1. Effects of Inductance on the Propagation Delay and Repeater Insertion in VLSI Circuits Yehea I. Ismail and Eby G. Friedman, Fellow, IEEE

2. Agenda Introduction Propagation Delay Formula Comparison to an RC Model Dependence of Delay on wire Length Repeater Insertion for an RLC Interconnect Conclusions

3. Introduction

4. Propagation Delay Formula

5. Close Form

6. Interconnect Model-RCL

7. A gate driving an RLC transmission line

8. The transfer function- a lossy transmission line Z s is source impedance, Z L is load impedance γ is propagation constant, Z 0 is characteristic impedance

9. Laplace transform-1

10. Laplace transform-2

11. The parameter ζ Three parameter:ζ,R T,C T

12. Laplace transform-3 The scaled 50% propagation delay t` pd can be calculated by solving Thus, the propagation delay of an RCL line with a source Resistance R tr and a load capacitance C L has the form

13. AS/X [20] simulations the maximum error is 4.6% and the average error is 1.65%

14. AS/X [20] simulations Eq.(18):

15. Delay function is more accurate When C T And R T are high and ζ is low Eq(18) suffers high errors This case can only occur for unreasonably high values of the inductance per unit length of the line as compared to the resistance and capacitance per unit length Such a case does not exist in a practical VLSI circuit.

16. ζ characterize inductance effects

20. The Factor of Inductance

21. Comparison to an RC model

23. copper interconnect line

26. Dependence of Delay on Interconnect Length α asym is the asymptotic value at high frequencies of the attenuation per unit length of the signals L  0 t pd =0.37RCl 2 R  0 t pd = l(LC) 0.5

28. Repeater Insertion for an RLC Interconnect The buffer output impedance R tr is R 0 /h The input capacitance of the buffer C L is hC 0

30. Curve Fitting Method

31. highly accurate The error is less than 0.05% If L t  0 h opt (RLC) = h opt (RC) k opt (RLC) = k opt (RC)

32. Comparison with RC model

33. Total delay in RC & RLC When T L/R =3,t pdtotal increase by 10% When T L/R =5,t pdtotal increase by 20% When T L/R =10,t pdtotal increase by 30%

34. Repeater Insertion in RC & RLC

35. Area in RC & RLC A min is the area of a minimum size buffer = When T L/R =3,Area increase by 154% When T L/R =5,Area increase by 435%

36. conclusion Propagation delay --- 35% propagation delay has linear dependence on the length of the line.

37. Conclusion for repeater insertion Insert less buffer Reduce delay Reduce buffer area