Transmission Lines
-
Upload
chantale-fernandez -
Category
Documents
-
view
71 -
download
0
description
Transcript of Transmission Lines
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
108 -
Transmission LinesTransmission Lines
• Transmission line effects must be considered when length is comparable to ¼ wavelength
• We will ignore the energy loss on transmission lines
• Concentrate on time-domain description rather than frequency domain
eg
+
-
zg
zl
i
i
e
x dl
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
208 -
Transmission LinesTransmission LinesxR xL
xG xC
),( txxi ),( txi
),( txe
-
+
),( txxe
-
+
x
C: Capacitance / Unit Length [F/m]L: Inductance / Unit Length [H/m]R: Conductor Resistance / Unit Length [Ω/m]G: Insulation Conductance /Unit Length [ /m]
Note G ≠ 1/R !
Ω
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
308 -
Transmission LinesTransmission Lines
xR xL
),( txi),( txe ),( txxe
By KVL:
t
txiLtxiR
x
txetxxe
txxet
txixLtxixRtxe
),(),(
),(),(
0),(),(
),(,
In the limit as :
t
txiLtxiR
x
txe
),(),(
),(
0x
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
408 -
Transmission LinesTransmission Lines),( txi
By KCL:
t
txxeCtxxeG
x
txitxxi
txxit
txxexCtxxexGtxi
),(),(
),(),(
0),(),(
),(,
In the limit as :
t
txeCtxeG
x
txi
),(),(
),(
0x
xC
),( txxi
xG
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
508 -
Lossless CaseLossless Case
(2)
(1)
t
eC
x
it
iL
x
e
0
0
G
R
Take partial derivative w.r.t. x in (1) and partial derivative w.r.t. t in (2),then substitute.
2
2
2
2
2
2
2
2 1or
1
t
i
x
i
LCt
e
x
e
LC
Recognize as wave equations
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
608 -
Lossless CaseLossless Case
Show that the solution is in the form . txLCfe
Q.E.D.
obtain weequation, wavethe into ngSubstituti
Similarly,
Let
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
.
.
ds
fd
ds
fdLC
LC
ds
fd
dt
ed
ds
fdLC
x
s
ds
fdLC
ds
dfLC
xx
e
ds
dfLC
x
s
ds
df
x
e
txLCs
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
708 -
InterpretationInterpretation
• is a wave traveling to right with velocity
• is a wave traveling to left with velocity
• Solving for i, we obtain
• is characteristic of line
txLCf
txLCfCL
i /
1
LC/1
txLCf LC/1
CL /
line lossless the of impedancestic characteri is CLz /0
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
808 -
InterpretationInterpretation
txLCftxLCf
zi
txLCftxLCfe
210
21
1
Note:Each traveling wave direction, e and i are related by z0.
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
908 -
ReflectionsReflections
• Look at terminations with real impedances frequency independent
• Wave of voltage and current traveling to right
• At termination
zl
z0
e-
+el
+
-
il
0zi
e
lzi
e
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1008 -
ReflectionsReflections
• Hence, there must be reflected waves e- and i- such that
• In terms of voltage
l
ll
ll zii
ee
z
ei
and
0
l
ll
ll z
ze
ze
ee
00
kzz
zz
e
e
l
l
l
l
0
0 Reflection coefficient
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1108 -
ReflectionsReflections
• Special cases:– zl = z0 k=0
Matched, no reflections. Line looks infinite.
– zl = 0, short circuit k = -z0/z0 = -1
– zl = , open circuit k = 1
8
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1208 -
Multiple ReflectionsMultiple Reflections
10
0
00
z
zk
5.033
00
00
zz
zzkl
R=3z0z0
t=0
x=0 x=l
E
Hitting load, a wave of E/2 is produced
Et=.3T
Et=1.3T 3E/2
Et=2.3T 3E/2
Arriving wave of E/2 isreflected toward load
T 2T 3T 4T 5T
Voltage at load1.5E
.75E
1.125E.9375E
6T 7T
E
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1308 -
Time-Space (Bounce) DiagramTime-Space (Bounce) Diagram
• Mark reflection coefficients
• Write initial voltages• Write wave amplitudes• Update
– Wave amplitudes when reflected
– Voltages as waves cross
xk0=-1 kl=.5
e=E
e=0
e=E
e=E
E
E/2
-E/2
-E/4
E/4
E/8
T
2T
3T
4T
5T
e=3E/2
e=3E/4
e=9E/8
Tim
e
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1408 -
ExampleExample
3z0
E z0
Magnitude of the first wave:
43 00
00
E
zz
zEE
xk0=.5 kl=1
e=E/4
e=0
e=5E/8
e=13E/16
E/4
E/4
E/8
E/8
E/16
E/16
T
2T
3T
4T
5T
e=E/2
e=3E/4
e=7E/8
Tim
e
z0
t=0
k0=.5
E
R=3z0
lkl=1
E
t
At x=0
T 2T 3T 4T 5T
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1508 -
Reflections in Digital LinesReflections in Digital Lines
• Consider one source and one load
• Option 1:Do not terminate either end. Ringing will stop eventually.– Pro: Simple, no additional power loss– Con: Limited speed
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1608 -
Reflections in Digital LinesReflections in Digital Lines
• Option 2:Matched termination at the end
– Pro: No reflections– Con: Excessive power consumption
For z0=150Ω, power consumption 135mWReduce by duty factor
(0.5 for regular lines, 0.05 for floppy drives)Multiply by number of lines
z0 Open-collectordriver
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1708 -
Reflections in Digital LinesReflections in Digital Lines
• Option 3:Matched termination at the source end
– Pro: Can be run at the same speed as loadtermination If receiver has very high input impedance, fullvoltage appears at the receiverNo power dissipated at constant voltage level
– Con: Special, high input impedance line receiversrequired (not suited for standard TTL)Look at multiple terminations
z0
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1808 -
Reflection in Digital LinesReflection in Digital Lines
• Problem for multiple receivers
– Assume the ideal case, where taps are infinitely short and have infinite impedances
– Even for this case, intermediate taps do not get full signal immediately
z0
E
xk0=0 kl=1
e=.5E
e=0E/2
E/2
T
2T
e=E
E
t
At x=.5L
.5T T 1.5T 2T
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
1908 -
Reflection and Transmission at Reflection and Transmission at JunctionsJunctions
e1+, i1+
e1-, i1-
e2+ , i2
+
e3
+, i3 +
0
33
0
22
0
11
0
11
3211
32
211
, , ,z
ei
z
ei
z
ei
z
ei
iiii
ee
eee
320
3
0
2
0
1
0
1
211
, eez
e
z
e
z
e
z
e
eee
121
121
2 eee
eee
3
21
11
21
11
111
eee
1122 3
2
3
11
11
eeee
Winter 2005
ECE
ECE 766Computer Interfacing and Protocols
2008 -
Reflection and Transmission at Reflection and Transmission at JunctionsJunctions
3/2 L
2/3 L
Lz0 t=0T
2T
3T
4T
k=0
k=1
k=1E/2
-E/6
2E/9
E/3E/3
2E/9
E/3
E/3
-E/9
Many multiple reflections,eventually come to rest