The circuit topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram; similarly to the mathematical concept of topology, it is only concerned with what connections exist between the components. Numerous physical layouts and circuit diagrams may all amount to the same topology.
Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.
Electronic network topology is related to mathematical topology. In particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of graph theory. In a network analysis of such a circuit from a topological point of view, the network nodes are the vertices of graph theory, and the network branches are the edges of graph theory.
Standard graph theory can be extended to deal with active components and multi-terminal devices such as integrated circuits. Graphs can also be used in the analysis of infinite networks.
Circuit diagrams The circuit diagrams in this article follow the usual conventions in electronics;Tooley, pp. 258–264 lines represent conductors, filled small circles represent junctions of conductors, and open small circles represent terminals for connection to the outside world. In most cases, impedances are represented by rectangles. A practical circuit diagram would use the specific symbols for resistors, inductors, capacitors etc., but topology is not concerned with the type of component in the network, so the symbol for a general impedance has been used instead.
The Graph theory section of this article gives an alternative method of representing networks.
Topology names Many topology names relate to their appearance when drawn diagrammatically. Most circuits can be drawn in a variety of ways and consequently have a variety of names. For instance, the three circuits shown in Figure 1.1 all look different but have identical topologies.Guillemin, pp.5–6
This example also demonstrates a common convention of naming topologies after a letter of the alphabet to which they have a resemblance. Greek alphabet letters can also be used in this way, for example Π (pi) topology and Δ (delta) topology.
Series and parallel topologies A network with two components or branches has only two possible topologies: series and parallel.
Even for these simplest of topologies, the circuit can be presented in varying ways.
A network with three branches has four possible topologies.
Note that the parallel-series topology is another representation of the Delta topology discussed later.
Series and parallel topologies can continue to be constructed with greater and greater numbers of branches ad infinitum. The number of unique topologies that can be obtained from series or parallel branches is 1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624, .
Y and Δ topologies
Y and Δ are important topologies in linear network analysis due to these being the simplest possible three-terminal networks. A Y-Δ transform is available for linear circuits. This transform is important because some networks cannot be analysed in terms of series and parallel combinations. These networks arise often in 3-phase power circuits as they are the two most common topologies for 3-phase motor or transformer windings. An example of this is the network of figure 1.6, consisting of a Y network connected in parallel with a Δ network. Say it is desired to calculate the impedance between two nodes of the network. In many networks this can be done by successive applications of the rules for combination of series or parallel impedances. This is not, however, possible in this case where the Y-Δ transform is needed in addition to the series and parallel rules.Farago, pp.18–21Redifon, p.22
The Y topology is also called star topology. However, star topology may also refer to the more general case of many branches connected to the same node rather than just three.Redifon, p.22
Simple filter topologies
The topologies shown in figure 1.7 are commonly used for filter and attenuator designs. The L-section is identical topology to the potential divider topology. The T-section is identical topology to the Y topology. The Π-section is identical topology to the Δ topology.
All these topologies can be viewed as a short section of a ladder topology. Longer sections would normally be described as ladder topology. These kinds of circuits are commonly analysed and characterised in terms of a two-port network.Farago, pp.112–116Redifon, pp.45–48
Bridge topology
Bridge topology is an important topology with many uses in both linear and non-linear applications, including, amongst many others, the bridge rectifier, the Wheatstone bridge and the lattice phase equaliser. Bridge topology is rendered in circuit diagrams in several ways. The first rendering in figure 1.8 is the traditional depiction of a bridge circuit. The second rendering clearly shows the equivalence between the bridge topology and a topology derived by series and parallel combinations. The third rendering is more commonly known as lattice topology. It is not so obvious that this is topologically equivalent. It can be seen that this is indeed so by visualising the top left node moved to the right of the top right node.
It is normal to call a network bridge topology only if it is being used as a two-port network with the input and output ports each consisting of a pair of diagonally opposite nodes. The box topology in figure 1.7 can be seen to be identical to bridge topology but in the case of the filter the input and output ports are each a pair of adjacent nodes. Sometimes the loading (or null indication) component on the output port of the bridge will be included in the bridge topology as shown in figure 1.9.Farago, pp.117–118
Bridged T and twin-T topologies
Bridged T topology is derived from bridge topology in a way explained in the Zobel network article. Many derivative topologies are also discussed in the same article.
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. This may be, for instance, because the input and output connections are made with co-axial topology. Connecting an input and output terminal is not allowable with normal bridge topology, so Twin-T is used where a bridge would otherwise be used for balance or null measurement applications. The topology is also used in the twin-T oscillator as a sine-wave generator. The lower part of figure 1.11 shows twin-T topology redrawn to emphasise the connection with bridge topology.Farago, pp. 125–127
Infinite topologies
Ladder topology can be extended without limit and is much used in filter designs. There are many variations on ladder topology, some of which are discussed in the Electronic filter topology and Composite image filter articles.
The balanced form of ladder topology can be viewed as being the graph of the side of a prism of arbitrary order. The side of an antiprism forms a topology which, in this sense, is an anti-ladder. Anti-ladder topology finds an application in voltage multiplier circuits, in particular the Cockcroft–Walton generator. There is also a full-wave version of the Cockcroft–Walton generator which uses a double anti-ladder topology.Campbell, pp.5–6, Kind and Fesser, pp.29–30
Infinite topologies can also be formed by cascading multiple sections of some other simple topology, such as lattice or bridge-T sections. 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.Campbell, pp.5–6, 20
Components with more than two terminals Circuits containing components with three or more terminals greatly increase the number of possible topologies. Conversely, the number of different circuits represented by a topology diminishes and in many cases the circuit is easily recognisable from the topology even when specific components are not identified.
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.Farago, pp. 98–134
Graph theory Graph theory is the branch of mathematics dealing with graphs. In network analysis, graphs are used extensively to represent a network being analysed. The graph of a network captures only certain aspects of a network: those aspects related to its connectivity, or, in other words, its topology. This can be a useful representation and generalisation of a network because many network equations are invariant across networks with the same topology. This includes equations derived from Kirchhoff's laws and Tellegen's theorem.Suresh, pp.483–484, 530–532
History Graph theory has been used in the network analysis of linear, passive networks almost from the moment that Kirchhoff's laws were formulated. Gustav Kirchhoff himself, in 1847, used graphs as an abstract representation of a network in his loop analysis of resistive circuits.Kirchhoff, G. (1847) "Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird" (On the solution of the equations to which one is led during the investigation of the linear distribution of galvanic currents), Annalen der Physik und Chemie, 72 (12) : 497–508. This approach was later generalised to RLC circuits, replacing resistances with impedances. In 1873 James Clerk Maxwell provided the dual of this analysis with node analysis.James Clerk Maxwell, A Treatise on Electricity and Magnetism (Oxford, England: Clarendon Press, 1873), vol. 1, Part II, "On linear systems of conductors in general", pp. 333–336.Wataru Mayeda and Sundaram Seshu (November 1957) "Topological Formulas for Network Functions," University of Illinois Engineering Experiment Station Bulletin, no. 446, p. 5. 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,H. Poincaré (1900) "Second complément à l'Analysis Situs", Proceedings of the London Mathematical Society, 32 : 277–308. Available on-line at: Mocavo.com hence founding the field of algebraic topology. In 1916 Oswald Veblen applied the algebraic topology of Poincaré to Kirchhoff's analysis.Oswald Veblen, The Cambridge Colloquium 1916, (New York : American Mathematical Society, 1918-1922), vol 5, pt. 2 : Analysis Situs, "Matrices of orientation", pp. 25-27. Veblen is also responsible for the introduction of the spanning tree to aid choosing a compatible set of network variables.Cederbaum, p.64