Transcript Document 7357440

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
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Introduction
Problem formulation
Skew reduction effect of transmission
line shunts
Optimal sizing of multilevel network
Experimental results
Motivation
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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. 1553-1560,
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
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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
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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
• 0.5um wide 1.2
cm long copper wire
• Input skew 20ps
Rs
2
u(t-T)
C
f(GHz)
0.5
skew(ps) 3.9
1
1.5
2
3
3.5
4
5
4.2
5.8
7.5
9.9
13
17
26
Wavelength Long Transmission Line
Synchronizes Two Sources
U s1 
Rs
Us2 
Rs
sin( t )
sin( t  )
1

2
Differential Sine Waves
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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
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Process variations
Variations on wire width and transistor length
 Linear variation model
d = d0 + kx x+ky y
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Supply voltage fluctuations
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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
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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
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Formulation B:
Constraint: skew tolerance
Object function: minimize total routing area
Skew Reduction Effect of
Transmission Line Shunts
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Two sources case
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Circuit model and skew expression
Derivation of skew function
Spice validation
Multiple sources case
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Random skew model
Skew expression
Spice validation
Transmission line Shunt with Two
Sources
U s1 
Rs
Us2 
Rs
sin( t )
sin( t  )
1

2
• 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, Φ]
 
1 e
1 e

3R
L
3R

L

Configuration of Wires
• Coplanar copper transmission
line
w
w
240nm
CLK+
2um
3.5 um
Ground
CLK-
• 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
is a convex function on
where, k is a positive constant.
Lemma:
k
w  [ ,  ),
2
• 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 1
i
i


ki
wi
ki
wi
)  2 )
1  c2e
1  c2e
A
wi  mi , i  (1,2,..., n)


k2
w2
k2
w2
  3 )...   n )
1  cn e
1  cn e


kn
wn
kn
wn
Optimal Sizing of Spiral Wires
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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
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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
79
81
PM(mw) 0.4 0.5 0.7 0.9
reduce(%) 48 67 67
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
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Transmission line shunts demonstrate
its unique potential of achieving low
skew low jitter global clock distribution
under parameter variations
Future Directions
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Exploring innovative topologies of
transmission line shunts
Design clock repeaters and generators
Actual layout and fabrication of test chip
Derivation of Skew Function
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Assumptions
i) G=0;
ii) R 2 /  2 L2  1 ;
iii)    / 4
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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
Derivation of Skew Function
V2 , 2
2
V1, 2
V2
V1
Vi,j : Voltage of node 1
caused by source Vsj
independently
Φ : Initial phase shift
(skew)

  2  1 :

1
V2 ,1
V1,1
 V1  V1,1  V1, 2

V2  V2,1  V2, 2
Resulted skew
•
Loss causes skew
•
Lossless line: V1,2 =V2,2 ,
V2,1 =V1,1 Zero skew
Derivation of Skew Function
• Summing up all the incoming and reflected
waveforms to get Vi,j
• Using first order Taylor expansion
sin x  x
and e  1  x
x
to simplify the derivation
• Utilizing the geometrical relation in the previous
figure, we get
 
1 e
1

R
L
R

e L
