ESE 370 CircuitLevel Modeling Design and Optimization for

ESE 370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Day 33: November 29, 2010 Transmission Lines Penn ESE 370 Fall 2010 -- De. Hon 1

This Week • • 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 Penn ESE 370 Fall 2010 -- De. Hon 2

Wires • In general, our “wires” have distributed R, L, C components Penn ESE 370 Fall 2010 -- De. Hon 3

RC Wire • When R dominates L – We have the distributed RC Wires we saw on Day 27 – Typical of on-chip wires in ICs Penn ESE 370 Fall 2010 -- De. Hon 4

Transmission Line • When resistance is negligible – Have LC wire = Lossless Transmission Line – More typical of Printed Circuit Board wires Penn ESE 370 Fall 2010 -- De. Hon 5

Intuitive: Lossless • Pulses travel as waves without distortion – (up to a characteristic frequency) Penn ESE 370 Fall 2010 -- De. Hon 6

SPICE Simulation Penn ESE 370 Fall 2010 -- De. Hon 7

SPICE Simulation Penn ESE 370 Fall 2010 -- De. Hon 8

Contrast RC Wire Penn ESE 370 Fall 2010 -- De. Hon 9

Visualization • See: http: //www. research. ibm. com/people/r/r estle/Animations/DAC 01 top. html Penn ESE 370 Fall 2010 -- De. Hon 10

Setup Relations • • Vi-Vi-1 = Ldii/dt Vi+1 -Vi = Ldii+1/dt Ici=Cd. Vi/dt Ii-Ii+1=Ici Vi-1 Ii Vi Ii+1 Vi+1 Ici Penn ESE 370 Fall 2010 -- De. Hon 11

Reduce to Single Equation • • • Vi-Vi-1 = Ldii/dt Vi+1 -Vi = Ldii+1/dt Ici=Cd. Vi/dt d. Ici/dti=Cd 2 Vi/dt Ii-Ii+1=Ici d. Ii/dt-d. Ii+1/dt=d. Ici/dt Vi-Vi-1 -(Vi+1 -Vi )= Ldii/dt - Ldii+1/dt – d 2 V/dx = -Ld 2 I/dx=-Ld. Ici/dt=-LCd 2 Vi/dt • Vi+1 -Vi-1=-LCd 2 Vi/dt Penn ESE 370 Fall 2010 -- De. Hon 12

Implication • Vi+1 -Vi-1=-LCd 2 Vi/dt • Once Vi settles, settle to same value • d 2 V/dx = LCd 2 V/dt • Wave equation • V(x, t) = A+Be-(wt+x) • Be-(wt+x)=LCw 2 Be-(wt+x) • w=1/sqrt(LC) – Rate of propagation Penn ESE 370 Fall 2010 -- De. Hon 13

Propagation Rate in Example • L=1 u. H • C=1 p. F Penn ESE 370 Fall 2010 -- De. Hon 14

Signal Propagation Penn ESE 370 Fall 2010 -- De. Hon 15

Propagation • Be-(wt+x)=LCw 2 Be-(wt+x) • w=1/sqrt(LC) – Rate of propagation – Delay linear in length • Compare RC wire delay quadratic in length Penn ESE 370 Fall 2010 -- De. Hon 16

Contrast RC Wire Penn ESE 370 Fall 2010 -- De. Hon 17

Propagation • Be-(wt+x)=LCw 2 Be-(wt+x) • w=1/sqrt(LC) – Rate of propagation – Delay linear in length • Compare RC wire delay quadratic in length • From Day 31 we know for wire: CL = em – w=1/sqrt(em)=c 0/sqrt(ermr) – Where c 0=speed of light in vacuum=30 cm/ns Penn ESE 370 Fall 2010 -- De. Hon 18

Impedance • V(x, t) = A+Be-(wt+x) • Ici=Cd. Vi/dt • Ici=w. CBe-(wt+x) • Z 0 = Vi/Ii ~Vi/Ici = 1/w. C = 1/(C/sqrt(LC)) – (really Ii --- differs in phase) Vi-1 Ii Vi Ii+1 Vi+1 Ici Penn ESE 370 Fall 2010 -- De. Hon 19

Impedance • Z 0 = Vi/Ici = 1/w. C = 1/(C/sqrt(LC)) – (really Ii --- differs in phase) • Transmission line has a characteristic impedance Penn ESE 370 Fall 2010 -- De. Hon 20

Infinite Lossless Transmission Line • Transmission line looks like resistive load Z 0 • Input waveform travels down line at velocity – Without distortion Penn ESE 370 Fall 2010 -- De. Hon 21

End of Line • What happens at the end of the transmission line? – Open Circuit – Short Circuit – Terminate with R=Z 0 Penn ESE 370 Fall 2010 -- De. Hon 22

Open Penn ESE 370 Fall 2010 -- De. Hon 23

Short Penn ESE 370 Fall 2010 -- De. Hon 24

Terminate R=Z 0 Penn ESE 370 Fall 2010 -- De. Hon 25

Longer LC (open) • 40 Stages • L=100 n. H • C=1 p. F Stage delay? • Drive with 2 ns Pulse • No termination Penn ESE 370 Fall 2010 -- De. Hon 26

Pulse Travel RC • V 1, V 3, V 4, V 5, V 6 about 10 stages apart Penn ESE 370 Fall 2010 -- De. Hon 27

Analyze End of Line Penn ESE 370 Fall 2010 -- De. Hon 28

Analyze End of Line • • • Incident wave Vi=Ii×Z 0 Ii=Ir+It Vi+Vr=Vt Vr=Ir×Z 0 Ir=Vr/Z 0 Vt=It×R It=Vt/R Ii=Vi/Z 0 Penn ESE 370 Fall 2010 -- De. Hon 29

Analyze End of Line • Vi+Vr=Vt Penn ESE 370 Fall 2010 -- De. Hon 30

Analyze End of Line • Vi+Vr=Vt Penn ESE 370 Fall 2010 -- De. Hon 31

Analyze End of Line • Vi+Vr=Vt Penn ESE 370 Fall 2010 -- De. Hon 32

Reflection • Sanity check with previous – Open – Short – Matched Penn ESE 370 Fall 2010 -- De. Hon 33

Pulse Travel RC Penn ESE 370 Fall 2010 -- De. Hon 34

Next Time • • • (finish reflections) Termination Strategy Implications Were Transmission Line Arise Hand-wave Lossy Penn ESE 370 Fall 2010 -- De. Hon 35

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 Penn ESE 370 Fall 2010 -- De. Hon 36

Idea • Signal propagate as wave down transmission line – Delay linear in wire length – Speed – Impedance • Behavior at end of line depends on termination Penn ESE 370 Fall 2010 -- De. Hon 37