a = 10
b = 5
mn
f = 4, 5 109
·
λ = 2d = 20
β =
2π = 0, 1π λ
k=
kc2 = k 2
−1
ω = 0, 67π 67π c
2
−1
− β ⇒ k = 0,3π
2
(kc ) =
c
mπ a
2
+
nπ b
2
8 = m 2 + 4n 4n2
m = 2 n = 1 21
sen E Ez = 50 sen
− ∇
πx πy − j0 sen e j0,3πz 5 5
Et
= j
β kc2
E z t E
H t =
ε µ
1/2
k uz β
×E
t
− − − − − · − − · · · · · − · − − · − − · · − · · − − E x = j
E y = j
β ∂ β π πx πy − j0,3πz E = j E cos sen e z o kc2 ∂x kc2 5 5 5 β ∂ β π πx πy − j0,3πz E = j E sen cos e z o 5 5 kc2 ∂y kc2 5 β k E j = e − jπ/2 c
o
E x = 2,23 cos
πx πy − j(0,3z+0,5)π sen e 5 5
E y = 2,23sen
πx πy − j(0,3z+0,5)π cos e 5 5
Hx =
Hy =
ε µ
ε µ
1/2
1/2
k πx πy − j0,3πz E y = j 8,83 10−4 sen cos e β 5 5 k πx πy − j0,3πz cos Ex = j8,83 10−4 sen e β 5 5
j = e jπ/2
H x = 8,83 10−4 sen
πx πy − j(0,3z −0,5)π cos e 5 5
πx πy − j(0,3z+0,5)π H y = 8,83 10−4 sen cos e 5 5 e jω t
ω = 9π 109
πx πy sen cos(9π 109t j0,3πz ) 5 5 πx πy 2,23cos sen cos(9π 109 t 0,3πz 0,5π) 5 5 πx πy 2,23sen cos cos(9π 109 t 0,3πz 0,5π) 5 5 πx πy 8,83 10−4 sen cos cos(9π 109t 0,3πz + 0,5π) 5 5 πx πy 8,83 10−4 sen cos cos(9π 109t 0,3πz 0,5π) 5 5
E z = 50 sen E x = E x = H x = H y =
5
×
5
E z = E o sen(kx x) sen(ky y)e− jβz
H z = 0. Et = E t cos(ωt
β (z 2
− βz + ϕ)u
t
− z ) = π 1
βd m´ın = π dm´ın = λc /2,
β = 2π/λ ,
λ = λ c β = kc ω2 (β ) = 2 c 2
ω2 (kc ) = 2 2c 2
2
(kc ) = kc
a
2
− (k ) c
⇒ k = 800π c
mπ a
2
+
m2 + n2 = 2
nπ a
2
2
m = 1
n = 1
11
√
β = k c = 20π 2 −1 ω k = = 40π −1 c
E z11
E x11 Ey 11
− √ −
√ π π j20π 2z = E 0 sen x sen y e 5 5 √ β mπ π π π π j20π 2z = j 2 E 0 cos x sen y = j 2E 0 cos x sen y e kc a 5 5 5 5 √ π π j20π 2z = j 2E 0 sen x cos y e 5 5
√
−
Z T M
√
√
β 2 Z T M = Z o = 120π = 60 2π k 2
Hy 11
H x11
E x 1 = = j E 0 cos 60π Z T M E y 1 = = j E 0 sen Z T M 60π
−
−
1 β 2 P = 2Z T M kc2
P
=
√ 2
π π x sen y 5 5 π π x cos y 5 5
·
E z Ez ∗ dxdy
√ − √ −
480π
2 = 480π
5
0
0
5
0
5
E 02 sen 2
E 02 sen2
2 2 1 = E x 192π 0 2
π π x sen2 y dxdy 5 5
π 1 x dx y 5 2 5
5sen 2π x 5 4π 3
0
2 0
P = 5,86 · 10− E
5sen 2π y 5 4π
5
0
λ = 0, 1 10
f = 1, 3f c ). a
b
ε = 4εo a b
f = c/λ = 3 108 /0,1 = 3
·
10
f c =
f = 2,31 1, 3
c 3 108 λc = = = 0,13 f c 2,31 109
·
·
10
λc = 2a
⇒ a = 6,5 E 01,
λ01 c = 2b
f c01 = 1,3f = 3,9 λ01 c =
c 3,9 109
·
= 0,078
λ01 b = c = 3,9 2
×
ε = 4εo
v pho =
√ 1εµ = 2c
10
v pho = 6,49 f c10
λc10 =
λc10 a = = 3,25 2 E 01,
λc01 = 2b
f c01 = 3,9 λc01 =
v pho = 1,97 10−2 9 3,9 10
·
·
λ101 b = c = 0,99 2
a = 2, 29 b = 1, 02
f cmn
v pho = 2π
mπ a
2
f c10 =
c = 6,5 2a
f c01 =
c = 14,7 2b
c f c11 = = 2π
nπ + b
π a
2
π b
+
2
2
= 15,24
10
kc =
2
β =
π = 43,67π a
− ⇒ ω c
2
kc2
β = 25,81π
π H z10 = H o cos( x)e− jβz a
H x10 = j Ey 10 =
β π H o sen( x)e− jβz π/a a
ωµ π H sen( x)e− − j π/a a
jβ z
o
H z10 = H o cos(43,67πx)e− j25,81πz H x10 = j0,59H o sen(43,67πx)e− j25,81πz Ey 10 =
j25,81πz
− j146,56H sen(43,67πx)e− o
a = 10,16
5,08
b=
11
Z T M
11
E z
E o
11
c f c11 = 2π
π a
2
π + b
2
= 3,30
f = 1,3f c = 4,3
− π a
kc11 =
β =
ω c
2
λ =
Z
11
=
2
+
π b
2
−1
= 22π
(kc11 )2 = 18,38π
−1
2π = 10, 88 β
µ β 11 cβ 11 = 120π = 76,94π = 241, 71Ω εo k ω
(E x )11,m´ax =
β π E 0 = 0,37E 0 kc2 a
(E y )11,m´ax =
β π E 0 = 0,75E 0 kc2 b
(H x )11,m´ax =
E x = 1,53 Z 11
(H y )11,m´ax =
E y = 3,1 Z 11
3
× 10− E 3
0
× 10− E
0
7,62 3,81
×
10
π H z10 = H o cos( x)cos(ωt βz ) a β π H x10 = H o sen( x) sen(ωt βz ) π/a a ωµ π E y10 = H o sen( x)sen(ωt βz ) π/a a
−
−
−
−
sen(β 10 z ) = 0
10 z =
2 pπ
β = 10π
−1
kc10 =
ω c
2
⇒ ∆z = β π
⇒ β
10
π = 13,12π a
−1
= (β )2 + (kc )2 = (10π)2 + (13,12π)2 k=
ω = 16,5π c
−1
f = 2,47 109
·
k Z T E = Z o = 198π Ω β 10
Et = j10
3
sen(13,12πx)e− j10πz (
−u ) y
10
Ht = uz
×
Et
Z T E
103 = j sen(13,12πx)e− j10πz ux 198π
1 P = 2
× a
Et
0
1 106 P = 2 198π
b
0
a
0
b
sen2 (πx/a)dxdy
0
1 106 b x sen(2πx/a) P = + 2 198π 2π/a 52,48π
∗
Ht dxdy
a
= 2,83 10−2 W 0
·
7,62
× 3,81 10
H z = H o cos(41, 23x)e−30,5z
10
f c =
c 10 c kc = = 1,97 109 2π 2a
·
a = 2b
01
f c =
c 01 c kc = = 3,94 109 2π 2b
·
kc =
π = 13,12π a
−1
ω = 2πf = 4,9π 109 ω k = = 16,33π −1 c
·
β β =
− k2
kc2 = 9,72π
−1 sen(ωt
− βz )
βz = nπ
z = β π = 0,103
−1
a = 2,29 β,
11
c 11 c f c = k = 2π c 2
100 2,29
2
+
100 1,02
2
= 16,1 109
·
f = 2f c = 32
kc =
100 2,29
β 11 =
2
+
− ω c
2
100 1,02
2
= 107,32π
kc2 = 184,37π
−1
−1
b = 1,02
λ =
=
ph
Z
11
=
2π = 1, 08 β 11
ω = 3,47 108 β 11
−1
·
µo β 11 β 11 = Z o = 326, 19 Ω εo k k a = 2,29 10
β, E o = 104
kc10 =
k =
β =
π = 43,67 a ω = 60π c
− k2
−1
−1
kc2 = 41,15π
2π = 4,86 λ = β
v ph =
ω = 4,37 108 β
·
−1
b = 1,02
k Z T10E = Z o = 174,97π Ω β
Et
π = jE o cos( x)e− jβz ( a
4
−u ) = j 10
Ht =
uz
×E
Z T E
t
x
= j
a
0
1 104 P = 2 174,97π
a
0
104b 174,97π
∗
Ht dxdy
Et
b
sen2 (πx/a)dxdy
0
a2 = 0,3 10−6 2π
·
a = 5 α = 5,75
10
E y
Hx
× 10−
α
H z = H o cos(πx/a)e− jβ z
t
120π
b = 2,5
3
P E o
x
b
0
1 P = 2
−u )
E o π cos( x)e− jβz uy Z T E a
× 1 P = 2
π cos( x)e− jβz ( a
= E o2
ab 4Z o
1
−
f c
f c f
2
Z o =
µo /εo =
10 :
f cT E 10
c 10 c π 3 108 = = =3 k = 2π c 2π a 2 0, 5
· ×
f cT E = 3 10
01
f cT E 01
c 01 c π 3 108 = k = = =6 2π c 2π b 2 0, 25 f = 4, 5
· ×
dB
= 8, 69
neper
α = 0, 05
P = P + P D
L
P = P − P L P L
L
D
D
1 P D α = 2 P
3
3
P = 5, 75 · 10− × 1, 2 · 10 × 2 = 13, 8 D
P = P L = 11, 04 D
E y
H x
D
H z
Ht
− −
− jβ ∇H ⇒ H = − jβ = k k 10
2 c
E y =
z
Z T E H x =
x
10
2 c
k Z o Hx β
∂ H z ∂x
π a
kc =
2 β 10 = ω 2 /c2
−k
2 c
H z = H o cos(πx/a)e− jβ z
H x =
jβ 10 H o sen(πx/a)e− jβ z kc
E y = jZ o
−
− β γ
10
H x =
H o sen(πx/a)sen(ωt
− β
k H o sen(πx/a) sen(ωt γ 10
− β
10
E = Z o
k H o sen(πx/a)e− jβ z γ 10
10 z )ux
10 z )uy
E o E o = Z o
k H o γ 10
− β z )u β H = − E sen(πx/a)sen(ωt − β z )u Z k E = E o sen(πx/a)sen(ωt 10
o
o
P = =
10
10 o
β E ab P = 2Z k 10 o
y
10
2 o
× H) ds =
2 o
2
2 o
x
z
2
2
2 T
(E y H x ) (uy
x
× u ) · u ds β E sen (πx/a)sen (ωt − βz )bdx 2Z k β E ab sen (ωt − βz ) 2Z k (E
o
=
10
T /2
0
sin2(ωt
− β
10 z )dt
=
β 10 2 E ab 4Z o k o
β P = 4Z E ab k 10
2 o
o
k k = β 10 =
β 10 , 2πf ω = c c
− − 2πf c
2
2π λc
E o2ab P = 4Z o
2
1
2πf = c
−
f c f
1
2
f c f
2
