# Nodal Analysis of Electric Networks

On this page the system of equations for an electric network is set up, equations mainly being the sum of all currents to the nodes in the network.

Additional equations exist that express the voltage difference between nodes, between which there is no resistance as is with ideal voltage sources.

The system of equations is solved and the result presented.

The results are potentials at all nodes, currents through resistors and voltages over resistors.

Voltage sources can as well be ideal sources with internal resistance R_{i}=0 or real sources.

## About the model data

New *points* can be created via the table or by double-clicking in the drawing area.

Existing points can then also be moved by mouse dragging.

*Resistors* are set using 2 nodes and a resistance value.

Ideal connection cables are also defined here, via the two connection nodes and resistor 0.

Instead of entering the node numbers, mouse dragging is also possible to create new resistors or cables.

*Voltage sources* are determined via 2 nodes, the voltage between them and the internal resistance.

If the internal resistance is 0, it is an *ideal* voltage source.

## About the calculation method

The *Nodal Analysis* is based on Kirchhoff's current law: The algebraic sum of all currents at each node of the mesh is zero.

The conductance (reciprocal of the resistance) is therefore determined for each connection between 2 nodes.

The current through the resistance results from the product of the conductance with the potential difference of the associated two nodes.

This current goes through both nodes, so it has to be taken into account twice in the system of equations.

In the system of equations, which represents all node current balances, the unknowns art the node potentials **φ** in the mesh:

The right hand side **i**_{cs} of the system of equations is 0 except for the nodes that belong to a current source.

For every resistance R_{jk} between 2 nodes j and k the associated conductance 1/R_{jk} is added or subtracted at 4 positions in the conductance matrix **G**:

G_{j,j} = G_{j,j} + 1/R_{jk},
G_{j,k} = G_{j,k} - 1/R_{jk} (line j)

G_{k,j} = G_{k,j} - 1/R_{jk},
G_{k,k} = G_{k,k} + 1/R_{jk} (line k)

When all the conductances have been entered, line j (or k) of the overall system is the current balance for node j (or k).

Thus, the current going through a resistor R_{jk} is always taken into account in the 2 lines for the nodes j and k.

The *modified Nodal Analysis* (MNA) used here extends the system of equations as follows:

For *connecting cables* and *ideal voltage sources* an additional current is introduced as an unknown
and an additional equation is set up to formulate the associated potential difference.

As a result, a current flows through the connecting cables, which is displayed in the visualization of the results.

The system of equations then takes the following form:

**G** | **φ** | **+** | **D**^{T} | **i**_{a} | **=** | **i**_{cs} |

**D** | **φ** | **+** | **0** | **i**_{a} | **=** | **v** |

The unknowns in this system are the unknown potentials in **φ** and the unknown auxiliary currents through ideal voltage sources and cables in **i**_{a}.

The system matrix now is a block matrix with 4 submatrices **G**, **D**^{T}, **D** and **0**.

The matrix **D** uses **φ** to formulate the potential differences between ideal voltage sources and connecting cables.

With the matrix **D**^{T}, the associated currents from **i**_{a} are added to the node current balances for resistance-free connections.

Vector **v** contains the voltage of the ideal voltage sources and is 0 for connecting cables.

While with *ideal* voltage sources and connecting cables the associated current (in **i**_{a}) is unknown,

*real* voltage sources are processed differently. These are first converted into *current sources*.

Their internal resistance is introduced into the conductance matrix in the same way as normal resistances, and the current of the current source is then on the right hand side

in vector **i**_{cs} at 2 places where the corresponding voltage source is present.

In order for the system of equations to be solvable, one node must be grounded, i.e. its potential must be set to 0.

## To the results

The current in the network is visualized. The intensity of the blue color is a measure of the current in the respective resistance.
In addition, arrows on the resistor indicate the technical current direction.

The potentials are output on the right side for all nodes and the associated currents and voltages for all resistors.

If you click on a resistor, the current and voltage for this resistor are displayed locally.

If you click on a node, its potential is displayed.

In addition, at the very bottom, the system of equations is output, as it appears before one of the node potentials is set to zero.

The above-mentioned submatrices **G** and **D** are shown in different colors for better mutual differentiation.

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