CC
∗
Ybarra
∗
BY:
$ \
8, 8541878 10−12
µ0 = 4 π 10−7
·
∼ 0 , 5 mm2 =
ε0 =
8, 8541878 10−12
µ0 = 4 π 10−7
·
∼ 0 , 5 mm2 =
ε0 =
a b
a
b
Y = G + jB
G
B
R = ρ ρ
l S σ = 1/ρ l
S
mm2 /m
·
4
×
−1
g/cm3
/km20
r
/km75
I 0, 5 10−7
·
µ0 Lii = 2π
1 2h + ln 4 r
Lii
i r µ0 = 4π 10−7 H/m
h
·
hm hm = h t
− 23 f = hv + 13 f
ht
hv
f µ0 Lii = 2π
Lii = Lii =
2h ln e + ln r 1 4
µ0 2h ln 2π r e
1 4
µ0 2h ln 2π r
r
∼
µ = µ0 r = r e−1/4 = 0, 7788r µr µ0 r = r e− µr = µ/µ 4
µr
0
µ0
Lij = Dij
i i
µr ∼ = 50
j
µ0 D ij ln 2π dij j
dij
∅
29, 59
µ = µ 0 Laa =
µ0 2h 2 20 ln = 2 10−7 ln 0,02959 = 1, 6305 10−6 H/m 2π r /2 0, 7788
Lab =
·
·
µ0 D ab ln = 2 10−7 ln 2π dab
·
·
·
√ 402 + 82 8
= 1, 981 10−7 H/m
·
˙m Z
˙m Z
∝ ln Ddijij
Z m
Dij = 102 + 602 = 60, 8 dij = 10 Z m 1, 8055 Z m(a) 2,4884 1,8055
= 37, 8%
∝ √ ∝ ln 5 5+60
Z m(b) 1
− 10,,8047 8055 = 55, 4%
2
∝
ln
2
√
= 2, 4884
102 +202 10
= 0, 8047
1
−
n
n
RMG =
n
n2
dij =
i=1 j =1
n2
(d11 d12
dii
··· d1n )(d21 d22 . . . d2n) ··· (dn1 dn2 ··· dnn)
i
dij
i
ri
j
d
√ RMG2 = r d √ 2 RMG3 = r d √ 3
4
RMG4 = 1, 09 r d3 RMG2 RMG3 RMG4 N R RM G =
√
N
r N RN −1 r
√ 3 RMGZ 4 = 1, 09 r d √ 3 4
4
RMGY 4 = 1, 09 r d M
L
=
µ MZ 2π
2 hi ri D ij = ln dij
M Z ii = ln M Zij
Z
= R I
+ j
ωL
= R I
µ MZ + j ω 2π
Z
=
R 0 0 R 0 0
0 0 R
µ + j ω 2π
R
ri
ln 2rha a Dba ln dba ca ln D dca
ab ln D dab ln 2rhb b Dcb ln dcb
ac ln D dac bc ln D dbc ln 2rhc
c
I
RMGi
0,0733 4
= 0, 018325
RMG = 1, 09
2·16 3
4
0, 02959 0, 7788 0, 4573 = 0, 1985 m 2
·
= 23, 33
28
− 2·316 = 17, 33
2 17, 33 = 5, 16277 0, 1985 2 23, 33 M bb = ln = 5, 46002 0, 1985 M cc = M aa
M aa = ln
· ·
34
−
M ab = ln
·
52 + (23, 33 + 17, 33)2 52 + (23, 33
M bc = M ab M ac = ln
− 17, 33)2 = 1, 65731
(2 5)2 + (2 17, 33)2 = 1, 28298 2 5
·
·
x h xj )2 + (hi + hj )2 ] Dij = sqrt[(xi
dij = sqrt[(xi Mij = log[Dij /dij ]
−
−x ) j
2
+ (hi log[x]
2
−h ) ] j
ln(x) M
M
Z
Z
=
5, 1627716 1, 6573122 1, 2829804 1, 6573122 5, 4600231 1, 6573122 1, 2829804 1, 6573122 5, 1627716 L L
M
jω
µ0 2π
R
µ0 M 2π 0, 018325 + j0, 3892730 j0, 1249613 j0, 0967367 j0, 1249613 0, 018325 + j0, 4116857 j0, 1249613 j0, 0967367 j0, 1249613 0, 018325 + j0, 3892730
=R + j ω =
/km
V
0,457 4
5
4 3 8 2
V =
q 2h ln 2πε 0 r
2h q C = = 2 π ε0 ln V r n
−1
MY C
MZ
= 2 π ε0 MY −1 8, 85 10−12
ε0
·
2 hi ri D ij = ln dij
M Y ii = ln M Y ij MY
Y
= G + j ω C
= j ω C
Y
M
RMG = 1, 09
4
0, 02959 0, 4573 = 0, 2113 m 2
2 17, 33 = 5, 10027 0, 2113 2 23, 33 M bb = ln = 5, 39752 0, 2113 M cc = M aa
M aa = ln
· ·
M
M
−−
=
−1 = M
5, 1002713 1, 6573122 1, 2829804 1, 6573122 5, 3975229 1, 6573122 1, 2829804 1, 6573122 5, 1002713 0, 224171 0, 0572269 0, 0377949
−0, 0572269 −0, 0377949 0, 2204133 −0, 0572269 −0, 0572269 0, 224171 C
2π ε0
j ω Y
= j ω 2 π ε0 M−1 =
−−
j4, 6994162 j1, 1996776 j0, 7923144
− j1, 1996776 − j0, 7923144 j4, 6206408 − j1, 1996776 − j1, 1996776 j4, 6994162
S/km 10−9
Z
1 Z= 3 1 = 3
(1) Z
+Z
(2)
Z
(1)
Z
(2)
Z
(3)
=
(3) Z
+
Z(1) Z(2)
Y
Z aa + Z bb + Z cc Z ba + Z cb + Z ac Z ca + Z ab + Z bc
=
=
Z aa Z ba Z ca
Z ab Z bb Z cb
Z ac Z bc Z cc
Z bb Z cb Z ab
Z bc Z cc Z ac
Z ba Z ca Z aa
Z cc Z ac Z bc
Z ca Z aa Z ba
Z cb Z ab Z bb
Z ab + Z bc + Z ca Z bb + Z cc + Z aa Z cb + Z ac + Z ba
Z ac + Z ba + Z cb Z bc + Z ca + Z ab Z cc + Z aa + Z bb
1 Z p = (Z aa + Z bb + Z cc ) 3 Z ij = Z ji 1 Z m = (Z ab + Z bc + Z ca ) 3
Z
=
Z p Z m Z m
Z m Z p Z m
Z m Z m Z p
Z(3)
Y
=
Y p Y m Y m
Y m Y p Y m
Y m Y m Y p
1 Y p = (Y aa + Y bb + Y cc ) 3 1 Y m = (Y ab + Y bc + Y ca ) 3
Z p Z m Y p Y m
= 0, 018325 + j0, 3967439 = j0, 115553 /km = j4, 6731578 S/km = j1, 0638899 S/km
/km
−
Z A
A
a = 1 120
a2 = 1 Z012
=
−120
1 1 1 a2 1 a
1 a a2
Z
Y
Z012
Z012
= A−1 Z A =
=
Y012
Z p + 2Z m 0 0
= A−1 Y A =
=
Z 0 0 0
Y s + 2Y m 0 0
0 Z 1 0
0 0 Z 2
0 Z p
− Z m 0
Y 0 0 0
0 Y 1 0
− Y m 0
Z 1 = Z p Y 1 = Y p
0 0
Z p
− Z m
0 0 Y 2
0 Y s
0 0
Y s
− Y m
− Z m − Y m Z c
γ
Z 1 Z Y Z c =
Y012
Z Y
Z = R + j X l = R + j ω L Y = j Bc = j ω C
Y 1
Z c Z c R
∼ ∼ X l = Bc
Z c =
γ =
√
Y Z =
m−1 γ = α + j β
L C
(R + j ω L) j ωC
α
β λ λ =
2π β γ
β
√ ∼ γ = j ω L j ω C = j ω L C √ ∼ β = ω L C
θ = β l
Z c2
Z c0
f v = λ f v=
√ L1 C
P c
U 02 P c = Z c U 0
Z 1 = 0, 018325 + j0, 2811908 Y 1 = j5, 7370477 S/km
Z c = 221, 506
− j7, 2100622
/km
γ = (0, 0413645 + j1, 2707934) 10−6 Np/m Z c
γ Z c
·
Qc = V 2 Bc
Bc = 5, 7370477 10−9
·
π Y 1 Z e = Z l Yl Y e2 = 2
Z e = Z c sinh γ l 1 γ l Y e2 = tanh Z c 2 Y e2 Z c
γ Z e
Y e2
Z 1
Ze
I1
Ye2
V1
Z e
I2
Ye2
Y e2
Ybarra
z Z
y Y
˙ Z
Z e = 5, 2343219 + j82, 339206 Y e2 = 0, 6986822 + j871, 12182 S
V2
Z e = 221, 5 sinh j1, 2707936 10−6 300 103 = 221, 5 j
·
·
·
(0, 381238) = j82, 415911
3 1 6 300 10 − Y e2 = tanh j1, 2707936 10 221, 5 2 1 = j (0, 190619) = j871, 13386 S 221, 5
X L
·
·
600
500
400
300
200
100
l km 500
1000
1500
2000
Z e
BC mS
15
10
5
l km 500
1000
1500
2000
Y e2
V 1
I 1
V 2
√ √ 2 3
I 2
Vs = 1 pu
Vr
Sr
l
I 1
I 2 V 1 = A V 2 + B I 2 I 1 = C V 2 + D I 2
V 1 I 1
=
V 2 I 2
T
V 1 = V 1 =
=
A C
B D
V 2 I 2
Y V 2 + I 2 Z + V 2 2 Z Y + 1 V 2 + Z I 2 2
Y Y + V 2 + I 2 2 2 Z Y I 1 = V 2 Y 1 + + 4 I 1 = V 1
Z Y + 1 I 2 2
Z Y +1 2 B = Z A =
Z Y C = Y 1 + 4 D = A AD
− B C = 1
V 1 = V 2 cosh(γ l) + I 2 Z c sinh(γ l) V 2 I 1 = I 2 cosh(γ l) + sinh(γ l) Z c
A = cosh(γ l) B = Z c sinh(γ l) 1 C = sinh(γ l) Z c D = A
π Z
Z e
Y
V 1 =
Y e
Z e Y e + 1 V 2 + Z e I 2 2
Z e = Z c sinh(γ l)
Z e Y e + 1 = cosh(γ l) 2 Y e Z c sinh(γ l) + 1 = cosh(γ l) 2 Y e 1 cosh(γ l) 1 = 2 Z c sinh(γ l)
−
tanh
x cosh x 1 = 2 sinh x
−
Y e 1 γ l = tanh 2 Z c 2
˙ 2 = S φ S
V 2
I 2 U 0 ˙ 2 = U 0 V 3 S 2 I ˙2 = U 0 3
√
√ −φ
U 0
˙ 2 = P c S V 2 =
0
I 2 = 0
˙ D = ˙ A = 0, 9672 0, 23 B˙ = 75, 15 83, 2 ˙ = j8, 633 10−4 S C
·
V 2 =
345 kV I 2 = 669, 39 A 3
√
√
V 1 = 204, 98 3 = 355, 04 kV ˙ 1 = (412, 32 S ∆P = 12, 32
I 1 = 670, 55 A
− j5, 48)
sinh jβ = j β cosh jβ = cos β tanh jβ = j β sinh α = j jα cosh α = cos jα tanh α = j jα sinh(α + jβ ) = sinh α cos β + j cosh α cosh(α + jβ ) = cosh α cos β + j sinh α
− −
Q1
Q2
·
A2 C 2
β β
Q = Q 1 Q2
·
V 1 I 1
=
A1 C 1
B1 D1
=
A1 A2 + B1 C 2 C 1 A2 + D1 C 2
B2 D2
V 2 I 2
A1 B2 + B1 D2 C 1 B2 + D1 D2
=
V 2 I 2
Q = Q2 Q1 Q = Q
z = j0, 34
/
y = j4, 8
·
Z c = 266, 1453 γ = j1, 2775 10−3 Np/km
·
A = 0, 7203 B = j184, 6 C = j2, 606 10−3 S
·
A = 0, 9275 B = j99, 52 C = j1, 405 10−3 S
·
750 103 V 2 = = 433, 0 kV 3 2 109 = 1539, 6 A I 2 = 3 750 10 3
√ · · √ ·
744, 5 20, 88
V m =
I m = 1552, 1 23, 07 V 1 = 730, 9 42, 34
1582, 2 45, 50 A = 0, 9971 B = j20, 38 C = j2, 8772 10−4 S
·
I 1 =
V 1 I 1
=
C 1 +
A1 B2 +A2 B1 B1 +B2 2 )(D2 −D1 ) C 2 + (A1 −A B1 +B2
V 1 I 1
=
B 2
A 2C + D
B1 B2 B1 +B2 B2 D1 +B1 D2 B1 +B2
V 2 I 2
V 2 I 2
P=0
1200
P = 2 GW P = 4 GW 1100
1000
] V k [ o a s n e T
900
800
700
600
500 1
2
3
4
5
6
7
8
9
10
11
V 1
V 2
I 1
I 2
˙I = V 1 V 2 ˙ 3 Z
−√
I 1 = I + I Y 2 I 2 = I I Y 2
−
345 0
6 + j50 V 2 = 320 10
V 1 =
−
V 1
˙ 1 V ˙2 V ˙ I = = 723, 32 ˙ Z 3
−√
V 2
−21, 4
A
∆P = 3 R I 2 = 9, 4 (392, 93 + j79, 30)
P =
|V 1| · |V 2| X
∼
δ = 383, 4 MW
˙ 2 = 3 V ˙ 2 I ˙∗ = S
Z c
∼ L = C
Z c =
X l Bc
X c = n s X l ns
− ∼
Z c =
X l
Bc
X c
√ − n
= Z c 1
∼ √ − ns
β = β 1
X L
− X C
s
C s
A = 1 1 = j ns X l j ω C s C = 0 D=1 B=
−
L p A = 1 B=0 C =
1 = j n p Bc j ω L p
−
D = 1 n p
QLT Ql
Qc Q = Q c Ql QLT Ql Qc
· ·
· ·
I1
- j Xc
V1 - j Bl
Ze = R + j Xl
- j Xc
Ye = j Bc
- j Bl
I2
V2
Z = 10 + j 250 Y = j10
Z i =
j ω µ I 0 (ρ) 2 π ρ I 1 (ρ)
ρ = r I 0
I 1
−
j ω σ µ σ
µ
α σf = σ 0 [1 + α(θ0 σ0
− θ)]
θ0
Z
=
Z ia 0 0
0 Z ib 0
0 0 Z ic
+ j ω
µ M 2π
M
=
Z i
ln ln ln
2 ha ra Dba dba Dca dca
ln ln ln
Dab dab 2 hb rb Dcb dcb
ln ln ln
Dac dac Dbc dbc 2 hc rc
r
M
a b n
Z
n
=
Z a11
Z a12
Z a1n
Z a1b1
Z a21 Z an1
Z a22 Z an2
Z a2n Z ann
···
Z b1a1 Z c1a1 Z cna1
···
Z b11
Z b1cn
Z cnn
···
Z
=
Zaa
Zab
Zac
Zba
Zbb
Zbc
Zca
Zcb
Zcc
Z 1 = R 1 + j ω L1 Y 1 = j ω C 1 C 1
Z a1cn
Z c11
×n
R1 L1
Z a1c1
3
×3
c
R0 R1 2
10
) m k /
1
10
Ω (
a i c n e t s i s e R
0
10
−1
10
−2
10
−1
10
0
10
1
10
2
10 Frequencia (Hz)
3
10
4
10
5
10
6
10
7 L0 L1
6
5 ) m k / H 4 m ( a i c n a 3 t u d n I
2
1
0 −2 10
−1
10
0
1
10
2
10
10 Frequencia (Hz)
10.000
3
10
4
5
10
10
6
10
·
h h = h + d d = σ
1 = σjωµ
√
ρ j ω µ
ω
µ µ0
ρ1
ρ2
900 Zc0 Zc1 800 ) Ω ( a 700 c i t s i r e t c a r a 600 c a i c n a d e p 500 m I
400
300 −2
−1
10
0
10
1
10
10
2
10 Frequencia (Hz)
3
4
10
10
100 1 = 324, 87 7 − 1/100 j 2 π 60 4π10
5
6
10
10
·
d =
10
·
L =
·
− j324, 87 m
d = 102, 73 j102, 73 d = 25, 16 25, 16
−
10
−
·
µ0 2 10 ln = 1, 5202 10−6 H/m 2π 0, 01
·
·
100 µ0 2 (10 + 324, 87 j324, 87) L = ln = (2, 2887 j0, 1541) 10−6 H/m 2π 0, 01
·
·
−
−
Z L = j ω L = (0, 0581 + j0, 8629) 10−3 /m
·
·
10 L =
·
µ0 2 (10 + 102, 73 j102, 73) ln = (2, 0651 j0, 1478) 10−6 H/m 2π 0, 01
·
−
Z p Z p + 2Z m
−
·
− Z m
V g = 0 I g = 0
va vb vc vg1 vg2
=
zaa zba zca zg1a zg2a
zab zbb zcb zg1b zg2b
zac zbc zcc zg1c zg2c
vu vg
=
vu Zred
= Zuu
zag1 zbg1 zcg1 z g 1g 1 z g 2g 1
Zuu
Zug
Zgu
Zgg
= Zred
zag2 zbg2 zcg2 z g 1g 2 z g 2g 2
iu ig
·i
u
−Z ·Z ug
gg
−1
·Z
gu
ia ib ic ig1 ig2
M
MY MY red
=
= Muu C
Muu
Mug
Mgu
Mgg
−M ·M ug
gg
1 = 2 π ε0 MY − red
−1
·M
gu
µ0 µr = 100 r = r e−
100 4
= 1, 3888 10−11 r
·
Rca = 0, 0448
Rca = 4, 2324
Z
=
Zred
µ = 100µ0
0.01493 + j 0.33657
=
0.01493 + j 0.33657 0.01493 + j 0.33657 4.2324 + j 2.61155 4.2324 + j 2.61155
0.0123768 + j 0.3382288
.0019213 + j 0.1020155 −0.0024816 + j 0.0762942 −0
.0019213 + j 0.1020155 0.0134680 + j 0.3158326 −0.0019213 + j 0.1020155 −0
/km
.0024816 + j 0.0762942 −0.0019213 + j 0.1020155 0.0123768 + j 0.3382288 −0
dV /dx
n I 1 + I 2 + + I n = I f dv1 dv2 dvn = = . . . = dx dx dx
···
n
n n n
Z
= j
0, 9381
0, 6513
0, 6513
0, 4397
0, 4444
0, 4419
0, 3874
0, 3898
0, 3886
0, 6513
0, 9381
0, 6513
0, 4353
0, 4397
0, 4374
0, 3852
0, 3874
0, 3863
0, 6513
0, 6513
0, 9381
0, 4374
0, 4419
0, 4397
0, 3863
0, 3886
0, 3874
0, 4397
0, 4353
0, 4374
0, 9381
0, 6513
0, 6513
0, 4397
0, 4444
0, 4419
0, 4444
0, 4397
0, 4419
0, 6513
0, 9381
0, 6513
0, 4353
0, 4397
0, 4374
0, 4419
0, 4374
0, 4397
0, 6513
0, 6513
0, 9381
0, 4374
0, 4419
0, 4397
0, 3874
0, 3852
0, 3863
0, 4397
0, 4353
0, 4374
0, 9381
0, 6513
0, 6513
0, 3898
0, 3874
0, 3886
0, 4444
0, 4397
0, 4419
0, 6513
0, 9381
0, 6513
0, 3886
0, 3863
0, 3874
0, 4419
0, 4374
0, 4397
0, 6513
0, 6513
0, 9381
0, 9381
0, 6513
0, 6513
0, 4397
0, 4444
0, 4419
0, 3874
0, 0023
0, 0011
0, 6513
0, 9381
0, 6513
0, 4353
0, 4397
0, 4374
0, 3852
0, 0022
0, 0011
0, 6513
0, 6513
0, 9381
0, 4374
0, 4419
0, 4397
0, 3863
0, 0023
0, 0011
0, 4397
0, 4353
0, 4374
0, 9381
0, 6513
0, 6513
0, 4397
0, 0047
0, 0022
0, 4444
0, 4397
0, 4419
0, 6513
0, 9381
0, 6513
0, 4353
0, 0044
0, 0021
0, 4419
0, 4374
0, 4397
0, 6513
0, 6513
0, 9381
0, 4374
0, 0045
0, 0023
0, 3874
0, 3852
0, 3863
0, 4397
0, 4353
0, 4374
0, 9381
0, 3898
0, 3874
0, 3886
0, 4444
0, 4397
0, 4419
0, 6513
0, 3886
0, 3863
0, 3874
0, 4419
0, 4374
0, 4397
0, 6513
0, 9381
0, 6513
0, 6513
0, 4397
0, 4444
0, 4419
0, 3874
0, 0023
0, 0011
0, 6513
0, 9381
0, 6513
0, 4353
0, 4397
0, 4374
0, 3852
0, 0022
0, 0011
0, 6513
0, 6513
0, 9381
0, 4374
0, 4419
0, 4397
0, 3863
0, 0023
0, 0011
0, 4397
0, 4353
0, 4374
0, 9381
0, 6513
0, 6513
0, 4397
0, 0047
0, 0022
0, 4444
0, 4397
0, 4419
0, 6513
0, 9381
0, 6513
0, 4353
0, 0044
0, 0021
0, 4419
0, 4374
0, 4397
0, 6513
0, 6513
0, 9381
0, 4374
0, 0045
0, 0023
0, 3874
0, 3852
0, 3863
0, 4397
0, 4353
0, 4374
0, 9381
0, 0023
0, 0022
0, 0023
0, 0047
0, 0044
0, 0045
0, 0011
0, 0011
0, 0011
0, 0022
0, 0021
0, 0023
, 2868 0, 2868 0, 0000
−0
, 2868 0, 5736 0, 2868
−0
, 2868 −0, 2868 −0
7×7
0, 9381
0, 6513
0, 6513
0, 4397
0, 4444
0, 4419
0, 3886
0, 6513
0, 9381
0, 6513
0, 4353
0, 4397
0, 4373
0, 3863
0, 6513
0, 6513
0, 9381
0, 4373
0, 4419
0, 4397
0, 3874
0, 4397
0, 4353
0, 4373
0, 9381
0, 6513
0, 6513
0, 4420
0, 4444
0, 4397
0, 4419
0, 6513
0, 9381
0, 6513
0, 4375
0, 4419
0, 4373
0, 4397
0, 6513
0, 6513
0, 9381
0, 4397
0, 3886
0, 3863
0, 3874
0, 4420
0, 4375
0, 4397
0, 7469
Zred
= j
0, 7469
0, 4397
0, 3875
0, 4397
0, 7468
0, 4397
0, 3875
0, 4397
0, 7469
/km
, 2868 0, 0000 0, 2868
−0
, 2868 0, 2868 0, 5736
−0
V 1 V 2 V 3 V 4 V 5 V 6
=
z11 z21 z31 z41 z51 z61
z12 z22 z32 z42 z52 z62
z13 z23 z33 z43 z53 z63
z14 z24 z34 z44 z54 z64
z15 z25 z35 z45 z55 z65
z16 z26 z36 z46 z56 z66
I 1 I 2 I 3 I 4 I 5 I 6
V 1 = V 4 V b = V 2 = V 5 V c = V 3 = V 6 I a = I 1 + I 4 I b = I 2 + I 5 I c = I 3 + I 6
V 1
V 6
Z m
Z c0 =
Z 0 Y 0
Y m
V a =
Ybarra
V1 = 1 pu
ZLT
XT1 Xeq1
XCS XT2 Xeq2
XRS Xn
E =
1 λ 4 π ε0 r2
λ λ = C V r
E
E c = 3, 0 106 m δ 1 +
·
δ
m
0, 0308 δr
√
5 nA/m2
√ V m √ 2 3
√
V m 2 10−3
¯1 Z
V ff ff
P C C
Z 1
Y 1
Z c
¯1 Z
1× 1× 4× 4× 6×
Deq =
Z = R + R + j j ω
µ D eq ln 2π r
Y = j 2 π ω ε0
D eq ln r
−1 r
√ d 3
ab
dbc dac
x vt
z
y
il i
v
dv = i z dx di = v y dx x dx dz
dy vt (x) il (x)
dz
i l ( x ) vt ( x )
dy
dx
d2 v di = z dx2 dx d2 i dv = y dx2 dx
√
− √
v = A 1 exp(x y z) + A2 exp( x y z) ea
exp a
d2 v = y z [A1 exp(x y z) + A2 exp( x y z)] dx2
√
i =
− √
1 A1 exp(x y z) z/y
√ −
Z c
1 A2 exp( x y z) z/y
− √
z √ γ ≡ y z
≡
z/y
y x = 0 v = V R
V R = A 1 + A2 1 I R = (A1 A2 ) Z c
−
V R + I R Z c 2 V R I R Z c A2 = 2 A1 =
−
V R + I R Z c V R I R Z c exp(γ x) + exp( γ x) 2 2 V R/Z c + I V R/Z c I R R i(x) = exp(γ x) + exp( γ x) 2 2
−
v(x) =
−
v
i
i2 Ldx dE m = 2 v 2 Cdx dE e = 2 i2 Ldx v 2 Cdx v i d t = + 2 2
− −
i = I R
u = i Z c = i
dE e =
L C
i
r0 Z i =
j ω µ K 1 (ρ0 )I 0 (ρ1 ) + K 0 (ρ1 ) I 1 (ρ0 ) 2 π r0 I 1 (ρ1 )K 1 (ρ0 ) I 1 (ρ0 )K 1 (ρ1 )
ρ1 = r 1
ε G
µ
K 1
Cdx i2 Ldx = = dE m 2 2
r1
ρ0 = r 0
I 0 I 1 K 0
2
− −
j ω σ µ j ω σ µ
−
L C
t = 0
