressions for the reflected electric and magnetic fields.

1r 4r

1r 4r

mVeyE zxji /20ˆ~ 43

d) The time-domain expressions for the transmitted electric and magnetic fields.

mVeyE zxji /20ˆ~ 43

e) The average power density carried by the wave in the dielectric medium.

1r 4r