1. Optimal digital circuit design Mohammad Sharifkhani

2. Outline Introduction Optimization for speed –Logical effort Optimization considering power consumption

3. Optimization for speed Assume we want to design a multi-stage block that does a special function –What is the best sizes of the transistors to achieve optimal speed?

4. Logical effort What is the best circuit topology for a function? o How large should the transistors be? o How many stages of logic give least delay? Logical Effort is a method of answering these questions: o Uses a very simple model of delay o Back of the envelope calculations and tractable optimization o Gives new names to old ideas to emphasize remarkable symmetries Who cares about logical effort? o Circuit designers waste too much time simulating and tweaking circuits o High speed logic designers need to know where time is going in their logic o CAD engineers need to understand circuits to build better tools

5. Example Decoder specification: o 16 word register file o Each word is 32 bits wide o Each bit presents a load of 3 unit-sized transistors o True and complementary inputs of address bits a are available o Each input may drive 10 unit-sized transistors o How many stages to use? o How large should each gate be? o How fast can the decoder operate?

6. Delay of a gate

7. Normalized delays Delay is expressed in terms of a basic delay unit, τ = 3RC, the delay of an inverter driving an identical inverter with no parasitic capacitance In a typical 600-nm process τ is about 50 ps. For a 250-nm process, τ is about 20 ps. In modern 45 nm processes the delay is approximately 4 to 5 ps.

8. Normalized delays Parasitic delay, p : an intrinsic delay of the gate and can be found by considering the gate driving no load Stage effort, f = gh: dependent on the load Logical effort, g: the ratio of the input capacitance of a given gate to that of an inverter capable of delivering the same output current –a constant for a particular class of gate and can be described as capturing the intrinsic properties of the gate Electrical effort, h: the ratio of the input capacitance of the load to that of the gate d = f + p

9. Logical effort/Electrical effort

10. Computing logical effort

11. Logical effort and parasitic delay Note: for any inverter size!

12. Example

14. Multi-stage paths It can quickly be extended to circuits composed of multiple stages. Total normalised path delay D =overall path effort, F, plus path parasitic delay P (which is the sum of the individual parasitic delays): –D = F + P Path effort, F = G x H – path logical effort G : the product of the individual logical efforts of the gates –path electrical effort H :the ratio of the load of the path to its input capacitance

15. Multi-stage paths

16. Branching effort

18. branching effort, b: the ratio of total capacitance being driven by the gate to the capacitance on the path of interest

19. Branching effort

20. Total delay The total delay of a path is the sum of individual effort delays f i and parasitic delays p i

21. Key point How to relate individual effort delays to total path effort delay? Lowest possible path delay can be found without even calculating the sizes of each gate in the path.

22. Gate sizing Gate sizes can be found by starting at the end of the path and working backward. Check your work by verifying that the input capacitance specification is satisfied at the beginning of the path.

23. Example 1

25. Optimal number of stages

26. n1 F: Path effort

27. Optimal number of stages For inverter chain:

28. Example Decoder specification: o 16 word register file o Each word is 32 bits wide o Each bit presents a load of 3 unit-sized transistors o True and complementary inputs of address bits a are available o Each input may drive 10 unit-sized transistors o How many stages to use? o How large should each gate be? o How fast can the decoder operate?

29. Example

30. Three other inputs create 8 different paths, only one of which is active: C total /C active =8

31. Example

32. g of 4 input NAND

33. Example

34. Summary

35. Method of Logical Effort

36. Limitation Logical effort is not a panacea. Some limitations include: o Chicken & egg problem –how to estimate G and best number of stages before the path is designed o Simplistic delay model –neglects effects of input slopes o Interconnect –iteration required in designs with branching and non-negligible wire C or RC –same convergence difficulties as in synthesis / placement problem o Maximum speed only –optimizes circuits for speed, not area or power under a fixed speed constraint

37. Conclusion Logical effort is a useful concept for thinking about delay in circuits: o Facilitates comparison of different circuit topologies o Easily select gate sizes for minimum delay o Circuits are fastest when effort delays of each stage are equal and about 4 o Path delay is insensitive to modest deviations from optimal sizes Some further results from logical effort include: o Logical effort can be applied to domino, pass gate, and other logic families o Logic gates can be skewed to favor one input or edge at the cost of another o While the logical effort of a multiplexer is independent of the number of inputs, parasitic delay increases with size, so 4-way multiplexers are best o Circuits that fork should equalize delays between legs of the fork