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A Coupled Efficient and Systematic Full-Wave Time-Domain Macromodeling and Circuit Simulation Method for Signal Integrity Analysis of High-Speed Interconnects Er-Ping Li, Senior Member, IEEE, En-Xiao Liu, Le-Wei Li, Senior Member, IEEE, and Mook-Seng Leong, Senior Member, IEEE
Abstract—This paper presents an accurate and systematic approach for analysis of the signal integrity of the high-speed interconnects, which couples the full-wave finite difference time domain (FDTD) method with scattering ( ) parameter based macromodeling by using rational function approximation and the circuit simulator. Firstly, the full-wave FDTD method is applied to characterize the interconnect subsystems, which is dedicated to extract the parameters of the subnetwork consisting of interconnects with fairly complex geometry. Once the frequency-domain discrete data of the parameters of the interconnect subnetwork is constructed, the rational function approximation is carried out to establish the macromodel of the interconnect subnetwork by employing the vector fitting method, which provides a more robust and accurate solution for the overall problem. Finally, the analysis of the signal integrity of the hybrid circuit can be fulfilled by using the parameters based macromodel synthesis and simulation program with integrated circuits emphasis (SPICE) circuit simulator. Numerical experiments demonstrate that the proposed approach is accurate and efficient to address the hybrid electromagnetic (interconnect part) and circuit problems, in which the electromagnetic field effects are fully considered and the strength of SPICE circuit simulator is also exploited. Index Terms—FDTD method, high-speed interconnects, macromodeling, rational function approximation, signal integrity, parameter.
I. INTRODUCTION
W
ITH THE rapid advancements in modern very largescale integration (VLSI) technology, the trend in VLSI industry is moving toward more complex designs, higher operating frequencies, sharper rise times, shrinking device sizes and low power consumption. The electrical property of interconnects has become a key factor in determining the overall electrical performance of high-speed circuits and systems [1], [2]. The ever-increasing demand for high-speed applications has
Manuscript received March, 2003; revised December 15, 2003. E.-P. Li is with the Division for Computational Electronics and Electromagnetics, Institute of High Performance Computing, National University of Singapore 117528 (e-mail:
[email protected]). E.-X. Liu is with the both Institute of High Performance Computing, and Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail:
[email protected]). L.-W. Li and M.-S. Leong are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail:
[email protected]). Digital Object Identifier 10.1109/TADVP.2004.825448
exhibited the importance of signal integrity (SI) and electromagnetic interference problems on the overall electrical performance of the VLSI systems. If these effects are not addressed during early design stages, it will cause malfunction of a fabricated digital circuit, or distort an analogue signal such that it fails to meet the required specifications. Since the high cost for extra iterations in the design cycle, accurate and efficient simulation techniques for complex interconnect signal integrity and electromagnetic radiation analysis become imperative at high-speed regime. However, the modeling and simulation techniques of high-speed large complex interconnect subnetwork can not be readily realized [1]. Two major difficulties impede the efficient broadband modeling and simulation of interconnects. One is the mixed frequency/time domain problem, where at high-frequency regime, the dispersive nature of interconnect requires a representation in frequency domain, while the circuit components especially nonlinear ones are ready to be formulated in time domain. A traditional ordinary differential equation solver such as a simulation program with integrated circuits emphasis (SPICE)-like circuit simulator [5] can not efficiently handle this mixed domain problem. The other bottleneck is the central processing unit (CPU) expense. As the operation speed of the devices is increasing into the range of multiple gigahertz and the complexity of the interconnect systems continuously increases, the analysis of the signal integrity of the interconnect system at both chip and package levels become increasingly time consuming. Therefore, the concept of macromodeling is developed to illustrate such a process, in which a difficult and complex system is modeled by an approximate but fairly accurate system which is easily simulated [1], [11]. In particular, an interconnect macromodel characterizes the behavior of an -port interconnect subnetwork. This macromodeling approach is designed to alleviate the above-mentioned two difficulties in interconnect electromagnetic behavior modeling and simulation, and to finally achieve a balance between accuracy and efficiency. One large family of macromodeling approaches falls under the category of so-called model-order reduction (MOR), such as AWE-like (asymptotic waveform evaluation) method [4] and Krylov-subspace based methods [6]. The purpose of these model reduction methods is to generate a lower order model and simultaneously maintain the main characteristics of the original problem with considerable accuracy. There exists
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another kind of macromodeling approach which is rooted in the interconnect representation using frequency-sampling data [7]. One common way to generate frequency-sampling data is to utilize full-wave robust finite difference time domain (FDTD) method [8], [9]. As the lumped circuit models and distributed models based on quasi-transverse electromagnetic (TEM) assumption may result inaccurate in describing interconnect electrical property [12], [14], [15]. The FDTD method has been widely used in the analysis of interconnect structures [29]. This traditional FDTD method has been extended to include external lumped circuit elements [17], which has paved a new way for the simulation of hybrid electromagnetic and circuit systems, where both the circuit elements and the components with significant electromagnetic effects shall be considered concurrently [18], [30]. However, this approach is not efficient in dealing with the nonlinear circuit elements, where it is often necessary to reduce the simulation time step to even far below the upper limit imposed by the Courant stability criterion to ensure the convergence of the procedure [19]. In addition, the hybrid FDTD-SPICE method was proposed in order to handle more general lumped elements efficiently [20], but it also suffers from the CPU-efficiency and convergence problems. An attractive alternative to address this kind of interconnect problem is to establish a transfer function representation of the interconnect subnetwork with discrete frequency-domain sampling data by using full-wave FDTD analysis. The mixed frequency/time domain problem aroused in signal integrity analysis including nonlinear circuit elements can be efficiently tackled by rational function approximation method, where the pole-residue form of the transfer function can be easily obtained by least-square method. Thereafter, the macromodel synthesis approach can be applied to derive the time domain differential equation representing the interconnect subnetwork. The so-called state-space representation of the interconnect subnetwork can be efficiently linked to circuit solvers such as SPICE to perform the signal integrity analysis of the whole system. A similar approach was proposed in [26], which was based on the admittance ( ) parameter representation of the interconnect subnetwork by using FDTD method. However, calculating admittance parameters directly corresponds to an unloaded oscillator circuit in contrast to the calculation of scattering parameters [28], which causes slow convergence of the transient waveforms due to the mismatch of the terminations. Although lumped resistors can be inserted into each port to expedite the convergence of the transient waveforms in FDTD simulation [27], this expedition was discounted by the extra expenses of the matrix manipulation involved in removing the effect of the resistors. In practice, scattering parameters are stable parameters readily available from full-wave electromagnetic analysis. This paper presents an accurate and systematic approach for analysis of the high-speed interconnect signal integrity, which was based on the full-wave FDTD macromodeling of the interconnect subnetwork represented by the scattering parameters [31]. The method employs the full-wave FDTD to model the electromagnetic characteristics of the interconnect subsystems and to extract the scattering parameters of the subnetwork. Thereafter, to construct the frequency-domain discrete data of
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the parameters of the interconnect subnetwork, once it is constructed, the rational function approximation is applied to establish the macromodel of the interconnect subnetwork by using the vector fitting method [10]. The signal integrity of the hybrid circuit is therefore analyzed by using the coupled macromodel synthesis and the SPICE circuit simulator. This paper is organized that Section II describes the calculation of the scattering parameters of the interconnect subnetwork by using the three-dimensional (3-D) FDTD method. Section III illustrates the formulation of the rational function approximation of the interconnect transfer function by using the vector fitting method. Section IV presents the development of the equivalent circuit representation of the interconnect subnetwork for SPICE circuit simulation with macromodel synthesis. Numerical experimental results are provided in Section V to validate the accuracy and efficiency of the proposed method and Section VI draws the conclusion. II. INTERCONNECT SUBNETWORK SCATTERING PARAMETERS CALCULATION The FDTD method has been demonstrated its ability to efficiently handle the complex geometries with different materials. Moreover, its feature as a time domain method implies that one single simulation can produce a solution that gives the response of the system with a wide range of frequencies. This advantage enables the FDTD method to be well suited for high-speed interconnect simulation where a wide range of frequency is concerned. The Maxwell’s time-dependent equations governing the electromagnetic fields in the media are (1) (2) are the electric field and magnetic where the vectors and field, respectively. The constants and are the respective electrical permittivity and magnetic permeability. The FDTD algorithm is usually based on the discretization of the above differential form of Maxwell’s equations over a finite volume and approximating the derivatives with a central-difference approach with second-order accuracy. Using staggering field component arrangement and central-difference approximation, (1) and (2) can be easily expressed in an explicit finite difference form. Only two components of the electromagnetic fields are given
(3)
(4) where , and are the respective lattice space incre, and coordinate directions. is the time ments in the increment of leapfrog time-stepping. The subscript integers
LI et al.: COUPLED EFFICIENT AND SYSTEMATIC FULL-WAVE TIME-DOMAIN MACROMODELING
denote a space point rectangular lattice. The superscript step.
in a uniform, indicates the th time
A. Dispersive Boundary Condition Due to the limitation of computer resources, the FDTD approach can not be applied directly to unbounded problems. An absorbing boundary condition (ABC) must be introduced at the outer mesh boundary to simulate the extension of the computational domain to infinity. In this paper, a simple and efficient second-order dispersive boundary condition (DBC) introduced in [16] is applied to the FDTD simulation, which is designed to absorb any linear combination of plane waves propagating with two velocities and . Assumed that the normal direction of the boundary wall is along coordinate axis, the second-order dispersive boundary condition can be expressed in the following form [16]:
(5) (6) where represents the tangential electric field component on the boundary with the superscript and subscript being the and are time index and space index, respectively. the tangential electrical field components located at one and two denotes the spatial steps inside the boundary. The symbol velocity function dependent on the analyzed structure. B. Scattering Parameters of Interconnect Structures The transient results of the interconnect structures can be easily obtained from the 3-D FDTD simulation. However, our concerns on these interconnect structures are their frequency dependent property. The frequency dependent parameters can be obtained by the discrete Fourier transform (DFT) accelerated by using the fast Fourier transform algorithm (FFT). In general, the frequency dependent scattering parameters of an port interconnect subnetwork, can be obtained as (7) where and are the voltages at ports and , respecand are the characteristic impedances of the tively. line connected to these ports. The characteristic impedance may be computed by the following formula [8]:
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III. FORMULATION OF RATIONAL FUNCTION APPROXIMATION BY VECTOR FITTING METHOD By using the 3-D FDTD method, the frequency dependent characteristics of an interconnect subnetwork can be expressed by parameters. However, these frequency-dependent parameters can not be easily linked with nonlinear circuit elements to conduct efficient circuit simulation for the purposes of signal integrity analysis and electrical performance verification. Many approaches are proposed to solve this mixed frequency/time domain problem. A straightforward approach to address this problem is to employ the inverse fast Fourier transform (IFFT) and convolution method [4]. However, this approach suffers from the excessive computational cost of the convolution process. Another approach to solve this mixed domain problem is based on complex frequency hopping (CFH) method by moment matching [24]. The difficulty of this method is that for every moment, a corresponding derivative of each parameter must be computed using numerical integration across the entire time domain. This process has to be repeated on multiple frequency expansion points, which can be cumbersome for a high order of approximation, or networks with many ports. An efficient approach for macromodeling based on sampled frequency data is discussed in [11], [25]–[28], which involves using the direct rational function approximation instead of a moment-matching approach to solve the mixed domain problem. The macromodel obtained from direct rational function approximation can be used in conjunction with recursive convolution [21] to efficiently simulate the interconnects along with nonlinear devices. Alternatively, the resultant macromodel can be converted to an equivalent circuit, which can be incorporated into popular SPICE circuit simulators for signal integrity analysis [1]. The vector fitting method developed by Gustavsen and Semlyen [10] is a robust method for rational function approximation. Vector fitting method has some advantages over other fitting methodologies [22]. Most fitting methods rely on nonlinear optimization algorithms that tend to be slow and may converge to a local minimum. Instead, vector fitting method relies on the solution of two linear least-square problems, thus obtaining the optimal solution rather directly. At the same time, vector fitting method does not suffer much from the numerical stability problem even when the bandwidth of interest is wide. Furthermore, one single run of vector fitting method can achieve the rational function approximation of all the elements in a transfer function matrix with the same set of poles. This section proposed to employ the vector fitting method [10] to perform the rational function approximation of the interconnect transfer function with scattering parameters. A. Concept of Rational Function Approximation
(8)
where the symbol denotes Fourier transform. The symbols represent the FDTD lattice wall parallel to the port plane but with half-step offset in space. This equation takes into account the fact that the voltage and current values derived from corresponding and fields in the Yee’s lattice are offset from each other by one-half space cell and one-half time-step.
As mentioned in Section II, the interconnect subnetwork can be characterized through 3-D FDTD method in terms of the scattering parameters at discrete frequency points ’s covering the frequency range of interest, e.g., as is given in
.. .
.. .
..
.
.. .
(9)
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In order to facilitate the analysis of the signal integrity of a circuit system involving interconnect components, these frequency-dependent response data can be approximated by rational functions to obtain a lower order model for the interconnect subnetwork. Rational function approximation is to fit the frequency response of a network with a ratio of two polynomials with real coefficients in Laplace domain
Equation (13) is a nonlinear problem with the unknowns located in the denominator. Vector fitting method [10] solves this equation as a linear problem in two stages. First Stage—Computing Poles: Specify a set of starting in (13), and multiply with an unknown scaling poles , which is also approximated with the same set function of starting poles . The rational function approximation for and are
(10) (14) where ’s denote the coefficients of the numerator polynomial with the order of and ’s represent the coefficients of the denominator polynomial of the order . is normalized to one. The approximation in (10) through vector fitting method [10] is a two-step weighted least square fitting, which will be illustrated later
(15) Multiplying (15) with
and substituting (14) produce (16)
(11) or B. Numerical Stability Problem A common way of rational function approximation is to multiply the both sides of (10) with its denominator. For discrete frequency data, the resultant linear equation with respect to the unknowns ’s and ’s are written as (12), shown at the bottom of the page. It is obvious that when a higher order of the polynomial is needed for rational function approximation over a wide frequency range, (12) will suffer from the numerical stability problem. The large discrepancy among the entries of the matrix in the left-hand side of (12) results in this problem. The implementation of the robust vector fitting method will overcome this instability problem, and it is detailed in the following section.
Equation (16) is linear in its unknowns be written as
and can easily
(17) Substituting a given frequency point
for in (17) yields (18)
where
C. Vector Fitting Method (VFM) for Rational Function Approximation The pole-residue form of (10) can be expressed as follows, where the subscripts is dropped for simplicity
where is the direct coupling coefficient. , respectively. residues and poles of
’s and
For all the discrete frequency points, the overdetermined linear matrix equation of (18) reads
(13)
(19)
’s are the
The singular value decomposition (SVD) method [32] is employed to accurately solve this linear equation. To this end, the and are derived. rational function approximations for
.. . .. .
.. .
..
.
.. .
.. .
.. .
..
.
.. .
.. . .. .
(12)
LI et al.: COUPLED EFFICIENT AND SYSTEMATIC FULL-WAVE TIME-DOMAIN MACROMODELING
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Fig. 2. Lumped circuit example with nonlinear element.
Fig. 1. Illustration of the equivalent circuit realization of S -parameter based macromodel (30) and (31).
Now, we expand both pole-zero form:
and
into the following
(20)
Therefore
(21)
are equal to the Equation (21) reveals that the poles of . Following the solution of the linear equation (19), zeros of can be calculated as the eigenvalues of the the zeros of following matrix [10]: (22) where is a diagonal matrix containing the staring poles and is a column vector of ones. is a row vector comprising the . residues for Second Stage—Obtaining Residues: As stated before, the obtained from (22) are actually the new poles zeros of . Substituting these new poles into (13) and writing of it at a series of discrete frequency points, an overdetermined linear problem similar to (19) can be formulated with respect and . Solving this linear problem can lead to unknowns of to the solution of the residues.
.. .
Fig. 3. Inverter realized by two MOSFETs.
To this end, the poles and residues of the transfer function are finally established. In addition, the selecting of used in the first stage of vector fitting method is of starting importance for a successful rational function approximation. For transfer functions with many resonant peaks, the starting poles should be chosen as complex conjugate. Furthermore, the imaginary parts of these conjugate pairs shall be linearly distributed over the frequency range of interest and one hundred times larger than the real parts. To assure the stability of the fitting model, a basic requirement is that all the poles of the fitting model must be located in the left-hand side of . This constraint of the the complex plane, i.e., fitting model is often enforced by some simple treatments, e.g., directly deleting the unstable poles or flipping them to the left half-plane [10]. IV. DEVELOPMENT OF EQUIVALENT CIRCUITS WITH MACROMODEL SYNTHESIS A. Synthesis of the S-Parameter Based Macromodel Applying rational function approximation based on VFM to each entry of the transfer function in (9) in Laplace-domain, the -parameter based macromodel can be expressed in (23), shown at the bottom of the page, where ’s are the poles of the interconnect subnetwork, which are identical for all the entries ’s and ’s of the transfer function. is the number of poles. are the direct coupling constant and residues for the entries of the transfer function. The synthesis of the -parameter based macromodel is to derive a set of differential equations from the reduced-order interconnect model in (23) obtained by using vector fitting method (VFM). In general, a set of first-order differential equations,
.. .
..
.
.. .
(23)
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Fig. 4. Scattering parameters of the circuit enclosed in the dashed line of Fig. 2.
which is also called state-space equations, can be formulated as
(24) , and where is the number of ports of the interconnect subnetwork. is the total number of the state variables, which equals the product of the total number of the poles and the total number of the ports. and output vector The th element of the input vector represent the incident wave and the reflected wave at port , respectively. The incident and reflected waves at the th port are defined in terms of the port voltage , the port and the arbitrary reference impedance at port current (25) Given a matrix-transfer function described by (23), several forms of time-domain realization of (24) can be obtained. In this paper, the macromodel synthesis using Jordan-canonical [1] form of realization is employed. For a general -port net, and contain only work, assuming that the matrices , real poles and their corresponding residues, the matrices comprise only complex poles and their corresponding and residues, the Jordan-canonical realization of (24) takes the following form:
Fig. 5.
Transient simulation results at the output port (V
).
Fig. 6. (a) Transmission line circuit schematic and (b) cross section of the microstrip.
(26)
is identity matrix and equals Therefore, the final form of (24) is
.
where the asterisk denotes the complex conjugate. Since complex poles do not have a direct meaning in time domain, the similarity transform is introduced by (27) where (28)
(29)
LI et al.: COUPLED EFFICIENT AND SYSTEMATIC FULL-WAVE TIME-DOMAIN MACROMODELING
Fig. 7.
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S parameters of the microstrip.
Fig. 9. Circuit schematic of a microstrip low-pass filter.
Fig. 8.
Transient result of the diode voltage.
B. Equivalent Circuits SPICE is a powerful general-purpose circuit simulation program for nonlinear dc, nonlinear transient and linear ac analyses, which is used to verify circuit designs and to predict the circuit behavior. For signal integrity analysis of interconnect circuit system, it is a straightforward approach to exploit the many types of nonlinear circuit model embedded in the powerful SPICE circuit simulator. However, SPICE simulators may not directly accept the differential equations of (29) as input. In this case, the state-space representation (29) of the macromodel can be converted to an equivalent circuit consisting of passive elements and controlled voltage and current sources [1], [23]. For the purpose of illustration, a simple case of (24) is considered, which is the representation of a two-port network with two states characterized by parameters
Fig. 10.
S parameters obtained by FDTD method.
where and are the incident wave and the reflected wave at port , respectively. Additional equations relating the wave variables to the port voltages and currents need to be supplemented
(31) is the reference impedance of port . where An equivalent circuit network representing (30) and (31) can be realized as shown in Fig. 1. Its generalization for the case with more state variables or ports can be easily obtained. V. NUMERICAL EXPERIMENTS
(30)
In this section, four examples are presented to demonstrate the validity and accuracy of the method developed in this paper.
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Fig. 11.
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S parameters of the microstrip low-pass filter.
In order to verify the accuracy of the proposed parameter based rational function approximation and macromodel synthesis approach, a lumped element example is simulated and the results is compared with that from SPICE circuit simulator. The second example is presented to verify the validity and accuracy of the proposed systematic approach of FDTD macromodeling. The analysis of signal integrity is conducted on another two circuit examples with respective two and three ports to further demonstrate the validity of the proposed method. A. Lumped Circuit With Nonlinear Component A lumped circuit with nonlinear component [24] is analyzed by using the vector-fitting rational function approximation (VFM) and -parameter based macromodel synthesis approach. The structure of this circuit is shown in Fig. 2. The CMOS inverter used in Fig. 2 is realized with two MOSFET transistors, as indicated in Fig. 3. By definition, the scattering matrix of a multiport network with ports is
Fig. 12.
Transient waveform: V
Fig. 13.
Three-port microstrip circuit.
.
(32) where is scattering matrix, and and are vectors, respectively, of incident and outgoing wave variables at the ports. The wave variables and of the circuit in Fig. 2 can be analytically computed using (25). Therefore, the scattering parameters for the lumped two-port subnetwork enclosed in the dotted lines (Fig. 2) can be easily obtained with respect to the reference port impedance of 30 . Rational function approximation of the matrix of the circuit is carried out by vector fitting method. A model with six poles, i.e., two real poles and four complex poles is constructed to represent the original transfer function of the circuit. Fig. 4 shows excellent agreement between the analytical results and the results obtained by the macromodel based on VFM. Subsequently, the transient simulation results obtained by the vector fitting and macromodel synthesis method presented in this paper are showed in Fig. 5. It shows a very good agreement between the results obtained by using the direct SPICE simulation and the method described above. The circuit is excited with a pulse having rise/fall time of 0.5 ns and a pulse width of 5 ns.
B. Uniform Microstrip Circuit A uniform microstrip circuit example [3] as shown in Fig. 6 is simulated to verify the accuracy of the proposed method. The circuit comprises a uniform microstrip and a diode. By using full-wave FDTD simulation, the scattering parameters can be extracted. The parameters of the microstrip obtained from FDTD simulation are approximated by vector fitting method to construct their macromodel. The approximated parameters are compared with the original ones from FDTD simulation (Fig. 7). and high accuracy can be observed. Finally, macromodel synthesis technique converts the microstrip model into time domain. The equivalent circuit representation of the
LI et al.: COUPLED EFFICIENT AND SYSTEMATIC FULL-WAVE TIME-DOMAIN MACROMODELING
Fig. 14.
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Scattering parameters of the three-port circuit.
microstrip subnetwork is compatible with SPICE circuit simulator, which consequently facilitates the signal integrity analysis of the whole system. The transient result of the diode voltage is is applied ( shown in Fig. 8, where a voltage source and MHz). The good agreements between the result of the proposed method and that of the FDTD and convolution method [3] verify the accuracy of the proposed method. C. Simulation of a Microstrip Low-Pass Filter Circuit A circuit with a microstrip low-pass filter is analyzed in this example, where the configuration of the microstrip filter is taken from [13] and repeated here in Fig. 9. First, the microstrip low-pass filter is simulated by FDTD parameters. The detailed parameters method to obtain its of the FDTD simulation are given: the unit cell size (mm) is ; the time step is ps; total grid size is and total simulation time steps are 5000. The dispersive ABC and Gaussian pulse source are used in this FDTD simulation. Good agreements between the FDTD simulation results and the measurements [13] demonstrate the validity of our FDTD code (Fig. 10). However, small discrepancies can be observed between the FDTD simulated results and the measurement results, which are mainly caused by the inability of the FDTD method to accurately model the geometry of the microstrip and the measurement being not de-embedded [13]. If the PML (perfect matched layer) absorbing boundary condition is used, the FDTD method will yield more accurate results but need more CPU time. Twenty poles (two real poles and nine complex conjugate pole pairs) are extracted by vector fitting method to match the parameters of this two-port low-pass filter up to 20 GHz (Fig. 11). Once the partial fraction expression form of the frequency responses is obtained, the equivalent circuit model of the microstrip is ready to be created. A pulse with 0.1 ns rise/fall time is used as the exand a width of 2 ns as shown in Fig. 12 citation source in this computation and the transient waveform at the output port is also plotted in Fig. 12.
Fig. 15. Input voltage V , and the transient output voltages V the port 2 and port 3.
and V
at
The total CPU time taken by the FDTD simulation of this example is about 15 min in comparison to only less than 3 min consumed by using the rational function approximation and transient simulation on a PC. D. Microstrip Circuit With Three Ports A fairly complex circuit with three ports is simulated in this example to show the accuracy and efficiency of the interconnect macromodeling approach presented in this paper. The components of the circuit are shown in Fig. 13, where the dimensions of the microstrip in , , and directions are 20 mm 20 mm 0.5 mm and the width of the microstrip conductor is 0.8 mm. Similarly, the 3-D FDTD method is used to simulate the microstrip circuit and obtain its parameters. The unit cell size mm, in the FDTD simulation is mm; the time step is ps; the total grid size and the total simulation time steps are is 3000. Fig. 14 shows the parameters of the macromodel based on VFM with up to 20 GHz. Again, the vector fitting method can achieve high accuracy. The transient simulation results of the
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overall circuit of Fig. 13 are shown in Fig. 15, where a pulse excitation with 0.1 ns rise/fall time and a width of 2 ns is used. The total CPU time consumed by this example is about 12 min with only 3 min taken by the rational function approximation and the transient simulation on the same PC. It should be pointed out that this amount of time does not include the pre- and post-processing time. VI. CONCLUSION Full-wave FDTD method coupled with macromodeling by the rational function approximation is an accurate and efficient approach to address the hybrid electromagnetic (interconnect part) and circuit problem where the electromagnetic field effects are fully considered and the strength of SPICE circuit simulator is also exploited. The frequency-dependent nature of the interconnect subnetwork is well accounted for by 3-D FDTD simulation using parameters. In particular, the proposed approach in this paper employs the vector fitting method for the rational function approximation, which provides a robust and accurate solution for the analysis of the interconnect subnetwork. The time-domain state-space equations can be conveniently constructed by the macromodel synthesis with the -parameter model, which can be converted to an equivalent SPICE circuit. Therefore, the mixed frequency/time domain problem is thus overcome, which facilitates the signal integrity analysis of a circuit system containing both distributed and nonlinear components. It should be pointed out that the proposed approach in this paper is readily applied to interconnect structures characterized by measurement scattering parameters. ACKNOWLEDGMENT The authors would like to express their sincere thanks to Dr. Y. Wei-Liang, Institute of High Performance Computing, for his technical discussion. REFERENCES [1] R. Achar and M. S. Nakhla, “Simulation of high-speed interconnects,” Proc. IEEE, vol. 89, pp. 693–728, May 2001. [2] A. E. Ruhli and A. C. Cagellaris, “Progress in the methodologies for the electrical modeling of interconnects and electronic packages,” Proc. IEEE, vol. 89, pp. 740–771, May 2001. [3] Q. Chu, F. Chang, Y. Lzu, and O. Wing, “Time-domain mode synthesis of microstrip,” IEEE Microwave Guided Wave Lett., vol. 7, pp. 9–11, Jan. 1997. [4] E. Chiprout and M. Nakhla, Asymptotic Waveform Evaluation and Moment Matching for Interconnect Analysis. Boston, MA: Kluwer, 1993. [5] J. Vlach and K. Singhal, Computer Methods for Circuit Analysis and Design. New York: Van Nostrand, 1983. [6] A. Odabasioglu, M. Celik, and L. T. Pillage, “PRIMA: Passive reducedorder interconnect macromodeling algorithm,” IEEE Trans. ComputerAided Design, vol. 17, pp. 645–654, Aug. 1998. [7] C. W. Ho, A. E. Ruehli, and P. A. Brennan, “The modified nodal approach to network analysis,” IEEE Trans. Circuits Syst., vol. CAS-22, pp. 504–509, June 1975. [8] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech, 2000. [9] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302–307, May 1966. [10] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery, vol. 14, pp. 1052–1061, July 1999.
[11] M. Elzinga, K. L. Virga, and J. L. Prince, “Improved global rational approximation macromodeling algorithm for networks characterized by frequency-sampled data,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1461–1468, Sept. 2000. [12] X. Zhang and K. K. Mei, “Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip discontinuities,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 1775–1787, Dec. 1988. [13] D. M. Sheen, S. M. Ali, M. D. Abouzahra, and J. A. Kong, “Application of the 3-D finite-difference time-domain method to the analysis of planar microstrip circuits,” IEEE Trans. Microwave Theory Tech., vol. 38, pp. 849–857, July 1990. [14] J. E. Schutt-Aine and R. Mittra, “Scattering parameter transient analysis of transmission lines loaded with nonlinear terminations,” IEEE Trans. Microwave Theory Tech., vol. 36, pp. 529–536, Mar. 1988. [15] R. Mittra, W. D. Becker, and P. H. Harms, “A general purpose Maxwell solver for the extraction of equivalent circuits of electronic package components for circuit simulation,” IEEE Trans. Circuits Syst. I, vol. 39, pp. 964–973, Nov. 1992. [16] Z. Bi, K. Wu, C. Wu, and J. Litva, “A dispersive boundary condition for microstrip component analysis using the FDTD method,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 774–777, Apr. 1992. [17] W. Sui, D. A. Christensen, and C. H. Durney, “Extending the two-dimensional FDTD method to hybrid electromagnetic systems with active and passive lumped elements,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 724–730, Apr. 1992. [18] M. Picket-May, A. Taflove, and J. Baron, “FD-TD modeling of digital signal propagation in 3-D circuits with passive and active loads,” IEEE Trans. Microwave Theory Tech., vol. 42, pp. 1514–1523, Aug. 1994. [19] P. Ciampolini, P. Mezzanotte, L. Roselli, and R. Sorrentino, “Accurate and efficient circuit simulation with lumped-element FDTD technique,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 2207–2214, Dec. 1996. [20] V. A. Thomas, M. E. Jones, M. Piket-May, A. Taflove, and E. Harrigan, “The use of SPICE lumped circuits as subgrid models for FDTD analysis,” IEEE Microwave Guided Wave Lett., vol. 4, pp. 141–143, May 1994. [21] S. Lin and E. S. Kuh, “Transient simulation of lossy interconnects based on the recursive convolution formulation,” IEEE Trans. Circuits Syst. I, vol. 39, pp. 879–892, Nov. 1992. [22] W. Pinello, J. Morsey, and A. Cangelaris, “Synthesis of SPICE-compatible broadband electrical models for pins and vias,” in Proc. Electron. Comp. Technol. Conf., 2001. [23] R. Neumayer, F. Haslinger, A. Stelzer, and R. Wiegel, “Synthesis of SPICE-compatible broadband electrical models from n-port scattering parameter data,” in Proc. IEEE Symp. Electromagn. Compat., Aug. 2002, pp. 469–474. [24] R. Achar and M. S. Nakhla, “Efficient transient simulation of embedded subnetworks characterized by S -parameters in the presence of nonlinear elements,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 2356–2363, Dec. 1998. [25] L. M. Silveira, I. M. Elfadel, J. K. White, M. Chilukuri, and K. S. Kundert, “Efficient frequency-domain modeling and circuit simulation of transmission lines,” IEEE Trans. Comp., Packag., Manufact. Technol. B, vol. 17, pp. 505–513, Nov. 1994. [26] T. Watanabe and H. Asai, “Synthesis of time-domain models for interconnects having 3-D structure based on FDTD method,” IEEE Trans. Circuits Syst. II, vol. 47, pp. 302–305, Apr. 2000. , “Efficient synthesis technique of time-domain models for inter[27] connects having 3-D structures based on FDTD method,” in Proc. IEEE Symp. Circuits Syst., July 1999, pp. 266–269. [28] T. Mangold and P. Russer, “Full-wave modeling and automatic equivalent-circuit generation of millimeter-wave planar and multilayer structures,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 851–858, June 1999. [29] C. Ye, E. P. Li, and Y. S. Gan, “Crosstalk and reflection for curvilinear conductors by utilizing a nonuniform transmission line approach,” IEEE Trans. Adv. Packag., vol. 25, pp. 302–306, May 2002. [30] W. Yuan and E. P. Li, “FDTD simulations for hybrid circuits with linear and nonlinear lumped elemtnts,” Microwave Opt. Technol. Lett., vol. 32, pp. 408–412, Mar. 2002. [31] E. X. Liu, E. P. Li, and L. W. Li, “High-speed interconnect simulation using FDTD macromodeling,”, Singapore, IHPC Res. Rep. CEE/030 301, Feb. 2003.
LI et al.: COUPLED EFFICIENT AND SYSTEMATIC FULL-WAVE TIME-DOMAIN MACROMODELING
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Er-Ping Li (M’93–SM’01) received the M.Eng. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1986 and the Ph.D. degree in electrical engineering from Sheffield Hallam University, Sheffield, U.K., in 1992. He worked as a Research Fellow from 1989 to 1990 and then as a Lecturer from 1991 to 1992 at Sheffield Hallam University, U.K. Between 1993 and 1999, he was a Senior Research Fellow and Technical Manager with the Singapore Research Institute and Industry. Since 2000, he has been with the Institute of High Performance Computing, National University of Singapore, where he is currently a Senior Scientist and R&D Manager for the Computational Electromagnetics and Electronics Division. He has published over 60 technical papers in international referred journals and conferences. His research interests include fast and efficient computational electromagnetics, EMC/EMI, high speed electronic modeling, and computational nanotechnology. Dr. Li is a Deputy Chairman for the IEEE EMC Chapter in Singapore.
En-Xiao Liu received the B.Eng. and M.Eng. degrees in energy and power engineering from Xi’an Jiaotong University, Xi’an, China, in 1996 and 1999, respectively, and is currently pursuing the Ph.D. degree in electrical and computer engineering at the National University of Singapore and the Institute of High Performance computing, Singapore. From September 1999 to June 2001, he was with the North China Electrical Power Design Institute, Beijing, China, as an Automation Control Design Engineer. His research interests include computational electromagnetics and high-speed interconnect modeling and simulation.
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Le-Wei Li (S’91–M’92–SM’96) received the B.Sc. degree in physics from Xuzhou Normal University, Xuzhou, China, in 1984, the M.Eng.Sc. degree in electrical engineering from the China Research Institute of Radiowave Propagation (CRIRP), Xinxiang, China, in 1987, and the Ph.D. degree in electrical engineering from Monash University, Melbourne, Australia, in 1992. In 1992, he worked at La Trobe University (jointly with Monash University), Melbourne, Australia, as a Research Fellow. Since 1992, he has been with the Department of Electrical and Computer Engineering, National University of Singapore where he is currently a Professor. Since 1999, he has also been with High Performance Computations on Engineered Systems (HPCES) Programme, Singapore-MIT Alliance (SMA), where he is an SMA Fellow. His current research interests include electromagnetic theory, radio wave propagation and scattering in various media, microwave propagation and scattering in tropical environment, and analysis and design of various antennas. In these areas, he has co-authored the book Spheroidal Wave Functions in Electromagnetic Theory (New York: Wiley, 2001), 35 book chapters, over 190 international refereed journal papers, 25 regional refereed journal papers, and over 200 international conference papers. He is an Editor of the Journal of Electromagnetic Waves and Applications, an Associate Editor of Radio Science, and an Editorial Board Member of Electromagnetics and the Chinese Journal of Radio Science. Dr. Li is an Editorial Board Member of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, a member of The Electromagnetics Academy (based at MIT), and is currently Chairman of IEEE Singapore Section MTT/AP Joint Chapter.
Mook-Seng Leong (M’81–SM’98) received the B.Sc. degree in electrical engineering and the Ph.D. degree in microwave engineering from the University of London, London, U.K., in 1968 and 1971, respectively. He is currently a Professor of electrical and computer engineering at the National University of Singapore. His main research interests include antenna and waveguide boundary-value problems, electromagnetic modeling, and semiconductor characterization using the SRP technique. He is a member of the Editorial Board for Microwave and Optical Technology Letters and Wireless Mobile Communication. Dr. Leong received the Defense Science Organization R&D Award from the DSO National Laboratories, Singapore, in 1996. He is a member of the MITbased Electromagnetic Academy and a Fellow of the Institution of Electrical Engineers, UK.
