Transcript chap10_propagation-reflection-of-plane

CHAPTER 10
PROPAGATION & REFLECTION
OF PLANE WAVES
10.0 PROPAGATION & REFLECTION OF PLANE WAVES
10.1 ELECTRIC AND MAGNETIC FIELDS FOR PLANE WAVE
10.2 PLANE WAVE IN LOSSY DIELECTRICS – IMPERFECT DIELECTRICS
10.3 PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS
10.4 PLANE WAVE IN FREE SPACE
10.5 PLANE WAVE IN CONDUCTORS
10.6 POWER AND THE POYNTING VECTOR
1
10.0 PROPAGATION & REFLECTION OF
PLANE WAVES
Will discuss the effect of propagation of EM wave in four medium :
Free space ; Lossy dielectric ; Lossless dielectric (perfect
dielectric) and Conducting media.
Also will be discussed the phenomena of reflections at interface between
different media.
Ex : EM wave is radio wave, TV signal, radar radiation and optical wave in
optical fiber.
Three basics characteristics of EM wave :
- travel at high velocity
- travel following EM wave characteristics
- travel outward from the source
These propagation phenomena for a type traveling wave called
plane wave can be explained or derived by Maxwell’s equations.
10.1 ELECTRIC AND MAGNETIC FIELDS FOR
PLANE WAVE
From Maxwell’s equations :
∂
B
∂
H
∇ E   -
∂
t
∂
t
∂
D
∂
E
∇ H  J 
 J 
∂
t
∂
t
∇ D   v
∇ B  0
Assume the medium is free of charge :
∂
H
∂
t
∂
E
∇ H  
∂
t
∇ D  0
∇ E  - 
∇ B  0
From vector identity and taking
the curl of (1)and substituting
(1) and (2)
∇ (∇ E )  ∇(∇ E ) -∇2 E
where ∇(∇ E )  0
 v  0, J  0
(1)
( 2)
(3)
( 4)

∂
H
  -∇2 E
 ∇  - 
∂
t 

∂
→ ∇2 E   (∇ H )
∂
t
∂2 E
2
∴ ∇ E  
∂
t2
ie Helmholtz ' s equation for electric field
∂2 E
2
3
∇ E  
Vm
∂
t2
In Cartesian coordinates :
 2 E ∂2 E ∂2 E
∂2 E
 2  2   2
2
x
∂
y
∂
z
∂
t
Assume that :
(i) Electric field only has x component
(ii) Propagate in the z direction
2
2
∂
Ex
∂
Ex
 
2
∂
z
∂
t2
Similarly in the same way, from
vector identity and taking the
curl of (2)and substituting (1)
and (2)
∂2 H
3
∇ H  
A
m
∂
t2
2
2
2
∂
Ex
∂
Ex
 
2
∂
z
∂
t2
The solution for this equation :
E x  E x cos(t - z )  E x- cos(t  z )
Incidence wave propagate
in +z direction
To find H field :
Reflected wave propagate
in -z direction
∂
H
∇ E  -
∂
t
∂
Ex
∂
Ex
∇ E 
yˆ zˆ
∂
z
∂
y


 E x sin( t - z ) - E x- sin( t  z ) yˆ
On the right side
equation :
∂
Hy
∂
∂
H
Hx
∂
Hz
-
 - 
xˆ 
yˆ 
∂
t
t
∂
t
∂
t
 ∂

zˆ 

Equating components on both side = y component
-
∂
Hy
∂
t


  E x sin( t -  z ) - E x- sin( t   z )
E x
 E x- H y  ∫ sin( t -  z )dt - 
sin( t   z ) dt


 
 E x cos(t -  z ) 
E x cos(t   z )


 

Hy 
E x cos(t -  z ) E x - cos(t   z )


 H y cos(t -  z ) - H y- cos(t   z )
Hence :
E x  E x cos(t - z )  E x- cos(t  z )
H y  H y cos(t - z) - H y- cos(t  z)
These equations of EM wave are called PLANE WAVE.
Main characteristics of EM wave :
(i) Electric field and magnetic field always perpendicular.
(ii) NO electric or magnetic fields component in the direction of
propagation.
(iii) E  H will provides information on the direction of
propagation.
10.2 PLANE WAVE IN LOSSY DIELECTRICS –
IMPERFECT DIELECTRICS
  0 ;   0 r ;    0 r
Assume a media is charged free , ρv =0
∂
D
∇ H  J 
   j E
∂
t
∂
B
(2)
∇ E   - jH
∂
t
Taking the curl of (2) :
(1)
∇∇ E  - j ∇ H 
From vector
identity :
∇∇ A  ∇∇ A -∇2 A
∇∇ E  -∇2 E  - j   j E
∇2 E - j   j E  0
∇2 E -  2 E  0
Where :
∇∇ E  - j ∇ H 
Equating (4) and (5) for Re
and Im parts :
  j   j 
 2 -  2  - 2  (Re) (6)
2
 -   j
  propagatio n constant
2
Define :
∂
D
   j E (1)
∂
t
∂
B
∇ E   - jH
(2)
∂
t
∇ H  J 
(4)
    j
 2   2 -  2   2 j
(5)
2   (Im) (7)
Magnitude for (5) ;
   
2
2
2
Add (10) and (6) :
(8)
Hence :
2 2    2   2 -  2 
Magnitude for (4) ;
 
2
- 
2
    
2
   2   2
2
(9)
Equate (8) and (9) :
 2   2    2   2
 2 -  2  - 2  (Re) (6)
(10)
2

  2  1  2 2 -  2 

2


2 
2


2
 1  2 2 - 1




 
2


2
 1  2 2 - 1 Np / m (11)



 is known as attenuation constant as a
measure of the wave is attenuated while
traveling in a medium.
Substract (10) and (6) :
2 2    2   2   2 
 
 

2
 1  2 2  1
2 
 

rad / m (12)
 is phase constant
If the electric field propagate in +z direction and has component x,
the equation of the wave is given by :
E ( z , t )  E0 e
-z
cost -  z xˆ
(13)
And the magnetic field :
H ( z, t )  H 0e-z cost - z -  yˆ
(14)
H0 
where ;
E0
E ( z , t )  E0 e -z cost - z xˆ
H ( z, t )  H 0e-z cost - z -  yˆ (15)
(15)

Intrinsic impedance :
j
j

  ∠   e , ()
  j
where ;
 
 /
2 1/ 4
    
 
1  
    
, tan 2 
(16)

, 0 ≤  ≤450

(17)
Conclusions that can be made for the wave propagating in lossy
dielectrics material :
-z
(i) E and H fields amplitude will be attenuated by e
(ii) E leading H by

(14)
Wave velocity ;
 
u   /  ;   2 / 
 /
   2 
 
1  
    
Jd
E



 tan 

jE
, tan 2 

, 0 ≤  ≤450 (17)

From (17) and (18)
Loss tangent ;
J
1/ 4
  2
(18)
Loss tangent values will determine types of media :
tan θ small (σ / ωε < 0.1) – good dielectric – low loss
tan θ large (σ /ωε > 10 ) - good conductor – high loss
Another factor that determined the characteristic of the media is operating
frequency. A medium can be regarded as a good conductor at low
frequency might be a good dielectric at higher frequency.
E ( z, t )  E0 e -z cost - z xˆ
 E0 e -z e jt - z  xˆ
H0 
(14)
H ( z , t )  H 0 e -z cost - z -  yˆ

E0

e -z e
jt - z -

(15)
yˆ
Graphical representation of E field in lossy dielectric
E0 x
E ( z , t )  E0 e -z cost -  z xˆ
e z
z
y
E0

10.3 PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS
Characteristics:
  0,    0 r ,   0 r
Substitute in (11) and (12) :
  0,    
u

1
2

, 



 o

0

(20)
(21)
(19)

 
2


2
1

1

 Np / m (11)
2 2




 
2


2
 1  2 2  1 rad / m (12)



j
j

  ∠   e  , ()
  j
(22)
The zero angle means that E and H fields are in phase at each
fixed location.
10.4 PLANE WAVE IN FREE SPACE
Free space is nothing more than the perfect dielectric media :
Characteristics:
  0,    0 ,   0
(23)
Substitute in (20) and (21) :
  0 ,    0 0   / c

1
2
u 
c ,  


0 0
where
u
(24)

(25)
u  c  3 108 m / s
  0 
0
 120 
0
  0,    
(20)

1
2

, 
(21)




 0o

(22)
  0  4 10 7 H / m
(26)
   0  8.854 10 12 
1
10 9 F / m
36
The field equations for E and H obtained :
E  E0 cos(t - z ) xˆ
H 
E0
0
cos(t - z ) yˆ
E ( z , t )  E0 e -z cost -  z xˆ
(27)
H ( z, t )  H 0e-z cost - z -  yˆ (15)
(28)
E and H fields and the direction of propagation :
x
Ex+
Generally :
ˆ  zˆ
k
kos(-z)
Eˆ  Hˆ  kˆ
z
(at t = 0)
y
Hy+ kos(-z)
(14)
10.5 PLANE WAVE IN CONDUCTORS

In conductors :    or
→∞

With the characteristics :  ~  ∞,    0 ,   0  r (29)
Substitute in (11 and (12) :
 


2
 f

 
2
 
1

1

 Np / m (11)
2 2
2 


(30)

 45o E leads H by 450


 
2
(31)

The field equations for E and H obtained :
E  E0e-z cos(t - z ) xˆ
H 
E0
0
(32)
e-z cos(t - z - 45o ) yˆ
(33)


2
 1  2 2  1 rad / m (12)



j
j
  ∠   e  , ()
  j
It is seen that in conductors
E and H waves are attenuated by e
-z
From the diagram  is referred to as the skin depth. It refers to the
amplitude of the wave propagate to a conducting media is reduced to
e-1 or 37% from its initial value.
In a distance :
E0 e -  E0 e -1
∴  1 /  
1
f
It can be seen that at higher
frequencies  is decreasing.
(34)
x
E0
0.368E0
z

Ex.10.1 : A lossy dielectric has an intrinsic impedance of 200∠30  at the
particular frequency. If at that particular frequency a plane wave that propagate
in a medium has a magnetic field given by :
o
H  10e-x cos(t - x/2 ) yˆ A / m. Find E
Solution :
Eˆ  Hˆ  kˆ
→ Eˆ  yˆ  xˆ
∴E  - zˆ
and  .
From intrinsic impedance, the magnitude
of E field :

E0
 200∠30o
H0
→ E0  2000∠30o
It is seen that E field leads H field :
  300   / 6
Hence :
E  -2000e-x cos(t - x / 2   / 6) zˆ (V / m)
E  -2000e-x cos(t - x / 2   / 6) zˆ (V / m)
To find
:




1






 1 




 - 1 

  1
2
1/ 2
2
 tan 2  tan 600  3
  2 -1 
1
; and we know   1 / 2
∴ 


  2  1
3

→ 
 0.2887 Np / m
1/ 2
3
Hence:
E  -2000e-0.2887x cos(t - x / 2   / 6) zˆ (V / m)
10.6 POWER AND THE POYNTING VECTOR
∂
H
∇ E  -
∂
t
(35)
∂
E
∇ H  E  
∂
t
Dot product (36) with
(36)
E:
∂
E
E  ∇ H   E  E  
∂
t
2
(37)
From vector identity:
∇ A  B   B  ∇ A - A  ∇ B 
Change
A  H, B  E
(38)
in (37) and use (38) , equation (37) becomes :
∂
E
H  ∇ E   ∇ H  E   E  E  
∂
t
2
(39)
∂
E
H  ∇ E   ∇ H  E   E  E  
∂
t
2
(39)
H
∂
∇ E  -
t
∂
(35)
And from (35):
 ∂
H
 ∂
  H  ∇ E   H   - 
H H
∂
t 
2∂
t

(40)
Therefore (39) becomes:
-
 ∂H 2
2 ∂
t
-∇ E  H   E 2  E  
∂
E
∂
t
(41)
where:
∇ H  E   -∇ E  H 
Integration (41) throughout volume v :
∂ 1 2 1
2
2


∇

E

H
dv


E


H
dv

E
dv
∫
∫
∫


∂
t v 2
2

v
v
(42)
∂ 1 2 1
2
2


∇

E

H
dv


E


H
dv

E
dv
∫
∫
∫


∂
t v 2
2

v
v
(42)
Using divergence theorem to (42):
∂ 1 2 1
2
2


E

H

d
S


E


H
dv

E
dv
∫
∫
∫


∂
t v 2
2

s
v
Total energy flow
leaving the volume
The decrease of the energy
densities of energy stored
in the electric and magnetic
fields
Dissipated
ohmic power
Equation (43) shows Poynting Theorem and can be
written as :
 E  H W / m
2
(43)
Poynting theorem states that the total power flow leaving the volume
is equal to the decrease of the energy densities of energy stored in
the electric and magnetic fields and the dissipated ohmic power.
The theorem can be explained as shown in
the diagram below :
Output power
σ
Ohmic losses
J
E
Stored electric
field
H
Stored magnetic
field
Input power
Given for lossless dielectric, the electric and magnetic fields are :
E  E0 cos(t - z ) xˆ
H 
E0

cos(t - z ) yˆ
The Poynting vector becomes:
  E  H W / m2

E 20

cos 2 (t - z ) yˆ
To find average power density :
Integrate Poynting vector and divide with interval T = 1/f :
Pave

1

T
T
E02
∫

cos 2 (t   z ) dt
0
2 T
0
1 E
2T 
1  cos( 2t - 2  z )dt
∫
0
T


1 E
1

sin( 2t - 2  z ) 
t 
2T  
2
0 


1 E02
 P ave 
W / m2
2 
2
0
Average power
through area S :
1 E02
P ave 
S (W )
2 
Given for lossy dielectric, the electric and magnetic fields are :
E  E0e -z cos(t - z ) xˆ
H 
E0
0
e -z cos(t - z -  ) yˆ
The Poynting vector becomes:

E 20

e  2z cos(t - z ) cos(t - z   )
Average power :
1 E02 2z
P ave 
e
cos
2 
Ex.10.2: A uniform plane wave propagate in a lossless dielectric in the
+z direction. The electric field is given by :
E ( z, t )  377 cost  4 / 3z   / 6xˆ (V / m)
2
The average power density measured was 377 W / m . Find:
(i) Dielectric constant of the material if
  0
(ii) Wave frequency
(iii) Magnetic field equation
Solution:
(i) Average power :
1 E2
Pave 
 377
2 
1 (377) 2
 377
2 
   377 / 2  188.5
For lossless dielectric :
0



 r 0

r 
1

0
 1.9986
0
  r  4.0
(ii) Wave frequency :
E ( z, t )  377 cost  4 / 3z   / 6xˆ (V / m)
  4 / 3    0

4
3  0
2f  3.9946 1016
 f  99.93 106  (100 MHz )
E ( z, t )  377 cost  4 / 3z   / 6xˆ (V / m)
(iii) Magnetic field equation :
H ( z, t ) 
377
cos(t  ( 4 / 3) z   / 6) yˆ

 2 cos(t  (4 / 3) z   / 6) yˆ ( A / m)