Download ELEC 401 – Microwave Electronics Plane Electromagnetic Waves

ELEC 401 – Microwave Electronics
ELEC 401
MICROWAVE ELECTRONICS
Lecture 3
Instructor: M. İrşadi Aksun
Acknowledgements:
1. An art work on the illustration of uniform plane wave was taken from
http://en.wikipedia.org/wiki/Image:Onde_plane_3d.jpg
2. Animation on the visualization of EM waves was taken from the following
web page:
http://web.mit.edu/~sdavies/MacData/afs.course.lockers/8/8.901/2007/TuesdayFeb
20/graphics/
M. I. Aksun
Koç University
1/32
ELEC 401 – Microwave Electronics
Outline
 Chapter 1: Motivation & Introduction
 Chapter 2: Review of EM Wave Theory
 Chapter 3: Plane Electromagnetic Waves
 Chapter 4: Transmission Lines (TL)
 Chapter 5: Microwave Network Characterization
 Chapter 6: Smith Chart & Impedance Matching
 Chapter 7: Passive Microwave Components
M. I. Aksun
Koç University
2/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
 So far, we have only considered waves in the form of planes, that is, the field
components of the wave are traveling in the z direction and have no x and y
dependence.
 To visualize such a wave, it would look like a uniform plane of electric
and magnetic fields on the x-y plane moving in the z direction, and hence
such waves are called uniform plane waves.
M. I. Aksun
Koç University
3/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
The source of a plane wave is supposed to be uniform over an infinite
plane in order to generate uniform fields over a plane parallel to the source
plane.
 There is no actual uniform plane wave in nature.
 However, if one observes an incoming wave far away from a finite extent
source, the constant phase surface of the fields (wavefront) becomes almost
spherical. Hence, the wave looks like a uniform plane wave over a small
area of a gigantic sphere of wavefront, where the observer is actually
located.
M. I. Aksun
Koç University
4/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
- It is rather easy to visualize the plane waves when the fields and
propagation direction coincide with the planes of the Cartesian
coordinate system:
Ex  E0e jk z ; H y  H 0 e  jk z
Constant phase plane
x
E  E0e
H  H0e
k  k n̂
r
r
z

 j k x x k y y k z z


 j k x xk y y kz z

 E0e j kr
 H0e j kr
where E0 and H0 are, in general, complex
constant vectors of the electric and magnetic
fields, respectively, r is the position vector (or
radius vector), and k is a real propagation
vector for the lossless medium
k  k x xˆ  k y yˆ  k z zˆ  k nˆ
y
- Note that the expressions of the uniform plane waves are written
based on the physical interpretation of the visualized waves.
M. I. Aksun
Koç University
5/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
- Since these fields have to satisfy Maxwell’s equations, some
relations connecting the amplitudes and propagation vectors of such
fields may exit.
 j kr
 j kr
E  E0e
H  H 0e
- Substitute the mathematical descriptions of the uniform plane
waves into Maxwell’s equations.
- Let us start with Gauss’s law in a source-free, homogeneous and
anisotropic medium:
  D    E  0    E0 e  j k r  0
 j k r
e  j k  r   E0  
e
 E0  0
 
jk r
0
 jk e
Using the vector identity
  f A  f   A  f  A
k  E0  0
M. I. Aksun
Koç University
6/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
- Additional relation can be obtained by implementing the FaradayMaxwell law as follows:
  E   j H
  E0e  j k r  e  j k r   E0  e  j k r  E0

0
 jke  j k r  E0   j H0 e  j k r
H0 
k

nˆ  E0 
1

nˆ  E0
Wave number: k   
Intrinsic Impedance:    
 Propagation vector k and the electric and magnetic field vectors E0 and H0
are all orthogonal to each other.
k
1
ˆ
H0 
n  E0  nˆ  E0
k  E0  0
M. I. Aksun
Koç University


7/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
 Propagation vector k and the electric and magnetic field vectors E and H are
all orthogonal to each other.
E
H
k
M. I. Aksun
Koç University
8/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
 Example (from “Fundamentals of Applied Electromagnetics” by F. T. Ulaby): An xpolarized electric field of a 1 GHz plane wave travels in the +z-direction in free-space,
and has its maximum at (t=0, z=5 cm) with the value of 1.2p (mV/cm). Using these
information, find the expressions of Electric and Magnetic field in time-domain.
~
Ez, t   xˆE0 cos(t  k0 z  )

2pf
2p
2p
k0   0 0  


c
c
c / f 0
c 3  1010 cm / s
0  
 30cm
9
f
1 10 / s
2p
~
E z , t   xˆ1.2p cos(2p  109 t 
z  )
30

For maximum  n 2p for n 0,1,
 p /3
2p
p
~
E z, t   xˆ1.2p cos(2p 109 t 
z )
30
3
M. I. Aksun
Koç University
9/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
Remember that electric and magnetic fields
in a uniform plane wave are related as
H0 
k

nˆ  E0 
1

nˆ  E0
Substituting the electric field expression
2p
p
~
9
E z, t   xˆ1.2p cos(2p 10 t 
z )
30
3
results in the following magnetic field
expression:
2p
p
~
H z, t    zˆ  xˆ (0.01) cos(2p 109 t 
z )
30
3
Direction of propagation
nˆ  zˆ
Intrinsic
  0 /  0  377   120p
impedance
M. I. Aksun
Koç University
10/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
 Example: Arbitrarily directed uniform plane wave. With E0 and k
being the appropriate constants, which of the following expressions
represents the complex electric field vector of a uniform plane timeharmonic electromagnetic wave propagating in free space along the main
diagonal of the first octant of the Cartesian coordinate system, so that its
direction of propagation makes equal angles with all three coordinate axes?
M. I. Aksun
Koç University
11/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves
 Example: Magnetic field from electric field in time domain. The
electric field of an electromagnetic wave propagating through free space is
given by E = ẑ 100 cos(3 × 108t + x) V/m (t in s ; x in m). The magnetic field
intensity vector of the wave is
M. I. Aksun
Koç University
12/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 So far, it was assumed that the free charge density  f and the free current
density J f are zero.
 Such a restriction is perfectly reasonable when wave propagation is
taking place in vacuum or insulating materials.
 In cases of conducting materials, the flow of charge, and in general J f , is
certainly not zero. Ohm’s law J f   F q   (E  v  B)  E
E
  (  B     E )
t
E 
 f



1 t

 f t    f 0e /  t

 Any initial free charge density dissipates in a characteristic time t/.
That is, free charges on a conductor will flow out to the edges.
 Not interested in transient, and wait for free charge to disappear.
M. I. Aksun
Koç University
13/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 Hence,  f  0 and we have
  E   jB

  B  jE  E
 

  B  j   
E
j 

B
t
E
  B  
 E
t
E  
 2E
E
 E   2  
0
t
t
2

  j
t
  B  j  c E
 2 E   2  c E  0
 Then, a plane wave solution will look like
E  E0e
M. I. Aksun
Koç University
 j  c  nˆ r
H  H 0e
 j  c  nˆ r
14/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 In general,
E  E 0 e 
nˆ r
H  H 0e  nˆr
  j  c     j
  e{ }
Attenuation constant in Neper/m
  m{ }
Propagation constant in rad/m
 2 E   2  c E  0
 2E   2 E  0
M. I. Aksun
Koç University
15/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 As an example, let us consider a uniform plane wave as
E( z )  xˆE0e
z
 xˆE0e
z  jz
e
H( z )  yˆ
z
E( z , t )  xˆ E0e cos(t  z   0 )

E0
c
e z e  jz
Em ( z )
c 




c
j

  c e j
H( z, t )  yˆ
E0
c
e z cos(t  z   0   )
In lossy media, the magnetic field lags the electric field by a phase lag .
M. I. Aksun
Koç University
16/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 Explicit expressions for the basic parameters:
  j  c     j
 2   2 c    2  j   2   2  j 2
1/ 2
2  
2
c  c e j
M. I. Aksun
Koç University
2

 
 
 
 1 
  1
2 
  


c 
1/ 2
2

 
 
;  
 1 
  1
2 
  


 /
1
 
;


arctan
 
1/ 4
2
2
  
   
1    
    
17/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 In many applications, we deal with either good insulators or good
conductors. Therefore, it is usually not necessary to use the exact
expressions of . . and c.
 For good dielectrics:  << 
 

  j  1  j 
 

1/ 2
c 
   

 j  1  j
 j 

2  2 


 /
1
 
;


arctan
 
1/ 4
2
2
  
   
1    
    


c 

 In practice, the losses in dielectrics are defined by the loss tangent of the

material rather than conductivity:
tan  d 
M. I. Aksun
Koç University

18/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 For good conductors:  >> 

  j  1  j 
 

1/ 2
 j  j


  j 
(1  j )

2
  
c 
1
 
;


arctan
 
1/ 4
2
2
  
   
1    
    
M. I. Aksun
Koç University
 /
c 
pf
(1  j )

19/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 Example: Fields and Poynting vector of a wave in a lossy case
The instantaneous magnetic field intensity vector of a wave propagating in a
lossy non-magnetic medium of relative permittivity r=10 is given by
~
H x, t   5e x cos(2.4 108 t  2.83x) zˆ A / m
Determine (a) the attenuation coefficient, (b) the instantaneous electric field
intensity vector, and (c) the time-average Poynting vector of the wave.
1/ 2
a)   2.4  108 rad / s
2



 



1



  1
  2.83 rad / s
2 
  
 

 
u
2 2c 2
 r
2
 1  1.5
u
 
M. I. Aksun
Koç University
 
1/ 2

 
 1 
  1
2 
  


2

u 1
 1.26 Np / m
u 1
20/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
b)
c 
0
1

 97.34;   arctan u 2  1  24.1
u
 ru
2
 /
~
E x, t   486.7 e 1.26 x cos(2.4 108 t  2.83x  24.1 ) yˆ V / m
c)


2
E
1
1
0 e  2x cos x  1.11e  2.52 x x kW / m 2
P  e E  H 
ˆ
ˆ
2
2 c
M. I. Aksun
Koç University
21/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves in Lossy Media
 Example: Shielding effectiveness of an aluminum foil vs. frequency.
In order to prevent the electric and magnetic fields from entering or leaving
a room, the walls of the room are shielded with a 1-mm thick aluminum foil.
The best protection is achieved at a frequency of
(A) 1 kHz.
(B) 10 kHz.
(C) 100 kHz.
(D) 1 MHz.
(E) No difference.
M. I. Aksun
Koç University
 
 
1/ 2

 
 1 
  1
2 
  


2
22/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 The polarization of a plane wave is defined as
- the locus of the tip of the electric field
- as a function of time
- at a given space point
- in the plane perpendicular to the direction of propagation
Ex
E
Ex
M. I. Aksun
Koç University
Ex
Circular
E
y
y
y
Linear
E
Elliptical
23/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Let us assume a uniform plane wave propagating in z-direction:
- no z-polarized E or H field ( E  H  k for uniform plane waves)
- electric field may have both x- and y-components;
~
~
~
E( z, t )  xˆEx ( z, t )  yˆE y ( z, t )
- Assume a sinusoidal time variation
~
E( z, t )  xˆA cos(t  kz  x )  yˆB cos(t  kz   y )
- Since wave polarization depends on the relative position of Ey with respect
to Ex at a constant position in the direction of propagation, it must be
independent of the absolute phases of Ex and Ey
~
E( z , t )  xˆA cos(t  kz)  yˆB cos(t  kz   )
 y  x
M. I. Aksun
Koç University
24/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 The same field is represented in frequency domain (phasor form) as
j  jkz
 jkz


ˆ
ˆ
E( z )  xˆ 
A e jkz  yˆ Be
e

x
E

y
E
e
x0
y0








Ex 0
E
y0
E0
 Question: How can we get the locus of the tip of the electric field as a
function of time at a fixed propagation distance (z=0 is usually chosen for
convenience)?
 Answer:
- write the electric field in time domain
- calculate the magnitude of the field
- calculate the direction of the field
M. I. Aksun
Koç University
E( z, t )  e[E( z )e jt ]
E( z , t )
1  E y ( z , t ) 

 ( z, t )  tan 
 Ex ( z, t ) 
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ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Linear Polarization: A wave is linearly polarized if both components
are in phase, that is 0.
A and B are real
j  jkz
 jkz

ˆ
ˆ

E( z )  xˆ 
A e  jkz  yˆ Be
e

x
A

y
B
e

constants
Ex 0
Ey0
Step 1: Write the time-domain representation
E( z, t )  e[E( z )e jt ]  xˆA cos(t  kz)  yˆB cos(t  kz)
Step 2: Calculate the magnitude of the electric field vector

 A
E( z , t )  A cos (t  kz)  B cos (t  kz)
2
2
2
2 1/ 2
B
2

2

1/ 2
cos(t  kz)
Step 3: Find the direction of the electric field vector
B
 ( z, t )  tan 1   Constant
 A
M. I. Aksun
Koç University
26/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization

 Linear Polarization: E( z, t )  A  B
2

2 1/ 2
B
cos(t  kz); ( z, t )  tan 1 
 A
A=0; B=1
A=ax; B= -ay
M. I. Aksun
Koç University
27/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Circular Polarization: A wave is circularly polarized if both
components of the electric field have the same magnitude (A=B) with
90-degree phase difference (   p / 2 ).
 jp / 2  jkz
 jkz

ˆ
ˆ

E( z )  xˆ 
A e jkz  yˆ 
Ae
e

A
x

j
y
e


Ex 0
Ey0
Step 1: Write the time-domain representation
E( z, t )  e[E( z )e jt ]  xˆA cos(t  kz)  yˆAsin(t  kz)
Step 2: Calculate the magnitude of the electric field vector

E( z, t )  A2 cos2 (t  kz)  A2 sin 2 (t  kz)

1/ 2
A
Step 3: Find the direction of the electric field vector
  A sin(t  kz) 
 ( z , t )  tan 1 
  tan 1  tan(t  kz)   (t  kz)
 A cos(t  kz) 
M. I. Aksun
Koç University
28/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Circular Polarization:
E( z , t )  A;  ( z, t )  (t  kz) for   p/2
A=a; B=a; 900
Left-Hand Circularly Polarized (LHCP)
M. I. Aksun
Koç University
A=a; B=a;  900
Right-Hand Circularly Polarized (RHCP)
29/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Elliptic Polarization: A wave is elliptically polarized if both
components of the electric field have different magnitudes (A B  0)
with arbitrary phase difference   0 .


 j  jkz
E( z )  xˆ 
A e  jkz  yˆ 
Be
 xˆ A  yˆ Be j e  jkz
e
Ex 0
Ex
E
y
Ey0
E( z , t )  e[E( z )e jt ]
 xˆA cos(t  kz )  yˆ B cos(t  kz  )
Elliptical
M. I. Aksun
Koç University
30/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Example: Determination of polarization state of a plane wave.
Determine the type (linear, circular, or elliptical) of the polarization of an
electromagnetic wave whose instantaneous electric field intensity vector is
given by E(x, t) = [2 cos(t + x) ŷ − sin(t + x) ẑ] V/m, where  and 
are the angular frequency and phase coefficient of the wave. The
polarization of the wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information
M. I. Aksun
Koç University
31/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Example: Determination of polarization state of a plane wave.
Determine the type (linear, circular, or elliptical) of the polarization of an
electromagnetic wave whose instantaneous electric field vector is given by
E(x, t) = [2 cos(t + x) ŷ − cos(t + x) ẑ] V/m. The polarization of the
wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information
M. I. Aksun
Koç University
32/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Example: Determination of polarization state of a plane wave.
Determine the type (linear, circular, or elliptical) of the polarization of an
electromagnetic wave whose instantaneous electric field vector is given by
E(x, t) = [ cos(t + x) ŷ − sin(t + x) ẑ] V/m. The polarization of the
wave is
A) Linear
B) Circular
C) Elliptical
D) Need more information
M. I. Aksun
Koç University
33/32
ELEC 401 – Microwave Electronics
Plane Electromagnetic Waves - Polarization
 Example: Polarization handedness for the magnetic field vector.
Polarization handedness (right- or left-handed) of an elliptically polarized
wave determined by considering the magnetic field vector of the wave is
opposite to that obtained by viewing the electric field vector.
A) True
B) False
M. I. Aksun
Koç University
34/32