ELEG 648 Summer 2012 Lecture #1 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware...
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Transcript of ELEG 648 Summer 2012 Lecture #1 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware...
ELEG 648Summer 2012
Lecture #1Mark Mirotznik, Ph.D.
Associate ProfessorThe University of Delaware
Tel: (302)831-4221Email: [email protected]
Vector Analysis Review:
A
Aa
AaA A
a
= unit vector
AaA A
1. Dot Product (projection)
)cos( ABBABA
2. Cross Product
)sin( ABn BAaBA
AaA A
A
BaB B
ABna
Orthogonal Coordinate Systems:
23
22
21
213
312
321
332211
uuu
uuu
uuu
uuu
Auuuuuu
AAAA
aaa
aaa
aaa
aAAaAaAaA
332211 uuuuuu BABABABA
)(
)()(
12213
3113223321
uuuuu
uuuuuuuuuu
BABAa
BABAaBABAaBA
321
321
321
uuu
uuu
uuu
BBB
AAA
aaa
BA
Orthogonal Coordinate Systems:
dl332211 dladladlald uuu
Sd
na
dSaSd n
321 dldldldv
dl1dl2
dl3
Cartesian Coordinate Systems:
yxz
zxy
zyx
Azzyyxx
aaa
aaa
aaa
aAAaAaAaA
zzyyxx BABABABA
zyx
zyx
zyx
BBB
AAA
aaa
BA
x
y
z
Cartesian Coordinate Systems (cont):
dzdydxdv
dydxads
dzdxads
dzdyads
dzdydxdl
dzadyadxald
zz
yy
xx
zyx
222
Cylindrical Coordinate Systems:
Azzrr aAAaAaAaA
dzdrdrdv
dzrdads
dzdrads
dzrdads
dzardadrald
zz
rr
zr
x
y
z
r
z
(r,z)
Spherical Coordinate Systems:
ARR aAAaAaAaA
dddRRdv
dRdRads
ddRRads
ddRads
dRaRdadRald
RR
R
)sin(
)sin(
)sin(
)sin(
2
2
x
y
z
R
(R)
Vector Coordinate Transformation:
z
r
z
y
x
A
A
A
A
A
A
100
0)cos()sin(
0)sin()cos(
A
A
A
A
A
A R
z
y
x
0)sin()cos(
)cos()sin()cos()sin()sin(
)sin()cos()cos()cos()sin(
Gradient of a Scalar Field:
Assume f(x,y,z) is a scalar fieldThe maximum spatial rate of change of f at some locationis a vector given by the gradient of f denoted byGrad(f) or f
)sin(R
fa
r
fa
R
faf
z
fa
r
fa
r
faf
z
fa
y
fa
x
faf
R
zr
zyx
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 thevolume goes to zero. It is denoted by E
E
R
ER
ERRR
E
z
EE
rrE
rrE
z
E
y
E
x
EE
R
zr
zyx
)sin(
1
))(sin()sin(
1)(
1
1)(
1
22
Curl of a Vector Field:
Assume E(x,y,z) is a vector field. The curl of E is measureof the circulation of E also called a “vortex” source. Itis denoted by E
ERREER
aRaRa
RE
ErEEzr
aara
rE
EEEzyx
aaa
E
R
R
zr
zr
zyx
zyx
)sin(
)sin(
)sin(
1
1
2
Laplacian of a Scalar Field:
)( VAssume f(x,y,z) is a scalar field. The Laplacian is defined as and denoted by V2
2
2
22
22
2
2
2
2
2
22
2
2
2
2
2
22
)(sin
1
))(sin()sin(
1)(
1
1)(
1
V
R
VR
VR
RRR
V
z
VV
rV
rr
rrV
z
V
y
V
x
VV
Examples:
1. Given the scalar functionzeyxzyxV )2/sin()2/sin(),,(
Find the magnitude and direction of the maximum rate of chance atlocation (xo,yo,zo)
2. Determine )( V
3. Determine )( V
3. The magnetic field produced by a long wire conducting a constant currentIs given by
r
IarB o
)(
Find B
Basic Theorems:
1. Divergence Theorem or Gauss’s Law
sdEdvEsv
2. Stokes Theorem
cs
ldEsdE
)(
Examples:
1. Verify the Divergence Theorem for
zarazrA zr 2),( 2
on a cylindrical region enclosed by r=5, z=0 and z=4
r = 5
z = 0
z = 4
Odds and Ends:
1. Normal component of field
n
E
nEEn
2. Tangential component of field
tEEn
Maxwell’s Equations in Differential Form
m
ic
B
D
JJt
DH
Mt
BE
Faraday’s Law
Ampere’s Law
Gauss’s Law
Gauss’s Magnetic Law
Faraday’s Law
sdBt
ldE
t
BE
c s
S
C
t
B
E
Ampere’s Law
sc ssdJsdD
tldH
t
DJH
t
D
J
J
H
H
Gauss’s Law
v totsQdvsdD
D
totQ
D
Gauss’s Magnetic Law
0
0
ssdB
B
B
“all the flow of B entering the volume V must leave the volume”
CONSTITUTIVE RELATIONS
EJ
HB
ED
c
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)
r o=permeability (H/m)
o=4 x 10-7 (H/m)
=conductivity (S/m)
POWER and ENERGY
Ji
E, H
V
S
n
icdi
d
JJJJEt
EHeq
Mt
HEeq
)2(
)1(
)2()1( eqEeqH
)()3( icdd JJJEMHHEEHeq
take
Using the vector identity )()()( BAABBA
0)()()4( icdd JJJEMHHEeq
Integrate eq4 over the volume V in the figure
v icddv
dvJJJEMHdvHEeq )]([)()5(
Applying the divergence theorem
0)][)()6(
v is
dvJEEEt
EE
t
HHdsHEeq
POWER and ENERGY (continued)
0)][)()6(
v is
dvJEEEt
EE
t
HHdsHEeq
222,
2
1,
2
1EEEw
tE
tt
EEw
tH
tt
HH em
0][][)()7(2
vv iv
ems
dvEdvJEdvt
w
t
wdsHEeq
0][][)()8(2
vv iv mes
dvEdvJEdvwwt
dsHEeq
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWt
Wt
P
Stored magnetic power (W)
Stored electric power (W)
Supplied power (W)
Dissipated power (W)
What is this term?
POWER and ENERGY (continued)
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWt
Wt
P
Stored magnetic power (W)
Stored electric power (W)
Supplied power (W)
Dissipated power (W)
What is this term?
Ps = power exiting the volume through radiation
HES
W/m2 Poynting vector
TIME HARMONIC EM FIELDS
]),,(~
Re[),,,(
)),,(cos(),,(),,,(tj
o
ezyxEtzyxE
zyxtzyxEtzyxE
Assume all sources have a sinusoidal time dependence and all materialsproperties are linear. Since Maxwell’s equations are linear all electricand magnetic fields must also have the same sinusoidal time dependence.They can be written for the electric field as:
),,(~
zyxE is a complex function of space (phasor) called the time-harmonic electricfield. All field values and sources can be represented by their time-harmonic form.
]),,(~Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
]),,(~
Re[),,,(
tj
tj
tj
tj
tj
tj
ezyxtzyx
ezyxJtzyxJ
ezyxBtzyxB
ezyxHtzyxH
ezyxDtzyxD
ezyxEtzyxE
)sin()cos( tjte tj Euler’s Formula
PROPERTIES OF TIME HARMONIC FIELDS
]),,(~
[Re[]]),,(~
[Re[ tjtj ezyxEjezyxEt
]),,(~
[Re[1
]),,(~
[Re[ tjtj ezyxEj
dtezyxE
Time derivative:
Time integration:
TIME HARMONIC MAXWELL’S EQUATIONS
tj
mtj
tjtj
tjtjtj
tjtjtj
eeB
eeD
eJeDt
eH
eMeBt
eE
~Re
~Re
~Re~
Re
~Re
~Re
~Re
~Re
~Re
~Re
mB
D
Jt
DH
Mt
BE
mB
D
JDjH
MBjE
~~
~~
~~~
~~~
Employing the derivative property results in the following set of equations:
TIME HARMONIC EM FIELDSBOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES
The constitutive properties and boundary conditions are very similarfor the time harmonic form:
0)~~
(ˆ
~)~~
(ˆ
~)
~~(ˆ
0)~~
(ˆ
12
12
12
12
BBn
DDn
JHHn
EEn
s
s
EJ
HB
ED
c~~
~~
~~
Constitutive Properties
General Boundary Conditions
0~
ˆ
~~ˆ
~~ˆ
0~
ˆ
2
2
2
2
Bn
Dn
JHn
En
s
s
PEC Boundary Conditions
TIME HARMONIC EM FIELDSIMPEDANCE BOUNDARY CONDITIONS
If one of the material at an interface is a good conductor but of finiteconductivity it is useful to define an impedance boundary condition:
HnjHnZJZE
jjXRZ
ssst
sss
~ˆ
2)1(
~ˆ
~~
2)1(
1,
2,
1>> 2
POWER and ENERGY: TIME HARMONIC
0~
2
1,0]
~~2
1[
]~
4
1[,]
~4
1[
)~~
(
2*
22
*
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
diems PPWWjP )(2
Time average magneticenergy (J)
Time average electric energy (J)
Supplied complex power (W)
Dissipated real power (W)Time average exiting power
CONTINUITY OF CURRENT LAW
JDt
Jt
DH
][][)(
0
B
D
Jt
DH
t
BE
0)( A
vector identity
JDt
][0
Jt
][0
tJ
jJ
time harmonic
SUMMARY
mBD
Jt
DHM
t
BE
mBD
JDjHMBjE
~~~~
~~~~~~
0)~~
(ˆ~)~~
(ˆ
~)
~~(ˆ0)
~~(ˆ
1212
1212
BBnDDn
JHHnEEn
s
s
0)(ˆ)(ˆ
)(ˆ0)(ˆ
1212
1212
BBnDDn
JHHnEEn
s
s
EJ
HB
ED
c~~
~~
~~
EJ
HB
ED
c
2
)1( jjXRZ sss
0,0][
]2
1[,]
2
1[
)(
2
22
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
0~
2
1,0]
~~2
1[
]~
4
1[,]
~4
1[
)~~
(
2*
22
*
vdv ii
vevm
ss
dvEPdvJEP
dvEWdvHW
dsHEP
Frequency DomainTime Domain
Electromagnetic Properties of Materials
Primary Material Properties
r o=permeability (H/m)
o=4 x 10-7 (H/m)
r o=permittivity (F/m)
o=8.854 x 10-12 (F/m)
=conductivity (S/m)
Electrical Properties
Magnetic Properties
Secondary Material Properties
Electrical Properties
rn Index of refraction
e Electric susceptibility
Magnetic Properties
m Magnetic susceptibility
Electric Properties of Materials
+
Eext
Eext
-
+
li
Qi
No external field
Applied external field
iii lQp
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
Bulk material (N molecules)
N
iii
N
ii lQpP
11
Electric dipole momentof individual atom ormolecule:
Net dipole moment or polarization vector:
Eext
Eext
Electric Properties of Materials (continued)
N
iii
N
ii lQpP
11
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
Bulk material (N molecules)
Eext
Eext
E
E
EE
PED
ro
eo
eoo
o
)1(EP eo
?
What are the assumptionshere? er 1 Static permittivity
or relative permittivity
Electric Properties of Materials (continued)Conductivity
x
y
z E
E
J
J=current density=qvvz whereqv=volume charge density andvz= charge drift velocity
When subjected to an external electric field E the charge velocity is increased and is given by
e v eq v
Where e is called the electron mobility. The current density is thus given by
E
E q Je v
Where is called the conductivity. Its units are S/m
m
1
Where is called the resistivity.Material Conductivity (S/m)Silver 6.1 x 107
Glass 1.0 x 10-12
Sea Water 4
Electric Properties of Materials (continued)
1. Orientational Polarization: molecules have a slight polarization even in theabsence of an applied field. However each polarization vector is orientated randomlyso the net P vector is zero. Such materials are known as polar; water is a good example.
2. Ionic Polarization: Evident in materials ionic materials such as NaCl. Positive andnegative ions tend to align with the applied field.
3. Electronic Polarization: Evident in most materials and exists when an appliedfield displaces the electron cloud of an atom relative to the positive nucleus.
812 OHr
Magnetic Properties of Materials
No magnetic field: randomoriented magnetic dipoles
N
iiii
N
ii ndsIMM
11
ˆ
Net magnetic dipolemoment or magnetization vector:
Ii Mi
I iM
i
I iM
i
IiM
i
Applied external magnetic field
BextBext
I i
M i
I i
M i
I i
M i
I i
M i
N
iiii ndsIM
1
0ˆ
Magnetic dipoles randomly orientedresulting in zero net magnetization vector:
Magnetic dipoles tend to align withexternal magnetic field resulting in non-zeronet magnetization vector:
N
iiii ndsIM
1
0ˆ
Magnetic Properties of Materials (continued)
H
H
HH
MHB
ro
mo
moo
o
)1(
HM mo
?
What are the assumptionshere? mr 1 Static permeability
or relative permeability
Applied external magnetic field
BextBext
I i
M i
I i
M i
I i
M i
I i
M i
N
iiii
N
ii ndsIMM
11
ˆ
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 1.0Examples: 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 1.0Examples: 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 1.0Examples: Iron (r=5000)
.
Classification of Materials
1. Homogenous or Inhomogenous: If the material properties are independent of spatiallocation then the material is homogenous, otherwise it is called inhomogenous
2. Isotropic or Anisotropic: If the material properties are independent of the polarizationof the applied field then the material is isotropic, otherwise it is called anisotropic.
3. Linear or non-Linear: If the material properties are independent on the magnitudeand phase of the electric and magnetic fields, otherwise it is called non-linear
),,( zyx Inhomogenous
ED
E
E
E
D
D
D
z
y
x
zzzyzx
yzyyyx
xzxyxx
z
y
x
anisotropic
...33
21 EEEED oo
Classification of Materials
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+
-+-
+-+-+-+-
+-
+-
+
-+
-+
-+
-+
-+
-+
-+
-+
-+-
+-
+-
+
-
+
-
+
-
+
-
+
-
+-
+-
+-
+
-+
-
+
-
+
-
+
t=t1t=t2
t=t3
t=t4 t=t5
t=t6A 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 themolecules will be a function of the frequency of the applied field (i.e. frequency dependantmaterial properties)
4. Dispersive or non-dispersive: If the material properties are independent of frequencythen the material is non-dispersive, otherwise it is called dispersive
Electric Properties of Materials Frequency Behavior (Complex Permittivity)
0~
~~
~~~~
~~
B
D
JJDjH
BjE
ci
EJ
HB
ED
sc
o~~
~~
~)(
~ *
0)~
(
~)~
(
~~~~
~~
*
*
H
E
EJEjH
HjE
si
o
)()()(* j
is called the complex permittivity
0)~
(
~)~
(
~~~)(
~
~~
*
H
E
EJEjjH
HjE
si
o
loss termDielectric constant
0)~
(
~)~
(
~~)(
~
~~
*
H
E
JEjH
HjE
ieff
o
s
eff jj )()()(
Frequency Behavior of Sea Water
Electric Properties of Materials Frequency Behavior (Complex Permittivity)
0)~
(
~)~
(
~)(
~~~
~~
*
H
E
EJEjH
HjE
si
o
0)~
(
~)~
(
~~~)(
~
~~
*
H
E
EJEjjH
HjE
si
o
0)~
(
~)~
(
~~~~
~~
*
H
E
EJEjH
HjE
effi
o
saseff )(
0)~
(
~)~
(
~~~~
~~
*
H
E
JJJH
HjE
effid
o
EJ
J
EjJ
effeff
i
d
~~
~
~~
Displacement current
Source current
Effective electric conduction current
Electric Properties of Materials Frequency Behavior (Complex Permittivity)
0)~
(
~)~
(
~~~~
~~
*
H
E
EJEjH
HjE
effi
o
saseff )(0)
~(
~)~
(
~~)1(
~
~~
*
H
E
JEjjH
HjE
ieff
o
0)~
(
~)~
(
~~))tan(1(
~
~~
*
H
E
JEjjH
HjE
ieff
o
0)~
(
~)~
(
~~~
~~
*
H
E
JJH
HjE
icd
o
EjjJ effcd
~)1(
~
effeff )tan(
Electric Properties of Materials Frequency Behavior (Complex Permittivity)
saseff )(
0)~
(
~)~
(
~~~~
~~
*
H
E
JJJH
HjE
effid
o
EJ
J
EjJ
effeff
i
d
~~
~
~~
Displacement current
Source current
Effective electric conduction current
)1(~~
eff
effd JJ
Good Dielectric
)1(~~
eff
deff JJ
Good Conductor
Wave Equation
0
B
E
JEt
EH
t
HE
Ht
E
t
HE
)(][
t
J
t
E
t
EE
JEt
E
tE
2
2
AAA
2)( Vector Identity
t
J
t
E
t
EEE
2
22)(
t
J
t
E
t
EE
2
221
Time Dependent Homogenous Wave Equation (E-Field)
1
2
22
t
J
t
E
t
EE
Wave EquationSource-Free Time Dependent Homogenous Wave Equation (E-Field)
1
2
22
t
J
t
E
t
EE
0,0 J
Source Free
02
22
t
E
t
EE
Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)
0Lossless
02
22
t
EE
Wave EquationSource-Free Time Dependent Homogenous Wave Equation (H-Field)
1
2
22 J
t
H
t
HH
0,0 J
Source Free 02
22
t
H
t
HH
0,0,0 J
Source Free and Lossless 02
22
t
HH
Wave Equation: Time Harmonic
1
2
22
t
J
t
E
t
EE
0,0 J
Source Free
02
22
t
E
t
EE
0Lossless
02
22
t
EE
Time Domain Frequency Domain
~1~~~~ 22 JEjEE
0,0 J
Source Free
0~~~ 22 EjEE
0Lossless
0~~ 22 EE
“Helmholtz Equation”
General Solution: Point Source
02
22
t
EE
)(1
)(1
),( rtfr
crtfr
ctrE
01
2
22
t
EE
rr
rr
x
y
z
r
(r)
pointsourcerr atrEatrEatrEtrE ˆ),(ˆ),(ˆ),(),(
01
01
2
22
2
22
t
EE
rr
rr
t
EE
rr
rr
Solution:
Inward traveling spherical waveOutward traveling spherical wave
0
General Solution: Point Source
y
z
r
(r)
pointsource
)(1
)(1
),(c
rtf
rcrtf
rctrE
sec
1 mc
Wave speed
In free space:
sec103
1085.8104
11 8
127
mc
oo
Same as the speed of light!
General Solution Case: Time HarmonicRectangular Coordinates
0~~ 22 EE 0
~~ 22 EE
Wave Number
0~
~~~2
2
2
2
2
2
2
Ez
E
y
E
x
E
0
0
0
22
2
2
2
2
2
22
2
2
2
2
2
22
2
2
2
2
2
zzzz
yyyy
xxxx
Ez
E
y
E
x
E
Ez
E
y
E
x
E
Ez
E
y
E
x
E
Separation of Variable Solutions
022
2
2
2
2
2
xxxx E
z
E
y
E
x
E
Assume Solution of the form: )()()(),,( zhygxfzyxEx
0)()()()()()()()()()()()( 2 zhygxfzhygxfzhygxfzhygxf
0)]()()()()()()()()()()()([)()()(
1 2 zhygxfzhygxfzhygxfzhygxfzhygxf
0)(
)(
)(
)(
)(
)( 2
zh
zh
yg
yg
xf
xf
function of x
function of y
function of z
constant
Separation of Variable Solutions
0)(
)(
)(
)(
)(
)( 2
zh
zh
yg
yg
xf
xf
function of x
function of y
function of z
constant
2222
2
2
2
)(
)(
)(
)(
)(
)(
zyx
z
y
x
zh
zh
yg
yg
xf
xf
2222
2
2
2
0)()(
0)()(
0)()(
zyx
z
y
x
zhzh
ygyg
xfxf
Separation of Variable Solutions
2222
2
2
2
0)()(
0)()(
0)()(
zyx
z
y
x
zhzh
ygyg
xfxf
Solutions:
zzjzzj
yyjyyj
xxjxxj
eBeAzh
eBeAyg
eBeAxf
33
22
11
)(
)(
)(
purely real purely imaginary
complex
xxjxxj eBeA 11
Traveling andstanding waves Evanesent waves
xx eBeA 11
Exponentially modulatedtraveling wave
xxjxxxjx eeBeeA 11
xxjxxxjx eeBeeA 11
oror
)sin()cos( 11 xDxC xx
Separation of Variable Solutions: Examples
case a. x, y, z all real (forward traveling waves)
zzjyyjxxjxox eeeEzhygxfzyxE )()()(),,( Plane Waves
case b. xreal (forward traveling wave), x(real standing wave), z imaginary (evanesent wave)
zyy
xxjxox eyDyCeEzhygxfzyxE ))sin()cos(()()()(),,( 11
Surface Waves
z
x
y
Separation of Variable Solutions: Examples
case c. xreal (forward standing wave), x(real standing wave), z real (traveling wave)
))sin()cos())(sin()cos((),,(
))sin()cos())(sin()cos((),,(
))sin()cos())(sin()cos((),,(
1111
1111
1111
yDyCxBxAeEzyxE
yDyCxBxAeEzyxE
yDyCxBxAeEzyxE
yyxxzzj
zoz
yyxxzzj
yoy
yyxxzzj
xox
Guided Waves
Unknown constants A1, B1, C1 , D1, x, y, z
Found by applying boundary conditions anddispersion relation. Namely:
z
x
y
b
hPEC Walls
22222
0),2/,2/(
0),,2/(
0),2/,(
zyx
z
y
x
zhybxE
zybxE
zhyxE
Stay tuned we will solve the complete solution for modes in a rectangular waveguide in a later lecture.