Interconnect Efficient Time-domain Model Order Reduction Algorithms

Interconnect has become a key factor in determining the performance of the integrated circuits with the rapid increase of the signal frequency and the decrease of the feature size of VLSI. Since the equivalent circuits of interconnects have reached the scale of tens to hundreds of thousands, it is very time-consuming or even impossible for interconnects to be simulated by traditional simulators. Model order reduction (MOR) techniques are proposed to reduce the complexity of interconnect analysis remarkably, by which the input-output behavior of the interconnect circuits can be approximated accurately. Besides, some important physical properties of the original system such as passivity and stability should be preserved.MOR can be performed either in frequency domain or time domain. The frequency-domain MOR approaches have been fully developed, but we are more interested in the time domain response of the circuits for the frequency domain approximation error can be magnified greatly by the frequency-time conversion. So some time-domain MOR approaches have been proposed to improve the reduction accuracy. However, it is less efficient for the time-domain reduction approaches to be applied for very large scale circuits for its high computational complexity.An efficient time-domain trapezoidal difference based MOR method is proposed to further improve the accuracy and the efficiency of the existing MOR methods. A recursive relation which is derived from the trapezoidal difference formulation of the time-domain equation of interconnect circuits formulates a non-homogenous Krylov subspace. A non-homogenous Arnoldi method is employed to construct the orthonormal basis of this non-homogenous Krylov subspace. The orthononnal basis is then used to project the original system to reduced-order models. The accuracy of the reduced-order models can be guaranteed while the MOR procedure is numerically stable and the passivity is also preserved. Compared with the existing frequency-domain methods, the proposed method can achieve higher accuracy due to the elimination of the frequency-time conversion error. In comparison with the existing time-domain methods, the computational cost of the proposed method is reduced remarkably.Time-domain trapezoidal difference based MOR method is input-dependent. So an efficient time-domain step-by-step integral based MOR method is proposed, which is input-independent by taking impulse response into account. Assuming impulse input, we integrate the time-domain equation of the interconnect circuits step by step. The recursive relation of state variables formulates a Krylov subspace. The projection matrices of the Krylov subspace which are generated by the numerically stable Arnoldi processes are used to transform the large-scale original system to reduced-order models. The reduced model is accuracy, numerically stable, passive preserving and input-independent. Compared with the existing time-domain methods, the proposed method is much more efficient. In comparison with the existing frequency-domain methods, the accuracy of the proposed method is much higher. Comparing with the time-domain trapezoidal difference based MOR method, the computational cost of the proposed method is reduced further.