Transcript ENE 429 Antenna and Transmission Lines
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ENE 428 Microwave Engineering
Lecture 1
Introduction, Maxwell’s equations, fields in media, and boundary conditions 1
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Syllabus
•Asst. Prof. Dr. Rardchawadee Silapunt, [email protected]
•Lecture: 9:30pm-12:20pm Tuesday, CB41004 12:30pm-3:20pm Wednesday, CB41002 •Office hours : By appointment •Textbook: Microwave Engineering by David M. Pozar (3 rd edition Wiley, 2005) • Recommended additional textbook: Applied Electromagnetics by Stuart M.Wentworth (2 nd edition Wiley, 2007) 2
Grading
Homework 10% Quiz 10% Midterm exam 40% Final exam 40%
Vision
Providing opportunities for intellectual growth in the context of an engineering discipline for the attainment of professional competence, and for the development of a sense of the social context of technology.
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Course overview
• Maxwell’s equations and boundary conditions for electromagnetic fields • Uniform plane wave propagation • Waveguides • Antennas • Microwave communication systems RS 10-11/06/51 4
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Introduction
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52 • Microwave frequency range (300 MHz – 300 GHz) • Microwave components are distributed components.
• Lumped circuit elements approximations are invalid.
• Maxwell’s equations are used to explain
H
and ) 5
Introduction (2)
• From Maxwell’s equations, if the electric field
E
is changing with time, then the magnetic field
H
varies spatially in a direction normal to its orientation direction • Knowledge of fields in media and boundary conditions allows useful applications of material properties to microwave components • A
uniform plane wave
, both electric and magnetic fields lie in the
transverse plane
, the plane whose normal is the direction of propagation 6 RS
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Maxwell’s equations
H
D
v B
0
t B
D
t M
(1) (2) (3) (4) 7
Maxwell’s equations in free space • = 0,
r
= 1,
r
= 1
H
0 0
E
t
H
t
Amp ère’s law Faraday’s law
0
0
= 4 x10 -7 Henrys/m = 8.854x10
-12 farad/m 8 RS
RS Integral forms of Maxwell’s equations
C
S
C
S
t
S
t
S
V
dv
Q
I
0 (1) (2) (3) (4) 9
Fields are assumed to be sinusoidal or harmonic, and time dependence with steady-state conditions • Time dependence form:
E
)
a x
• Phasor form:
E s
j
A x y z e a x
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Maxwell’s equations in phasor form
E S
H S
D
v B
0
M
(1) (2) (3) (4) 11 RS
Fields in dielectric media (1)
atoms or molecules of the material to create electric flux,
D
0
E
P e
/ 2
P e
where is the electric polarization. • In the linear medium, it can be shown that
P e
0
e E
.
D
0 (1
e
)
E
0
r E
E
.
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Fields in dielectric media (2)
• may be complex then
e
expressed as can be complex and can be
j
'' • Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. • The loss of dielectric material may be considered as an equivalent conductor loss if the material has a conductivity . Loss tangent is defined as tan '' ' .
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Anisotropic dielectrics
• The most general linear relation of anisotropic dielectrics can be expressed in the form of a tensor which can be written in matrix form as
D x D y D z
xx yx zx
xy
yy
zy
xz
yz
zz
E E
E E
.
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Analogous situations for magnetic media (1) polarization of by aligned magnetic dipole moments
B
0 (
H
P m
) / 2
P
• In the linear medium, it can be shown that
P m
m H
.
• Then we can write
B
0 (1
m
)
H
0
r H
H
.
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Analogous situations for magnetic • media (2) may be complex then
m
expressed as
j
can be complex and can be '' • Imaginary part is counted for loss in the medium due to damping of the vibrating dipole moments. 16 RS
Anisotropic magnetic material
• The most general linear relation of anisotropic material can be expressed in the form of a tensor which can be written in matrix form as
z
xx
yx
zx
xy
yy
zy
xz
yz
zz
H H H x y z
H H H x y z
.
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Boundary conditions between two media
B n2 H t2 D n2 E t2 n
2
D
1
S
2 1
B n1 H t1 D n1 E t1
E
2
n
H E
1 2
H
1
M S J S
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Fields at a dielectric interface
• Boundary conditions at an interface between two lossless dielectric materials with no charge or current densities can be shown as 2 1 2 1 1 1 2 2 .
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Fields at the interface with a perfect conductor • Boundary conditions at the interface between a dielectric with the perfect conductor can be shown as 0 0 0.
M S
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General plane wave equations (1) • Consider medium free of charge • For linear, isotropic, homogeneous, and time invariant medium, assuming no free magnetic current,
E
t E
H t
(1) (2) 21 RS
General plane wave equations (2) Take curl of (2), we yield (
E
t
t H
)
E t
)
E
t
2
E
t
2 From
A A
2
A
then 2
E
E
t
2
E
t
2 For charge free medium 0 RS 22
Helmholtz wave equation
For electric field 2
E
E
t
2
E
t
2 For magnetic field 2
H
H
t
2
H
t
2 RS 23
Time-harmonic wave equations
• Transformation from time to frequency domain
t
j
Therefore 2
E s
j
j
)
E s
2
E s
j
j
)
E s
0 2
E s
2
E s
0 24 RS
Time-harmonic wave equations
or 2
H s
2
H s
0 where
j
j
) This term is called
propagation constant
or we can write
=
+j
RS where = attenuation constant (Np/m)
=
phase constant (rad/m) 25
Solutions of Helmholtz equations • Assuming the electric field is in x-direction and the wave is propagating in z- direction • The instantaneous form of the solutions
E
E e
0
z
cos(
x
E e
0
z
cos(
x
• Consider only the forward-propagating wave, we have
E
E e
0
z
cos(
x
• Use Maxwell’s equation, we get
H
H e
z
cos( RS
y
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Solutions of Helmholtz equations in phasor form • Showing the forward-propagating fields without time harmonic terms.
E s
z e
a x H s
z e
a y
• Conversion between instantaneous and phasor form Instantaneous field = Re(e j t phasor field) RS 27
Intrinsic impedance
• For any medium,
E x H y
j
j
• For free space
E x H y
E
0
H
0 0 0 120 RS 28
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Propagating fields relation
H s
1
a
E s E s
a
H s a
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