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 Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

Where Transmission Lines Arise Penn ESE370 Fall2014 -- DeHon

Transmission Lines Cable: coaxial PCB Twisted Pair (Cat5) Strip line Microstrip line Twisted Pair (Cat5) Penn ESE370 Fall2014 -- DeHon

Transmission Lines How did the coaxial cables behave in lab on Friday? How differ from Ideal equipotential? RC-wire on chip? Penn ESE370 Fall2014 -- 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 Penn ESE370 Fall2014 -- DeHon

Transmission Lines Need theory/model to support Reason about behavior Understand what can cause noise Engineer high performance communication Penn ESE370 Fall2014 -- DeHon

Wire Formulation Penn ESE370 Fall2014 -- DeHon

Wires In general, our “wires” have distributed R, L, C components Penn ESE370 Fall2014 -- DeHon

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? Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

Build Intuition from LC What did one LC do? What will chain do? Penn ESE370 Fall2014 -- DeHon

Intuitive: Lossless Pulses travel as waves without distortion (up to a characteristic frequency) Penn ESE370 Fall2014 -- DeHon

SPICE Simulation Penn ESE370 Fall2014 -- DeHon

SPICE Simulation Penn ESE370 Fall2014 -- DeHon

Pulse Response SPICE Penn ESE370 Fall2014 -- DeHon

Contrast RC Wire Penn ESE370 Fall2014 -- DeHon

Contrast Penn ESE370 Fall2014 -- DeHon

Visualization See: http://www.research.ibm.com/people/r/restle/Animations/DAC01top.html Penn ESE370 Fall2014 -- DeHon

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) Penn ESE370 Fall2014 -- DeHon

Setup Relations i is a position Position: x=i × Δx So Vi is V(x=iΔx) Ici Penn ESE370 Fall2014 -- DeHon

Setup Relations Vi-Vi-1 = Ici= Ii-Ii+1= Ii+1 Ii Vi-1 Vi Vi+1 Ici Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

Setup Relations Vi-Vi-1 = -Ldii/dt Ici=CdVi/dt Ii-Ii+1=Ici Ii+1 Ii Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

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? Penn ESE370 Fall2014 -- DeHon

Light Cycle Context http://www.youtube.com/watch?v=GNfs6v7i7eY Penn ESE370 Fall2014 -- DeHon

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) Penn ESE370 Fall2014 -- DeHon

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=? Penn ESE370 Fall2014 -- DeHon

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? Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

Propagation Rate in Example L=1uH C=1pF What is w ? Penn ESE370 Fall2014 -- DeHon

Signal Propagation Penn ESE370 Fall2014 -- 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 Penn ESE370 Fall2014 -- DeHon

Contrast RC Wire Penn ESE370 Fall2014 -- DeHon

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 Penn ESE370 Fall2014 -- DeHon

Idea Signal propagate as wave down transmission line Delay linear in wire length Speed Penn ESE370 Fall2014 -- DeHon

Admin HW8 out Back here on Monday Includes writeup for previous and this lab Also three questions Back here on Monday Penn ESE370 Fall2014 -- DeHon