Download Lecture EM Overview #1 - ECE/CIS

ELEG 648
Summer 2012
Lecture #1
Mark Mirotznik, Ph.D.
Associate Professor
The University of Delaware
Tel: (302)831-4221
Email: [email protected]
Vector Analysis Review:
 
A  aA A

 A
a
A
A
 
A  aA A

a = unit vector
1. Dot Product (projection)
 

A  B  A B cos( AB ) an
2. Cross Product
  
A  B  an A B sin( AB )
 
B  aB B
 AB
 
A  aA A
Orthogonal Coordinate Systems:
 



A  au1 Au1  au2 Au2  au3 Au3  A a A



au1  au 2  au 3



au2  au1  au 3



au3  au1  au 2
A  Au1 2  Au 2 2  Au 32
 
A  B  Au1Bu1  Au 2 Bu 2  Au 3Bu 3
  

A  B  au1 ( Au 2 Bu 3  Au 3 Bu 2 )  au 2 ( Au 3 Bu1  Au1Bu 3 )




 au 3 ( Au1Bu 2  Au 2 Bu1 )   au1 au 2 au 3
A  B  Au1 Au 2 Au 3
Bu1
Bu 2
Bu 3
Orthogonal Coordinate Systems:
 


dl  au1dl1  au 2dl2  au 3dl3
 
dS  an dS

an

dS
dv  dl1 dl2 dl3 dl3
dl1
dl2
dl
Cartesian Coordinate Systems:
 



A  ax Ax  a y Ay  az Az  A a A
  
ax  a y  az

 
a y  ax  az
  
az  ax  a y
 
A  B  Ax Bx  Ay By  Az Bz

ax
 
A  B  Ax
Bx

ay
Ay
By

az
Az
Bz
z
y
x
Cartesian Coordinate Systems (cont):
 


dl  ax dx  a y dy  az dz
dl  dx  dy  dz

ds x  ax dy dz

ds y  a y dx dz

ds z  az dx dy
2
dv  dx dy dz
2
2
Cylindrical Coordinate Systems:
 



A  ar Ar  a A  az Az  A a A
 


dl  ar dr  a rd  az dz

z
dsr  ar rd dz

ds  a dr dz

ds z  a z rd dz
dv  rdr d dz

(r,,z)
z
r
y
x
Spherical Coordinate Systems:
 



A  aR AR  a A  a A  A a A
 


dl  aR dR  a Rd  a R sin( )d
 2
z
dsR  aR R sin( )d d
(R,,)

ds  a R sin( )dR d


R
ds  a RdR d
dv  R 2 sin( )dR d d

y
x
Vector Coordinate Transformation:
Ax cos( )  sin( ) 0 Ar
Ay   sin( ) cos( ) 0 A


Az  0
0
1 Az
Ax sin( ) cos( ) cos( ) cos( )  sin( ) AR
Ay   sin( ) sin( ) cos( ) sin( ) cos( )  A


Az  cos( )
 sin( )
0  A
Gradient of a Scalar Field:
Assume f(x,y,z) is a scalar field
The maximum spatial rate of change of f at some location
is a vector given by the gradient of f denoted by
Grad(f) or f
 f  f  f
f  a x  a y  a z
x
y
z
 f  f
 f
f  ar
 a
 az
r
r
z
f
 f  f

f  a R
 a
 a
R
r
R sin( )
Divergence of a Vector Field:
Assume E(x,y,z) is a vector field. The divergence of E is
defined as the net outward flux of E in some
 volume as the
volume goes to zero. It is denoted by   E
 E x E y E z
E 


x
y
z
 1
1 E E z
E 
(rEr ) 

r r
r 
z
 1 
1

2
E  2
( R ER ) 
(sin( ) E )
R sin( ) 
R R
E
1

R sin( ) 
Curl of a Vector Field:
Assume E(x,y,z) is a vector field. The curl of E is measure
of the circulation ofE also called a “vortex” source. It
is denoted by   E

 ax
 
 E  
 x
 E x

 ar
 1 
 E  
r  r
 Er

ay

y
Ey

az 


z 
E z 


ra a z 



 z 
rE E z 


 aR Ra


1

 E  2

R sin( )  R 
 ER RE

R sin( )a 





R sin( ) E 
Laplacian of a Scalar Field:
Assume f(x,y,z) is a scalar field. The Laplacian is
defined as   (V ) and denoted by  2V
V V V
V 2  2  2
x
y
z
2
2
2
2
2
2
1


1

V

V
2
V
(r V )  2 2  2
r r r
r 
z
1 
1


2
2 
V 2
(R
V)
(sin( ) V )
R
R sin( ) 

R R
1
 2V
 2 2
R sin ( )  2
Examples:
1. Given the scalar function
V ( x, y, z )  sin( / 2 x) sin( / 2 y ) e z
Find the magnitude and direction of the maximum rate of chance at
location (xo,yo,zo)
2. Determine   (V )
3. Determine   (V )
3. The magnetic field produced by a long wire conducting a constant current
 Io
Is given by 
B(r )  a

Find   B
r
Basic Theorems:
1. Divergence Theorem or Gauss’s Law

 
  E dv   E  ds
v
s
2. Stokes Theorem
 
 
 (  E )  ds   E  dl
s
c
Examples:
1. Verify the Divergence Theorem for

 2 
A(r , z )  ar r  az 2 z
on a cylindrical region enclosed by r=5, z=0 and z=4
r=5
z=4
z=0
Odds and Ends:
1. Normal component of field

E
 
n  E  En
2. Tangential component of field
 
n  E  Et

n
Maxwell’s Equations in Differential Form


B 
Faraday’s Law
 E  
M
t

 D  
 H 
 J c  J i Ampere’s Law
t

Gauss’s Law
D  

  B  m
Gauss’s Magnetic Law
Faraday’s Law

B
t
C
S

E


B
 E  
t
 
 

c E  dl   t s B  ds
Ampere’s Law

J

D
t

J

H

H

  D
 H  J 
t
  
 
 
c H  dl  t s D  ds  s J  ds
Gauss’s Law
Qtot

D

D  
 
s D  ds v  dv  Qtot
Gauss’s Magnetic Law

B  0
 
s B  ds 0
“all the flow of B entering the
volume V must leave the volume”

B
CONSTITUTIVE RELATIONS


D  E
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)


BH
r o=permeability (H/m)
o=4 x 10-7 (H/m)


Jc   E
=conductivity (S/m)
POWER and ENERGY


H
(eq1)   E   
 M d
t


  
 
E
(eq2)   H  
  E  Ji  J d  Jc  Ji
t


take H  (eq1)  E  (eq2)

 


 
 
(eq3) H    E  E    H   H  M d  E  ( J d  J c  J i )
Using the vector identity   ( A  B)  B  (  A)  A  (  B)
 

 
 
(eq4)   ( E  H )  H  M d  E  ( J d  J c  J i )  0
n
E, H
Ji
V
, , 
Integrate eq4 over the volume V in the figure
(eq5)
 

 
 


(
E

H
)
dv


[
H

M

E

(
J

J
d
d
c  J i )] dv
v
v
Applying the divergence theorem


 
 H
 E
   
(eq6) s ( E  H )  ds  v [  H 
 E
  E  E  E  J i )] dv  0
t
t
S
POWER and ENERGY (continued)


 
 H
 E
   
(eq6) s ( E  H )  ds  v [  H 
 E
  E  E  E  J i )] dv  0
t
t


 H   1
 E   1
 


2
2
2
H
   H   wm ,  E 
   E   we ,  E  E   E
t t  2
t t  2
 t
 t
 
 
w
w
2
(eq7) s ( E  H )  ds  v [ m  e ] dv  v [ E  J i ] dv  v  E dv  0
t
t
 
 

2
(eq8) s ( E  H )  ds  v [ we  wm ] dv  v [ E  J i ] dv  v  E dv  0
t
 
Ps  s ( E  H )  ds


1
1
2
2


Wm  v [  H ] dv , We v [  E ] dv
2
2


 
2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
Stored magnetic power (W)
Ps 
What is this term?
Supplied power (W)


Wm  We  Pi  Pd
t
t
Dissipated power (W)
Stored electric power (W)
POWER and ENERGY (continued)
 
Ps  s ( E  H )  ds
1
1
2
2


Wm  v [  H ] dv , We v [  E ] dv
2
2


 
2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
Stored magnetic power (W)
Ps 
What is this term?
Supplied power (W)


Wm  We  Pi  Pd
t
t
Dissipated power (W)
Stored electric power (W)
Ps = power exiting the volume through radiation
  
S  E  H W/m2 Poynting vector
TIME HARMONIC EM FIELDS
Assume all sources have a sinusoidal time dependence and all materials
properties are linear. Since Maxwell’s equations are linear all electric
and magnetic fields must also have the same sinusoidal time dependence.
They can be written for the electric field as:


E ( x, y, z, t )  Eo ( x, y, z ) cos( t   ( x, y, z ))
Euler’s Formula

~
e jt  cos(t )  j sin(t )
E ( x, y, z, t )  Re[ E ( x, y, z ) e jt ]
~
E ( x, y, z ) is a complex function of space (phasor) called the time-harmonic electric
field. All field values and sources can be represented by their time-harmonic form.

~
E ( x, y, z , t )  Re[ E ( x, y, z ) e jt ]

~
D ( x, y, z , t )  Re[ D( x, y, z ) e jt ]

~
H ( x, y, z , t )  Re[ H ( x, y, z ) e jt ]

~
B ( x, y, z , t )  Re[ B ( x, y, z ) e jt ]

~
J ( x, y, z , t )  Re[ J ( x, y, z ) e jt ]

 ( x, y, z , t )  Re[ ~ ( x, y, z ) e jt ]
PROPERTIES OF TIME HARMONIC FIELDS
Time derivative:
Time integration:

~
~
[Re[E ( x, y, z ) e jt ]]  j[Re[E ( x, y, z ) e jt ]
t
1
~
~
jt
jt
[Re[
E
(
x
,
y
,
z
)
e
]
dt

[Re[
E
(
x
,
y
,
z
)
e
]

j
TIME HARMONIC MAXWELL’S EQUATIONS


B 
 E  
M
t

 D 
 H 
J
t

D  

  B  m










~
~
~
  Re E e jt   Re B e jt  Re M e jt
t

~
~
~
  Re H e jt  Re D e jt  Re J e jt
t
~
  Re D e jt  Re ~ e jt
~
  Re B e jt  Re ~ e jt





m





Employing the derivative property results in the following set of equations:
~
~ ~
  E   j B  M
~
~ ~
  H  j D  J
~
  D  ~
~
  B  ~m

TIME HARMONIC EM FIELDS
BOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES
The constitutive properties and boundary conditions are very similar
for the time harmonic form:
General Boundary Conditions
Constitutive Properties
~
~
D  E
~
~
BH
~
~
Jc   E
~ ~
nˆ  ( E2  E1 )  0
~
~
~
nˆ  ( H 2  H1 )  J s
~
~
nˆ  ( D2  D1 )  ~s
~ ~
nˆ  ( B2  B1 )  0
PEC Boundary Conditions
~
nˆ  E2  0
~
~
nˆ  H 2  J s
~
nˆ  D2  ~s
~
nˆ  B2  0
TIME HARMONIC EM FIELDS
IMPEDANCE BOUNDARY CONDITIONS
If one of the material at an interface is a good conductor but of finite
conductivity it is useful to define an impedance boundary condition:
2,2,2
1,1,1
1>> 2

Z s  Rs  jX s  (1  j )
2

~
~
~
~
Et  Z s J s  Z s nˆ  H  (1  j )
nˆ  H
2
POWER and ENERGY: TIME HARMONIC
~ ~
Ps  s ( E  H * )  ds
1 ~2
1 ~2


Wm  v [  H ] dv , We v [  E ] dv
4
4


1~ ~
1 ~2
Pi  v [ E  J i* ] dv  0, Pd  v  E dv  0
2
2
Time average magneticenergy (J)
Supplied complex power (W)
Ps  j 2 (Wm  We )  Pi  Pd
Time average electric energy (J)
Time average exiting power
Dissipated real power (W)
CONTINUITY OF CURRENT LAW


B
 E  
t

 D 
 H 
J
t

D  

B  0




D  
  (  H )    [
 J ]  [  D]    J
t
t

vector identity   (  A)  0



0  [  D]    J
t


0  []    J
t


 J  
t

time harmonic   J   j
SUMMARY
Time Domain


B 
 E    M
t

D  

 D 
 H 
J
t

  B  m





nˆ  ( E2  E1 )  0 nˆ  ( H 2  H1 )  J s




nˆ  ( D2  D1 )   s nˆ  ( B2  B1 )  0
Frequency Domain
~
~ ~
  E   j B  M
~
  D  ~
~ ~
~
~
~
nˆ  ( E2  E1 )  0 nˆ  ( H 2  H1 )  J s
~
~
~ ~
nˆ  ( D2  D1 )  ~s nˆ  ( B2  B1 )  0
Z s  Rs  jX s  (1  j )


D  E


BH


Jc   E
 
Ps  s ( E  H )  ds
1
1
2
2


Wm  v [  H ] dv , We v [  E ] dv
2
2


 
2
Pi  v [ E  J i ] dv  0, Pd  v  E dv  0
~
~ ~
  H  j D  J
~
  B  ~m

2
~
~
D  E
~
~
BH
~
~
Jc   E
~ ~
Ps  s ( E  H * )  ds
1 ~2
1 ~2


Wm  v [  H ] dv , We v [  E ] dv
4
4


1~ ~
1 ~2
Pi  v [ E  J i* ] dv  0, Pd  v  E dv  0
2
2
Electromagnetic Properties of Materials
Primary Material Properties
Electrical Properties
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)
=conductivity (S/m)
Magnetic Properties
r o=permeability (H/m)
o=4 x 10-7 (H/m)
Secondary Material Properties
Electrical Properties
n  r
e
Index of refraction
Electric susceptibility
Magnetic Properties
m
Magnetic susceptibility
Electric Properties of Materials
Qi Eext
+
++++
Eext
li
+
-
- - - ++++
Eext
- - - ++++
No external field
Eext
- - - Bulk material (N molecules)
Applied external field
Electric dipole moment
of individual atom or
molecule:


pi  Qili
Net dipole moment or
polarization vector:
N

 N 
P   pi   Qili
i 1
i 1
Electric Properties of Materials (continued)
++++
Eext
- - - ++++
Eext
- - - ++++
- - - Bulk material (N molecules)
?

P o e E
What are the assumptions
here?
N

 N 
P   pi   Qili
i 1
i 1

 
D  oE  P


 o E  o eE

  o (1   e ) E

  o r E
 r 1   e
Static permittivity
or relative permittivity
Electric Properties of Materials (continued)
Conductivity
E
J
y
z
J=current density=qvvz where
qv=volume charge density and
vz= charge drift velocity
x
E
Where e is called the electron
mobility. The current density is
thus given by


1
m
When subjected to an external
electric field E the charge
velocity is increased and is
given by
ve  qv e
 E

v
J  q e E


Where  is called the
conductivity. Its units are S/m
Material
Where  is called the resistivity. Silver
Glass
Sea Water
Conductivity (S/m)
6.1 x 107
1.0 x 10-12
4
Electric Properties of Materials (continued)
1.
Orientational Polarization: molecules have a slight polarization even in the
absence of an applied field. However each polarization vector is orientated randomly
so the net P vector is zero. Such materials are known as polar; water is a good example.
 rH 2O  81
2.
Ionic Polarization: Evident in materials ionic materials such as NaCl. Positive and
negative ions tend to align with the applied field.
3.
Electronic Polarization: Evident in most materials and exists when an applied
field displaces the electron cloud of an atom relative to the positive nucleus.
Magnetic Properties of Materials
Ii
Mi
Bext
Bext
Net magnetic dipole
moment or
magnetization vector:
N 
N

M   M i   I i dsi nˆi
No magnetic field: random
oriented magnetic dipoles
i 1
Magnetic dipoles randomly oriented
resulting in zero net magnetization vector:
 N
M   I i dsi nˆi  0
i 1
i 1
Applied external magnetic field
Magnetic dipoles tend to align with
external magnetic field resulting in non-zero
net magnetization vector:
 N
M   I i dsi nˆi  0
i 1
Magnetic Properties of Materials (continued)
Bext
Bext
Applied external magnetic field
 ?

M  o  m H
What are the assumptions
here?
N
 N 
M   M i   I i dsi nˆi
i 1
i 1

 
B  o H  M


 o H  o  m H

  o (1   m ) H

 o r H
r 1   m
Static permeability
or relative permeability
Magnetic Properties of Materials (continued)
1.
Diamagnetic: Net magnetization vector tends to appose the direction of the
applied field resulting in a relative permeability slightly less than 1.0
Examples: silver (r=0.9998)
2.
Paramagnetic: Net magnetization vector tends to align in the direction
of the applied field resulting in a relative permeability slightly greater than 1.0
Examples: Aluminum (r=1.00002)
3
Ferromagnetic: Net magnetization vector tends to align strongly in the direction
of the applied field resulting in a relative permeability much greater than 1.0
Examples: Iron (r=5000)
.
Classification of Materials
1.
Homogenous or Inhomogenous: If the material properties are independent of spatial
location then the material is homogenous, otherwise it is called inhomogenous
 ( x, y, z )  Inhomogenous
2.
Isotropic or Anisotropic: If the material properties are independent of the polarization
of the applied field then the material is isotropic, otherwise it is called anisotropic.
 Dx   xx
 D   
 y   yx
 Dz   zx
 xy
 yy
 zy
 xz   E x 



 yz   E y   D   E
 
 zz   E z 
anisotropic
3.
Linear or non-Linear: If the material properties are independent on the magnitude
and phase of the electric and magnetic fields, otherwise it is called non-linear



2
3
D   o E   o E  1E   3 E  ...
Classification of Materials
4.
Dispersive or non-dispersive: If the material properties are independent of frequency
then the material is non-dispersive, otherwise it is called dispersive
+++ +
++++
---++++
-- -++++
----
---+++ +
---+++ +
----
+ + + ++
+- +- +
+-- +
+
- -- - +-
+- + + ++- +-- +- +
----
t=t4
t=t5
t=t3
++++
-+ -+ -+ -+
-+ -+ -+ -+
----
+ + + +-
t=t2
--++++
+
+
- - - +-
t=t1
t=t6
A material’s atoms or molecules attempt to keep up with a changing electric field. This results
in two things: (1) friction causes energy loss via heat and (2) the dynamic response of the
molecules will be a function of the frequency of the applied field (i.e. frequency dependant
material properties)
Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
~
~
  E   j B
~
~ ~ ~
  H  j D  J i  J c
~
  D  ~
~
B  0
~
~
D   * ( ) E
~
~
B  o H
~
~
Jc   s E
Dielectric
constant
loss term
 * ( )   ( )  j ( )
is called the complex permittivity
~
~
  E   j  o H
~
~ ~
~
  H  j  * E  J i   s E
~
  ( * E )  ~
~
  (H )  0
~
~
  E   j  o H
~
~ ~
  H  j  eff ( ) E  J i
~
  ( * E )  ~
~
  (H )  0
~
~
  E   j  o H
~
~ ~
~
  H  j (   j ) E  J i   s E
~
  ( * E )  ~
~
  (H )  0
 eff ( )   ( )  j ( )  j
s

Frequency Behavior of Sea Water
Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
~
~
  E   j  o H
~
~ ~
~
  H  j (   j ) E  J i   s E
~
  ( * E )  ~
~
~
  E   j  o H
~
~ ~
~
  H  j   E  J i  ( s    ) E
~
  ( * E )  ~
~
  (H )  0
~
  (H )  0
~
~
  E   j  o H
~
~ ~
~
  H  j   E  J i   eff E
~
  ( * E )  ~
~
~
  E   j  o H
~ ~ ~ ~
  H  J d  J i  J eff
~
  ( * E )  ~
~
  (H )  0
~
  (H )  0
~
~
J d  j  E
~
Ji
~
~
J eff   eff E
 eff ( )  s   a   s   
Displacement current
Source current
Effective electric
conduction current
Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
~
~
  E   j  o H
~
~ ~
~
  H  j   E  J i   eff E
~
  ( * E )  ~
~
  (H )  0
 eff ( )  s   a   s   
~
~
  E   j  o H
 eff ~ ~
~
  H  j   (1  j
)E  Ji


~
  ( * E )  ~
~
  (H )  0
~
~
  E   j  o H
~
~ ~
  H  j   (1  j tan( eff )) E  J i
~
  ( * E )  ~
~
~
  E   j  o H
~ ~
~
  H  J cd  J i
~
  ( * E )  ~
~
  (H )  0
~
  (H )  0
tan( eff ) 
 eff
 

~
~
J cd  j (1  j eff ) E
 
Electric Properties of Materials
Frequency Behavior (Complex Permittivity)
~
~
  E   j  o H
~ ~ ~ ~
  H  J d  J i  J eff
~
  ( * E )  ~
~
  (H )  0
~
~
J d  j  E
~
Ji
~
~
J eff   eff E
Good Dielectric

~
~
J d  J eff ( eff  1)
 
Displacement current
Source current
Effective electric
conduction current
 eff ( )  s   a   s   
Good Conductor
~
~ 
J eff  J d ( eff  1)
 
Wave Equation
Time Dependent Homogenous Wave Equation (E-Field)


H
  E  
t


 
E
 H  
 E  J
t
 
E 


B  0


H
  [  E ]    ( 
)
t



    E     H
t


 
  E
    E    
 E  J 
t  t




Vector Identity
2





E

E

J
    E    2  

    A  (  A)  2 A

t
t
t



2


 E
E
J
(  E )   2 E    2  

t
t

 t 
2

1
 E
E
J
   2 E    2  


t
t
t



2

 E
E
J 1
 2 E   2  

 
t
t 
t
Wave Equation
Source-Free Time Dependent Homogenous Wave Equation (E-Field)



2

 E
E
J 1
 2 E   2  

 
t
t 
t

Source Free J  0,   0


2

 E
E
 2 E   2  
0
t
t
Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)
Lossless
 0

2

 E
 2 E   2  0
t
Wave Equation
Source-Free Time Dependent Homogenous Wave Equation (H-Field)


2

 1
 H
H
 2 H   2  
   J  
t

t
Source Free

J  0,   0
Source Free and Lossless


2


H

H
 2 H   2  
0
t
t

J  0,   0,   0

2


H
 2 H   2  0
t
Wave Equation: Time Harmonic
Time Domain



2

 E
E
J 1
 2 E   2  

 
t
t 
t

Source Free J  0,   0


2


E

E
 2 E   2  
0

t
t
Lossless
 0

2

 E
 2 E   2  0
t
Frequency Domain
~
~
~
~ 1
2 E   2  E  j E   J  ~

Source Free

J  0,   0
~
~
~
 2 E   2  E  j E  0
Lossless
 0
~
~
2 E   2  E  0
“Helmholtz Equation”
General Solution: Point Source

2


E
 2 E   2  0
t

1   2  
2E
E    2  0
r
r r  r 
t
0

E (r , t )  E (r , t )aˆ  E (r , t )aˆ  Er (r , t )aˆr
point
source

y
2
Outward traveling spherical wave
Solution:
E (r , t )  c
(r,,)
r
 2 E
1 2 

E    2  0
r
r r  r 
t
1 2 
 E

E    2  0
r
r r  r 
t
z
x
Inward traveling spherical wave
1
1
f (t   r )  c f (t   r )
r
r
General Solution: Point Source
E (r , t )  c
1
1
1
r
f (t   r )  c f (t  )
r
r
c
m
c
 sec
point
source
Wave speed

m
c

 3  10
 o o
sec
4  107  8.85  1012
Same as the speed of light!
(r,,)
r
In free space:
1
z
1
8
y
General Solution Case: Time Harmonic
Rectangular Coordinates
~
~
2 E   2  E  0
   
~
~
2 E   2 E  0
Wave Number
~
~
~
2E 2E 2E
2 ~




E 0
2
2
2
x
y
z
 2 Ex  2 Ex  2 Ex
2




Ex  0
2
2
2
x
y
z
2Ey
x
2

2Ey
y
2

2Ey
z
2
  2 Ey  0
 2 Ez  2 Ez  2 Ez
2




Ez  0
2
2
2
x
y
z
Separation of Variable Solutions
 2 Ex  2 Ex  2 Ex


  2 Ex  0
2
2
2
x
y
z
Assume Solution of the form:
Ex ( x, y, z)  f ( x) g ( y)h( z)
f ( x) g ( y)h( z )  f ( x) g( y)h( z )  f ( x) g ( y)h( z )   2 f ( x) g ( y)h( z )  0
1
[ f ( x) g ( y)h( z )  f ( x) g ( y)h( z )  f ( x) g ( y)h( z )   2 f ( x) g ( y)h( z )]  0
f ( x ) g ( y ) h( z )
constant
f ( x) g ( y) h( z )


2 0
f ( x ) g ( y ) h( z )
function
of x
function
of y
function
of z
Separation of Variable Solutions
constant
f ( x) g ( y) h( z )


2 0
f ( x ) g ( y ) h( z )
function
of x
f ( x)
2
  x
f ( x)
g ( y )
  y2
g ( y)
h( z )
  z2
h( z )
 2   x2   y2   z2
function
of y
function
of z
f ( x)   x f ( x)  0
2
g ( y )   y 2 g ( y )  0
2
h( z )   z h( z )  0
 2   x2   y2   z2
Separation of Variable Solutions
f ( x)   x f ( x)  0
2
g ( y )   y 2 g ( y )  0
2
h( z )   z h( z )  0
 2   x2   y2   z2
Solutions:
f ( x)  A1e j x x  B1e j x x
g ( y )  A2 e
 j y y
 B2 e
j y y
h( z )  A3 e j z z  B3 e j z z
 purely real
Traveling and
standing waves
A1e j x x  B1e j x x
or
C1 cos(  x x )  D1 sin(  x x )
 purely
imaginary
Evanesent waves
A1e
 x
x
 B1e
 complex
Exponentially modulated
traveling wave
A1e xe j x x  B1e x e j x x
or
 x  j x x
A1e e
 B1e x e j x x
Separation of Variable Solutions: Examples
case a. x, y, z all real (forward traveling waves)
Ex ( x, y, z )  f ( x) g ( y)h( z )  Exo e j x xe
 j y y  j z z
e
Plane Waves
case b. xreal (forward traveling wave), x(real standing
wave), z imaginary (evanesent wave)
Ex ( x, y, z )  f ( x) g ( y )h( z )  Exo e j x x (C1 cos(  y y )  D1 sin(  y y ))e z
z
y
x
Surface Waves
Separation of Variable Solutions: Examples
case c. xreal (forward standing wave), x(real standing
wave), z real (traveling wave)
Guided Waves
E x ( x, y, z )  E xo e  j z z ( A1 cos(  x x)  B1 sin(  x x))(C1 cos(  y y )  D1 sin(  y y ))
E y ( x, y, z )  E yo e  j z z ( A1 cos(  x x)  B1 sin(  x x))(C1 cos(  y y )  D1 sin(  y y ))
E z ( x, y, z )  E zo e  j z z ( A1 cos(  x x)  B1 sin(  x x))(C1 cos(  y y )  D1 sin(  y y ))
Unknown constants A1, B1, C1 , D1, x, y, z
Found by applying boundary conditions and
dispersion relation. Namely:
y
E x ( x, y   h / 2, z )  0
E y ( x  b / 2, y, z )  0
E z ( x  b / 2, y   h / 2, z )  0
 x   y   z     
2
2
2
2
2
h
x
, 
PEC Walls
z
b
Stay tuned we will solve the complete
solution for modes in a rectangular
waveguide in a later lecture.