Electrical Network Topology | Electrical Network Graph

Topology is the branch of mathematics in which we study those properties of the geometrical configuration of shapes that are unaltered under certain transformations as bending or stretching. In network topology, we study the geometrical configuration of networks irrespective of the components present in the network (circuit).  The concept will explain later in this post. This is a very easy topic and usually a part of the first year of ECE discipline. It is important to understand this topic because it helps solve complex circuits. This will help in later courses.

At this point, I would like to clarify the difference between a circuit and a network. The network is an interconnection of elements (resistor, capacitors, sources, batteries etc). While a circuit is also a type of network providing a closed path or loop for the flow of current. In this context words network and circuit are used interchangeably.

Key Concepts:

• What is meant by network topology in electrical circuits?
• What are the components of electric circuits? Why it is used?
• Explanation of branches, nodes, loop, graphs etc
• Relationship between branches, loops, and nodes
• Concept of graphs
• Concept of trees

Network Topology:

Topology: Topology is the branch of geometry that is concerned with those properties of a geometrical figure which are unchanged.

Network Topology: Network topology is an application of Graph Theory, a discipline of mathematics with special reference to electrical circuits. In network topology, we study the placement of elements in a network and the geometrical configuration of networks. It is a graphical representation of electrical circuits. It is useful for analysing complex circuits by converting them into network graphs.

For example, low pass filters and high pass filters have the same topology. By interchanging, inductors and capacitors in low pass filters will result in high pass filters. Due to the interchanging of these two components, the entire function of the circuit is changed but topology remains the same.

Some common terminologies used in network topology is as under:

Branch:

A branch represents a single element either passive or active. For example a voltage source or a resistor. In the context of network topology, it is a line segment. While drawing a network graph every two-terminal element present in a network (circuit ) is replaced by a line segment. Two or more branches are connected by nodes.

Node:

A node is a point of connection between two or more branches. A node is usually shown by a dot in a circuit. See Fig 1. Look at the figure where I marked b. Here are three dots joined together by a wire (short circuit) these three nodes constitute a single node. Similarly, look at Fig 1 where I marked c. Here are 4 dots (nodes) joined together by a wire (short circuit). These 4 nodes constitute a single node. So in Fig 1, there are 3 nodes.

Fig 1 is redrawn such that it clearly shows three different nodes.

Loop:

A loop is any closed path in a circuit or network. A loop is formed by passing through sets of nodes and returning to its starting point (node). A loop is passed through a node once. For example in Fig 1 loop is formed by passing through nodes abca. In loops the direction of flow of current matters.

A loop is said to be independent if it contains a single branch. That branch should not be a part of any other independent loop. Independent loops have independent equations.

Fig 1 Branches, nodes and loops

Relationship Between Branch, Loop & Node:

A fundamental theorem of network topology is stated as

B = L+N-1

Where B = branches, N = nodes, L = loops

Graphs:

In the context of network topology graph is a circuit model which is an interconnection of branches and nodes.

To draw the graph of a circuit we replace every element with a line segment (branch) and nodes remain the same. While drawing a graph a two-terminal element is replaced by a line segment (branch) and suppress the nature of that element as we do not interested in the type and nature of the element. Each passive element and voltage source is replaced by a short circuit/line segment/, branch. While the current source is replaced with an open circuit. The number of branches in a graph may be equal to or less than branches present in a network. The voltage source in series with passive elements is not included in the graph. While a passive element in parallel with a voltage source is not included in the graph.

Oriented Graph: A directed graph. In an oriented graph, each branch has an arrowhead indicating the direction of the current in that branch.

Subgraphs and their types:

Subgraphs or a subset of a graph has two types.

• Tree
• Cotree

Trees:

It is a subgraph or a subset of a graph. We can get trees either by removing some branches or by removing some nodes. If a graph has N nodes then the tree has N-1 branches. A tree doesn't have any closed loop. The branches of a tree are called twigs.

• A tree contains all the nodes present in a network
• For tree number of branches=n-1
• A tree doesn't contain any closed path
• The number of equivalent current equations form for a network is equal to the number of twigs
• Number of KCL equations = N-1=twigs
• There are many possible trees for a single graph depending upon the number of nodes and branches
• The rank of a tree is the same as the rank of a graph that is (n-1)

Cotrees:

A cotree is a set of branches that are not part of a tree. Dotted lines show the cotrees of the corresponding tree. It is also called a complement of a tree. A branch in a cotree is called a link. The number of branches (links) in a cotree is given by:

L = B - (N-1)

Where

L = number of links in a cotree

B = number of branches present in a given graph

N = number of nodes present in a given graph

There is a unique loop associated with every link in a cotree. By rearranging one of the links it forms a loop. The number of loops is equal to the number of independent voltage equations. To summarize

• Tree and cotree are complements of each other. The number of branches in the tree (twigs) and number of branches in cotree (links) is equal to the total number of branches in a graph
• The number of independent voltage equations is equal to the number of links
• Number of KVL equations = number of links = L = B + (N-1)
• 
  

Fig 2 Graph of the circuit in Fig 1, trees and Cotrees
Trees are drawn with blue and cotrees are drawn with red

Example#1: Indicate the number of branches loops and nodes in the following circuit.  Draw its graph, trees and cotrees.

Fig 3 Example Circuit

Fig 4 Graph, trees and cotrees of the example circuit

Conclusion:

Network topology is an application of graph theory. It is helpful in the analysis of the electrical circuit. It is just an introductory article on network topology. In later posts, I will discuss matrices associated with network topology. These matrices are

• Incidence Matrix
• Fundamental Loop or Tie Set Matrix
• Fundamental Cut Set Matrix

Reference: 
Network Analysis And Synthesis
Electrical circuit analysis and design by Noel M Morris