Passive Synthesis of Compact FrequencyDependent Interconnect Model via Quadrature Spectral Rules

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Passive Synthesis of Compact Frequency-Dependent
Interconnect Model via Quadrature Spectral Rules

• A. Woo, A. C. Cangellaris
• Center for Computational Electromagnetics
• EM Laboratory
• ECE Department, University of Illinois
• T. Yioultsis
• ECE Department
• Aristotle University of Thessaloniki, Greece
• Research funded by SRC and IBM

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Outline

• Objective
• Overview of Interconnect Reduced-Order Modeling
  Techniques
• A New Methodology
• Implementation in SPICE
• Samples of Numerical Studies
• Summary

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Objectives

• Develop SPICE-compatible, reduced-order
  macromodels for interconnects such that
• They exhibit the simplicity and versatility of
  the popular concatenated ? or T lumped
  circuit-based models
• Allow for dispersive interconnect modeling (R(f),
  L(f), C(f), G(f))
• They are passive by construction
• They are optimal
• Their order is the minimum possible for the given
  accuracy and a given bandwidth of validity

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Shortcomings of concatenated lumped-circuit models

• Number of segments 10 ? (length/?min)
• Interconnect SPICE models become very large
• Model order reduction techniques used to contain
  complexity
• PRIMA-based reduction is typically used to reduce
  their order
• Maintaining passivity of the reduced-order model
  is a challenge
• Especially when dealing with dispersive
  (frequency-dependent) models
• Correlation of order of reduced model and desired
  accuracy is difficult

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Alternative Approach Passive rational function
approximations of the transmission-line transfer
functions

• Several techniques have been proposed
• They tend to depart from the lumped-circuit model
  type of representation
• Special schemes for their incorporation in SPICE

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Proposed Methodology

• A hybrid of the two aforementioned classes of
  techniques
• It combines
• Padé approximations of the transmission line
  transfer functions
• With equivalent circuit representation in terms
  of concatenated lumped circuits constructed from
  the per-unit-length R(f), L(f), C(f) and G(f)
  matrices
• It is passive by construction
• Its netlist is similar to that used for the
  popular concatenated ? lumped circuit
  representations
• Direct SPICE compatibility
• It is optimal for the given accuracy and
  frequency bandwidth of validity

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A Useful Result for Uniform MTL 2N-Ports
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Approximation of the frequency-dependent input
impedance of an MTL

• Consider the input impedance of a short-circuited
  transmission line of length L
• Input impedance available in analytic form
• Seek an optimal approximation of Z(?) over a
  desired bandwidth in terms of the input impedance
  of a discrete model of the short-circuited line

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The discrete input impedance may also be cast
in closed (pole-residue) form
Functions of the grid lengths
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All analytic poles in the bandwidth of interest
are kept in the fit
Analytic poles in the bandwidth of interest are
kept in the fit
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Error minimization through moment matching
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The synthesized grid is non-uniform
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The Synthesis Process

• Multi-conductor system of length d
• Calculated minimum modal velocity vmin
• From fmax and vmin obtain ?max
• Synthesize optimal grids over the bandwidth 0,
  ?max for
• A short-circuited line of length d/2
• An open-circuited line of length d/2

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Optimality of the Grids

• Equivalent circuit is realized using the minimum
  number of degrees of freedom
• For accurate modeling of electromagnetic
  retardation the sampling of the minimum useful
  wavelength needs to satisfy the Nyquist criterion
  
• ? number of elements per ?min ? 2
• Uniform mesh ? 10 number of segments per
  wavelength
• Proposed non-uniform mesh 2.2 segments per
  wavelength

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Exponential reduction of fitting error with
increasing of segments
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Synthesized Model is Passive by Construction

• All that is needed is the synthesis of passive
  representations of the lumped
• per-unit-length line-impedance (R(?)j?L(?)) and
• Per-unit-length line-admittance (G(?)j?C(?))
• multiports
• SPICE-Compatible Realization in terms of
  SPICE-native elements

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Summary of the Synthesis Process

• Given N-conductor MTL of length d and described
  in terms of p.u.l. R(f ), L(f ), C(f ), G(f )
  matrices
• Using L(f??), C(f??) and neglecting loss generate
  optimal grids for two MTLs
• The synthesized equivalents are directly
  compatible with SPICE-like simulators

Optimal SPICE Equivalents
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SPICE Implementation

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Numerical Study 1
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Numerical Study 1
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Numerical Study 2

• Physical Dimensions
• w 125 µm, t 25 µm, s 200 µm, h 68.8 µm
• Substrate dielectric Lossless with ?r 4.0
• Bandwidth of interest 5 GHz
• 5 V step pulse with rise time of 0.5 ns

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Numerical Study 2
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Numerical Study 2
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Summary

• Methodology for the synthesis of optimal,
  passive, SPICE-compatible MTL models for coupled
  interconnects
• Number of segments is the minimum possible
• 10-fold reduction over traditional uniform
  segmentation
• Synthesis based on Chebyshev-Padé fitting of
  input port impedances of short-circuited and
  open-circuited half-length sections of the MTL
• Passivity is ensured by construction
• Synthesized netlist is in terms of SPICE-native
  elements