Network analysis (electrical circuits)

As network analysis is called to compute (see picture) from the known values ​​of the switching elements and the given source sizes all currents and voltages in electrical engineering, the approach in a network. By hand and by analytical methods only linear systems can be studied with realistic effort. The computer-aided circuit simulation, however, is mainly based on iterative approximation method requires a lot of computation steps, but can also deal with non-linear components.

Generally

In a network, the relationships between all occurring currents or voltages appearing all are described by the Kirchhoff's rules, named after the German physicist Gustav Robert Kirchhoff. The relationship between current and voltage is described by Ohm's law, which describes the components of equation resistors. Requirement are real linear switching elements, i.e., no capacitive or inductive component and a straight-line curve in contrast for example to the diode.

For non- ohmic resistances, the so-called complex calculation is required. This allows an analysis for AC voltage to perform, each considered frequency is individually calculated.

Thus, a network analysis is possible, nodes, branches and loops are defined in the network. Using the Kirchhoff's rules and the current-voltage relations of the two poles of equations can be assigned to them then. Thus, the mathematical equations lead to a unique solution, the respective equations must be independent of each other.

A node is a point in the network, in which a current branching occurs. If a network has nodes, then there are only independent node equations. One of these in the example shown,

A branch is the connection between two nodes by two-terminal elements. If the network has branches, then there is also a total independent branch equations.

In the example shown, the branch equations by utilizing the component equations:

As a (complete) tree is called a frame of branches, which connects all nodes, no node is allowed to be touched twice. To put it clearly, the structure formed may not provide opportunities to walk in circles. For the tree different variants are possible. In total there are in a fully meshed network ( each node has one branch to every other node ) variants are conceivable. In this example there are three different trees, as more than a branch two nodes together.

The individual branches in the tree are called tree twigs or branches. Because of the structure of the tree, there are branches.

All branches that do not belong to the tree, is referred to as tendons or connecting branches. Corresponding in number to the number of independent equations mesh which can be positioned by means of the mesh set.

A loop is a closed circulation through branches. For a simple analysis, a circulation should always be chosen over only a chord or a branch. For the Consecutive this way will be used.

The sense of rotation of the independent loops can be set arbitrarily, but is relevant for subsequent calculations.

In the example, the following two Maschenumläufe are selected.

Thus, there are independent node equations, mesh equations and independent for each branch a Zweipolgleichung, so Zweipolgleichungen. With this system of equations from equations therefore allow all " unknown quantities ", ie identify all branch currents and branch voltages all unique.

Branch current analysis

To solve by branch current analysis, all independent node equations and the independent mesh equations are set up. These are then sorted by stringing it on the other hand, according to current / resistance, on the one side of the equation, and stress. The result is a linear equation system.

In the illustrated example above, hereby then follows in an ordered sequence (node ​​Equation 1, Equation 1 mesh and mesh equation 2):

In matrix notation, now is the system of equations:

To the solution of the linear system of equations, there are standard techniques that can be used for this purpose. Smaller systems of equations can be solved analytically "by hand", for larger circuits, numerical methods (computer programs ) can be used. ( An example can be found on the talk page )

Superposition method according to Helmholtz

The overlay method is based on the principle of superposition for linear systems.

Procedure:

• Except for one source all others are removed. Voltage sources are replaced by short circuits or power sources seen as an interruption. However, the internal resistances of the sources remain in the circuit.
• The desired substreams with the remaining source are calculated.
• The procedure is repeated for any other source.
• Finally, the addition of the correct sign of the calculated partial streams is performed for the considered branches.

Result: The desired partial flow was determined.

Mesh current method

With increasing complexity of the effort increases for the calculation of the network with the branch current analysis. A reduction of the computational effort is given by the mesh current method.

Approach (short form):

Node potential method

As with the mesh current method arises when node potential method a reduced linear system of equations.

Approach (short form):

Resistance cube

A problem of network analysis is the so-called resistance cube which is formed by twelve equal cubical soldered together ohmic resistors. For this construction, the total resistance of the space diagonal A'C is to be determined.

The problem can be simplified by symmetry considerations. The die is composed of twelve resistance equal resistances and is thus not electrically balanced. By the three resistors connected together at a corner to flow because of the symmetry in the application of a voltage and equal flows. The opposite ends of the corner under consideration, therefore, these three resistors with one another are also at the same potential. By connecting points of equal potential, Äquipotentialpunkte, conducting manner, no current flows through these connections in the absence of tension between them. It follows that these additional compound does not adversely affect the circuit. Further, it is apparent that the same current must flow through the resistors, as drops to the same voltage to them. Thus, for the resistance dice when determining the total resistance across the corners of the space diagonal ( in the graph the vertices A 'and C ) two times three potential same points ( in the diagram the points A, B ', D ', and B, C', D) find the without causing an electrical change, each of which may be interconnected. After making these connections, the clear series gives a parallel circuit of three resistors to a parallel circuit of six resistors and a parallel circuit of three resistors.