ESE370:
Circuit-Level
Transmission Line Agenda
Modeling, Design, and Optimization
for Digital Systems
Day 37: December 1, 2014
Transmission Lines
Modeling and Termination
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Penn ESE370 Fall2014 -- DeHon
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Wire “Resistance”
•  What is the resistance at Vi ?
•  Signal propagate as wave down
transmission line
•  Q to charge Vi?
•  Ii given velocity w?
•  R = Vi/Ii?
•  V(x,t) = A+Be(x-wt)
w=
2
Penn ESE370 Fall2014 -- DeHon
From Day 36
–  Delay linear in wire length
–  Speed
See in action in lab
Where arise?
General wire formulation
Lossless Transmission Line
Impedance
End of Transmission Line?
Termination
Discuss Lossy
Implications
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c0
=
LC
ε r µr
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
€
Penn ESE370 Fall2014 -- DeHon
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Penn ESE370 Fall2014 -- DeHon
Wire “Resistance”
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Q=CV
I = dQ/dt
Moving at rate w
I=wCV
R=V/I=1/(wC)
Vi-1
Ii
Vi
Ici
Penn ESE370 Fall2014 -- DeHon
w=
1
LC
•  Z0 =R= 1/wC = 1/(C/sqrt(LC))
Z0 =
LC
R=
C
€
Ii+1
Impedance
Vi+1
Vi-1
€
Ii
Vi
Ii+1
R=
LC
C
L
C
€
Vi+1
Ici
5
€
Penn ESE370 Fall2014 -- DeHon
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1
Impedance
Impedance
•  Assuming infinitely long wire,
how look different at Vi, Vi+1, Vi+2 ?
Z0 =
Vi-1
Ii
Ii+1
Vi
•  Transmission line has a characteristic
impedance
L
C
–  Looks to driving circuit like a resistance
Vi+1
Z0 =
Ici
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Penn ESE370 Fall2014 -- DeHon
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L
C
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Penn ESE370 Fall2014 -- DeHon
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Infinite Lossless
Transmission Line
•  Transmission line looks like
resistive load
Z0 =
End of Line
L
C
•  What happens at the end of the
transmission line?
–  Short Circuit
•  Hint: what must happen in steady state?
Z0
–  Terminate with R=Z0
–  Open Circuit
€
•  Input waveform travels down line at
velocity
1
–  Without distortion
Penn ESE370 Fall2014 -- DeHon
w=
LC
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Penn ESE370 Fall2014 -- DeHon
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€
Short
Penn ESE370 Fall2014 -- DeHon
Terminate R=Z0
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Penn ESE370 Fall2014 -- DeHon
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2
Open
Longer LC (open)
w=
•  40 Stages
•  L=100nH
•  C=1pF
1
c0
=
LC
ε r µr
Stage delay?
€
Penn ESE370 Fall2014 -- DeHon
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•  Drive with 2ns Pulse
•  No termination (open circuit)
Penn ESE370 Fall2014 -- DeHon
Pulse Travel RC
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Analyze End of Line
•  V1,V3,V4,V5,V6 about 10 stages apart
Penn ESE370 Fall2014 -- DeHon
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Penn ESE370 Fall2014 -- DeHon
Analyze End of Line
Analyze End of Line
•  Incident wave Vi=Ii×Z0
•  @ T-ε Vi is voltage on line
•  Vt is what goes forward
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–  Voltage seen by end of line
at T
Incident wave Vi=Ii×Z0
KCL @ end
KVL @ end
V=IR relationships?
–  Which resistances go
with each V?
•  Vr is the delta voltage
that starts moving back
towards source at T+ε
Penn ESE370 Fall2014 -- DeHon
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•  Relate all three V’s
using R, Z0
Penn ESE370 Fall2014 -- DeHon
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Analyze End of Line
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Incident wave Vi=Ii×Z0
KCL: Ii=Ir+It
KVL: Vi+Vr=Vt
Vr=Ir×Z0 Ir=Vr/Z0
Vt=It×R
It=Vt/R
Ii=Vi/Z0
V
i
Z0
Analyze End of Line
•  Vi+Vr=Vt
Eliminate Vt
=
V r Vt
+
Z0 R
Penn ESE370 Fall2014 -- DeHon
Vi Vr Vi + Vr
=
+
Z0 Z0
R
€V V V V
i
− i = r + r
Z0 R Z0 R
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€
€
Penn ESE370 Fall2014 -- DeHon
Analyze End of Line
Vi Vi Vr Vr
− =
+
Z0 R Z0 R
•  Vi+Vr=Vt
•  Vi+Vr=Vt
⎛ R − Z 0 ⎞
Vi ⎜
⎟ = Vr
⎝ R + Z 0 ⎠
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Penn ESE370 Fall2014 -- DeHon
⎛ R − Z 0 ⎞
Vi ⎜
⎟ = Vr
⎝ R + Z 0 ⎠
⎛ R − Z 0 ⎞
Vi ⎜
+1⎟ = Vt
⎝ R + Z 0 ⎠
€
⎛ 2R ⎞
Vi ⎜
⎟ = Vt
⎝ R + Z 0 ⎠
RVi − Z 0Vi = RVr + Z 0Vr
€
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€
Analyze End of Line
€
Vi V r Vt
=
+
Z0 Z0 R
Penn ESE370 Fall2014 -- DeHon
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€
€
Pulse Travel RC
Reflection
•  Sanity check with
previous
–  Short
–  Matched
–  Open
⎛ R − Z 0 ⎞
Vi ⎜
⎟ = Vr
⎝ R + Z 0 ⎠
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Penn ESE370 Fall2014 -- DeHon
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⎛ 2R ⎞
Vi ⎜
⎟ = Vt
⎝ R + Z 0 ⎠
Penn ESE370 Fall2014 -- DeHon
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€
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Visualization
•  http://www.williamson-labs.com/xmission.htm
Back to Source
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Penn ESE370 Fall2014 -- DeHon
Back at Source?
R≠Z0
•  What happens at source?
•  What happens?
–  Depends on how terminated
–  Looks like at sink end
–  75 Ω termination on 50 Ω line
–  “Short-Circuit” source?
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Penn ESE370 Fall2014 -- DeHon
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Penn ESE370 Fall2014 -- DeHon
Simulation
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Transmission Line Symbole
•  For these, with direct drive from voltage
source
–  Source looks like short circuit (not typical of CMOS)
•  Specify delay and impedance
•  Need reference
•  Source cannot be changed
Penn ESE370 Fall2014 -- DeHon
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Penn ESE370 Fall2014 -- DeHon
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50Ω line, 75Ω termination
Simulation
•  For these, with direct drive from voltage
source
–  Source looks like short circuit (not typical of CMOS)
•  Source cannot be changed
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Penn ESE370 Fall2014 -- DeHon
⎛ R − Z 0 ⎞
⎛ 75 − 50 ⎞
Vr = Vi ⎜
⎟ = 0.2Vi
⎟ = Vi ⎜
⎝ 75 + 50 ⎠
R
+
Z
Penn ESE370 Fall2014 -- DeHon ⎝
0 ⎠
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€
Idea
Step Response
•  Signal propagate as wave down
transmission line
–  Delay linear in wire length
–  Speed
–  Impedance
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Penn ESE370 Fall2014 -- DeHon
1
c0
=
LC
ε r µr
L
Z0 =
C
w=
•  Behavior at end of line
depends on termination
•  Both src and sink€
⎛ R − Z 0 ⎞
are “ends”
V = V ⎜
⎟
r
Penn ESE370 Fall2014 -- DeHon
i
€
⎝ R + Z 0 ⎠
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Admin
•  HW8 Thursday
•  Finals 2010—2013 online
–  Good, but maybe too long
•  Lectures Wednesday and Friday
•  Andre out next week (12/8-17)
–  Last office hour tomorrow
•  Ron topic review Monday Dec. 8th
classtime (last day of classes)
Penn ESE370 Fall2014 -- DeHon
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