Embed Size (px)
Citation preview
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ceV
ario
us
equ
atio
ns
[]
22
22
2(
,)
1(
,)
0(
)xt
xt
xt
cx
ψψ
∂∂
−=
∂∂
1D w
ave
equa
tion
22
22
2(
,)
(,
)xy
gxy
xy
ψψ
ψ∂
∂∇
=+
=∂
∂2D
Poi
sson
equ
atio
n
22
2
22
22
(,
,)
(,
,)
1(
,,
)0
xyt
xyt
xyt
xy
ct
ψψ
ψ∂
∂∂
+−
=∂
∂∂
2D w
ave
equa
tion
22
2
22
2
2
22
(,
,,
)(
,,
,)
(,
,,
)
1(
,,
,)
0
xyzt
xyzt
xyzt
xy
z xyzt
ct
ψψ
ψ
ψ
∂∂
∂+
+∂
∂∂
∂−
=∂
3D w
ave
equa
tion
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n
rr
22
22
1(
,)
(,
)0
ttc
tψψ
∂∇
−=
∂
x
y
Orig
inal
wav
e eq
uatio
n
2D w
ave
equa
tion
Reg
ion
of in
tere
st
a
b[0
,],
[0,
]
[0,
]
xay
b
tT
∈∈
∈
Dis
cret
izat
ion
rate
xyh
t
∆=
∆=
∆
x∆
y∆
22
2
22
22
(,
,)
(,
,)
1(
,,
)0
xyt
xyt
xyt
xy
ct
ψψ
ψ∂
∂∂
+−
=∂
∂∂
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
disc
reti
zati
ons
(,
,)
l mn
mnl
xyt
ψψ
=
x
y
Dis
cret
e un
know
ns
xh
∆=
yh
∆=
m =
0m
= M
n =
0
n =
N
0,1,
2,,
0,1,
2,,
0,1,
2,,t
mM
nN
lN
= = =
… … …
22
2
22
22
10
0m n l
xx
yy
tt
xy
ct
ψψ
ψ= = =
∂∂
∂
+
−=
=
∂
∂∂
Test
ing
the
PDE
Use
cen
tral
diff
eren
ce
2
2m n l
xx
yy
tt
xψ= = =
∂
∂
2
2m n l
xx
yy
tt
yψ= = =
∂
∂
2
2m n l
xx
yy
tt
tψ= = =
∂
∂
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
[]
11
1,,
1,,
1,
,1
22
22
22
12
0(
)
ll
ll
ll
ll
lm
nmn
mn
mn
mn
mn
mn
mn
mn
mn
hh
tc
ψψ
ψψ
ψψ
ψψ
ψ+
−+
−+
−−
+−
+−
++
−=
∆
[]
[]
22
11
1,,
1,2
22
,1
,,
12
()
22
()
2
mn
ll
ll
ll
mn
mn
mn
mn
mn
mn
mn
ll
lmn
mn
mn
ct
hc
th
ψψ
ψψ
ψψ
ψψ
ψ
+−
+−
+−
∆
=
−+
−+
∆
+
−+
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
tim
e m
arch
ing
Mar
chin
g on
in t
ime
Dis
cret
e eq
uatio
n
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
BC
x
y
x∆
y∆
m =
0m
= M
n =
0
n =
NTh
e bo
unda
ry v
alue
s ar
e as
sum
ed k
now
n
Two
initi
al c
ondi
tions
in
time
are
also
nee
ded
[]
[]
22
11
1,,
1,2
22
,1
,,
12
()
22
()
2
mn
ll
ll
ll
mn
mn
mn
mn
mn
mn
mn
ll
lmn
mn
mn
ct
hc
th
ψψ
ψψ
ψψ
ψψ
ψ
+−
+−
+−
∆
=
−+
−+
∆
+
−+
01
,mnmn
ψψ
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
(,
,)
()p
qjxjy
ab
pqp
qxyt
Ate
eπ
πψ
+∞
+∞
=−∞=−∞
=∑
∑(
)(
)p
qj
mxj
ny
ll
ab
mn
pqp
qAe
eπ
πψ
+∞
+∞
∆∆
=−∞=−∞
=∑
∑
()
l pqpq
AAlt
=∆
pp
kaπ
=
[]
[]
22
11
1,,
1,2
22
,1
,,
12
()
22
()
2
mn
ll
ll
ll
mn
mn
mn
mn
mn
mn
mn
ll
lmn
mn
mn
ct
hc
th
ψψ
ψψ
ψψ
ψψ
ψ
+−
+−
+−
∆
=
−+
−+
∆
+
−+
[]
[]2
21
12
22
2()
22
()
2
pp
jkx
jkx
mn
ll
ll
ll
pqpq
pqpq
pqpq
jky
jky
mn
ll
lpq
pqpq
ct
AA
AAe
AAe
hc
tAe
AAe
h
∆−
∆+
−
∆−
∆
∆
=
−+
−+
∆
+
−+
kbπ
=
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
stab
ility
Two-
dim
ensi
onal
Fou
rier
mod
e ex
pans
ion
For
one
mod
e
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
[]
[]2
21
12
2
22
22(
)2
4si
n2
()
4si
n2p
mn
ll
ll
pqpq
pqpq
qmn
l pq
kx
ct
AA
AA
h
ky
ctA
h
+−
∆∆
=−
−
∆∆
−
Def
ine
1
1
ll
pqpq
pql
lpq
pq
AA
gA
A
+
−=
= []
[]
22
22
22
22
2(
)(
)(
)2
12
sin
2si
n1
22
pq
mn
mn
pqpq
kx
ky
ct
ct
gg
hh
∆∆
∆∆
=−
−−
2(
)2
10
pqpqpq
gg
χ−
+=
[]
[]
22
22
22
22
()
()
12
sin
2si
n2
2p
qmn
mn
pqkx
ky
ct
ct
hh
χ
∆
∆∆
∆
=
−−
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
stab
ility
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
2(
)2
10
pqpqpq
gg
χ−
+=
[]2
22
22(
)1
2si
nsi
n2
2p
qmn
pqkx
ky
ct
hχ
∆∆
∆
=
−+
Not
e
Solu
tion
2(
)1
pqpq
pqg
χχ
=±
−
If2
()
1pqχ<
21
()
pqpq
pqg
jχ
χ=
±−
max
{}
1pqχ=
[]2
2
2()
min
{}
14mn
pqc
th
χ∆
=−
[]2
2
2()
21
mn
ct
h∆
<
||
1pqg=
(sta
ble)
2(
)1
pqχ<
/2
cth
∆<
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
stab
ility
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
[]
[]2
21
12
2
22
22(
)2
4si
n2
()
4si
n2p
mn
ll
ll
pqpq
pqpq
qmn
l pq
kx
ct
AA
AA
h
ky
ctA
h
+−
∆∆
=−
−
∆∆
−
Assu
me
time-
harm
onic
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
phas
e di
sper
sion
err
or
(,
,)
()
pq
pq
jxjy
jkxjky
jt
ab
pqpq
pq
pq
xyt
Ate
eeee
ππ
ωψ
α+∞
+∞
+∞
+∞
=−∞=−∞
=−∞=−∞
==
∑∑
∑∑
()
jt
pqpq
At
eω
α=
For
each
mod
e, a
nd a
ssum
e co
nsta
nt w
ave
spee
d
22
2
22
22
(,
,)
(,
,)
1(
,,
)0
xyt
xyt
xyt
xy
ct
ψψ
ψ∂
∂∂
+−
=∂
∂∂
22
()
()
pq
ck
kω=
+
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
1 1
lljt
pqpq
ll
jt
pqpq
AAe
AAe
ω ω
+∆
−−
∆
= =
22
22
4si
nsi
n2
2p
qjt
jt
kx
ky
ct
ee
hω
ω∆
−∆
∆∆
∆
−+
=−
+
[]
[]2
21
12
2
22
22(
)2
4si
n2
()
4si
n2p
mn
ll
ll
pqpq
pqpq
qmn
l pq
kx
ct
AA
AA
h
ky
ctA
h
+−
∆∆
=−
−
∆∆
−
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
phas
e di
sper
sion
err
or
Assu
me
time-
harm
onic
()
jt
pqpq
At
eω
α=
For
each
mod
e, a
nd a
ssum
e co
nsta
nt w
ave
spee
d
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
22
22
4si
nsi
n2
2p
qjt
jt
kx
ky
ct
ee
hω
ω∆
−∆
∆∆
∆
−+
=−
+
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
phas
e di
sper
sion
err
or
22
sin
sin
sin
22
2p
qkh
kh
tcth
ω
∆
∆
=
+
Assu
me
his
sm
all
12
2 22
33
1
2
sin
sin
sin
22
2
11
sin
23
22
32
()
(2
pq
pp
pq
kh
kh
tcth
kh
kh
kh
kh
cth
t ck
k
ω− −
∆∆
=+
∆
=+
++
∆=
+4
42
22
22
22
2
()
()
()
)1
()
()
24(
)(
)24
pq
pq
pq
kk
hc
tk
kk
k
+∆
−+
+
+
3
31
sin
3
sin
6
xxx
xxx
−
≈+
≈+
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ce2
D w
ave
equ
atio
n:
phas
e di
sper
sion
err
or
Anal
ytic
al r
elat
ion
44
22
22
22
22
2
44
22
22
22
22
2
()
()
()
()
()
1(
)(
)2
224
()
()
24
()
()
()
()
()
1(
)(
)24
()
()
24
pq
pq
pq
pq
pq
pq
pq
pq
kk
tt
hc
tck
kk
kk
k
kk
hc
tck
kk
kk
k
ω ω
+∆
∆∆
=+
−+
+
+
+∆
=+
−+
+
+
22
()
()
pq
ck
kω=
+
Phas
e di
sper
sion
err
or in
fin
ite d
iffer
ence
met
hod
–Rel
ativ
e er
ror
in t
he w
ave
prop
agat
ion
–
The
wav
e tr
avel
s th
roug
h de
tour
ed s
tairc
ase
path
s–
Abso
lute
pha
se e
rror
for
a p
robl
em w
ith d
imen
sion
L–
With
the
incr
ease
of
prob
lem
siz
e, h
has
to b
e re
duce
d: in
effic
ient
2/
(/
)kk
sλ
∆=
∆
2(
/)
()
skL
λ∆
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Exam
ple
of F
init
e D
iffe
ren
ce3
D w
ave
equ
atio
n
Dis
cret
izat
ion
and
time
mar
chin
gBo
unda
ry c
ondi
tions
Stab
ility
con
ditio
n
Phas
e di
sper
sion
err
or
22
22
22
22
2(
,,
,)
(,
,,
)(
,,
,)
1(
,,
,)
0xyzt
xyzt
xyzt
xyzt
xy
zc
tψ
ψψ
ψ∂
∂∂
∂+
+−
=∂
∂∂
∂
/3
cth
∆<
2(
/)
ks
kλ
∆∝
∆
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
1D
wav
e eq
uat
ion
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
V g (t)
Z L
z= 0
z
z= -d
+ -
Lum
ped
v.s.
dis
trib
uted
: de
term
ined
by
elec
tric
al s
ize
Tran
smis
sion
line
: ci
rcui
t de
scrip
tion
of e
lect
rical
ly
larg
e de
vice
sM
odel
ed b
y 1D
wav
e eq
uatio
n
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Tele
grap
her
’s e
quat
ion
s
R’∆z
L’∆z
C’∆
zG
’∆z
+ -
V(z
, t)
V(z
+ ∆
z, t)
I(z,
t)I(
z + ∆
z, t)
+ -
(,
)(
,)
(,
)(
,)
(,
)(
,)
(,
)(
,)
Iz
tV
zz
tV
zt
Iz
tR
zL
zt
Vz
zt
Iz
zt
Iz
tV
zz
tG
zC
zt
∂′
′+∆
=−
∆−
∆∂∂
+∆
′′
+∆
=−
+∆
∆−
∆∂
Whe
n 0
z∆→
(,
)(
,)
(,
)
(,
)(
,)
(,
)
Vz
tI
zt
Iz
tR
Lz
tI
zt
Vz
zt
Vz
tG
Cz
t
∂∂
′′
−=
+∂
∂∂
∂+∆
′′
−=
+∂
∂
Circ
uit
para
met
ers
per
unit
leng
th
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Tele
grap
her
’s e
quat
ion
s in
FD
TD:
PDE
(,
)(
,)
(,
)
(,
)(
,)
(,
)
Vz
tI
zt
Iz
tR
Lz
tI
zt
Vz
zt
Vz
tG
Cz
t
∂∂
′′
−=
+∂
∂∂
∂+∆
′′
−=
+∂
∂
FD:
OD
E(
)
()
()
()
()
()
dVz
Iz
Rj
Ldz
dIz
Vz
Gj
Cdz
ω ω
′′
−=
+ ′′
−=
+
%%
%%
1D w
ave
equa
tion
22
2()
()
0d
Vz
Vz
dzγ
−=
%%
()(
)2
Gj
CR
jL
γω
ω′
′′
′=
++
Eige
nso
lutio
ns0
0
00
00
()
11
()
zz
zz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
γγ
γγ
+−
−+
+−
−+
=+
=−
% %
00
00
0
VV
Rj
LZ
Gj
CI
Iω ω
+−
++
′′
−+
==
=′
′+
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Wav
es o
n a
n in
fin
ite
line
1D w
ave
equa
tion
22
2()
()
0d
Vz
Vz
dzγ
−=
%%
()(
)2
Gj
CR
jL
jγ
ωω
γα
β
′′
′′
=+
+
=+
Eige
nso
lutio
ns
00
00
00
()
11
()
zz
zz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
γγ
γγ
+−
−+
+−
−+
=+
=−
% %0
Rj
LZ
Gj
Cω ω
′′
+=
′′
+
Loss
less
cas
e2
2
0
/1/
/p
LC
LC
vL
C
ZL
C
γω
βω ω
β
′′
=−
′′
=
′′
==
′′
=
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Wav
es o
n t
erm
inat
ed li
ne
00 0
0
L L
VZ
ZZ
ZV
− +−
Γ=
=+
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
V g (t)
Z L
z= 0
z
z= -d
+ -
Pass
ive
load
Ope
nSh
ort
Mat
ched
Pure
ly r
esis
tive
Pure
ly r
eact
ive |
|1
Γ≤ 1
Γ=
1Γ=− 0
Γ=
Im{
}0
Γ=
||
1Γ=
00
00
00
()
11
()
zz
zz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
γγ
γγ
+−
−+
+−
−+
=+
=−
% %
00
00
00
()
11
()
jz
jz
jz
jz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
ββ
ββ
+−
−+
+−
−+
=+
=−
% %
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Wav
es o
n t
erm
inat
ed li
ne
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
V g (t)
Z L
z= 0
z
z= -d
+ -
00 0
0
L L
VZ
ZZ
ZV
− +−
Γ=
=+
00
00
00
()
11
()
jz
jz
jz
jz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
ββ
ββ
+−
−+
+−
−+
=+
=−
% %
0
20
|(
)||
||
|
|
|1
||
2|
|cos
(2)
jz
jz
r
Vz
Ve
e
Vz
ββ
βθ
+−
+
=⋅
+Γ
=+
Γ+
Γ+
%Stan
ding
wav
e pa
tter
n | V
|
λ/ 2
z
1|
|1
||
VSW
R+
Γ=
−Γ
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Wav
es o
n t
erm
inat
ed li
ne
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
V g (t)
Z L
z= 0
z
z= -d
+ -
00 0
0
L L
VZ
ZZ
ZV
− +−
Γ=
=+
00
00
00
()
11
()
jz
jz
jz
jz
Vz
Ve
Ve
Iz
Ve
Ve
ZZ
ββ
ββ
+−
−+
+−
−+
=+
=−
% %
2
02
00
00
0
00
0
1(
)1 (
)(
)(
)(
)co
s()
(sin
)co
s()
(sin
)
jz
inj
z
jd
jd
LL
jd
jd
LL
L
L
eZ
dZ
eZ
Ze
ZZ
eZ
ZZ
eZ
Ze
Zd
jZd
ZZ
djZ
d
β β
ββ
ββ
ββ
ββ
− −
+Γ
−=
−Γ +
+−
=+
−−
+=
+
Inpu
t im
peda
nce
(loss
less
cas
e)
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Wav
es o
n t
erm
inat
ed li
ne
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
V g (t)
Z L
z= 0
z
z= -d
+ -
00 0
0
L L
VZ
ZZ
ZV
− +−
Γ=
=+
00
0
cos(
)(s
in)
()
cos(
)(s
in)
Lin
L
Zd
jZd
Zd
ZZ
djZ
dβ
ββ
β+
−=
+
Sour
ce c
urre
nt
Z g
z= -d
+ -Z in
gin
ing
VI
ZZ
=+%
%
0(
)j
dj
dV
dV
ee
ββ
+−
−=
+Γ
%
can
be s
olve
d
()
gin
inin
g
VZ
VV
dZ
Z=
−=
+
%%
%
0V+
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
TDR:
Wire
d ra
dar
To d
etec
t di
scon
tinui
ties
on g
uidi
ng s
truc
ture
sTh
ere
is a
lso
FDR
Tran
smis
sion
Lin
es (
Pro
ject
2)
Tim
e do
mai
n r
efle
ctom
eter
Uni
vers
ity o
f Tex
as a
t Arli
ngto
nEl
ectri
cal E
ngin
eerin
g D
epar
tmen
t
Sept
embe
r 23,
200
8EE
533
9 C
ompu
tatio
nal M
etho
ds
Tran
smis
sion
Lin
es (
Pro
ject
2)
Tim
e do
mai
n r
efle
ctom
eter
Tran
smis
sion
line
(L’,
C’,
R’, G
’)
Z g
Z L
z= 0
z
z= -d
+ -
Sour
ce is
a t
ime
dom
ain
puls
eRef
lect
ed p
ulse
due
to
vario
us lo
ads
Anal
ytic
al r
esul
ts:
thro
ugh
IFFT
Tim
e do
mai
n an
alys
is:
solv
ing
the
PDE
–W
hich
one
to
solv
e?
V g (t)
(,
)(
,)
(,
)
(,
)(
,)
(,
)
Vz
tI
zt
Iz
tR
Lz
tI
zt
Vz
tV
zt
GC
zt
∂∂
′
′−
=+
∂
∂
∂∂
′
′−
=+
∂
∂
2
2
2
2
(,
)(
,)
(,
)(
,)
(,
)
(,
)(
,)
(,
)(
)
Vz
tV
zt
Vz
tV
zt
GC
RV
zt
GC
Lt
tt
zV
zt
Vz
tV
zt
GR
RC
GL
CL
tt
∂∂
∂∂
′′
′′
′′
=+
++
∂∂
∂∂
∂∂
′′
′′
′′
′′
=+
++
∂∂
