1. EE 4271 VLSI Design, Fall 2013 Static Timing Analysis and Gate Sizing Optimization

2. Delay Evaluation 1. Gate delay 2. Interconnect delay 2015/12/18Circuit Delay PJF- 2

3. Circuit Model For an inverter 2015/12/18Circuit Delay PJF- 3 Csink … …

4. Gate Resistance Pull-up and pull-down resistors are not a constant. Which value should we choose? Gate delay also depends on its input signal 2015/12/18Circuit Delay PJF- 4

5. K-Factor Gate Delay More accurate than RC Consider input transition time tr (transition rising time) or tf (transition falling time) Transition time is signal rising/falling time from 10% to 90% K-factor equation – Delay td=k(tr/f, Ctotal) – Output transition time t ’ r/f=k ’ (tr/f, Ctotal) Synopsis K-factor form: – Delay= a*tr+b*Ctotal+c*tr*Ctotal+d – Obtained from SPICE simulation Widely used 2015/12/18Circuit Delay PJF- 5 rising time

6. Circuit Delay Evaluation - Two Components Cell delay + interconnect delay – Cell delay is computed using RC – Interconnect delay is computed using Elmore delay 2015/12/18Circuit Delay PJF- 6 Cell Interconnect

7. Wire and Gate Models 7

8. Step by Step 2015/12/18Circuit Delay PJF- 8 Model combinational circuit using the previous slide Model combinational circuit using the previous slide Starting from primary input gates, compute the arrival time (AT) at each gate, i.e., compute gate delay and interconnect delay Starting from primary input gates, compute the arrival time (AT) at each gate, i.e., compute gate delay and interconnect delay In order to compute the AT at a gate, the ATs of all its input gates need to be computed In order to compute the AT at a gate, the ATs of all its input gates need to be computed Repeat the above process until the ATs at all primary output gates are computed Repeat the above process until the ATs at all primary output gates are computed

9. Example of Static Timing Analysis Arrival time (AT): input -> output, take max 2015/12/18Circuit Delay PJF- 9 1 3 2 4 5

10. Timing Optimization Arrival time (AT): input -> output, take max 2015/12/18Circuit Delay PJF- 10 Should we size up this gate to improve timing?

11. Timing Optimization- II Suppose that we have a gate (with same gate type) doubling its width. We roughly have C=10, R=1. If we change the gate with this new one, what is the new delay? Does not change 2015/12/18Circuit Delay PJF- 11

12. Timing Optimization- III Suppose that we have a gate (with same gate type) doubling its width. We roughly have C=8, R=1. If we change the gate with this new one, what is the new delay? 2015/12/18Circuit Delay PJF- 12

13. Gate Sizing This optimization is called gate sizing. Change the gate size (width) in optimization. 1. Given multiple choices (implementations) per gate type, find a gate implementation at each gate such that the circuit timing is minimized. 2. Given multiple choices per gate type, find a gate implementation at each gate such that the circuit timing satisfies the target and the total gate area/power is minimized – This formulation is widely used. 2015/12/18Circuit Delay PJF- 13

14. Delay due to Gate Sizing Suppose that unit width gate capacitance is c and unit width gate resistance is r. Given gate size w i, – Gate size w i : R  r/w i, C  cw i Delay is a function of RC – Delay   R i C j   w i /w j 14

15. Combinatorial Circuit Model Gate size variables x 1, x 2, x 3 Delay on each gate depends on x 15 DriversLoads x2x2 x3x3 x1x1 a3a3 a4a4 a5a5 a1a1 a2a2 D1D1 D3D3 D2D2 D5D5 D4D4 D7D7 D6D6 D9D9 D8D8 D 10 a6a6 a7a7

16. Path Delay Express path delay in terms of component delay A component can be a gate or a wire Delay D for each component Arrival time a for some components 16

17. Gate Sizing Power/area minimization under delay constraints: This can be solved efficiently using gpsolve 17

18. Gate Sizing using GPSOLVE Follow the steps in gatesizing.m for the example in the slides of timing analysis and optimization 18