Transient Analysis of Three-Phase Non-linear Circuits by means of Homotopy Method G. Acciani, E. Chiarantoni, G. Fornarelli and S. Vergura Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Bari, Italy. {acciani, chiarantoni, fornarelli, vergura}@deemail.poliba.it Abstract-In this paper a class of homotopy functions for transient analysis of non-linear circuits is described. This class of homotopy is based on the hypothesis that the Modified Nodal Analysis is used to solve the circuit. The method is employed for the simulation of a three-phase converter. The results show the reduction of the iteration number to trace the waveforms of the circuital variables preserving the accuracy of numerical results. I. INTRODUCTION The computation of transient analysis of non-linear circuits is yet under investigation [1]-[2]. Particularly the simulation of some circuit classes as such the three-phase converters is very complex because of the number of switching devices. The first issue in circuit simulation is the formulation of Non-linear Differential Algebraic Equations (NDAE), providing the behavior of the interconnected devices constituting the circuit. The most of circuit simulators use the Modified Nodal Analysis (MNA) to write NDAE [3]-[4]. Then the NDAE must be transformed in Non-linear Algebraic Equations (NAE). This is obtained discretizing the time derivatives in the differential equations by means of the integration formulas. The subsequent issue is to solve the NAE at each discretized time point; Newton-Raphson Method (NRM) is usually utilized for this aim [5]. In this algorithm the last solution point is also the starting point to find the new solution, which could be not found if it is not enough close to the starting point. Obviously, in this case how much these two points must be close to reach the convergence depends on the smoothing of the characteristic of the circuital devices. In fact, the NAEs describe a resistive circuit associate with the original one for a fixed time-point of integration, because at each integration step the circuital dynamic elements are represented in the circuital simulators by companion models. If the characteristics of the nonlinear elements are not sufficiently smooth, the NRM can fail to converge. For this reason, the characteristics of the circuital elements play a fundamental role in the possibility to reach the new solution point [5]. There are many different ways to tackle this problem, which could be summarized in two basic approaches: algorithmic optimization and circuital modeling. It is also clear that there is a strong link between the two approaches. If the characteristics of circuital elements are very smooth the NRM always holds the convergence; vice-versa the more the characteristic is sharp, the more the employed solution algorithm is complex. In the general purpose circuit simulators such as Spice, the switches are represented by a simplified model and the efforts to overcome the convergence problems are focused on the switch model. It is represented by a time-variant conductance between two limit values corresponding to ON and OFF switch states. The variation between these two values is given by a differentiable function as smooth as possible with reference to the behavior of the circuital elements. In some cases this trick is not sufficient and the NRM does not solve efficiently the NAEs. These considerations suggest that an appropriate solution method of NAEs must be chosen on the basis of the assumed model for the switching elements. The quasi-newton methods and the homotopy methods are often used in DC operating point calculation [6]-[7]-[8]-[9], when a robust method to solve the NAEs is needed. In fact, under proper conditions, the zero set of some homotopy functions contains a smooth curve that leads to the subsequent time solution. In other words this means that the homotopy methods can find the solution of the NAEs even if the circuital element under test has not a very smoothing characteristic. The transient analysis needs the solution of the nonlinear algebraic equations after the discretization of NDAEs and it can be seen as a sequence of operating point solutions. Therefore, each algorithm which solves efficiently NAEs in DC analysis can be successfully utilized to tackle the transient analysis. As a consequence of these considerations Melville proposes to apply the fixed point homotopy for the steady- state computation in time domain [9], while in [6] it has been proposed an hybrid approach to simulate the switching circuits. In the last case the method is based on transient windows analysis joined by the DC analysis and the fixed point homotopy is used to joint the circuit solutions before and after the commutation. The homotopy function have been used also in [10] to improve the performance of switching circuits. The proposed homotopy functions is based on a specific resistive time-variant switch model under the assumption that the solution equations are obtained by means of the MNA. In this paper a generalization of this method under the same hypotheses is given and the application to the three- phase converter is shown. The paper is organized as follows. Section II explains the proposed class of homotopy functions, whereas the