Electrical network/System of linear equations/Introduction/Example

An electrical network consists of several connected wires, which we call the edges of the network in this context. In every edge ${}K_{j}$, there is a certain (depending on the material and the length of the edge) resistance ${}R_{j}$. The points ${}P_{n}$, 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 ${}I_{j}$. The goal is to determine the currents from the data of the network and the voltages.

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; therefore, their sum must be ${}0$. In an edge ${}K_{j}$, there is a voltage drop ${}U_{j}$, determined by Ohm's law to be

${}U_{j}=R_{j}\cdot I_{j}\,.$

We call a closed, directed alignment of edges in a network a mesh. For such a mesh, the sum of voltages is ${}0$, unless a certain voltage is enforced from "outside“.

We list these Kirchhoff's laws again.

1. In every vertex, the sum of the currents equals ${}0$.
2. In every mesh, the sum of the voltages equals ${}0$.
3. If in a mesh, a voltage ${}V$ is enforced, then the sum of the voltages equals ${}V$.

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 ${}K_{1},\ldots ,K_{5}$ (with the resistances ${}R_{1},\ldots ,R_{5}$) are directed from left to right and that the connecting edge ${}K_{0}$ from ${}A$ to ${}C$ (where the voltage ${}V$ is applied) is directed upwards. The four vertices and the three meshes ${}(A,D,B),\,(D,B,C)$ and ${}(A,D,C)$ yield the system of linear equations

${\begin{matrix}I_{0}&+I_{1}&&-I_{3}&&&=&0\\&&&I_{3}&+I_{4}&+I_{5}&=&0\\-I_{0}&&+I_{2}&&-I_{4}&&=&0\\&-I_{1}&-I_{2}&&&-I_{5}&=&0\\&R_{1}I_{1}&&+R_{3}I_{3}&&-R_{5}I_{5}&=&0\\&&-R_{2}I_{2}&&-R_{4}I_{4}&+R_{5}I_{5}&=&0\\&-R_{1}I_{1}&+R_{2}I_{2}&&&&=&-V\,.\end{matrix}}$

Here the ${}R_{j}$ and ${}V$ are given numbers, and the ${}I_{j}$ are the unknowns we are looking for.