Model order reduction of parameterized circuit equations based on interpolation
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Model order reduction of parameterized circuit equations based on interpolation Nguyen Thanh Son1 · Tatjana Stykel2
Received: 15 January 2014 / Accepted: 14 April 2015 © Springer Science+Business Media New York 2015
Abstract In this paper, the state-of-the-art interpolation-based model order reduction methods are applied to parameterized circuit equations. We analyze these methods in great details, through which the advantages and disadvantages of each method are illuminated. The presented model reduction methods are then tested on two circuit models. Keywords Parameterized circuit equations · Parametric model order reduction · Interpolation Mathematics Subject Classification (2010) 65D05 · 65F30 · 93C05 · 94C
1 Introduction Model order reduction is a crucial tool for the simulation of very large integrated circuits and interconnects. This issue attracted the attention of many researchers. It can be seen through numerous published works [21, 23, 25, 27, 32, 43, 44], just to name a few. Communicated by: Editors of Special Issue on MoRePas Nguyen Thanh Son
[email protected] Tatjana Stykel [email protected] 1
Department of Mathematics and Informatics, Thai Nguyen University of Sciences, Tan Thinh Ward, 23000 Thai Nguyen, Vietnam
2
Institute of Mathematics, University of Augsburg, Universit¨atsstr. 14, 86159 Augsburg, Germany
N.T. Son, T. Stykel
Using the modified nodal analysis (MNA), see, e.g., [51], linear RLC circuits can be modeled by a system of differential-algebraic equations (DAEs) E x(t) ˙ = Ax(t) + Bu(t), y(t) = B T x(t),
(1)
where E, A ∈ RN ×N , B ∈ RN ×m , x(t) ∈ RN is the state vector containing the node potentials and currents through inductors and voltage sources, u(t) ∈ Rm is the input vector consisting of the currents of current sources and the voltages of voltage sources, and y(t) ∈ Rm stands for the output vector containing the negative of the voltages of current sources and the currents of voltage sources. The system matrices in Eq. 1 have the form ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ −AR G ATR −AL −AV −AI 0 AC C ATC 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎦, E=⎣ 0 L 0 ⎦, A = ⎣ ATL 0 0 ⎦, B = ⎣ 0 0 −I 0 0 0 0 0 ATV where AC , AL , AR , AV and AI are the incidence matrices describing the circuit topology, and G , L and C are the conductance, inductance and capacitance matrices, respectively. For practical models, the state space dimension N depends on the number of circuits components and is usually huge. The simulations of such largescale circuit equations are unfeasibly time consuming. The main task of model order reduction is to approximate system (1) by a model of lower dimension which inherits some important physical properties of Eq. 1 such as passivity and reciprocity. System (1) is passive if its transfer function H (s) = B T (sE − A)−1 B is positive real, i.e., H (s) is analytic in the open right half-plane C+ and H (s) + H (s)∗ ≤ 0 for all s ∈ C+ . Furthermore, system (1) is reciprocal if H (s) = SH (s)T S for all s ∈ C, where S ∈ Rm×m is a diagonal signature matrix satisfying S 2 = I . Note that if the circuit
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