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 Outline Introduction Problem formulation Skew reduction effect of transmission line shunts Optimal sizing of multilevel network Experimental results Motivation Clock skew caused by parameter variations consumes increasingly portion of clock period in high speed circuits RC shunt effect diminishes in multipleGHz range Transmission line can lock the periodical signals Difficult to analysis and synthesis network with explicit non linear feedback path Related Work I Reference Clock Phase Detector G s a s Pullable VCO C0 Pullable VCO C1 Pullable VCO Cn 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 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 15531560 vol 35 No 11 Nov 2000 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 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 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 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 Inductance Diminishes Shunt Effects Vs1 Rs 1 u t C L R Vs2 Rs 2 u t T f GHz C 0 5 1 1 5 2 0 5um wide 1 2 cm long copper wire Input skew 20ps 3 skew ps 3 9 4 2 5 8 7 5 9 9 3 5 4 5 13 17 26 Wavelength Long Transmission Line Synchronizes Two Sources 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 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 Multilevel Transmission Line Spiral Network clock drivers a H tree b Spirals driven by H tree 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 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 Transmission line Shunt with Two Sources Transmission Line with exact multiple wave length long Large driving resistance to increase reflection 1 e 1 R L R e L Spice Validation of Skew Equation Multiple Sources Case Random model Infinity long wire Input phases uniformly distribution on 0 1 e 1 e 3 R L 3 R L 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 Optimal Sizing of Spiral Wires 1 ce k w f w 1 ce k w Lemma is a convex where k is a positive constant k w function 2on Impose the minimal wire width constraint for each level spiral such that the cost function is convex Min 1 1 c1e 1 c1e S t n l w i i ki wi ki wi 2 1 c2e 1 c2e A i 1 wi mi i 1 2 n k2 w2 k2 w2 3 n 1 cne 1 cn e kn wn kn wn 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 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 Optimized Wire Width Total Area W1 um W2 um W3 um Skew M ps Skew S ps Impr 0 0 0 0 23 15 23 15 0 0 5 1 7 0 0 17 796 20 50 13 1 1 9308 1 0501 0 12 838 14 764 13 3 2 5751 1 3104 1 3294 8 6087 8 7309 15 5 2 9043 3 7559 2 3295 6 2015 6 3169 16 10 3 1919 4 5029 6 8651 4 2755 5 2131 18 15 3 6722 6 1303 10 891 2 4917 3 5182 29 20 4 0704 7 5001 15 072 1 7070 2 6501 37 25 4 4040 8 6979 19 359 1 2804 2 1243 40 Simulated Output Voltages Transient response of 16 nodes on transmission line Signals synchronized in 10 clock cycles Simulated Output voltages Steady state response skew reduced from 8 4ps to 1 2ps Power Consumption Area 3 4 5 7 10 15 20 25 1 0 1 4 1 5 1 6 PS mw 0 83 1 5 2 1 2 64 3 04 4 7 7 2 8 3 reduce 48 67 67 79 81 PM mw 0 4 0 5 0 7 0 9 66 67 PM power consumption of multilevel mesh PS power consumption of single level mesh 70 Skew with supply fluctuation Area 0 3 5 10 15 25 Skew S Skew M Ave Worst Ave Worst Impr 28 4 36 5 28 4 9 75 12 33 8 75 7 32 9 06 6 55 6 31 805 4 41 5 03 7 33 2 81 3 83 4 61 1 72 36 5 9 07 6 91 5 41 4 93 3 06 0 11 12 30 44 55 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 …
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