1 ENE 325 Electromagnetic Fields and Waves Lecture 11 Uniform Plane Waves.

1

ENE 325ENE 325Electromagnetic Electromagnetic Fields and WavesFields and Waves

Lecture 11Lecture 11 Uniform Plane Waves Uniform Plane Waves

2

IntroductionIntroduction

From Maxwell’s equations, if the electric field is changing with time, then the magnetic field varies spatially in a direction normal to its

orientation direction

A uniform plane wave, both electric and magnetic fields lie in the transverse plane, the plane whose normal is the direction of propagation

Both fields are of constant magnitude in the transverse plane, such a wave is sometimes called a transverse electromagnetic (TEM) wave.

��E��H

http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=52

3

Maxwell’s equationsMaxwell’s equations

0

��

��

��

v

DH J

t

BE

t

D

B

(1)

(2)

(3)

(4)

4

Maxwell’s equations in free Maxwell’s equations in free space space

= 0, = 0, rr = 1, = 1, rr = 1 = 1

0

0

��

��

EH

t

HE

t

Ampère’s law

Faraday’s law

5

General wave equationsGeneral wave equations

Consider medium free of charge where For linear, isotropic, homogeneous,

and time-invariant medium,

(1)

(2)

�� EH E

t

�� HE

t

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General wave equationsGeneral wave equations

Take curl of (2), we yield

From

then

For charge free medium

( )

�� HE

t

2

2

( )

��

��E

E E EtEt t t

2 ��

A A A

22

2

��

E EE E

t t

0 ��E

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Helmholtz wave equationHelmholtz wave equation

22

2

�� E EE

t t

22

2

�� H HH

t t

For electric field

For magnetic field

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Time-harmonic wave Time-harmonic wave equationsequations

Transformation from time to frequency domainTransformation from time to frequency domain

ThereforeTherefore

j

t

2 ( ) ��s sE j j E

2 ( ) 0 ��s sE j j E

2 2 0 ��s sE E

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Time-harmonic wave Time-harmonic wave equationsequations

or

where

This term is called propagation constant or we can write

= +j

where = attenuation constant (Np/m) = phase constant (rad/m)

2 2 0 ��s sH H

( ) j j

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Solutions of Helmholtz Solutions of Helmholtz equationsequations

Assuming the electric field is in x-direction and the wave is propagating in z- direction

The instantaneous form of the solutions

Consider only the forward-propagating wave, we have

Use Maxwell’s equation, we get

0 0cos( ) cos( )

��z z

x xE E e t z a E e t z a

0 cos( )

��z

xE E e t z a

0 cos( )

��z

yH H e t z a

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Solutions of Helmholtz equations Solutions of Helmholtz equations in phasor formin phasor form

Showing the forward-propagating fields without time-harmonic terms.

Conversion between instantaneous and phasor form

Instantaneous field = Re(ejtphasor field)

0

��

z j zs xE E e e a

0

��

z j zs yH H e e a

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Intrinsic impedance Intrinsic impedance

For any medium,

For free space

x

y

E jH j

0 0

0 0

120 x

y

E EH H

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Propagating fields Propagating fields relationrelation

1

��

��s s

s s

H a E

E a H

where represents a direction of propagation.a

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Propagation in lossless-charge Propagation in lossless-charge free mediafree media

Attenuation constant = 0, conductivity = 0

Propagation constant

Propagation velocity

for free space up = 3108 m/s (speed of light)

for non-magnetic lossless dielectric (r = 1),

1

pu

p

r

cu

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Propagation in lossless-Propagation in lossless-charge free mediacharge free media intrinsic impedance

Wavelength

2

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Ex1Ex1 A 9.375 GHz uniform plane wave is A 9.375 GHz uniform plane wave is propagating in polyethelene (propagating in polyethelene (rr = = 2.26). 2.26). If the amplitude of the electric field If the amplitude of the electric field intensity is 500 V/m and the material is intensity is 500 V/m and the material is assumed to be lossless, findassumed to be lossless, finda) phase constant

b) wavelength in the polyethelene

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c) propagation velocity

d) intrinsic impedance

e) amplitude of the magnetic field intensity

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Propagation in Propagation in dielectricsdielectrics

Cause finite conductivity polarization loss ( = ’-j” )

Assume homogeneous and isotropic medium

' "( ) ��H E j j E

" '[( ) ] ��H j E

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",eff Define

from2 ( ) j j

and 2 2( ) j

20


We can derive2

( 1 1)2

2

( 1 1)2

and 1

.1 ( )

j

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Loss tangentLoss tangent

A standard measure of lossiness, used to classify a material as a good dielectric or a good conductor

"

' 'tan

eff

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Low loss material or a good Low loss material or a good dielectric (tandielectric (tan «« 1) 1)

If , consider the material

‘low loss’ , then

1

2

(1 ).2

jand

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Low loss material or a good Low loss material or a good dielectric (tandielectric (tan «« 1) 1)

propagation velocity

wavelength

1

pu

2 1

f

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High loss material High loss material or a good or a good conductor conductor (tan(tan »» 1) 1)

In this case , we can

approximate

1

2 f

45 .

jje

therefore

2

1 1)

and

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High loss material High loss material or a or a good conductor good conductor (tan(tan »» 1) 1)

depth of penetration or skin depth, is a distance

where the field decreases to e-1 or 0.368 times of

the initial field

propagation velocity

wavelength

1 1 1m

f

pu

22

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Ex2Ex2 Given a nonmagnetic material Given a nonmagnetic material having having rr = 3.2 and = 3.2 and = 1.5= 1.51010-4-4

S/m, at S/m, at ff = 3 MHz, find = 3 MHz, find a) loss tangent

b) attenuation constant

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c) phase constant

d) intrinsic impedance

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Ex3Ex3 Calculate the followings for Calculate the followings for the wave with the frequency the wave with the frequency ff = 60 = 60 Hz propagating in a copper with Hz propagating in a copper with the conductivity, the conductivity, = 5.8 = 5.8101077 S/m: S/m:

a) wavelength

b) propagation velocity

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c) compare these answers with the same wave propagating in a free space

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1 ENE 325 Electromagnetic Fields and Waves Lecture 11 Uniform Plane Waves.

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