# Logical Effort: Designing for Speed on the Back of an Envelope David Harris Harvey Mudd College Claremont, CA

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1 Logical Effort: Designing for Speed on the Back of an Envelope David Harris Harvey Mudd College Claremont, CA

2 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 2 of 56

3 Introduction Chip designers face a bewildering array of choices. o 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 Logical Effort David Harris Page 3 of 56

4 Example Ben Bitdiddle is the memory designer for the Motoroil 68W86, an embedded processor for automotive applications. Help Ben design the decoder for a register file: a<3:0> a<3:0> 32 bits Decoder specification: o 16 word register file o Each word is 32 bits wide 16 Register File o Each bit presents a load of 3 unit-sized transistors o True and complementary inputs of address bits a<3:0> are available o Each input may drive 10 unit-sized transistors 4:16 Decoder 16 words Ben needs to decide: o How many stages to use? o How large should each gate be? o How fast can the decoder operate? Logical Effort David Harris Page 4 of 56

5 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 5 of 56

6 Delay in a Logic Gate Let us express delays in a process-independent unit: Delay of logic gate has two components: d d abs τ τ 12 ps in 0.18 µm technology d f + p Effort delay again has two components: effort delay, a.k.a. stage effort parasitic delay f gh logical effort electrical effort C out /C in electrical effort is sometimes called fanout o Logical effort describes relative ability of gate topology to deliver current (defined to be 1 for an inverter) o Electrical effort is the ratio of output to input capacitance Logical Effort David Harris Page 6 of 56

7 Delay Plots Normalized delay: d input NAND inverter g p d effort delay g p d parasitic delay How about a 2-input NOR? Electrical effort: h C out / C in o d f + p gh + p o Delay increases with electrical effort o More complex gates have greater logical effort and parasitic delay Logical Effort David Harris Page 7 of 56

8 Computing Logical Effort DEF: Logical effort is the ratio of the input capacitance of a gate to the input capacitance of an inverter delivering the same output current. o Measured from delay vs. fanout plots of simulated or measured gates o Or estimated, counting capacitance in units of transistor width: a Inverter: C in 3 g 1 (def) 2 1 x a b NAND2: C in 4 g 4/3 x a b NOR2: C in 5 g 5/3 1 x Logical Effort David Harris Page 8 of 56

9 A Catalog of Gates Table 1: Logical effort of static CMOS gates Gate type Number of inputs n inverter 1 NAND 4/3 5/3 6/3 7/3 (n+2)/3 NOR 5/3 7/3 9/3 11/3 (2n+1)/3 multiplexer XOR, XNOR Table 2: Parasitic delay of static CMOS gates Gate type inverter n-input NAND n-input NOR n-way multiplexer 2-input XOR, XNOR Parasitic delay p inv np inv np inv 2np inv 4np inv p inv 1 parasitic delays depend on diffusion capacitance Logical Effort David Harris Page 9 of 56

10 Example Estimate the frequency of an N-stage ring oscillator: Logical Effort: Electrical Effort: Parasitic Delay: Stage Delay: Oscillator Frequency: g h p d F Logical Effort David Harris Page 10 of 56

11 Example Estimate the delay of a fanout-of-4 (FO4) inverter: d Logical Effort: Electrical Effort: Parasitic Delay: Stage Delay: g h p d Logical Effort David Harris Page 11 of 56

12 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 12 of 56

13 Multi-stage Logic Networks Logical effort extends to multi-stage networks: 10 x y z 20 g 1 1 h 1 x/10 g 2 5/3 h 2 y/x g 3 4/3 h 3 z/y g 4 1 h 4 20/z o Path Logical Effort: o Path Electrical Effort: o Path Effort: G H F g i C out (path) C in (path) f i g i h i Don t define H h i because we don t know h i until the design is done Can we write F GH? Logical Effort David Harris Page 13 of 56

14 Branching Effort No! Consider circuits that branch: G H GH h 1 h 2 F GH? Logical Effort David Harris Page 14 of 56

15 Delay in Multi-stage Networks We can now compute the delay of a multi-stage network: o Path Effort Delay: D F f i o Path Parasitic Delay: P F p i o Path Delay: D F d i D F + P We can prove that delay is minimized when each stage bears the same effort: Therefore, the minimum delay of an N-stage path is: ˆ f g i h i NF 1 N + P F 1 N o This is a key result of logical effort. Lowest possible path delay can be found without even calculating the sizes of each gate in the path. Logical Effort David Harris Page 15 of 56

16 Determining Gate Sizes Gate sizes can be found by starting at the end of the path and working backward. o At each gate, apply the capacitance transformation: C outi g i C ini ˆ f o Check your work by verifying that the input capacitance specification is satisfied at the beginning of the path. Logical Effort David Harris Page 16 of 56

17 Example Select gate sizes y and z to minimize delay from A to B Logical Effort: Electrical Effort: G H A C y y z z z 9C 9C B 9C Branching Effort: B Path Effort: F Best Stage Effort: Delay: ˆ f D Work backward for sizes: z y Logical Effort David Harris Page 17 of 56

18 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 18 of 56

19 Choosing the Best Number of Stages How many stages should a path use? o Delay is not always minimized by using as few stages as possible o Example: How to drive 64 bit datapath with unit-sized inverter Initial driver Datapath load N: f: D: Fastest D NF 1 N + P N64 ( ) 1 N + N assuming polarity doesn t matter Logical Effort David Harris Page 19 of 56

20 Derivation of the Best Number of Stages Suppose we can add inverters to the end of a path without changing its function. o How many stages should we use? Let Nˆ be the value of N for least delay. Logic Block: n 1 stages Path effort F N-n 1 extra inverters F 1 Nˆ NF 1 N p i D + + ( N n 1 )p inv o Define ρ to be the best stage effort. Substitute and simplify: n 1 1 D F 1 N ln( F 1 N ) + F 1 N + p N inv 0 p inv + ρ( 1 lnρ) 0 Logical Effort David Harris Page 20 of 56

21 Best Number of Stages (continued) p inv + ρ( 1 lnρ) 0 has no closed form solution. o Neglecting parasitics (i.e. p inv 0), we get the familiar result that ρ (e) o For p inv 1, we can solve numerically to obtain ρ 3.59 How sensitive is the delay to using exactly the best number of stages? D(N) / D(N) (ρ2.4) (ρ6) I like to use ρ N / N o 2.4 < ρ < 6 gives delays within 15% of optimal -> we can be sloppy Logical Effort David Harris Page 21 of 56

22 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 22 of 56

23 Example Let s revisit Ben Bitdiddle s decoder problem using logical effort: a<3:0> a<3:0> 32 bits Decoder specification: Register File o 16 word register file 16 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<3:0> are available o Each input may drive 10 unit-sized transistors 4:16 Decoder 16 words Ben needs to decide: o How many stages to use? o How large should each gate be? o How fast can the decoder operate? Logical Effort David Harris Page 23 of 56

24 Example: Number of Stages How many stages should Ben use? o Effort of decoders is dominated by electrical and branching portions H B o Electrical Effort: o Branching Effort: If we neglect logical effort (assume G 1), o Path Effort: F Remember that the best stage effort is about ρ 4 o Hence, the best number of stages is: N Logical Effort David Harris Page 24 of 56

25 Example: Gate Sizes & Delay G Lets try a 3-stage design using 16 4-input NAND gates with a 0 a 0 a 1 a 1 a 2 a 2 a 3 a 3 10 unit input capacitance y z out 0 96 unit wordline capacitance y z out 15 o Actual path effort is: o Therefore, stage effort should be: o Gate sizes: o Path delay: F f z y D Logical Effort David Harris Page 25 of 56

26 Example: Alternative Decoders Table 3: Comparison of Decoder Designs Design Stages G P D NAND4; INV INV; NAND4; INV INV; NAND4; INV; INV NAND2; INV; NAND2; INV 4 16/ INV; NAND2; INV; NAND2; INV 5 16/ NAND2; INV; NAND2; INV; INV; INV 6 16/ INV; NAND2; INV; NAND2; INV; INV; INV 7 16/ NAND2; INV; NAND2; INV; INV; INV; INV; INV 8 16/ We underestimated the best number of stages by neglecting the logical effort. o Logical effort facilitates comparing different designs before selecting sizes o Using more stages also reduces G and P by using multiple 2-input gates o Our design was about 10% slower than the best Logical Effort David Harris Page 26 of 56

27 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 27 of 56

28 Asymmetric Gates Asymmetric logic gates favor one input over another. Example: suppose input A of a NAND gate is most critical. o Select sizes so pullup and pulldown still match unit inverter o Place critical input closest to output a 2 2 4/3 4 x b o Logical Effort on input A: o Logical Effort on input B: o Total Logical Effort: g A g B g tot g A + g B Logical Effort David Harris Page 28 of 56

29 Symmetry Factor In general, consider gates with arbitrary symmetry factor s: o s 1/2 in symmetric gate with equal sizes o s 1/4 in previous example a 2 2 1/(1-s) x Logical effort of inputs: s s s1 ( s) 4 g A g 3 B g 3 tot o Critical input approaches logical effort of inverter 1 for small s o But total logical effort is higher for asymmetric gates 1/s b Logical Effort David Harris Page 29 of 56

30 Skewed Gates Skewed gates favor one edge over the other. Example: suppose rising output of inverter is most critical. o Downsize noncritical NMOS transistor to reduce total input capacitance a 2 1/2 x a 2 1 x a 1 1/2 x HI-Skewed inverter Unskewed w/ Unskewed w/ equal rise equal fall Compare with unskewed inverter of the same rise/fall time to compute effort. o Logical Effort for rising (up) output: o Logical Effort for falling (down) output: o Average Logical Effort: g u g d g avg ( g u + g d ) 2 Logical Effort David Harris Page 30 of 56

31 HI- and LO-Skewed Gates DEF: Logical effort of a skewed gate for a particular transition is the ratio of the input capacitance of that gate to the input capacitance of an unskewed inverter delivering the same output current for the same transition. Skew gates by reducing size of noncritical transistors. o HI-Skewed gates favor rising outputs by downsizing NMOS transistors o LO-Skewed gates favor falling outputs by downsizing PMOS transistors o Logical effort is smaller for the favored input due to lower input capacitance o Logical effort is larger for the other input Logical Effort David Harris Page 31 of 56

32 Catalog of Skewed Gates Inverter NAND2 NOR2 HI-Skew 2 ½ g u 5/6 g d 5/3 g 5/4 avg g u 1 g d 2 g 3/2 avg 4 4 ½ g u 3/2 g d 3 g 9/4 avg ½ LO-Skew 1 1 g u 4/3 g d 2/3 g 1 avg g u 2 g d 1 g 3/2 avg g u 2 g d 1 g 3/2 avg 1 Logical Effort David Harris Page 32 of 56

33 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 33 of 56

34 Pseudo-NMOS Pseudo-NMOS gates replace fat PMOS pullups on inputs with a resistive pullup. o Resistive pullup must be much weaker than pulldown stack (e.g. 4x) o Reduces logical effort because inputs must only drive the NMOS transistors o However, NMOS current reduced by contention with pullup o Unequal rising and falling efforts o Quiescent power dissipation when output is low Example: Pseudo-NMOS inverter o Logical Effort for falling (down) output: o Logical Effort for rising (up) output: o Average Logical Effort: g d a g u g avg ( g u + g d ) 2 2/3 4/3 x Logical Effort David Harris Page 34 of 56

35 Pseudo-NMOS Gates Inverter NAND2 NOR2 a 2/3 4/3 x a 2/3 8/3 x a 2/3 4/3 b 4/3 x b 8/3 g d 4/9 g u 4/3 g avg 8/9 g d 8/9 g u 8/3 g avg 16/9 g d 4/9 g u 4/3 g avg 8/9 Tradeoffs exist between power and effort by varying P/N ratio. Logical Effort David Harris Page 35 of 56

36 Dynamic Logic Dynamic logic replace fat PMOS pullups on inputs with a clocked precharge. o Reduces logical effort because inputs must only drive the NMOS transistors o Eliminates pseudo-nmos contention current and power dissipation o Only the falling ( evaluation ) delay is critical o Downsize noncritical precharge transistors to reduce clock load and power Example: Footless dynamic inverter o Logical Effort for falling (down) output: g d φ a 1 1 x Robust gates may require keepers and clocked pulldown transistors ( feet ). o Feet prevent contention during precharge but increase logical effort o Weak keepers prevent floating output at cost of slight contention during eval Logical Effort David Harris Page 36 of 56

37 Dynamic Gates Inverter NAND2 NOR2 Footless φ 1 φ 1 φ 1 x x x a 1 a 2 a 1 1 b g d 1/3 b 2 g d 2/3 g d 1/3 Footed φ 1 φ 1 φ 1 x x x a 2 a 3 a 2 2 b g d 2/3 b 3 g d 1 g d 2/3 φ 2 φ 2 φ 3 Logical Effort David Harris Page 37 of 56

38 Domino Gates Dynamic gates require monotonically rising inputs. o However, they generate monotonically falling outputs o Alternate dynamic gates with HI-skew inverting static gates o Dynamic / static pair is called a domino gate Example: Domino Buffer o Constraints: maximum input capacitance 3, load 54 o Logical Effort: G o Branching Effort: B o Electrical Effort: H φ 3 o Path Effort: F a o Stage Effort: f 3 o HI-Skew Inverter: size o Transistor Sizes: n p p n g 1 g 2 HI-Skew 54 Logical Effort David Harris Page 38 of 56

39 Comparison of Circuit Families Assumptions: o PMOS transistors have half the drive of NMOS transistors o Skewed gates downsize noncritical transistors by factor of two o Pseudo-NMOS gates have 1/4 strength pullups Table 4: Summary of Logical Efforts Circuit Style Inverter g n-input NAND g n-input NOR g g u g d g u g d g u g d Static CMOS 1 (n+2)/3 (2n+1)/3 HI-Skew 5/6 5/3 (n/2+2)/3 (n+4)/3 (2n+.5)/3 (4n+1)/3 LO-Skew 4/3 2/3 2(n+1)/3 (n+1)/3 2(n+1)/3 (n+1)/3 Pseudo-NMOS 4/3 4/9 4n/3 4n/9 4/3 4/9 Footed Dynamic 2/3 (n+1)/3 2/3 Footless Dynamic 1/3 n/3 1/3 Adjust these numbers as you change your assumptions. Logical Effort David Harris Page 39 of 56

40 Outline o Introduction o Delay in a Logic Gate o Multi-stage Logic Networks o Choosing the Best Number of Stages o Example o Asymmetric & Skewed Logic Gates o Circuit Families o Summary Logical Effort David Harris Page 40 of 56

41 Summary Table 5: Key Definitions of Logical Effort Term Stage expression Path expression Logical effort (seetable 1) Electrical effort Branching effort Effort Effort delay Number of stages Parasitic delay Delay g G g i n/a C out h H C in B C out (path) C in (path) b i f gh F GBH f D F 1 N f i p (seetable 2) P p i d f + p D D F + P Logical Effort David Harris Page 41 of 56

42 Method of Logical Effort Logical effort helps you find the best number of stages, the best size of each gate, and the minimum delay of a circuit with the following procedure: o Compute the path effort: o Estimate the best number of stages: F GBH Nˆ log 4 F o Estimate the minimum delay: o Sketch your path using the number of stages computed above o Compute the stage effort: o Starting at the end, work backward to find transistor sizes: C outi g i C ini ˆ D ˆ f Nˆ F 1 Nˆ F 1 N f + P Logical Effort David Harris Page 42 of 56

43 Limitations of Logical Effort 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 Logical Effort David Harris Page 43 of 56

44 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 o Logic gates can be skewed to favor one input or edge at the cost of another o Logical effort can be applied to domino, pseudo-nmos, and other logic families Logical effort provides a language for engineers to discuss why circuits are fast. o Like any language, requires practice to master A book on Logical Effort is available from Morgan Kaufmann Publishers o o Discusses P/N ratios, gate characterization, pass gate logic, forks, wires, etc. Logical Effort David Harris Page 44 of 56

45 Delay Plots Normalized delay: d input NAND inverter effort delay g 4/3 p 2 d (4/3)h + 2 g 1 p 1 d h + 1 parasitic delay Electrical effort: h C out / C in o d f + p gh + p o Delay increases with electrical effort o More complex gates have greater logical effort and parasitic delay Logical Effort David Harris Page 45 of 56

46 Example Estimate the frequency of an N-stage ring oscillator: Logical Effort: g 1 Electrical Effort: h C out C in Parasitic Delay: Stage Delay: Oscillator Frequency: p p inv 1 d gh + p 2 F Nd abs Nτ A 31 stage ring oscillator in a 0.18 µm process oscillates at about 670 MHz. Logical Effort David Harris Page 46 of 56

47 Example Estimate the delay of a fanout-of-4 (FO4) inverter: d Logical Effort: Electrical Effort: Parasitic Delay: Stage Delay: g 1 h C out C in p p inv 1 d gh + p 5 The FO4 inverter delay is a useful metric to characterize process performance. 1 FO4 delay 5τ This is about 60 ps in a 0.18 µm process. Logical Effort David Harris Page 47 of 56

48 Branching Effort No! Consider circuits that branch: G H GH h 1 h 2 F 1 90 / (15+15) / / , not 18! Introduce new kind of effort to account for branching within a network: o Branching Effort: b C on path + C off path C on path o Path Branching Effort: B Now we can compute the path effort: o Path Effort: F b i GBH Note: h BH H i in circuits that branch Logical Effort David Harris Page 48 of 56

49 Example Select gate sizes y and z to minimize delay from A to B Logical Effort: G ( 4 3) 3 A C y y z z z 9C 9C B 9C Electrical Effort: H C out C in Branching Effort: Path Effort: Best Stage Effort: Delay: B F GHB 128 ˆ f F D Work backward for sizes: 9C ( 4 3) z C 5 3z ( 4 3) y C 5 Logical Effort David Harris Page 49 of 56

50 Example: Number of Stages How many stages should Ben use? o Effort of decoders is dominated by electrical and branching portions 32 3 o Electrical Effort: H o Branching Effort: B 8 because each address input controls half the outputs If we neglect logical effort, o Path Effort: F GBH Remember that the best stage effort is about ρ 4 o Hence, the best number of stages is: o Let s try a 3-stage design N log Logical Effort David Harris Page 50 of 56

51 Example: Gate Sizes & Delay Lets try a 3-stage design using 16 4-input NAND gates with G a 0 a 0 a 1 a 1 a 2 a 2 a 3 a 3 10 unit input capacitance y z out 0 96 unit wordline capacitance y z out 15 o Actual path effort is: o Therefore, stage effort should be: f ( 154) 1 3 z y D 3f + P o o F Close to 4, so f is reasonable Logical Effort David Harris Page 51 of 56

52 Asymmetric Gates Asymmetric logic gates favor one input over another. Example: Suppose input A of a NAND gate is most critical: o Select sizes so pullup and pulldown still match unit inverter o Place critical input closest to output a 2 2 4/3 4 x b o Logical Effort on input A: o Logical Effort on input B: o Total Logical Effort: g A 10 9 g B 2 g tot g A + g B 28 9 Effort on A goes down at expense of effort on B and total gate effort Logical Effort David Harris Page 52 of 56

53 Skewed Gates Skewed gates favor one edge over the other. Example: suppose rising output of inverter is most important. o Downsize noncritical NMOS transistor to reduce total input capacitance a 2 1/2 x a 2 1 x a 1 1/2 x Skewed inverter Unskewed w/ Unskewed w/ equal rise equal fall Compare with unskewed inverter of the same rise/fall time o Logical Effort for rising (up) output: o Logical Effort for falling (down) output: o Average Logical Effort: g u 5 6 g d 5 3 Critical rising effort goes down at expense of noncritical and average effort g avg ( g u + g d ) Logical Effort David Harris Page 53 of 56

54 Pseudo-NMOS Pseudo-NMOS gates replace fat PMOS pullups on inputs with a resistive pullup. o Resistive pullup must be much weaker than pulldown stack (e.g. 4x) o Reduces logical effort because inputs must only drive the NMOS transistors o However, NMOS current reduced by contention with pullup o Unequal rising and falling efforts o Logical effort can be applied to domino, pseudo-nmos, and other logic families Example: Pseudo-NMOS inverter o Logical Effort for falling (down) output: o Logical Effort for rising (up) output: o Average Logical Effort: g d 4 9 g u 4 3 a 2/3 4/3 g avg ( g u + g d ) x Logical Effort David Harris Page 54 of 56

55 Dynamic Logic Dynamic logic replace fat PMOS pullups on inputs with a clocked precharge. o Reduces logical effort because inputs must only drive the NMOS transistors o Eliminates pseudo-nmos contention current and power dissipation o Critical pulldown ( evaluation ) delay independent of precharge size Example: Footless dynamic inverter o Logical Effort for falling (down) output: g d 1 3 φ a 1 1 x Robust gates may require keepers and clocked pulldown transistors ( feet ). o Feet prevent contention during precharge but increase logical effort o Weak keepers prevent floating output at cost of slight contention during eval Logical Effort David Harris Page 55 of 56

56 Domino Gates Dynamic gates require monotonically rising inputs. o However, they generate monotonically falling outputs o Alternate dynamic gates with HI-skew inverting static gates o Dynamic / static pair is called a domino gate Example: Domino Buffer o Constraints: maximum input capacitance 3, load 54 o Logical Effort: G (1/3) * (5/6) 5/18 o Branching Effort: B 1 o Electrical Effort: H 54/3 18 φ 3 o Path Effort: F (5/18) * 1 * 18 5 o Stage Effort: f o HI-Skew Inverter: size 54 * (5/6) / o Transistor Sizes: n 4 p 16 a 3 p n g 1 1/3 g 2 5/6 HI-Skew 54 Logical Effort David Harris Page 56 of 56

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