Nodal analysis

Analysing an electrical network can sometimes be a complex task. As the network grows, so does the complexity of the analysis. In some cases, the analysis methods presented so far for node and mesh analysis are not sufficient to determine all the variables of an electrical network. In this case, the node potential method offers a possible solution for analysis. In the nodal analysis, all other potentials are determined based on a reference potential, with an assignment of 0V. Using the node potential method , the electrical network from Figure 1 is analysed as follows.

1 Preparation of the network

The nodal analysis method works with current sources and conductance values. In addition to the current source, the network shown has a voltage source and resistance values. These need to be converted. As shown in Figure 2, the voltage source is transformed into a current source. The series resistance of the voltage source becomes a parallel resistance for the current source.

When converting voltage sources into current sources, the voltage values of the voltage sources must also be converted into current values. This is done according to equation 1.

\begin {equation} I_\mathrm {q} = U_\mathrm {q} \cdot G_\mathrm {i} \label {GleichungUmrechnungSpannungsquelle} \end {equation}

Since the nodal analysis method does not use the resistance values of the components, but rather the conductance values, all resistance values are converted into conductance values according to equation 2.

\begin {equation} G_\mathrm {i} = \frac {1}{R_\mathrm {i}} \label {GleichungUmrechnungWiderstand} \end {equation}

Figure 3 shows the conversion of the voltage sources from the specified network into equivalent current sources. The voltage source \(U_0\) and the resistor \(R_0\) become the current source \(I_\mathrm {q0}\) and the parallel conductance \(G_0\). The remaining resistors are also converted into conductances.

2 Determination of nodes and node potentials

Now the nodes must be determined. To do this, we define a reference node with the index 0, i.e. \(K_0\). Next, all nodes are numbered consecutively. The electrical network with the numbered nodes is shown in Figure 4. The nodes drawn without components between the numbered nodes have the same potential and thus form a node.

To determine the node potentials, all potentials of nodes \(K_1\) and \(K_2\) are referenced to the reference node. \(k-1\) equations are required to perform the node potential method. Referring to the specified network, this results in two equations, which are written in vector notation according to equation 3. Here, the potential for node \(K_1\) is defined by the voltage from node \(K_1\) to the reference node. This results in the notation \(U_\mathrm {K1-K0}\). The same is done for node \(K_2\). Here, the voltage \(U_\mathrm {K2-K0}\) is defined.

\begin {equation} U_\mathrm {K} = \begin {bmatrix} U_\mathrm {K1-K0} \\ \\ U_\mathrm {K2-K0} \\ \end {bmatrix} \label {GleichungKnotenpotentiale} \end {equation}

3 Allocation of source currents

For the specified nodes, apart from the reference node, the outgoing and incoming currents must be recorded. Each current source at the nodes is noted. Here, incoming current sources are given a positive sign and outgoing current sources are given a negative sign. The current from the current source \(-I_\mathrm {q0}\) flows out of node \(K_1\), and this current is assigned a negative sign. The current from the current source \(-I_\mathrm {q2}\) flows out of node \(K_2\), and this is also assigned a negative sign. According to equation 4, this results in the vector for the source currents.

\begin {equation} I_\mathrm {K} = \begin {bmatrix} -I_\mathrm {q0} \\ \\ -I_\mathrm {q2} \\ \end {bmatrix} \label {GleichungQuellströme} \end {equation}

4 Conductance matrix

After converting the resistance values into conductance values, the conductance matrix is created using the conductance values. To do this, the nodes apart from the reference node are examined again. For this purpose, the electrical network presented is shown once more in Figure 5 with coloured branches. The conductance values of the nodes are noted along the main diagonal. For each node, the conductance values of the directly adjacent components are written down. For node \(K_1\), these would be the conductance values \(G_0\), \(G_1\) and \(G_3\) adjacent to the branch highlighted in blue, and for node \(K_2\), the conductance values \(G_2\) and \(G_3\). Away from the main diagonal, the conductance values that lie directly between the nodes are recorded on the other elements of the conductance matrix. Between nodes \(K_1\) and \(K_2\), the branch highlighted in green with the conductance value \(G_3\) lies directly between them. If there are other nodes between the nodes under consideration, these are not directly connected to each other and a 0 is entered in the element of the conductance matrix. The conductance matrix created in this way is illustrated in equation 5.

\begin {equation} G_\mathrm {K} = \begin {bmatrix} \color {blue}{G_0 + G_1 + G_3} & \color {green}{-G_3} \\ \\ \color {green}{-G_3} & G_2 + G_3 \\ \end {bmatrix} \label {GleichungLeitwertmatrix} \end {equation}

5 Set up a system of equations

After the vector of node potentials, the vector of source currents, and the conductance matrix have been determined, these equations are used to set up the system of equations for the node potential method, which is shown in Equation 5.

\begin {equation} \begin {bmatrix} G_0 + G_1 + G_3 & -G_3 \\ \\ -G_3 & G_2 + G_3 \\ \end {bmatrix} \cdot \begin {bmatrix} U_\mathrm {K1-K0} \\ \\ U_\mathrm {K2-K0} \\ \end {bmatrix} = \begin {bmatrix} -I_\mathrm {q0} \\ \\ -I_\mathrm {q2} \\ \end {bmatrix} \end {equation}

The system of equations of the node potential method can then be solved using the Gauss elimination method as an example. By rearranging the equation according to the node potentials sought, these can be identified. If, for example, the voltage across resistor \(R_1\) has been calculated in the network under investigation, the current through the resistor can be determined. As an alternative to the Gaussian elimination method, Cramér’s rule can also be used to solve linear systems of equations.

Node potential method

Analysis of the electrical network from Figure 6 using the node potential method.

The following steps should be carried out for the node potential method:

• Preparation of the network
• Determination of nodes and node potentials
• Allocation of source currents
• Preparation of the conductivity matrix
• Set up a system of equations

a)

Preparing the network:

b)

Determination of nodes and node potentials:

\begin {equation} U_\mathrm {K} = \begin {bmatrix} U_\mathrm {K1-K0} \\ \\ U_\mathrm {K2-K0} \\ \\ U_\mathrm {K3-K0} \\ \end {bmatrix} \nonumber \end {equation}

c)

Allocation of source currents: \begin {equation} I_\mathrm {K} = \begin {bmatrix} 0 \\ \\ -I_\mathrm {q0}+I_\mathrm {q1} \\ \\ -I_\mathrm {q1}-I_\mathrm {q2} \\ \end {bmatrix} \nonumber \end {equation}

d)

Determination of the conductance matrix: \begin {equation} G_\mathrm {K} = \begin {bmatrix} G_3 + G_4 + G_5 & -G_3 & -G_5 \\ \\ -G_3 & G_0 + G_1 + G_3 & -G_1 \\ \\ -G_5 & -G_1 & G_1 + G_2 + G_5 \\ \end {bmatrix} \nonumber \end {equation}

e)

Set up a system of equations: \begin {equation} \begin {bmatrix} G_3 + G_4 + G_5 & -G_3 & -G_5 \\ \\ -G_3 & G_0 + G_1 + G_3 & -G_1 \\ \\ -G_5 & -G_1 & G_1 + G_2 + G_5 \\ \end {bmatrix} \cdot \begin {bmatrix} U_\mathrm {K1-K0} \\ \\ U_\mathrm {K2-K0} \\ \\ U_\mathrm {K3-K0} \\ \end {bmatrix} = \begin {bmatrix} 0 \\ \\ -I_\mathrm {q0}+I_\mathrm {q1} \\ \\ -I_\mathrm {q1}-I_\mathrm {q2} \\ \end {bmatrix}\nonumber \end {equation}