MultiLevel Approach for Integrated Spiral Inductor Optimization

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Multi-Level Approach for Integrated Spiral
Inductor Optimization

• Arthur Nieuwoudt and Yehia Massoud
• Rice Automated Nanoscale Design Group (RAND)
• Rice University
• www.rand.rice.edu

RICE AUTOMATED NANOSCALE DESIGN GROUP
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Spiral Inductors A Key to System-on-Chip
Integration

• Analog synthesis crucial for fully automated
  system-on-chip (SoC) design
• Integrated spiral inductors key SoC component
• Utilized in key Analog/RF circuits including
  LNAs, VCOs, and on-chip filters
• Quality factor limits circuit performance
• Integrated inductor modeling is complex
• Design space complexity limits the effectiveness
  of ad hoc design techniques

• Efficient inductor optimization crucial
• Maximize inductor performance (Q, L, area)
• Efficiently enable fully automated SoC design

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Previous Optimization Techniques

• Exhaustive Enumeration Niknejad and Meyer, 1998
• Enumerate over a range of possible parameters
• Computationally inefficient

• Geometric Programming Hershenson et al., 1999
• Convex optimization technique that exploits
  models consisting of geometric functions
• Requires specific model formulation based on
  monomials or posynomials

• Sequential Quadratic Programming (SQP) Zhan and
  Sapatnekar 2004
• General nonlinear constrained convex optimization

Need Generalized Inductor Optimization
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Inductor Optimization Problem Formulation
Maximize
Subject to
Bound Constraints on (n,s,w,d)
Lw desired inductance ? tolerance

• Also optimizes over other constraints
• Self-resonant frequency
• Parasitic capacitance

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Design Space Characterization

• Consider inductor turns (n) as continuous
  variable
• Avoids expensive mixed-integer nonlinear
  optimization techniques
• Provides more flexibility in the design domain
• Produces designs with larger quality factors
• Must explore design space convexity to
  efficiently optimize the inductor

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Quality Factor
Initial Start Point W 12 D 325 Optimized
Quality Factor 0.2
Initial Start Point W 5 D 240 Optimized
Quality Factor 2
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Non-Convex Effective Inductance Constraint

• Effective inductance
• Positive in inductive region
• Zero at self-resonance frequency
• Negative in capacitive regime

• Can impact the location of feasible values

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Multi-Level Inductor Optimization

• The idea combine global optimization with convex
  local optimization to exploit design space
  properties
• Want global optimization algorithm that
  approaches the neighborhood of the optimal value
  quickly
• Statistical methods may not quickly approach the
  neighborhood of the optimal value
• Simulated annealing and genetic algorithms
  require significant computational effort
• May not approach optimal value initially due to
  statistical nature
• Need deterministic global optimization algorithms
• Enumeration
• Pattern search algorithms

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Mesh-Adaptive Direct Search

• Belongs to a class of pattern search algorithms
• Intelligently constructs and refines a set of
  possible design points (mesh)
• Algorithm executes 2 steps at each iteration
• Search step Searches mesh using deterministic
  sampling
• Poll step If search is unsuccessful, algorithm
  polls nearby mesh points
• Based on results in search and poll steps, the
  mesh is either coarsened or reduced
• Monotonically converges to optimal values

C. Audet and J.E. Dennis, 2004.
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Optimization Methodology

• Global Phase
• Can use Mesh-Adaptive Direct Search (MADS)
• Does not require convexity
• May use several start points for better
  performance

• Local Phase
• Utilizes SQP to exploit local convexity
• Rapidly converges to optimal value

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Comparison of Optimization Techniques

• Single-level approaches
• Enumeration over uniform grid
• Sequential Quadratic Programming
• 200 runs with random start points
• Calculated the average number of function
  evaluations to reach within 95 of optimal
  quality factor
• New multi-level approach
• Mesh-Adaptive Direct Search coupled with SQP
• Selected Design Problems

Examples
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Comparison with Existing Techniques
16 nH, 4 GHz, Low ?
10 nH, 2.4 GHz, High ?
16 nH, 4 GHz, Low ?
10 nH, 2.4 GHz, High ?
35x Speedup Over Enumeration
25x Speedup Over Convex
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Conclusions

• Multi-level approach for spiral inductor
  optimization efficiently optimizes inductor
  designs
• Does not require convex objective or constraint
  functions
• Appropriate for use with any general inductor
  model
• Number of turns treated as continuous variable
• Avoids mixed-integer optimization
• Generates higher quality factors
• Will enable the efficient automated design of
  integrated spiral inductors in SoC

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Comparison with Other New Multi-Level Methods
Example 1
Example 1
Example 3
Example 3
19x Maximum Speedup Over Enumeration Convex