# ESE370: Circuit-Level Modeling, Design, and Optimization for Digital

ESE370: Circuit-Level Modeling, Design, and Optimization for Digital Systems Day 21: October 28, 2011 Distributed RC Delay Penn ESE370 Fall2011 -- DeHon 1

Today Estimate delay in RC Network Elmore delay calculation Wire Delay Penn ESE370 Fall2011 -- DeHon 2 What is response?

Penn ESE370 Fall2011 -- DeHon 3 What is response? Penn ESE370 Fall2011 -- DeHon 4 What is response?

Penn ESE370 Fall2011 -- DeHon 5 SPICE Response Penn ESE370 Fall2011 -- DeHon 6 What is response?

Penn ESE370 Fall2011 -- DeHon 7 SPICE Response Penn ESE370 Fall2011 -- DeHon 8 Intuition Look at series of Rs on path

Must move Q=V(C) across each R Not as much as if both Rs precede Cs Penn ESE370 Fall2011 -- DeHon 9 Elmore Delay For each resistor Ri in path Compute CRi = sum of all Cs downstream of Ri Delay through Ri is RiCRi

Delay = Ri path i Penn ESE370 Fall2011 -- DeHon C j path j 10

Superposition Penn ESE370 Fall2011 -- DeHon 11 Superposition R1 C1

R2 C2 R2 R1 C1 Penn ESE370 Fall2011 -- DeHon R1

C2 12 Superposition R1*C1 (R1+R2)*C2 R1

C1 R2 C2 R2 R1 C1 Penn ESE370 Fall2011 -- DeHon

R1 C2 R1*(C1+C2)+R2*C2 13 Superposition not concurrent Dont happen concurrently since must divide current

Penn ESE370 Fall2011 -- DeHon 14 Superposition For R1=R2=R, C1=C2=C Delay = 3RC R2 R1

C1 Penn ESE370 Fall2011 -- DeHon C2 R1*(C1+C2)+R2*C2 15 SPICE Response

Penn ESE370 Fall2011 -- DeHon 16 Apply to Y Penn ESE370 Fall2011 -- DeHon 17 Apply Y 10003pF

+10001pF =4ns Penn ESE370 Fall2011 -- DeHon 18 SPICE Response Penn ESE370 Fall2011 -- DeHon 19

Elmore Delay For each resistor Ri in path Compute CRi = sum of all Cs downstream of Ri Delay through Ri is RiCRi Delay = Ri path i Penn ESE370 Fall2011 -- DeHon

C j path j 20 Wire Penn ESE370 Fall2011 -- DeHon 21

Wire Capacitance Penn ESE370 Fall2011 -- DeHon A C r 0 d 22 Wire Resistance

Penn ESE370 Fall2011 -- DeHon L R A 23 Wire as RC Ladder Penn ESE370 Fall2011 -- DeHon

24 Wire Delay as f(L) Measure wire length in units Say Each lambda have Cunit, Runit Capacitance and resistance of wire of length Penn ESE370 Fall2011 -- DeHon 25

Wire Delay Wire N units long: Runit*(N*Cunit) +Runit((N-1)*Cunit +Runit*(N-2)*Cunit+ +Runit*Cunit =(Runit*Cunit)*(N+N-1+N-2+.1) Penn ESE370 Fall2011 -- DeHon 26

Sum of integers Whats the sum of the integer 1 to N? N+N-1+N-2+1 N k k =0 Penn ESE370 Fall2011 -- DeHon 27

Sum of integers Whats the sum of the integer 1 to N? N+N-1+N-2+1 N N(N +1) 2 k = 2 0.5N k =0 Penn ESE370 Fall2011 -- DeHon

28 Wire Delay Wire N units long: Runit*(N*Cunit)+Runit((N-1)*Cunit +Runit*(N-2)*Cunit++Runit*Cunit =(Runit*Cunit)*(N+N-1+N-2+.1) =Runit*Cunit*N2/2 Penn ESE370 Fall2011 -- DeHon

29 Wire Delay Rwire = N*Runit Cwire=N*Cunit Wire delay = Runit*Cunit*N2/2 Wire delay = 0.5 * Rwire*Cwire

Half the delay of lumped RC product Quadratic in length of wire Penn ESE370 Fall2011 -- DeHon 30 Branching Wire What is delay of: Drive L/2

L/2 Receive L/2 Penn ESE370 Fall2011 -- DeHon 31 Branching Wire Drive Wire of length L

L2/2 (Runit*Cunit) Charge L/2 Cunit through R=L/2 Runit L2/4 (Runit*Cunit) Total 0.75 L2 Drive L/2 L/2 Receive

L/2 Penn ESE370 Fall2011 -- DeHon 32 Branching Wire Direct calc like ladder for first half 3L 2 5 3L 3L

2 RC + 1 + ...L RC 0.5 0.5L = RCL2 2 2 2 8 Then wire of length L/2 0.5(L/2)2=L2/8

Total 0.75 L2 Drive L/2 L/2 Receive L/2 Penn ESE370 Fall2011 -- DeHon

33 Admin Project Due next Friday Baseline done; many ideas on what to do Starting to optimize. Now dig into optimization and design-space exploration Midterm 2: Nov. 9th Wednesday, 1.5 week, in evening

Penn ESE370 Fall2011 -- DeHon 34 Idea Lumped wiring calculation is pessimistic Not all capacitance at end of wire Elmore delay calculation allows us to estimate Wires are distributed RC Half delay lumped calculation

Still quadratic in length Penn ESE370 Fall2011 -- DeHon 35

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