Mathematical Modeling of Energy Efficiency

based on the use of basic concepts of topology and provide automatic generation of models of electrical circuits.
The basis of electrical circuits are active two-terminal networks, which have equivalent resistances and EMF, and passive ones, which have only resistance, and the EMF is zero.
Auto two-pole networks are sources of EMF and current, batteries, generators, electric motors;
Passive two-pole networks are transformers, load resistance lines;

Introduction. Basic notions. Graphs. Incidence matrixes

Lecturer: Masheyeva R.U.

a node or a node is the point of connection between two or more branches.
Links between nodes are called branches.
A branch represents a single element such as a voltage source or a resistor. In other words, a branch represents any two-terminal element.
The branches form loops. A loop is any closed path in a circuit.

Introduction. Basic notions. Graphs. Incidence matrixes

Lecturer: Masheyeva R.U.

a short circuit (a connecting wire) connects two nodes, the two nodes constitute a single node. The circuit in Figure 1 has three nodes a, b, and c.
Notice that the three points that form node b are connected by perfectly conducting wires and therefore constitute a single point.

The same is true of the four points forming node c. We demonstrate that the circuit in Fig. 1 has only three nodes by redrawing the circuit in Fig. 2. The two circuits in Figs. 1 and 2 are identical.
However, for the sake of clarity, nodes b and c are spread out with perfect conductors as in Fig. 1.

Introduction. Basic notions. Graphs. Incidence matrixes

Lecturer: Masheyeva R.U.

passing through a set of nodes, and returning to the starting node without passing through any node more than once. A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop. Independent loops or paths result in independent sets of equations.
It is possible to form an independent set of loops where one of the loops does not contain such a branch. In Fig. 2, abca with the 2Ω resistor is independent. A second loop with the 3Ω resistor and the current source is independent. The third loop could be the one with the 2Ω resistor in parallel with the 3Ω resistor. This does form an independent set of loops.
A network with b branches, n nodes, and I independent loops will satisfy the fundamental theorem of network topology
b= I+n-1

Introduction. Basic notions. Graphs. Incidence matrixes

Lecturer: Masheyeva R.U.

that is, a structural diagram of a circuit, is presented in the form of a directed graph (Граф), the tops of which correspond to the nodes of the circuit, and the edges correspond to its branches.
Sources of EMF, current, resistance do not show in this graphs but only take into account the nodes and their circuits.
For each branch, its orientation (positive direction) is set, in accordance with which the positive directions of the current and voltage of the branch are taken. We will take the direction of the current in the branch to the node as a positive direction. For loop currents, we take the clockwise direction as positive. Under these conditions, any electrical circuit can be represented in the form of a graph and P matrix, connections that uniquely reflects the structural diagram of the graph.

Introduction. Basic notions. Graphs. Incidence matrixes

Lecturer: Masheyeva R.U.

is a conditional image of an electrical circuit diagram, in which the branches of the circuit are represented by segments - the branches of the graph, and the nodes by points are the nodes of the graph. The nodes and branches of the graph correspond to the nodes and branches of the electrical circuit. A schema subgraph is part of a schema graph.

Electrical circuit’s (network) and it’s graph

relate connects all the nodes of the graphs. It is not form a closed loop. The edges of the graph that are not part of the tree are called chords. (1,3,4) –chords, (2,4,5) – tree of the graph.

A directed graph can be expressed in a compact matrix form. The branches are connected with each other with the help of nodes. A directed branch connected to a node is called incidence. It represents the orientation of the branches and the number of branches incident to a node. The number of branches incident to a node is called degree of node.

there are 'n’ nodes and 'b’ branches in a network graph, then incidence matrix have 'n’ rows and 'b’ columns. So order of the incidence matrix in nxb.
Reduced incidence matrix (A): When any one row is completely deleted from the matrix then this is called reduced incidence matrix. The order of this matrix is (n-1)xb. This reduction results from mathematical manipulation.