Dual networks

Linear electrical networks
Ideal element
Electrical component
Series and parallel connection
Network transformations
Generator sets	Network sentences
Network analysis methods
Two-port parameters

Dual networks or dual circuits are electrical networks in which currents and voltages are interchanged.

Dual topology

dual graphs

In the context of graph theory , a concept of duality can be grasped for the case of plane graphs , which is closely related to the relationship between the cycle space and intersection space of a graph. As shown in the adjacent sketch, the edges of both graphs correspond to one another uniquely , while the nodes of one graph appear as cycles in the other and vice versa. In terms of electrical engineering, the cycles are referred to as meshes, the edges as branches with a two-pole each , the graph as a network or, with a focus on the graph-theoretical background, as a topology.

Dual elements and laws

Bipoles are dual to one another if their current-voltage characteristics are structurally retained when voltage and current are interchanged. This is particularly impedances with the case. This means that basic elements such as sources and impedances are dually opposed as follows. ${\displaystyle Z'=1/Z}$

element	${\displaystyle Z}$	dual	${\displaystyle Z'}$
Power source	${\displaystyle \infty }$	Voltage source	${\displaystyle 0}$
resistance	${\displaystyle R={\frac {1}{G}}}$	Conductance	${\displaystyle G={\frac {1}{R}}}$
Inductance	${\displaystyle \mathrm {j} \omega L}$	capacity	${\displaystyle {\frac {1}{\mathrm {j} \omega C}}}$

In addition, the duality can also be specified in the case of legal relationships and the sizes that occur:

network	Dual network
Series connection	Parallel connection
Short circuit	Neutral
1. Kirchhoff's law	2. Kirchhoff's law
Norton's theorem	Thévenin's theorem
mesh	node
T-link	π term
Delta connection	Star connection

Note: Of course, the correspondences also apply in the opposite direction, i.e. the dual of what is listed in the right column can be found in the left column.

Dual networks

Two networks are dual to each other, though

• their topology is dual.
• all corresponding bipoles are dual.

Examples

Duality of series and parallel connection

Series and parallel connection

The adjacent sketch shows the graphic construction of the duality of series and parallel connection of impedances. Since the circuit is not closed, the connections have been connected with the dashed lines. Make sure that every bipolar has its corresponding dual and that every node lies exactly in one mesh of the dual circuit. Furthermore, each of the dual circuits is individually planar, that is to say does not contain any intersecting connections.

Individual evidence

1. ^ Kories, Schmidt-Walter: Pocket book of electrical engineering, basics and electronics, 9th edition, 2010, ISBN 978-3-8171-1858-8 . P. 145ff.
2. ^ Diestel: Graph theory. 3rd edition, 2006, ISBN 978-3-540-21391-8 . P. 113ff