2005 Delay-extraction-based sensitivity analysis of multiconductor transmission lines with nonlinear

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3520IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Delay-Extraction-Based Sensitivity Analysis of Multiconductor Transmission Lines With Nonlinear TerminationsNatalie M. Nakhla, Student Member, IEEE, Anestis Dounavis, Member, IEEE, Michel S. Nakhla, Fellow, IEEE, and Ramachandra Achar, Senior Member, IEEEAbstract—An ef cient approach is presented for time-domain sensitivity analysis of lossy multiconductor transmission lines with respect to electrical and/or physical parameters. The proposed method employs delay extraction prior to approximating the matrix exponential stamp of the line and guarantees macromodel passivity. A delay-extraction-based equivalent circuit for sensitivity analysis is derived in a closed form, leading to signi cant computational advantages. Index Terms—Delay extraction, matrix rational approximation, passive macromodels, sensitivity analysis, transmission lines.I. INTRODUCTIONTHE RAPID increase in operating speeds, density, and complexity of modern integrated circuits and microwave applications has made interconnect analysis and optimization a challenging task for high-frequency circuit designers. Effects such as re ections, crosstalk, and propagation delays associated with the interconnects have become critical factors, and improperly designed interconnects can result in increased signal delay, ringing, inadvertent, and false switching [1]. Interconnections are present at various levels of the design hierarchy such as on-chip, packaging structures, multichip modules (MCMs), printed circuit boards (PCBs), and backplanes. Moreover, in high-frequency and multilevel designs, interconnects play a major role in determining important design requirements such as power consumption, density, clock frequency, etc. For example, increasing the circuit’s density leads to shorter interconnects, which reduce the problem of delay and re ections. However, this leads to greater crosstalk between adjacent interconnects. Designers must make the proper tradeoffs, often between con icting design requirements, to obtain the best possible performance. Therefore, ef cient and accurate sensitivity analysis of circuit response with respect to interconnect parameters is of signi cant importance in identifying critical design components, in tolerance assignments, and in optimizing the overall interconnect network performance [2]–[6]. Several techniques have been presented in the literature for modeling and sensitivity analysis of distributed interconnectsManuscript received April 2, 2005; revised July 18, 2005. N. M. Nakhla, M. S. Nakhla, and R. Achar are with the Department of Electronics, Carleton University, Ottawa, ON, Canada K1S 5B6 (e-mail: nnakhla@doe.carleton.ca; msn@doe.carleton.ca; achar@doe.carleton.ca). A. Dounavis is with the Department of Electronics and Communications Engineering, University of Western Ontario, London, ON, Canada N6A 5B9 (e-mail: adounavis@eng.uwo.ca). Digital Object Identi er 10.1109/TMTT.2005.857344[7]–[29]. Among the most commonly used algorithms are those based on the method of characteristics (MoC) [7], [23]. The MoC model is based on the extraction of the line propagation delay and is exact if applied to lossless transmissions lines. Several approximation techniques have been proposed to extend the applicability of MoC to lossy lines [8], [10], [22]. Also, a fast algorithm for sensitivity analysis based on MoC was described in [3]. Although MoC provides fast solutions for long low-loss lines, it does not guarantee the passivity of resulting macromodels. However, passivity of the macromodel is important since nonpassive but stable models when coupled with arbitrary nonlinear elements can lead to unstable systems [1]. To address this issue, algorithms based on matrix rational approximations (MRA) were introduced in [13] and [24], and extension to sensitivity analysis was described in [27]. MRA guarantees the macromodel passivity by construction. However, in the presence of large delay lines (e.g., long lines with small losses), it requires high-order approximations (to accurately capture the at delay portion) leading to inef cient transient simulation [22]. This limits its usefulness to short lines (on-chip interconnects). Recently, a delay-extraction-based passive compact multiconductor transmission line model (DEPACT) was introduced in [25]. It combines the merits of both MoC and MRA and is based on delay extraction [23] prior to performing the MRA. In contrast to the MRA technique, delay effects are represented by lossless transmission lines instead of lumped circuits (i.e., by pure delay elements instead of rational function approximations). This results in signi cantly lower-order macromodels for long lossy-coupled lines, leading to fast transient simulation for both on-chip and off-chip interconnects. In this paper, the delay-extraction-based passive macromodeling algorithm is extended to time-domain sensitivity analysis. A major advantage of the proposed method is that the sensitivity of the circuit matrices is obtained analytically in terms of the per-unit-length (p.u.l.) interconnect parameters and predetermined coef cients, leading to signi cant computational cost advantage. The method enables sensitivity analysis with respect to both electrical and physical parameters and can handle transmission lines with lossy as well as frequency-dependent parameters. The rest of this paper is organized as follows. In Section II, an overview of the delay-extraction-based macromodel is presented. The new sensitivity algorithm is described in Section III and its application to sensitivity analysis and optimization of interconnects with nonlinear terminations is demonstrated in Section IV.0018-9480/$20.00 © 2005 IEEE

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

NAKHLA et al.: DELAY-EXTRACTION-BASED SENSITIVITY ANALYSIS OF MTL WITH NONLINEAR TERMINATIONS3521Fig. 1. Realization of an MLP cell in (5).II. REVIEW OF DELAY-EXTRACTION-BASED MACROMODELS Consider distributed transmission lines represented by Telegrapher’s equations [7] aswhere the associated erroris expressed as (4)(1) where , , , and are the p.u.l. parameters of the transmission line, and represent the voltage and current vectors of the transmission line subnetwork as a function of position and time , and is the number of coupled lines. It is to be noted that the p.u.l. parameters are, in general, frequency-dependent due to skin and proximity effects. Equation (1) can be expressed in matrix-exponential form in the frequency domain asIn the case of lines with frequency-dependent parameters, a maximum delay is extracted without affecting transmission line causality conditions. In this case, (3) can be modi ed as (5) where (6) such that , are positive de nite matrices. The products represented in (5) can be viewed as a cascade of transmission line subnetworks. In addition, each of these subnetworks can be viewed as a cascade of lossy and lossless transin (5) can be realized as mission lines. For example, each shown in Fig. 1. Here, the lossy terms are macromodeled using the passive MRA [14]. They are later combined with the lossless sections using the MoC approach along with an algorithm to decouple the multiconductor transmission line (MTL) equations [25]. III. SENSITIVITY ANALYSIS From the previous section, it is clear that in order to calculate the network sensitivity with respect to any interconnect parameter , the sensitivity of both the MRA-based lossy subsections and the delay-based lossless subsections with respect to is required. The following sections describe obtaining these sensitivities. A. Sensitivity Analysis of Lossy Subsections The lossy subsection (subnetwork ) in Fig. 1 can be realized using the passive MRA [14], [25]. Using MRA, the exponential , can be approximatrix , where mated as (7) where , are polynomial matrices expressed in terms of closed-form Padé rational functions [13].(2) where is the length of the line. Equation (2) does not have a direct representation in the time domain, which makes it dif cult to interface with nonlinear simulators. In order to address this dif culty, a new passive macromodel was proposed [25]. A brief review of the algorithm as relevant to this paper is given in this section. Examining the terms in (2), it can be noted that the delay part is essentially contributed by the matrix . However, extracting the delay from is not a simple task due to the fact that do not commute, i.e., . the matrices and in terms Hence, it is desired to nd an approximation of of a product of exponential matrices. The modi ed Lie product (MLP) formula [25] provides a systematic approach to obtain and is such an approximation with an error estimation given by(3)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3522IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Equation (7) for can be represented in terms of subsections, described by pole-zero pairs asB. Sensitivity Analysis of Lossless Subsections The terminal voltages and currents of the lossless subsections in Fig. 1 are related by(12) (8) for even values of and where and . Next, (12) can be decoupled in terms of propagating modal voltages and currents as [7](13) are diagonal matrices of the form and , and can be calculated in terms of modal transformation matrices and , where where (9) It can be shown that represents the unity matrix and for odd values of . Here, are predetermined coef cients [14]. The symbol represents the complex conjugate operation. The p.u.l. frequency-dependent parameters of the lines are modeled in the form of positive-real rational functions using a suitable tting technique [30]–[33]. Next, the resulting rational functions are synthesized in the form of passive RLC circuits [14]. Equations (8) and (9) for the lossy subsection represented in Fig. 1 can be translated into a set of ordinary differential equations in the form [14] and (14) can be obtained by [25] (15) and is a where is a matrix of the right eigenvectors of , i.e., diagonal matrix with the eigenvalues of . Matrix contains the right eigenvectors of the product (i.e., ). The diagonal matrix is a normalizing factor such that the Euclidean norm is unity. The diagonal elements are of the columns of given by (16) (10) , contain the stamps of the lumped where memory and memoryless elements of subnetwork , and can be obtained in a closed form in terms of the p.u.l. parameters is the node space of the subnetwork ; of the line; is a selector matrix (with entries 0 or 1); and represent the terminal currents and voltages of subnetwork , respectively; and superscript denotes the transpose. Using (10), the sensitivity equations can be written as where and(17) The solution to the modal equations in (13) can be written as [7], [25](11) where represents any interconnect parameter of interest. and in terms of A detailed computation of the predetermined coef cients and the sensitivity of the p.u.l. line parameters was described in [27]. We will later combine these equations with the sensitivity equations of the lossless subsections and the rest of the nonlinear circuit. where(18)(19)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

NAKHLA et al.: DELAY-EXTRACTION-BASED SENSITIVITY ANALYSIS OF MTL WITH NONLINEAR TERMINATIONS3523Fig. 2. Equivalent circuit for analysis of lossless subsections.The modal parameters in (18) are given by(20) and the vectors that are functions of are de ned as (21) Using (18), the solution to the modal equations can be represented with the equivalent circuit shown in Fig. 2, where the subscript denotes the th component of the associated vector. Differentiating (14) with respect to , we can express the sensitivity of the actual voltages and currents in terms of the sensitivity of the modal voltages and currents by the relationand , and the sensimodal voltages and currents and . The tivity of the modal transformation matrices details involved in determining these sensitivities are presented in the following sections. 1) Sensitivity of Modal Voltages and Currents: It can be proved (Appendix I) that the sensitivity equations for the lossless subsections can be written as(23) where the independent sources and the solution of (18) and are given by are obtained from(22) Examining (22), it can be seen that and can be determined based on the sensitivity of the (24)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3524IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Fig. 3.Equivalent circuit for sensitivity analysis of lossless subsections.2) Sensitivity of Modal Transformation Matrices: The next step is to calculate the derivatives of the modal transformaand tion matrices with respect to . In order to obtain , we use (15) to getrespectively. Once the solution for (26) is obtained, be computed from the relationshipcan(27) where Similarly, since of the product . contains the right eigenvectors can be determined by,(25) Recall that is a matrix of the right eigenvecand is a diagonal matrix with the tors of eigenvalues of . The derivatives and can be calculated by solving the following system of equations [28]:(28) in (28) can be obtained in Note that using (26) and (27). terms of Next, the derivative of the normalizing diagonal matrix can be computed using (16) as(26)where is the identity matrix, and and represent the th eigenvalue and its corresponding eigenvector,(29)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

NAKHLA et al.: DELAY-EXTRACTION-BASED SENSITIVITY ANALYSIS OF MTL WITH NONLINEAR TERMINATIONS3525Fig. 4.MLP cell for a line with constant p.u.l. parameters.where is de ned in (17). Differentiating (17) with respect to and combining with (26)–(29), the sensitivity of the modal and can be determined. transformation matrices Next, knowing the sensitivity of , , , and with respect to , the sensitivity of the terminal voltages and currents of the lossless subsections can be computed using (22). An equivalent circuit representation for the sensitivity analysis of the lossless subsections is shown in Fig. 3. Finally, the sensitivity of the terminal voltages and currents of the MLP cell shown in Fig. 1 can be obtained by combining the sensitivity equations of the lossless subsections with the sensitivity equations of the MRA lossy subsections. C. Special Case: Frequency-Independent Parameters For the case of frequency-independent p.u.l. parameters, the lossy subsections can be simply realized by passive purely resistive networks. In this case, the delay-extracted exponential matrix representing a lossy subsection can be expressed as (30) where the resistive matrix is de ned in (2). For the case of single lines, the MLP cell can be represented by a resistive circuit and two lossless subsections as shown in Fig. 4 where, and and represent the terwhere can be minal currents and voltages, respectively. The matrix written as(33)Differentiating (32) with respect to , the sensitivity equations for the resistive network can be written as (34) where can be expressed as(35) and can be obUsing (31), the derivatives tained in terms of the sensitivities of the p.u.l. parameters with respect to . D. Uni ed Sensitivity Equations In general, distributed networks in the presence of nonlinear elements can be expressed as [1] (36) (37) is a vector of the unknown voltages and currents, , and describe the lumped components, and is a selector matrix. is a matrix representing the interconnect subnetwork, where is and represent the number of coupled conductors. the Laplace terminal voltages and currents of the interconnect subnetwork, respectively. To simplify notation, (36) and (37) assume only one set of coupled lines; however, extension to multiple sets of coupled lines is straightforward. Using the proposed delay-extraction-based macromodel, (37) can be replaced with an equivalent time-domain representation using MLP cells as described in (5). where , ,(31) For the case of coupled lines, a decoupling method similar to that described in the previous section is used prior to realizing the resistive network. Thus, to avoid repetition, as an illustration, we will only consider here the case of single lines. Extension to coupled lines is straightforward. To calculate the sensitivity of the resistive subcircuit in Fig. 4, the nodal equations for the subcircuit can be written as (32)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3526IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Next, the sensitivity of the nonlinear network with respect to a parameter can be obtained using (36) as(38) where(39) Equations (11) and (22)–(24) represent an implicit relationship between the sensitivities of the terminal currents and terminal voltages of the th MLP ). Combining these equations with cell (for (38), we get a consistent set of ordinary differential equations that can be solved simultaneously with the original circuit equations to compute the required sensitivity information. These equations can be easily interfaced with commercial circuit simulators (such as HSPICE) by converting them to an equivalent circuit representation that can be directly embedded in the simulator netlist. Several ef cient techniques have been previously proposed to perform this conversion [1], [34]. E. Sensitivity With Respect to Physical Parameters When studying the sensitivity of distributed networks, the design parameters of interconnects are usually required with respect to physical parameters (such as width and spacing of conductors). The sensitivities of electrical parameters are often intermediate steps to the calculation of sensitivities of physical parameters. In the case where represents a physical parameter of an interconnect, the sensitivity of the output nodes can be obtained asFig. 5. Transmission line network with nonlinear terminations (Example 1).Fig. 6.Physical/geometrical parameters for the MTL subnetworks in Fig. 5.(40) where , , , and are the p.u.l. parameters and the subscripts and are matrix indices. IV. COMPUTATIONAL RESULTS In this section, two numerical examples are presented to demonstrate the validity and computational ef ciency of the proposed sensitivity computation algorithm. The rst example presents the sensitivity analysis results for MTL networks with frequency-dependent p.u.l. parameters. The second example presents interconnect performance optimization using the sensitivity analysis results from the rst example. Example 1: In this example, a circuit with three MTL subnetworks (each with two coupled conductors) is considered (Fig. 5). Fig. 6 shows the physical parameters of the three interconnect structures. The frequency-dependent p.u.l. parameters were computed from the physical description of the transmission line using HSPICE.1 A sample comparison of the1HSPICEtransient responses corresponding to a 3-V step pulse input with a rise time of 0.5 ns is shown in Fig. 7. Fig. 8 shows the and with respect to (where sensitivity of node voltages is the width of the conductors as shown in Fig. 6). The sensitivity results of the proposed algorithm are compared with the perturbation of the conventional lumped segment model [7]. As seen, the results from both methods are in good agreement. It is important to note here that perturbation-based techniques can lead to inaccurate results, depending on the magnitude of the perturbation. In addition, the perturbed network must be solved separately for every parameter of interest. However, in the proposed approach, the sensitivity information with respect to all the parameters can essentially be obtained from the solution of the original network. Table I shows a comparison of savings in the main computational cost (in terms of the total number of LU decompositions) using the proposed approach versus the perturbation approach for the above example. It is to be noted that the advantage in the computational cost increases as we increase the number of parameters. Example 2: This example demonstrates the use of the proposed sensitivity computation algorithm in a network performance optimization process. For the purpose of illustration, the MTL subnetworks in the circuit of Fig. 5 are optimized to achieve the following performance speci cations. 1) The delays of V1 and V3 should not exceed 2.0 and 2.5 ns, respectively, based on a threshold voltage of 1.5 V. V for ns and 2) It is desired that V for ns. V 3) It is desired that the crosstalk components and V subject to the following constraints: cm mil mil mil mil mil (41)ver. 2004.03, Synopsis Inc., Mountain View, CA, 2004.

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

NAKHLA et al.: DELAY-EXTRACTION-BASED SENSITIVITY ANALYSIS OF MTL WITH NONLINEAR TERMINATIONS3527Fig. 7. Comparison of transient response (Example 1): (a) at node V1 and (b) at node V2.Fig. 8. Sensitivity with respect to conductor width w (Example 1): (a) at node V1 and (b) at node V2.The initial values of the design variables are taken as mil mil mil cm (42)TABLE I COMPUTATIONAL COST COMPARISONAs seen from Fig. 9, the speci cations are clearly violated before optimization. Using a constrained optimization technique [35] coupled with the proposed sensitivity analysis algorithm, the optimized design variables mil mil mil milcm cm(43)were obtained after nine iterations. Fig. 9 shows the comparison between circuit responses before and after optimization. As seen from the gure, the node voltages meet all the circuit speci cations without violating any design constraints.

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3528IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Fig. 9. Comparison of circuit responses before and after optimization (Example 2).V. CONCLUSION A new approach for time-domain sensitivity analysis of lossy MTLs in the presence of nonlinear terminations is described. Sensitivity information is derived using the recently developed closed-form delay-extraction-based macromodel. The method enables sensitivity analysis of interconnect structures with respect to both electrical and physical parameters while providing signi cant computational cost advantages. APPENDIX I DERIVATION OF SENSITIVITY EQUATIONS FOR MODAL VOLTAGES AND CURRENTS In order to obtain the sensitivity equations in (23), we rst consider the solution to the modal equations in the Laplace domain (44) (45) where . (46) (47) where is de ned as and the modal propagation constant(48) Differentiating (46) with respect to , we get(49)

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

NAKHLA et al.: DELAY-EXTRACTION-BASED SENSITIVITY ANALYSIS OF MTL WITH NONLINEAR TERMINATIONS3529Using the relation , (49) can be rewritten as(50) Converting (50) to the time domain yields(51) Similarly, differentiating (47) with respect to , we obtain(52) Differentiating (44) and (45), we can write (53) (54) Combining (51)–(54), and with some simple mathematical manipulation, the sensitivity equations in (23) can be obtained. REFERENCES[1] R. Achar and M. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, no. 5, pp. 693–728, May 2001. [2] C. Jiao, A. C. Cangellaris, A. M. Yaghmour, and J. L. Prince, “Sensitivity analysis of multiconductor transmission lines and optimization for highspeed interconnect circuit design,” IEEE Trans. Adv. Packag., vol. 23, no. 2, pp. 132–141, May 2000. [3] M. Jun-Fa and E. S. Kuh, “Fast simulation and sensitivity analysis of lossy transmission lines by the method of characteristics,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 44, no. 5, pp. 391–401, May 1997. [4] Q. Zhang, S. Lum, and M. Nakhla, “Minimization of delay and crosstalk in high-speed VLSI interconnects,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1555–1563, Jul. 1992. [5] S. Lum, M. Nakhla, and Q. Zhang, “Sensitivity analysis of lossy coupled transmission lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 4, pp. 607–615, Apr. 1994. [6] N. Nakhla, A. Dounavis, M. Nakhla, and R. Achar, “Delay-extraction based sensitivity analysis of multiconductor transmission lines with nonlinear terminations,” presented at the IEEE MTT-S Int. Microwave Symp., Long Beach, CA, Jun. 2005. [7] C. R. Paul, Analysis of Multiconductor Transmission Line. New York: Wiley, 1994. [8] S. Grivet-Talocia and F. Canavero, “Topline: A delay pole-residue method for simulation of dispersive interconnects,” in Proc. Electrical Performance Electronic Packaging (EPEP), Monterey, CA, Oct. 2002, pp. 359–362. [9] M. Nakhla and R. Achar, Multimedia Book Series on Signal Integrity. Nepean, ON, Canada: OMNIZ Global Knowledge Corporation, 2002. [Online]. Available: . [10] F. Y. Chang, “The generalized method of characteristics for waveform relaxation analysis of lossy coupled transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 12, pp. 2028–2038, Dec. 1989.[11] M. Celik, A. C. Cangellaris, and A. Yaghmour, “An all purpose transmission line model for interconnect simulation in SPICE,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 127–138, Oct. 1997. [12] A. C. Cangellaris, S. Pasha, J. L. Prince, and M. Celik, “A new discrete time domain model for passive model order reduction and macromodeling of high-speed interconnections,” IEEE Trans. Compon. Packag. Technol., vol. 22, no. 3, pp. 356–364, Aug. 1999. [13] A. Dounavis, X. Li, M. Nakhla, and R. Achar, “Passive closed-loop transmission line model for general-purpose circuit simulators,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2450–2459, Dec. 1999. [14] A. Dounavis, R. Achar, and M. Nakhla, “Ef cient passive circuit models for distributed networks with frequency-dependent parameters,” IEEE Trans. Adv. Packag., vol. 23, no. 3, pp. 382–392, Aug. 2000. [15] A. Dounavis, E. Gad, R. Achar, and M. Nakhla, “Passive model reduction of multiport distributed interconnects,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2325–2334, Dec. 2000. [16] W. T. Beyene and J. E. Schutt-Ainé, “Ef cient transient simulation of high-speed interconnects characterized by sampled data,” IEEE Trans. Compon., Packag., Manuf. Technol., B, vol. 21, no. 1, pp. 105–113, Feb. 1998. [17] E. Chiprout and M. Nakhla, “Analysis of interconnect networks using complex frequency hopping,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 14, no. 2, pp. 186–200, Feb. 1995. [18] M. Celik and A. C. Cangellaris, “Simulation of dispersive multiconductor transmission lines by Padé via Lanczos process,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pp. 2525–2535, Dec. 1996. , “Ef cient transient simulation of lossy packaging interconnects [19] using moment-matching techniques,” IEEE Trans. Compon., Packag., Manuf. Technol., B, vol. 19, no. 1, pp. 64–73, Feb. 1996. [20] A. Odabasioglu, M. Celik, and L. T. Pilleggi, “PRIMA: Passive reducedorder interconnect macromodeling algorithm,” IEEE Trans. Comput.Aided Des. Integr. Circuits Syst., vol. 17, no. 8, pp. 645–653, Aug. 1998. [21] Q. Yu, J. L. Wang, and E. Kuh, “Passive multipoint moment matching model order reduction algorithm on multiport distributed interconnect networks,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 46, no. 1, pp. 140–160, Jan. 1999. [22] I. Elfadel, H. Huang, A. Ruehli, A. Dounavis, and M. Nakhla, “A comparative study of two transient analysis algorithms for lossy transmission lines with frequency-dependent data,” in Proc. Electrical Performance Electronic Packaging, Cambridge, MA, Oct. 2001, pp. 255–258. [23] F. H. Branin, “Transient analysis of lossless transmission lines,” Proc. IEEE, vol. 55, no. 11, pp. 2012–2013, Nov. 1967. [24] A. Dounavis, R. Achar, and M. Nakhla, “Addressing transient errors in passive macromodels of distributed transmission line networks,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2759–2768, Dec. 2002. [25] N. Nakhla, A. Dounavis, R. Achar, and M. Nakhla, “DEPACT: Delay extraction-based passive compact transmission line macromodeling algorithm,” IEEE Trans. Adv. Packag., vol. 28, no. 1, pp. 13–23, Feb. 2005. [26] A. Dounavis, R. Achar, and M. Nakhla, “A general class of passive macromodels for lossy multiconductor transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1686–1696, Oct. 2001. , “Ef cient sensitivity analysis of lossy multiconductor transmis[27] sion lines with nonlinear terminations,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2292–2299, Dec. 2001. [28] S. Lum, M. Nakhla, and Q. Zhang, “Sensitivity analysis of lossy coupled transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 2089–2099, Dec. 1991. [29] S. Grivet-Talocia, H. Huang, A. Ruehli, F. Canavero, and I. Elfadel, “Transient analysis of lossy transmission lines: An ef cient approach based on the method of characteristics,” IEEE Trans. Adv. Packag., vol. 27, no. 1, pp. 45–56, Feb. 2004. [30] D. Saraswat, R. Achar, and M. Nakhla, “A fast algorithm and practical considerations for passive macromodeling of measured/simulated data,” IEEE Trans. Adv. Packag., vol. 27, no. 1, pp. 57–70, Feb. 2004. [31] S. Grivet-Talocia, “Passivity enforcement via perturbation of Hamiltonian matrices,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 9, pp. 1755–1769, Sep. 2004. [32] G. Antonini, A. Ruehli, and C. Yang, “PEEC modeling of dispersive and lossy dielectrics,” in IEEE 12th Topical Electrical Performance Electronic Packaging Meeting, Princeton, NJ, 2003, pp. 349–352. [33] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector tting,” IEEE Trans. Power Del., vol. 14, no. 3, pp. 1052–1061, Jul. 1999. [34] T. Palenius and J. Roos, “Comparison of reduced-order interconnect macromodels for time-domain simulation,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2240–2250, Sep. 2004. [35] J. W. Bandler, W. Kellermann, and K. Madsen, “A superlinearly convergent minimax algorithm for microwave circuit design,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 12, pp. 1519–1530, Dec. 1985.

传输线、多导体传输线论文,Transmission lines,Multiconductor Transmission lines,MTL

3530IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 11, NOVEMBER 2005Natalie M. Nakhla (S’04) was born in Ottawa, ON, Canada, in September 1980. She received the B.Eng. degree in telecommunications engineering and M.A.Sc. degree in electrical engineering from Carleton University, Ottawa, ON, Canada, in 2003 and 2005, respectively, and is currently working toward the Ph.D. degree at Carleton University. Her research interests include computer-aided design of very large scale integration (VLSI) circuits, simulation and modeling of high-speed interconnects, packaging characterization, and numerical techniques. Ms. Nakhla was the recipient of the National Science and Engineering Research Council (NSERC) Postgraduate Award (2003–2005) and the Canada Graduate Scholarship at the doctoral level (2005–2007). She was also the recipient of the Carleton University Senate Medal for outstanding academic achievement at the undergraduate level.Anestis Dounavis (S’99–M’04) received the B.Eng. degree in electrical engineering from McGill University, Montreal, QC, Canada, in 1995, and the M.Eng. and Ph.D. degrees in electrical engineering from Carleton University, Ottawa, ON, Canada, in 2000 and 2004, respectively. He is currently an Assistant Professor with the Department of Computer and Electrical Engineering, University of Western Ontario, London, ON, Canada. His research interests are in electronic design automation, simulation of high-speed networks, signal integrity, and numerical algorithms. Dr. Dounavis was the recipient of the Ottawa Centre for Research and Innovation (OCRI) Futures Award—student Researcher of the Year in 2004 and the INTEL Best Student Paper Award presented at the 2003 Electrical Performance of Electronic Packaging Conference. He was also the recipient of the Carleton University Medal for outstanding graduate work at the master and Ph.D. levels in 2000 and 2004, respectively.Ramachandra Achar (S’95–M’99–SM’04) received the B.Eng. degree in electronics engineering from Bangalore University, Bangalore, India, in 1990, the M.Eng. degree in microelectronics from the Birla Institute of Technology and Science, Pilani, India, in 1992, and the Ph.D. degree from Carleton University, Ottawa, ON, Canada, in 1998. He is currently an Associate Professor with the Department of Electronics, Carleton University. Prior to joining the Carleton University faculty, he served in various capacities in leading research laboratories including the T. J. Watson Research Center, IBM, Yorktown Heights, NY (1995), the Central Electronics Engineering Research Institute, Pilani, India (1992), the Indian Institute of Science, Bangalore, India (1990), and Larsen and Toubro Engineers Ltd., Mysore, India (1992). He is a consultant for several leading industries focused on high-frequency circuits, systems, and tools. He has authored or coauthored over 100 peer-reviewed papers in international journals/conferences, six multimedia books on signal integrity, and four chapters in different books. His research interests include signal integrity analysis, numerical algorithms and development of computer-aided design tools for modeling and simulation of high-frequency interconnects, nonlinear circuits, microwave/RF networks, opto-electronic devices, MEMS and electromagnetic compatibility/electromagnetic interference (EMC/EMI). Dr. Achar is a practicing Professional Engineer in the Province of Ontario. He serves on the Technical Program Committee of several leading IEEE conferences. He is a member of the Canadian Standards Committee on Nanotechnology and is chair of the joint chapters of Circuits and Systems (CAS)/Electron Devices (ED)/Solid-State Circuits (SSC) societies of the IEEE Ottawa Section. He was the recipient of several prestigious awards including the University Medal for the outstanding doctoral work (1998), Natural Science and Engineering Research Council (NSERC) Doctoral Medal (2000), Strategic Microelectronics Corporation (SMC) Award (1997), Canadian Microelectronics Corporation (CMC) Award (1996), and the Best Student Paper Award in the 1998 Micronet (a Canadian Network of Centres of Excellence on Microelectronics) Annual Workshop. His students/coauthors have also been the recipient of Best Student Paper Awards in international forums including the Electrical Performance of Electronic Packages (EPEP) Conference (1998, 2000, 2002–2004) and the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) (2003).Michel S. Nakhla (S’73–M’75–SM’88–F’98) received the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1973 and 1975, respectively. From 1976 to 1988, he was with Bell-Northern Research, Ottawa, ON, Canada, as the Senior Manager of the Computer-Aided Engineering Group. In 1988, he joined Carleton University, Ottawa, ON, Canada, as a Professor and the Holder of the Computer-Aided Engineering Senior Industrial Chair established by Bell-Northern Research and the Natural Sciences and Engineering Research Council (NSERC) of Canada. He is the Founder of the High-Speed Computer-Aided Design (CAD) Research Group, Carleton University. He is a Chancellor’s Professor of Electrical Engineering with Carleton University. He serves as a technical consultant for several industrial organizations and is the Principal Investigator for several major sponsored research projects. His research interests include CAD of very large scale integration (VLSI) and microwave circuits, modeling and simulation of high-speed interconnects, nonlinear circuits, multidisciplinary optimization, thermal and electromagnetic emission analysis, microelectromechanical system (MEMS), and neural networks. Dr. Nakhla was associate editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS and guest editor of the IEEE TRANSACTIONS ON COMPONENTS, PACKAGING, AND MANUFACTURING TECHNOLOGY (Advanced Packaging) and the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART II: ANALOG AND DIGITAL SIGNAL PROCESSING.

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