Transcript ppt

ESE370:
Circuit-Level
Modeling, Design, and Optimization
for Digital Systems
Day 36: November 26, 2013
Transmission Line
Introduction and Analysis
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Penn ESE370 Fall2014 -- DeHon
Next few Lectures/Lab
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See in action in lab (last Friday)
Where arise?
General wire formulation
Lossless Transmission Line
End of Transmission Line?
Termination
Discuss Lossy
Implications
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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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Transmission Lines
• How did the coaxial cables behave in
lab on Friday?
• How differ from
– Ideal equipotential?
– RC-wire on chip?
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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 theory/model to support
– Reason about behavior
– Understand what can cause noise
– 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 24
– Typical of on-chip wires in Ics
– What is RC response to step?
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Transmission Line
• When resistance is negligible
– Have LC wire = Lossless Transmission Line
• No energy dissipation (loss) through R’s
– 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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Pulse Response SPICE
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Contrast RC Wire
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Contrast
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Visualization
• See:
http://www.research.ibm.com/people/r/r
estle/Animations/DAC01top.html
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Model
• Need to understand Voltage as a
function of position and time
– Position along wire
• Want to get V(x,t)
– Also I(x,t)
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Setup Relations
• i is a position
• Position: x=i × Δx
• So Vi is V(x=iΔx)
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Setup Relations
• Vi-Vi-1 =
• Ici=
• Ii-Ii+1=
Vi-1
Ii
Vi
Ii+1
Vi+1
Ici
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Setup Relations
• Vi-Vi-1 = -Ldii/dt
• Ici=CdVi/dt
• Ii-Ii+1=Ici
Vi-1
Ii
Vi
i is spatial dimension
Vi at different positions
Ii+1
Vi+1
Ici
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Setup Relations
V
I
L
x
t
I
V
 Ici  C
x
t
• Vi-Vi-1 = -Ldii/dt
• Ici=CdVi/dt
• Ii-Ii+1=Ici

V
Ii
Vi-1
i
Ii+1
Vi+1

Ici
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Reduce to Single Equation
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Eliminate Ici?
Ii-Ii+1=Ici=CdVi/dt
Take derivative with respect to time
dii/dt - dii+1/dt=Cd2Vi/dt
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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
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Implication
• d2V/dx = LCd2V/dt
• Wave equation
• V(x,t) = A+Be(x-wt)
• Be(x-wt)=LCw2Be(x-wt)
• w=1/sqrt(LC)
– What is w?
Penn ESE370 Fall2014 -- DeHon

V
V
 LC
x
t
2
2
w
1
LC
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Light Cycle Context
• http://www.youtube.com/watch?v=GNfs
6v7i7eY
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Light Cycle Example
• What is the position of a cycle at time t
– Start at x=0
– Travel at v
• Light Cycle is step function at x=0
– Cycle creates trail of height 1
– F(0, t=0)=1, F(x>0,t=0)=0  F(x,t=0)=1-u(x)
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Light Cycle Example
• Light Cycle is step function at x=0
– Cycle creates trail of height 1
– F(0, t=0)=1, F(x>0,t=0)=0  F(x,t=0)=1-u(x)
– When does cycle reach position x>0?
– What is F(x,t)?
t=0
x=0
t=t1
x=0
x=?
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Implication
• d2V/dx = LCd2V/dt
• Wave equation
• V(x,t) = A+Be(x-wt)
• Be(x-wt)=LCw2Be(x-wt)
• w=1/sqrt(LC)
– What is w?
Penn ESE370 Fall2014 -- DeHon

V
V
 LC
x
t
2
2
w
1
LC
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Implication
• 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 Fall2014 -- DeHon

V
V
 LC
x
t
2
2
w
1
LC
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Propagation Rate in Example
• L=1uH
• C=1pF
• What is w ?
w
1
LC

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Signal Propagation
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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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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 35 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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Idea
• Signal propagate as wave down
transmission line
– Delay linear in wire length
– Speed
1
c0
w

LC
e r mr

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Admin
• HW8 out
– Includes writeup for previous and this lab
– Also three questions
• Back here on Monday
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