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

2. Today Estimate delay in RC Network –Elmore delay calculation Wire Delay Penn ESE370 Fall2011 -- DeHon 2

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

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

5. What is response? Penn ESE370 Fall2011 -- DeHon 5

6. SPICE Response Penn ESE370 Fall2011 -- DeHon 6

7. What is response? Penn ESE370 Fall2011 -- DeHon 7

8. SPICE Response Penn ESE370 Fall2011 -- DeHon 8

9. Intuition Look at series of R’s on path –Must move Q=V(  C) across each R Not as much as if both R’s precede C’s Penn ESE370 Fall2011 -- DeHon 9

10. Elmore Delay For each resistor Ri in path –Compute C Ri = sum of all C’s downstream of Ri –Delay through Ri is Ri×C Ri Penn ESE370 Fall2011 -- DeHon 10

11. Superposition Penn ESE370 Fall2011 -- DeHon 11

12. Superposition Penn ESE370 Fall2011 -- DeHon 12 R1 C1 R1 R2 C2 R1 R2 C1 C2

13. Superposition Penn ESE370 Fall2011 -- DeHon 13 R1 C1 R1 R2 C2 R1 R2 C1 C2 R1*C1(R1+R2)*C2 R1*(C1+C2)+R2*C2

14. Superposition not concurrent Don’t happen concurrently since must divide current Penn ESE370 Fall2011 -- DeHon 14

15. Superposition For R1=R2=R, C1=C2=C –Delay = 3RC Penn ESE370 Fall2011 -- DeHon 15 R1 R2 C1 C2 R1*(C1+C2)+R2*C2

16. SPICE Response Penn ESE370 Fall2011 -- DeHon 16

17. Apply to Y Penn ESE370 Fall2011 -- DeHon 17

18. Apply Y 1000  ×3pF +1000  ×1pF =4ns Penn ESE370 Fall2011 -- DeHon 18

19. SPICE Response Penn ESE370 Fall2011 -- DeHon 19

20. Elmore Delay For each resistor Ri in path –Compute C Ri = sum of all C’s downstream of Ri –Delay through Ri is Ri×C Ri Penn ESE370 Fall2011 -- DeHon 20

21. Wire Penn ESE370 Fall2011 -- DeHon 21

22. Wire Capacitance Penn ESE370 Fall2011 -- DeHon 22

23. Wire Resistance Penn ESE370 Fall2011 -- DeHon 23

24. Wire as RC Ladder Penn ESE370 Fall2011 -- DeHon 24

25. 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

26. 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

27. Sum of integers What’s the sum of the integer 1 to N? N+N-1+N-2+…1 Penn ESE370 Fall2011 -- DeHon 27

28. Sum of integers What’s the sum of the integer 1 to N? N+N-1+N-2+…1 Penn ESE370 Fall2011 -- DeHon 28

29. 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*N 2 /2 Penn ESE370 Fall2011 -- DeHon 29

30. Wire Delay Rwire = N*Runit Cwire=N*Cunit Wire delay = Runit*Cunit*N 2 /2 Wire delay = 0.5 * Rwire*Cwire Half the delay of lumped RC product Quadratic in length of wire Penn ESE370 Fall2011 -- DeHon 30

31. Branching Wire What is delay of: Penn ESE370 Fall2011 -- DeHon 31 L/2 DriveReceive

32. Branching Wire Drive Wire of length L – L 2 /2 (Runit*Cunit) Charge L/2 Cunit through R=L/2 Runit –L 2 /4 (Runit*Cunit) Total 0.75 L 2 Penn ESE370 Fall2011 -- DeHon 32 L/2 DriveReceive

33. Branching Wire Direct calc like ladder for first half Then wire of length L/2 – 0.5(L/2) 2 =L 2 /8 Total 0.75 L 2 Penn ESE370 Fall2011 -- DeHon 33 L/2 DriveReceive

34. 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. 9 th –Wednesday, 1.5 week, in evening Penn ESE370 Fall2011 -- DeHon 34

35. 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