Circuit topology (electrical)

In particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory.

Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits.

[2] This example also demonstrates a common convention of naming topologies after a letter of the alphabet to which they have a resemblance.

These networks arise often in 3-phase power circuits as they are the two most common topologies for 3-phase motor or transformer windings.

There is also a twin-T topology, which has practical applications where it is desirable to have the input and output share a common (ground) terminal.

The balanced form of ladder topology can be viewed as being the graph of the side of a prism of arbitrary order.

Anti-ladder topology finds an application in voltage multiplier circuits, in particular the Cockcroft-Walton generator.

Such infinite chains of lattice sections occur in the theoretical analysis and artificial simulation of transmission lines, but are rarely used as a practical circuit implementation.

With more complex circuits the description may proceed by specification of a transfer function between the ports of the network rather than the topology of the components.

Gustav Kirchhoff himself, in 1847, used graphs as an abstract representation of a network in his loop analysis of resistive circuits.

[15][16] Maxwell is also responsible for the topological theorem that the determinant of the node-admittance matrix is equal to the sum of all the tree admittance products.

In 1900 Henri Poincaré introduced the idea of representing a graph by its incidence matrix,[17] hence founding the field of algebraic topology.

[18] Veblen is also responsible for the introduction of the spanning tree to aid choosing a compatible set of network variables.

[19] Comprehensive cataloguing of network graphs as they apply to electrical circuits began with Percy MacMahon in 1891 (with an engineer-friendly article in The Electrician in 1892) who limited his survey to series and parallel combinations.

[note 1] Ronald M. Foster in 1932 categorised graphs by their nullity or rank and provided charts of all those with a small number of nodes.

This work grew out of an earlier survey by Foster while collaborating with George Campbell in 1920 on 4-port telephone repeaters and produced 83,539 distinct graphs.

[20] For a long time topology in electrical circuit theory remained concerned only with linear passive networks.

Enormous increases in circuit complexity have led to the use of combinatorics in graph theory to improve the efficiency of computer calculation.

In fact, the incidence matrix is an alternative mathematical representation of the graph which dispenses with the need for any kind of drawing.

[26] A tree is a graph in which all the nodes are connected, either directly or indirectly, by branches, but without forming any closed loops.

[note 3] Mesh analysis can only be applied if it is possible to map the graph onto a plane or a sphere without any of the branches crossing over.

[31] There is an approach to choosing network variables with voltages which is analogous and dual to the loop current method.

Rank and nullity are dual concepts and are related by:[35] Once a set of geometrically independent variables have been chosen the state of the network is expressed in terms of these.

The result is a set of independent linear equations which need to be solved simultaneously in order to find the values of the network variables.

In some cases the minimum number possible may be less than either of these if the requirement for homogeneity is relaxed and a mix of current and voltage variables allowed.

A major aim of topological methods of network synthesis has been to eliminate the need for these mutual inductances.

One theorem to come out of topology is that a realisation of a driving-point impedance without mutual couplings is minimal if and only if there are no all-inductor or all-capacitor loops.

Certainly all early studies of infinite networks were limited to periodic structures such as ladders or grids with the same elements repeated over and over.

For instance Kirchhoff's laws can fail in some cases and infinite resistor ladders can be defined which have a driving-point impedance which depends on the termination at infinity.

Other examples are launching waves into a continuous medium, fringing field problems, and measurement of resistance between points of a substrate or down a borehole.

Figure 1.1 . T,Y and Star topologies are all identical.
Figure 1.2 . Series and parallel topologies with two branches
Figure 1.3 . All these topologies are identical. Series topology is a general name. Voltage divider or potential divider is used for circuits of that purpose. L-section is a common name for the topology in filter design.
Figure 1.4 . Series and parallel topologies with three branches
Figure 1.5 . Y and Δ topologies
Figure 1.6
Figure 1.7 . Common balanced and unbalanced filter topologies
Figure 1.8
Figure 1.9 . Bridge circuit with bridging output load shown
Figure 1.10 . Bridged T topology
Figure 1.11
Figure 1.12
Figure 1.12
Figure 1.13 . Anti-ladder topology
Figure 1.14 . Basic amplifier topology such as common emitter bipolar junction transistor amplifier
Figure 1.15 . Balanced amplifier such as a long-tailed pair amplifier
Figure 2.1. Circuit diagram of a ladder network low-pass filter: a two-element-kind network
Figure 2.2 . Graph of the ladder network shown in figure 2.1 with a four rung ladder assumed.
Figure 2.3 . One possible tree of the graph in figure 2.2. Links are shown as dotted lines.
Figure 2.4 . A cut set of the graph in figure 2.2 derived from the tree of figure 2.3 by cutting branch 3.
Figure 2.5 . The dual graph of the graph in figure 2.2.
Figure 2.6 . A double-tuned circuit frequently used to couple stages of tuned amplifiers. A , the graph of the double-tuned circuit. B , an equivalent graph with the disjoint parts combined.
Figure 2.7 . An example of a hypergraph. Regular edges are shown in black, hyperedges are shown in blue, and tentacles are shown in red.