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Guias de Onda Circulares
V F R d i EV. F. Rodriguez Esquerre
x
ρa
cσ
φy⊗z
ε, , dμ ε σ
Si t i ilí d i d d ilí d iSimetria cilíndrica: coordenadas cilíndricas
ˆˆ ˆE E E Eφ
ˆ
zE E E E zρ φρ φ= + +
ˆˆ ˆzH H H H zρ φρ φ= + +
2 2 0z zE k E∇ + =
2 2 0H k H∇ + = 0z zH k H∇ +
2 2 22
2 2 2 2
1 1zρ ρ ρ ρ φ
⎛ ⎞∂ ∂ ∂ ∂∇ = + + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠zρ ρ ρ ρ φ∂ ∂ ∂ ∂⎝ ⎠
MODOS TE
0zE = 2 2 0z zH k H∇ + =
2 2 22
2 2 2 2
1 1zρ ρ ρ ρ φ
⎛ ⎞∂ ∂ ∂ ∂∇ = + + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
( ) ( ) ( ) ( ), ,zH z R P Z zρ φ ρ φ=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 21 1R R P
P Z z P Z z R Z zρ ρ φ
φ φ ρ∂ ∂ ∂
+ +( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2 2 2
22 0
P Z z P Z z R Z z
Z zR P k R P Z z
φ φ ρρ ρ ρ ρ φ
ρ φ ρ φ
+ +∂ ∂ ∂
∂+ +
( ) ( ) ( ) ( ) ( ) ( )2 0R P k R P Z zz
ρ φ ρ φ+ + =∂
( ) ( ) ( ) ( ), ,zH z R P Z zρ φ ρ φ=Dividindo cada termo por:
MODOS TE
( )( )
( )( )
( )( )
( )( )2 2 2
22 2 2 2
1 1 1 1 0R R P Z z
kρ ρ φ
φ φ∂ ∂ ∂ ∂
+ + + + =∂ ∂ ∂ ∂( ) ( ) ( ) ( )2 2 2 2R R P Z z zρ ρ ρ ρ ρ ρ φ φ∂ ∂ ∂ ∂
Os termos envolvendo uma única variável devem serOs termos envolvendo uma única variável devem ser constantes
( )221 Z z
γ∂
=( ) 2Z z z
γ=∂
( ) ( ) ( ) zZ A γ−( ) 1 2z zZ z Ae A eγ γ−= + ( )Z finito+∞ = ( ) 1
zZ z Ae γ=
( )( )
( )( )
( )( )2 22
2 2 2 22 2
1 0R R P
kR R P
ρ ρ φρ ρ ρ γ ρρ ρ ρ ρ φ φ
∂ ∂ ∂+ + + + =
∂ ∂ ∂
MODOS TE( )2
( )( )2
22
1 Pk
P φ
φφ φ
∂= −
∂( ) ( )
22
2
Pk Pφ
φφ
φ∂
= −∂( )
( ) 3 4sin cosP A k A kφ φφ φ φ= +
φ
( ) ( )2P P mφ φ π= ± k nφ = inteiron =
( ) ( )2
( ) 3 4sin cosP A n A nφ φ φ= +
( )( )
( )( )22
2 2 2 2 22 0
R Rn k
R Rρ ρρ ρ ρ γ ρ
ρ ρ ρ ρ∂ ∂
+ − + + =∂ ∂
2 2 2k hγ+ =
( ) ( )22 R R∂ ∂
( )( )
( )( )22
2 2 22 0
R Rh n
R Rρ ρρ ρ ρ
ρ ρ ρ ρ∂ ∂
+ + − =∂ ∂
( )( )
( )( )22
2 2 22 0
R Rh n
R Rρ ρρ ρ ρ
∂ ∂+ + − =
∂ ∂( ) ( )2R Rρ ρ ρ ρ∂ ∂
Equação diferencial de Bessel ( ) ( ) ( )5 6n nR A J h A Y hρ ρ ρ= +( ) ( ) ( )
( )0R finito= ( ) ( )5 nR A J hρ ρ=
0 8
1.0 ( )0J ρ
0.6
0.8
( )1J ρ( )2J ρ
( )3J ρ
0.2
0.4( )3J ρ
2 4 6 8 10 12
-0.2
ρ
-0.4
( ) ( ) ( ) ( ), ,zH z R P Z zρ φ ρ φ=
( ) zZ A γ−( ) ( )R A J hρ ρ ( ) sin cosP A Aφ φ φ+ ( ) 1zZ z Ae γ=( ) ( )5 nR A J hρ ρ= ( ) 3 4sin cosP A n A nφ φ φ= +
( ) ( )( )( )( )( ) ( )( )( )( )5 3 4 1, , sin cos zz nH z A J h A n A n Ae γρ φ ρ φ φ −= +
βjγ α β= +
Considerando meios dielétricos sem perdas e pcondutores perfeitos
jγ β= jγ β=
( ) ( ) ( ), , sin cos j zz nH z A n B n J h e βρ φ φ φ ρ −= +( ) ( ) ( )
As demais componentes são obtidas usando:p
E j Hωμ∇× = − H j Eωε∇× = −
ˆˆ ˆzE E E E zρ φρ φ= + +
ˆˆ ˆzH H H H zρ φρ φ= + +
1 ˆˆ zρ φ∂ ∂ ∂∇ = + +
zρ φ
ρ ρ φ∂ ∂ ∂
2z zE HjE
hρωμβ
ρ ρ φ⎛ ⎞∂ ∂
= − +⎜ ⎟∂ ∂⎝ ⎠h ρ ρ φ∂ ∂⎝ ⎠
z zE HjE β⎛ ⎞∂ ∂⎜ ⎟2
z zjEhφ
β ωμρ φ ρ
⎛ ⎞= − −⎜ ⎟∂ ∂⎝ ⎠
2z zE HjH
hρωε βρ φ ρ
⎛ ⎞∂ ∂= −⎜ ⎟∂ ∂⎝ ⎠h ρ φ ρ∂ ∂⎝ ⎠
E Hj β⎛ ⎞∂ ∂2
z zE HjHhφ
βωερ ρ φ
⎛ ⎞∂ ∂= − +⎜ ⎟∂ ∂⎝ ⎠
MODOS TE 0zE =
( ) ( )sin cos j zH A n B n J h e βφ φ ρ −= +( ) ( )sin cosz nH A n B n J h eφ φ ρ= +
( ) ( )2 cos sin j zn
j nE A n B n J h eh
βρ
ωμ φ φ ρρ
−= − −
( ) ( )2 sin cos n j zJ hjE A n B n e
hβ
φ
ρωμ φ φ −∂= +
∂( )2hφ φ φ
ρ∂
( )J hj ρβ ∂( ) ( )2 sin cos n j zJ hjH A n B n e
hβ
ρ
ρβ φ φρ
−∂= − +
∂
( ) ( )2 cos sin j zn
j nH A n B n J h eh
βφ
β φ φ ρρ
−= − −
Condições de contorno: campos elétricos tangenciais nulos sobre o condutor
( ) 0E aφ ρ = = 0zE =
( ) ( )2 sin cos n j zJ hjE A n B n e
hβ
φ
ρωμ φ φ −∂= +
∂
( )
( )2hφ ρ∂
( ) ( )2 sin cos 0n j z
a
J hj A n B n eh
β
ρ
ρωμ φ φρ
−
=
∂+ =
∂
( )0nJ hρ∂
= ( ) 0nJ ha∂=0
aρρ
=
=∂
0ρ
=∂
( )J ρ∂
0.4
( )1J ρρ
∂∂
( )J ρ∂
0.2
( )3J ρρ
∂∂
2 4 6 8 10 12ρ
2 4 6 8 10 12
-0.2
-0.4 ( )2J ρ∂
-0.6 ( )J ρ∂
( )2 ρρ∂
( )0 0; 3,832 7,016 10,174J ρ
ρρ
∂= =
∂
( ) 0nJ ha∂=
∂( )'
0n nmJ ρ∂=
∂'nmha ρ= 'nmh ρ
=ρ∂ ρ∂ a
Valores correspondentes ao modo TEρ'n0 ρ'n1 ρ'n2 ρ'n3
n = 0 0 3,832 7,016 10,174
Valores correspondentes ao modo TE
n 0 0 3,832 7,016 10,174
n = 1 0 1,841 5,331 8,536
n = 2 0 3,054 6,706 9,970
Modos TE n índica a variação em φModos TEnm n índica a variação em φm índica a variação em ρ
Frequencia de corte fnm do modo TEnm
2'ρ⎛ ⎞2 2 2 nmnm k h
aρβ ω με ⎛ ⎞= − = − ⎜ ⎟
⎝ ⎠
22 ' 0nm
aρω με ⎛ ⎞− =⎜ ⎟
⎝ ⎠a⎝ ⎠
'nmf ρ=
2nmcf aπ με
Analisando a tabela o menor valor corresponde aosAnalisando a tabela, o menor valor corresponde aos valores m=1 n=1
TE11 é o modo dominante e mais usado
Campos dos Modos TE
( ) ( )2 cos j zn
j nE A n J h eh
βρ
ωμ φ ρρ
−= −h ρ
( ) ( )2 sin n j zJ hjE A n e
hβ
φ
ρωμ φρ
−∂=
∂( )2hφ ρ∂
0E =
( ) ( )sin n j zJ hjH A n e βρβ φ −∂
= −
0zE =
( )2 sinH A n ehρ φ
ρ= −
∂
j nβ ( ) ( )2 cos j zn
j nH A n J h eh
βφ
β φ ρρ
−= −
( ) ( )sin j zz nH A n J h e βφ ρ −=
Impedância para Modos TE
TE
E E kH H
ρ φ
φ ρ
ωμ ηηβ β
= = − = =
( ) ( )2 cos j zn
j nE A n J h eh
βρ
ωμ φ ρρ
−= −
( ) ( )2 cos j zn
j nH A n J h eh
βφ
β φ ρρ
−= −
Campos dos Modos TE11
( ) ( )12 cos j zjE A J h eh
βρ
ωμ φ ρρ
−= −h ρ
( ) ( )2 sin n j zJ hjE A e
hβ
φ
ρωμ φρ
−∂=
∂( )2hφ ρ∂
0E =
( ) ( )1sin j zJ hjH A e βρβ φ −∂= −
0zE =
( )2 sinH A ehρ φ
ρ= −
∂
jβ ( ) ( )12 cos j zjH A J h eh
βφ
β φ ρρ
−= −
( ) ( )1sin j zzH A n J h e βφ ρ −=
Potencia Modo TE112
01 ˆRe2
a
P E H z d dρ φ π
ρ φ ρ= =
∗= × ⋅∫ ∫0 0
2ρ φ= =∫ ∫2
1 Ra
P E H E H d dρ φ π
φ= =
∗ ∗⎡ ⎤∫ ∫0
0 0
1 Re2
P E H E H d dρ φ φ ρ
ρ φ
ρ φ ρ∗ ∗
= =
⎡ ⎤= −⎣ ⎦∫ ∫⎡ ⎤
( ) ( )2 2212 2 2 2
0 14 2
0 0
1Re cos sin2
aJ hA
P J h h d dh
ρ φ π
φ
ρωμ βφ ρ φ ρ φ ρ
ρ ρ
= = ⎡ ⎤⎛ ⎞∂⎢ ⎥= + ⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦
∫ ∫0 0ρ φ= = ⎝ ⎠⎣ ⎦
( ) ( ) 2212 21Re
aJ hA
P J h h dρ
ρπωμ βρ ρ ρ
= ⎡ ⎤⎛ ⎞∂⎢ ⎥= + ⎜ ⎟∫ ( )0 14
0
Re2
P J h h dh
ρ
ρ ρ ρρ ρ
=
⎢ ⎥= + ⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦∫
2
( ) ( )2
2 20 11 14 ' 1
4A
P J hah
πωμ βρ= −
Perdas nos condutores no modo TE11
.2
sc
RP H H dρ φ∗= ∫ ( )2
2 2
2s
c zRP H H ad
φ π
φ φ=
= +∫2C∫ ( )
02
φ=∫
( )22 2
2 2 2isA RP J h d
φ πβ φ φ φ
=⎛ ⎞⎜ ⎟∫ ( )2 2 2
14 2
0
cos sin2
scP J ha ad
h aφ
β φ φ φ=
⎛ ⎞= +⎜ ⎟
⎝ ⎠∫
( )2 2
214 21
2s
c
A R aP J ha
h aπ β⎛ ⎞
= +⎜ ⎟⎝ ⎠⎝ ⎠
( )2 2
21sA R aJ h
π β⎛ ⎞⎜ ⎟ ( )
( )
214 2 2
22
0 11
12
2 ' 1
s
c sc
J hah aP R kh
P akA
β
αηβ ρπωμ β
+⎜ ⎟ ⎛ ⎞⎝ ⎠= = = +⎜ ⎟−⎝ ⎠( ) ( )0 112 211 14
2 12 ' 1
4P akA
J hah
ηβ ρπωμ βρ ⎝ ⎠−
MODOS TM 0zH =
2 2 0z zE k E∇ + =
( ) ( )sin cos j zz nE C n D n J h e βφ φ ρ −= +
( ) ( )2 sin cos n j zJ hjE C n D n e
hβ
ρ
ρβ φ φρ
−∂= − +
∂h ρ∂
( ) ( )2 cos sin j zj nE C n D n J h e βφ
β φ φ ρ −= − −
( ) ( )cos sin j zj enH C D J h βω φ φ ρ −
( ) ( )2 cos sin nE C n D n J h ehφ φ φ ρρ
( ) ( )2 cos sin j zn
jH C n D n J h eh
βρ φ φ ρ
ρ= −
( )J hj ρωε ∂( ) ( )2 sin cos n j zJ hjH C n D n e
hβ
φ
ρωε φ φρ
−∂= − +
∂
Condições de contorno: campos elétricos tangenciais nulos sobre o condutor
( ) 0zE aρ = = ( ) 0E aφ ρ = =
( ) ( )sin cos j zz nE C n D n J h e βφ φ ρ −= +
( ) ( )sin cos 0j zC n D n J h e βφ φ ρ −+ =
( ) ( )
( ) ( )sin cos 0jn a
C n D n J h e β
ρφ φ ρ
=+ =
( ) 0J hρ = ( ) 0J ha =( ) 0n aJ h
ρρ
== ( ) 0nJ ha =
1.0
( )0 0, 2, 405 5,520 8,654J ρ ρ= =
0 6
0.8
( )1J ρ( )J ρ
0.4
0.6 ( )2J ρ( )3J ρ
2 4 6 8 10 12
0.2
ρ
-0.4
-0.2
nmha ρ= nmh ρ=( ) 0nJ ha = ( ) 0n nmJ ρ =
a
Valores correspondentes ao modo TMρn0 ρn1 ρn2 ρn3
n = 0 ‐ 2,405 5,520 8,654
Valores correspondentes ao modo TM
n 0 2,405 5,520 8,654
n = 1 0 3,832 7,016 10,174
n = 2 0 5,135 8,417 11,620
Modos TM n índica a variação em φModos TMnm n índica a variação em φm índica a variação em ρ
Frequencia de corte fnm do modo TMnm
2ρ⎛ ⎞2 2 2 nmnm k h
aρβ ω με ⎛ ⎞= − = − ⎜ ⎟
⎝ ⎠
22 0nm
aρω με ⎛ ⎞− =⎜ ⎟
⎝ ⎠a⎝ ⎠
nmf ρ=
2nmcf aπ με
Analisando a tabela o menor valor corresponde aosAnalisando a tabela, o menor valor corresponde aos valores n=0 m=1
TE11 é o modo dominante e mais usado
Impedância para Modos TM
E Eρ φ β ηβTE H H k
ρ φ
φ ρ
β ηβηωε
= = − = =
( ) ( )2 sin cos n j zJ hjE C n D n e
hβ
ρ
ρβ φ φ −∂= − +
∂( )2hρ ρ∂
( ) ( )2 sin cos n j zJ hjH C n D n e
hβ
φ
ρωε φ φρ
−∂= − +
∂
Perdas nos Modos TMProblema Proposto: Analisar perdas do modo TM
Perdas num guia circular de cobre com a =2,54 cm
EXEMPLO
Determinar a freqüência de corte dos dois primeirosmodos num guia circular com raio a=0 5 cm cujo interiormodos num guia circular com raio a=0,5 cm cujo interioré revestido em ouro e está preenchido com teflon.Calcule as perdas devido a 30 cm do guia operando emCalcule as perdas devido a 30 cm do guia, operando em14 GHz. Faça um esboço da distribuição dos campos.
7 2 08 0 0004δ74,1 10 S/mcσ = × 2,08rε = tan 0,0004δ =
( )2nm
nmcf TM
aρ
π με=' ( )
2nm
cf TEρ=
Das tabelas os menores valores correspondem aos
2 aπ με( )2nmcf aπ με
Das tabelas, os menores valores correspondem aosmodos TE11 e TM01
81 841 3 10× ×11 2
1,841 3 10( ) 12,19 GHz2 0,5 10 2,08cf TEπ −
× ×= =
× ×
8
01 2
2, 405 3 10( ) 15,92 GHz2 0 10 2 08cf TM × ×
= =01 2( )
2 0,5 10 2,08cf π −× ×
91
8
2 2 14 10 2,08 422,9 m3 10
rfk
π ε π −× × ×= = =83 10c ×
22111' 1,831422 9 208k ρβ −⎛ ⎞⎛ ⎞
⎜ ⎟ ⎜ ⎟111
2
,422,9 208 m0,5 10
kaρβ −
⎛ ⎞⎛ ⎞= − = − =⎜ ⎟ ⎜ ⎟×⎝ ⎠ ⎝ ⎠
( )2( )22 422,9 0,0004tan= = = 0,172 Np/m = 1,49 dB/m2 2 208d
k δαβ
××
74,1 10 S/mcσ = × 0 0,0367 2sR ωμσ
= = Ω2 cσ
22 0 672 Np/m 0 583 dB/msR khα
⎛ ⎞+⎜ ⎟
2
11
0,672 Np/m = 0,583 dB/m' 1
sc h
akα
ηβ ρ= + =⎜ ⎟−⎝ ⎠
( ) ( )Perdas 1,49 0,583 0,3 0,62 dBd c Lα α= + = + × =
TE11H EzH Eρ
Eφ
