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CSE 464 Transmission Line Equations (Dally/Poulton Notation) January 2009. Based on Notes by Fred U. Rosenberger David M. Zar Washington University in St. Louis [email protected]. Transmission Line Equations. Superposition (if linear). General Equations. For Parallel Plates:. - PowerPoint PPT Presentation
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CSE 464 Transmission Line
Equations(Dally/Poulton Notation)
January 2009Based on Notes by Fred U.
RosenbergerDavid M. Zar
Washington University in St. [email protected]
T-Line Equations 2 - David M. Zar - 04/19/23
Transmission Line Equations
)( and )(v
xtV
v
xtV rf
rf VxV anddirection positive in the gpropagatin with waves traveling twoare
rf VVV
.city with velomoveBoth direction. negative in the gpropagatin vx
C
LZ 0 LC
v1
rf III v
ZL 0
vZC
0
1
ff IZV 00
0
ZZZZK
T
TrT
rr IZV 00
0
ZZZZK
S
SrS
2
1
vCL
Superposition (if linear)
T-Line Equations 3 - David M. Zar - 04/19/23
General Equations
cmpsinpsftns
cmECLv
CLZ
mCEmCmLvCL
rrr
r
r
rr
/)33( /)83( /)1(Or
sec/)/103(/1
1) (s/w squareper /377/
)//(17112.1)//(/
/1
0
0
2
For Parallel Plates:
T-Line Equations 4 - David M. Zar - 04/19/23
Transmission Line Equations
)()( 0 v
ltVtV ffl
)()(0 v
ltVtV rlr
0
00
0
00 )()()(
ZZ
ZZtV
ZZ
ZtVtV
S
Sr
SSf
0
0
0
0 )()()(ZZ
ZZtV
ZZ
ZtVtV
T
Tfl
TTrl
TT
T
Tfll ZZ
ZV
ZZ
ZVV
0
0
0
2
00
0
00 2
ZZ
ZV
ZZ
ZVV
S
Sr
SS
Z0
Zs
ZT
V0+-
+
-Vs
L0
+
-Vl
(ZS-Z0)/(ZS + Z0)= KrS
(ZT-Z0)/(ZT + Z0)= KrT
T-Line Equations 5 - David M. Zar - 04/19/23
General Equations (VT = 0)Use Superposition for Multiple Sources
SSrSrf ZZ
ZtVKtVtV
0
000 )()()( 1)
rTfr Kv
ltVtV )
2()( 00 2)
2) into 1),S
SrSrTff ZZ
ZtVKK
v
ltVtV
0
000 )()
2()(
4)S
SrSrTf
ffx
ZZ
Z
v
xtVKK
v
x
v
ltV
v
xtVtV
0
00
0
)()2
(
)()(
SrTS
rSrTfrx
ZZ
ZK
v
x
v
ltV
KKv
x
v
ltVtV
0
0
20
)2
(
)4
()(
3)
5)
At arbitrary x,
From 2) & 3),
2l since we have to go “there and back” to see it at the start of the line
T-Line Equations 6 - David M. Zar - 04/19/23
Equations Continued
Repeat 3) into 4), 3) into 5),
0
1
0
0 ))1(2
()2
()(n
nrS
nrTS
nrT
nrSS
Sx KK
v
x
v
lntVKK
v
nl
v
xtV
ZZ
ZtV
fVrV 100
At x = 0,
1
1
0
00 )()
2()()(
n
nrS
nrS
nrTSS
S
KKKv
nltVtV
ZZ
ZtV
fVrV fV
00
0 )2
()1()(n
nrS
nrTSrT
Sl KK
v
l
v
nltVK
ZZ
ZtV
fVrV
Now can find V0, Vl at any time.
At x = l,
T-Line Equations 7 - David M. Zar - 04/19/23
Example Driving T-Line
Apply to problem,
01
00)(
t
ttVS
00
0 )2
()1()(n
nrS
nrTSrT
Sl KK
v
l
v
nltVK
ZZ
ZtV
0)12(
0
0)12(
1
v
lnt
v
lnt
v
lNt
v
lNKKK
ZZ
Z N
n
nrS
nrTrT
S
)32()12(for )1(
00
0
Z0
Zs
ZT
This is the time to go from l0 l for the last (N) time.
T-Line Equations 8 - David M. Zar - 04/19/23
Steady State Value and Error Term, Example
With algebra,
))(1( 1
NrSrT
ST
T KKZZ
Z
Attenuation in steady state
Error 0 as N .
For 1rSK,0SZ We get
9.0 ,1K ,0 ,1900 ,100 :Example rS0 rTST KZZZ
))9.0(1()( 1
N
ST
Tl ZZ
ZtV
(l/v).N 642 t,8.21)9.0ln(/)1.0ln(1 :10% error For
v
lNt
v
lN )32()12(for
T-Line Equations 9 - David M. Zar - 04/19/23
Example Continued For 2 foot line, this could be on the order of
85ns (or even 3.55 ns for a 1 inch line!) What to do?
»Wait a long time»Real wire will have some resistance (Zs≠0)»CS solution: use 1900 Ohm transmission Line»Reduce resistance of receiver to match t-line
– Increased power– May be a problem with driver resistance (e.g. 10 to 50
Ohms)»Increase resistance of driver to match t-line
– 100 Ohms in series with driver gives 2.5% error after only l/v,or after 2ns for one foot line (this is the minimum possible)
T-Line Equations 10 - David M. Zar - 04/19/23
Transmission Line Equations for Wire
)/ln()/60(
)/ln(/)/6.55()/ln(/2Coax
120
1212
rrZ
rrmpFrrC
r
r
)/2ln()/60(
)/2ln(/)/(6.55)/2ln(/2/
0 rsZ
rsmpFrsmC
r
r
r
r2
r1
s
Wire over ground plane(warning, not accurate for r close to s)
Between 2 Wires: 1/2 capacitance of wire over ground plane
T-Line Equations 11 - David M. Zar - 04/19/23
PC Board Equations
w
3770 w
sZ
r s
Parallel Plates(ignoring fringing)
0
2
2
0rw
sws
swsw
C
LZ
T-Line Equations 12 - David M. Zar - 04/19/23
PC Board Equations
67.0475.0017.1)/(
)8.0
98.5ln(
41.1
870
rpd
r
ftnst
tw
hZ
w
t
h dielectric
air
wt
h2=h1rpd
r
ftnst
bttbw
wtw
bZ
017.1)/(
25.0/ ,35.0)/(for
))/8.0(67.0
4ln(
600
b
Stripline
Microstrip
h1
(Careful here, approximations grossly wrong for some values: large w/h)