Penn ESE370 Fall DeHon 1 ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Day 33: November 30, 2011 Transmission Line Introduction and Analysis

Next few Lectures/Lab Where arise? General wire formulation Lossless Transmission Line See in action in lab (Friday) End of Transmission Line? Termination Discuss Lossy Implications Penn ESE370 Fall DeHon 2

Where Transmission Lines Arise Penn ESE370 Fall DeHon 3

Transmission Lines Cable: coaxial PCB –Strip line –Microstrip line Twisted Pair (Cat5) Penn ESE370 Fall DeHon 4

Transmission Lines How do these wires behave? –25m of category-5 cable? –VGA cable? (Analog?) How differ from –Ideal equipotential? –RC-wire on chip? Penn ESE370 Fall DeHon 5

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 Fall DeHon 6

Transmission Lines Need to understand –How to model  how to reason about –What can cause noise –How to engineer high performance communication Penn ESE370 Fall DeHon 7

Wire Formulation Penn ESE370 Fall DeHon 8

Wires In general, our “wires” have distributed R, L, C components Penn ESE370 Fall DeHon 9

RC Wire When R dominates L –We have the distributed RC Wires we saw on Day 21 –Typical of on-chip wires in ICs Penn ESE370 Fall DeHon 10

Transmission Line When resistance is negligible –Have LC wire = Lossless Transmission Line –More typical of Printed Circuit Board wires Penn ESE370 Fall DeHon 11

Build Intuition from LC What did one LC do? What will chain do? Penn ESE370 Fall DeHon 12

Intuitive: Lossless Pulses travel as waves without distortion –(up to a characteristic frequency) Penn ESE370 Fall DeHon 13

SPICE Simulation Penn ESE370 Fall DeHon 14

SPICE Simulation Penn ESE370 Fall DeHon 15

Penn ESE370 Fall DeHon 16 Contrast RC Wire

Visualization See: estle/Animations/DAC01top.html estle/Animations/DAC01top.html Penn ESE370 Fall DeHon 17

Setup Relations Penn ESE370 Fall DeHon 18 ViVi V i+1 V i-1 IiIi I i+1 I ci

Setup Relations V i -V i-1 = I ci = I i -I i+1 = Penn ESE370 Fall DeHon 19 ViVi V i+1 V i-1 IiIi I i+1 I ci

Setup Relations V i -V i-1 = Ldi i /dt I ci =CdV i /dt I i -I i+1 =I ci Penn ESE370 Fall DeHon 20 ViVi V i+1 V i-1 IiIi I i+1 I ci i is spatial dimension V i at different positions

Setup Relations V i -V i-1 = Ldi i /dt I ci =CdV i /dt I i -I i+1 =I ci Penn ESE370 Fall DeHon 21 ViVi V i+1 V i-1 IiIi I i+1 I ci Maybe sign wrong on LdI/dt

Reduce to Single Equation Eliminate I ci ? I i -I i+1 =I ci  dI i /dt-dI i+1 /dt=dI ci /dt I ci =CdV i /dt  dI ci /dt i =Cd 2 V i /dt di i /dt - di i+1 /dt=Cd 2 V i /dt Penn ESE370 Fall DeHon 22

Reduce to Single Equation di i /dt - di i+1 /dt=Cd 2 V i /dt V i -V i-1 = Ldi i /dt V i+1 -V i = Ldi i+1 /dt Eliminate Is ? V i -V i-1 -(V i+1 -V i )= Ldi i /dt - Ldi i+1 /dt d 2 V/dx =-LCd 2 V/dt V i+1 -V i-1 =-LCd 2 V i /dt Penn ESE370 Fall DeHon 23 Multiple sign problems

Implication V i+1 -V i-1 =LCd 2 V i /dt Once V i settles, settle to same value d 2 V/dx = LCd 2 V/dt Wave equation V(x,t) = A+Be (x-wt) Be (x-wt) =LCw 2 Be (x-wt) w=1/sqrt(LC) –Rate of propagation Penn ESE370 Fall DeHon 24

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 ? Penn ESE370 Fall DeHon 25

Propagation V(x,t) = A+Be (x-wt) If V(x0,t0)=Va And V(x0+  x,t0+  t)=Va What is w? Penn ESE370 Fall DeHon 26

Propagation Rate in Example L=1uH C=1pF What is w ? Penn ESE370 Fall DeHon 27

Signal Propagation Penn ESE370 Fall DeHon 28

Propagation Be (x-wt+x) =LCw 2 Be (x-wt) w=1/sqrt(LC) –Rate of propagation –Delay linear in length Compare RC wire delay quadratic in length Penn ESE370 Fall DeHon 29

Contrast RC Wire Penn ESE370 Fall DeHon 30

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 32 we know for wire: CL =  –w=1/sqrt  c 0 /sqrt  r  r  –Where c 0 =speed of light in vacuum=30cm/ns Penn ESE370 Fall DeHon 31

Class Ended Here Penn ESE370 Fall DeHon 32

Wire “Resistance” What is the resistance at V i ? Penn ESE370 Fall DeHon 33 ViVi V i+1 V i-1 IiIi I i+1 I ci

Wire “Resistance” Q=CV I = dQ/dt Moving at rate w I=wCV R=V/I=1/(wC) Penn ESE370 Fall DeHon 34 ViVi V i+1 V i-1 IiIi I i+1 I ci

Impedance Z 0 =R= 1/wC = 1/(C/sqrt(LC)) Penn ESE370 Fall DeHon 35 ViVi V i+1 V i-1 IiIi I i+1 I ci

Impedance Assuming infinitely long wire, how look different at V i, V i+1, V i+2 ? Penn ESE370 Fall DeHon 36 ViVi V i+1 V i-1 IiIi I i+1 I ci

Impedance Transmission line has a characteristic impedance –Looks to driving circuit like a resistance Penn ESE370 Fall DeHon 37

Infinite Lossless Transmission Line Transmission line looks like resistive load Input waveform travels down line at velocity –Without distortion Penn ESE370 Fall DeHon 38 Z0Z0

End of Line What happens at the end of the transmission line? –Short Circuit –Terminate with R=Z 0 –Open Circuit Experimentally in Lab Friday Mathematically in Class Monday Penn ESE370 Fall DeHon 39

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 Penn ESE370 Fall DeHon 40

Idea Signal propagate as wave down transmission line –Delay linear in wire length –Speed –Impedance Penn ESE370 Fall DeHon 41