Nodal analysis

The node potential method (also node voltage analysis or Knotenadmittanzverfahren ) is a method of network analysis in electrical engineering. This method allows the node potentials of an electrical network of linear components determine.

• 2.1 Define node potentials and the reference node
• 2.2 Conversion of the resistors and voltage sources
• Up 2.3 system of equations
• 2.4 Calculate Wanted potentials

Application of the method

The process is usually used to determine a current into a branch. With respect to the branch flow analysis as many equations can be saved in this method, as the network has independent loops. In the following all the steps for this value are shown. This procedure also applies to complex and magnetic networks, where only linear components occur.

Specify node potentials and the reference node

For a network with k nodes, there are k-1 independent node equations. No equation has to be built up for a node, since its equation set from the equations of the other nodes and thus could be linearly dependent. This node is therefore the reference node with zero potential (ground ) and can be chosen arbitrarily. Appropriately, the node should be on a branch with sought, voltage drop, as has already been established as a required potential and the system of equations must be solved even less. All other potentials are still unknown and are designated with a unique variable name.

Conversion of the resistors and voltage sources

The branch currents are expressed as the product of Zweigleitwert and node potential difference. Therefore, the branch resistances are replaced by their conductance and reshaped the voltage sources after Norton 's theorem in alternative power sources.

Ideal voltage sources without resistance in the branch can not be reshaped. More to the point treatment of ideal voltage sources.

Set up the matrix of the linear system

The conductance matrix is set up as follows:

• On the main diagonals is the sum of the conductances of all branches that are connected to node i.
• At the other places with is the negative sum of the conductances between the neighboring nodes i and j ( Koppelleitwerte ). There is no direct connection between two nodes, a zero is entered here.

The conductance matrix is a symmetric matrix. Consequently, the opposite Koppelleitwerte (with respect to the main diagonal ) are identical. This must be so, because these are Koppelleitwerte in both cases between the same nodes. In contrast to the positive Summenleitwerten on the main diagonal are all Koppelleitwerte negative.

The vector of nodal potentials in the same order as on the diagonal of the conductance matrix must be adhered to.

In the vector of the current node on the other side of the system of equations the sum of the compensation current source which is connected to the respective nodes. Hinfließende streams of positive currents flowing away leave negative in the sum of one ( there is another way around, it must be done uniformly for all nodes ). If there are no sources connected to the node, a zero is entered.

Treatment ideal voltage sources

In very rare cases, an ideal voltage source can be located (without Innen-/Zweigwiderstand ) in a branch between two nodes. Characterized the voltage difference between the two nodes is known, and the potential can be calculated directly by using the constant value of the source voltage from the other. It should be noted, the direction in which drops the voltage from the source:

The equation is converted according to the potential to be replaced and inserted into the equation system. If one is the node of the reference node, the potential of the other needs to be replaced, of course. In the system of equations, the term used is multiplied in each row with the guide values ​​in the corresponding column. With the terms are moved to the side of the power sources.

The procedure is now dependent on the position of the reference node. All lines of the superseded potentials whose ideal voltage source to the reference node is directly connected must be deleted. This reduces the degree of the equation system to each ideal voltage source at the reference node by one. For all other ideal voltage sources, an unknown branch stream is introduced into their branch. These are first entered on the side of the current sources in the same way as the current sources. Hinfließende added, subtracted flowing away. Finally the unknown branch streams are brought to the left side. For ideal voltage sources with no direct connection to the reference node, the degree of the equation system is not reduced, consequently, there is added an unknown stream for each Eliminated potential.

Calculate Wanted potentials

Before calculating a branch current, the potential of the two adjacent nodes (? I and φj ) must be known. To the system of equations for one of the potentials is solved. This is done either by using Cramer's rule or Gaussian elimination. Should one of them be the reference node, only one potential has to be calculated. The branch voltage is calculated as the difference of the node potentials usually that the resulting branch voltage drops in the presumed direction of the desired current. The value of any existing voltage source in the branch must be subtracted when their voltage is added, if it runs in the opposite direction falls in the direction of branch voltage, or after the stitches set of the branch voltage. The result is then divided by the resistance branch and multiplied by the Zweigleitwert to receive this stream. A positive branch current flowing in the direction of the voltage drop of the node potential difference, and a negative shunt current in the opposite direction.

Example

Wanted is in the circuit shown on the right. This will be calculated step by step with the help of the node potential method.

Specify node potentials and the reference node

For faster computation is a node to which the branch is connected to the reference node with zero potential. In this example, the decision fell on the lower nodes. The remaining three nodes with, and inscribed. Like here in the case of the reference node is to be noted that several nodes are represented practically only one node, if there are on the branches between them no circuit elements.

Conversion of the resistors and voltage sources

In the circuit there are two voltage sources and a power source. The voltage sources are formed as described above in backup power sources.

Note that the correct direction of current is drawn at the current sources. Also, now the current through not the same, because this is now divided on the branches of and. After replacing the resistors with their guiding values ​​, the lower circuit as shown in the picture.

Set of equations

According to the rules above, the equation system is now set up in matrix form.

Calculate Wanted potentials

Since the reference potential is already known, the potential is only needed. For this calculation, a variety of solution methods are available.

Using the calculated potential is followed by the determination of the required current. The zero potential is expressed by. The potential difference is formed in the assumed direction of. The value of the voltage source needs to be added after the above-mentioned control to the difference.

Application

The node potential method is ideal for computer-aided calculation of the solution vector, since its linear equation system can be set up through a simple to program algorithm as the mesh current method, in which initially the network must be scanned graph theory after one whole tree. It therefore forms the basis of most computer programs for analysis of linear electrical networks. However, the optimal selection of the network to be used in the analytical method will depend on the structure of the network (the number of sectors is compared with the number of nodes) and in practice, individually for each network.