Download Plane Electromagnetic Wave

Plane Electromagnetic Wave
The Helmhotz Equation:
In source free linear isotropic medium, Maxwell equations in phasor form are,
  E   j H
  H  j E
 E  
 H  0
Therefore,     E      E    2 E = -j  H
Or,
2 E   j ( j E )
Or,
2 E   2  E  0
Or,
2 E  k 2 E  0
where k   k
An identical equation can be derived for H.
i.e.,  2 H  k 2 H  0
These equations
 2 E  k 2 E  0................( a) 
 ........(1)
&  2 H  k 2 H  0................(b) 
are called homogeneous vector Helmholtz’s equation.
k    is called the wave number or propagation constant of the medium.
Plane waves in Lossless medium:
In a lossless medium,  and  are real numbers so k is real.
In Cartesian coordinates each of the equations 1 (a) and 1(b) are equivalent to three scalar
Helmholtz’s equations, one each in the components Ex , E y & Ez or H x , H y & H z .
For example if we consider Ex component we can write
 2 Ex  2 Ex  2 Ex
 2  2  k 2 Ex  0 ……………………………………………………… (2)
2
x
y
z
A uniform plane wave is a particular solution of Maxwell’s equation assuming electric
field (and magnetic field) has same magnitude and phase in infinite planes perpendicular
to the direction of propagation. It may be noted that in the strict sense a uniform plane
wave doesn’t exist in practice as creation of such waves are possible with sources of
infinite extent. However, at large distances from the source, the wavefront or the surface
of the constant phase becomes almost spherical and a small portion of this large sphere
can be considered to plane. The characteristics of plane waves are simple and useful for
studying many practical scenarios.
Let us consider a plane wave which has only Ex component and propagating along z.
Since the plane wave will have no variation along the plane perpendicular to z i.e., xy
E
E
plane, x  x  0 . The Helmholtz’s equation (2) reduces to,
x
y
d 2 Ex
 k 2 Ex  0 ………………… (3)
dz 2
The solution to this equation can be written as
Ex ( z )  E x  ( z )  E x  ( z )
 E0  e jkz  E0 e jkz
…………………….. (4)
E0 & E0 are the amplitude constants (can be determined from boundary conditions).
In the time domain,  X ( z, t )  Re( Ex ( z )e jwt )
 X ( z, t )  E0 cos t  kz   E0 cos t  kz  ……………… (5)
assuming E0 & E0 are real constants.
Here,  X  ( z , t )  E0  cos( t   z ) represents the forward traveling wave. The plot of
 X  ( z , t ) for several values of t is shown in the figure -1.
Figure 1: Plane wave traveling in the +z direction
As can be seen from the figure, at successive times, the wave travels in the +z direction.
If we fix our attention on a particular point or phase on the wave (as shown by the dot)
i.e. ,
t  kz =constant
Then we see that as t is increased to t  t , z also should increase to z  z so that
 (t  t )  k ( z  z )  constant  t   z
Or, t  k z
z 

Or,
t k
When t  0
z dz
 = phase velocity vP .
t  0  t
dt
We write lim
 vP 

k
………………………….. (6)
If the medium in which the wave is propagating is free space i.e.,   t0 ,   0

1
Then vP 

C
 0 0
0 0
Where ‘C’ is the speed of light. That is plane EM wave travels in free space with this
speed of light.
The wavelength  is defined as the distance between two successive maxima (or minima
or any other reference points).
i.e., t  kz   t  k  z      2
or, k  2
2
or,  
k
Substituting k 

vP
2 vP vP

2 f
f
or,  f  vP ……………………………..… (7)

Thus wavelength  also represents the distance covered in one oscillation of the wave.
Similarly,    z, t   E0 cos t  kz  represents a plane wave traveling in the –z
direction.
The associated magnetic field can be found as follows:
From (A),
Ex  ( z )  E0  e  jkz
1
H 
 E
j
ax
a4

=
k
1
j
0
0
E0  e jkz
0
E  e  jkz a
a3

z
0
4
 0
E
= 0 e jkz a4  H 0  e jkz a4

where  




is the intrinsic impedance of the medium.


 
k
When the wave travels in free space
0 

0
 120  377 is the intrinsic impedance of the free space.
0
H ( z, t )  a y
E0

cos t   z  ………………….. (9)
Which represents the magnetic field of the wave traveling in the +z direction.
For the negative traveling wave,

H ( z, t )  a4
E0 

cos t   z  ………………(10)
For the plane waves described, both the E & H fields are perpendicular to the direction of
propagation, and these waves are called TEM (transverse electromagnetic) waves.
Fig 2: E & H fields of a perpendicular plane wave at time t.
The E & H field components of a TEM wave is shown in Fig-2.
TEM Waves:
So far we have considered a plane electromagnetic wave propagating in the z-direction.
Let us now consider the propagation of a uniform plane wave in any arbitrary direction
that doesn’t necessarily coincides axis.
For a uniform plane wave propagating in z-direction
E ( z )  E0 e  jkz , E0 is a constant vector………. (11)
The more general form of the above equation is
E  x, y,z   E0e
 jk x x  jk y y  jk z z
…………………... (12)
This equation satisfies Helmholtz’s equation  2 E  k 2 E  0 provided,
kx 2  k y 2  kz 2  k 2   2  …………………… (13)




We define wave number vector k  ax kx  ay k y  az kz  k ax ……………..(14)
And radius vector from the origin



r  ax x  ay y  az z ………………………… (15)
Therefore we can write

E (r )  E0e jk .r  E0e jk ax .r …………………. (16)

Here a x .r =constant is a plane of constant phase and uniform amplitude just in the case
of E ( z )  E0 e jkz , z = constant denotes a plane of constant phase and uniform amplitude.
If the region under consideration is charge free,
.E  0 , therefore
.  E0 e jk .r   0 .
Using the vector identity .  fA  A.f  f . A and noting that E0 is constant we can
write,

E0 .  e jk ax .r   0


         j  k x  k y  k z  
or , E0 .  ax  a y  az  e x y z   0
y
z 
 x




or , E0 .   jk ax e  jk ax .r   0




E . an  0 ……………………… (17)
i.e., E0 is transverse to the direction of the propagation.
The corresponding magnetic field can be computed as follows:
H r   
1
j
 E  r   
1
j
  E0e jk .r 
Using the vector identity,
  A   A    A
Since E0 is constant we can write,
H (r )  


H (r ) 
1
j


1 
 jk an . r 

jk
a

E
e
n
0

j 

k

1

e  jk .r  E0
an  E  r 

an  E  r  …...……...……… (18)
Where  is the intrinsic impedance of the medium. We observe that H ( r ) is

perpendicular to both an and E  r  . Thus the electromagnetic wave represented by
E ( r ) and H ( r ) is a TEM wave.
Plane waves in a lossy medium:
In a lossy medium, the EM wave looses power as it propagates. Such a medium is
conducting with conductivity  and we can write:
  H  J  j E    j  E

 
 j   
E
j 

 j c E
Where  c    j

  ' j '' is called the complex permittivity.

We have already discussed how an external electric field can polarize a dielectric and
give rise to bound charges. When the external electric field is time varying, the
polarization vector will vary with the same frequency as that of the applied field. As the
frequency of the applied filed increases, the inertia of the charge particles tend to prevent
the particle displacement keeping pace with the applied field changes. This results in
frictional damping mechanism causing power loss.
In addition, if the material has an appreciable amount of free charges, there will be ohmic
losses. IT is customary to include the effect of damping and ohmic losses in the
imaginary part of  c . An equivalent conductivity    '' represents all losses.
The ratio
 ''
is called loss tangent as this quantity is a measure of the power loss.
'
With reference to the Fig,
J
  ''
tan   c 

.......................... (20)
J d   '
Where J c is the conduction current density and J a is displacement current density. The
loss tangent gives a measure of how much lossy is the medium under consideration. For a
good dielectric medium     tan  is very small and the medium is a good
conductor at low frequencies but behave as lossy dielectric at higher frequencies.
For a source free lossy medium we can write
 H    j  E
 E   j H
.H  0
 ........................... (21)
.E  0 
    E    .E    2 E   j  H = - j   j  E
or,  2 E  r 2 E  0
Where r 2  j   j 
.................... (22)
Proceeding in the same manner we can write,
2 H   2 H  0
1/ 2

 
    i  j   j   j  1 

j 

is called the propagation constant.
The real and imaginary parts  and  of the propagation constant  can be computed
as follows:
 2    i   j   j 
2
or ,  2   2   2 

And  
2
  
2
  
   
2



4
2 2
or , 4  4     2  2 2
2
2
or , 4 4  4 2 2    4  2 2   2  2 2   4  2 2
2

2 
or ,  2 2   2     4  2 2 1  2 2 
  
 
2

 

 1 


1
2 
 




or ,   
Similarly,   
 
2

 
 1  


1
2 
 




 23(a )







 23(b)
Let us now consider a plane wave that has only x-component of electric field and
propagate along z.

 Ex ( z )   E0  e z  E0 e z  ax ....... (24)
Considering only the forward traveling wave

  z, t   Re  E0  e z e jt  ax

 E0  e z cos t   z  ax ................................... (25)
Similarly, from H  
H ( z, t ) 
E0

Where  
H 
1
j
 E , we can find

e z cos t   z  a4 .................................. (26)
j
 n e jn
  j

E0  z
e cos t   z  n  a4 ................................. (27)
n
From (25) and (26) we find that as the wave propagates along z, it decreases in amplitude
by a factor e z . Therefore  is known as attenuation constant. Further E and H are out
of phase by an angle  n .

 1 i.e.,  ''   ' .

Using the above condition approximate expression for  and  can be obtained as
follows:
For low loss dielectric,
 '' 

    i   j  ' 1  j 
 '

1/ 2
2

1  '' 1   ''  
 j  ' 1  j
   
2  ' 8   '  

 '' 
2 '



 .........(28)


 1   '' 2 
&     ' 1    
 8   '  

 '' 
1  j 
'
'

1/ 2

 '' 
1  j
 ................................ (29)
'
2 ' 
& phase velocity
vp 

1  1   '' 

1   

 '  8   ' 
For good conductors
2

 ............... (30)


 1


 
  j  1 
  j 
j 

=

j
1 j
 ............................ (31)
2
We have used the relation
1/ 2
1
j   e j / 2   e j / 4 
1  j 
2
From (31) we can write
  i   f   j  f 
     f  ................................................. (32)
j



 
j 1 
j 

 j

 
j

 (1  j )
 (1  j )
 f


................................ (33)

And phase velocity
vp 

2

........................................ (34)

