1. Day 36: November 26, 2013 Transmission Line Introduction and Analysis
ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems
Day 36: November 26, 2013
Transmission Line
Introduction and Analysis
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2. Next few Lectures/Lab 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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3. Where Transmission Lines Arise
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4. Transmission Lines Cable: coaxial PCB Twisted Pair (Cat5) Strip line
Microstrip line
Twisted Pair (Cat5)
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5. 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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6. 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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7. Transmission Lines Need theory/model to support Reason about behavior
Understand what can cause noise
Engineer high performance communication
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8. Wire Formulation
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9. Wires In general, our “wires” have distributed R, L, C components
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10. 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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11. 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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12. Build Intuition from LC
What did one LC do?
What will chain do?
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13. Intuitive: Lossless Pulses travel as waves without distortion
(up to a characteristic frequency)
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14. SPICE Simulation
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15. SPICE Simulation
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16. Pulse Response SPICE
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17. Contrast RC Wire
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18. Contrast
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19. Visualization
See:
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20. 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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21. Setup Relations i is a position Position: x=i × Δx So Vi is V(x=iΔx)
Ici
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22. Setup Relations Vi-Vi-1 = Ici= Ii-Ii+1= Ii+1 Ii Vi-1 Vi Vi+1 Ici
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23. Setup Relations Vi-Vi-1 = -Ldii/dt Ici=CdVi/dt i is spatial dimension
Ii-Ii+1=Ici
i is spatial dimension
Vi at different positions
Ii+1 Ii
Vi-1 Vi
Vi+1
Ici
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24. Setup Relations Vi-Vi-1 = -Ldii/dt Ici=CdVi/dt Ii-Ii+1=Ici Ii+1 Ii
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25. Reduce to Single Equation
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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26. Reduce to Single Equation
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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27. Implication d2V/dx = LCd2V/dt V(x,t) = A+Be(x-wt)
Wave equation
V(x,t) = A+Be(x-wt)
Be(x-wt)=LCw2Be(x-wt)
w=1/sqrt(LC)
What is w?
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28. Light Cycle Context http://www.youtube.com/watch?v=GNfs6v7i7eY
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29. 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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30. 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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31. Implication d2V/dx = LCd2V/dt V(x,t) = A+Be(x-wt)
Wave equation
V(x,t) = A+Be(x-wt)
Be(x-wt)=LCw2Be(x-wt)
w=1/sqrt(LC)
What is w?
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32. Implication d2V/dx = LCd2V/dt V(x,t) = A+Be(x-wt)
Wave equation
V(x,t) = A+Be(x-wt)
Be(x-wt)=LCw2Be(x-wt)
w=1/sqrt(LC)
Rate of propagation
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33. Propagation Rate in Example
L=1uH
C=1pF
What is w ?
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34. Signal Propagation
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35. 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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36. Contrast RC Wire
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37. Propagation From Day 35 we know for wire: CL = em
Be(wt+x)=LCw2Be(wt+x)
w=1/sqrt(LC)
Rate of propagation
Delay linear in length
Compare RC wire delay quadratic in length
From Day 35 we know for wire: CL = em
w=1/sqrt(em)=c0/sqrt(ermr)
Where c0=speed of light in vacuum=30cm/ns
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38. Idea Signal propagate as wave down transmission line
Delay linear in wire length
Speed
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39. Admin HW8 out Back here on Monday
Includes writeup for previous and this lab
Also three questions
Back here on Monday
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