Kirchhoff's circuit laws

The two Kirchhoff's rules were formulated in 1845 by Gustav Robert Kirchhoff. ( See more discoveries cal Kirchhoff 's radiation law of 1859). Each of them describes the relationship between a number of electric currents and between multiple voltages in electrical networks.

The node set ( usually node ) - first Kirchhoff 's law

In a node of an electric network, the sum of the currents flowing in is equal to the sum of the currents flowing.

Bepfeilt you all adjacent branch currents so that all the related arrows to node down or all of its arrows from node to show off, so you can write down the node set for a node with n branch currents as follows:

This rule is initially valid for DC networks.

For AC networks it is valid under the condition that you have only concentrated components and thus fail, for example, charge storage effects in the nodes and lines due to the existing capacity there.

Instead of the values ​​you can also consider the vector representation of the current:

For a network with nodes linearly independent node equations can be set up.

Extension

If one starts from lumped elements, the node rule applies not only to individual nodes, but also for all the circuits. However, it is assumed that the node remains electrically neutral. If you want to, for example, only one capacitor plate look ( and not the entire capacitor ), this requirement is no longer met. One takes the viewing in this case the so-called displacement current which flows between the capacitor plates to expand. For a description of these no longer source-free fields the Ampère law must be used.

Example of a node

As can be seen in the picture, the currents I1 and I3 into the node and the currents I2, I4 and I5 from the node out. After the node is normally derived from the following formula:

Or transformed

For example a network node

Also complete networks can be viewed as nodes. In the example, the alternating currents I1 and I2 into the node and the current I3 from the node out.

Thus:

Are the incoming currents following complex rms values ​​given ( with the usual in electrical engineering imaginary unit j ):

Thus, for the effluent stream from the node rule:

The mesh set (mesh rule) - second Kirchhoff 's law

All part of a cycle or a loop voltages in an electrical network add up to zero. The direction of the circulation can be chosen arbitrarily; she then places but determines the sign of the partial voltages. As far as arrows pointing opposite to the direction of rotation, the voltages of opposite sign should be used.

In a round with n partial voltages of an electrical network, the following formula applies:

Also, this rule applies to any time-dependent flows and for networks with non-linear components.

In AC networks, the sum of the complex rms values ​​or complex amplitudes of the voltage can be considered:

However, the mesh equation is valid in this case, only the terminal voltages. The stresses in the components themselves ( for example, along the coil wire ) are not captured properly by the mesh equation.

A network with z branches and n independent node equations has (z - n ) independent mesh equations.

The mesh is usually a special case of the induction law and may be used only in the absence of time- varying magnetic flux.

Background

Both Kirchhoff's rules are conclusions from physical conservation laws, the 1st and 3rd Maxwell's equation:

• The junction rule describes the conservation of electric charge, and states that are cached in the nodes neither destroyed nor charges.

• The mesh rule describes the conservation of electric energy and indicates that a total charge Q does not work on the electric field performs during a single revolution of the circuit. To move in the opposite simple circuit which charges within the resistance to the electric field, and within the voltage source they move against the field.

The mesh is usually a formal conclusion from the induction law. It applies only to the case that in the loop does not change the magnetic flux is carried out ( ), and thus also fed to a magnetic path in the network, no power or is removed from there. In the absence of alternating magnetic fields the induction law provides

,

What exactly the statement of the mesh rule. The term denotes the circulation voltage, ie, the " sum of all voltages in a full circle around".

In the application of Kirchhoff's equations is well to note that all connections between the individual circuit elements are assumed to be ideal conductive. In addition, the components are regarded as concentrated components. Concentrated components can be designed in their electrical behavior described completely by the flowing currents at the terminals and externally applied voltages. If the circuit to be examined not concentrated components occur, these concentrated by equivalent circuits, circuit elements must be replaced.

Common misconception

An occasionally encountered in scientific literature misconception assumes that the Kirchhoff's loop rule can be used to explain induction processes. For example, writes Wolfgang Demtröder in connection with the operations of the self-induction of an unloaded transformer:

The underlying idea starts from two misconceptions:

In fact, the Kirchhoff's loop rule for induction processes has no explanatory power, since it was derived just under the assumption that the existing magnetic fluxes are constant over time and is used no power to the field structure ( conservative field ).

Instead of Kirchhoff's loop rule, the law of induction should be used in induction processes in general. The result for the depicted coil in a clockwise rotation:

The ohmic resistance R of the winding coil. For an open loop, this results in the equation:

The voltage across the coil wire is thus identically zero at idle and increases in current flow according to Ohm's law proportional to the coil current.

In network theory the said incorrect assumptions are still widely used for the calculation of networks with coils. This is therefore admissible, because the two mistaken assumptions " In the application of the induction rule is valid mesh " largely cancel each other and lead outside the coil windings for correct calculation of currents and voltages "In the coil a voltage source exists". It is a terminal equivalent modeling the induction using a network model. The advantage of this model is that the calculation of the calculation method for the usual networks DC networks, so mesh and node equation can be used. Only within the coil itself gives this model the physics not properly again and leads to contradictions.