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7/23/2019 notes 15 3317
1/31
Prof. David R. Jackson
Notes 15Notes 15Plane WavesPlane Waves
ECE 3317
[Chapter 3]
7/23/2019 notes 15 3317
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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=
7/23/2019 notes 15 3317
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http+,,www.ipression-.or&,solarener&(,isc,espectr!.htl
The Electromagnetic Spectrum
7/23/2019 notes 15 3317
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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.
7/23/2019 notes 15 3317
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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/
7/23/2019 notes 15 3317
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Intrinsic Impedance
or
/ence
$H Exk
y
=
E
H
x
y k
= = =
where
= 8ntrinsic ipedance of the edi!
H Ey xk
=
00
0
376.730313 [ ]
= = B0reespace+
7/23/2019 notes 15 3317
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Poynting 2ector
'he cople# Po(ntin& vector is &iven ?(+
0
0
E
1H
jkz
jkz
x E e
y E e
=
=
( )*1
S E H2
=
/ence we have
( )
( ) ( )
*
00
*
0 0
2
0
1S
2
1
2
1
2
jkz jkz
jkz jkz
Ez E e e
z E e E e
z E
=
=
=
2
0 2S [VA/m ]
2
Ez
=
( ) ( )2
0 2= Re S [W/m ]2
ES t z
=
7/23/2019 notes 15 3317
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Phase 2elocity
( )
0
0 0
E
cos
jkz
x
x
E e
E E t kz
=
= +
pv
k
= =
1p d
v c
= =
0ro o!r previo!s disc!ssion on phase velocit( for transission lines* we know that
/ence we have
5speed of li&ht in the dielectric aterial6
pv
=
so
0
0 0
jE E e
=
>ote+ all plane wave travel at the sae speed in a lossless edi!*
re&ardless of the fre%!enc(. 'his iplies that there is no dispersion* which
in t!rn iplies that there is no distortion of the si&nal.
7/23/2019 notes 15 3317
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Wavelength
2 2
k
= =
2 2 2 1
2
dc
k ff f
= = = = =
dc
f=
0ro o!r previo!s disc!ssion on wavelen&th for transission lines* we know that
/ence
Also* we can write
0
0
2
k
=0ree space+
0ree space+ 0c
f =
( )
0
0 0
E
cos
jkz
x
x
E e
E E t kz
=
= +
0
0 0
jE E e
=
7/23/2019 notes 15 3317
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Summary ,)ossless "ase/
0
0
2
0
E
1H
S2
jkz
x
jkz
y
z
E e
E e
E
=
=
=
y
z
x E
H
S
( )0 0cosxE E t kz = +
00 0
jE E e =
( )Re E j tx xE e
=
'ie doain+
Denote
1pvk
= =
2
k
=
7/23/2019 notes 15 3317
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)ossy +edium
E HH J E
jj
= = +
J E=
Ret!rn to "a#well$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
7/23/2019 notes 15 3317
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)ossy +edium
E HH Ec
jj
= =
"a#well$s e%!ations+ then ?ecoe+
'he for is e#actl( the sae as we had for the lossless case* with
c
/ence we have
0
0
E
1H
jkz
x
jkz
y
E e
E e
=
=
!ck k jk = =
j
c
e
= =
5cople#6
5cople#6
7/23/2019 notes 15 3317
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)ossy +edium ,cont/
E#aine the waven!?er+
!k k jk =
!
0 0
jkz jk z k z
xE E e E e e
= =
c k
ck = c j
=
0
0
k
k
Denote+
8n order to ens!re deca(* the waven!?er k!st ?e in the fo!rth %!adrant.
/ence+ k
k
Copare with loss( '+
) + di , /
7/23/2019 notes 15 3317
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)ossy +edium ,cont/
!
0
jk z k z
xE E e e =
z
0
k z
E e
2
k
=
) + di , t /
7/23/2019 notes 15 3317
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)ossy +edium ,cont/
!
0
jk z k z
xE E e e =
1 / !pd k=
'he depth of penetration dpis defined.
!
k ze
z
xE 1
1 0.37e B
pd
) + di , t /
7/23/2019 notes 15 3317
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)ossy +edium ,cont/
2 2
0 0* 2 ! 2 !
*1S E H2 2 2
k z j k z
x yE Ez z e z e e
= = =$ $ $
!
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
7/23/2019 notes 15 3317
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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 /
7/23/2019 notes 15 3317
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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
7/23/2019 notes 15 3317
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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
7/23/2019 notes 15 3317
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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+ +
7/23/2019 notes 15 3317
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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 /
7/23/2019 notes 15 3317
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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 /
7/23/2019 notes 15 3317
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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