ENE 428
Microwave
Engineering
Lecture 1 Introduction, Maxwell’s
equations, fields in media, and
boundary conditions
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Syllabus
•Assoc. Prof. Dr. Rardchawadee Silapunt (Ann),
rardchawadee.sil@kmutt.ac.th
•Dr. Ekapon Siwapornsathain (Eric), sie4129@hotmail.com,
Tel: 0814389024
•Lecture: 9:00am-12:00pm Wednesday, AIT
•Instructors at King Mongkut’s University of Technology
Thonburi, BKK, Thailand
•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
• Transmission lines
• Matching networks
• Waveguides
• Two-port networks
• Resonators
• 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) ( = 1 mm – 1 m in free space)
• Microwave components are distributed
components.
• Lumped circuit elements approximations are
invalid.
• Maxwell’s equations are used to explain
circuit behaviors ( H and E )
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Lumped circuit model and
distributed circuit model
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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
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Point forms of Maxwell’s equations
B
E  
H 
t
D
t
M
J
(1)
(2)
  D  v
(3)
B  0
(4)
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The magnetic north
can never be
isolated from the
south.
Magnetic field lines
always form closed
loops.
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Maxwell’s equations in free space
•  = 0, r = 1, r = 1
  H  0
E
t
  E  0
H
t
Ampère’s law
Faraday’s law
0 = 4x10-7 Henrys/m
0 = 8.854x10-12 Farads/m
 = conductivity (1/ohm)
(“constitutive parameters”)
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Integral forms of Maxwell’s equations

 E  d l   t  B  d S
(1)
S
 H dl 

D d S  I

t
(2)
S
 D  d S    dV
S
 B d S
 Q enc
(3)
V
0
(4)
S
Note: To convert from the point forms to the integral forms, we need to apply Stoke’s
Theorem (for (1) and (2)) and Divergence theorem (for (3) and (4)), respectively.
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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 ) co s(  t   ) a x
• Phasor form:
E s  A ( x , y , z )e
j
ax
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Maxwell’s equations in phasor form
  E S   j B  M
(1)
  H S  j D  J
(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 D
dipole moments that complements the total displacement
flux,
D  0 E  Pe
C /m
2
where P e is the electric polarization.
• In the linear medium, it can be shown that
Pe  0e E.
• Then we can write
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D   0 (1   e ) E   0  r E   E .
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Fields in dielectric media (2)
•  may be complex then  can be complex and can be
e
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
 D x    xx

 
D y    yx


 D z    zx

 xy
 yy
 zy
 xz   E x 
 Ex 
 
 
 yz  E y     E y .
 
 
 E z 
 zz   E z 
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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  P m )
Wb / m
2
where P m is the magnetic polarization or magnetization.
• 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
 B x    xx
  
B y    yx
 
 B z    zx

 xy
 yy
 zy
 xz   H x 
H x 




 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
n H
2
S

 H1  JS
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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,
H  E 
  E  
H
t
E
t
(1)
(2)
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General plane wave equations (2)
Take curl of (2), we yield
    E  
    E  
From
then
 (  H )
t
E
2
)

E

E
 t   
 
2
t
t
t
 ( E  
 A  A A
2
  E   E   
2
E
t
 E
2
 
t
2
For charge free medium
E  0
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Helmholtz wave equation
For electric field
For magnetic field
 E  
2
E
t
 H  
2
 E
2
 
H
t
t
2
 H
2
 
t
2
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Time-harmonic wave equations
• Transformation from time to frequency domain

t
 j
Therefore
 E s  j  (  j  ) E s
2
 E s  j  (  j  ) E s  0
2
 Es  Es  0
2
2
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Time-harmonic wave equations
or
 H s  H
2
where
 
2
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  E 0 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
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 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
H s  H 0e
e
 z
 j z
e
 j z
ax
ay
• Conversion between instantaneous and phasor form
Instantaneous field = Re(ejtphasor field)
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Intrinsic impedance
• For any medium,
 
Ex
Hy

j 
  j 
• For free space
 
Ex
Hy

E0
H0

0
 120 
0

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Propagating fields relation
Hs 
1

a  Es
E s   a   H s
where a  represents a direction of propagation
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