ENE 429
Antenna and
Transmission lines
Theory
Lecture 1 Uniform plane waves
Syllabus
Dr.
Rardchawadee Silapunt,
rardchawadee.sil@kmutt.ac.th
Lecture: 1:30pm-4:20pm Monday, CB41002
Office hours :By appointment
Textbook: Applied Electromagnetics by Stuart M.
Wentworth (Wiley, 2007)
Grading

Homework
20%
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.
Course overview
Uniform plane waves
 Transmission lines
 Waveguides
 Antennas

Introduction
http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52

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

A uniform plane wave, both electric and magnetic fields
lie in the transverse plane, the plane whose normal is the
direction of propagation

Both fields are of constant magnitude in the transverse
plane, such a wave is sometimes called a transverse
electromagnetic (TEM) wave.
Maxwell’s equations
D
 H  J 
t
B
 E  
t
 D  v
(2)
 B0
(4)
(1)
(3)
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
General wave equations
Consider medium free of charge where
 For linear, isotropic, homogeneous, and
time-invariant medium,

E
 H   E  
t
(1)
H
 E   
t
(2)
General wave equations
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
 E 0
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
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
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)
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
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)
Intrinsic impedance

For any medium,
Ex
j


Hy
  j

For free space
Ex
E0



H y H0
0
 120
0

Propagating fields relation
Hs 
1

a  E s
E s   a   H s
where a  represents a direction of propagation
Propagation in lossless-charge free
media

Attenuation constant  = 0, conductivity  = 0

Propagation constant


1
Propagation velocity u p  


   

for free space up = 3108 m/s (speed of light)

for non-magnetic lossless dielectric (r = 1),
c
up 
r
Propagation in lossless-charge free
media

intrinsic impedance




Wavelength

2

Ex1 A 9.375 GHz uniform plane wave is
propagating in polyethelene (r = 2.26). If the
amplitude of the electric field intensity is 500 V/m
and the material is assumed to be lossless, find
a) phase constant
b) wavelength in the polyethelene
c) propagation velocity
d) Intrinsic impedance
e) Amplitude of the magnetic field intensity
Propagation in dielectrics

Cause
 finite conductivity
 polarization loss ( = ’-j” )

Assume homogeneous and isotropic medium
  H   E  j ( '  j " ) E
  H  [(   " )  j ' ]E
Propagation in dielectrics
Define
 eff     "
From
 2  j (  j )
and
 2  (  j  )2
Propagation in dielectrics
We can derive
 
 
and

 
( 1 
 1)

2
  

2
 
( 1 
 1)

2
  
2

1

.
 1  j (  )
Loss tangent

A standard measure of lossiness, used to
classify a material as a good dielectric or a good
conductor
   "  eff
tan  

'

 '
Low loss material or a good
dielectric (tan « 1)

 If

1
or < 0.1 , consider the material
‘low loss’ , then


2


   
and



(1  j
).

2
Low loss material or a good
dielectric (tan « 1)

propagation velocity
up 


1



wavelength

2


1
f 
High loss material or a good
conductor (tan » 1)

 In this case

1 or
> 10, we can
approximate

 
1 
 1) 


  
2
therefore
  
and


j

2
  f 
 j 45

e .

High loss material or a good
conductor (tan » 1)

depth of penetration or skin depth,  is a distance where
the field decreases to e-1 or 0.368 times of the initial field
1
1 1

 
 f   

propagation velocity

u p   


wavelength

2

 2
m
Ex2 Given a nonmagnetic material having r
= 3.2 and  = 1.510-4 S/m, at f = 3 MHz,
find
a) loss tangent 
b) attenuation constant 
c) phase constant 
d) intrinsic impedance
Ex3 Calculate the followings for the wave
with the frequency f = 60 Hz propagating in
a copper with the conductivity,  = 5.8107
S/m:
a) wavelength
b) propagation velocity
c) compare these answers with the same
wave propagating in a free space