Transcript ppt

ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 33: November 29, 2010
Transmission Lines
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Penn ESE370 Fall2010 -- DeHon
This Week
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General wire formulation
Lossless Transmission Line
End of Transmission Line?
Termination
See in action in lab
What cover today and
Discuss Lossy
order of week somewhat
unclear.
Where arise?
Implications
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Wires
• In general, our “wires” have distributed
R, L, C components
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RC Wire
• When R dominates L
– We have the distributed RC Wires we saw
on Day 27
– Typical of on-chip wires in ICs
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Transmission Line
• When resistance is negligible
– Have LC wire = Lossless Transmission Line
– More typical of Printed Circuit Board wires
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Intuitive: Lossless
• Pulses travel as waves without
distortion
– (up to a characteristic frequency)
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SPICE Simulation
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SPICE Simulation
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Contrast RC Wire
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Visualization
• See:
http://www.research.ibm.com/people/r/r
estle/Animations/DAC01top.html
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Setup Relations
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V
I
L
x
t
I
 I ci
x
Vi-Vi-1 = Ldii/dt
Vi+1-Vi = Ldii+1/dt
Ici=CdVi/dt
Ii-Ii+1=Ici
Vi-1

V
Ii
i
Ici
Ii+1
Vi+1

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Reduce to Single Equation
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Vi-Vi-1 = Ldii/dt
Vi+1-Vi = Ldii+1/dt
Ici=CdVi/dt  dIci/dti=Cd2Vi/dt
Ii-Ii+1=Ici  dIi/dt-dIi+1/dt=dIci/dt
Vi-Vi-1 -(Vi+1-Vi )= Ldii/dt - Ldii+1/dt
– d2V/dx = -Ld2I/dx=-LdIci/dt=-LCd2Vi/dt
• Vi+1-Vi-1=-LCd2Vi/dt
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Implication
• Vi+1-Vi-1=-LCd2Vi/dt
• Once Vi settles, settle to same value
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• d2V/dx = LCd2V/dt
• Wave equation
• V(x,t) = A+Be-(wt+x)
• Be-(wt+x)=LCw2Be-(wt+x)
• w=1/sqrt(LC)
– Rate of propagation
Penn ESE370 Fall2010 -- DeHon

V
V
 LC
x
t
w
2
1
LC
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Propagation Rate in Example
• L=1uH
• C=1pF
w
1
LC

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Signal Propagation
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Propagation
• Be-(wt+x)=LCw2Be-(wt+x)
• w=1/sqrt(LC)
– Rate of propagation
– Delay linear in length
• Compare RC wire delay quadratic in length
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Contrast RC Wire
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Propagation
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Be-(wt+x)=LCw2Be-(wt+x)
w=1/sqrt(LC)
Rate of propagation
Delay linear in length
w
1
LC
• Compare RC wire delay quadratic in length
• From Day 31 weknow for wire: CL = em
c0
w
– w=1/sqrt(em)c0/sqrt(ermr)
e r mr
– Where c0=speed of light in vacuum=30cm/ns
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Impedance
• V(x,t) = A+Be-(wt+x)
• Ici=CdVi/dt
• Ici=wCBe-(wt+x)
• Z0 = Vi/Ii ~Vi/Ici = 1/wC = 1/(C/sqrt(LC))
– (really Ii --- differs in phase)
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Impedance
• Z0 = Vi/Ici = 1/wC = 1/(C/sqrt(LC))
– (really Ii --- differs in phase)
L
Z0 
C
• Transmission line has a characteristic
impedance
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Infinite Lossless
Transmission Line
• Transmission line looks like
resistive load
L
Z0 
C
Z0

• Input waveform travels down line at
velocity
1
– Without distortion
Penn ESE370 Fall2010 -- DeHon
w
LC
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End of Line
• What happens at the end of the
transmission line?
– Open Circuit
– Short Circuit
– Terminate with R=Z0
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Open
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Short
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Terminate R=Z0
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Longer LC (open)
• 40 Stages
• L=100nH
• C=1pF
Stage delay?
• Drive with 2ns Pulse
• No termination
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Pulse Travel RC
• V1,V3,V4,V5,V6 about 10 stages apart
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Analyze End of Line
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Analyze End of Line
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Incident wave Vi=Ii×Z0
Ii=Ir+It
Vi+Vr=Vt
Vr=Ir×Z0 Ir=Vr/Z0
Vt=It×R
It=Vt/R
Ii=Vi/Z0
V
V r Vt


Z0 Z0 R
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Analyze End of Line
• Vi+Vr=Vt
Vi V r Vt


Z0 Z0 R
Vi Vr Vi  Vr


Z0 Z0
R
V V V V
i
i
r
r
 

Z0 R Z0 R
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Analyze End of Line
• Vi+Vr=Vt
Vi Vi Vr Vr
 

Z0 R Z0 R
RVi  Z0Vi  RVr  Z0Vr

R  Z 0 
Vi 
 Vr
R  Z0 
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Analyze End of Line
• Vi+Vr=Vt
R  Z 0 
Vi 
 Vr
R  Z0 
R  Z 0 
Vi 
1 Vt
R  Z 0 

 2R 
Vi 
 Vt
R  Z 0 
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Reflection
• Sanity check with
previous
– Open
– Short
– Matched
R  Z 0 
Vi 
 Vr
R  Z0 
 2R 
Vi 
 Vt
R  Z 0 
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Pulse Travel RC
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Next Time
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(finish reflections)
Termination Strategy
Implications
Were Transmission Line Arise
Hand-wave Lossy
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Admin
• Project 3 out (due Friday 12/10)
– Lab portion split off, separate due date
• André out Tuesday
– Won’t be around for office hours
• Lab on Friday
– Lecture Wednesday
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Idea
• Signal propagate as wave down
transmission line
– Delay linear in wire length
– Speed
– Impedance
• Behavior at end of line
depends on termination
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c0
w

LC
e r mr
L
Z0 
C
R  Z 0 
Vr  Vi 

 R  Z 0 

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