Node potential method

The node potential method (also node voltage analysis or node admittance method) is a method for network analysis in electrical engineering . This method can be used to determine the node potentials of an electrical network made up of linear components .

Application of the procedure

The method is usually used to determine a current in a branch. Compared to branch flow analysis, this method saves as many equations as the network has independent meshes. All steps to the value you are looking for are shown below. This procedure also applies to complex and magnetic networks, provided that only linear components are used.

Define node potentials and reference nodes

In a network with k nodes, there are k − 1 independent node equations. No equation has to be set up for a node, since its equation could be set up from the equations of the other nodes and would therefore be linearly dependent. This node is therefore the reference node with zero potential (ground) and can be freely selected. The node should expediently be on a branch with the sought-after voltage drop, since a required potential is already established and the system of equations does not have to be solved one time. All other potentials are still unknown and are identified with a unique variable name.

Conversion of the resistors and voltage sources

Thevenin-Norton equivalent conversion

The branch currents are expressed as the product of the branch conductance and the node potential difference. The branch resistances are therefore replaced by their conductance values ​​and the voltage sources are converted into substitute current sources according to Norton's theorem .

${\displaystyle G_{ij}={\frac {1}{R_{ij}}}}$

${\displaystyle I_{q_{ij}}=U_{q_{ij}}\cdot G_{ij}}$

Ideal voltage sources without resistance in the branch cannot be transformed. More on this in the section on treatment of ideal voltage sources .

Establish the matrix of the linear equation system

The conductance matrix is ​​set up as follows:

• On the main diagonal with is the sum of the conductance values ​​of all branches that are connected to node i. ${\displaystyle \forall \,i,j}$ ${\displaystyle \!\,i=j}$
• The other places with the negative sum of the conductance values ​​between the neighboring nodes i and j (coupling conductance values). If there is no direct connection between two nodes, a zero is entered here. ${\displaystyle \forall \,i,j}$ ${\displaystyle i\neq j}$

${\displaystyle {\begin{pmatrix}G_{11}&-G_{12}&...&-G_{1n}\\-G_{21}&G_{22}&...&-G_{2n}\\...&...&...&...\\-G_{n1}&-G_{n2}&...&G_{nn}\end{pmatrix}}\cdot {\begin{pmatrix}\varphi _{1}\\\varphi _{2}\\...\\\varphi _{n}\end{pmatrix}}={\begin{pmatrix}I_{1}\\I_{2}\\...\\I_{n}\end{pmatrix}}}$

The conductance matrix is a symmetrical matrix . Consequently, the opposite coupling conductor values ​​(with regard to the main diagonal) are identical to . This has to be the case because these coupling values ​​are located between the same nodes in both cases. In contrast to the positive total conductance on the main diagonal, all coupling conductors are negative. ${\displaystyle \left(G_{ij}=G_{ji}\,\forall \,i,j\right.}$ ${\displaystyle \left.i\neq j\right)}$

In the vector of the node potentials, the same sequence must be adhered to as on the main diagonal of the conductance matrix.

In the vector of the nodal currents on the other side of the system of equations is the sum of the equivalent power sources with which the respective node is connected. Incoming currents are positive, outgoing currents are included in the total negatively (it also works the other way around, it just has to be uniform for all nodes). If no sources are connected to the node, a zero is entered.

Treatment of ideal voltage sources

In very rare cases an ideal voltage source (without internal / branch resistance) can be located in a branch between two nodes. This means that the voltage difference between the two nodes is known and one potential can be calculated directly from the other with the aid of the constant value of the source voltage. It should be noted in which direction the voltage of the source drops:

${\displaystyle \varphi _{\text{high}}-\varphi _{\text{low}}=U_{q_{\text{ideal}}}}$ with voltage drop from "high" node to "low" node

The equation is transformed according to the potential to be replaced and inserted into the equation system. If one of the nodes is the reference node, the potential of the other must of course be replaced. In the system of equations, the term used in each line is multiplied by the leading values ​​in the associated column. The terms with are shifted to the side of the current sources. ${\displaystyle U_{q_{\text{ideal}}}}$

The further procedure now depends on the position of the reference node. All lines of the replaced potentials whose ideal voltage source is directly connected to the reference node must be deleted. This reduces the degree of the equation system by one with every ideal voltage source at the reference node. For all other ideal voltage sources, an unknown branch current is introduced into their branch. These are first entered on the side of the power sources according to the same scheme as the power sources. Inflowing added, outflowing subtracted. Finally, the unknown branch streams are brought to the left. For ideal voltage sources without a direct connection to the reference node, the degree of the equation system is consequently not reduced, since an unknown current is added for each potential that is lost.

Calculate wanted potentials

Branch with voltage source

Before calculating a branch current, the potentials of the two adjacent nodes (φ _i and φ _j ) must be known. To do this, the system of equations is solved for one of the potentials. This is done either with the help of Cramer's rule or by the Gaussian elimination method . If one of them is the reference node, only one potential needs to be calculated. The branch voltage is usually calculated from the difference in the node potentials in such a way that the resulting branch voltage drops in the assumed direction of the current sought. The value of any voltage source present in the branch must be subtracted from the branch voltage according to the mesh principle if its voltage drops in the direction of the branch voltage, or added if it runs in the opposite direction. The result is then divided by the branch resistance or multiplied by the branch conductance in order to obtain the current sought. A positive branch current flows in the direction of the voltage drop of the node potential difference, a negative branch current in the opposite direction.

${\displaystyle I_{ij}={\frac {\varphi _{i}-\varphi _{j}\pm U_{q_{ij}}}{R_{ij}}}=\left(\varphi _{i}-\varphi _{j}\pm U_{q_{ij}}\right)\cdot G_{ij}}$

example

Circuit to demonstrate the node potential method

Look for in the circuit shown on the right. This is now calculated step by step using the node potential method. ${\displaystyle I_{6}}$

Define node potentials and reference nodes

For faster calculation, a node to which the branch of is connected becomes the reference node with zero potential. In this example, the decision was made on the lower node. The remaining three nodes are labeled , and . As in the case of the reference node, it should be noted that several nodes shown are practically only one node if there are no circuit elements on the branches between them. ${\displaystyle I_{6}}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \varphi _{2}}$ ${\displaystyle \varphi _{3}}$

Conversion of the resistors and voltage sources

There are two voltage sources and one current source in the circuit. The voltage sources are converted into backup power sources as described above.

${\displaystyle I_{q1}={\frac {U_{q1}}{R_{3}}}}$ and ${\displaystyle I_{q3}={\frac {U_{q3}}{R_{6}}}}$

It must be ensured that the correct current direction is drawn in at the current sources. In addition, the current through is no longer the same because it is now divided between the branches of and . After replacing the resistors with their conductance values, the lower circuit in the picture results. ${\displaystyle G_{6}}$ ${\displaystyle I_{6}}$ ${\displaystyle G_{6}}$ ${\displaystyle I_{q3}}$

Establish a system of equations

The equation system is now set up in matrix form according to the rules mentioned above.

${\displaystyle {\begin{pmatrix}G_{1}+G_{2}+G_{3}+G_{4}&-G_{4}&-G_{3}\\-G_{4}&G_{4}+G_{5}+G_{6}&-G_{5}\\-G_{3}&-G_{5}&G_{3}+G_{5}+G_{7}\end{pmatrix}}\cdot {\begin{pmatrix}\varphi _{1}\\\varphi _{2}\\\varphi _{3}\end{pmatrix}}={\begin{pmatrix}-I_{q1}+I_{q2}\\-I_{q2}+I_{q3}\\I_{q1}\end{pmatrix}}}$

Calculate wanted potentials

Since the reference potential is already known, only the potential is required. A large number of solution methods are available for this calculation . ${\displaystyle \varphi _{2}}$

With the help of the calculated potential, the required current is determined . The zero potential is expressed by. The potential difference is formed in the assumed direction of . The value of the voltage source must be added to the difference according to the rule mentioned above. ${\displaystyle I_{6}}$ ${\displaystyle 0V}$ ${\displaystyle I_{6}}$ ${\displaystyle U_{q3}}$

${\displaystyle I_{6}={\frac {0V-\varphi _{2}+U_{q3}}{R_{6}}}}$

application

The node potential method is ideal for the computer-aided calculation of the solution vector, since its linear system of equations can be set up using an algorithm that is easier to program than with the mesh flow method , in which the network must first be searched for a complete tree using graph theory. It therefore forms the basis of most computer programs for analyzing linear electrical networks. However, the optimal selection of the network analysis method to be used depends on the structure of the network (number of branches compared to the number of nodes) and in practice is individual for each network.

See also

• Mesh flow method
• Kirchhoff's rules
• Network analysis (electrical engineering)

literature

• Oliver Haas, Christian Spieker: Electrical engineering tasks 1 . Oldenbourg, Munich 2012, ISBN 3-486-71680-8 , p. 81–103 ( limited preview in Google Book search).