Transmission Line Agenda
ESE370: Circuit-Level Modeling, Design,
and Optimization for Digital Systems
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Lec 32: November 30, 2015
Transmission Lines
Modeling and Termination
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Penn ESE 370 Fall 2015 – Khanna
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Wire “Resistance”
Signal propagates as wave down transmission line
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Open, short, matched
Termination
Discuss Lossy
Implications
Penn ESE 370 Fall 2015 – Khanna
Reminder: Propogation
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See in action in lab
Where transmission lines arise?
General wire formulation
Lossless Transmission Line
Impedance
End of Transmission Line?
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Delay linear in wire length, if resistance negligible
Solution:
What is the resistance at Vi ?
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Q to charge Vi?
V (x, t) = A + Be x−wt
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Rate of propogation (characteristic frequency), w:
1
c
=
LC
εr µr
w=
Penn ESE 370 Fall 2015 – Khanna
Vi-1
3
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Vi
Ii+1
Vi+1
Q to charge Vi?
Ii given velocity w?
R = Vi/Ii?
Vi-1
Ii
Ici
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What is the resistance at Vi ?
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Ii
Vi+1
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Q to charge Vi?
Ii given velocity w?
Vi-1
Ii+1
Wire “Resistance”
What is the resistance at Vi ?
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Vi
Ici
Wire “Resistance”
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Ii
Vi
Ii+1
Vi+1
Ici
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1
Wire “Resistance”
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Wire “Resistance”
Q=CVi
Ii = dQ/dt
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Vi-1
Ii
Ii+1
Vi
Q=CVi
Ii = dQ/dt
Moving at rate w
Ii=wCVi
Vi+1
Vi-1
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Wire “Resistance”
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Q=CVi
Ii = dQ/dt
Moving at rate w
Ii=wCVi
R=Vi/Ii=1/(wC)
Vi-1
Ii
Vi
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Characteristic Impedance
w=
€
Ii+1
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1
LC
Z0 = R
R=
LC
R=
C
Vi+1
Vi-1
Ii
LC
L
=
= Z0
C
C
Vi
Ii+1
Vi+1
Ici
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Impedance
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Impedance
Assuming infinitely long wire, how does the
impedance look at Vi, Vi+1, Vi+2 ?
Z0 =
Vi-1
Vi+1
Penn ESE 370 Fall 2015 – Khanna
€
Ici
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Vi
Ii+1
Ici
Penn ESE 370 Fall 2015 – Khanna
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1
LC
€
Ii
Ici
w=
Ii
€
Penn ESE 370 Fall 2015 – Khanna
Vi
Ii+1
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Transmission lines have a characteristic impedance
L
C
Z0 =
L
C
Vi+1
Ici
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€
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2
Infinite Lossless Transmission Line
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Transmission line looks like
resistive load
Z0 =
End of Line
L
C
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What happens at the end of the transmission line?
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Short Circuit
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Z0
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€
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Input waveform travels down line at velocity
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Without distortion
w=
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Hint: what must happen in steady state?
Terminate with R=Z0
Open Circuit
What did the reflections look like in each case in
lab?
1
LC
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€
Short
Penn ESE 370 Fall 2015 – Khanna
Terminate R=Z0
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Open
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Longer LC (open)
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40 Stages
L=100nH
C=1pF
w=
1
c0
=
LC
ε r µr
Stage delay? How long to propogate?
€
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Drive with 2ns Pulse
No termination (open circuit)
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What reflection do we expect?
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Pulse Travel RC
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Analyze End of Line
V1,V3,V4,V5,V6 about 10 stages apart
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Analyze End of Line
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Voltage seen by end of line
at T
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Analyze End of Line
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Relate all three V’s
using R, Z0
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KCL: Ii=Ir+It
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Incident wave Vi=Ii×Z0
KCL: Ii=Ir+It
KVL @ end of line
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Penn ESE 370 Fall 2015 – Khanna
Which resistances go
with each V?
Analyze End of Line
Incident wave Vi=Ii×Z0
KCL @ end of line
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Incident wave Vi=Ii×Z0
KCL @ end of line
KVL @ end of line
V=IR relationships?
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Vr is the delta voltage
that starts moving back
towards source at T+ε
Penn ESE 370 Fall 2015 – Khanna
Penn ESE 370 Fall 2015 – Khanna
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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Incident Wave, Reflected Wave, Transverse Wave
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KVL: Vi+Vr=Vt
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Analyze End of Line
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Analyze End of Line
Incident wave Vi=Ii×Z0
KCL: Ii=Ir+It
KVL: Vi+Vr=Vt
V=IR relationships?
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Which resistances go
with each V?
Vr=Ir×Z0 Ir=Vr/Z0
Vt=It×R It=Vt/R
Vi=Ii×Z0 Ii=Vi/Z0
Penn ESE 370 Fall 2015 – Khanna
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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
Vi=Ii×Z0 Ii=Vi/Z0
Vi V r Vt
=
+
Z0 Z0 R
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Analyze End of Line
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Vi+Vr=Vt
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Eliminate Vt
Analyze End of Line
Vi V r Vt
=
+
Z0 Z0 R
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Vi+Vr=Vt
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Eliminate Vt
€
€
Vi V r Vt
=
+
Z0 Z0 R
Vi Vr Vi +Vr
=
+
Z0 Z0
R
RVi = RVr + Z 0 (Vi +Vr )
Vi (R − Z 0 ) = Vr (R + Z 0 )
" R − Z0 %
Vr = $
'Vi
# R + Z0 &
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Analyze End of Line
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Vi+Vr=Vt
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Eliminate Vt
€
Penn ESE 370 Fall 2015 – Khanna
Analyze End of Line
Vi V r Vt
=
+
Z0 Z0 R
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Vi+Vr=Vt
Vi Vr Vi +Vr
=
+
Z0 Z0
R
RVi = RVr + Z 0 (Vi +Vr )
Vi (R − Z 0 ) = Vr (R + Z 0 )
" R − Z0 %
Vr = $
'Vi Reflection coefficient
# R + Z0 &
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Penn ESE 370 Fall 2015 – Khanna
" R − Z0 %
Vi $
' = Vr
# R + Z0 &
" R − Z0 %
Vi $
' = Vt −Vi
# R + Z0 &
" R − Z0 %
Vi $
+1' = Vt
# R + Z0 &
" 2R %
Vi $
' = Vt
# R + Z0 &
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Reflection
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Pulse Travel RC
Sanity check with
previous
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Short
Matched
Open
# R − Z0 &
Vi %
( = Vr
$ R + Z0 '
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€
Visualization
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http://www.williamson-labs.com/xmission.htm
Back to Source
Matched
Open
Short
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Back at Source?
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Depends on how terminated
Looks like at sink end
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R≠Z0
What happens at source?
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What happens?
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75 Ω termination on 50 Ω line
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Transmission Line Symbol
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Simulation
Specify delay of full Tline and characteristic
impedance
Need reference
Penn ESE 370 Fall 2015 – Khanna
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For these, with direct drive from voltage source
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Source cannot be changed
Penn ESE 370 Fall 2015 – Khanna
50Ω line, 75Ω termination
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Idea
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Signal propagate as wave down transmission line
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# R − Z0 &
# 75 − 50 &
Vr = Vi %
( = 0.2Vi
( = Vi %
$ 75 + 50 '
$ R + Z0 '
Penn ESE 370 Fall 2015 – Khanna
Source looks like short circuit (not typical of CMOS)
Delay linear in wire length
Speed
Impedance
Behavior at end of line
depends on termination
Both src and sink
are “ends” with reflections
w=
€# R − Z &
0
Vr = Vi %
(
$ R + Z0 '
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€
Penn ESE 370 Fall 2015 – Khanna
€
1
c0
=
LC
ε r µr
Z0 =
L
C
40
€
Admin
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Project due Friday
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Allowed to submit in canvas until 11:59pm
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Submit SINGLE PDF or word DOC (can still submit in class)
Only one person needs to submit
Make sure division of labor and partner contributions are
included
Final (T 12/15) is cumulative
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Finals 2010—2014 online
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Good, but too long
Quantitative understanding with analysis and design from lec
1-28
Qualitative understanding from lec 29-34
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