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Prof. David R. Jackson

Notes 15Notes 15Plane WavesPlane Waves

ECE 3317

[Chapter 3]

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Introduction to Plane Waves

A plane wave is the siplest sol!tion to "a#well$s e%!ations for a wave that

travels thro!&h free space.

'he wave does not re%!ires an( cond!ctors ) it e#ists in free space.

A plane wave is a &ood odel for radiation fro an antenna* if we are far

eno!&h awa( fro the antenna.

x

z

E

H

S

S E H=

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http+,,www.ipression-.or&,solarener&(,isc,espectr!.htl

The Electromagnetic Spectrum

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Source Frequency Wavelength

US !" Po#er $% &' 5%%% (m

E)F Su*marine"ommunications 5%% &' $%% (m

!+ radio ,-T.&/ 0% (&' %5 m

T2 ch 3 ,2&F/ $% +&' 5 m

F+ radio ,Sunny 441/ 441 +&' m

T2 ch 6 ,2&F/ 16% +&' 10 m

T2 ch 4 ,U&F/ $3% +&' 6 cm

"ell phone ,P"S/ 65% +&' 5 cm

"ell Phone ,P"S 14%%/ 145 7&' 15 cm

-#ave oven 35 7&' 13 cm

Police radar ,89*and/ 1%5 7&' 365 cm

mm #ave 1%% 7&' mm

)ight 51%1

:&'; %$% m89ray 1%16 :&'; otational chan&e+

>ote+ for a lossless transission line* we have+ LC = =

A transission line filled with a dielectric aterial has the sae waven!?er asdoes a plane wave travelin& thro!&h the sae aterial.

Plane Wave Field ,cont/

A wave travels with the sae velocit( on a transission line

as it does in space* provided the aterial is the sae.

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'he Hfield is fo!nd fro+

so

E Hj =

$ ( )( )

$

$

1H E

E1

1( ) E

x

x

x

x zj

d

yj dz

jk yj

=

=

=

E =

E E Ex y z

x y z

x y z

0E ( ) jkz

x z E e=

Plane Wave Field ,cont/

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Intrinsic Impedance

or

/ence

$H Exk

y

=

E

H

x

y k

= = =

where

= 8ntrinsic ipedance of the edi!

H Ey xk

=

00

0

376.730313 [ ]

= = B0reespace+

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Poynting 2ector

'he cople# Po(ntin& vector is &iven ?(+

0

0

E

1H

jkz

jkz

x E e

y E e

=

=

( )*1

S E H2

=

/ence we have

( )

( ) ( )

*

00

*

0 0

2

0

1S

2

1

2

1

2

jkz jkz

jkz jkz

Ez E e e

z E e E e

z E

=

=

=

2

0 2S [VA/m ]

2

Ez

=

( ) ( )2

0 2= Re S [W/m ]2

ES t z

=

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Phase 2elocity

( )

0

0 0

E

cos

jkz

x

x

E e

E E t kz

=

= +

pv

k

= =

1p d

v c

= =

0ro o!r previo!s disc!ssion on phase velocit( for transission lines* we know that

/ence we have

5speed of li&ht in the dielectric aterial6

pv

=

so

0

0 0

jE E e

=

>ote+ all plane wave travel at the sae speed in a lossless edi!*

re&ardless of the fre%!enc(. 'his iplies that there is no dispersion* which

in t!rn iplies that there is no distortion of the si&nal.

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Wavelength

2 2

k

= =

2 2 2 1

2

dc

k ff f

= = = = =

dc

f=

0ro o!r previo!s disc!ssion on wavelen&th for transission lines* we know that

/ence

Also* we can write

0

0

2

k

=0ree space+

0ree space+ 0c

f =

( )

0

0 0

E

cos

jkz

x

x

E e

E E t kz

=

= +

0

0 0

jE E e

=

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Summary ,)ossless "ase/

0

0

2

0

E

1H

S2

jkz

x

jkz

y

z

E e

E e

E

=

=

=

y

z

x E

H

S

( )0 0cosxE E t kz = +

00 0

jE E e =

( )Re E j tx xE e

=

'ie doain+

Denote

1pvk

= =

2

k

=

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)ossy +edium

E HH J E

jj

= = +

J E=

Ret!rn to "a#well$s e%!ations+

Ass!e Bh$s law+

( )

H E E

E

j

j

= +

= +

Apere$s law

e define an effective 5cople#6 perittivit( cthat acco!nts for cond!ctivit(+

cj j = + c j

=

z

xE

ocean

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)ossy +edium

E HH Ec

jj

= =

"a#well$s e%!ations+ then ?ecoe+

'he for is e#actl( the sae as we had for the lossless case* with

c

/ence we have

0

0

E

1H

jkz

x

jkz

y

E e

E e

=

=

!ck k jk = =

j

c

e

= =

5cople#6

5cople#6

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)ossy +edium ,cont/

E#aine the waven!?er+

!k k jk =

!

0 0

jkz jk z k z

xE E e E e e

= =

c k

ck = c j

=

0

0

k

k

Denote+

8n order to ens!re deca(* the waven!?er k!st ?e in the fo!rth %!adrant.

/ence+ k

k

Copare with loss( '+

) + di , /

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)ossy +edium ,cont/

!

0

jk z k z

xE E e e =

z

0

k z

E e

2

k

=

) + di , t /

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)ossy +edium ,cont/

!

0

jk z k z

xE E e e =

1 / !pd k=

'he depth of penetration dpis defined.

!

k ze

z

xE 1

1 0.37e B

pd

) + di , t /

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)ossy +edium ,cont/

2 2

0 0* 2 ! 2 !

*1S E H2 2 2

k z j k z

x yE Ez z e z e e

= = =$ $ $

!

0

!

01

jk z k z

x

jk z k z

y

E E e e

H E e e

=

=

'he cople# Po(ntin& vector is+

j

c

e

= =

( )

2

0 2 !ReS cos2k zz z

ES t e

= = 2 !k ze

z

Sz1

1 / !p

d k=

2 0.1"e =

E l

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E=ample

Bcean water+

0

#1

" [S/m]

r

=

==

Ass!efF 2.: G/4

0

0

c rj j

= =

( )( )0 #1 3$.%$ [&/m]c j =

0 0 0c rck = = ( )3#6.022 #1.#16 [1/m]k j=

1/pd k= 0.013 [m]pd =

2 /k = 0.016 [m]=

5'hese val!es are fairl( constant !p thro!&hicrowave fre%!encies.6

E l , t /

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E=ample ,cont/

f dp [m]

1 [/4] 2-1.

1: [/4] 7=.

1:: [/4] 2-.2

1 [k/4] 7.=1: [k/4] 2.-2

1:: [k/4] :.7=

1 ["/4] :.22

1: ["/4] :.:9:

1:: ["/4] :.:22

1.: [G/4] :.:131:.: [G/4] :.:12

1:: [G/4] :.:12

'he depth of penetration into the ocean water is shown for vario!s fre%!encies.

) T t

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)oss Tangent

c j

=

Denote+ c c cj =

'he loss tan&ent is defined as+ 'a( c

c

=

e then have+ 'a(

=

'he loss tan&ent characteri4es the nat!re of the aterial+

'a HH 1+ lowloss edi! 5atten!ation is sall over a wavelen&th6'a II 1+ hi&hloss edi! 5atten!ation is lar&e over a wavelen&th6

( ) ( )2 / 2 /k k k k ke e e

= =>ote+

) ) )i it

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( )

0 0

0

0

1

1 'a

ck j

j

j

= =

=

=

)o#9)oss )imit

( )( )0 1 'a / 2k j owloss liit+ 1 1 / 2 1( )z z z+ +

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Polari'ation )oss

'he perittivit( can also ?e cople#* d!e to olec!lar and atoic polari4ationloss5friction at the olec!lar and atoic levels6 .

c j

=

j =

( )c j j

=

c j

= +

E#aple+ distilled water+ 0 5?!t heats !p well in a icrowave oven6.

'a

= + or

>ote+ 8n practice* it is !s!all( diffic!lt to deterine how !ch of the loss

tan&ent coes fro cond!ctivit( and how !ch coes fro polari4ation loss.

P l i ti ) , t /

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Polari'ation )oss ,cont/

c j

= +

( )1 'ac j =

0 r =

Re&ardless of where the loss coes fro* 5cond!ctivit( or polari4ation loss6* we

can write

where

1c

j

= +

or

>ote+ 8f there is no polari4ation loss* then r r =

Polari'ation )oss ,cont /

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Polari'ation )oss ,cont/

0or odelin& p!rposes* we can l!p all of the losses into an

effective cond!ctivit(ter+

eff

c j

=

c j

= +

0 r =where