Research On Math Model And Some Structural Properties Of Electrical Networks Over F(z)

In field of circuit theory, the conclusion about structure properties of electrical network based on the field of real number is determined by both network structure and the value of element, but structure properties based on the field of rational functions in multi-parameters is only determined by network structure and not by the value of elements (because these elements do not have values). We just need to observe the network structure and know whether the network can be controlled and observed or not. The method is simple and straightforward, and especially is suitable for analysis and synthesis of electrical network. The current study of linear passive RLC network gets some conclusion. Even the passive network, the output equations and structure observability of network also need further research. Active networks is a more realistic, broader application of the electrical network than passive RLC network, so research on structure properties of active network is inherited and developed from passive RLC network, and has more significant meaning. The analysis of controllability and observability of electrical network over F (z) requires the application of the algebraic criterion, so firstly we derive state equation and output equation of electrical network. Manual analysis and calculation is not accurate enough, so the computer-aided software development is of practical value.The paper studies topology structure of linear electrical network, and study the system structure properties by establishing mathematical model over F(z) (state equation and output equation). The paper consists of four parts as follows:1. The paper introduce development of relevant topics at home and abroad, and analysis two natures of characteristic polynomials of linear systems and reducibility condition of a class of rational function matrices, and moreover apply reducibility condition to analysis RLC network over F(z) of rational function in multi-parameters.2. According to electrical network theory over F(z), the relationship between voltage and current of each branch of electrical network is expressed as algebraic equation form, and transforms these equations, and gets state equations of passive equation and active equation. The paper analyzes existence condition and topology condition of solvability of linear active state equation over F(z).3. The paper use matrix F(z) over the field of rational functions in multi-parameters to describe coefficient matrix X|·=AX+BU .By use of graph theory and matrix theory with electrical theory, the paper gets structure properties and the conclusions of reducibility and separability. Combining these conclusions with system theory over F(z), we arrive structure controllability condition of linear network over F(z) and further analysis sufficient condition of structure controllability of active network. Finally we only observe topology structure of network and determine whether network can be controlled or not.4. The paper derives output equations of passive and active network, and analysis arbitrary branch current and the voltage expression between any two nodes, and also analysis sufficient condition of structure observability condition of active network over F(z). These results lay the foundation for the output equation and analysis of network observability by computer-aided analysis.5. The paper develops computer software by VC++ and MATLAB, and analysis structure property of active network by computer software (include network graph protract, analysis topology structure and structure controllability and observability), and finally apply computer software to analysis example. User interface of the computer software is friendly, intuitive, and easy to operate.