Method of Equivalent of the Scheme Using the Methodology of Equilibrium Balance

Abstract

A new method of equivalence has been developed using the proposed methodology of equilibrium balancing in order to simplify the calculation of equivalent mutual resistances between the nodes of the electrical circuit using matrix transformations of the network structure. The developed method allows, based on the use of the first incidence matrix, to calculate the equivalents of any connections of two circuit nodes. Mathematical models have been created for calculating the full resistance matrix of the existing and virtual circuit branches, which represents all existing interconnections between nodes, taking into account the effects of the adjacent network circuit.

The calculation of equivalent resistances by this method is carried out based on the analysis of the electrical network diagram. Thus, a single input driving current of the initial node and a single output driving current of the end node are set, the remaining driving currents are set to zero. So, the equivalent current in this case will be equal to unity, and the value of the equivalent resistance between the nodes of the circuit will be the sum of the voltage drops at such driving currents. The currents in the branches are determined by multiplying the inverse incidence matrix by the augmented matrix. Finding the equivalent resistances between branches that do not have a direct connection is carried out by supplementing the incidence matrix with additional virtual branches, which make it possible to calculate the value of the mutual equivalent resistances for these branches as well.

The operation of ordinal multiplication of the current by the resistance can be performed before subtracting the columns of the inverse matrix, that is, before the product of the inverse matrix by the augmented incidence matrix, which will allow calculating additional mutual equivalent resistances for the virtual branches. As a result, models for calculating the full matrix of the interdependence of equivalent resistances on branch voltages were obtained.

The values of the equivalent resistances are located on the diagonal of the complete resistance interconnection matrix and can be calculated using a simplified formula as the sum of the products of currents and branch resistances. This method of equivalent circuitry greatly simplifies the calculations of all equivalent circuit resistances due to the use of the equilibrium balancing methodology, it allows calculations for all circuit nodes, establishing the dependence of equivalent resistances on voltage drops.

The aim of the work is to increase the efficiency of the method of equivalence of electrical networks through the use of equilibrium balancing for calculating the steady state of electrical systems, which allows to optimize the calculation processes.