A Cancellation-Free Symbolic Sensitivity Technique Based on Network Determinant Expansion

A new approach to the problem of symbolic circuit analysis of large-scale circuits is presented. The methodology has been implemented in a computer program called SCAPP (symbolic circuit analysis program with partitioning). The method solves the problem by utilizing a hierarchical network approach and the sequence of expressions concept rather than a topological approach and the single expression idea which have dominated symbolic analysis in the past. The result is a linear growth in the number of terms in the symbolic solutions for the network approach versus the exponential growth exhibited by traditional methods. The analysis methodology introduces the concept of the RMNA (reduced modified nodal analysis) matrix. This allows the characterization of symbolic networks in terms of only a small subset of the network variables (external variables) rather than the complete set of variables. The analysis algorithm is most efficient when network partitioning is used. Partitioning results in a reduction in the number of terms in the symbolic solutions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Symbolic circuit analysis using Mathematics A method of symbolic circuit analysis using Mathematica is reported. With the powerful symbolic manipulation capacity of Mathematica, symbolic circuit analysis is significantly simplified. The analysed circuits can be passive or active, with active devices modelled by nullors. Non-trivial circuit analysis examples are presented and the program list is given.

Engineering analysis has undergone a dramatic change in the last 5 years. Computer programs with symbolic analysis, advanced interfaces, and high-level languages have brought advanced mathematics to everyone. Not only have these programs made mathematics more accessible, the programs have also made it easier to communicate the results of our analyses. This presentation will demonstrate several aspect of engineering design: symbolic circuit analysis; inverting analysis into design; simple field computation; complex algebra; documentation and presentation. These new programs significantly multiply an engineer's productivity. >

Parallel to algebraic methods, graphical circuit analysis methods have the advantage of cancellation-free. This paper proposes a graph reduction method for hierarchical symbolic circuit analysis by applying a binary decision diagram (BDD) for data sharing. This method is extended from the Graph-Pair Decision Diagram (GPDD) method which was developed for two-port dependent sources. New graph construction rules for multiple-port dependent sources are introduced, with which large analog circuits can be analyzed hierarchically. The new hierarchical method guarantees the cancellation-free property at each layer of hierarchy. The BDD-based hierarchical analysis method can greatly reduce the analysis complexity of the entire circuit, while the software construction and circuit partition remain easy. The new method is compared to the algebraic hierarchical method based on DDD (Determinant Decision Diagram) which does not have the cancellation-free property. Comparable performance can be achieved with the new method which has the extra cancellation-free property.

A sequential method of analysis of nullor-based active network is presented. The symbolic circuit analysis can be performed easily with the knowledge of properties of circuit elements including nullor elements. The practical circuit examples are given to demonstrate the feasibility of the proposed approach.

Symbolic circuit analysis involves deriving symbolic expressions for performance measures, such as voltage gain, input impedance, and evaluating them to obtain more insight into the behaviour of a circuit. In modern semiconductor technologies, it is more useful to evaluate the symbolic expressions using interval methods in order handle variations in parameter values. We compare the performance of different interval methods in evaluating symbolic expressions. Our experiments show that Generalised Interval Arithmetic is the most efficient method in affne form for our application. However, this method should be modified to suit long chains of computation. Our modification yields tighter interval bounds compared with other interval methods.

Symbolic circuit analysis

A symbolic linear analyzer that integrates the analysis of both lumped and distributed networks is presented in this paper. This is achieved by implementing a consistent set of notations and a unified mathematical framework. The merits of sparse symbolic matrices are utilized to develop a complete symbolic sparse toolbox that includes all basic elements of electrical circuits. A new storage scheme called Row-Indexed Semi-Symetric (RISS) sparse is proposed and implemented for symbolic network matrices. Several existing and newly developed matrix solvers are implemented and their performances are compared. These solvers are used in the proposed symbolic solver to provide a multi-path analysis root that can be optimized for a particular problem. The overall performance of the proposed symbolic analyzer is assessed practically using several application examples. The obtained results are found in agreement with the results obtained from previously reported analyses and thus confirm validity of the adopted formula approximation method. A polynomial fitting shows that the time complexity of a variable size RC network is super-quadratic with the number of network sections. A substantial reduction in the analysis time of a large ladder RC network is demonstrated with the implementation of macromodeling techniques.KeywordsSymbolic analysissymbolic matrix storagelumped and distributed network analysisoptimal computer solution

ABSTRACTCurriculum typically addresses classical electrodynamic field theory and circuit analysis (with resistors, capacitors, and inductors) separately. As such, there exists a gap relating the two. This work provides a strong tie between these two disciplines that is useful for bridging this gap. Prior work establishes a relationship between electrodynamic fields and circuit elements, but application to circuit analysis remains ambiguous because of the path-dependent node voltages formulated. This paper develops a model resulting in path-independent node voltages that are necessary for circuit analysis techniques. The model is based on the electric field and the magnetic vector potential. Both fields have components associated with the circuit elements as well as external or excitation sources. The model is applicable for determining the coupling of excitation electrodynamic fields to transmission lines. This paper provides a rigor and collective work not found elsewhere in electrodynamic or circuit theory literature.

Introduction to Circuit Analysis and Design

This paper demonstrates the use of the genetic algorithm to simplify both the final symbolic transfer functions and the initial equivalent circuits so that meaningful and understandable results can be obtained. The resulting simplified circuits are only as complicated as necessary to give the required accuracy in the results, which will also help in giving a better graphical insight into the circuits under consideration. This highly efficient algorithm has also been integrated into a symbolic circuit analysis package.

This chapter gives a description of the extension of Generalized Parameter Extraction Method (GPEM) for symbolic analysis of large-scale analog circuits containing pathological elements. The brief overview of the parameter extraction approach is included. An algorithm implementing the concept of Higher Order Summative Cofactors (HOSC) for determinants computation of the pathological element-based circuits is proposed. In this chapter, we also present the hierarchical decomposition techniques of upward and downward analysis of electronic circuits by GPEM. The proposed techniques are used in freeware symbolic analyzer CirSym. Several examples are presented to illustrate the advantages of the GPEM applications.

The general framework of a symbolic analysis environment for linear time-invariant circuits is presented. Such a framework consists of two major parts: a simplification at circuit level and a simplifying expression-generation. An overview of the known techniques for both circuit simplification and expression generation is given, and in addition a new nonhierarchical expression generation technique based on the concept of determinant decision diagrams (DDDs) is presented. The DDD-based technique is evaluated on a set of benchmark circuits together with a known fast symbolic analysis algorithm, and turns out to be the most efficient. The DDD-based expression generation algorithm is, therefore, the best solution for nonhierarchical linear symbolic expression generation to the authors' knowledge.

SMT solving is often a major source of cost in a broad range of techniques such as symbolic program analysis. Thus, speeding up SMT solving is still an urgent requirement. A dominant approach, which is known as eager SMT solving, is to reduce a first-order formula to a pure Boolean formula, which is handed to an expensive SAT solver to determine the satisfiability. We observe that the SAT solver can utilize the knowledge in the first-order formula to boost its solving efficiency. Unfortunately, despite much progress, it is still not clear how to make use of the knowledge in an eager SMT solver. This paper addresses the problem by introducing a new and fast method, which utilizes the interval and data-dependence information learned from the first-order formulas. We have implemented the approach as a tool called Trident and evaluated it on three symbolic analyzers (Angr, Qsym, and Pinpoint). The experimental results, based on seven million SMT solving instances generated for thirty real-world software systems, show that Trident significantly reduces the total solving time from 2.9X to 7.9X over three state-of-the-art SMT solvers (Z3, CVC4, and Boolector), without sacrificing the number of solved instances. We also demonstrate that Trident achieves end-to-end speedups for three program analysis clients by 1.9X, 1.6X, and 2.4X, respectively.

Analog integrated circuit designers always use closed-form design equations in design reasoning. However, most of the time, they have to derive all equations by hand. When high-order effects are of interest, manual derivation would become much harder. Although the art of symbolic circuit analysis has been making steady progress, symbolically generating closed-form equations for high-order effect is still a less well-studied subject. This paper proposes a novel symbolic computation method for generating the second-order characterization of differential cells. It is a perturbation approach that takes the advantage of structural symmetry in differential cells, and is formulated as a systematic computation procedure comprised of several intermediate steps, the computation of each step gives a compact partial expression. Examples are provided to illustrate that closed-form common-mode rejection ratio equations in pole-zero form (unknown before) for two classical operational amplifiers can be generated by applying the symbolic perturbation procedure.