Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 4 ECE 6340 Intermediate EM Waves 1.

1

Prof. David R. JacksonDept. of ECE

Fall 2013

Notes 4

ECE 6340Intermediate EM Waves

2

Debye Model

Molecule:

Explains molecular effects

molecule at rest

molecule withapplied field xE

0x p

0xp

We consider an electric field applied in the x direction.

The dipole moment px of this

single molecule represents the average dipole moment in the x direction for all of the dipoles in a little volume.

q

-qx

y

xE

3

Debye Model (cont.)

2

2

E S F

dT I

dtT T T T

cosE x

S

F

T qd

T s

dT c

dt

E

s = spring constant

c = friction constant

Note: T = -Tz

xE

ˆET qr qr q p d E E p E

ˆ sin2E xT z

pE

4

Hence

Assume

2

2cosx

d dq d s c I

dt dt

E

1, cos 1

2

2x

d dq d s c I

dt dt

E

Debye Model (cont.)

(small fields)

5

sinx q d

q d

pNote that

2

2x

d dq d s c I

dt dt

E

Hence: x

q d

pInsert this into the top equation.

Debye Model (cont.)

xE

6

Then we have:

Assume sinusoidal steady state:

or

2

2

1 1x x xx

d dq d s c I

q d q d dt q d dt

p p pE

2

2

2

x xx x

d ds c I q d

dt dt

p pp E

22x x x xsp j cp I p q d E

Debye Model (cont.)

7

Hence

Then we have

2

2x x

q dp E

s I j c

3

#moleculesmN

m

Mx m xP N p

The term P denotes the total

dipole moment per unit volume.

The M superscript reminds us that we are talking

about molecules

Debye Model (cont.)

Denote

8

Also, for a linear material,

Hence

0M M

x e xP E

0 0

MM x m xe

x x

P N p

E E

Therefore 2

20

1M me

Nq d

s I j c

Assume2I s (The frequency is fairly low relative to

molecular resonances. The frequency is at millimeter wave frequencies and below.)

Debye Model (cont.)

9

Denote the time constant as:

2

0

1 1

1

M me

Nq d

cs js

c

s

2

0

10M m

e

Nq d

s

0

1

MeM

e j

Denote the zero-frequency value as:

Debye Model (cont.)

(real constant)

Then we have

10

0

1

MeM

e j

Debye Model (cont.)

This would imply that

01

1

Me

r j

At high frequency the molecules cannot respond to the field, so the relative permittivity due to the molecules tends to unity.

This equation gives the wrong result at high frequency, where atomic effects become important.

11

0

1

MeM

e j

Include BOTH molecule and atomic effects:

Debye Model (cont.)

Molecule effects:

Atomic effects:

0 0

0

M Ax x x

M Ae x e x

e x

P P P

E E

E

Ae constant (real)

Atoms can respond much faster than molecules, so the atomic susceptibility

is almost constant (unless the frequency is very high, e.g., at THz

frequencies).

12

We then have thatM A

e e e

0

1

Me A

e ej

Debye Model (cont.)

Hence:

13

Permittivity formula: 1r e

21

01

1

1

MeA

r e j

aa

j

1

2

1

0

Ae

Me

a

a

where

Debye Model (cont.)

(a1 and a2 are real constants)

14

1 2

1

0r

r

a a

a

so

1

2 0

r

r r

a

a

0

1r r

r r j

Note that:

Hence:

Debye Model (cont.)

21 1r

aa

j

15

2

2

0

1

0

1

r rr r

r rr

r

0r

r

1/

r

r

Debye Model (cont.)

16

Frequency for maximum loss:

Let x

20

1r r r

x

x

A maximum occurs at 1x

1

or

Debye Model (cont.)

17

Water obeys the Debye model quite well.

Water obeys the Debye model quite well.

Debye Model (cont.)Complex relative permittivity for pure (distilled) water

18

Water obeys the Debye model quite well.

Ocean water: = 4 [S/m]

Example

Calculate the complex relative permittivity rc for ocean water at 10.0 GHz.

0 0 0

1crc rj j

9 12

460 35

2 10.0 10 8.854 10rc j j

from previous plot for distilled water

60 42.19rc j 60 35 7.19rc j j Hence

or

19

Cole-Cole Model

This is a modification of the Debye model.

10

1r r

r rj

When = 0, the model reduces to the Debye model.

This model has often been used to describe the permittivity of some polymers, as well as biological tissues.

20

Cole-Cole Model (Cont.)Parameters for Some Biological Tissues

21

Havriliak–Negami Model

This is another modification of the Debye model.

0

1

r rr r

j

When = 1 and = 1, the model reduces to the Debye model.

This has been used to describe the permittivity of some polymers.

22

Lorentz ModelExplains atom and electron resonance effects (usually observed

at high frequencies, such as THz frequencies and optical frequencies, respectively).

Dipole effect:

Atom:

xE

qn= - qe

qnqe

0xE

electrons

23

Lorentz Model (cont.)

Model:

Electron equation of motion for electrons in atom:

xqn

qe

m

xE

2

2x

E S Fx x x x

e x

d xm

dt

dxq sx c

dt

E

F

F F F F

The heavy positive nucleus is fixed.

24

so

Hence

2

2e x

dx d xq sx c m

dt dt E

x n eq x q x p

22

2 x x

e x x

d dq s c m

dt dt

p pE p

Sinusoidal Steady State:

2 2e x x x xq E s p j c p m p

For a single atom, = /x ex qpor

Lorentz Model (cont.)

25

Therefore,

2

2e

x x

qp E

s m j c

0

Ax a x

Ae x

P N p

E

0

A a xe

x

N p

E

The term P denotes the total

dipole moment per unit volume.

The A superscript reminds us that we are talking about

atoms.

Lorentz Model (cont.)

26

Denote:

2

20

2

20

1

A a ee

a e

N q

s m j c

N qs cm jm m

220

0

a ef

N q s cA c

m m m

Hence

(real constants)

Lorentz Model (cont.)

27

We then have

2 20

Ae

f

A

j c

2 20

2 22 22 2 2 20 0

1 fA Ar r

f f

A c A

c c

2 20

1Ar

f

A

j c

Permittivity formula: 1A Ar e

Lorentz Model (cont.)

Hence

Real and imaginary parts:

28

A sharp resonance occurs at 0

Lorentz Model (cont.)

Low frequency value for Ae

Ar

0

1

Ar

Ar

201 /A

(This is still "high" frequency in the Debye model.)

29

Total Response

From the dielectric spectroscopy page of the research group of Dr. Kenneth A. Mauritz.

frequency (Hz)

Response of a hypothetical material

103 106 109 1012 1015

MW THz UVVis

0c j

Low frequency:

30

Atmospheric Attenuation

60 GHz

90 GHz

31

Atmospheric Attenuation (cont.)

Atmosphericabsorption (% power

absorbed) formillimeter-wave

frequenciesover a 1-km path

2 (1000)100 1 e

% power absorbed

10

2 (1000)10

dB / km 10 log

10 log

out

in

P

P

e

Terabeam Document Number: 045-1038-0000Revision: A / Release Date: 09-03-2002Copyright © 2002 Terabeam Corporation. All Rights Reserved.

Attenuation in dB/km:

32

Plasma

Electrically neutral plasma medium (positive ions and electrons):

ion (+)

electron (-)

We assume that only the electrons are free to move when an electric field is applied. This causes a current to flow.

33

Plasma (cont.)

Equation of motion for average electron:

dvm e mv

dt F E

= collision frequency (rate of collisions per second of average electron)

Notes: (1) The last term assumes perfect inelastic collisions (loss mechanism). (2) We neglect the force due to the magnetic field.

Force due to electric field Force due to collisions with ions (loss of momentum)

There is no “spring” force now.

34

Plasma (cont.)

Sinusoidal steady state:

m j v e E mv

ev E

m j

Current:

ve

ve

eJ v E

m j

35

Plasma (cont.)

Amperes’ law:

0

0

0

ve

ve

c

H J j E

eE j E

m j

ej E

m j

j E

veve

eJ v E

m j

We assume that there is no polarization current – only conduction

current. Hence we use 0.

36

Plasma (cont.)

Hence:

0ve

c

ej j

m j

so 0

1 vec

e

j m j

or 0

1vec

e

m j

37

Plasma (cont.)

Define:

20

vep

e

m

2

0 1 pc j

(p plasma frequency)

We then have

0

1vec

e

m j

38

Plasma (cont.)

Lossless plasma:

2

0 1 pc

0

(Drude equation)

0

0

: 0, ( )

: 0, ( )

p c c c

p c c c

k

k j

propagation

attenuation

Plane wave in lossless plasma:

39

Plasma (cont.)

Complex relative permittivity of silver at optical frequencies

Rel

ativ

e pe

rmitt

ivity

The Drude model is an approximate model for how metals behave at optical frequencies.

“plasmonic behavior”

40

Plasma (cont.)

At microwave frequencies, a plasma-like medium can be simulated by using a wire medium.

d

d

2a

y

x

1

2 ln 0.52752

p da

c

d

0 0

1c

41

Plasma (cont.)

An example of a directive antenna using a wire medium:

Geometrical optics (GO):Refraction towards broadside

y1r hsh

f

x

yzx

, 0 1p p r