The alternating direction implicit (ADI) method is proposed for low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. The low-rank solution is expressed by Cholesky factor that is similar to that of Cholesky factorization for linear system of equations. The Cholesky factor is represented in a real form so that it is useful for balanced truncation of sparsely connected RLC networks. Moreover, we show how to determine the shift parameters which are required for the ADI iterations, where Krylov subspace method is used for finding the shift parameters that reduce the residual error quickly. In the illustrative examples, we confirm that the real Cholesky factor certainly provides low-rank solution of projected generalized continuous-time algebraic Lyapunov equations. Effectiveness of the shift parameters determined by Krylov subspace method is also demonstrated.

The cooperative and competitive network suitable for circuit realization is presented, based on the network proposed by Amari and Arbib. To ensure WTA process, the output function of the original network is replaced with the piecewise linear function and supplying the inputs as pulse waveforms is obtained. In the SPICE simulations, it is confirmed that the network constructed by operational amplifiers attains WTA process, even if the scale of the network becomes large.

An efficient reciprocity and passivity preserving balanced truncation for RLC networks is presented in this paper. Reciprocity and passivity are fundamental principles of linear passive networks. Hence, reduction with preservation of reciprocity and passivity is necessary to simulate behavior of the circuits including the RLC networks accurately and stably. Moreover, the proposed method is more efficient than the previous balanced truncation methods, because sparsity patterns of the coefficient matrices for the circuit equations of the RLC networks are fully available. In the illustrative examples, we will show that the proposed method is compatible with PRIMA, which is known as a general reduction method of RLC networks, in efficiency and used memory, and is more accurate at high frequencies than PRIMA.

This paper presents the selective orthogonal matrix least-squares (SOM-LS) method for representing a multiport network characterized by sampled data with the rational matrix, improving the previous works, and providing new criteria. Recently, it is needed in a circuit design to evaluate physical effects of interconnects and package, and the evaluation is done by numerical electromagnetic analysis or measurement by network analyzer. Here, the SOM-LS method with the criteria will play an important role for generating the macromodels of interconnects and package in circuit simulation level. The accuracy of the macromodels is predictable and controllable, that is, the SOM-LS method fits the rational matrix to the sampled data, selecting the dominant poles of the rational matrix. In examples, simple PCB models are analyzed, where the rational matrices are described by Verilog-A, and some simulations are carried out on a commercial circuit simulator.

There are many kinds of transmission lines such as uniform, nonuniform and nonlinear ones terminated by linear and/or nonlinear subnetworks. The nonuniform transmission lines are crucial in integrated circuits and printed circuit boards, because these circuits have complex geometries and layout between the multi layers, and most of the transmission lines possess nonuniform characteristics. On the other hand, the nonlinear transmission line have been focused in the fields of communication and instrumentation. Here, we present a new numerical method for analyzing nonuniform and nonlinear transmission lines with linear and/or nonlinear terminations. The waveforms at any points along the lines are described by the Fourier expansions. The partial differential equations representing the circuit are transformed into a set of ordinary differential equations at each frequency component, where for nonlinear transmission line, the perturbation technique is applied. The method is efficiently applied to weakly nonlinear transmission line. The nonuniform transmission lines terminated by a nonlinear subnetwork are analyzed by hybrid frequency-domain method. The stability for stiff circuit is improved by introducing compensation element. The efficiency of our method is illustrated by some examples.

A fast time-domain simulation technique of plane circuits via two-layer Cellular Neural Network (CNN)-based modeling, which is necessary for power/signal integrity evaluation in VLSIs, printed circuit boards, and packages, is presented. Using the new notation expressed by the two-layer CNN, 1,553 times faster simulation is achieved, compared with Berkeley SPICE (ngspice). In CNN community, CNNs are generally simulated by explicit numerical integration such as the forward Euler and Runge-Kutta methods. However, since the two-layer CNN is a stiff circuit, we cannot analyze it by using an explicit numerical integration method. Hence, to analyze the two-layer CNN and reduce the computational cost, the leapfrog method is introduced. This procedure would open an application of CNN to electronic design automation area.

This paper presents generating stable and sparse reluctance/inductance matrix from the inductance matrix which is extracted under insufficient discretization. To generate the sparse reluctance matrix with guaranteed stability, the original matrix has to be (strictly) diagonally dominant M matrix. Hence, the repeated inductance extractions with a smaller grid size are necessary in order to obtain the well-defined matrix. Alternatively, this paper provides some ideas for generating the sparse reluctance matrix, even if the extracted reluctance matrix is not diagonally dominant M matrix. These ease the extraction tasks greatly. Furthermore, the sparse inductance matrix is also generated by using double inverse methods. Since reluctance components are not still supported in SPICE-like simulators, generating the sparse inductance matrix is more useful than the sparse reluctance one.

An efficient balanced truncation for RC and RLC networks is presented in this paper. To accelerate the balanced truncation, sparse structures of original networks are considered. As a result, Lyapunov equations, the solutions of which are necessary for making the transformation matrices, are efficiently solved, and the reduced order models are efficiently obtained. It is proven that reciprocity of original networks is preserved while applying the proposed method. Passivity of the reduced RC networks is also guaranteed. In the illustrative examples, we will show that the proposed method is compatible with PRIMA in efficiency and is more accurate than PRIMA.

This paper describes a sparse realization of passive reduced-order interconnect models via PRIMA to provide the SPICE compatible models. It is demonstrated that, if the SPICE models are directly realized so that the reduced-order equations obtained via PRIMA are stamped into the MNA matrix, the simulations of networks containing the macromodels become computationally inefficient when size of the reduced-order equations is relatively large. This is due to dense coefficient matrices of the reduced-order equations resulting from congruent transformations in PRIMA. To overcome this disadvantage, we propose a sparse realization of the reduced-order models. Since the expression is equivalent to the reduced-order equations, the passivity of the SPICE models generated is also guaranteed. Computational efficiency on SPICE is demonstrated in performing the transient analysis of circuits containing the proposed macromodels.

This paper discusses pulse responses of multi-conductor transmission lines terminated by linear and nonlinear subnetworks. At first step, the circuit is partitioned into a linear transmission lines and nonlinear subnetworks by the substitution voltage sources. Then, the linear subnetworks are solved by a well-known phasor technique, and the nonlinear subnetworks by a numerical integration technique. The variational value at each iteration is calculated by a frequency domain relaxation method to the associated linearized time-invariant sensitivity circuit. Although the algorithm can be efficiently applied to weakly nonlinear circuits, the convergence ratio for stiff nonlinear circuits becomes very small. Hence, we recommend to introduce a compensation element which plays very important role to weaken the nonlinearity. Thus, our algorithm is very simple and can be efficiently applied to wide classes of nonlinear circuits.

This paper describes the matrix order reduction method by the nodal analysis formulation and the application of relaxation-based simulation technique to interconnect and plane networks. First, the characteristics of the power/ground plane networks are considered. Next, the formulation of the plane network by nodal analysis (NA) method is suggested. Furthermore, application and estimation results of the relaxation-based numerical analyses are shown. Finally, it is confirmed that the relaxation-based methods improved by the suggested formulation are much more efficient than the conventional direct-based methods.

Nonuniform transmission lines are crucial in integrated circuits and printed circuit boards, because these circuits have complex geometries and layout between the multi layers, and most of the transmission lines possess nonuniform characteristics. In this article, an efficient numerical method for analyzing nonuniform transmission lines has been presented by using the Chebyshev expansion method and moment techniques. Efficiency on computational cost is demonstrated by numerical example.

Fast simulation techniques of large scale RLC networks with nonlinear devices are presented. Generally, when scale of nonlinear part in a circuit is much less than the linear part, matrix or circuit partitioning approach is known to be efficient. In this paper, these partitioning techniques are used for the conventional transient analysis using an implicit numerical integration and the circuit-based finite-difference time-domain (FDTD) method, whose efficiency and accuracy are evaluated developing a prototype simulator. It is confirmed that the matrix and circuit partitioning approaches do not degrade accuracy of the transient simulations that is compatible to SPICE, and that the circuit partitioning approach is superior to the matrix one in efficiency. Moreover, it is demonstrated that the circuit-based FDTD method can be efficiently combined with the matrix or circuit partitioning approach, compared with the transient analysis using an implicit numerical integration.

This paper presents a fast transient simulation method for power distribution networks (PDNs) of the PCB/Package. Because these PDNs are modeled as large-scale linear circuits consisting of a large number of RLC elements, it takes large costs to solve by conventional circuit simulators, such as SPICE. Our simulation method is based on the leapfrog algorithm, and can solve RLC circuits of PDNs faster than SPICE. Actual PDNs have frequency-dependent dispersions such as the skin-effect of conductors and the dielectric loss. To model these dispersions, more number of RLC elements are required, and circuit structures of these dispersion models are hard to solve by using the leapfrog algorithm. This paper shows that the circuit structures of dispersion models can be converted to suitable structures for the leapfrog algorithm. Further, in order to reduce the simulation time, our proposed method exploits parallel computation techniques. Numerical results show that our proposed method using single processing element (PE) enables a speedup of 20-100 times and 10 times compared to HSPICE and INDUCTWISE with the same level of accuracy, respectively. In a large-scale example with frequency-dependent dispersions, our method achieves over 94% parallel efficiency with 5PEs.

An equivalent circuit of Yee's cells is proposed for mixed electromagnetic and circuit simulations. Using the equivalent circuit, a mixed electromagnetic and circuit simulator can be developed, in which the electromagnetic field and circuit responses are simultaneously analyzed. Representing the electromagnetic system as a circuit, active and passive device models in a circuit simulator can be used for the mixed simulations without any modifications. Hence, the propose method is very useful for designing various electronic systems. To evaluate the mixed simulations with the equivalent circuit, two implementations with shared or distributed memory computer system are presented. In the numerical examples, we evaluate the performances of the prototype simulators to demonstrate the effectiveness.

In this paper, we show the generalized method of the time-domain circuit simulation based on LIM. Our method is applicable to any structure of circuits by combination with the SPICE-like method. In order to show the validity and efficiency of our method, an example circuit is simulated and the proposed method is compared with the conventional ones.

In this paper, we propose a new algorithm for calculating the exact poles of the admittance matrix of RLCG interconnects. After choosing dominant poles and corresponding residues, each element of the exact admittance matrix is approximated by partial fraction. A procedure to obtain the residues that guarantee the passivity is also provided, based on experimental studies. In the procedure the residues are calculated by using the least squares method so that the partial fraction matches each element of the exact admittance matrix in the frequency-domain. From the partial fraction representation, the asymptotic equivalent circuit models which can be easily simulated with SPICE are synthesized. It is shown that an efficient model-order reduction is possible for short-length interconnects.

The interconnect analysis of on- and off-chips is very important in the design of high-speed signal processing, digital communication, and microwave electronic systems. When the interconnects are characterized by sampled data via electromagnetic analysis, the circuit-level simulation of the network requires rational approximation of the sampled data. Since the frequency band of the sampled data is more than 10 GHz, the rational function must fit into it at many frequency points. The rational function is approximated using the orthogonal least-squares method. With an increase in the number of the fitting data, the least-squares method suffers from a singularity problem. To avoid this, the sampled data are hierarchically approximated in this paper. Moreover, to reduce the computational cost of the circuit-level simulation, the parameter matrix of the interconnects is approximated by a rational matrix with one common denominator polynomial, and the selective orthogonalization procedure is presented.

Analysis of frequency-dependent lossy transmission lines is very important for designing the high-speed VLSI, MCM and PCB. The frequency-dependent parameters are always obtained as tabulated data. In this paper, a new curve fitting technique of the tabulated data for the moment matching technique in the interconnect analysis is presented. This method based on Chebyshev interpolation enhances the efficiency of the moment matching technique.

The passive and sparse reduced-order modeling of a RLC network is presented, where eigenvalues and eigenvectors of the original network are used, and thus the obtained macromodel is more accurate than that provided by the Krylov subspace methods or TBR procedures for a class of circuits. Furthermore, the proposed method is applied to low pass filtering of a reduced-order model produced by these methods without breaking the passivity condition. Therefore, the proposed eigenspace method is not only a reduced-order macromodeling method, but also is embedded in other methods enhancing their performances.