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ECE 546 – Jose Schutt‐Aine 1
ECE 546 Lecture 02
Review of ElectromagneticsSpring 2018
Jose E. Schutt-AineElectrical & Computer Engineering
University of Illinoisjesa@illinois.edu
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Printed Circuit Board
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High-Density Package
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Electromagnetic Quantities
E
H
D
B
Electric field (Volts/m)
Magnetic field (Amperes/m)
Electric flux density (Coulombs/m2)
Magnetic flux density (Webers/m2)
J
Current density (Amperes/m2)
Charge density (Coulombs/m2)
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Faraday’s Law of Induction
Ampère’s Law
Gauss’ Law for electric field
Gauss’ Law for magnetic field
BEt
H J D
t
D
0B
Maxwell’s Equations
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B H
D E
Constitutive Relations
Permittivity : Farads/m
Permeability : Henries/m
Free Space128.85 10 /o F m
74 10 /o H m
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Continuity EquationDH Jt
0DH J J Dt t
D
0Jt
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0E
with E
ElectrostaticsAssume no time dependence 0
t
Since 0, such thatE
where is the scalar potentialE
2we get
Poisson’s Equation 2we get
2 0 Laplace’s Equationif no charge is present
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Integral Form of ME
C S
E dl B dSt
C S S
H dl J dS D dSt
S V
D dS dv
0S
B dS
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Boundary Conditions
1 2ˆ 0n E E
1 2ˆ Sn H H J
1 2ˆ Sn D D
1 2ˆ 0n B B
n̂
1 1 1 1, , ,E H D B
2 2 2 2, , ,E H D B
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Faraday’s Law of Induction
Ampère’s Law
Gauss’ Law for electric field
Gauss’ Law for magnetic field
oHEt
oEHt
0E
0B
Free Space Solution
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oE Ht
2o o
EE Et t
Wave Equation
22
2o oEEt
Wave Equation
can show that2
22o oHHt
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Wave Equation2 2 2 2
2 2 2 2o oE E E E
x y z t
separating the components
2 2 2 2
2 2 2 2x x x x
o oE E E Ex y z t
2 2 2 2
2 2 2 2y y y y
o o
E E E Ex y z t
2 2 2 2
2 2 2 2z z z z
o oE E E Ex y z t
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Wave Equation Plane Wave
0x y
(a) Assume that only Ex exists Ey=Ez=0
2 2
2 2x x
o oE Ez t
(b) Only z spatial dependence
This situation leads to the plane wave solution
In addition, assume a time‐harmonic dependencej t
xE e then jt
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Plane Wave Solution
j zx o
j zx o
E E e forward waveE E e backward wave
solution
where
In the time domain
22
2x
o o xE Ez
o o propagation constant
( ) cos( ) cosx o
x o
E t E t z forward traveling waveE t E t z backward traveling wave
solution
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Plane Wave Characteristics
where
In free space
o o propagation constant
1 v propagation velocity
81 3 10 /o o
v c m s
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HE E j Ht
Solution for Magnetic Field
ˆ j zoE xE e
If we assume that
ˆ ˆ ˆ1 1
0 0x
x y z
H Ej j x y z
E
then ˆ j zoEH y e
ˆ j zoE xE e
If we assume that then ˆ j zoEH y e
intrinsic impedance of medium
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( )P t E t H t
Time-Average Poynting Vector
We can show that
Poynting vector W/m2
time‐average Poynting vector W/m2
0 0
1 1( )T T
P P t dt E t H t dtT T
*1 Re2
P E H
where and are the phasors of ( ) and ( ) respectivelyE H E t H t
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: conductivity of material medium (‐1m‐1)
HEt
EH Jt
J E
Material Medium
E j H
H J j E
1H E j E E j j Ej
2 2 1E Ej
1
j
or
since then
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Wave in Material Medium2 2 21E E E
j
is complex propagation constant
2 2 1j
1j jj
associated with attenuation of wave
associated with propagation of wave
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Wave in Material Medium
ˆ ˆz z j zo oE xE e xE e e
1/22
1 12
decaying exponentialSolution:
1/22
1 12
ˆ z j zoEH y e e
jj
Magnetic field
Complex intrinsic impedance
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Wave in Material Medium1/2
22 1 1pv
Phase Velocity:
0
0
1. Perfect dielectric
Special Cases
Wavelength:1/2
22 2 1 1f
air, free space
and
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Wave in Material Medium2. Lossy dielectric
Loss tangent: 1
2
2 2
318 2
j
2
2 218
2
2 212 8 2
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Wave in Material Medium
3. Good conductors
Loss tangent: 1
j j j
f f
1j f jj
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attenuation
propagation
dp H, E Examples
PEC ‐ 0 0 0 supercond
Good conductor finite copper
Poor conductor finite
Ice
Perfectdielectric 0 finite
air
2
2 1f j
1f
2
/
12
j
2
/
Material Medium
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Radiation - Vector Potential
E j H
H J j E
/D
0B
(1)
(2)
(3)
(4)
Assume time harmonicity ~ j te
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Radiation - Vector Potential
0 such thatB A A B
A
: vector potential
0A
0E j A E j A
Using the property: 0any vector
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0vector vector
where is the scalar potentialE j A
Since a vector is uniquely defined by its curl and its divergence, we can choose the divergence of A
choose such thatA
0A j Lorentz
condition
Vector Potential
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B J j E
A J j j A
2 2A A J A j
2 2A j J A j
2 2A A J
D’Alembert’sequation
Vector Potential
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'
, '4 '
j r reG r rr r
'
'
''
4 '
j r r
V
J r eA r dv
r r
From A, get E and H using Maxwell’s equations
Three-dimensional free-space Green’s function
Vector potential
Vector Potential
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ˆ' ( ') ( ') ( ')oJ r zI dl x y z
For infinitesimal antenna, the current density is:
ˆ4
j roI dlA r z er
ˆˆˆ cos sinz r
Calculating the vector potential,
In spherical coordinates,
Vector Potential
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Resolving into components,
ˆˆˆ cos sin4
j roI dlA r z r er
cosˆ4
j ror
I dlA z er
sin4
j roI dlA er
Vector Potential
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Calculate E and H fields
1H A
1 sin 0sinr
AH A
1 1 0sin
rAH rAr r
E and H Fields
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1 rAH rAr r
11 sin4
j roI dlH j er j r
1E Hj
1 1sinsinrE H
r j
E and H Fields
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2
2cos 1 114
j ror
I dlE j er j r j
1 1E rHr r j
2 2
1 1 1sin 14
j roj I dlE j er j r r j
E and H Fields
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21 1sin 1
4j roj I dlE e
r j r j r
2
2cos 1 114
j ror
I dlE j er j r j
11 sin4
j roI dlH j er j r
E and H Fields
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Far field : 1 orr r 2
1then, 0r
, whereE H
0rE
sin4
j roj I dlH er
Note that:
sin4
j roj I dlE er
Far Field Approximation
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• Uniform constant phase locus is a plane• Constant magnitude• Independent of • Does not decay
Characteristics of plane waves
Similarities between infinitesimal antenna far field radiated and plane wave
(a) E and H are perpendicular(b) E and H are related by (c) E is perpendicular to H
Far Field Approximation
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P t E t H t
Time-average Poynting vector or TA power density
1 Re *2
P E H
E and H here are PHASORS
22 2ˆ ˆRe sin
2 2 4oI dlrP H rr
Poynting Vector
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Total power radiated (time-average)
2
0 0P ds
P = 2ˆwith sinds rr d d
22 2 2
0 0sin sin
2 4oI dl
r d dr
P =
23
0
2 sin2 4
oI dl d
P =
Time-Average Power
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243 4
oI dl
P =
Poynting Power DensityDirective Gain =Average Poynting Power Densityover area of sphere with radius r
2Directive Gain =/ 4
P
r
P
Time-Average Power
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22
22
sin2 4
Directive Gain4 / 43 4
o
o
I dlr
I dlr
For infinitesimal antenna,
23Directive Gain sin2
Directivity
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Directivity: gain in direction of maximum value
Radiation resistance:
From 212 oRIP we have: 2
2rad
o
RI
P
For infinitesimal antenna:
2
2 2
2 4 23 4 3
orad
o
I dl dlRI
Directivity
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2280rad
dlR
For free space, 120
The radiation resistance of an antenna is the value of a fictitious resistance that would dissipate an amount of power equal to the radiated power Prwhen the current in the resistance is equal to the maximum current along the antenna
(for Hertzian dipole)
A high radiation resistance is a desirable property for an antenna
Radiation Resistance