Research And Improvement Of Model Order Reduction Method In The Field Of Integrated Circuit Analysis

With the feature dimension of IC descreasing and the clock frequency of IC rising, the interconnect circuit is also much more complex, which results in very large dimension of equivalent RCL or RCS equations. Thus, we have the problem of Model Order Reduction. Usually use the reduction method based on matrix projection to solve this problem. There are generally two categories:balanced truncation method based on controllable and observable Gram matrix, and moment matching method based on Krylov subspace. The former is more suitable for analyzing small dimension dymanical systems which need to strictly control errors. However, the latter has the advantages of numerical stability and computational complexity. The computational complexity of latter method is O(nr2) or O(n2r), which is much more suitable for large order model to be reduced. Moreover, with the development of Krylov subspace method applied to IC, many special and signifigant physical features could be maintained during this reduction method.Therefore, the discussion and research of this paper is basically about the moment matching method based on Krylov subspace.Beyond the basic reduction method, this paper further introduces Stable Asymmetric Lanczos Algorithm, C-Orthogonal Arnoldi Algorithm and Global Arnoldi Algorithm. For second-order systems, this paper introduces SOAR and Q-Arnoldi, and discusses contraction and interrupt, Generalized Krylov subspace, and implicitly restarted technique. On the basis of the previous discussions, this paper also discusses the MIMO second-order system, high-order linear system and multi-point moment matching. This paper introduces the SVD-Krylov subspace method which is combined with controllable and observable Gram matrix, and then presents alternate Krylov subspace to ensure the stability of SVD-Krylov subspace method. This paper also describes that the geometric background of reduction method based on matrix projection is Grassmann manifold, and then studies the relation between model order reduction and tangent interpolation from the view of rational interpolation.This paper illustrates in detail the establishment of differential equation system and its specific model order reduction method during the IC analyzation. This paper describes Maxwell equations and related simplified form, PEEC model, and the RLC and RCS equations of interconnect. For the differential equation system of RLC and RCS model, this paper gradually elaborates the general and specific requirements during reduction, such as moment matching, stability, passivity, preservation of second-order structure, preservation of input and output structure, elimination of inductance loop, large amount of ports, and process parameters.This paper then inllustrates the approaches and thoughts of AWE, PVL, PRIMA, SPRIM and ENOR, SMOR, SAPOR, when these methods solve the above requirements.This paper presents the applications of Second-Order Biorthogonal method (SOB) for the reduction of second-order circuit equation with unequal input and output correlation matrices. To solve the calculation problems, this paper then presents Generalized Second-Order Biorthogonal method (GSOB). For MIMO second-order circuit model, this paper presents using the Global algorithms based on Frobenius Norm to replace Block algorithms. Then, this paper analyzes accessible benchmark circuit examples, and chooses and modifies the appropriate models. At the end, this paper conpares the computational accuracy and time of our original Global second-order algorithms, including GSPRIM, GSAPOR and GGSOB, with the widely-used SPRIM and BSAPOR by numerical experiments.