PROPAGATION & REFLECTION OF PLANE WAVES

1

PROPAGATION & REFLECTION OF PLANE WAVES

CHAPTER 10

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

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 :

These propagation phenomena for a type traveling wave called plane wave can be explained or derived by Maxwell’s equations.

- travel at high velocity

- travel following EM wave characteristics

- travel outward from the source

10.1 ELECTRIC AND MAGNETIC FIELDS FOR PLANE WAVE

From Maxwell’s equations :

0∇

∇∂

∂J

∂

∂J∇

∂

∂-

∂

∂-∇

B

Dt

E

t

DH

t

H

t

BE

v

0,0 JvAssume the medium is free of charge :

)4( 0∇

)3( 0∇

)2( ∂

∂∇

)1( ∂

∂-∇

B

Dt

EH

t

HE

From vector identity and taking the curl of (1)and substituting (1) and (2)

EEE 2∇-)∇(∇)∇(∇

0)∇(∇ Ewhere

fieldelectricforequationsHelmholtziet

EE

Ht

E

Et

H

' ∂

∂∇∴

)∇(∂

∂∇→

-∇∂

∂-∇

2

22

2

2

2

2

2

2

2

2

2

2

∂

∂

∂

∂

∂

∂

t

E

z

E

y

E

x

E

32

22

∂

∂∇ Vm

t

EE

In Cartesian coordinates :

Assume that :

(i) Electric field only has x component

(ii) Propagate in the z direction

2

2

2

2

∂

∂

∂

∂

t

E

z

E xx

Similarly in the same way, from vector identity and taking the curl of (2)and substituting (1) and (2)

32

22 A

∂

∂∇ m

t

HH

)cos()-cos( - ztEztEE xxx

2

2

2

2

∂

∂

∂

∂

t

E

z

E xx

The solution for this equation :

Incidence wave propagate in +z direction

Reflected wave propagate in -z direction

To find H field :

t

HE

∂

∂-∇

yztEztE

zy

Ey

z

EE

xx

xx

ˆ)sin(-)-sin(

ˆ∂

∂-ˆ

∂

∂∇

-

zt

Hy

t

Hx

t

H

t

H zyx ˆ∂

∂ˆ

∂

∂ˆ

∂

∂-

∂

∂-

On the right side equation :

)cos(-)-cos(

)cos(-)-cos(

)cos()-cos(-

)sin(-)-sin(-

)sin(-)-sin(∂

∂-

-

-

-

-

-

∫

ztHztH

ztEztEH

ztEztE

dtztE

dtztE

H

ztEztEt

H

yy

xxy

xx

xxy

xxy

Equating components on both side = y component

)cos()-cos( - ztEztEE xxx

)cos(-)-cos( - ztHztHH yyy

Hence :

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) will provides information on the direction of propagation.

HE

10.2 PLANE WAVE IN LOSSY DIELECTRICS – IMPERFECT DIELECTRICS

rr 00 ; ; 0

Assume a media is charged free , ρv =0

Ejt

DJH

∂

∂∇

Hjt

BE -

∂

∂-∇

(1)

(2)

HjE ∇-∇∇ Taking the curl of (2) :

AAA 2∇-∇∇∇∇ From vector identity : Ej

t

DJH

∂

∂∇

Hjt

BE -

∂

∂-∇

(1)

(2)

HjE ∇-∇∇ EjjEE -∇-∇∇ 2

0-∇

0-∇22

2

EE

EjjE

j

jj

2

2

-

Where :

(4)

constantn propagatio

jDefine :

j2- 222 (5)

(6) (Re) -- 222

(7) (Im) 2

Equating (4) and (5) for Re and Im parts :

)8( 222

)9(

-

22

2222

Magnitude for (5) ;

Magnitude for (4) ;

(6) (Re) -- 222

)10( 2222

Equate (8) and (9) :

Hence :

222

22

2222

-1

-2

)11( / 1-12

1-12

22

2

22

222

mNp

is known as attenuation constant as a measure of the wave is attenuated while traveling in a medium.

Add (10) and (6) :

Substract (10) and (6) :

22222

)12( /112 22

2

mrad

is phase constant

xzteEtzE z ˆ-cos),( -0

If the electric field propagate in +z direction and has component x, the equation of the wave is given by :

yzteHtzH z ˆ--cos),( -0

And the magnetic field :

(13)

(14)

0

0

EH where ; (15) yzteHtzH z ˆ--cos),( -

0

xzteEtzE z ˆ-cos),( -0 (14)

(15)

Conclusions that can be made for the wave propagating in lossy dielectrics material :

(i)   E and H fields amplitude will be attenuated by

(ii) E leading H by

ze -

)(,∠

je

j

j

Intrinsic impedance :

04/12

45≤ ≤0,2tan,

1

/

where ;

(16)

(17)

/2 ; / u

Wave velocity ;

tanEj

E

J

J

d

Loss tangent ;

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

2(18)

)17( 45≤ ≤0,2tan,

1

/ 04/12

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.

From (17) and (18)

x

z

y

ze

0E

Graphical representation of E field in lossy dielectric

xzteEtzE z ˆ-cos),( -0

0

0

EH

xeeE

xzteEtzEztz

z

ˆ

ˆ-cos),(-j-

0

-0

(14)

(15)

yee

E

yzteHtzH

ztz

z

ˆ

ˆ--cos),(

--j-0

-0

10.3 PLANE WAVE IN LOSSLESS (PERFECT) DIELECTRICS

)11( / 1-12 22

2

mNp

)12( /112 22

2

mrad

rr 00 ,,0 Characteristics:

,0

2

,1

u

o0

)(,∠

je

j

j

Substitute in (11) and (12) :

The zero angle means that E and H fields are in phase at each fixed location.

(19)

(20)

(21)

(22)

10.4 PLANE WAVE IN FREE SPACE

00 ,,0 Characteristics:

Free space is nothing more than the perfect dielectric media :

c/, 0 00

2

, 1

00

cu

smcu /103 8

Substitute in (20) and (21) :

,0

2

,1

u

o0

(20)

(21)

(22)

where mH / 104 70

mF / 1036

110854.8 912

0

1200

00

(23)

(24)

(25)

(26)

xztEE ˆ)-cos(0

yztE

H ˆ)-cos(0

0

xzteEtzE z ˆ-cos),( -0 (14)

(15) yzteHtzH z ˆ--cos),( -0

The field equations for E and H obtained :

x

y

z

(at t = 0)

Ex+ kos(-z)

Hy+ kos(-z)

zk ˆˆ

(27)

(28)

E and H fields and the direction of propagation :

kHE ˆˆˆ

Generally :

10.5 PLANE WAVE IN CONDUCTORS

)11( / 1-12 22

2

mNp

)12( /112 22

2

mrad

)(,∠

je

j

j

∞→

In conductors : or

r 00 ,,∞~ With the characteristics : (29)

f2

Substitute in (11 and (12) :

o45 E leads H by 450

xzteEE z ˆ)-cos(-0

yzteE

H oz ˆ)45--cos(-

0

0

The field equations for E and H obtained :

(30)

(32)

(33)

(31)

E H ze -It is seen that in conductors and waves are attenuated by

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.

0.368E0

E0

x

z

f

eEeE

1/1∴

1-0

-0

In a distance :

It can be seen that at higher frequencies is decreasing.

(34)

Ex.10.1 : A lossy dielectric has an intrinsic impedance of at the particular frequency. If at that particular frequency a plane wave that propagate in a medium has a magnetic field given by :

o30∠200

./ˆ)x/2-cos(10 - mAyteH x Find and .E

zE

xyE

kHE

ˆ-∴

ˆˆˆ→

ˆˆˆ

Solution :

o

o

E

H

E

30∠2000→

30∠200

0

0

0

From intrinsic impedance, the magnitude of E field :

6/300 It is seen that E field leads H field :

)/( ˆ)6/2/-cos(2000- - mVzxteE x Hence :

To find :

)/( ˆ)6/2/-cos(2000- - mVzxteE x

2/1

2

2

11

1-1

360tan2tan 0

mNp /2887.03

→

3

1

12

1-2∴2/1

2/1; and we know

)/( ˆ)6/2/-cos(2000- 2887.0- mVzxteE x Hence:

10.6 POWER AND THE POYNTING VECTOR

From vector identity:

EDot product (36) with :

t

HE

∂

∂-∇

t

EH

∂

∂E∇

(35)

(36)

t

EEEHE

∂

∂∇ 2 (37)

EBHA ,Change in (37) and use (38) , equation (37) becomes :

BAABBA ∇-∇∇ (38)

t

EEEEHEH

∂

∂∇∇ 2 (39)

HHtt

HHEH

∂

∂

2-

∂

∂-∇

t

HE

∂

∂-∇ (35)

And from (35):

t

EEEHE

t

H

∂

∂∇-

∂

∂

2- 2

2

t

EEEEHEH

∂

∂∇∇ 2 (39)

(40)

Therefore (39) becomes:

HEEH -∇∇

where:(41)

Integration (41) throughout volume v :

dvEdvHEt

dvHEvvv∫∫∫ 222 -

2

1

2

1

∂

∂-∇

(42)

)43( -2

1

2

1

∂

∂- ∫∫∫ 222 dvEdvHE

tSdHE

vvs

Using divergence theorem to (42):

dvEdvHEt

dvHEvvv∫∫∫ 222 -

2

1

2

1

∂

∂-∇

(42)

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 :

2/ mWHE

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.

Stored electric field

Stored magnetic field

Ohmic losses

Output power

Input power

J HE

σ

The theorem can be explained as shown in the diagram below :

Given for lossless dielectric, the electric and magnetic fields are :

xztEE ˆ)-cos(0

yztE

H ˆ)-cos(0

yztE

ˆ)-(cos202

The Poynting vector becomes:

2/ mWHE

220

0

20

0

20

2

0

20

/ 2

1

)2-2sin(2

1

2

1

)2-2cos(12

1

)(cos1

∫

∫

mWE

P

zttE

T

dtztE

T

dtztE

TP

ave

T

T

T

ave

To find average power density :

Integrate Poynting vector and divide with interval T = 1/f :

)( 2

1 20 W S

EPave

Average power through area S :

xzteEE z ˆ)-cos(-0

yzteE

H z ˆ)--cos(-

0

0

Given for lossy dielectric, the electric and magnetic fields are :

)-cos()-cos(202

ztzte

E z

The Poynting vector becomes:

cos

2

1 220 z

ave eE

P

Average power :

Ex.10.2: A uniform plane wave propagate in a lossless dielectric in the +z direction. The electric field is given by :

)/(ˆ6/3/4cos377),( mVxzttzE 2/ 377 mWThe average power density measured was . Find:

(i) Dielectric constant of the material if

(ii) Wave frequency

(iii)Magnetic field equation

0

Solution:

5.1882/377

377)377(

2

1

3772

1

2

2

E

Pave

(i) Average power :

0.4

9986.11

0

0

0

0

r

r

r

For lossless dielectric :

)100(1093.99

109946.32

3

4

3/4

6

16

0

0

MHzf

f

(ii) Wave frequency : )/(ˆ6/3/4cos377),( mVxzttzE

)/(ˆ)6/)3/4(cos(2

ˆ)6/)3/4(cos(377

),(

mAyzt

yzttzH

(iii) Magnetic field equation :

)/(ˆ6/3/4cos377),( mVxzttzE