2015 10 19 2015 9 15
Study on Homotopy Methods for MOS Nonlinear Circuit DC Analysis
Dan NIU
Study on Homotopy Methods for MOS Nonlinear Circuit DC Analysis
(1) Chapter 1, Introduction section, page 2-3;
The integrated circuit productions are so universal and critical at present, which provide us a high living quality and make the world operate smoothly. From the birth of the integrated circuit by Jack Kilby in 1958, a great development is made till now. Now the integrated circuits contain several hundred millions of components in order to provide various functions and a high performance. These factors force the emergence of the electronic design automation (EDA) technologies to evaluate the integrated circuit designs [1].
Traditionally, the designs of most electronic circuits were designed by hand by an engineer who was armed with a wealth of design charts, tables, and monograms. The designer relies very heavily on his intuition, past experience, and knowledge to make reasonable approximations. Then the design come to the “breadboard stage” [1]. The advent of integrated circuits, however, has greatly changed the picture. Not only are the circuits much larger but the specifications are also much tighter. The paper-and-pencil method is no longer adequate when we consider the required accuracy of the results and the time to complete a design. Breadboard is also of little help because it is impossible to duplicate an integrated circuit with discrete components [2]. It is in such an environment that the digital computer emerges as an important design tool. By the mid-70s, developers are starting to automate the design, and not just the drafting. A computer program is developed to simulate and analyze the circuit [2].
Currently EDA technology can be found everywhere in the automated design flow chart of the integrated circuit, from design to mask generation [2]. Therefore, the integrated circuit designs become more and more computer-dependent. The continuous scaling of semiconductor technology pushes EAD technologies to increase rapidly. At the same time the advanced EDA technologies also benefit the integrated circuit designs to be much more efficient and excellent.
There are many sub-areas in EDA, such as Design and architecture, Behavioral synthesis, Simulation, Floor planning, Mask data preparation, and so on [2]. In this
research, we focused on the Simulation sub-area, in more detail, the DC (Direct Current) analysis of the nonlinear circuit simulation, which is a basic but difficult task in analog/mixed circuit simulation.
In the following part of this chapter, much more detailed backgrounds focused on the circuit simulation are introduced in Section 1.1. Then the purpose of this dissertation is presented in Section 1.2. Finally the dissertation organization is shown.
Recently, integrated circuit technology is developing rapidly. Not only is circuit scale continuously increased to achieve various functions but also feature of element is becoming minimizing. The rapid progress makes the circuits become complex. The complex parasitic effects and the quite a lot of parameters of the device models must be considered [1]. However, traditional paper-and-pencil calculation method relies heavily on designers’ past experiences and knowledge to make some approximations. Thus the practical circuit parameters can’t be calculated just by hand when the accuracy of the results and the time to finish a design is considered [2]. In this case, the EDA (Electronic Design Automation) technologies are proposed to assess the integrated circuit designs and the integrated circuit designs become more and more computer dependent [1].
Currently EDA technology is widely used in the design flow chart, from design to mask generation [2]. It is important and indispensable for modern engineers to check and verify the design of electrical and electronic circuits prior to manufacturing and deployment [1]. Moreover, EDA technology is developing rapidly along with the continuous scaling of semiconductor technology. Some EDA softwares are developed and one of the most widely used EDA software is SPICE, first developed by L. W. Nagel from the University of California, Berkeley. At present, there are many commercial SPICE-based simulators such as HSPICE, Spectre, PSPICE and so on [2].
EDA technology mainly includes behavioral synthesis, design and architecture, simulation, floor planning, and so on [2]. In this dissertation, simulation sub-area is selected. Moreover, in circuit simulation there are many analysis modes for designers to assess their circuits, including DC (Direct Current) analysis, AC small-signal analysis, transient analysis and so on. In this thesis, the DC analysis of the nonlinear circuit simulation is focused. Since DC analysis is not only a first basic check of circuit operation, but also a precondition for further analyses, such as AC small-signal analysis, noise analysis and transient analysis [2].
This chapter is organized as follows. In Section 1.1, more detailed backgrounds on circuit simulation are introduced, followed by the purpose of this dissertation in Section
1.2. At last, the dissertation organization is shown.
(2) Chapter 1, Section 1.1.2, page 7;
It was developed at the integrated circuits group of the Electronics Research Laboratory of the University of California, Berkeley by Larry Nagel with direction from his research advisor, Prof. Donald Pederson [5]-[8]. Today, SPICE is used worldwide as an essential computer-aid for circuit design, which can simulate circuits prior to their fabrication and predict detailed performance.
SPICE evolved from the CANCER program. The first version of SPICE essentially was finished in 1972 [5]. The real popularity of SPICE started with SPICE2 in 1975 [6].
SPICE3 was developed by Thomas Quarles (with A. Richard Newton as advisor) in 1989.
It was written in C and used the same netlist syntax [6]. It was superset of SPICE2.
SPICE3 demonstrates superior convergence characteristics to the SPICE2 algorithms.
Today, the latest version is SPICE3f5.
It was developed by L. W. Nagel in the Electronics Research Laboratory of the University of California, Berkeley, in the early 1970s, under the supervision of Prof. D.
O. Pederson [1] [5]-[8]. At present, SPICE is used widely to simulate circuits before their fabrication and forecast circumstantial performance for circuit design.
SPICE evolved from the CANCER program. The first version of SPICE essentially was finished in 1972 [5]. The real popularity of SPICE started with SPICE2 in 1975 [6]. In 1989, SPICE3 was developed by Thomas Quarles under the supervision of A. Richard Newton. It uses the same netlist syntax, however, it is superset of SPICE2 and has better convergence performances than SPICE2 program [2] [6]. Moreover, it was written using C language. Today, SPICE3f5 is the latest version of the program.
(3) Chapter 2, Section 2.5, page 36;
The solution curve of the homotopy equation can be traced by the BDF (Backward Differentiation Formula) curve-tracing algorithm [2] [6] [26]. In the BDF algorithm, the solution curve is parameterized by its arc-length s, and the following system of differential-algebraic equations
The BDF (Backward Differentiation Formula) curve-tracing algorithm can be used to trace the solution curve of the homotopy equation [2] [6] [26]. In this algorithm, the solution curve can be parameterized by its arc-length s, and the following differential-algebraic equations [6]
(4) Chapter 2, Section 2.6, page 37;
As we introduced in the above sections, we can have a summary about these continuation methods to overcome the non-convergence of Newton-Raphson method.
The source stepping method is the first way brought out to make up the problem of Newton-Raphson method. This works well for switching or oscillatory circuits, and is especially useful for multi-stable circuits, where it helps the network converge on the same operation point under similar conditions. Unfortunately, this technique doesn’t work for all case; in particular, when the input voltage reaches the level necessary to turn on transistors in digital circuits. The solution can change sufficiently rapidly to cause converge failure.
The idea behind the Gmin stepping method is to swamp out the strong nonlinearities presented in transistor models [28]. Currently it is the default method to work when Newton-Raphson method fails in most commercial SPICE-like simulators. It works well for highly nonlinear designs, such as high-gain amplifiers or long inverter chains. Users can simulate this technique by manually altering Gmin values, but the effort involved in managing more than a few steps in prohibitive. However, as mentioned in [29], there is one circuit which did not converge. It is a fifteen stage CMOS ring oscillator. Moreover, any nodes which can have multiple values should first be constrained with an initial setting since Gmin stepping does not ordinarily resolve the ambiguity.
As we presented in the above sections, a summary about how continuation methods to work out the non-convergence problem of Newton-Raphson method can be made.
Firstly, the source stepping method is brought out to deal with the problem of Newton-Raphson method. This method has good performance for oscillatory or switching circuits. Especially for multi-stable circuits under similar conditions, this method can make the circuit converge on the same operating point [7]. However, it doesn’t work for all cases. For example, the solution can change rapidly enough to bring about convergence fail in digital circuits, when the input voltage achieves enough level to turn on transistors.
Secondly, the Gmin stepping method wants to weaken the strong nonlinearities shown in transistor models [28]. At present, in most commercial SPICE-like simulators, it is the acquiescent method to be used when Newton-Raphson method fails. This method performs well for highly nonlinear circuits, such as long inverter chains or high-gain amplifiers. However, some circuits such as a fifteen stage CMOS ring oscillator does not converge [29]. Moreover, for the nodes having multiple values, an initial setting should first be constrained, since Gmin stepping does not ordinarily work out the ambiguity [7].
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