1. Logical Effort A Method to Optimize Circuit Topology Swarthmore College E77 VLSI Design Adem Kader David Luong Mark Piper December 6, 2005

2. Current Issues Facing Circuit Designers Wanting to optimize circuits for faster performance, inexperienced designers often encounter… –“Simulate-and-Tweak” loops –Incomplete intuition in design process –Uncertainty in decision-making

3. Logical Effort as a Solution Quick method of circuit analysis –Circuit topology –Transistor sizing –Delay estimation Easy way to compare multi-stage designs “Back-of-the-envelope” calculation Provides intuition of circuit timing characteristics in complex circuitry

4. How does it work? Assumes RC model of a transistor d = gh + p d = propagation delay gh = effort delay g = logical effort h = electrical effort = Cout/Cin p = parasitic delay

5. Defining Logical Effort Ratio of the input capacitance of the gate to the input capacitance of an inverter that can deliver the same output current Measure of a gate to drive a particular fan-out relative to an inverter

6. Visualizing Logical Effort

7. Application of Logical Effort Estimating Delay Propagation d = g h + p INVERTER NAND

8. Multi-Stage Design and Logical Effort Often circuits are more complicated than an inverter or a NAND gate Same framework applies with the modification…

9. Logical Effort and Transistor Sizing Interested in choosing transistor sizing to minimize stage and overall delay f (min) = g(i) * h(i) = F 1/N Delay equation becomes… In the end…

10. Application of Transistor Sizing How do we choose stage capacitances given we want to minimize propagation delay?

11. Optimal Number of Gates Path Effort F Optimal NMinimum Delay, D Stage Effort, f 0-5.8311.0-6.80-5.8 5.82-22.326.8-11.42.4-4.7 22.3-82.2311.4-16.02.8-4.4 82.2-300416.0-20.73.0-4.2 300-1090520.7-25.33.1-4.1 1090-3920625.3-29.83.2-4.0 Rule of thumb is … Note that single gate does not always translate to minimized delay

12. Example: The Implementation Problem Which do you choose?

13. Using Logical Effort… Option 1: Path logic effort G = 1 * 6/3 * 1 = 2 Path Branch Effort B = 1 Path electrical effort H = C out /C in = 8C/C = 8 Path Stage effort = F = GBH = 2*1*8 = 16 D min = N*F1/N+P = 3*(16)1/3 + (1+4*1 + 1) = 3*3.25 + 6 = 13.5

14. Using Logical Effort… Option 2: Path logic effort G = 1 * 4/3 * 5/3 = 20/9 Path Branch Effort B = 1 Path electrical effort H = C out /C in = 8C/C = 8 Path Stage effort = F = GBH = 20/9*1*8 = 160/9 D min = N*F1/N+P = 3*(160/9)1/3 + (1+2*1 + 2) = 3*3.25 + 5 = 12.8

15. Using SPICE…

16. Example: Choosing the Optimal N The Buffer Problem Must drive 64 parallel inverters Choose 1, 3, or 5 series inverter stages to drive the load?

17. finding optimal #of stages N531 f2.3464 D16.51565

18. 1 inverter

19. 3 inverters

20. 5 inverters

21. all together

22. Problems with logical effort It’s only an approximation –But a good one It does not guarantee optimal solution –but gets quite close Chicken and egg problem –chicken Built for speed –Does not account for power consumption and physical size

23. So What Have We Learned? Logical Effort… –Provides method to quickly determine speed of design topologies for comparison –Displays changes to parameter tweaking

24. I agree with stupid  It’s so… logical! now that makes sense!