Homotopy methods are known to be effective methods for finding DC operating points of nonlinear circuits with the theoretical guarantee of global convergence. There are several types of homotopy methods; as one of the most efficient methods for solving bipolar transistor circuits, the variable-gain homotopy (VGH) method is well-known. In this paper, we propose an efficient VGH method for solving bipolar and MOS transistor circuits. We also show that the proposed method converges to a stable operating point with high possibility from any initial point. The proposed method is not only globally convergent but also more efficient than the conventional VGH methods. Moreover, it can easily be implemented in SPICE.

Finding DC operating points of nonlinear circuits is an important and difficult task. The Newton-Raphson method adopted in the SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. However, the previous studies are mainly focused on the bipolar transistor circuits. Also the efficiencies of the previous homotopy methods for MOS transistor circuits are not satisfactory. Therefore, finding a more efficient homotopy method for MOS transistor circuits becomes necessary and important. This paper proposes a Newton fixed-point homotopy method for MOS transistor circuits and proposes an embedding algorithm in the implementation as well. Moreover, the global convergence theorems of the proposed Newton fixed-point homotopy method for MOS transistor circuits are also proved. Numerical examples show that the efficiencies for finding DC operating points of MOS transistor circuits by the proposed MOS Newton fixed-point homotopy method with the two embedding types can be largely enhanced (can larger than 50%) comparing with the conventional MOS homotopy methods, especially for some large-scale MOS transistor circuits which can not be easily solved by the SPICE3 and HSPICE simulators.

Finding DC operating points of nonlinear circuits is an important and difficult task. The Newton-Raphson method adopted in the SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. However, most previous studies are mainly focused on the bipolar transistor circuits and no paper presents the global convergence theorems of homotopy methods for MOS transistor circuits. Moreover, due to the improvements and advantages of MOS transistor technologies, extending the homotopy methods to MOS transistor circuits becomes more and more necessary and important. This paper proposes two nonlinear homotopy methods for MOS transistor circuits and proves the global convergence theorems for the proposed MOS nonlinear homotopy method II. Numerical examples show that both of the two proposed homotopy methods for MOS transistor circuits are more effective for finding DC operating points than the conventional MOS homotopy method and they are also capable of finding DC operating points for large-scale circuits.

As a powerful computational test for nonexistence of a DC solution of a nonlinear circuit, the LP test is well-known. This test is useful for finding all solutions of nonlinear circuits; it is also useful for verifying the nonexistence of a DC operating point in a given region where operating points should not exist. However, the LP test has not been widely used in practical circuit simulation because the programming is not easy for non-experts or beginners. In this paper, we propose a new LP test that can be easily implemented on SPICE without programming. The proposed test is useful because we can easily check the nonexistence of a solution using SPICE only.

Recently, an efficient homotopy method termed the variable gain Newton homotopy (VGNH) method has been proposed for finding DC operating points of nonlinear circuits. This method is not only very efficient but also globally convergent for any initial point. However, the programming of sophisticated homotopy methods is often difficult for non-experts or beginners. In this paper, we propose an effective method for implementing the VGNH method on SPICE. By this method, we can implement a "sophisticated VGNH method with various efficient techniques" "easily" "without programming," "although we do not know the homotopy method well."

Recently, efficient algorithms have been proposed for finding all characteristic curves of one-port piecewise-linear (PWL) resistive circuits. Using these algorithms, a middle scale one-port circuit can be represented by a PWL resistor that is neither voltage nor current controlled. By modeling often used one-port subcircuits by such resistors (macromodels), large scale circuits can be analyzed efficiently. In this paper, an efficient method is proposed for finding DC operating points of nonlinear circuits containing such neither voltage nor current controlled resistors using the SPICE-oriented approach. The proposed method can be easily implemented on SPICE without programming.

We consider oscillators consisting of a reactance circuit and a negative resistor. They may happen to have multi-mode oscillations around the anti-resonant frequencies of the reactance circuit. This kind of oscillators can be easily synthesized by setting the resonant and anti-resonant frequencies of the reactance circuits. However, it is not easy to analyze the oscillation phenomena, because they have multiple oscillations whose oscillations depend on the initial guesses. In this paper, we propose a Spice-oriented solution algorithm combining the harmonic balance method with Newton homotopy method that can find out the multiple solutions on the homotopy paths. In our analysis, the determining equations from the harmonic balance method are given by modified equivalent circuit models of "DC," "Cosine" and "Sine" circuits. The modified circuits can be solved by a simulator STC (solution curve tracing circuit), where the multiple oscillations are found by the transient analysis of Spice. Thus, we need not to derive the troublesome circuit equations, nor the mathematical transformations to get the determining equations. It makes the solution algorithms much simpler.

Finding DC operating points of nonlinear circuits is an important problem in circuit simulation. The Newton-Raphson method employed in SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. There are several types of homotopy methods, one of which succeeded in solving bipolar analog circuits with more than 20000 elements with the theoretical guarantee of global convergence. In this paper, we propose an improved version of the homotopy method that can find DC operating points of practical nonlinear circuits smoothly and efficiently. Numerical examples show the effectiveness of the proposed method.

Finding DC operating points of transistor circuits is a very important and difficult task. The Newton-Raphson method employed in SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. For efficiency of homotopy methods, it is important to construct an appropriate homotopy function. In conventional homotopy methods, linear auxiliary functions have been commonly used. In this paper, a homotopy method for solving transistor circuits using a nonlinear auxiliary function is proposed. The proposed method utilizes the nonlinear function closely related to circuit equations to be solved, so that it efficiently finds DC operating points of practical transistor circuits. Numerical examples show that the proposed method is several times more efficient than conventional three homotopy methods.

Path following circuits (PFC's) are circuits for solving nonlinear problems on the circuit simulator SPICE. In the method of PFC's, formulas of numerical methods are described by circuits, which are solved by SPICE. Using PFC's, numerical analysis without programming is possible, and various techniques implemented in SPICE will make the numerical analysis very efficient. In this paper, we apply the PFC's of the homotopy method to various nonlinear problems (excluding circuit analysis) where the homotopy method is proven to be globally convergent; namely, we apply the method to fixed-point problems, linear programming problems, and nonlinear programming problems. This approach may give a new possibility to the fields of applied mathematics and operations research. Moreover, this approach makes SPICE applicable to a broader class of scientific problems.

Finding DC operating points of transistor circuits is an important and difficult task. The Newton-Raphson method adopted in SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. For efficiency of globally convergent homotopy methods, it is important to give an appropriate initial solution as a starting point. However, there are few studies concerning such initial solution algorithms. In this paper, initial solution problems in homotopy methods are discussed, and an effective initial solution algorithm is proposed for globally convergent homotopy methods, which finds DC operating points of transistor circuits efficiently. Numerical examples using practical transistor circuits show the effectiveness of the proposed algorithm.

Finding DC operating points of nonlinear circuits is an important and difficult task. The Newton-Raphson method adopted in the SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. For the global convergence of homotopy methods, it is a necessary condition that a given initial solution is the unique solution to the homotopy equation. According to the conventional criterion, such an initial solution, however, is restricted in some very narrow region. In this paper, considering the circuit interpretation of homotopy equations, we prove theorems on the uniqueness of an initial solution for globally convergent homotopy methods. These theorems give new criteria extending the region wherein any desired initial solution satisfies the uniqueness condition.

Finding DC operating-points of nonlinear circuits is an important and difficult task. The Newton-Raphson method employed in the SPICE-like simulators often fails to converge to a solution. To overcome this convergence problem, homotopy methods have been studied from various viewpoints. The fixed-point homotopy method is one of the excellent methods. However, from the viewpoint of implementation, it is important to study it further so that the method can be easily and widely used by many circuit designers. This paper presents a practical method to implement the fixed-point homotopy method. A special circuit called the solution-tracing circuit for the fixed-point homotopy method is proposed. By using this circuit, the solution curves of homotopy equations can be traced by performing the SPICE transient analysis. Therefore, no modification to the existing programs is necessary. Moreover, it is proved that the proposed method is globally convergent. Numerical examples show that the proposed technique is effective and can be easily implemented. By the proposed technique, many SPICE users can easily implement the fixed-point homotopy method.

This paper presents the several bifurcation phenomena generated in nonlinear three-phase circuit with symmetry. The circuit consists of delta-connected nonlinear inductors, capacitors and three-phase symmetrical voltage sources. Particular attention is paid to the subharmonic oscillations of order 1/2. We analyze the bifurcations of the oscillations from both theoretical and experimental points. As a tool of analysis, we use the homotopy method. Additionally, by comparing with single-phase and single-phase-like circuits, the special feature of the three-phase circuit is revealed.

Related with accuracy, computational complexity and so on, quality of computing for the so-called homotopy method has been discussed recently. In this paper, we shall propose an estimation method with interval analysis of region in which unique solution path of the homotopy equation is guaranteed to exist, when it is applied to a certain class of uniquely solvable nonlinear equations. By the estimation, we can estimate the region a posteriori, and estimate a priori an upper bound of the region.

In circuit simulation, dc operating points of nonlinear circuits are obtained by solving circuit equations. In this paper, we consider "hybrid equations" as the circuit equations and discuss the stability of dc operating points obtained by solving hybrid equations. We give a simple criterion for identifying unstable operating points from the information of the hybrid equations. We also give a useful criterion for identifying initial points from which homotopy methods coverge to stable operating points with high possibility. These results are derived from the theory of dc operating point stability developed by M. M. Green and A. N. Willson, Jr.

This paper presents the several bifurcation phenomena of harmonic oscillations occurred in nonlinear three-phase circuit. The circuit consists of delta-connected nonlinear inductors, capacitors and three-phase symmetrical voltage sources. We analyze the bifurcations of the oscillations by the homotopy method. Additionally, we confirm the bifurcation phenomena by real experiments. Furthermore, we reveal the effect of nonlinear couplings of inductors by the comparison of harmonic oscillations in a single-phase circuit.

In this paper a priori estimation method is presented for calculating solution of convex optimization problems (COP) with some equality and/or inequality constraints by so-called Newton type homotopy method. The homotopy method is known as an efficient algorithm which can always calculate solution of nonlinear equations under a certain mild condition. Although, in general, it is difficult to estimate a priori computational complexity of calculating solution by the homotopy method. In the presented papers, a sufficient condition is considered for linear homotopy, under which an upper bound of the complexity can be estimated a priori. For the condition it is seen that Urabe type convergence theorem plays an important role. In this paper, by introducing the results, it is shown that under a certain condition a global minimum of COP can be always calculated, and that computational complexity of the calculation can be a priori estimated. Suitability of the estimation for analysing COP is also discussed.

Tracing solution curves of nonlinear equations is an important problem in circuit simulation. In this paper, simple techniques are proposed for improving the computational efficiency of the spherical method, which is a method for tracing solution curves. These techniques are very effective in circuit simulation where solution curves often turn very rapidly. Moreover, they can be easily performed with little computational effort.

An efficient algorithm is presented for solving nonlinear resistive networks. In this algorithm, the techniques of the piecewise-linear homotopy method are introduced to the Katzenelson algorithm, which is known to be globally convergent for a broad class of piecewise-linear resistive networks. The proposed algorithm has the following advantages over the original Katzenelson algorithm. First, it can be applied directly to nonlinear (not piecewise-linear) network equations. Secondly, it can find the accurate solutions of the nonlinear network equations with quadratic convergence. Therefore, accurate solutions can be computed efficiently without the piecewise-linear modeling process. The proposed algorithm is practically more advantageous than the piecewise-linear homotopy method because it is based on the Katzenelson algorithm that is very popular in circuit simulation and has been implemented on several circuit simulators.