ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 33: November 30, 2011
Transmission Line
Introduction and Analysis
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Penn ESE370 Fall2011 -- DeHon
Next few Lectures/Lab
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Where arise?
General wire formulation
Lossless Transmission Line
See in action in lab (Friday)
End of Transmission Line?
Termination
Discuss Lossy
Implications
Penn ESE370 Fall2011 -- DeHon
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Where Transmission Lines
Arise
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Transmission Lines
• Cable: coaxial
• PCB
– Strip line
– Microstrip line
• Twisted Pair (Cat5)
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Penn ESE370 Fall2010 -- DeHon
Transmission Lines
• How do these wires behave?
– 25m of category-5 cable?
– VGA cable? (Analog?)
• How differ from
– Ideal equipotential?
– RC-wire on chip?
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Penn ESE370 Fall2011 -- DeHon
Transmission Lines
• This is what wires/cables look like
– Aren’t an ideal equipotential
– Signals do take time to propagate
– Maintain shape of input signal
• Within limits
– Shape and topology of wiring effects how
signals propagate
• …and the noise effects they see
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Transmission Lines
• Need to understand
– How to model  how to reason about
– What can cause noise
– How to engineer high performance
communication
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Wire Formulation
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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 21
– 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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Build Intuition from LC
• What did one LC do?
• What will chain do?
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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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Penn ESE370 Fall2011 -- DeHon
Setup Relations
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Penn ESE370 Fall2011 -- DeHon
Setup Relations
• Vi-Vi-1 =
• Ici=
• Ii-Ii+1=
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Penn ESE370 Fall2011 -- DeHon
Setup Relations
• Vi-Vi-1 = Ldii/dt
• Ici=CdVi/dt
• Ii-Ii+1=Ici
Vi-1
Ii
i is spatial dimension
Vi at different positions
Vi
Ii+1
Vi+1
Ici
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Penn ESE370 Fall2011 -- DeHon
Setup Relations
V
• Vi-Vi-1 = Ldii/dt
• Ici=CdVi/dt
• Ii-Ii+1=Ici
x
I
x
Maybe sign wrong on LdI/dt
Vi-1
Vi
Ii
Ici
Ii+1
L
I
t
 I ci
Vi+1

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Penn ESE370 Fall2011 -- DeHon
Reduce to Single Equation
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Eliminate Ici?
Ii-Ii+1=Ici  dIi/dt-dIi+1/dt=dIci/dt
Ici=CdVi/dt  dIci/dti=Cd2Vi/dt
dii/dt - dii+1/dt=Cd2Vi/dt
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Penn ESE370 Fall2011 -- DeHon
Reduce to Single Equation
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dii/dt - dii+1/dt=Cd2Vi/dt
Vi-Vi-1 = Ldii/dt
Vi+1-Vi = Ldii+1/dt
Eliminate Is ?
Vi-Vi-1 -(Vi+1-Vi )= Ldii/dt - Ldii+1/dt
d2V/dx =-LCd2V/dt
Vi+1-Vi-1=-LCd2Vi/dt
Multiple sign
problems
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Penn ESE370 Fall2011 -- DeHon
Implication
• Vi+1-Vi-1=LCd2Vi/dt
• Once Vi settles, settle to same value
• d2V/dx = LCd2V/dt
• Wave equation
• V(x,t) = A+Be(x-wt)
• Be(x-wt)=LCw2Be(x-wt)
• w=1/sqrt(LC)
– Rate of propagation
Penn ESE370 Fall2011 -- DeHon

 V
2
x
 V
2
 LC
w 
t
1
LC
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Propagation
• V(x,t) = A+Be(x-wt)
• If V(1cm,1ns)=Va
• and w = 10cm/ns
• for what t does
• V(2cm,t)=Va ?
w 
1
LC

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Penn ESE370 Fall2011 -- DeHon
Propagation
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V(x,t) = A+Be(x-wt)
If V(x0,t0)=Va
And V(x0+Dx,t0+Dt)=Va
What is w?
w 
1
LC

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Penn ESE370 Fall2011 -- DeHon
Propagation Rate in Example
• L=1uH
• C=1pF
• What is w ?
w 
1
LC

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Penn ESE370 Fall2011 -- DeHon
Signal Propagation
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Penn ESE370 Fall2011 -- DeHon
Propagation
• Be(x-wt+x)=LCw2Be(x-wt)
• w=1/sqrt(LC)
– Rate of propagation
– Delay linear in length
• Compare RC wire delay quadratic in length
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Penn ESE370 Fall2011 -- DeHon
Contrast RC Wire
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Penn ESE370 Fall2011 -- DeHon
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 32 we 
know for wire: CL = em
c0
w 
– w=1/sqrt(em)c0/sqrt(ermr)
e rmr
– Where c0=speed of light in vacuum=30cm/ns
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Class Ended Here
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Wire “Resistance”
• What is the resistance at Vi ?
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Penn ESE370 Fall2011 -- 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
w 
1
LC
R 

Ii+1
LC
C
Vi+1

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Penn ESE370 Fall2011 -- DeHon
Impedance
• Z0 =R= 1/wC = 1/(C/sqrt(LC))
Z0 
Vi-1
Ii
Vi
Ii+1
R 
LC
C
L
C

Vi+1
Ici

Penn ESE370 Fall2011 -- DeHon
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Impedance
• Assuming infinitely long wire,
how look different at Vi, Vi+1, Vi+2 ?
Z0 
Vi-1
Ii
Vi
Ii+1
L
C
Vi+1
Ici
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Penn ESE370 Fall2011 -- DeHon
Impedance
• Transmission line has a characteristic
impedance
– Looks to driving circuit like a resistance
Z0 
L
C
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Penn ESE370 Fall2011 -- DeHon
Infinite Lossless
Transmission Line
• Transmission line looks like
resistive load
Z0 
L
C
Z0

• Input waveform travels down line at
velocity
1
– Without distortion
Penn ESE370 Fall2011 -- DeHon
w 
LC
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End of Line
• What happens at the end of the
transmission line?
– Short Circuit
– Terminate with R=Z0
– Open Circuit
• Experimentally in Lab Friday
• Mathematically in Class Monday
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Penn ESE370 Fall2011 -- DeHon
Admin
• In Lab on Friday
– Lab instructions online
• HW6
– Includes writeup for previous and this lab
– Also two questions
– Due Monday
• Project 3
– Should have tools to attack
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Penn ESE370 Fall2011 -- DeHon
Idea
• Signal propagate as wave down
transmission line
– Delay linear in wire length
w 
– Speed
– Impedance
1

LC
Z0 
c0
e rmr
L
C

Penn ESE370 Fall2011 -- DeHon

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