Structure Preserving Reduction of Frequencydependent Interconnect

1
Structure Preserving Reduction of
Frequency-dependent Interconnect

• Quming Zhou, Kartik Mohanram and Athanasios C.
  Antoulas
• Dept. of Electrical and Computer Engineering Rice
  University , Houston

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Purpose

• Frequency-dependent interconnect
• Resistance R(f)
• Rational Arnoldi model-order reduction (MOR)
• Passivity-preserving
• Implicit multi-point matching
• Two types of moments w.r.t s and sqrt(f)
• Preserve structure of original system

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Frequency-dependent Resistance
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State Space Model

• MOR preserves structure H(s)

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New Matrix K

• AAT negative semi-definite
• E symmetric positive-definite
• K diagonal negative semi-definite

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Congruence Transform

• V is n x nr projection matrix
• Reduced transfer function is passive given that
  the original system is passive

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Moments

• Interpolate the frequency response and its
  moments at multiple points (sjw)
• Two types of moments
• Coefficient of s in G(s)
• Coefficient of sqrt(f) in G(s)

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Krylov Subspace
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Jointed Krylov Space

• To obtain real matrix A, B, E and K in reduced
  model, interpolate at complex conjugate
  frequencies s and s
• Full-rank projection matrix V

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Multiple-point Rational Arnoldi
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Simulation

• R(f) couples time domain andfrequency domain
• FFT and IFFT for simulation
• Harmonic frequencies of inputs
• Response at each harmonic frequency
• IFFT to get time-domain response
• 0.1 GHz and 1GHz

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Example I
Skin effect 24 delay
n100?12
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Example II
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Conclusions

• Rational Arnoldi MOR based on multiple
  interpolation to handle R(f)
• Preserve H(s) sE A Ksqrt(f)
• Significant reduction in system complexity
• High accuracy
• Computational efficiency