438:. There are several ways that bridge topology is rendered in circuit diagrams. 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.
442:
1692:. Topological methods, on the other hand, do not start from a given canonical form. Rather, the form is a result of the mathematical representation. Some canonical forms require mutual inductances for their realisation. 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.
417:
217:
625:
1717:
be defined which have a driving-point impedance which depends on the termination at infinity. Another unphysical property of theoretical infinite networks is that, in general, they will dissipate infinite power unless constraints are placed on them in addition to the usual network laws such as Ohm's and
Kirchhoff's laws. There are, however, some real-world applications. The transmission line example is one of a class of practical problems that can be modelled by infinitesimal elements (the
601:
319:
235:
546:
203:
522:
1616:
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338:
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difference between the number of variables in a loop analysis to a node analysis. 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 result from Kishi and
Katajini in 1967 is that the absolute minimum number of variables required to describe the behaviour of the network is given by the maximum distance between any two spanning
1137:. Ability to map onto a plane or a sphere are equivalent conditions. Any finite graph mapped onto a plane can be shrunk until it will map onto a small region of a sphere. Conversely, a mesh of any graph mapped onto a sphere can be stretched until the space inside it occupies nearly all of the sphere. The entire graph then occupies only a small region of the sphere. This is the same as the first case, hence the graph will also map onto a plane.
369:
474:
492:
1093:
925:
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168:
773:
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865:, to capture the uniqueness of the branches and nodes. For instance, a graph consisting of a square of branches would still be the same topological graph if two branches were interchanged unless the branches were uniquely labelled. In directed graphs, the two nodes that a branch connects to are designated the source and target nodes. Typically, these will be indicated by an arrow drawn on the branch.
1584:. Signal-flow graphs are weighted, directed graphs. He used these to analyse circuits containing mutual couplings and active networks. The weight of a directed edge in these graphs represents a gain, such as possessed by an amplifier. In general, signal-flow graphs, unlike the regular directed graphs described above, do not correspond to the topology of the physical arrangement of components.
1149:. A tie set is formed by allowing all but one of the graph links to be open circuit. A cut set is formed by allowing all but one of the tree branches to be short circuit. The cut set consists of the tree branch which was not short-circuited and any of the links which are not short-circuited by the other tree branches. A cut set of a graph produces two disjoint
916:, that is, a graph with two sets of nodes which have no branches incident to a node in each set. Two such separate parts are considered an equivalent graph to one where the parts are joined by combining a node from each into a single node. Likewise, a graph that can be split into two separate parts by splitting a node in two is also considered equivalent.
1561:
would, however, make a practical difference if a circuit were to be implemented this way in that it would destroy the isolation between the parts. An example would be a transformer earthed on both the primary and secondary side. The transformer still functions as a transformer with the same voltage ratio but can now no longer be used as an
666:. 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
1588:
Passive components will have identical branches in both trees but active components may not. The method relies on identifying spanning trees that are common to both graphs. An alternative method of extending the classical approach which requires only one graph was proposed by Chen in 1965. Chen's method is based on a
348:
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. 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.
1634:
which connect the hyperedge to the nodes. The graphical representation of a hyperedge may be a box (compared to the edge which is a line) and the representations of its tentacles are lines from the box to the connected nodes. In a directed hypergraph, the tentacles carry labels which are determined
1587:
The second approach is to extend the classical method so that it includes mutual couplings and active components. Several methods have been proposed for achieving this. In one of these, two graphs are constructed, one representing the currents in the circuit and the other representing the voltages.
1560:
with more than one separate part. For convenience of analysis, a graph with multiple parts can be combined into a single graph by unifying one node in each part into a single node. This makes no difference to the theoretical behaviour of the circuit so analysis carried out on it is still valid. It
1477:
Duals cannot be formed for every graph. Duality requires that every tie set has a dual cut set in the dual graph. This condition is met if and only if the graph is mappable on to a sphere with no branches crossing. To see this, note that a tie set is required to "tie off" a graph into two portions
911:
In the field of electrical networks, there are two additional transforms that are considered to result in equivalent graphs which do not produce congruent graphs. The first of these is the interchange of series connected branches. This is the dual of interchange of parallel connected branches which
1716:
Infinite networks are largely of only theoretical interest and are the plaything of mathematicians. Infinite networks that are not constrained by real-world restrictions can have some very unphysical properties. For instance
Kirchhoff's laws can fail in some cases and infinite resistor ladders can
1659:
Classical network analysis develops a set of network equations whose network variables are homogeneous in either current (loop analysis) or voltage (node analysis). The set of network variables so found is not necessarily the minimum necessary to form a set of independent equations. There may be a
1646:
Hypergraphs can be characterised by their incidence matrices. A regular graph containing only two-terminal components will have exactly two non-zero entries in each row. Any incidence matrix with more than two non-zero entries in any row is a representation of a hypergraph. The number of non-zero
1732:
Transfinite networks extend the idea of infinite networks even further. A node at an extremity of an infinite network can have another branch connected to it leading to another network. This new network can itself be infinite. Thus, topologies can be constructed which have pairs of nodes with no
1363:
The dual of a graph is another graph. For a given tree in a graph, the complementary set of branches (i.e., the branches not in the tree) form a tree in the dual graph. The set of current loop equations associated with the tie sets of the original graph and tree is identical to the set of voltage
592:
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
1516:
The inverse of this transform is the Δ-Y transform which analytically corresponds to the elimination of a mesh current and topologically corresponds to the elimination of a mesh. However, elimination of a mesh current whose mesh has branches in common with an arbitrary number of other meshes will
1346:
and provides a direct expression for the unknown variable in terms of determinants. This is useful in that it provides a compact expression for the solution. However, for anything more than the most trivial networks, a greater calculation effort is required for this method when working manually.
1144:
to the loop current method. Here the voltage associated with pairs of nodes are the primary variables and the branch voltages are found in terms of them. In this method also, a particular tree of the graph must be chosen in order to ensure that all the variables are independent. The dual of the
883:
on that vertex. The incidence of a graph can be captured in matrix format with a matrix called an incidence matrix. In fact, the incidence matrix is an alternative mathematical representation of the graph which dispenses with the need for any kind of drawing. Matrix rows correspond to nodes and
347:
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
1076:
links removed and there can be no currents in a tree. Since the remaining branches of the tree have zero current they cannot be independent of the link currents. The branch currents chosen as a set of independent variables must be a set associated with the links of a tree: one cannot choose any
759:
For a long time topology in electrical circuit theory remained concerned only with linear passive networks. The more recent developments of semiconductor devices and circuits have required new tools in topology to deal with them. Enormous increases in circuit complexity have led to the use of
1712:
in 1881. 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. It was not until the late 20th century that tools for analysing infinite networks with an arbitrary topology became available.
1127:
It is possible to choose a set of independent loop currents without reference to the trees and tie sets. A sufficient, but not necessary, condition for choosing a set of independent loops is to ensure that each chosen loop includes at least one branch that was not previously included by loops
1107:
rather than branch currents. The branch currents are then found in terms of the loop currents. Again, the set of loop currents cannot be chosen arbitrarily. To guarantee a set of independent variables the loop currents must be those associated with a certain set of loops. This set of loops
1067:
of the elements of which they are composed. A complete solution of the network can therefore be either in terms of branch currents or branch voltages only. Nor are all the branch currents independent from each other. The minimum number of branch currents required for a complete solution is
333:
is available for linear circuits. This transform is important because there are some networks that 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
1153:, that is, it cuts the graph into two parts, and is the minimum set of branches needed to do so. The set of network equations are formed by equating the node pair voltages to the algebraic sum of the cut set branch voltages. The dual of the special case of mesh analysis is
163:
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.
132:, 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
73:
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
49:
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
509:. Connecting together an input and output terminal is not allowable with normal bridge topology and for this reason Twin-T is used where a bridge would otherwise be used for balance or null measurement applications. The topology is also used in the
61:
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
1485:. A tie set that passes through a hole in a torus will fail to tie the graph into two parts. Consequently, the dual graph will not be cut into two parts and will not contain the required cut set. Consequently, only planar graphs have duals.
579:
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
1108:
consists of those loops formed by replacing a single link of a given tree of the graph of the circuit to be analysed. Since replacing a single link in a tree forms exactly one unique loop, the number of loop currents so defined is equal to
939:
is a graph in which all the nodes are connected, either directly or indirectly, by branches, but without forming any closed loops. Since there are no closed loops, there are no currents in a tree. In network analysis, we are interested in
833:
of the network. Elements are represented as the edges of the graph. An edge is drawn as a line, terminating on dots or small circles from which other edges (elements) may emanate. In circuit analysis, the edges of the graph are called
695:
provided the dual of this analysis with node analysis. 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
1695:
Graph theory is at its most powerful in network synthesis when the elements of the network can be represented by real numbers (one-element-kind networks such as resistive networks) or binary states (such as switching networks).
1088:
branch voltages. This is a consequence the fact that short-circuiting all the branches of a tree results in the voltage being zero everywhere. The link voltages cannot, therefore, be independent of the tree branch voltages.
884:
matrix columns correspond to branches. The elements of the matrix are either zero, for no incidence, or one, for incidence between the node and branch. Direction in directed graphs is indicated by the sign of the element.
1322:
in order to find the values of the network variables. This set of equations can be expressed in a matrix format which leads to a characteristic parameter matrix for the network. Parameter matrices take the form of an
690:
himself, in 1847, used graphs as an abstract representation of a network in his loop analysis of resistive circuits. This approach was later generalised to RLC circuits, replacing resistances with impedances. In 1873
1857:
between trees is defined as the number of edges that are in one tree but not in the other. That is, it is the number of edges which must be changed in order to transform one tree into the other (Kishi and
Kajitani,
1132:
in which the loops are all chosen to be meshes. Mesh analysis can only be applied if it is possible to map the graph on to a plane or a sphere without any of the branches crossing over. Such graphs are called
1062:
The goal of circuit analysis is to determine all the branch currents and voltages in the network. These network variables are not all independent. The branch voltages are related to the branch currents by the
828:
Conversely, topology is concerned only with the geometric relationship between the elements of a network, not with the kind of elements themselves. The heart of a topological representation of a network is the
460:
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
1779:(Guillemin, p.xv). Guillemin says the name was chosen because if the branches of the tie set were reduced to zero length the graph would become "tied off" as a fishnet with a drawstring (Guillemin, p.17).
178:
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
1504:
of a node voltage from a set of network equations corresponds topologically to the elimination of that node from the graph. For a node connected to three other nodes, this corresponds to the well known
1355:
Two graphs are dual when the relationship between branches and node pairs in one is the same as the relationship between branches and loops in the other. The dual of a graph can be found entirely by a
1272:
Rank plays the same role in nodal analysis as nullity plays in mesh analysis. That is, it gives the number of node voltage equations required. Rank and nullity are dual concepts and are related by;
1568:
More recent techniques in graph theory are able to deal with active components, which are also problematic in conventional theory. These new techniques are also able to deal with mutual couplings.
1318:
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
391:
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.
787:
making them up. In a circuit diagram these element-kinds are specifically drawn, each with its own unique symbol. Resistive networks are one-element-kind networks, consisting only of
1224:
The nullity of a graph represents the number of degrees of freedom of its set of network equations. For a planar graph, the nullity is equal to the number of meshes in the graph.
278:
1635:
by the hyperedge's label. A conventional directed graph can be thought of as a hypergraph with hyperedges each of which has two tentacles. These two tentacles are labelled
1492:
since there is no corresponding capacitive element. Equivalent circuits can be developed which do have duals, but the dual cannot be formed of a mutual inductance directly.
58:, it is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.
2129:
Wataru Mayeda and
Sundaram Seshu (November 1957) "Topological Formulas for Network Functions," University of Illinois Engineering Experiment Station Bulletin, no. 446, p. 5.
2150:
1805:
This term is another coining by
Guillemin (Guillemin, p.xv). So named because the spaces in a graph traversed by passing through the links has the form of a puzzle maze.
1041:
1517:
not, in general, result in a realisable graph. This is because the graph of the transform of the general star is a graph which will not map on to a sphere (it contains
1219:
1308:
1267:
1000:
298:
2748:
Applications of Graph
Transformations with Industrial Relevance: international workshop, AGTIVE'99, Kerkrade, The Netherlands, September 1–3, 1999: proceedings
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unless otherwise stated. A given network graph can contain a number of different trees. The branches removed from a graph in order to form a tree are called
465:
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.
1626:
In a conventional representation components are represented by edges, each of which connects to two nodes. In a hypergraph, components are represented by
1478:
and its dual, the cut set, is required to cut a graph into two portions. The graph of a finite network which will not map on to a sphere will require an
113:
854:
are terms that can be used interchangeably when discussing graphs of networks. Figure 2.2 shows a graph representation of the circuit in figure 2.1.
1759:. Yokes are branches in parallel, chains are branches in series.(MacMahon, 1891, p.330) A single branch can be considered either a yoke or a chain.
534:
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
1815:
1722:
904:; bending and stretching the branches; and crossing or knotting the branches. Two graphs which are equivalent through deformation are said to be
398:. Longer sections would normally be described as ladder topology. These kinds of circuits are commonly analysed and characterised in terms of a
1552:
In conventional graph representation of circuits, there is no means of explicitly representing mutual inductive couplings, such as occurs in a
1521:
and hence multiple crossovers). The dual of such a graph cannot exist, but is the graph required to represent a generalised mesh elimination.
2636:
2621:
2599:
686:
Graph theory has been used in the network analysis of linear, passive networks almost from the moment that
Kirchhoff's laws were formulated.
752:
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
944:, that is, trees that connect every node present in the graph of the network. In this article, spanning tree is meant by an unqualified
513:
as a sine wave generator. The lower part of figure 1.11 shows twin-T topology redrawn to emphasise the connection with bridge topology.
426:
Bridge topology is an important topology with many uses in both linear and non-linear applications, including, amongst many others, the
307:
501:
There is also a twin-T topology which has practical applications where it is desirable to have the input and output share a common (
2094:"Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird"
912:
can be achieved by deformation without the need for a special rule. The second is concerned with graphs divided into two or more
2096:(On the solution of the equations to which one is led during the investigation of the linear distribution of galvanic currents),
2147:
1501:
329:
Y and Δ are important topologies in linear network analysis due to these being the simplest possible three-terminal networks. A
90:
46:
1124:. The set of network equations are formed by equating the loop currents to the algebraic sum of the tie set branch currents.
1622:. An example of a hypergraph. Regular edges are shown in black, hyperedges are shown in blue, and tentacles are shown in red.
1781:
Guillemin was a leading figure in the development and teaching of linear network analysis (Wildes and
Lindgren, pp.154–159).
712:
applied the algebraic topology of
Poincaré to Kirchhoff's analysis. Veblen is also responsible for the introduction of the
85:. In particular, for networks which contain only two-terminal devices, circuit topology can be viewed as an application of
1643:
and usually indicated by an arrow. In a general hypergraph with more tentacles, more complex labelling will be required.
1576:
There are two basic approaches available for dealing with mutual couplings and active components. In the first of these,
892:
Graphs are equivalent if one can be transformed into the other by deformation. Deformation can include the operations of
227:
or potential divider is used for circuits of that purpose. L-section is a common name for the topology in filter design.
2694:
MacMahon, Percy A., "Yoke-chains and multipartite compositions in connexion with the analytical forms called “Trees”",
1500:
Operations on a set of network equations have a topological meaning which can aid visualisation of what is happening.
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2821:
2806:
2791:
2776:
2755:
2673:
1474:
It consists of spaces connected by links in the same way that the tree consists of nodes connected by tree branches.
1612:
have many more connections than this. This problem can be overcome by using hypergraphs instead of regular graphs.
2859:
2854:
576:. There is also a full-wave version of the Cockcroft-Walton generator which uses a double anti-ladder topology.
740:
in 1892) who limited his survey to series and parallel combinations. MacMahon called these graphs yoke-chains.
17:
1150:
1141:
1704:
Perhaps, the earliest network with an infinite graph to be studied was the ladder network used to represent
2588:
1892:
830:
663:
557:
248:
Series and parallel topologies can continue to be constructed with greater and greater numbers of branches
196:
144:
etc., but topology is not concerned with the type of component in the network, so the symbol for a general
2117:(Oxford, England: Clarendon Press, 1873), vol. 1, Part II, "On linear systems of conductors in general",
879:
Incidence is one of the basic properties of a graph. An edge that is connected to a vertex is said to be
1557:
671:
615:
568:
forms a topology which, in this sense, is an anti-ladder. Anti-ladder topology finds an application in
245:
Note that the parallel-series topology is another representation of the Delta topology discussed later.
108:
Standard graph theory can be extended to deal with active components and multi-terminal devices such as
2680:
573:
535:
363:
255:
1882:
1718:
213:
Even for these simplest of topologies, there are variations in the way the circuit can be presented.
2568:
Brittain, James E., The introduction of the loading coil: George A. Campbell and Michael I. Pupin",
2169:, (New York : American Mathematical Society, 1918-1922), vol 5, pt. 2 : Analysis Situs,
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388:
2170:
1681:
1011:
667:
435:
1509:. The transform can be extended to greater numbers of connected nodes and is then known as the
1357:
1689:
1577:
1537:
1183:
1099:. A cut set of the graph in figure 2.2 derived from the tree of figure 2.3 by cutting branch 3.
897:
893:
753:
745:
539:
505:) terminal. This may be, for instance, because the input and output connections are made with
2714:
2118:
2093:
1651:
of the corresponding branch, and the highest branch rank is the rank of the incidence matrix.
1608:
has three connection points, but a normal graph branch may only connect to two nodes. Modern
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1562:
1319:
1278:
1237:
970:
839:
675:
457:
98:
30:
This article is about the topology of electrical circuits. For the topology of polymers, see
2799:
BTEC First Engineering: Mandatory and Selected Optional Units for BTEC Firsts in Engineering
2483:
Samuel J. Mason (September 1953) "Feedback theory — Some properties of signal flow graphs,"
1684:
by Cauer's canonical ladder network or Foster's canonical form or Brune's realisation of an
1600:
Another way of extending classical graph theory for active components is through the use of
730:
Comprehensive cataloguing of network graphs as they apply to electrical circuits began with
283:
1335:
502:
145:
129:
2812:
Wildes, Karl L.; Lindgren, Nilo A., "Network analysis and synthesis: Ernst A. Guillemin",
8:
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692:
352:
109:
102:
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There is an approach to choosing network variables with voltages which is analogous and
1609:
784:
705:
581:
569:
195:
For a network with two components or branches, there are only two possible topologies:
42:
1930:
521:
2832:
2817:
2802:
2787:
2772:
2751:
2740:
2669:
2655:
1946:
1870:. A forest of trees in which every node of the graph is visited by one of the trees.
1705:
1673:
1581:
1489:
1328:
1084:
In terms of branch voltages, a complete solution of the network can be obtained with
1064:
644:
510:
431:
384:
66:
697:
2736:
2703:
2577:
1983:
1942:
1887:
1737:
between them. Such networks of infinite networks are called transfinite networks.
1709:
1676:. Classical network synthesis realises the required network in one of a number of
1343:
1324:
874:
741:
701:
687:
648:
632:
561:
488:
article. There are many derivative topologies also discussed in the same article.
453:
441:
427:
399:
376:
75:
31:
1506:
779:. Graph of the ladder network shown in figure 2.1 with a four rung ladder assumed.
330:
2154:
1776:
1726:
1334:
These equations can be solved in a number of well-known ways. One method is the
949:
843:
822:
736:
531:
395:
224:
125:
94:
51:
931:. One possible tree of the graph in figure 2.2. Links are shown as dotted lines.
791:
elements. Likewise capacitive or inductive networks are one-element-kind. The
416:
2707:
1677:
1604:. Some electronic components are not represented naturally using graphs. The
1367:
The following table lists dual concepts in topology related to circuit theory.
1154:
862:
858:
731:
726:
Circuit diagram of a ladder network low-pass filter: a two-element-kind network
612:
506:
411:
184:
670:
across networks with the same topology. This includes equations derived from
624:
216:
155:
section of this article gives an alternative method of representing networks.
2848:
1756:
1129:
941:
825:
can have many elements but is another example of a two-element-kind network.
761:
713:
709:
643:
With more complex circuits the description may proceed by specification of a
485:
484:
Bridged T topology is derived from bridge topology in a way explained in the
452:
It is normal to call a network bridge topology only if it is being used as a
128:
in this article follow the usual conventions in electronics; lines represent
1518:
1134:
1104:
756:
in 1920 on 4-port telephone repeaters and produced 83,539 distinct graphs.
659:
600:
223:. All these topologies are identical. Series topology is a general name.
86:
2814:
A Century of Electrical Engineering and Computer Science at MIT, 1882–1982
2576:, no. 1, pp. 36–57, The Johns Hopkins University Press, January 1970
1987:
1479:
280:
series or parallel branches is 1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624,
1968:
1589:
1553:
1339:
811:
318:
180:
2637:"Maximum output networks for telephone substation and repeater circuits"
234:
2659:
1685:
1605:
1601:
1128:
already chosen. A particularly straightforward choice is that used in
1120:
in graph theory. The set of branches forming a given loop is called a
804:
799:
793:
545:
231:
For a network with three branches, there are four possible topologies.
2782:
Suresh, Kumar K. S., "Introduction to network topology" chapter 11 in
202:
2600:"Some applications of graph theory to network analysis and synthesis"
1627:
1615:
1529:
1327:
if the equations have been formed on a loop-analysis basis, or as an
608:
565:
337:
141:
70:
2581:
1630:
which can connect to an arbitrary number of nodes. Hyperedges have
1364:
node-pair equations associated with the cut sets of the dual graph.
473:
368:
2681:"Maximally distant trees and principal partition of a linear graph"
764:
in graph theory to improve the efficiency of computer calculation.
137:
133:
82:
63:
55:
491:
1721:). Other examples are launching waves into a continuous medium,
1146:
1092:
924:
719:
556:
The balanced form of ladder topology can be viewed as being the
93:
of such a circuit from a topological point of view, the network
857:
Graphs used in network analysis are usually, in addition, both
772:
167:
2641:
Transactions of the American Institute of Electrical Engineers
2626:
Transactions of the American Institute of Electrical Engineers
1540:
circuit frequently used to couple stages of tuned amplifiers.
1836:
Fifth Annual Allerton Conference on Circuit and System Theory
1834:
Kishi, Genya; Kajitani, Yoji, "On maximally distinct trees",
1370:
584:, but are rarely used as a practical circuit implementation.
252:. The number of unique topologies that can be obtained from
2746:
Minas, M., "Creating semantic representations of diagrams",
1331:
if the equations have been formed on a node-analysis basis.
861:, to capture the direction of current flow and voltage, and
2138:
H. Poincaré (1900) "Second complément à l'Analysis Situs",
651:
of the network rather than the topology of the components.
394:
All these topologies can be viewed as a short section of a
302:
587:
27:
Form taken by the network of interconnections of a circuit
383:
The topologies shown in figure 1.7 are commonly used for
2443:
2441:
1793:
A mesh is a loop which does not enclose any other loops.
1680:. Examples of canonical forms are the realisation of a
1116:
in this context is not the same as the usual meaning of
1727:
measurement of resistance between points of a substrate
1548:, an equivalent graph with the disjoint parts combined.
716:
to aid choosing a compatible set of network variables.
2438:
1969:"The Number of Two-terminal Series-parallel Networks"
1281:
1240:
1186:
1072:. This is a consequence of the fact that a tree has
1014:
973:
286:
258:
241:. Series and parallel topologies with three branches
1488:
Duals also cannot be formed for networks containing
1845:
section for the full paper published later in 1969.
1005:An important relationship for circuit analysis is;
700:introduced the idea of representing a graph by its
209:. Series and parallel topologies with two branches
2470:
2468:
2466:
2464:
2462:
2182:
2180:
2178:
1302:
1261:
1213:
1035:
994:
468:
292:
272:
101:of graph theory, and the network branches are the
2735:, iss.Iss.2–3, pp. 225–228, 17 October 1994
1054:is the number of links removed to form the tree.
817:is the simplest three-element-kind network. The
448:. Bridge circuit with bridging output load shown
190:
2846:
1313:
952:, the branches remaining in the tree are called
783:Networks are commonly classified by the kind of
2459:
2175:
1831:A summary of this work was first presented at;
2696:Proceedings of the London Mathematical Society
2595:, November 1922, vol. 1, no. 2, pp. 1–32.
2140:Proceedings of the London Mathematical Society
1966:
767:
734:in 1891 (with an engineer friendly article in
112:. Graphs can also be used in the analysis of
2622:"Geometrical circuits of electrical networks"
2589:"Physical theory of the electric wave-filter"
174:. T,Y and Star topologies are all identical.
1654:
1495:
1377:. The dual graph of the graph in figure 2.2.
960:nodes, the number of branches in each tree,
2668:, translator Y. Narayana Rao, Newnes, 2001
2617:, The English Universities Press Ltd, 1961.
2056:Campbell, pp.5–6, Kind and Fesser, pp.29–30
1967:Riordan, John; Shannon, C.E. (April 1942).
1822:, vol.13, iss.4, pp.438–439, December 1966.
1103:A common analysis approach is to solve for
1050:is the number of branches in the graph and
810:are simple two-element-kind networks. The
357:
2615:An Introduction to Linear Network Analysis
2146: : 277–308. Available on-line at:
1816:"Topological analysis for active networks"
662:is the branch of mathematics dealing with
81:Electronic network topology is related to
2604:IEEE Transactions on Circuits and Systems
1544:, the graph of the double-tuned circuit.
1470:The dual of a tree is sometimes called a
266:
2654:, New York: John Wiley & Sons, 1953
2647:, iss.1, pp. 230–290, January 1920.
2635:Foster, Ronald M.; Campbell, George A.,
1928:
1614:
1528:
1369:
1091:
1057:
923:
771:
718:
623:
599:
544:
490:
472:
440:
415:
367:
336:
317:
233:
215:
201:
166:
2691:, iss.3, pp. 323–330, August 1969.
2115:A Treatise on Electricity and Magnetism
588:Components with more than two terminals
14:
2847:
2764:, William Collins Sons & Co, 1969.
2610:, iss.1, pp. 64–68, January 1984.
1556:, and such components may result in a
1338:. Another method involves the use of
516:
2632:, iss.2, pp. 309–317, June 1932.
2171:"Matrices of orientation", pp. 25-27.
313:
2816:, pp. 154–159, MIT Press, 1985
1699:
1667:
1571:
744:in 1932 categorised graphs by their
564:of arbitrary order. The side of an
2831:, Cambridge University Press, 1991
2750:, pp. 209–224, Springer, 2000
2685:IEEE Transactions on Circuit Theory
1929:MacMahon, P.A. (17 October 1994) .
1820:IEEE Transactions on Circuit Theory
1336:systematic elimination of variables
1160:
119:
24:
1976:Journal of Mathematics and Physics
1524:
919:
520:
405:
25:
2871:
2725:, pp. 601–602, 8 April 1892.
2547:Zemanian, pp.vii-ix, 17–18, 24–26
1842:
1708:developed, in its final form, by
821:ladder network commonly used for
273:{\displaystyle n\in \mathbb {N} }
158:
152:
2786:, Pearson Education India, 2010
1931:"The combination of resistances"
631:. Balanced amplifier such as a
56:mathematical concept of topology
2562:
2550:
2541:
2530:
2521:
2512:
2503:
2494:
2477:
2450:
2429:
2418:
2409:
2398:
2387:
2376:
2367:
2358:
2349:
2340:
2331:
2320:
2311:
2300:
2291:
2282:
2273:
2262:
2251:
2242:
2233:
2220:
2209:
2200:
2189:
2159:
2132:
2123:
2107:
2086:
2077:
2068:
2059:
2050:
1861:
1848:
1825:
1808:
1796:
1784:
1762:
1746:
1672:Graph theory can be applied to
842:of the graph and represent the
654:
469:Bridged T and twin-T topologies
2784:Electric Circuits And Networks
2679:Kishi, Genya; Kajitani, Yoji,
2041:
2032:
2021:
2012:
2001:
1960:
1922:
1913:
1904:
1595:
887:
704:, hence founding the field of
191:Series and parallel topologies
13:
1:
2715:"Combinations of resistances"
2593:Bell System Technical Journal
2167:The Cambridge Colloquium 1916
2098:Annalen der Physik und Chemie
1898:
1314:Solving the network variables
2829:Infinite Electrical Networks
2741:10.1016/0166-218X(94)90024-8
2729:Discrete Applied Mathematics
2666:High-voltage Test Techniques
1947:10.1016/0166-218X(94)90024-8
1935:Discrete Applied Mathematics
1893:Topological quantum computer
868:
572:circuits, in particular the
7:
2769:The Algorithm Design Manual
2664:Kind, Dieter; Feser, Kurt,
2652:Introductory Circuit Theory
2083:Suresh, pp.483–484, 530–532
1876:
1755:. A terminology coined by
1036:{\displaystyle b=\ell +t\ }
838:. The dots are called the
768:Graphs and circuit diagrams
616:bipolar junction transistor
10:
2876:
2197:Foster and Campbell, p.232
1350:
1231:of a graph is defined by;
872:
681:
574:Cockcroft-Walton generator
536:Electronic filter topology
456:with the input and output
409:
364:Electronic filter topology
361:
350:
29:
2762:Redifon Radio Diary, 1970
2485:Proceedings of the I.R.E.
2328:Suresh, p.518, pp.523–528
1883:Symbolic circuit analysis
1719:distributed-element model
1655:Non-homogeneous variables
1496:Node and mesh elimination
1214:{\displaystyle N=b-n+s\ }
45:is the form taken by the
2708:10.1112/plms/s1-22.1.330
1740:
1647:entries in a row is the
1383:Summary of dual concepts
1177:branches is defined by;
358:Simple filter topologies
2239:Suresh, pp.485, 487–489
2074:Farago, pp. 98–134
1690:positive-real functions
1682:driving-point impedance
1303:{\displaystyle R+N=b\ }
1262:{\displaystyle R=n-s\ }
995:{\displaystyle t=n-1\ }
552:. Anti-ladder topology
436:lattice phase equaliser
377:balanced and unbalanced
148:has been used instead.
2860:Electronic engineering
2855:Electrical engineering
2570:Technology and Culture
2491:(9) : 1144–1156.
2447:Guillemin, pp. 127–132
1664:of the network graph.
1623:
1578:Samuel Jefferson Mason
1549:
1378:
1304:
1263:
1215:
1100:
1081:branches arbitrarily.
1037:
996:
932:
780:
727:
636:
619:
553:
540:Composite image filter
528:
498:
481:
449:
423:
380:
344:
326:
294:
293:{\displaystyle \dots }
274:
242:
228:
210:
175:
2650:Guillemin, Ernst A.,
2373:Guillemin, pp.116–120
2364:Guillemin, pp.112–116
2113:James Clerk Maxwell,
2104:(12) : 497–508.
2092:Kirchhoff, G. (1847)
1988:10.1002/sapm194221183
1618:
1563:isolation transformer
1532:
1373:
1320:solved simultaneously
1305:
1264:
1216:
1095:
1058:Tie sets and cut sets
1038:
997:
927:
775:
722:
627:
603:
548:
524:
494:
480:. Bridged T topology
476:
444:
419:
371:
340:
325:. Y and Δ topologies
321:
295:
275:
237:
219:
205:
170:
83:mathematical topology
2827:Zemanian, Armen H.,
2713:MacMahon, Percy A.,
2065:Campbell, pp.5–6, 20
1729:or down a borehole.
1342:. This is known as
1279:
1238:
1184:
1012:
971:
956:. For a graph with
284:
256:
2767:Skiena, Steven S.,
2702:(1891), pp.330–346
2620:Foster, Ronald M.,
2415:Guillemin, pp.51–53
2404:Guillemin, pp.43–44
2393:Guillemin, pp.49–50
2355:Guillemin, pp.64–81
2317:Guillemin, pp.17–23
2306:Guillemin, pp.23–27
2297:Guillemin, pp.10–17
2047:Farago, pp. 125–127
1910:Tooley, pp. 258–264
1838:, pp.635–643, 1967.
1610:integrated circuits
1580:in 1953 introduced
1511:star-mesh transform
1453:Parallel connection
1385:
1173:separate parts and
785:electrical elements
693:James Clerk Maxwell
517:Infinite topologies
353:Star-mesh transform
197:series and parallel
110:integrated circuits
54:; similarly to the
2801:, Routledge, 2010
2771:, Springer, 2008,
2384:Suresh, pp.516–517
2346:Foster, pp.312–313
2337:Foster, pp.310–311
2288:Guillemin, pp.9–10
2153:2014-11-01 at the
2038:Farago, pp.117–118
2027:Farago, pp.112–116
1706:transmission lines
1624:
1582:signal-flow graphs
1558:disconnected graph
1550:
1490:mutual inductances
1456:Series connection
1381:
1379:
1300:
1259:
1211:
1169:, of a graph with
1101:
1033:
992:
933:
781:
728:
706:algebraic topology
676:Tellegen's theorem
637:
620:
582:transmission lines
570:voltage multiplier
554:
529:
499:
482:
450:
424:
381:
345:
327:
314:Y and Δ topologies
290:
270:
243:
229:
211:
183:) topology and Δ (
176:
43:electronic circuit
2587:Campbell, G. A.,
2509:Minas, pp.213–214
2456:Guillemin, pp.6–7
2279:Guillemin, pp.8–9
2228:Minas, pp.213–214
2215:Guillemin, pp.5–6
2029:Redifon, pp.45–48
1919:Guillemin, pp.5–6
1700:Infinite networks
1674:network synthesis
1668:Network synthesis
1572:Active components
1468:
1467:
1329:admittance matrix
1299:
1258:
1210:
1065:transfer function
1032:
991:
846:of the network.
645:transfer function
641:
640:
611:topology such as
560:of the side of a
511:twin-T oscillator
507:co-axial topology
432:Wheatstone bridge
379:filter topologies
114:infinite networks
105:of graph theory.
76:prototype network
16:(Redirected from
2867:
2557:
2554:
2548:
2545:
2539:
2534:
2528:
2525:
2519:
2516:
2510:
2507:
2501:
2498:
2492:
2481:
2475:
2472:
2457:
2454:
2448:
2445:
2436:
2435:Guillemin, p.536
2433:
2427:
2424:Guillemin, p.535
2422:
2416:
2413:
2407:
2402:
2396:
2391:
2385:
2380:
2374:
2371:
2365:
2362:
2356:
2353:
2347:
2344:
2338:
2335:
2329:
2324:
2318:
2315:
2309:
2304:
2298:
2295:
2289:
2286:
2280:
2277:
2271:
2266:
2260:
2257:Guillemin, p.6-7
2255:
2249:
2246:
2240:
2237:
2231:
2224:
2218:
2213:
2207:
2204:
2198:
2193:
2187:
2184:
2173:
2163:
2157:
2136:
2130:
2127:
2121:
2111:
2105:
2090:
2084:
2081:
2075:
2072:
2066:
2063:
2057:
2054:
2048:
2045:
2039:
2036:
2030:
2025:
2019:
2016:
2010:
2007:Farago, pp.18–21
2005:
1999:
1998:
1996:
1994:
1973:
1964:
1958:
1957:
1955:
1953:
1941:(2–3): 225–228.
1926:
1920:
1917:
1911:
1908:
1888:Network topology
1871:
1865:
1859:
1852:
1846:
1829:
1823:
1814:Chen, Wai-Kai.,
1812:
1806:
1800:
1794:
1788:
1782:
1766:
1760:
1750:
1710:Oliver Heaviside
1386:
1380:
1358:graphical method
1325:impedance matrix
1309:
1307:
1306:
1301:
1297:
1268:
1266:
1265:
1260:
1256:
1220:
1218:
1217:
1212:
1208:
1161:Nullity and rank
1042:
1040:
1039:
1034:
1030:
1001:
999:
998:
993:
989:
875:incidence matrix
823:low-pass filters
742:Ronald M. Foster
702:incidence matrix
688:Gustav Kirchhoff
672:Kirchhoff's laws
633:long-tailed pair
596:
595:
593:not identified.
454:two-port network
428:bridge rectifier
400:two-port network
305:
299:
297:
296:
291:
279:
277:
276:
271:
269:
126:circuit diagrams
120:Circuit diagrams
91:network analysis
39:circuit topology
32:Circuit topology
21:
2875:
2874:
2870:
2869:
2868:
2866:
2865:
2864:
2845:
2844:
2726:
2719:The Electrician
2613:Farago, P. S.,
2598:Cederbaum, I.,
2582:10.2307/3102809
2565:
2560:
2555:
2551:
2546:
2542:
2538:Zemanian, p.vii
2537:
2535:
2531:
2527:Cederbaum, p.67
2526:
2522:
2517:
2513:
2508:
2504:
2499:
2495:
2482:
2478:
2474:Cederbaum, p.65
2473:
2460:
2455:
2451:
2446:
2439:
2434:
2430:
2425:
2423:
2419:
2414:
2410:
2405:
2403:
2399:
2394:
2392:
2388:
2383:
2382:Guillemin, p.44
2381:
2377:
2372:
2368:
2363:
2359:
2354:
2350:
2345:
2341:
2336:
2332:
2327:
2326:Guillemin, p.43
2325:
2321:
2316:
2312:
2307:
2305:
2301:
2296:
2292:
2287:
2283:
2278:
2274:
2269:
2268:Guillemin, p. 7
2267:
2263:
2258:
2256:
2252:
2247:
2243:
2238:
2234:
2229:
2227:
2225:
2221:
2216:
2214:
2210:
2205:
2201:
2196:
2194:
2190:
2186:Cederbaum, p.64
2185:
2176:
2165:Oswald Veblen,
2164:
2160:
2155:Wayback Machine
2137:
2133:
2128:
2124:
2112:
2108:
2091:
2087:
2082:
2078:
2073:
2069:
2064:
2060:
2055:
2051:
2046:
2042:
2037:
2033:
2028:
2026:
2022:
2017:
2013:
2008:
2006:
2002:
1992:
1990:
1971:
1965:
1961:
1951:
1949:
1927:
1923:
1918:
1914:
1909:
1905:
1901:
1879:
1874:
1868:Spanning forest
1866:
1862:
1853:
1849:
1830:
1826:
1813:
1809:
1801:
1797:
1789:
1785:
1780:
1777:Ernst Guillemin
1767:
1763:
1751:
1747:
1743:
1702:
1678:canonical forms
1670:
1657:
1598:
1574:
1527:
1525:Mutual coupling
1498:
1353:
1316:
1280:
1277:
1276:
1239:
1236:
1235:
1185:
1182:
1181:
1163:
1145:tie set is the
1060:
1013:
1010:
1009:
972:
969:
968:
922:
920:Trees and links
890:
877:
871:
863:labelled graphs
859:directed graphs
770:
754:George Campbell
737:The Electrician
684:
657:
590:
532:Ladder topology
519:
471:
414:
408:
406:Bridge topology
396:ladder topology
366:
360:
355:
316:
301:
285:
282:
281:
265:
257:
254:
253:
225:Voltage divider
193:
161:
122:
52:circuit diagram
35:
28:
23:
22:
15:
12:
11:
5:
2873:
2863:
2862:
2857:
2841:
2840:
2825:
2810:
2797:Tooley, Mike,
2795:
2780:
2765:
2759:
2744:
2711:
2692:
2677:
2662:
2648:
2633:
2618:
2611:
2596:
2585:
2564:
2561:
2559:
2558:
2549:
2540:
2536:Brittain, p.39
2529:
2520:
2511:
2502:
2493:
2476:
2458:
2449:
2437:
2428:
2417:
2408:
2397:
2386:
2375:
2366:
2357:
2348:
2339:
2330:
2319:
2310:
2299:
2290:
2281:
2272:
2270:Suresh, p. 486
2261:
2250:
2241:
2232:
2226:Guillemin, p.5
2219:
2208:
2206:Guillemin, p.5
2199:
2188:
2174:
2158:
2131:
2122:
2106:
2085:
2076:
2067:
2058:
2049:
2040:
2031:
2020:
2011:
2000:
1982:(1–4): 83–93.
1959:
1921:
1912:
1902:
1900:
1897:
1896:
1895:
1890:
1885:
1878:
1875:
1873:
1872:
1860:
1847:
1840:
1839:
1824:
1807:
1795:
1783:
1775:was coined by
1761:
1744:
1742:
1739:
1725:problems, and
1723:fringing field
1701:
1698:
1669:
1666:
1656:
1653:
1597:
1594:
1573:
1570:
1526:
1523:
1497:
1494:
1466:
1465:
1462:
1458:
1457:
1454:
1450:
1449:
1446:
1442:
1441:
1438:
1434:
1433:
1430:
1426:
1425:
1422:
1418:
1417:
1414:
1410:
1409:
1406:
1402:
1401:
1398:
1394:
1393:
1390:
1352:
1349:
1315:
1312:
1311:
1310:
1296:
1293:
1290:
1287:
1284:
1270:
1269:
1255:
1252:
1249:
1246:
1243:
1222:
1221:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1162:
1159:
1155:nodal analysis
1059:
1056:
1044:
1043:
1029:
1026:
1023:
1020:
1017:
1003:
1002:
988:
985:
982:
979:
976:
942:spanning trees
921:
918:
914:separate parts
889:
886:
873:Main article:
870:
867:
769:
766:
732:Percy MacMahon
698:Henri Poincaré
683:
680:
656:
653:
639:
638:
621:
613:common emitter
589:
586:
518:
515:
470:
467:
412:Bridge circuit
410:Main article:
407:
404:
362:Main article:
359:
356:
315:
312:
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159:Topology names
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121:
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26:
18:Filter section
9:
6:
4:
3:
2:
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2837:0-521-40153-4
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2822:0-262-23119-0
2819:
2815:
2811:
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2807:1-85617-685-1
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2800:
2796:
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2792:81-317-5511-8
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2777:1-84800-069-3
2774:
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2757:
2756:3-540-67658-9
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2727:Reprinted in
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2674:0-7506-5183-0
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2597:
2594:
2590:
2586:
2583:
2579:
2575:
2571:
2567:
2566:
2556:Zemanian, p.x
2553:
2544:
2533:
2524:
2518:Skiena, p.382
2515:
2506:
2497:
2490:
2486:
2480:
2471:
2469:
2467:
2465:
2463:
2453:
2444:
2442:
2432:
2426:Suresh, p.517
2421:
2412:
2406:Foster, p.313
2401:
2395:Suresh, p.517
2390:
2379:
2370:
2361:
2352:
2343:
2334:
2323:
2314:
2303:
2294:
2285:
2276:
2265:
2259:Foster, p.310
2254:
2248:Foster, p.310
2245:
2236:
2230:Suresh, p.485
2223:
2217:Suresh, p.485
2212:
2203:
2195:Foster, p.309
2192:
2183:
2181:
2179:
2172:
2168:
2162:
2156:
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2110:
2103:
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2035:
2024:
2018:Redifon, p.22
2015:
2009:Redifon, p.22
2004:
1989:
1985:
1981:
1977:
1970:
1963:
1948:
1944:
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1757:Arthur Cayley
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1543:
1539:
1535:
1531:
1522:
1520:
1519:star polygons
1514:
1512:
1508:
1507:Y-Δ transform
1503:
1493:
1491:
1486:
1484:
1482:
1475:
1473:
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1459:
1455:
1452:
1451:
1448:Open circuit
1447:
1445:Short circuit
1444:
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1344:Cramer's rule
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1165:The nullity,
1158:
1156:
1152:
1148:
1143:
1138:
1136:
1135:planar graphs
1131:
1130:mesh analysis
1125:
1123:
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1111:
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1105:loop currents
1098:
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762:combinatorics
757:
755:
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714:spanning tree
711:
710:Oswald Veblen
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378:
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331:Y-Δ transform
324:
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69:results in a
68:
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40:
33:
19:
2842:
2828:
2813:
2798:
2783:
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2699:
2695:
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2665:
2651:
2644:
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2614:
2607:
2603:
2592:
2573:
2569:
2563:Bibliography
2552:
2543:
2532:
2523:
2514:
2505:
2500:Minas, p.213
2496:
2488:
2484:
2479:
2452:
2431:
2420:
2411:
2400:
2389:
2378:
2369:
2360:
2351:
2342:
2333:
2322:
2313:
2308:Suresh p.514
2302:
2293:
2284:
2275:
2264:
2253:
2244:
2235:
2222:
2211:
2202:
2191:
2166:
2161:
2143:
2139:
2134:
2125:
2119:pp. 333–336.
2114:
2109:
2101:
2097:
2088:
2079:
2070:
2061:
2052:
2043:
2034:
2023:
2014:
2003:
1991:. Retrieved
1979:
1975:
1962:
1950:. Retrieved
1938:
1934:
1924:
1915:
1906:
1867:
1863:
1854:
1850:
1843:Bibliography
1835:
1827:
1819:
1810:
1802:
1798:
1790:
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1772:
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1764:
1752:
1748:
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1631:
1625:
1619:
1599:
1586:
1575:
1567:
1551:
1545:
1541:
1538:double-tuned
1533:
1515:
1499:
1487:
1480:
1476:
1471:
1469:
1432:Tree branch
1382:
1374:
1366:
1362:
1354:
1340:determinants
1333:
1317:
1271:
1228:
1226:
1223:
1174:
1170:
1166:
1164:
1139:
1126:
1121:
1113:
1112:. The term
1109:
1102:
1096:
1085:
1083:
1078:
1073:
1069:
1061:
1051:
1047:
1045:
1004:
961:
957:
953:
945:
934:
928:
913:
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905:
891:
880:
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851:
847:
835:
827:
818:
812:
805:
798:
792:
788:
782:
776:
758:
735:
729:
723:
685:
660:Graph theory
658:
655:Graph theory
647:between the
642:
628:
604:
591:
578:
555:
549:
530:
525:
500:
495:
483:
477:
462:
451:
445:
425:
420:
393:
382:
372:
346:
341:
328:
322:
250:ad infinitum
249:
247:
244:
238:
230:
220:
212:
206:
194:
187:) topology.
177:
171:
162:
153:Graph theory
150:
123:
107:
87:graph theory
80:
60:
38:
36:
1753:Yoke-chains
1602:hypergraphs
1596:Hypergraphs
1590:rooted tree
1554:transformer
1502:Elimination
1483:-fold torus
964:, must be;
894:translation
888:Equivalence
724:Figure 2.1.
708:. In 1916
629:Figure 1.15
605:Figure 1.14
550:Figure 1.13
526:Figure 1.12
496:Figure 1.11
478:Figure 1.10
2849:Categories
2148:Mocavo.com
1899:References
1686:immittance
1628:hyperedges
1620:Figure 2.7
1606:transistor
1534:Figure 2.6
1424:Node pair
1375:Figure 2.5
1227:The rank,
1097:Figure 2.4
929:Figure 2.3
902:reflection
777:Figure 2.2
542:articles.
446:Figure 1.9
421:Figure 1.8
389:attenuator
375:. Common
373:Figure 1.7
351:See also:
342:Figure 1.6
334:windings.
323:Figure 1.5
300:(sequence
239:Figure 1.4
221:Figure 1.3
207:Figure 1.2
172:Figure 1.1
142:capacitors
130:conductors
1771:The term
1688:from his
1632:tentacles
1251:−
1197:−
1151:subgraphs
1022:ℓ
984:−
906:congruent
869:Incidence
668:invariant
635:amplifier
618:amplifier
609:amplifier
607:. Basic
566:antiprism
288:…
263:∈
146:impedance
138:inductors
134:resistors
71:high-pass
2151:Archived
1877:See also
1855:Distance
1841:See the
1769:Tie set.
1440:Cut set
1392:Voltage
898:rotation
881:incident
840:vertices
836:branches
808:circuits
463:adjacent
434:and the
99:vertices
97:are the
89:. In a
64:low-pass
2574:vol. 11
1993:22 July
1952:22 July
1858:p.323).
1773:tie set
1733:finite
1662:forests
1461:Nullity
1437:Tie set
1408:Branch
1389:Current
1351:Duality
1147:cut set
1122:tie set
815:circuit
746:nullity
682:History
306:in the
303:A000084
47:network
2835:
2820:
2805:
2790:
2775:
2754:
2733:vol.54
2723:vol.28
2700:vol.22
2689:vol.16
2672:
2660:535111
2658:
2645:vol.39
2630:vol.51
2608:vol.31
1641:target
1637:source
1405:Branch
1298:
1257:
1209:
1046:where
1031:
990:
852:vertex
664:graphs
503:ground
430:, the
385:filter
67:filter
41:of an
1972:(PDF)
1803:Maze.
1791:Mesh.
1741:Notes
1464:Rank
1416:Node
1400:Maze
954:twigs
950:links
844:nodes
831:graph
649:ports
562:prism
558:graph
458:ports
185:delta
103:edges
95:nodes
2833:ISBN
2818:ISBN
2803:ISBN
2788:ISBN
2773:ISBN
2752:ISBN
2670:ISBN
2656:OCLC
1995:2023
1954:2023
1735:path
1649:rank
1639:and
1536:. A
1472:maze
1429:Link
1421:Loop
1413:Mesh
1397:Tree
1142:dual
1118:loop
1114:loop
946:tree
937:tree
900:and
850:and
848:Node
803:and
750:rank
674:and
538:and
387:and
308:OEIS
151:The
124:The
37:The
2737:doi
2704:doi
2578:doi
1984:doi
1943:doi
813:RLC
748:or
310:).
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2731:,
2721:,
2717:,
2698:,
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2606:,
2602:,
2591:,
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2489:41
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2440:^
2177:^
2144:32
2142:,
2102:72
2100:,
1980:21
1978:.
1974:.
1939:54
1937:.
1933:.
1818:,
1592:.
1565:.
1513:.
1360:.
1157:.
935:A
908:.
896:,
819:LC
806:LC
800:RL
797:,
794:RC
678:.
402:.
199:.
181:pi
140:,
136:,
116:.
78:.
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2824:.
2809:.
2794:.
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2758:.
2743:.
2739::
2710:.
2706::
2676:.
2584:.
2580::
1997:.
1986::
1956:.
1945::
1546:B
1542:A
1481:n
1295:b
1292:=
1289:N
1286:+
1283:R
1254:s
1248:n
1245:=
1242:R
1229:R
1206:s
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1200:n
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1191:=
1188:N
1175:b
1171:s
1167:N
1110:l
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1070:l
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1019:=
1016:b
987:1
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978:=
975:t
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958:n
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267:N
260:n
179:(
34:.
20:)
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