Kirchhoff’s laws

The component equations introduced in the previous module, which show the relationship between voltage and current at the individual components, are not sufficient to calculate all voltages and currents within a network. Kirchhoff’s rules, also known as mesh or node rules, provide the equations required for this.

1 Node rule I (1. Kirchhoff rule)

The node rule states that the sum of all currents flowing into a node is identical to the sum of all currents flowing out of it:

Mathematically expressed, this results in the following relationship: \begin {equation} \sum I_\mathrm {zu} = \sum I_\mathrm {ab} \end {equation}

It should be noted that the currents are evaluated according to their counting arrow direction. Currents flowing into the node are counted as positive, currents flowing out of the node are counted as negative. If the actual direction of a current is not known in advance, the counting arrow direction can be set arbitrarily. The actual current direction is determined from the numerical value as the result of the calculation. If this is negative for a current, i.e. \(I < 0\), the real current flows in the opposite direction.

Alternatively, this relationship can also be expressed by the fact that the sum of all currents in a node is equal to 0:

\begin {equation} \sum _{i=1}^{n} = 0 \end {equation}

Applied to the example nodes in Figure 1, this results in the following equations:

\begin {equation*} I_1 = I_2 + I_3 \end {equation*} \begin {equation*} I_1 - I_2 - I_3 = 0 \end {equation*}

However, the node to which the rule refers does not have to consist of just one point. Instead, it is possible to define so-called envelope nodes, which completely enclose an area within a circuit. Applied to the envelope node shown in Figure 3.2 this results in the equation

\begin {equation*} I_1 + I_2 - I_3 - I_4 = 0 \end {equation*}

The node rule is not only applicable to discrete components due to the generalisation of the charge conservation law for source-free flow fields. Rather, it can be applied to any real structure. In Figure 3, the current flowing into the wire \(I\) is therefore equal to the total current flowing out of the envelope surface of the sheet metal cut-out.

In general, this relationship can be described by the closed surface integral over the envelope surface \(\vec {A}\):

\begin {equation} \iint _A \vec {J} \cdot d\vec {A} = 0 \end {equation}

2 Application case node rule: Parallel connection of resistors

A basic application for the node rule is the parallel connection of resistors. The current divides at a common node, and a partial current flows through each of the individual resistors. After passing through the respective resistors, the partial currents rejoin and continue to flow as a total current. This is illustrated in example ??.

Parallel connection of resistorsknoten

In a parallel circuit, the currents split at the common node. How large is the current \(I_3\) in relation to the currents \(I_0\), \(I_1\) and \(I_2\)?

\begin {equation*} I_0-I_1-I_2 = 0 \end {equation*} \begin {equation*} \rightarrow I_0 = I_1 + I_2 \end {equation*} \begin {equation*} -I_3+I_1+I_2 = 0 \end {equation*} \begin {equation*} \rightarrow I_1+I_2 = I_3 \end {equation*} \begin {equation*} \rightarrow I_3 = I_0 \end {equation*}

If one of the resistors is replaced by an ideally conductive connection (conductance approaches infinity, resistance consequently approaches 0), the current flow in the area between the two nodes changes (example ??)..

Parallel connection with conductorknoten2

A resistor is replaced by a conductive connection. How large are the currents \(I_1\) and \(I_2\)?

\begin {equation*} I_0=I_1+I_2 \end {equation*} \begin {equation*} \mathrm {mit} \, U_1 = R_1 \cdot I_1 \end {equation*} \begin {equation*} U_2 = R_2 \cdot I_2 =0 \stackrel {!}{=} U_1 \end {equation*} \begin {equation*} \rightarrow I_1 = 0 \end {equation*} \begin {equation*} \rightarrow I_0 = I_2 = I_3 \end {equation*}

3 Mesh rule (Kirchhoff’s second law)

Kirchhoff’s second law (mesh rule) states that the sum of all voltages in a mesh is zero. Analogous to the direction of the currents, the direction of the individual partial voltages must also be taken into account here. If the direction arrow of a partial voltage points in the opposite direction to the direction of circulation of the mesh, then this partial voltage must be given a negative sign. If the direction of an applied voltage is not known, then an arbitrary counting arrow direction can also be assumed here. An oppositely applied voltage is also expressed in calculations with a negative sign.

Equivalent to the above definition, it can be stated that the sum of all voltages connected in the same direction at voltage sources corresponds to the voltage that drops across the consumers.

Applied to the example network shown in Figure 4, the following mesh equations result:

\begin {equation*} U_1+U_2-U_3-U_4 =0 \end {equation*} respectively

\begin {equation*} U_1+U_2 = U_3+U_4 \end {equation*}

The mesh equation also applies if additional currents are fed into the mesh, or if individual two-poles are passed through several times during the circulation around a closed mesh.

The general relationship, whereby any integrated voltage along a closed contour equals 0, can be described as follows according to Maxwell’s equations:

\begin {equation} \oint _{s} \vec {E} d\vec {s} = 0 \end {equation}

4 Application case mesh rule: Series connection of resistors

While the nodes in electrical networks are primarily important for current, the branches and thus also the meshes are primarily of interest for calculating voltages. In a series connection of resistors in a branch, all partial voltages add up to a total voltage with the correct sign. An application case for determining a partial voltage is shown in Example ??.

Series connections in networksreihe

In a series connection, the voltages add up to a total voltage. How large is the voltage \(U_0\)?

Set up the mesh counterclockwise:

\begin {equation*} U_2-U_0-U_3 =0 \end {equation*}

\begin {equation*} \rightarrow U_0 = U_2- U_3 \end {equation*}

The mesh rule can also be applied if the mesh to which it is applied has an interruption point (see example ??).

Series connections with interruption pointreihe2

A connection is interrupted. How large is the voltage \(U_1\) at the point of interruption?

Set up the mesh counterclockwise:

\begin {equation*} U_2+U_1-U_0-U_3 = 0 \end {equation*}

Determine the voltage \(U_2\) using Ohm’s law:

\begin {equation*} U_2=I_2 \cdot R_2 \end {equation*}

\begin {equation*} \rightarrow I_2 = 0 \end {equation*}

\begin {equation*} \rightarrow U_2 = 0 \end {equation*}

Insert into the initial equation and solve for \(U_1\):

\begin {equation*} U_1 = U_0 + U_3 \end {equation*}