Electrical network/System of linear equations/Introduction/Example

From Wikiversity
Jump to navigation Jump to search

An electrical network consists of several connected wires, which we call the edges of the network in this context. In every edge , there is a certain (depending on the material and the length of the edge) resistance . The points , where the edges meet, are called the vertices of the network. If we put to some edges of the network a certain electric tension (voltage), then we will have in every edge a certain current . It is helpful to assign to each edge a fixed direction, in order to distinguish the direction of the current in this edge (if the current is in the opposite direction, it gets a minus sign). We call these directed edges. In every vertex of the network, the currents of the adjacent edges come together, their sum must be . In an edge , there is a voltage drop , determined by Ohm's law to be

We call a closed directed alignment of edges in a network a mesh. For such a mesh, the sum of voltages is , unless from "outside“ a certain voltage is enforced.

We list these Kirchhoff's laws again.

  1. In every vertex, the sum of the currents equal .
  2. In every mesh, the sum of the voltages equals .
  3. If in a mesh, a voltage is enforced, then the sum of the voltages equals .

Due to "physical reasons“, we expect that, given voltages in every edge, there should be a well-defined current in every edge. In fact, these currents can be computed, if we translate the stated laws into a system of linear equations and solve this system.

In the example given by the picture, suppose that the edges (with the resistances ) are directed from left to right, and that the connecting edge from to (where the voltage is applied) is directed upwards. The four vertices and the three meshes and yield the system of linear equations

Here the and are given numbers, and the are the unknowns we are looking for.