Transmission Line Network For Multi-GHz Clock Distribution

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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
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Outline

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

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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

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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
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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
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Related work (III)

• Combined clock generation and distribution using
  standing wave oscillator
• Placing lamped transconductors along the wires
  to compensate wire loss

F. OMahony, 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
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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
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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

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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

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Inductance Diminishes Shunt Effects

• 0.5um wide 1.2 cm long copper wire
• Input skew 20ps

f(GHz) 0.5 1 1.5 2 3 3.5 4 5
skew(ps) 3.9 4.2 5.8 7.5 9.9 13 17 26
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Wavelength Long Transmission Line Synchronizes
Two Sources
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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

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Model of parameter variations

• Process variations
• Variations on wire width and transistor length
• Linear variation model
• d d0 kx xky y
• Supply voltage fluctuations
• Random variation (?10)
• Easy to change to other more sophisticated
  variation models in our design framework

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Multilevel Transmission Line Spiral Network
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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

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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

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Transmission line Shunt with Two Sources

• Transmission Line with exact multiple wave
  length long
• Large driving resistance to increase reflection

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Spice Validation of Skew Equation
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Multiple Sources Case

• Random model
• Infinity long wire
• Input phases uniformly distribution on 0, F

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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/Lw Relation
• R/L a/wb

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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

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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

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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

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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
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Simulated Output Voltages
Transient response of 16 nodes on transmission
line Signals synchronized in 10 clock cycles
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Simulated Output voltages
Steady state response skew reduced from 8.4ps to
1.2ps
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Power Consumption
Area 3 4 5 7 10 15 20 25
PM(mw) 0.4 0.5 0.7 0.9 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 66 67 70 79 81
PM power consumption of multilevel mesh PS
power consumption of single level mesh
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Skew with supply fluctuation
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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

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Derivation of Skew Function

• Assumptions
• i) G0
• 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

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Derivation of Skew Function

• Vi,j Voltage of node 1 caused by source Vsj
  independently
• F Initial phase shift (skew)
• Resulted skew
• Loss causes skew
• Lossless line V1,2 V2,2 , V2,1 V1,1 Zero skew

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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