Doctoral Dissertation

A contribution to algorithms in computer-aided symbolic analysis of linear electric circuits and systems

University of Belgrade, School of Electrical Engineering, Belgrade, Serbia, Yugoslavia, 1996

Advisor: Dr Branimir D. RELJIN, full professor

Summary

Matrix approach to the analysis of linear time-invariant electric circuits and systems reveals a unique final form of the circuit/system equations - the matrix system of linear equations. What is specific for certain circuits and systems is the algorithm of forming the required matrices and matrix equations, and identification of the circuit/system variables. This research presents a new and unique concept and formulation of general symbolic analysis of linear electric circuits and systems (CAS - Circuit And/or System).

The most general concept is proposed for

• representing linear circuits and systems,
• representing the symbolic analysis knowledge, and
• formulating the required matrices to solve a circuit or system.

The package SALECAS (Symbolic Analysis of Linear Circuits and Systems) has been developed in Mathematica to implement this general concept of symbolic analysis.

At the highest level of abstraction, from the viewpoint of the automated computer-aided symbolic analysis, a linear circuit and system have identical representations in the form of a list

system = {"class", "label", components, knowledge}.

The elements of the list are

• class - the CAS sort/category,
• label - the CAS individual name unique for each instance of the CAS,
• components - the CAS constitutive elements or building blocks,
• knowledge - entities, rules and procedures for the CAS analysis.

The following major CAS classes can be identified:

• LEC - linear time-invariant electric circuits,
• MWC - linear time-invariant microwave circuits,
• CLS - continuos-time linear time-invariant systems,
• DLS - discrete-time linear time-invariant systems, and
• PS - linearized power systems.

The CAS components represent ideal circuit elements, microwave network ports, functional blocks like integrators and differentiators, digital filter multipliers and delays, equivalent networks for modeling electronic devices, multiport networks whose matrix parameters are known, etc. A CAS component is described by a list:

component = {"type", "name", connection, parameters, energy}.

The list items are

• type - the component type/sort,
• name - the component's unique name,
• connection - the component's terminals or ports whose order of appearance includes the reference direction assumptions of associated variables,
• parameters - the component's parameters,
• energy - the initial conditions for capacitors, inductors and integrators.

The required knowledge to analyze a CAS is a list of the form

knowledge = {library, variables, matrices, rules, procedures, presentation}.

The corresponding items are

• library - the list of the component types and their definitions,
• variables - voltages, currents, signals, average powers, etc. appearing in the analysis and requested as output quantities,
• matrices - matrices involved in the analysis,
• rules - rules to check the system description for syntax/semantics, to count variables, to determine the order of the system, to initialize the matrices, rules for reserved symbols (the complex frequencies s and z are special symbols), etc.,
• procedures - are sets of actions and algorithms to formulate CAS equations, to reduce/compact this system of equation to solve it, to find the response or transfer functions, to approximate the result, etc.,
• presentation - conventions of post-processing the result of the analysis.

A CAS library is a list of types:

library = {type1, type2, type3, ...}.

Each type is a list of the form

type = {symbol, terminals, definition, stamp},

whose items are

• symbol - a graphical icon for schematic entry or displaying of a component,
• terminals - a list of terminals of a component,
• definition - constitutive equations of a component,
• stamp - rules for automated modification of the CAS matrices according to the component description.

Generally, symbolic analysis can be viewed as a scan of the CAS net-list, and systematic update of the CAS matrices according to the components' stamps.

Following the previously exposed system of abstractions, several symbolic simulators have been developed to analyze various classes of CASs. They constitute the package SALECAS. The simulators are:

• SALEC - symbolic analysis of lumped linear time-invariant electric circuits,
• SALTIS - symbolic analysis of linear stationary continuous-time dynamic systems,
• SALDTIS - symbolic analysis of linear stationary discrete-time systems (digital filters),
• SALF - symbolic analysis of DC load flow in electric power systems.

Except the simulators, a set of specialized functions has been developed within SALECAS:

• CGC - adding auxiliary voltage sources to help SALEC solve a circuit whose graph is disconnected,
• SPAR - find scattering parameters of a circuit symbolically by calling SALEC,
• OPTPARAM - find one parameter symbolically to achieve the prescribed steady-state response in dynamic systems by calling SALTIS,
• BUMP - automated symbolic synthesis of one class of second-order filters and amplitude equalizers calling SALEC,
• VQNR - symbolic derivation of variance of quantization noise, due to rounding the output of a multiplier, in digital filters by calling SALDTIS. A new model of the multiplier is proposed in SALDTIS for efficient symbolic derivation of noise transfer functions.

A novel framework for general symbolic analysis of linear time-invariant circuits and systems is presented. A detailed decomposition of the problem is offered. Linear circuits and systems are viewed as lists of items representing their identification, components and symbolic analysis knowledge.

A consequent system of abstractions is developed to specify linear CASs and to provide encapsulation mechanisms of analysis procedures and entities. It can serve as a starting point for object-oriented decomposition in symbolic analysis.

A unique conceptual engine is proposed for symbolic analysis of physically different systems. These systems are essentially different in appearance and physical background, but they have the same mathematical matrix formulation from the symbolic analysis viewpoint.

Last updated 1998/10/27 16:00.