Dissertation Conclusions

Homotopy method is thought as a continuous deformation of functions and shows the great ability to work out the computational difficulties, which are often encountered in transistor circuit simulation. Most importantly, its global convergence can be guaranteed in principle. Now many computer-aided design (CAD) researchers have been studying the globally convergent homotopy methods from various viewpoints for many years. These studies include how to construct a homotopy function, how to trace the solution curve of homotopy function and how to set an initial solution. However, the previous studies for homotopy methods are mainly focused on the bipolar transistor circuits. Considering that the MOS/Bi-MOS transistor circuits are quite popular in current analog circuit designs, extending homotopy methods to MOS transistor circuits is important and urgent. Besides, no paper presents the global convergence theorems of homotopy methods for MOS transistor circuits.

Therefore, study on the homotopy methods for finding the dc solutions of MOS transistor circuits is considered as the subject of this dissertation. Two globally convergent and effective homotopy methods for MOS transistor circuits are proposed in this dissertation, respectively.

Firstly, a nonlinear homotopy (NLH) method with two different auxiliary functions (I and II) for MOS transistor circuits is proposed. The detailed auxiliary functions for the two proposed nonlinear homotopy methods and the two initial homotopy circuits are also presented. Most importantly, we have firstly proved the global convergence theorems for the proposed MOS nonlinear homotopy method II. The two proposed nonlinear homotopy method I and II are implemented by the netlist implementation method and source code implementation method, respectively. Numerical examples show that the efficiencies of two proposed MOS nonlinear homotopy methods (I and II) are higher than the conventional MOS ATAN-SH homotopy method (the number of total iterations can be reduced by as large as 63%). Moreover, the proposed MOS nonlinear homotopy method II can even solve the dc solutions of large-scale MOS

analog circuits (the number of MOSFETs > 1000) that both SPICE3 and HSPICE simulators are failed.

Secondly, a MOS Newton fixed-point homotopy (NFPH) method is proposed to enhance the simulation efficiency further. Moreover, the idea of embedding algorithm is firstly introduced to the MOS homotopy methods and two quite efficient baseG and baseD embedding types are proposed and employed. Moreover, the global convergence theorems of the proposed MOS NFPH method are proved. Numerical examples show that the efficiencies to find the dc solutions of MOS circuits by the proposed MOS Newton fixed-point homotopy method with the two embedding types can be largely enhanced (the number of total iterations can be reduced by at least 50%) comparing with the conventional MOS homotopy methods. Moreover, the proposed MOS NFPH method can also efficiently find the dc operating points of some large-scale MOS analog circuits (the number of MOSFETs>1000) which cannot be solved with the SPICE3 and HSPICE simulators. Besides, some large-scale MOS digital circuits are also tested and the proposed MOS NFPH method with the baseD embedding type is also more effective than the MOS NLH homotopy method to find the dc operating points of MOS digital circuits. Moreover, the proposed MOS NFPH method can even effective solve the dc operating points of large-scale MOS digital circuits (the number of MOSFETs> 10000).

The dissertation is organized as follows:

In Chapter 1, the background for circuit simulation is shown. The overview of circuit analysis types and DC analysis algorithms are given. Since finding the dc operating points of nonlinear circuits is so important but difficult task and NR algorithm in the SPICE-like simulators is not globally convergent, various continuation methods including Gmin stepping method, source stepping method, pseudo-transient analysis method and homotopy methods are studied. Homotopy methods are promising for its global convergence can be guaranteed in principle.

Moreover, the previous studies for homotopy methods are mainly focused on the bipolar transistor circuits and the efficiencies of the conventional MOS homotopy

methods are not satisfactory. Therefore, this research is focused on the homotopy methods for MOS transistor DC analysis. At last the organization of this dissertation is described.

In Chapter 2, as preliminaries of this dissertation, some kinds of technique for dc analysis of nonlinear circuits are overviewed. Modified nodal equation and circuit models especial the EKV model for MOS transistor are presented. Some homotopy methods including Newton fixed-point homotopy method for BJT transistor circuits, nonlinear homotopy (NLH) for BJT transistor circuits are showed. Besides, the convergence theorems and proofs of these homotopy methods for BJT transistor circuits are also discussed. Moreover, the conventional MOS ATAN-SH homotopy method is introduced. Furthermore, the algorithms that are used to trace the solution curve of homotopy methods and the criterion that to set the initial solutions are also shown. From the above introductions, the reasons why the MOS homotopy method is selected to be our research topic are detailed presented.

In Chapter 3, a globally convergent nonlinear homotopy method with two different auxiliary functions (I and II) for MOS transistor circuits is proposed. Most previous studies for homotopy methods are mainly focused on the bipolar transistor circuits.

The efficiency of the conventional ATAN-SH MOS homotopy method is not satisfactory due to two homotopy parameters and two computing phases are needed.

The detailed auxiliary functions for the two proposed MOS nonlinear homotopy methods (I and II) and the two initial homotopy circuits are presented, respectively. In the proposed MOS NLH method I, the bodies of the additional MOS diodes (diode-connected MOS transistor) are connected together with the body of the original MOS transistor. Comparing with it, the bodies of the additional MOS diodes (diode-connected MOS transistor) are connected with their own source nodes in the proposed MOS NLH method II, where the conductance between gate node and body node is also added considering the global convergence. In Section 3.3, the global convergence theorems of the proposed MOS NLH method II are proved. The EKV model (in Chapter 2) is employed. Firstly, the passivity of a single MOS transistor is

proved. Secondly, two important theorems are proved to present the proposed MOS NLH method II can satisfy the uniqueness condition and boundary free condition.

Lastly, the global convergence theorem of the proposed MOS NLH method II is proved (the detailed proof is shown in Appendix A). The numerical examples and efficiency comparisons are shown in Section 3.4. Two implementation methods including the netlist implementation method and the source code implementation method are discussed firstly. Five numerical examples are tested and some simulation efficiency indexes, including total iteration number, step number and arc-length are shown. It is clear that the proposed two MOS nonlinear homotopy methods with both of the two implementation methods are more effective than the conventional MOS ATAN-SH homotopy method (the number of total iterations can be reduced by as large as 63%). Besides, the proposed MOS NLH method II can even solve the dc operating points of large-scale MOS circuits (the number of MOSFETs>1000) that both SPICE3 and HSPICE simulators are failed. At last, in Section 3.5 the conclusions of this chapter are presented.

In Chapter 4, for further enhancing the simulation efficiency, an effective and globally convergent Newton fixed-point homotopy method for MOS transistor circuits is proposed. Although the efficiencies of the MOS nonlinear homotopy methods are not bad, the efficiencies are not satisfactory for some large-scale MOS analog circuits. In Section 4.2, the proposed MOS Newton fixed-point homotopy method is shown, where conductances are additional elements. Moreover, the embedding algorithm (baseG and baseD embedding types) for the proposed homotopy method is also proposed, which is the way how to connect conductances to the MOS transistor. The auxiliary function and the initial homotopy circuits with the baseG and baseD embedding types are also shown. In Section 4.3, the global convergence theorems of the proposed MOS Newton fixed-point homotopy method are proved.

Two important theorems are proved to satisfy the uniqueness condition and boundary free condition, respectively. Then the global convergence theorem of the proposed MOS Newton fixed-point homotopy method is proved (the detailed proof is shown in

Appendix B). Numerical examples are shown in Section 4.4. The proposed method with the two embedding types of baseG and based are implemented by the source code implementation method, which is to directly modify the SPICE3 C language source codes. Three simulation efficiency indexes are considered, which are number of total iterations, arc-length and number of steps. Five MOS analog circuit examples and six digital large-scale MOS circuit examples are tested. From the simulation results, the efficiency by the proposed MOS Newton fixed-point homotopy method can be enhanced largely (the number of total iterations can be reduced by at least 50%) for some large scale MOS analog circuits compared with the MOS nonlinear homotopy methods. Moreover, the proposed MOS NFPH method with the baseD embedding type is more effective than that with the baseG embedding type for some MOS analog circuits. Furthermore, the proposed MOS NFPH method with the baseG and baseD embedding types can also find the DC operating points of some large-scale MOS analog circuits (the number of MOSFETs>1000), which cannot be solved with the SPICE3 and even HSPICE simulators. Besides, the proposed MOS NFPH method with the baseD embedding type is also more effective than the MOS NLH homotopy method to find the dc operating points of MOS digital circuits. Moreover, the proposed MOS NFPH method can even effective solve the dc solutions of large-scale MOS digital circuits (the number of MOSFETs> 10000). Then, in Section 4.5, the conclusions of this chapter are summarized.

In Chapter 5, the conclusions of the dissertation are presented.

At last, in the circuit verification research area, finding the dc solutions of the nonlinear circuit is studied in this dissertation. The proposed MOS nonlinear homotopy method and its global convergence theorem and the proposed MOS globally convergent Newton fixed-point homotopy method and the embedding algorithm contribute to the development of the circuit verification technologies as a pioneering work.