# 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 ${\displaystyle {}K_{j}}$, there is a certain (depending on the material and the length of the edge) resistance ${\displaystyle {}R_{j}}$. The points ${\displaystyle {}P_{n}}$, 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 ${\displaystyle {}I_{j}}$. 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 ${\displaystyle {}0}$. In an edge ${\displaystyle {}K_{j}}$, there is a voltage drop ${\displaystyle {}U_{j}}$, determined by Ohm's law to be

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

We call a closed directed alignment of edges in a network a mesh. For such a mesh, the sum of voltages is ${\displaystyle {}0}$, 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 ${\displaystyle {}0}$.
2. In every mesh, the sum of the voltages equals ${\displaystyle {}0}$.
3. If in a mesh, a voltage ${\displaystyle {}V}$ is enforced, then the sum of the voltages equals ${\displaystyle {}V}$.

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

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

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