chung-ang university field & wave electromagnetics ch 8. plane electromagnetic waves 8-4 group...

Chung-Ang University Field & Wave Electromagnetics

CH 8. Plane Electromagnetic Waves

8-4 Group Velocity8-5 Flow of Electromagnetic Power and the Poynting Vector


Dispersion - All information-bearing signals consist of a band of frequencies.

8.4 Group Velocity

- Waves of the component frequencies travel with different velocities, causing a distortion in the signal wave shape.- This phenomenon is called dispersion.

+

+=

1f

2f

3f+

+

1f

2f

3f

Group velocity

A group velocity is the velocity of propagation of the wave-packet envelope (of a group of frequencies)

Consider a case of a wave packet that consists of two traveling waves having equal amplitude and slightly different angular frequencies and .

0 0 0( )

0 0 0( , ) cos[( ) ( ) ]E z t E t z

0 0 0cos[( ) ( ) ]E t z 0 02 cos( )cos( ).E T Z t z

Rapidly oscillating wave having an angular frequency 0

Slowly oscillating wave having an angular frequency


8.4 Group VelocityPhase velocity of the wave inside the envelope

0

0p

dzu

dt

The velocity of the envelope : The group velocity

1 1

/ /gdz

udt d d

0 0( , ) 2 cos( )cos( ).E z t E T Z t z


8.4 Group Velocity

p

pSlope: g

du

d

Slope: pu

For , wave propagation is possible.p

Phase velocity

pu

20 0

1

1 ( )p

21 ( )p

c

2 2

0 01 1p pfj j j

f

2

0 1 ,p

For the wave propagating in an ionized medium,

Group velocity

1

/gu d d

2( 1 ( / ) )p

c

d

d

2 2

2 2

( )1 12

p

p

c

d

d

2 2

p

c

21 ( / )pc (8-75)


Group velocity1

/gu d d

2( 1 ( / ) )p

c

d

d

2 2

2 2

( )1 12

p

p

c

d

d

2 2

p

c

General relation between the group and phase velocity can be obtained as

8.4 Group Velocity

2

1( ) p

p p p

dud d

d d u u du

g

du

d

Here,

21 ( / )pc (8-75)

So,1

pg

p

p

uu

du

u d

(8-76)

a) No dispersion :

(lossless medium)

0pdu

d

b) Normal dispersion : 0pdu

d

.g pu u

c) Anomalous dispersion : 0pdu

d

( up independent of ω , β a linear function of ω )

( increasing with )pu .g pu u

( decreasing with )pu .g pu u


8.5 Flow of Electromagnetic Power and the Poynting Vector

Electromagnetic waves carry with them electromagnetic power. Energy is transported through space to distant receiving points by electromagnetic waves. We will now derive energy of the electric and magnetic field.

We begin with the curl equations.

The verification of the following identity of vector operations is straightforward:

Substitution of Eqs. (8-77) and (8-78) in Eq.(8-79) yields

(8 77)B

Et

��

(8 78)D

H Jt

��

( ) ( ) ( ) (8 79)E H H E E H ��

( ) (8 80)B D

H E E Jt t

E H

��

��



In a simple medium, whose constitutive parameters , , and do not change with time, we have

Equation (8-80)can then be written as

2( ) 1 ( ) 1( )

2 2

B H H HH H H

t t t t

��

2( ) 1 ( ) 1( )

2 2

D E E EE E E

t t t t

��

2 2 21 1( ) ( ) (8 81)

2 2E H E H E

t

��

2( )E J E E E �� So, it is possible that

Here, suppose that =f

than .

H��

dHf

dt

��

2( ) 2f f f As we know ,

( ) 1 ( )

2

H H HH

t t

��



The time-rate of change of the energy stored in the electric and magnetic fields

The ohmic power dissipated in the volume as a result of the flow of conduction current density σE in the presence of the electric field E.

2 2 21 1( ) (8 82)

2 2S V VE H ds E H dv E dv

t

2 2 21 1( ) ( ) (8 81)

2 2E H E H E

t

��

Poynting vector : To be consistent with the law of conservation of energy , the quantity E ×H is a vector representing the power flow per unit area.

Poynting’s theorem The surface integral of Poynting vector over a closed s

urface equals the power leaving the enclosed volume.



2 *1 1(8 85)

2 2e Electric energy densE E ityw E ��

2 *1 1(8 86)

2 2m Magnetic energy dew H nsityH H ��

2 2 * 2/ / (8 87)p E J E E J Ohmic power dJ ensity ��

Equation (8-82) may be written in another form.

It states that the total power flowing into a closed surface.

The Poynting vector is in a direction normal to both and .P��

E��

H��

The Poynting vector as the power density vector at every point on the surface is an arbitrary , albeit useful, concept.

( ) (8 84)e ms V VP ds w w dv p dv

t


8.5.1 Instantaneous and Average Power Density

In dealing with time-harmonic electromagnetic waves, it is convenient to use phasor notation.

The example of phasor notation: ( )0( ) ( ) (8 88)j z

x x xE z a E z a E e ��

The instantaneous expression: ( , ) Re[ ( ) ]j tE z t E z e ��

For uniform plane wave propagating in a lossy medium in the +z-direction, the associated magnetic field intensity phasor is :

( )0

( )0

( ) ( )

(8 90)

z j zy y y

j zzy

j

EH z a H z a e e

Ea e

e

e

��

��

je Intrinsic impedance of a lossy medium is the phase angle of the intrinsic impedance of the medium.je

( )0Re[ ]j z j t

xa E e e ��

0 cos( ) (8 89)zxa E e t z ��



The corresponding instantaneous expression for is:( )H z��

0( , ) Re[ ( ) ] cos( ) (8 91)j t zy

EH z t H z e a e t z

��

The instantaneous expression for the Poynting vector or power density vector, from Eqs. (8-88) and (8-90) .

( , ) ( , ) ( , )P z t E z t H z t ��

average value is zero.

From Eq. (8-92), we obtain the time-average Poynting vector,

22 20

0

1( ) ( , ) cos ( / ) (8 94)

2

Tz

av z

EP z P z t dt a e W m

T

��

Re[ ( ) ] Re[ ( ) ]j t j tE z e H z e ��

220 cos( )cos( )z

z

Ea e t z t z

��

220 [cos cos(2 2 )] (8 92)

2z

z

Ea e t z

��



On the other hand,

220Re[ ( ) ( ) ] cos( 2 )j t z

z

EE z H z e a e t z

��

average value is zero.

incorrect because of

Re[ ( ) ] Re[ ( ) ] Re[ ( ) ( ) ]j t j t j tE z e H z e E z H z e ��

* *

* * * *

*

1 1Re( ) Re( ) ( ) ( )

2 21

[( ) ( )]41

Re( ) (8 93)2

A B A A B B

A B A B A B A B

A B A B

��

��

��

* 2

( , ) Re[ ( ) ] Re[ ( ) ]

1Re[ ( ) ( ) ] (8 9( ) )( 5)

2

j t j

j

t

t

P z t E z e H z e

E z H z E z H z e

��

��

Periodic component.So, average value is zero.

* 21( ) Re[ ] ( / ) (8 96)

2avP z E H W m ��

chung-ang university field & wave electromagnetics ch 8. plane electromagnetic waves 8-4 group...

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