ENE 428
Microwave
Engineering
Lecture 1 Introduction, Maxwell’s
equations, fields in media, and
boundary conditions
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Syllabus
•Asst. Prof. Dr. Rardchawadee Silapunt,
rardchawadee.sil@kmutt.ac.th
•Lecture: 9:30pm-12:20pm Tuesday, CB41004
12:30pm-3:20pm Wednesday, CB41002
•Office hours : By appointment
•Textbook: Microwave Engineering by David M. Pozar (3rd
edition Wiley, 2005)
• Recommended additional textbook: Applied
Electromagnetics by Stuart M.Wentworth (2nd edition Wiley,
2007)
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Grading
Homework
Quiz
Midterm exam
Final exam
10%
10%
40%
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
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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
circuit behaviors
( Hand )
E
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Introduction (2)
• From Maxwell’s equations, if the electric field E
is changing with time, then the magnetic fieldH
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
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Maxwell’s equations
B
 E  
M
t
D
 H  J 
t
 D  v
 B0
(1)
(2)
(3)
(4)
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Maxwell’s equations in free space
•  = 0, r = 1, r = 1
E
  H  0
t
H
  E   0
t
Ampère’s law
Faraday’s law
0 = 4x10-7 Henrys/m
0 = 8.854x10-12 farad/m
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Integral forms of Maxwell’s equations

 E  dl   t  B  d S
C
S
(1)

 H  dl  t  D  d S  I
C
S
(2)
 D  d s    dv  Q
(3)
S
V
 Bds  0
(4)
S
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Fields are assumed to be sinusoidal or
harmonic, and time dependence with
steady-state conditions
• Time dependence form:
E  A( x, y, z)cos(t   )a x
• Phasor form:
j
E s  A( x, y, z )e a x
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Maxwell’s equations in phasor form
  E S   j B  M
(1)
  H S  J  j D
(2)
 D  v
(3)
 B0
(4)
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Fields in dielectric media (1)
• An applied electric field E causes the polarization of the
atoms or molecules of the material to create electric
dipole moments that complements the total displacement
D
flux,
D   0 E  Pe
C / m2
Pe
where
is the electric polarization.
• In the linear medium, it can be shown that
Pe   0 e E.
• Then we can write
D   0 (1  e ) E   0 r E   E.
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Fields in dielectric media (2)
•  e may be complex then  can be complex and can be
expressed as
   ' 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
 Dx   xx
 D   
 y   yx
 Dz   zx
 xy  xz   Ex 
 Ex 
 
 yy  yz   E y      E y  .
 Ez 
 zy  zz   Ez 
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Analogous situations for magnetic
media (1)
• An applied magnetic field H causes the magnetic
polarization of by aligned magnetic dipole moments
B  0 ( H  Pm )
Wb / m2
where P m is the electric polarization.
• In the linear medium, it can be shown that
Pm  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)
•  m may be complex then  can be complex and can be
expressed as
   ' j  ''
• Imaginary part is counted for loss in the medium due to
damping of the vibrating dipole moments.
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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
 Bx    xx
B   
 y   yx
 Bz    zx
 xy
 yy
 zy
 xz   H x 
Hx 
 
 yz   H y       H y  .
 H z 
 zz   H z 
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Boundary conditions between two
media
n
Dn2
Bn2
Ht2
Et1
Ht1
Bn1
Et2
Dn1


n  D 2  D1   S
n  B 2  n  B1
E
2


 E1  n  M S

n H 2  H1  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
n  D 2  n  D1
n  B 2  n  B1
n  E1  n  E 2
n  H 1  n  H 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
nD  0
nB  0
n  E  M S
n  H  0.
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General plane wave equations (1)
• Consider medium free of charge
• For linear, isotropic, homogeneous, and timeinvariant medium, assuming no free magnetic
current,
E
 H   E  
t
(1)
H
 E   
t
(2)
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General plane wave equations (2)
Take curl of (2), we yield
( H )
 E  
t
E
2
 ( E  
)

E

E
t   
    E  
  2
t
t
t
From
then
 A   A 2 A
2

E

E
 E   2 E   
  2
t
t
For charge free medium
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 E 0
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Helmholtz wave equation
For electric field
For magnetic field
2

E

E
2
 E  
  2
t
t
2

H

H
2
 H  
  2
t
t
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Time-harmonic wave equations
• Transformation from time to frequency domain

 j
t
Therefore
 2 E s  j (  j ) E s
 2 E s  j (  j ) E s  0
2 E s   2 E s  0
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Time-harmonic wave equations
or
where
2 H s   2 H s  0
  j (  j )
This  term is called propagation constant or we can write
 = +j
where  = attenuation constant (Np/m)
 = phase constant (rad/m)
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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  E0 e z cos(t   z )a x  E0e z cos(t   z )a x
• Consider only the forward-propagating wave, we have
E  E0e z cos(t   z )a x
• Use Maxwell’s equation, we get
H  H 0e z cos(t   z )a y
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Solutions of Helmholtz equations in phasor
form
• Showing the forward-propagating fields without timeharmonic terms.
E s  E0e z e j z a x
H s  H 0e z e j z a y
• Conversion between instantaneous and phasor form
Instantaneous field = Re(ejtphasor field)
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Intrinsic impedance
• For any medium,
Ex
j


Hy
  j
• For free space
Ex
E0



H y H0
0
 120
0

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Propagating fields relation
1
H s  a  E s

E s   a   H s
where a  represents a direction of propagation
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