Efficient Time-Domain Macromodeling of Complex Interconnection Structures Bart Haegeman ∗ , Dirk Deschrijver ∗† ,Tom Dhaene ∗† ∗ Dept. of Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium † Dept. of Information Technology (INTEC), Ghent University, Gent, Belgium e-mail: bart.haegeman@gmail.com, {dirk.deschrijver,tom.dhaene}@ua.ac.be Abstract—The Vector Fitting algorithm [1] is an iterative procedure to compute rational approximations of frequency- domain responses. It was shown that the robustness of this technique can be enhanced by using a set of orthonormal rational basis functions, leading to the Orthonormal Vector Fitting method [2]. In this paper, a time-domain implemen- tation of this method is proposed for the macromodeling of transient port responses. It is shown that this method is more robust towards the initial pole specification, when compared to the classical time-domain Vector Fitting method [3]. Keywords— Time domain macromodeling, System identi- fication, Orthonormal Vector Fitting. I. I NTRODUCTION Compact and accurate macromodels are crucial for accurate system-level simulations. The derivation of such models from measurements or first-principle simulators is numerically not a trivial task, even for linear systems. In the frequency-domain, the Vector Fitting (VF) method [1] was proposed to calculate broadband transfer functions from a given frequency-domain response. It was shown in [3] that this method can be extended to model transient port responses in the time-domain. Initially, both methods start from a prescribed set of poles which define the basis functions of the approxi- mation problem. Using an iterative least-squares method, these poles are relocated in a two-step procedure to minimize the global fitting error. It was shown in [2] that the numerical robustness of the technique can be improved by applying a Gram-Schmidt orthonormalization procedure on the basis functions, lead- ing to the Orthonormal Vector Fitting (OVF) method. In this paper, an analogous reasoning is applied to build macromodels in the time-domain. This way, the condi- tioning of the system equations becomes less sensitive to the initial pole specification, and accurate models can be computed in fewer iterations. This improves the efficiency of the method and reduces the overall computation time. II. MODEL REPRESENTATION Frequency-domain macromodelling tools are used to build a rational transfer function R(s), based on the spectral response (s,H(s)) of a physical structure. R(s)= N (s) D(s) = P  p=1 c p Φ p (s, a) ˜ c 0 + P  p=1 ˜ c p Φ p (s, a) (1) In the frequency-domain OVF technique, numerator and denominator are expanded as a linear combination of orthonormal rational functions Φ p (s, a), which are based on a prescribed set of stable poles a = {-a 1 , ..., -a P } [2]. If -a p is a real pole, then the basis functions are defined as Φ p (s, a)=  2 ℜe(a p ) s + a p   p-1  j=1 s - a ∗ j s + a j   (2) and a linear combination is formed when two poles -a p = -a ∗ p+1 form a complex conjugate pair Φ p (s, a)=  2 ℜe(a p )(s -|a p |) (s + a p )(s + a p+1 )   p-1  j=1 s - a ∗ j s + a j   (3) Φ p+1 (s, a)=  2 ℜe(a p )(s + |a p |) (s + a p )(s + a p+1 )   p-1  j=1 s - a ∗ j s + a j   (4) It can be shown that these basis functions are orthonormal with respect to the following inner product (1 ≤ m, n ≤ P ) 〈Φ m (s), Φ n (s)〉 s = 1 2πi  iR Φ m (s)Φ ∗ n (s)ds (5) III. TRANSFER FUNCTION I DENTIFICATION The goal of the identification process, is to identify the coefficients c p and ˜ c p in (1), such that the difference between R(s) and H(s) is minimized in a least-squares sense. A linear approximation of this non-linear identifi- cation problem is obtained by solving the following set of equations (˜ c 0 =1) P  p=1 c p Φ p (s, a) - H(s) P  p=1 ˜ c p Φ p (s, a) ≈ H(s). (6) In successive iterations, a Sanathanan-Koerner iteration [5], [6] can be applied which iteratively relocates the EUROCON 2007 The International Conference on “Computer as a Tool” Warsaw, September 9-12 1-4244-0813-X/07/$20.00 2007 IEEE. 85 Authorized licensed use limited to: Norges Teknisk-Naturvitenskapelige Universitet. Downloaded on November 9, 2008 at 12:10 from IEEE Xplore. Restrictions apply.