Uniform Plane Waves in
Lossless Media
UCF
Phasor Form Representation
u ( z, t )  u  ( z, t )  u  ( z, t )
 A cos(  t  kz   A )  B cos(  t  kz   B )
 Re (Ae j A e  jkz e j t )  Re( Be j B e jkz e j t )
U   Ae j A e  jkz  U 0 e  jkz (propagating in +z direction)
U   Be j B e jkz  U 0 e jkz
(propagating in -z direction)


 0,
0
x
y
z
Uniform Plane Wave (UPW)
UCF
Definition: For all the points in the plane which is perpendicular
to the direction of propagation, the electric and
magnetic fields are the same (or uniform).
For an electromagnetic UPW propagating in +z direction, we have
E  E 0 e  jkz
H  H 0 e  jkz
If the wave is propagating in +direction, we replace z by .
E  E 0 e  jk 
H  H 0 e  jk 
a  or a k
k  k r  a   ( k a  )  r  k  r

r
0
  r  a
Define k  ka   ka k as wave vector.
In the following write a  as a k .
Del Operator in UPW Solution
UCF
For UPW solution:
E  E 0 e  j k r
H  H 0 e  j k r
k  r  ( k x a x  k y a y  k z a z )  ( xa x  ya y  z a z )
 kx x  k y y  kz z
  jk r
  j(kx xk y ykz z)
(e
)
[e
]   jk x e  jk r
x
x
This means that we can replace  / x by  jk x .
Likewise, we can replace  / y by  jk y and  / z by  jk z .
Since



ax 
ay 
az
z
x
y
  jk x a x  jk y a y  jk z a z   jk

Lossless UPW Equations
UCF
In lossless, isotropic, source free media
  E   jωH
  H  j ω E
   jk
 jk  E   jωH
 jk  H  jωE
E  0
 jk  E  0
H  0
 jk  H  0
k  E  ωH
k  H   ω E
k E  0
k H  0
UCF
(1)
Properties of UPW in Lossless
Isotropic Media (1)
k E  0  k  E
k H  0  k  H
This means that there is no electromagnetic field components
in the direction of propagation. We can define the direction of
propagation as longitudinal direction and the directions that
are perpendicular to the direction of propagation as transverse
directions. For a wave, when there is no longitudinal electric and
magnetic field components or all the field components are in the
transverse directions, it is called “transverse electromagnetic”
(TEM) wave. The UPW in lossless, isotropic media is a TEM
wave.
Properties of UPW in Lossless
Isotropic Media (2)
UCF
E
k  E  ωH
k  H   ω E
(2)
EH
H
k
E, H and k form a right handed system.
(3)
k  k  E  ω k  H  ω ( ω )E  ω 2  E
k  k  E  k ( k  E )  E (k  k )   k 2 E
k E  ω  E
2
2

ω
or k  ω  
vp
k  ω  dispersion relationship
2
2
UCF
(4)
Properties of UPW in Lossless
Isotropic Media (3)
ω
k  E  ωH  a k  E 
H
k
Since k is real in lossless media, E and H have
the same phase.
(5) E  ω  ω  
H
k

ω 
0

Define  


0
as wave impedance.
1
H  a k  E,

r
r
 377

r
r
E   ak  H
UCF
Average Power Density of UPW
1
S av  Re( E  H * )
2
For UPW in lossless, isotropic media
k E
k  E  ωH  H 
ω
1
k E *
S av  Re[ E  (
) ]
2

1
 Re[
2

E
2
2
2
k E  E* (k  E)

or
ak 
H
2
2
ak
]
Example 1
UCF
An electric field of UPW in free space is given as
z
E  800 cos(108 t  ky)a z V/m
Find : (a) k , (b)  , (c)H at point (0.1,1.5,0.4) m and t  8 ns.
Solution :
E  800e  jky a z V/m
x
propagation direction : a y , electric field direction : a z ,
magnetic field direction : a x ,   108 rad/s.

108
(a) k  
 0.333 rad/m
8
c 3 10
2
2
(b)  

 18.84 m
0.333
k
800  jky
800  jky
e ax 
e a x  2.12 e  jky a x V/m
(c) H 
0
377
H ( x, y, z , t )  2.12 cos(108 t  ky)a x
H (0.1, 1.5, 0.4, 8ns)  2.12 cos(108  8 10 9  0.333 1.5)a x  2.03a x V/m
y
Example 2
UCF
The EM fields of UPW in lossless,isotropic and uniform dielectricare given as
E  500 cos(107 t  kz)a x V/m, H  1.1cos(107 t  kz)a y A/m, v p  0.5c
Find : (a) r , (b)  r , (c)k , (d)  , and (e).
z
Solution :
propagation direction : a z , electricfield direction : a x ,
magneticfield direction : a y ,   107 rad/s.
500

 454.54 ,   0 r /  r  377 r /  r
1.1
 r /  r  454.54 / 377  r /  r  1.455 (1)
v p  c / r  r  0.5c 
x
r r  2  r r  4 (2)
From (1) and (2), we have r  2.41 and  r  1.66.


107
2
2

 0.0667 rad/m,  

 94.2 m
k 
8
v p 0.5c 0.5  3 10
k 0.0667
y
Example 3 (1)
UCF
The electric fields of UPW in lossless,isotropic and uniform dielectricis given as
z
E  (5a y  j10a z )e j 2 x V/m, f  50 MHz, r  1
Find : (a) k , (b) , (c)v p , (d)  r , (e) , and
(f) H at origin (0,0,0) and t   / 2.
Solution :
x
(a) k  2 rad/m, propagation direction is  a x , k  2a x rad/m
(b)   2f  2  50 106  3.14 108 rad/s
(c) k 

vp

3.14 108 rad/s
 vp  
 1.57 108 m/s
k
2 rad/m
2
2
c
1  c  1  3 108 m/s 
  3.65
(d) v p 
 r 
 
8


r  v p  1  1.57 10 m/s 
r  r
(e)  0
r
1
 377
 197 
r
3.65
y
Example 3 (2)
UCF
z
(f) E  (5a y )e j 2 x  ( j10a z )e j 2 x V/m
H
5

(a z )e j 2 x 
j10

a y e j 2 x A/m
5
10


H  Re[He jt ]   cos(t  2 x)a z 
at origin (0,0,0) and t 
sin(t  2 x)a y

2
5
10


cos(
2
0
)
sin(




a

 2  0)a y

z
t 
2
2


2
H (0,0,0) |

10

a y  0.051a y .
x
y