2016 1 25 2016 1 13
Study on the Pseudo-Transient Analysis Algorithm for Nonlinear Circuit DC Analysis
Hong Yu
Study on the Pseudo-Transient Analysis Algorithm for Nonlinear Circuit DC Analysis
(1) Chapter 1, “1.1.1 SPICE Simulator” from page 5 to page 6
All of them share the same SPICE core.
The first SPICE version is SPICE1 which was largely a derivative of the CANCER program [4]. During the early 1970s, Larry Nagel had worked on under Prof. Ronald Rohrer. CANCER was an acronym for "Computer Cadence Design Systems). The academic spinoffs of SPICE include XSPICE, developed at Georgia Tech, which added mixed analog/digital "code models" for behavioral simulation, and Cider (previously CODECS, from UC Berkeley/Oregon State Univ.) which added semiconductor device simulation [3].
All of them share the same SPICE core.
The first SPICE version is SPICE1 which was largely a derivative of the CANCER program, which was an acronym for "Computer Analysis of Nonlinear Circuits, Excluding Radiation”. During the early 1970s, the United States Department of Defense contracts request to evaluate the hardness of the circuit’s radiation and the capability to it. Therefore many simulators for circuits were developed. Larry Nagel named as “The father of SPICE”, joined CANCER as his master’s project under Prof. Ronald Rohrer as his original advisor. After Prof. Rohrer, left Berkeley, he followed with the super advisor Prof. Don Pederson to star his PHD program. And at that time CANCER was renamed SPICE as version 1 with Prof. Pederson’s insisting that it had been rewritten enough to be put in the public domain with removing the restrictions.
It was really popular for SPICE as version SPICE2 from 1975 [4]. SPICE2 in FORTRAN language, was much-improved in a bigger capacity for circuit elements, equation formulation with modified nodal analysis which avoid nodal analysis limitation, variable timestep transient analysis with either Gear integration or trapezoidal, and an new FORTRAN-based system for memory allocation developed by Ellis Cohen as graduate student.
Thomas Quarles developed SPICE3 with A. Richard Newton in 1989. It was written in language C and used the same syntax for netlist. It was superset of SPICE2, including all of the analysis types and device models of SPICE2 as well as new features such as improved device models, voltage- and current-controlled switches, pole-zero analysis, and a graphical postprocessor for viewing simulation. Today, the latest version is SPICE3f.
SPICE widely spreaded and was used as an early open source program. In academia, in industry, and in commercial products, many other simulation programs for circuits were developed based on it. Its ubiquity became such that "to SPICE a circuit" remains synonymous with circuit simulation. As mentioned before, due to CANCER program, UC Berkeley firstly distributed the SPICE source code. Presently, both commercial and academic offshoots of the program were influent from Berkeley SPICE. As the industry standard, HSPICE owned by Synopsys was early commercial SPICE version. Also PSpice owned by Cadence was one of early commercial one. XSPICE was the representative of academic spinoffs, which was developed at Georgia Tech and included behavioral simulation using mixed digital/analog code models [3].
(2) Chapter 1, “1.1.2 Basic Conception of Nonlinear Circuit Simuation” from page 12 to page 14
The device models in SPICE-like simulators require many parameter values to simulate the circuit design.
Moreover, many devices in a circuit often are defined by the same set of device models parameters. For these reasons, a set of device model parameters is defined on a separate MODEL statement and assigned a unique mode
All sources that are not time-dependent, for example power supplies, are set to their DC value. We can find that there are two types of the DC analysis. Firstly the initial value is from a DC analysis where it may fail without a good initial guess. The second one is during the time analysis where the iteration usually converges rapidly
The device models in SPICE-like simulators require many parameter values to simulate the circuit design.
Furthermore, in a circuit there are usually a set of device models parameters used by many devices. Due to this reason, we defined the same set of device model parameter, which is based on a assigned unique name and a separate statement for MODEL in a circuit netlist. In this way, each statement for device element includes the name, the model name and the nodes to which the device is connected. Additionally, we can specify other optional parameters for each device: geometric factors and initial conditions [38].
Circuit Analysis Types
There are many analysis types in SPICE-like simulators for designers to evaluate their circuits from various viewpoints before tape out [1],[2],[15]-[17].
DC Analysis
The DC operating point of the circuit with inductors shorted and capacitors opened can be obtained by the DC analysis in SPICE-like simulators. Moreover, a DC analysis is a simulation, which is performed automatically before a circuit analysis for transient to calculate the initial conditions for transient, and to calculate the nonlinear devices’ small-signal, linearized models before AC analysis.
In DC analysis, a transfer function for DC small-signal value, output resistance, and input resistance, would also be able to be computed if they are requested. We can also use the DC analysis to generate DC transfer curves, which is a specified independent voltage or current source is stepped over a user-specified range, and the DC output variables are stored for each sequential source value. Therefore, the DC analysis solution, DC operating point, is so important for the circuit simulation.
Small Signal AC Analysis
The small signal AC analysis is to compute AC output as a function of frequency [15]. After the DC operating point of the circuit is computed by the program firstly, small-signal and linearized models are determined for all circuit nonlinear devices. The outcome linearized circuit is then analyzed based on the frequency customer specified range. The flowchart is shown in Fig.1-7 [15].
A transfer function, such as voltage gain, transimpedance, and so forth is usually the AC small signal analysis’s output. It is convenient to set the input to phase zero and unity if there is only an AC input in the circuit, thus output has the same value as the transfer function of the output variable according to the input.
Transient Analysis
Based on the time over a user-specified time interval, the circuit output variables could be considered as a function of time, which is regarded as transient analysis [15]. It is usually used to determine the large signal time domain behavior of circuits containing nonlinear elements. The flowchart is shown in Fig. 1-8 [15]. DC analysis results are automatically regarded as the initial conditions. All sources that are not time-dependent, for example power supplies, are set to their DC value. We can find that there are two types of the DC analysis. Firstly the initial value is from a DC analysis where it may fail without a good initial guess. The second one is during the time analysis where the iteration usually converges rapidly.
(3) Chapter 2 page 32
As discussed in Chapter 1, solving the DC operating point of the nonlinear circuit is a basic and important task. The Newton-Raphson method in the SPICE simulator is employed to do it, which is guaranteed to converge to a solution. However, this guarantee has some conditions [1]:
easily found. This easy circuit is then continuously deformed into the original hard circuit. The analysis is stepped during this deformation with the operating point found at the previous step as the initial condition for the iterations ate the current step.
As discussed in Chapter 1, solving the DC operating point of the nonlinear circuit is a basic and important task. The Newton-Raphson method in the SPICE simulator is employed to do it, whose convergence could be guaranteed under the conditions below [1]:
The nonlinear function must have a solution, at least one;
The function must be continuous;
The algorithm needs the equations’ derivatives;
The initial approximation must be close enough to the solution.
In order to overcome this non-convergence problem, continuation methods are one means of dealing with a major limitation of Newton-Raphson methods [1]-[3].
Continuation methods [2]-[6] provide a means of obtaining a point “close enough” to the solution to allow the Newton-Raphson iteration to succeed. Continuation methods accomplish this by converting the task from the solution of a single problem to the solution of a continuous set of problems. One member of the set of problems is easily computable, and the other members proceed continuously as a function of a single variable to the original problem. By varying this control variable slowly enough, the successive points taken are each close enough to the previous one for the Newton-Raphson iteration converging.
Namely, the thinking of a continuation method here is to firstly find an easily found solution of the circuit which is related to the one being simulated, then to take advantage of the local convergence property [6] of the Newton-Raphson algorithm. Secondly the found easy circuit is deformed to the original circuit continuously. With the initial condition for the iterations ate the current step, the circuit analysis is stepped at the time of this deformation with the operating point found at the previous step.
(4) Chapter 2, “2.4.1 Various Homotopy Methods” from page 43 to page 47
As an effective method to overcome the non-convergence of Newton Raphson method, homotopy methods are regarded as continuous deformation of functions and show the promising in resolving the computational difficulties often encountered in transistor network simulations. Based on the understanding of this basic concept, we introduce homotopy methods as follow.
An effective initial solution algorithm for globally convergent homotopy methods is proposed in [28], where an initial solution is the forward active operation region of bipolar transistors are considered as the start point. In the practical transistor circuits, we may assume that the intrinsic transistor has an extrinsic resistor.
As an effective method to overcome the non-convergence of Newton Raphson method, homotopy methods are regarded as continuous deformation of functions and show the promising in resolving the computational difficulties which are usually come across in simulations of network with transistors. Base on the understanding of this basic concept, we introduce homotopy methods as follow.
A "continuation parameter" is employed into in the circuit nonlinear equations through a continuation method. If the continuation parameter is set to zero, the complex system could be simplified to the one that equations can be solved easily, or the one that solution is already known. Using such a simple solution as the continuation path starting point, the equations which are augmented are able to be deformed continuously. With this various parameter, the equations finally changed to the originally posed difficult problem. In more detail, take an instance, suppose the nonlinear equation to be solved as [34]
Homotopy methods could be converged globally and bifurcated freely if certain coercivity conditions are satisfied with the nonlinear equations; that is to say, they would me made convergent to the solution which is from an arbitrary starting point. With this exploration of the passivity and no-gain properties [27] of the circuit elements, equations of transistor circuit can be displayed to satisfaction for such conditions. It also has been proved that other forms of equations describing transistor circuits also satisfy such conditions. [27]
Here we want to describe some continuation methods which are based on these continuous and uninterrupted characteristics of elements of the nonlinear circuit. Simplicial methods are provided by an alternate method of solving circuit equations [25]. It solves circuit’s elements with piecewise-liners characteristics. And the underlying homotopy methods are very similar with such methods with alternative techniques to follow the path of homotopy.where an insertion of the parameter , and an initial value (random vector) are embedded. At = 0, which means the homotopy path’s starting point, the inserted branches include a voltage source and let the nodal voltages be equal to the elements of the initial value (random vector) . When = 1, the augmented circuit reverts back to the original circuit since the inserted branches get disconnected from the circuit. [34]
2.4.2 Some Algorithms about Homotopy Methods
Choosing a good starting point for the homotopy method is essential to assure fast convergence of the algorithm.
In circuit terms, a good starting point may decide the uniquity of the solution to a nonlinear or linear circuit.
Furthermore, in homotopy methods, choosing the point to star is always important since the homotopy path length and the iterations number for reaching the destination would be greatly influenced by it [16] [39]. It is high priority to choose a starting point with easy to computation and physically relationship to the final state of the circuit [39].
An effective initial solution algorithm for globally convergent homotopy methods is proposed in [28], where an initial solution is the forward active operation region of bipolar transistors are considered as the start point. In the practical transistor circuits, we may assume that the intrinsic transistor has an extrinsic resistor. For such practical
circuits, we give an initial solution directly as the branch voltages v0 to guarantee the transistors work in active region. m to all MOS’s, which guarantees each MOS transistor is in the saturation region so that the algorithm is expected to be efficient.
(5) Chapter 2, Reference page 55
[33] J.Roychowdhury and R. Melville, “Delivering global DC convergence for large mixed-signal circuits via homotopy/continuation methods,” IEEE Trans.Comput.-Aided Des. Integr. Circuits Syst., vol.25, no.1, pp.66-78, Jan. 2006.
[33] J.Roychowdhury and R. Melville, “Delivering global DC convergence for large mixed-signal circuits via homotopy/continuation methods,” IEEE Trans.Comput.-Aided Des. Integr. Circuits Syst., vol.25, no.1, pp.66-78, Jan. 2006.
Electronics Engineering, J. G. Webster, Ed., New York: John Wiley & Sons, Inc., vol. 9, pp. 171-176, 1999.
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