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# FormSecoOnd

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```Formulário de Ondas
R = R\ufffdz ; L = L\ufffdz ; C = C\ufffdz ; G = G\ufffdz
\ufffd@v(z;t)@z = L@i(z;t)@t +Ri(z; t) \ufffd @i(z;t)@z = C @v(z;t)@t + Gv(z; t)
@2v(z;t)
@z2 = LC @
2v(z;t)
@2t
@2i(z;t)
@z2 = LC @
2i(z;t)
@2t v(z; t)) v(z \ufffd ut) u =
q
1
LC
f!(z; t) = f+(z; t) = f(z \ufffd ut) f (z; t) = f\ufffd(z; t) = f(z + ut)
@2v(z;t)
@z2 = LC @
2v(z;t)
@2t + (LG +RC) @v(z;t)@t +RGv(z; t) @
2i(z;t)
@z2 = LC @
2i(z;t)
@2t + (LG +RC) @i(z;t)@t +RGi(z; t)
k =
p
(j!L+R) (j!C + G) = kR + jkI ;
f+(z; t) = Ae
j(\ufffdkIz+!t)e\ufffdkRz; f\ufffd(z; t) = Aej(kIz+!t)e+kRz; Zlinha =
q
j!L+R
j!C+G =
q
L
C
r
1\ufffdj R!L
1\ufffdj G!C
kR = 0) Z0 =
q
L
C ; kI = !
pLC; v(z; t) = v+0e\ufffdkRzej(\ufffdkIz+!t) + v\ufffd0ekRzej(kIz+!t);
i(z; t) = i+0e
\ufffdkRzej(\ufffdkIz+!t) + i\ufffd0ekRzej(kIz+!t) =
v+0
Z0
e\ufffdkRzej(\ufffdkIz+!t) \ufffd v\ufffd0Z0 ekRzej(kIz+!t)
hP i (z) = 12 Re (v(z) \ufffd i\ufffd(z)) = hP i+ (z) + hP i\ufffd (z) =
jvj2+0
2 e
\ufffd2kRz Re( 1Z\ufffdlinha )\ufffd
jvj2\ufffd0
2 e
2kRz Re( 1Z\ufffdlinha
);
vfase =
!
kI
; vgrupo =
@!
@kI
= 1@kI
@!
; \ufffd0 = j\ufffd0j ej\ufffdr = ZL\ufffdZ0ZL+Z0 ; ZL = Z0
1+j\ufffd0jej\ufffdr
1\ufffdj\ufffd0jej\ufffdr ;
kR = 0) \ufffd(z) = v\ufffd(z)v+(z) =
v\ufffd0ej(kIz)
v+0e
j(\ufffdkIz)
= Z(z)\ufffdZ0Z(z)+Z0 = \ufffd0e
j2kIz = j\ufffd0j ej\ufffdrej2kIz
v(z) = v+0
\ufffd
ej(\ufffdkIz) + j\ufffd0j ej\ufffdrej(kIz)
\ufffd
i(z) = v+0Z0
\ufffd
ej(\ufffdkIz) \ufffd j\ufffd0j ej\ufffdrej(kIz)
\ufffd
;
ROE = jvmaxjjvminj =
1+j\ufffd0j
1\ufffdj\ufffd0j j\ufffd0j = ROE\ufffd1ROE+1 ; vrms(\ufffdl) =
jv+0jp
2
q
1 + j\ufffd0j2 + 2 j\ufffd0j cos(2kI l \ufffd \ufffdr)
irms(\ufffdl) = jv0+jp2Z0
r\ufffd
1 + j\ufffd0j2
\ufffd
\ufffd 2 j\ufffd0j cos(2kI l \ufffd \ufffd);Z(\ufffdl) = Z0 1+j\ufffd0je
j\ufffdr ej(\ufffd2kI l)
1\ufffdj\ufffd0jej\ufffdr ej(\ufffd2kI l)
Z(\ufffdl) = Z0 ZL+jZ0 tan(kI l)Z0+jZL tan(kI l)
ZL
Z0
= RZ0 + j
X
Z0
= r + jx; YLZ0 = GZ0 + jBZ0 = g + jb
v\ufffd(0)
v+(0)
= \ufffd0;
v0+(0)
v+(0)
= \ufffd0; 1 + \ufffd0 = \ufffd0; 1\ufffd \ufffd0 =
\ufffd
Z01
ZL
+ Z01Z02
\ufffd
\ufffd0; \ufffd0 =
1\ufffd
\ufffd
Z01
ZL
+
Z01
Z02
\ufffd
1+
\ufffd
Z01
ZL
+
Z01
Z02
\ufffd ; \ufffd0 = 2
1+
\ufffd
Z01
ZL
+
Z01
Z02
\ufffd
hP i+ = hP i\ufffd + hP idiss + hP i0+ ) 1 = j\ufffd0j2 +
\ufffd
Z01Re(
1
Z\ufffdL
) + Z01Z02
\ufffd
j\ufffd0j2
\ufffdl =
Rl
Z0
\ufffd1
Rl
Z0
+1
\ufffdg =
Rg
Z0
\ufffd1
Rg
Z0
+1
VT = V
+
0
\ufffd 1P
i=0
(\ufffdl\ufffdg)
i
+ \ufffdl
1P
i=0
(\ufffdl\ufffdg)
i
\ufffd
V +0 = Vg
Z0
Z0+Rg
VT = Vg
Rl
Rl+RgH
S
\ufffd!
D \ufffd d\ufffd!s = R \ufffddV ;\ufffd!r \ufffd \ufffd!D = \ufffd; H
S
\ufffd!
B \ufffd d\ufffd!s = 0;\ufffd!r \ufffd \ufffd!B = 0;H \ufffd!
E \ufffd d\ufffd!l = \ufffd @@t
R \ufffd!
B \ufffd d\ufffd!s ;\ufffd!r \ufffd\ufffd!E = \ufffd @@t
\ufffd!
B ;
H \ufffd!
H \ufffd d\ufffd!l = R \ufffd!J \ufffd d\ufffd!s + @@t R \ufffd!D \ufffd d\ufffd!s ;\ufffd!r \ufffd\ufffd!H = \ufffd!J + @@t\ufffd!D\ufffd!
D = \ufffd
\ufffd!
E ;
\ufffd!
B = \ufffd
\ufffd!
H ;
\ufffd!
J = \ufffd
\ufffd!
E r \ufffd \ufffd!E = \ufffd\ufffd ; r \ufffd
\ufffd!
H = 0; r\ufffd\ufffd!E = \ufffd\ufffd @@t
\ufffd!
H ; r\ufffd\ufffd!H = \ufffd\ufffd!E + \ufffd @@t
\ufffd!
E
r \ufffd \ufffd!E = 0 r \ufffd \ufffd!H = 0 r\ufffd\ufffd!E = \ufffd\ufffd @@t
\ufffd!
H r\ufffd\ufffd!H = \ufffd @@t
\ufffd!
E\ufffd!
k == bz =) @@zEx(z; t) = \ufffd\ufffdr\ufffd0 @@tHy(z; t); @@zHy(z; t) = \ufffd\ufffdr\ufffd0 @@tEx(z; t);\ufffd!
k == bz =) @@zEy(z; t) = \ufffdr\ufffd0 @@tHx(z; t); @@zHx(z; t) = \ufffdr\ufffd0 @@tEy(z; t)
k2 = \ufffde\ufffdr\ufffd0e\ufffdr\ufffd0!2 =) k = kR + jkI ; vf = !kI ; kI = 2\ufffd\ufffd ; kR = 1daten ;
Ex;y(z; t) = E
+
x e
\ufffdjkIzej!t; Hy;x(z; t) = H+y e
\ufffdjkIzej!t
E+x =
q
\ufffdr
\ufffdr
q
\ufffd0
\ufffd0
H+y ; E
+
y = \ufffd
q
\ufffdr
\ufffdr
q
\ufffd0
\ufffd0
H+x ; Z =
q
\ufffdr
\ufffdr
q
\ufffd0
\ufffd0
; Z0 =
q
\ufffd0
\ufffd0
\ufffd 377
;
\ufffd!hSi = 12 Re[
\ufffd!
E \ufffd\ufffd!H \ufffd];
D\ufffd!
S
E
= 12 jE0j2Re
\ufffd
1
Z\ufffd
\ufffd bk
\ufffd!
E (\ufffd!r ; t) == \ufffd!E 0ej!te\ufffdj
\ufffd\ufffd!
kI \ufffd\ufffd!r
\ufffd
=
\ufffd!
E 0e
j!te\ufffdj(kIxx+kIyy+kIzz);
\ufffd!
H (\ufffd!r ; t) = \ufffd!H 0ej!te\ufffdj
\ufffd\ufffd!
kI \ufffd\ufffd!r
\ufffd
=
\ufffd!
H 0e
j!te\ufffdj(kIxx+kIyy+kIzz);bE \ufffd bH = bkI ; Zprop; contraprop = \ufffdp\ufffd\ufffd\ufffd!
E 0 = E0;xbx+ E0;yby =) \ufffd!E (z; t) = jE0;xj\ufffdbx+ jE0;yjjE0;xjej\ufffdby\ufffd ej!t
\ufffd1 (E?;i + E?;r)\ufffd \ufffd2E?;t = 0; \ufffd1 (H?;i +H?;r)\ufffd \ufffd2H?;t = 0; Eq;i + Eq;r \ufffd Eq;t = 0; Hq;i +Hq;r \ufffdHq;t = 0
E?;i + E?;r = E?;total = 1\ufffd1
Q
A ; H?;i +H?;r = H?;total = 0; Eq;i + Eq;r = Eq;total = 0; Hq;i +Hq;r = Hq;total =
I
l
\ufffd = Z2\ufffdZ1Z2+Z1 ; \ufffd =
2Z2
Z2+Z1
; \ufffdi = \ufffdr;
1
Z1
sin(\ufffdi) =
1
Z2
sin(\ufffdt)
\ufffd? =
Z2
cos(\ufffdt)
\ufffd Z1
cos(\ufffdi)
Z2
cos(\ufffdt)
+
Z1
cos(\ufffdi)
; \ufffd? =
2
Z2
cos(\ufffdt)
Z2
cos(\ufffdt)
+
Z1
cos(\ufffdi)
; \ufffdq =
Z2 cos(\ufffdt)\ufffdZ1 cos(\ufffdi)
Z2 cos(\ufffdt)+Z1 cos(\ufffdi)
; \ufffd q =
Z2
Z1
2Z1 cos(\ufffdi)
Z2 cos(\ufffdt)+Z1 cos(\ufffdi)
tan(\ufffdi;B) =
Z1
Z2
= n2n1 tan(\ufffdt) =
Z22
Z21
tan(\ufffdi;1) tan(\ufffdi;2) =
Z21
Z22
tan(\ufffdt)
k2 = k2x + k
2
z ; k1x = k2x =) ki sin(\ufffdi) = kr sin(\ufffdr); ki sin(\ufffdi) = kt sin(\ufffdt)
kIi = kIr =
p
\ufffdr;1
!
c ; sin(\ufffdi) = sin(\ufffdr) =) \ufffdi = \ufffdr
kIt =
p
\ufffdr;2
!
c =) kIi sin(\ufffdi) = kIt sin(\ufffdt) =) sin(\ufffdt) = n1n2 sin(\ufffdi) cos(\ufffdt) =
r
1\ufffd
\ufffd
n1
n2
sin(\ufffdi)
\ufffd2
k2I2 = k
2
I2x + k
2
I2z;
\ufffd
n2
!
c
\ufffd2
=
\ufffd
n1
!
c
\ufffd2
sin2(\ufffdi) + k
2
I2z =) kI2z = !c
q
n22 \ufffd n21 sin2(\ufffdi)
1
kI2z = Re) \ufffd!E (x; z)2 =
\ufffd!
E 0e
\ufffdjkI2zze\ufffdjkI2xx; kI2z = Im) \ufffd!E (x; z)2 =
\ufffd!
E 0e
\ufffdkR2zze\ufffdjkI2xx
k2 = k2R + jk2I ; k2 = j
pe\ufffd2re\ufffd2rp\ufffd0\ufffd0!;e\ufffdr = 1; \ufffd jk2 = !c (n2 \ufffd j\ufffd2) ; sin(\ufffdt) = n1(n2\ufffdj\ufffd2) sin(\ufffdi); cos(\ufffdt) =
r
1\ufffd
\ufffd
n1
(n2\ufffdj\ufffd2) sin(\ufffdi)
\ufffd2
\ufffdk22 = \ufffdk22x \ufffd k22z =)
\ufffd
(n2 \ufffd j\ufffd2) !c
\ufffd2
=
\ufffd
n1
!
c
\ufffd2
sin2(\ufffdi)\ufffd k22z; k2z = j !c
q
(n2 \ufffd j\ufffd2)2 \ufffd n21 sin2(\ufffdi) = kz;R + jkz;I
k = j !c
r\ufffd
\ufffdr \ufffd j \ufffd\ufffd0!
\ufffd
; n (!)\ufffd j\ufffd (!) =
r\ufffd
\ufffdr \ufffd j \ufffd\ufffd0!
\ufffd
; kI =
!
c n (!) ; kR =
!
c \ufffd (!) ; Z = Z0
1p
\ufffdr\ufffdj \ufffd\ufffd0!\ufffd!
E (\ufffd!r ) = \ufffd!E 0e\ufffd
\ufffd!
k R\ufffd\ufffd!r e\ufffdj
\ufffd!
k I \ufffd\ufffd!r ;
\ufffd!
H (\ufffd!r ) =
\ufffd!
E 0
Z e
\ufffd\ufffd!k R\ufffd\ufffd!r e\ufffdj
\ufffd!
k I \ufffd\ufffd!r ;
\ufffd!
E (\ufffd!r ) = \ufffd!E 0e\ufffd!c \ufffd(!)bk\ufffd\ufffd!r e\ufffdj !c n(!)bk\ufffd\ufffd!r ; \ufffd!H (\ufffd!r ) = \ufffd!E 0Z e\ufffd!c \ufffd(!)bk\ufffd\ufffd!r e\ufffdj !c n(!)bk\ufffd\ufffd!r ;
D\ufffd!
S
E
(\ufffd!r ) = jE0j22Z0 n (!) e\ufffd2
!
c \ufffd(!)
bk\ufffd\ufffd!r bk
! << \ufffd\ufffd0 ; \ufffdr <<
\ufffd
\ufffd0!
=)
r\ufffd
\ufffdr \ufffd j \ufffd\ufffd0!
\ufffd
\ufffd
q
\ufffd
2\ufffd0!
(1\ufffd j) ; k =
q
\ufffd0!\ufffd
2 (1 + j) ; Z =
q
\ufffd0!
2\ufffd (1 + j) ;
\ufffd!
E (\ufffd!r ) = \ufffd!E 0e\ufffd
p
\ufffd0!\ufffd
2 (1+j)
bk\ufffd\ufffd!r ; \ufffd!H (\ufffd!r ) = \ufffd!E 0q \ufffd\ufffd0! e\ufffdj \ufffd4 e\ufffdp\ufffd0!\ufffd2 (1+j)bk\ufffd\ufffd!r ;
hSzi (z) = 12 jE0j2
q
\ufffd
2\ufffd0!
e\ufffd
p
2\ufffd0!\ufffd z; hSiJoule (\ufffd!r ) = 12 Re
h\ufffd!
E \ufffd \ufffd!J \ufffd
i
= 12 jE0j2 \ufffde\ufffd
p
2\ufffd0!\ufffd
bk\ufffd\ufffd!r
k2I = k
2
Ix + k
2
Iz; k
2
I =
\ufffd
n!c
\ufffd2
=
\ufffd
2\ufffd n\ufffd0
\ufffd2
=
\ufffd
2\ufffd 1\ufffdx
\ufffd2
+
\ufffd
2\ufffd 1\ufffdz
\ufffd2
; kIz = 0) fc;l = cn 12a l; \ufffdc;l = cn 1fc;l
vf;z =
!
kIz
= cn
1r
1\ufffd
\ufffd
fc;l
f
\ufffd2 ; vg;z = d!dkIz = cn
r
1\ufffd
\ufffd
fc;l
f
\ufffd2
k2I = k
2
Ix + k
2
Iy + k
2
Iz; kIz (!) =
q\ufffd
!n
c
\ufffd2 \ufffd \ufffd 2\ufffd2a l\ufffd2 \ufffd \ufffd 2\ufffd2bm\ufffd2; kIz = 0) fc;lm = cnq\ufffd 12a l\ufffd2 + \ufffd 12bm\ufffd2;
vf;z =
!
kIz
= cn
1r
1\ufffd
\ufffd
fc;lm
f
\ufffd2 ; vg;z = d!dkIz = cn
r
1\ufffd
\ufffd
fc;lm
f
\ufffd2
Hz (x; y) = H0;z cos(kIxx) cos(kIyy)
Ey (x; y) =
j
k2I?
\ufffd!
\ufffd
l\ufffda
\ufffd
H0;z sin(l
\ufffd
ax) cos(m
\ufffd
b y); Hx (x; y) = \ufffd jk2I? kIz
\ufffd
l\ufffda
\ufffd
H0;z sin(l
\ufffd
ax) cos(m
\ufffd
b y)
Ex (x; y) =
j
k2I?
\ufffd!
\ufffd
m\ufffdb
\ufffd
H0;z cos(l
\ufffd
ax) sin(m
\ufffd
b y); Hy (x; y) =
j
k2I?
kIz
\ufffd
m\ufffdb
\ufffd
H0;z cos(l
\ufffd
ax) sin(m
\ufffd
b y)
Zxy;TE =
Ex
Hy
\ufffd
TE
= \ufffd0!kIz =
Z0
n
1r
1\ufffd
\ufffd
fc;lm
f
\ufffd2 ; Zyx;TE = EyHx
\ufffd
TE
= \ufffd\ufffd0!kIz = \ufffdZ0n 1r
1\ufffd
\ufffd
fc;lm
f
\ufffd2
hSxyiTE = (\ufffd)
\ufffd!kIzH
2
0;z
2
\ufffd
(l\ufffda )
2
+(m\ufffdb )
2
\ufffd2 \ufffdm\ufffdb \ufffd2 cos2(l\ufffdax) sin2(m\ufffdb y); hSyxiTE = (\ufffd) \ufffd!kIzH20;z2\ufffd(l\ufffda )2+(m\ufffdb )2\ufffd2 \ufffdl\ufffda \ufffd2 sin2(l\ufffdax) cos2(m\ufffdb y)
PTE =
ZZ
hSiTE dxdy =) PTE;xy =
\ufffd!kIzH
2
0;z
8
\ufffd
(l\ufffda )
2
+(m\ufffdb )
2
\ufffd2 \ufffdm\ufffdb \ufffd2 ab; PTE;yx = \ufffd!kIzH20;z8\ufffd(l\ufffda )2+(m\ufffdb )2\ufffd2 \ufffdl\ufffda \ufffd2 ab
Ez (x; y) = E0;z sin(kIxx) sin(kIyy)
Ey (x; y) =
\ufffdj
k2I?
kIz
\ufffd
m\ufffdb
\ufffd
E0;z sin(l
\ufffd
ax) cos(m
\ufffd
b y); Hx (x; y) =
j
k2I?
\ufffd!
\ufffd
m\ufffdb
\ufffd
E0;z sin(l
\ufffd
ax) cos(m
\ufffd
b y)
Ex (x; y) =
\ufffdj
k2I?
kIz
\ufffd
l\ufffda
\ufffd
E0;z cos(l
\ufffd
ax) sin(m
\ufffd
b y); Hy =
\ufffdj
k2I?
\ufffd!
\ufffd
l\ufffda
\ufffd
E0;z cos(l
\ufffd
ax) sin(m
\ufffd
b y)
Zxy;TM =
Ex
Hy
\ufffd
TM
= kIz\ufffd! =
Z0
n
r
1\ufffd
\ufffd
fc;lm
f
\ufffd2
; Zyx;TM =
Ey
Hx
\ufffd
TM
= \ufffdkIz\ufffd! = \ufffdZ0n
r
1\ufffd
\ufffd
fc;lm
f
\ufffd2
hSxyiTM = (\ufffd)
\ufffd!kIzE
2
0;z
2
\ufffd
(l\ufffda )
2
+(m\ufffdb )
2
\ufffd2 \ufffdl\ufffda \ufffd2 cos2(l\ufffdax) sin2(m\ufffdb y); hSyxiTM = (\ufffd) \ufffd!kIzE20;z2\ufffd(l\ufffda )2+(m\ufffdb )2\ufffd2 \ufffdm\ufffdb \ufffd2 sin2(l\ufffdax) cos2(m\ufffdb y)
PTM =
ZZ
hSiTM dxdy =) PTM;xy =
\ufffd!kIzE
2
0;z
2
\ufffd
(l\ufffda )
2
+(m\ufffdb )
2
\ufffd2 \ufffdl\ufffda \ufffd2 ab PTM;yx = \ufffd!kIzE20;z2\ufffd(l\ufffda )2+(m\ufffdb )2\ufffd2 \ufffdm\ufffdb \ufffd2 ab
sin(\ufffdentr;crit) =
r
1\ufffd
\ufffd
n2
n1
\ufffd2
; sin(\ufffdentr;crit;0) =
n1
n0
r
1\ufffd
\ufffd
n2
n1
\ufffd2
X tan(X) = Y ; X2 + Y 2 = R2 kIy
d
2 = X; kRy
d
2 = Y ; R =
\ufffd
\ufffd0
d
p
n21 \ufffd n22
Y = 0 =) X = R =) R tan(R) = 0; R = l\ufffd; l = 0; 1:::
\ufffdX cot(X) = Y ; X2 + Y 2 = R2 kIy d2 = X; kRy d2 = Y ; R = \ufffd\ufffd0 d
p
n21 \ufffd n22
Y = 0 =) X = R =) R cot(R) = 0; R = \ufffdl + 12\ufffd\ufffd; l = 0; 1:::
Constantes de interesse:
\ufffd0 = 8:85pF=m \ufffd0 = 1260nH=m Z0 = 377
c = 3\ufffd 108m=s
2```