| 摘要(英) |
With the advancement of high-speed systems, signal integrity between packaging and printed circuit boards has become increasingly important. In order to effectively deal with these complex signal transmission behaviors, macro-modeling can be used to assist us in analysis.
Vector Fitting is a commonly used method for model construction, which utilizes input parameters obtained from simulation or measurement of a system, and represents the possible broadband response of the system using rational functions. The poles and zeros obtained from Vector Fitting can be used to construct a sparse state-space equation, where the number of ports (P) and poles (N) determines the system order (NP). For example, if the number of ports is 10 and the number of poles is 300, the system order would be 3000. In time-domain simulations, this can require longer computation time.
In this thesis, we employ Balanced Truncation (BT) and Passive Reduce-order Interconnect Macro-modeling Algorithm (PRIMA) for achieving model order reduction (MOR) of the system. BT is a technique that transforms system states into balanced states through balance transformation. It then truncates states with lower observability and controllability by observing the distribution of Hankel singular values, achieving the goal of model reduction.
PRIMA was originally a white box macro modeling method. After knowing the component content of the routing circuit, it establishes a state-space equation. Then, using the Block Arnoldi iteration method, it calculates the orthogonal basis of the Krylov subspace. By applying matrix projection while preserving moments, it achieves the goal of model reduction. This paper does not use white box macro modeling, but instead utilizes a black box macro model established using VF. The PRIMA process is applied to it.
In this thesis, the part about BT (Balanced Truncation) discusses the reduction of errors using the pole-residue model under different Hankel singular value ratios. The errors decrease as the Hankel singular value ratio decreases. The relationship between H_∞(H-infinity norm) and H_2 (H2 norm) with the Hankel singular values is thatH_∞ and H_2 are proportional to the Hankel singular values ratio. In terms of BT time-domain simulation, we propose a circuit synthesis method and apply it to the reduced model obtained through BT. This method converts the state-space model into SPICE-readable R,C,VCCS elements. We observe the time relationship consumed by different model reduction sizes and validate that after model reduction, faster simulation time can be achieved with an accuracy of 90% compared to the original model.
In the PRIMA section, we explore the model reduction capabilities of PRIMA under the same error criterion as BT. Through comparison, it is found that PRIMA has inferior model reduction capabilities compared to BT, ranging from 3 to 30 times less reduction ability. Additionally, for PRIMA, typical characteristics of moment matching can be observed. |
| 參考文獻 |
Reference
[1] J. E. Schutt-Ainé, P. Goh, Y. Mekonnen, J. Tan, F. Al-Hawari, P. Liu, and W. Dai, “Comparative study of convolution and order reduction techniques for blackbox macromodeling using scattering parameters,” IEEE Trans. Compon. Packag. Manuf. Technol., vol. 1, no. 10, pp. 1642-1650, Oct. 2011.
[2] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Aug. 1999.
[3] B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del., vol. 21, no. 3, pp. 1587-1592, July 2006.
[4] D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 6, pp. 383–385, 2008.
[5] P. K. Goh, “Broadband macromodeling via a fast implementation of vector fitting with passivity enforcement,” M.S. Thesis, Dept. Elect. Comput. Eng., Univ. Illinois Urbana-Champaign, Urbana, IL, USA, 2009.
[6] C. -C. Chou and J. E. Schutt-Ainé, “On the acceleration of the vector fitting for multiport large-scale macromodeling” IEEE Microw. Wireless Compon. Lett., vol. 31, no. 1, pp. 1-4, Jan. 2021.
[7] R. Gao, Y.S. Mekonnen, W.T. Beyene and J.E. Schutt-Aine,“Black-box modeling of passive systems by rational function approximation.” IEEE Transactions on advanced packaging., VOL. 28, NO. 2,pp 209-215 MAY 2005
[8] S. Lefteriu and A. C. Antoulas, “A new approach to modeling multiport systems from frequency-domain data,” IEEE Trans. Computer-Aided Design Int. Circuits Syst., vol. 29, no. 1, pp. 14–27, Jan. 2010.
[9] Y. Nakatsukasa, O. Sète, and L. N. Trefethen, “The AAA algorithm for rational approximation,” arXiv preprint arXiv:1612.00337, 2016.
[10] F. Ebert and T. Stykel, “Rational interpolation, minimal realization and model reduction,” Research Center MATHEON, Berlin, Germany, Tech. Rep. 371-2007, 2007.
[11] A. Odabasioglu, M Celik, and L. T. Pileggi “PRIMA: Passive Reduced-Order Interconnect Macromodeling Algorithm” IEEE ICCAD., vol. 17, no.8, pp. 645-654 Aug 1998
[12] C. -C. Chou and J. E. Schutt-Ainé, “Equivalent Circuit Synthesis of Multiport S Parameters in Pole Residue Form” IEEE Trans. Compon. Packag. Manuf. Technol., vol. 1, no. 10, pp. 1971-1979 Nov 2021
[13] M. Schierholz et al., “SI/PI-Database of PCB-Based Interconnects for Machine Learning Applications,” in IEEE Access, vol. 9, pp. 34423-34432, 2021.
[14] J. P. Hespanha, Linear Systems Theory: Second Edition, Princeton University Press,2009
[15] C. -T. Chen, Linear System Theory and Design, 4/e , Oxford University,2013
[16] Lyapunov equation. Accessed: Jun. 15, 2023 [Online]. Available:
https://en.wikipedia.org/wiki/Lyapunov_equation
[17] D. C. Lay, S.R. Lay and J. J. McDonald, Linear Algebra and Its Applications, Pearson FT Press, 2014
[18] Solve singular value problems. Accessed: Jun. 15, 2023 [Online]. Available::
https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html
[19] Array broadcasting. Accessed: Jun. 15, 2023 [Online]. Available:
https://numpy.org/doc/stable/user/basics.broadcasting.html
[20] S. Grivet-Talocia and B. Gustavsen, Passive macromodeling: Theory and applications, John Wiley & Sons, 2016.
[21] D. M. Pozar, Microwave Engineering, 4/e, John Wiley & Sons, 2012
[22] L. N. Trefethen, David Bau III, Numerical Linear Algebra, Cambridge, 1997 |
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