Download Module 5 - University of Illinois Urbana

Fundamentals of Electromagnetics:
A Two-Week, 8-Day, Intensive Course for
Training Faculty in Electrical-, Electronics-,
Communication-, and Computer- Related
Engineering Departments
by
Nannapaneni Narayana Rao
Edward C. Jordan Professor Emeritus
of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, USA
Distinguished Amrita Professor of Engineering
Amrita Vishwa Vidyapeetham, India
Amrita Viswa Vidya Peetham, Coimbatore
August 11, 12, 13, 14, 18, 19, 20, and 21, 2008
5-1
Module 5
Wave Propagation in
Material Media
Conductors and dielectrics
Magnetic materials
Wave equation and solution
Uniform waves in dielectrics and conductors
Boundary conditions
5-2
Instructional Objectives
17. Find the charge densities on the surfaces of infinite plane
conducting slabs (with zero or nonzero net surface charge
densities) placed parallel to infinite plane sheets of
charge
18. Find the displacement flux density, electric field intensity,
and the polarization vector in a dielectric material in the
presence of a specified charge distribution, for simple
cases involving symmetry
19. Find the magnetic field intensity, magnetic flux density,
and the magnetization vector in a magnetic material in
the presence of a specified current distribution, for simple
cases involving symmetry
5-3
Instructional Objectives (Continued)
20. Determine if the polarization of a specified electric/magnetic
field in an anisotropic dielectric/magnetic material of
permittivity/permeability matrix represents a characteristic
polarization corresponding to the material
21. Write expressions for the electric and magnetic fields of a
uniform plane wave propagating away from an infinite plane
sheet of a specified sinusoidal current density, in a material
medium
22. Find the material parameters from the propagation parameters
of a sinusoidal uniform plane wave in a material medium
23. Find the charge and current densities on a perfect conductor
surface by applying the boundary conditions for the electric
and magnetic fields on the surface
24. Find the electric and magnetic fields at points on one side of a
dielectric-dielectric interface, given the electric and magnetic
fields at points on the other side of the interface
5-4
Conductors
and Dielectrics
(FEME, Secs. 5.1; EEE6E, Secs. 4.1, 4.2)
5-5
Materials
Materials contain charged particles that under the
application of external fields respond giving rise
to three basic phenomena known as conduction,
polarization, and magnetization. While these
phenomena occur on the atomic or “microscopic”
scale, it is sufficient for our purpose to
characterize the material based on “macroscopic”
scale observations, that is, observations averaged
over volumes large compared with atomic
dimensions.
8
5-6
Material Media can be classified as
(1) Conductors
and Semiconductors
electric property
(2) Dielectrics
(3) Magnetic materials – magnetic property
Conductors and Semiconductors
Conductors are based upon the property of
conduction, the phenomenon of drift of free
electrons in the material with an average drift
velocity proportional to the applied electric field.
5-7
electron
cloud
free electrons
+ bound
elecrons
nucleus
In semiconductors, conduction occurs not only by
electrons but also by holes – vacancies created by
detachment of electrons due to breaking of
covalent bonds with other atoms.
The conduction current density is given by
J c  E
Ohm’s Law
at a point
5-8
  conductivity (S/m)
e Ne e
 
h N h e   e N e e
conductors
semiconductors
  Mobility
Nh,e = Density of holes ( h) or electrons ( e)
5-9
Ohm’s Law
V  El
Jc = E =
A
V
l
A
I  Jc A 
V
l
l
VI
A
V  IR Ohm’s Law
l
R=
A
l

E, Jc
I
V
5-10
D4.1
I
0.1
Jc   –4  10 3 A m 2
A 10
(a) For cu,
  5.8  10 7 S m
103
E

 17.24 V m
7
 5.8  10
(b)    h  e  Ne e
Jc
 1700  3600 104  2.5  1013  106
1.602 1019
 2.1229 S m
103
E 
 471.1 V m
 2.1227
Jc
5-11
l
(c) From R 
1
A
l
1
10 6



Sm
–6
RA   10

Jc
10 3
E
 6
 3.14 mV m
 10 
5-12
Conductor in a static electric field
E
E
5-13
–
–
–
–
– S = –0E0
E0az
z=d
z=d
+
–
z=0
z=0
+
+
+
+
+ S = 0E0
–
–
–
–
– S = –0E0
z=d
+
+
+
+
+
S = 0E0
z=0
–
–
–
–
–
S = –0E0
+
+
+
+
+ S = 0E0
E =0
S0
+
–
+
–
+
–
+
–
E = – S0 az
0
–S0
S0
–
a z  E0 a z  0
0
S0   0 E0
5-14
P4.3
(a) S0  S1  S2
S1
Ei = 0
z
S2
S1
S2
Ei  –
az 
az  0
2 0
2 0
1
  S1  S2   S0
2
5-15
(b)
S11
E i1 = 0
S12
S21
Ei2 = 0
S22
S11  S12  S1
S21  S22  S2
Write two more equations and solve for the four
unknowns.
5-16
Dielectrics
are based upon the property of polarization, which is the
phenomenon of the creation of electric dipoles within the
material.
Electronic polarization: (bound electrons are displaced to
form a dipole)
Q
E
+

+
d
Q

Dipole moment
p = Qd
5-17
Orientational polarization: (Already existing dipoles are acted
upon by a torque)
QE
Torque  QEd sin q
+
d
q
q

E
Direction into the paper.
 T  Qd × E
 p×E
QE
Ionic polarization: (separation of positive and negative ions
in molecules)
5-18
The Permittivity Concept
Applied
Field, E a
+
Total Field
+
E  Ea  Es
Dielectric
Secondary Field, E s
Polarization
D   0 E  P   0 E   0 e E
D   0 1   e E
  0  r E  E, Displacement Flux Density
11
5-19
The phenomenon of polarization results in a polarization
charge in the material which produces a secondary E.
S0
z=d







Ea
e  e 0
z=0
+
+
+
+
S0
z=d
z=0
+

+

+
+

+

+

+

+

+

+

+

+

+

+

+

+
 pS   pS 0
+
+

+
ES

 pS    pS 0

+


+

+


+
+

+

t

+
+

E+

+
+
Polarization Current
5-20
5-21
To take into account the effect of polarization, we define the
displacement flux density vector, D, as
D  0E  P
=  0 E   0 e E
=  0 1  e  E
= 0 r E
=  E C m2
 = permittivity, F m
 r = relative permittivity
 r and  vary with the material, implicitly taking into account
the effect of polarization.
5-22
As an example, consider
 S 0







z=d

+
+
+
+
S 0
z
+
Then, inside the material,
+
+
z=0
S 0
 S 0
E
az 
 az 
2
2
S 0

az

D   E  S 0 a z
5-23
D4.3
1 C m2







z=d
z
  4 0
+
+
+
+
+
+
1 C m2
For 0 < z < d,
6
2
D


a

10
a
C
m
(a)
S0 z
z
+
z=0
5-24
1
6
(b) E 

10
az 

 4 0
D
36
6


10
az
9
4  10
 9000 az V m
(c) P  D   0 E
= 106 az  0.25  106 az
 0.75  106 az C m2
5-25
Isotropic Dielectrics:
D is parallel to E for all E.
y
Dx   Ex
D
Dy   Ey
Dz   Ez
DE
E
x
Anisotropic Dielectrics:
D is not parallel to E in general. Only for certain directions
(or polarizations) of E is D parallel to E. These are known
as characteristic polarizations.
5-26
Dx   xx Ex   xy Ey   xz Ez
Dy   yx Ex   yy Ey   yz Ez
Dz   zx Ex   zy Ey   zz Ez
y
E
D
x
 Dx   xx
 D   
 y   yx
 Dz   zx

 xy  xz   Ex 
 
 yy  yz   Ey 
 zy  zz   Ez 
5-27
D4.4
8 2 0 
    2 5 0 
 0 0 9 
(a) E  E0 az
 Dx 
8 2 0   0 
 0 
 D     2 5 0  0     0 
0
0
 y

 
 
 Dz 
0 0 9   E0 
9 0 
D  90 E0 az  90 E
 eff  9 0 ,  reff  9
5-28

(b) E  E0 ax  2a y

 Dx 
8 2 0  E0 
 4 E0 
 D     2 5 0  2 E     8E 
0
0
0
0
 y





 Dz 
0 0 9   0 
 0 
D  4 0 E0  ax  2a y   4 0 E
 eff  4 0 ,  reff  4
5-29

(c) E  E0 2ax  a y

 Dx 
8 2 0   2 E0 
18E0 
 D    2 5 0  E     9E 
0
0
 y

 0 
 0
 Dz 
0 0 4  0 
 0 
D  9 0 E0  2ax  a y   9 0 E
 eff  9 0 ,  reff  9
Magnetic Materials
(FEME, Sec. 5.2; EEE6E, Sec. 4.3)
5-31
Magnetic Materials
are based upon the property of magnetization, which is the
phenomenon of creation of magnetic dipoles within the
material.
Diamagnetism:
A net dipole moment is induced by changing the angular
velocities of the electronic orbits.
I
e
+
A
I
Dipole moment
m = IA an
5-32
Paramagnetism
Already existing dipoles are acted upon by a torque.
I dl × B
I
B
I dl × B
I
5-33
The Permeability Concept
Applied
Field, Ba
+
+
Total Field
B  Ba  Bs
Magnetic Material
Secondary Field, Bs
Magnetization
m B
B  0 H   0 M  0 H  0
1  m  0
B
B
B
H

 , Magnetic Field Intensity
0 (1  m ) 0 r 

5-34
The phenomenon of magnetization results in a magnetization
current in the material which produces a secondary B.
 J S 0a y
z=d
Ba
m  m 0
z=0
J S 0a y
J mS   J mS 0a y
Bt
z=d
Bs
z=0
JmS  JmS 0a y
Magnetization Current
5-35
5-36
To take into account the effect of magnetization, we define the
magnetic field intensity vector, H, as
H
B
0
M
m B


 0 1  m  0
B
B

 01  m 


B
 0 r
B

  permeability, H m
r  relative permeability
A m
r and  vary with the material, implicitly taking into account
the effect of magnetization.
As an example, consider
5-37
 J S 0a y
z

y
x
J S 0a y
Then inside the material,
B=

2
J S 0a y × az 
  J S 0 ax
H
B

 J S 0 ax

J

2
a  ×  az 
S0 y
5-38
D4.6
0.1 a y
z=d
z
  100  0
y
z=0
0.1 a y
For 0 < z < d,
(a) H  0.1 a y × az  0.1 ax A m
x
5-39
(b) B =  H = 100  0 0.1 ax 
 10  0ax Wb m
2
 4  106 ax Wb m2
(c) M 
B
0
H
= 10 ax  0.1 ax
 9.9 ax A m
5-40
Materials and Constitutive Relations
Summarizing,
J c   E Conductors
D  E
B
H

Dielectrics
Magnetic materials
E and B are the fundamental field vectors.
D and H are mixed vectors taking into account the
dielectric and magnetic properties of the material
implicity through  and , respectively.

Wave Equation and Solution
(FEME, Sec. 5.3; EEE6E, Sec. 4.4)
5-42
Waves in Material Media
H y  z, t 
Ex  z, t 
 
z
t
H y  z, t 
Ex  z, t 
  Ex  z , t   
z
t
Ex
  j H y
z
H y
  Ex  j Ex     j  Ex
z
5-43
Combining, we get
 Ex
 j   j  Ex
2
z
2
Define
    j   j   j 
Then
2 Ex
2
Wave equation


E
x
2
z
5-44
Solution:
Ex  z   Ae
 z
z
 Be
Ex  z , t   Re  Ex  z  e jt 
 z
z
jt

 Re  Ae  Be  e 
 Re  Ae e

jq   z  j  z
 z
 Ae
e
e
jt
 Be e e j  z e jt 

cos t   z  q
jq   z


 Be z cos t   z  q  
5-45
Ae z cos t   z  q  

attenuation    wave
B e z cos t   z  q  

attenuation    wave
 = attenuation constant, Np/m
 = phase constant, rad/m
 = propagation constant, m1
5-46
f  z, t   e cos t   z 
 z
f
1
0


t 
4
t 0
2

1
t 
2
z
5-47
g  z, t   e z cos t   z 
g
t 
2
t 
4
1
t 0
z
2


0

1
5-48
Ex
  j H y
z
1 Ex
Hy  
j z
1 
 Ae z  Be  z 

j z
1
  Ae z  Be  z 

j
where  
 intrinsic impedance of the medium.
  j
5-49
Summarizing,
    j 
j   j 
j
  e 
  j
j
conversely,
1


j

  Re


  Im


1
5-50
Example:
For dry earth,   105 s/m,   50 , and   0 .
Let us compute  ,  , vp ,  , and  for f  100 kHz.
Solution:
 

j   j 



j j 1 

j



 j  1  j

2 f 
5
2


10

5
 j
1  j 0.36
8
3 10
5-51
 j 0.004683 1.0628  19.8
 j 0.004683 1.0309  9.9
 j 0.004683 1.0155  j0.1772
 0.00083  j 0.004756
  0.00083 Np/m
  0.004756 rad/m
5-52
5

2


10
8
vp  
 1.32110 m/s
 0.004756
2
  2 
 1321.05 m
 0.004756
j

  j

j
j
1
1   j
5-53



1  j  j
 120
5
1
1  j 0.36
 168.6
1
1
1.0309  9.9
 163.559.9
 161.1  j 28.1 
Uniform Plane Waves in Dielectrics
and Conductors
(FEME, Sec. 5.4; EEE6E, Sec. 4.5)
5-55
Special Cases:
1. Perfect dielectric:   0
  j j  j 
  0
no attenuation
   

j


, purely real
j

Behavior same as in free space except that 0  
and 0  .
5-56
2. Imperfect Dielectric:   0 but 
 
  j   j 

2

 j 

 
 

1

j


 
2 
Behavior essentially like in a perfect dielectric except
for attenuation.
3. Good Conductor: 
5-57
 
  j   j 
  f  1  j 
     f 
j
 f


1  j 
  j


2 f 

45
Behavior much different from that in a dielectric.
5-58
4. Perfect Conductor:    
Idealization of good conductor in the limit
that   .
  ,   0
No waves can penetrate into a perfect conductor.
No time-varying fields inside a perfect conductor.
Boundary Conditions
(FEME, Sec. 5.5; EEE6E, Sec. 4.6)
5-60
Why boundary conditions?
Medium
1
Inc.
wave
Ref.
wave
Medium
2
Trans.
wave
5-61
Maxwell’s equations in integral form must be satisfied
regardless of where the contours, surfaces, and volumes are.
Example:
C3
C1
Medium 1
C2
Medium 2
5-62
Boundary Conditions
Jn1
JS
Ht 2
Ht 1
an
Jn2
Medium 1, z > 0

Bn1
Dn1
s
Et1
Bn2 Dn2
Et2
z 0
z
Medium 2, z < 0
 
x
y

5-63
Example of derivation of boundary conditions
d
C E d l   dt S B d S
Medium 1
an
Lim
ad 0
bc 0

abcda
as
a
b
d
c
E dl  
Lim
ad 0
bc 0
Medium 2
d
area B d S

dt abcd
5-64
Eab  ab   Ecd  cd   0
Eab  Edc  0
aab
as × an
 E1  E2   0
 E1  E2   0
as an ×  E1  E2   0
an ×  E1  E2   0
or,
Et1  Et 2  0
5-65
Summary of boundary conditions
an ×  E1  E2   0
or
Et1  Et 2  0
an ×  H1  H2   JS or
Ht1  Ht 2  JS
 D1  D2   S
or
Dn1  Dn 2  S
 B1  B2   0
or
Bn1  Bn 2  0
an
an
5-66
Perfect Conductor Surface
(No time-varying fields inside a perfect conductor. Also
no static electric field; may be a static magnetic field.)
Assuming both E and H to be zero inside, on the surface,
an ×E = 0
or
Et  0
an × H = JS or
Ht  JS
an D  S
or
Dn  S
an B  0
or
Bn  0
5-67
an
E


E
 
an
JS
H
 
H
JS
5-68
Dielectric-Dielectric Interface
S  0, JS  0
an ×  E1  E2   0
or
Et1  Et 2
an ×  H1  H2   0
or
Ht1  Et 2
an
 D1  D2   0
or
Dn1  Dn 2
an
 B1  B2   0
or
Bn1  Bn 2
5-69
an
Medium0
Dn1
En1
Dn2
En2
Bn1
Hn1
Bn2
Hn2
Et1
Et2
Medium0
an
Medium0
Ht1
Ht2
Medium0
5-70
Example:
D4.11 At a point on a perfect conductor surface,


(a) D  D0 ax  2a y  2az and pointing away from
the surface. Find S . D0 is positive.
D D0  ax  2a y  2az 
an 

D
D0 ax  2a y  2az
S  an
D
D
D=
D=
D
D
2
 D  D0 ax  2a y  2az  3D0
5-71


(b) D  D0 0.6 ax  0.8 a y and pointing toward the
surface. D0 is positive.
D
an  
D
S  an
D
D
D=
D=
D
D
2
  D   D0 0.6 ax  0.8 a y
  D0
5-72
Example:
z
E1  E0 az for r < a.
r>a
0
(0, 0, a)
(a) At
 0, 0, a  ,
an  az
E1 is entirely normal.
 D2  D1  2 0 E1
E2 
D2
0
 2E1  2 E0 az
a a 

0,
,


2 2

(0, a, 0)
r<a
0
y
5-73
(b) At  0, a, 0  ,
an  a y
E1 is entirely tangential
E2  E1  E0 az
a a 

,
(c) At  0,
,
2 2

1
an 
a y  az 

2
an ×  E2  E1   0 

 Solve.
an  D2  D1   0 

The End