Transmission Line Network For Multi-GHz Clock Distribution Hongyu Chen and Chung-Kuan Cheng

1. Transmission Line Network For Multi-GHz Clock Distribution Hongyu Chen and Chung-Kuan Cheng Department of Computer Science and Engineering, University of California, San Diego January 2005

2. Outline • Introduction • Problem formulation • Skew reduction effect of transmission line shunts • Optimal sizing of multilevel network • Experimental results

3. Motivation • Clock skew caused by parameter variations consumes increasingly portion of clock period in high speed circuits • RC shunt effect diminishes in multiple-GHz range • Transmission line can lock the periodical signals • Difficult to analysis and synthesis network with explicit non-linear feedback path

4. Related Work (I) • Transmission line shunts with less than quarter wavelength long can lock the RC oscillators both in phase and magnitude I. Galton, D. A. Towne, J. J. Rosenberg, and H. T. Jensen, “Clock Distribution Using Coupled Oscillators,” in Prof. of ISCAS 1996, vol. 3, pp.217-220

5. Related work (II) • Active feedback path using distributed PLLs • Provable stability under certain conditions V. Gutnik and A. P. Chandraksan, “Active GHz Clock Network Using Distributed PLLs,” in IEEE Journal of Solid-State Circuits, pp. 1553-1560, vol. 35, No. 11, Nov. 2000

6. Related work (III) • Combined clock generation and distribution using standing wave oscillator • Placing lamped transconductors along the wires to compensate wire loss F. O’Mahony, C. P. Yue, M. A. Horowitz, and S. S. Wong, “Design of a 10GHz Clock Distribution Network Using Coupled Standing-Wave Oscillators,” in Proc. of DAC, pp. 682-687, June 2003

7. Related work (IV) • Clock signals generated by traveling waves • The inverter pairs compensate the resistive loss and ensure square waveform J. Wood, et al., “Rotary Traveling-Wave Oscillator Arrays: A New Clock Technology” in IEEE JSSC, pp. 1654-1665, Nov. 2001

8. Our contributions • Theoretical study of the transmission line shunt behavior, derive analytical skew equation • Propose multi-level spiral network for multi-GHz clock distribution • Convex programming technique to optimize proposed multi-level network. The optimized network achieves below 4ps skew for 10GHz rate

9. Problem Formulation • Inductance diminishes shunt effect • Transmission line shunts with proper tailored length can reduce skew • Differential sine waves • Variation model • Hybrid h-tree and shunt network • Problem statement

10. Inductance Diminishes Shunt Effects • 0.5um wide 1.2 cm long copper wire • Input skew 20ps

11. Wavelength Long Transmission Line Synchronizes Two Sources

12. Differential Sine Waves • Sine wave form simplifies the analysis of resonance phenomena of the transmission line • Differential signals improve the predictability of inductance value • Can convert the sine wave to square wave at each local region

13. Model of parameter variations • Process variations • Variations on wire width and transistor length • Linear variation model d = d0 + kx x+ky y • Supply voltage fluctuations • Random variation (10%) • Easy to change to other more sophisticated variation models in our design framework

14. Multilevel Transmission Line Spiral Network

15. Problem Statement • Formulation A: Given: model of parameter variations Input: H-tree and n-level spiral network Constraint: total routing area Object function: minimize skew Output: optimal wire width of each level spiral • Formulation B: Constraint: skew tolerance Object function: minimize total routing area

16. Skew Reduction Effect of Transmission Line Shunts • Two sources case • Circuit model and skew expression • Derivation of skew function • Spice validation • Multiple sources case • Random skew model • Skew expression • Spice validation

17. Transmission line Shunt with Two Sources • Transmission Line with exact multiple wave length long • Large driving resistance to increase reflection

18. Spice Validation of Skew Equation

19. Multiple Sources Case • Random model: • Infinity long wire • Input phases uniformly distribution on [0, Φ]

20. Configuration of Wires • Coplanar copper transmission line • height: 240nm, separation: 2um, distance to ground: 3.5um, width(w): 0.5 ~ 40um • Use Fasthenry to extract R,L • Linear R/L~w Relation • R/L = a/w+b

21. Min: S.t.: Optimal Sizing of Spiral Wires Lemma: is a convex function on , where, k is a positive constant. • Impose the minimal wire width constraint for each level spiral, such that the cost function is convex

22. Optimal Sizing of Spiral Wires • Theorem: The local optimum of the previous mathematical programming is the global optimum. • Many numerical methods (e.g. gradient descent) can solve the problem • We use the OPT-toolkit of MATLAB to solve the problem

23. Experimental Results • Set chip size to 2cm x 2cm • Clock frequency 10.336GHz • Synthesize H-tree using P-tree algorithm • Set the initial skew at each level using SPICE simulation results under our variation model • Use FastHenry and FastCap to extract R,L,C value • Use W-elements in HSpice to simulate the transmissionlin

24. Optimized Wire Width

25. Simulated Output Voltages Transient response of 16 nodes on transmission line Signals synchronized in 10 clock cycles

26. Simulated Output voltages Steady state response: skew reduced from 8.4ps to 1.2ps

27. Power Consumption PM: power consumption of multilevel mesh PS: power consumption of single level mesh

28. Area Skew-S Skew-M Ave. Worst Ave. Worst Impr(%) 0 28.4 36.5 28.4 36.5 0% 3 9.75 12.33 8.75 9.07 11% 5 7.32 9.06 6.55 6.91 12% 10 6.31 805 4.41 5.41 30% 15 5.03 7.33 2.81 4.93 44% 25 3.83 4.61 1.72 3.06 55% Skew with supply fluctuation

29. Conclusion and Future Directions • Transmission line shunts demonstrate its unique potential of achieving low skew low jitter global clock distribution under parameter variations • Future Directions • Exploring innovative topologies of transmission line shunts • Design clock repeaters and generators • Actual layout and fabrication of test chip

30. Derivation of Skew Function • Assumptions i) G=0; ii) ; iii) • Interpretation of assumptions i) ignores leakage loss ii) assumes impedance of wire is inductance dominant (true for wide wire at GHz) iii) initial skew is small

31. Derivation of Skew Function • Vi,j : Voltage of node 1 caused by source Vsj independently • Φ : Initial phase shift (skew) • : • Resulted skew • Loss causes skew • Lossless line: V1,2 =V2,2 , V2,1 =V1,1 Zero skew

32. Derivation of Skew Function • Summing up all the incoming and reflected waveforms to get Vi,j • Using first order Taylor expansion • and • to simplify the derivation • Utilizing the geometrical relation in the previous figure, we get