# 7. Combinational Circuits

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1 7. Combinational Circuits Jacob Abraham Department of Electrical and Computer Engineering The University of Texas at Austin VLSI Design Fall 2017 September 25, 2017 ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

2 Review: Decoder Example, Number of Stages 16 word, (32 bit) register file Each bit presents load of 3 unit-sized transistors True and complementary address inputs A[3:0] Each input may drive 10 unit-sized transistors Find: number of stages, sizes of gates, speed Decoder effort is mainly electrical and branching Electrical Effort: H = (32*3)/10 = 9.6 Branching Effort: B = 8 If we neglect logical effort (assume G = 1) Path Effort: F = GBH = 76.8 Number of Stages: N = log 4 F = 3.1 Try a 3-stage design ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

3 Decoder Review: Gate Sizes and Delay Logical Effort: G = 1 * 6/3 * 1 = 2 Path Effort: F = GBH = 154 Stage Effort: ˆf = F 1 3 = 5.36 Path Delay: D = 3 ˆf = 22.1 Gate sizes: z = 96*1/5.36 = 18 Gate sizes: y = 18*2/5.36 = 6.7 ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

4 Decoder Review: Comparison Compare many alternatives with a spreadsheet Design N G P D NAND4-INV NAND2-NOR2 2 20/ INV-NAND4-INV NAND4-INV-INV-INV NAND2-NOR2-INV-INV 4 20/ NAND2-INV-NAND2-INV 4 16/ INV-NAND2-INV-NAND2-INV 5 16/ NAND2-INV-NAND2-INV-INV-INV 6 16/ ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

5 Review: Logical Effort Example Find the logical efforts for the inputs, a, b, and c in the circuit below. Output a b c Rising Falling Suggest a way to reduce the parasitic delay of this circuit by modifying the structure (but keeping the same function). ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

6 Example, Cont d Find the logical efforts for the inputs, a, b, and c in the circuit below. Output Rising Falling a 55/18 22/15 b 55/18 22/15 c 35/18 7/9 To reduce the parasitic delay of this circuit, swap the parallel combination of pmos transistors with inputs a, b with the pmos transistor with input c ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

7 Example: Sizing Paths Size the path G1-G2-G3-G4 in the circuit below using logical effort Find the minimum delay and give the sizes of the P and N transistors to achieve this delay Assume that the off-path capacitance is the same as the on-path capacitance for each branch Input capacitance of Inverter G1 = 3 units. Load capacitance driven by Gate G4 = 52 units. ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

8 Sizing Paths, Cont d Size the path G1-G2-G3-G4 in the circuit Input capacitance of Inverter G1 = 3 units. Load capacitance driven by Gate G4 = 52 units. Delay = 4.8 FO4 units Sizes of transistors: Gate P N Gate G4 8 8 Gate G3 5 5 Gate G2 3 3 Gate G1 2 1 Delay of the path from A through gates G5 and G4 (assuming the input from G2=1 and D=0): 2.08 FO4 units ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

9 Example of Bubble Pushing Implement the circuit described by the code below module mux(input s, d0, d1, output y); assign y = s? d1 : d0; endmodule The specifications are easily met with a design using AND, OR and NOT gates ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

10 Convert to Design using NAND/NOR/NOT Gates Bubble Pushing Start with network of AND/OR gates Convert to NAND/NOR + inverters Push bubbles around to simplify logic Use DeMorgan s Law ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

11 Example, Continued Now, design the circuit with one compound gate and one inverter. Assume that S is available CE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

12 Compound Gates ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

13 Another Example A multiplexer has a maximum input capacitance of 16 units on each input. It must drive a load of 160 units Estimate the delay of the NAND and compound gate designs NAND Solution Compound Solution H = 160/16 = 10 B = 1 N = 2 P = = 4 G = (4/3) (4/3) = 16/9 F = GBH = 160/9 ˆf = N F = 4.2 D = N ˆf + P = 12.4τ H = 160/16 = 10 B = 1 N = 2 P = = 5 G = (6/3) (1) = 2 F = GBH = 20 ˆf = N F = 4.5 D = N ˆf + P = 14τ CE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

14 Example, Cont d Annotate the designs for the multiplexer with transistor sizes which achieve the minimum delay ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

15 Order of Inputs to a Transistor Stack Delay of CMOS gate is affected by input order Parasitic delay model used in logical effort calculations is too simple Example, calculate parasitic delay for Y falling If A arrives latest: 2τ If B arrives latest: 2.33τ Choosing inner and outer inputs Outer input is closest to rail (power or ground): B Inner input is closest to output: A If input arrival time is known Connect latest arriving input to inner terminal ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

16 Asymmetric Gates Asymmetric gates favor one input over another Example, suppose input A of a NAND gate is most critical Use smaller transistor on A (less capacitance) Boost size of noncritical input So total resistance is same Calculate logical effort g A = 10/9 g B = 2 g total = g A + g B = 28/9 Symmetric gate approaches g = 1 on critical input However, total logical effort goes up CE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

17 Symmetric Gates Inputs can be made perfectly symmetric Make both A and B behave like inner/outer inputs, and keep the P:N ratio 2:1 ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

18 Skewed Gates Favor One Edge Over Another Example, suppose rising output of inverter is most critical Downsize noncritical nmos transistor Calculate logical effort by comparing to unskewed inverter with same effective resistance on that edge g u = 2.5/3 = 5/6; g d = 2.5/1.5 = 5/3 Definition: 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 Skewed gates reduce size of noncritical transistors HI-skew gates favor rising output (small nmos) LO-skew gates favor falling output (small pmos) Logical effort is smaller for favored direction, but larger for the other direction ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

19 Catalog of Skewed Gates ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

20 Asymmetric Skew Combine asymmetric and skewed gates Downsize noncritical transistor on unimportant input Reduces parasitic delay for critical input ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

21 Example I Find the (worst case) logical efforts of the different inputs in the CMOS circuit below. ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

22 Example - II (a) Calculate the delay of the segment from B to S of the adder shown below (the AOI21 is a single stage, shown on the right), given that the output capacitance is 25 units (normalized), and the input capacitance at B is 6 units. ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

23 Example - II, Cont d (b) Calculate the capacitances of the nodes y, x and w, and the resulting widths of the transistors. ECE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

24 Best P/N Ratio We have selected P/N ratio for unit rise and fall resistance (µ = 2-3 for an inverter). Alternative: choose ratio for least average delay Example: inverter Delay driving identical inverter t pdf = (P + 1) t pdr = (P + 1)(µ/P ) t pd = (P + 1)(1 + µ/p )/2 = (P µ + µ/p )/2 Differentiating t pd w.r.t. P, we get, least delay for P = µ In general, best P/N ratio is sqrt of that giving equal delay Only improves average delay slightly for inverters But significantly decreases area and power CE Department, University of Texas at Austin Lecture 7. Combinational Circuits Jacob Abraham, September 25, / 23

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