Automatic circuit simplification for meaningful symbolic analysis using the genetic algorithm

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. >

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.

The generalization of Bode’s sensitivity analysis technique for all types of the transfer functions and circuit elements is presented in the paper. The proposed formulae for first- and second-order symbolic sensitivity calculation provide the compact size of obtained expression and have the advantages of cancellation-free sum-of-product terms and matrix-free computation. This is achieved by means of the concept of high order summative cofactors and the generalized parameter extraction method. The proposed technique is implemented in symbolic circuit analysis program Cirsym. Illustrative example on symbolic sensitivity circuit analysis and comparison of the presented technique with the transimpedance method and the method based on the modified Coates flow-graph are given.

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 vector analysis in plasma physics

A symbolic analyzer based on the use of Coates' flow-graphs is proposed. It allows full automatic generation of analog transfer functions. The program, which is implemented using MATLAB software, automatically computes symbolic transfer functions. It generates the symbolic network functions in a cancellation-free form. Besides, the numerator and denominator of the transfer function are generated separately without any sorting process.

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.

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.

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.

SSCNAP is a program developed on a microcomputer for symbolic analysis of ideal k -phase switched capacitor circuits. Symbolic transfer functions and sensitivity functions can be obtained by using SSCNAP. It can also calculate the poles and their sensitivities in an SC circuit. In the paper, an efficient computer method for symbolic analysis of linear SC circuits is described. No topological or duty-cycle constraints have been imposed on the circuits. An example is included to illustrate the capabilities of the SSCNAP program.

A symbolic approach to the frequency domain characterization of dc-dc converters is presented. The symbolic analysis is based on a sampled-data technique derived from a state variable approach, exploiting the numerical values of the state variable and output samples obtained by s-domain symbolic network functions of the circuit. A symbolic program developed by the authors is presented. It is shown that the computer symbolic analysis can be easily applied also to the averaged models of DC-DC converters. Possible developments to the frequency domain analysis of the whole DC-DC converter including the control system are also proposed.

This article presents a signal flow graph representation for a microwave MESFET equivalent circuit. The model has been found to provide accurate fits to NE71000 MESFET chip scattering parameters; typical results are shown. Four symbolic transfer functions are presented which allow device scattering parameters to be computed. The symbolic transfer functions can be differentiated in order to determine component or device behavior sensitivities; sample formula are presented. The principle advantage of the symbolic transfer function approach is that when used as the function evaluation engine in a numerical optimization scheme computational overhead is small. In addition, having analytical derivatives available allows for rapid gradient and sensitivity evaluations that are not prone to numerical approximation inaccuracies.

This study examines the function and significance of symbolism in three selected poems by William Wordsworth: "I Wandered Lonely as a Cloud," "My Heart Leaps Up," and "The World Is Too Much with Us." Through qualitative thematic analysis and quantitative descriptive statistics, the research identifies seventeen distinct symbols (eleven public and six private) and analyzes their contribution to thematic development and Romantic philosophy. Findings reveal that Wordsworth employs natural imagery—clouds, daffodils, rainbows, and mythological references—to symbolize emotional states, philosophical reflection, and moral values. These symbols function on multiple levels, expressing Wordsworth's belief that nature serves as a moral guide, spiritual companion, and mirror of human consciousness. The analysis demonstrates how Wordsworth's symbolic language bridges the relationship between humanity and the natural world, exemplifying core Romantic ideals. This study contributes to deeper understanding of Wordsworth's poetic techniques and provides valuable insights for literary scholars and English language educators.

Animation model of Krško nuclear power plant for RELAP5 calculations