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 

Voltage Sources And Current Sources

Series And Parallel Combination Of Current & Voltage Sources

This is my second article on voltage and current sources. The previous article was written on introduction of voltage and current sources and different types of voltage and current sources are discussed. You can view this article here.

Key Questions:

• Series connected voltage sources
• Under which conditions voltage sources can connect in parallel
• Parallel connected current sources
• Under which conditions current sources can connect in series

Series Connected Voltage Sources:

Voltage sources should be connected in series. The question is why are voltage sources connected in series?

The answer is more voltage. When voltage sources connected in series with same polarity they will be added.

DC voltage sources connected in series can be combined and replaced with a single equivalent DC voltage source. See Fig 1

AC voltage sources connected in series having same angular frequency can be combined and replaced with a single equivalent AC voltage source. See Fig 1

AC and DC voltage sources connected in series can be combined to get total voltage across them. See Fig 1

Fig 1 Series connected voltage sources

If voltage sources connected in series having same polarity then they are series aiding sources. Since it will add the total voltage. See Fig 2

If voltage sources connected in series with opposite polarity then they are series opposing sources. Since it will decrease the total voltage. See Fig 2

Fig 2 Series aiding and series opposing sources

Parallel Connected Voltage Sources:

Why are voltage sources connected in parallel??

The answer is more current. Let me explain this answer.

Have you seen toys or gadgets in which cells (mistakenly called batteries. Batteries are series connected cells) connected in parallel. Everyone definitely saw these types of gadgets.These cells are connected in parallel to meet the current requirements. To connect voltage sources in parallel each voltage source must have same voltage and polarity and frequency of operation (angular frequency). Under this condition current from each voltage source will be added.

For example 3 AAA size batteries (1.5V) connected in parallel. Voltage remains same in parallel, while the current from each source will be added. And the effective resistance of parallel connected sources would also decrease.

Keep in mind that the parallel connected voltage sources have same magnitude, same polarity and same angular frequency.

Fig 3 Parallel connected voltage sources

Parallel Connected Current Sources:

Current sources should be connected in parallel. The question is why are current sources connected in parallel?

The answer is more current.

According to Kirchhoff's Current Law in parallel connected current sources current will be added.

DC current sources connected in parallel can be combined and replaced with a single equivalent DC current source.

AC current sources connected in parallel and having same angular frequency can be combined and replaced with a single equivalent AC current source.

Fig 4

AC and DC current sources connected in parallel can be combined to get total current flowing across them.

Fig 5

Aiding Sources: If sources are connected in parallel such that all the currents from each source added. For example current sources connected in parallel and with similar polarity.

Opposing Sources: If sources are connected in parallel such that current from different sources are subtract from one another. Example current sources are connected in parallel but with opposite polarity.

Fig 6

Series Connected Current Sources:

Current sources with different current values can not be connected in series. Because it violates Kirchhoff's Current Law (KCL). According to this law current entering the node is equal to current leaving the node. So at a single node two current sources with different current values can not be connected.

For example a current source having 5A and another current source having 1A current are connected in series. Now according to KCL the current entering the node is equal to current leaving the node. But we have 5A current source connected in series with 1A current source. Current from 5A source and current from 1A source are at the same node. Which violates the KCL.

Fig 7

When are current sources connected in series?

Current sources can be connected in series if and only if they carry equal currents.

A 2A current source can be connected in series with 2A current source only. These two current sources can be replaced by single equivalent current source of 2A.

See Fig 7

Evaluation of Complement Numbers In Binary Number System

Complement In Binary Number System:

This post is in continuation of conversion of binary numbers into decimal numbers. The difference is these numbers are negative numbers. Before starting this article you should know how to convert a binary number into decimal number. Here is my article on binary numbers conversion. And the second thing you should know is the complement system, 1’s complement and 2’s complement. Here is my detailed article on representation of numbers.

Key Questions:

• Evaluation of Complement Numbers in binary number system using 1’s complement
• Evaluation of Complement Numbers in binary number system using 2’s can complement

Evaluation of Complement Numbers In Binary Numbers Using 1’s Complement:

There are few steps to follow

• First look at MSB, either a positive number or a negative number
• If the number is positive (MSB=0) then the conversion technique remains same
• If the number is negative (MSB=1) then you have to add (+1) while calculating sum of weights

Example#01:00111101)2

MSB=0; Number is positive.

00111101)2=(0*27)+(0*26)+(1*25)+(1*24)+(1*23)+(1*22)+(0*21)+(1*20)

                    =(0*128)+(0*64)+(1*32)+(1*16)+(1*8)+(1*4)+(0*2)+(1*1)

                    =0+0+32+16+8+4+0+1

                    =61

 Lets verify 

Take 1’s complement of this number 00111101, which is given below:

1’s complement=11000010)2

MSB=1; Number is negative. MSB shows both sign and magnitude of the number. We add (+1) also.

11000010)2=(-1*27)+(1*26)+(0*25)+(0*24)+(0*23)+(0*22)+(1*21)+(0*20)+1

                    =(-1*128)+(1*64)+(0*32)+(0*16)+(0*8)+(0*4)+(1*2)+(0*1)+1

                    =  -128+64+2+1

                    =-61

00111101)2=+61)10

11000010)2=-61)10

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Example#02:10110011)2

This time we have a negative number. MSB=1

10110011)2=(-1*27)+(0*26)+(1*25)+(1*24)+(0*23)+(0*22)+(1*21)+(1*20)+1

                    =(-1*128)+(0*64)+(1*32)+(1*16)+(0*8)+(0*4)+(1*2)+(1*1)+1

                    =-128+32+16+2+1+1

                    =-76

Lets verify

1’s complement of this number 10110011)2 which is given below:

1’s complement = 01001100)2

01001100)2=(0*27)+(1*26)+(0*25)+(0*24)+(1*23)+(1*22)+(0*21)+(0*20)

                   =(0*128)+(1*64)+(0*32)+(0*16)+(1*8)+(1*4)+(0*2)+(0*1)

                   =64+8+4

                   =76

10110011)2 = -76)10

01001100)2 = +76)10

Evaluation of Complement Numbers In Binary Numbers Using 2’s Complement:

The following steps to follow:

• First look at MSB, either a positive number or a negative number
• If number is positive (MSB=0), the conversion technique remains the same
• If number is negative you don't need to add (+1) as you did previously

Example#01:01110101)2

MSB=0; Number is positive

01110101)2=(0*27)+(1*26)+(1*25)+(1*24)+(0*23)+(1*22)+(0*21)+(1*20)

                    =(0*128)+(1*64)+(1*32)+(1*16)+(0*8)+(1*4)+(0*2)+(1*1)

                    =64+32+16+4+1

                    =117

Lets verify

Take 2’s complement of this number 01110101 which is given below

1’s complement = 10001010)2

2’s complement = 10001011)2

MSB=1; Number is negative. MSB shows both sign and magnitude of the number.

10001011)2=(-1*27)+(0*26)+(0*25)+(0*24)+(1*23)+(0*22)+(1*21)+(1*20)

                    =(-1*128)+(0*64)+(0*32)+(0*16)+(1*8)+(0*4)+(1*2)+(1*1)

                    =-128+8+2+1

                    =-117

01110101)2=117)10

10001011)2=-117)10

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Example#02: 10011100)2

This time we have a negative number. MSB=1

10011100)2=(-1*27)+(0*26)+(0*25)+(1*24)+(1*23)+(1*22)+(0*21)+(0*20)

                     =(-1*128)+(0*64)+(0*32)+(1*16)+(1*8)+(1*4)+(0*2)+(0*1)

                     = -128+16+8+4

                     =-100

           

1’s complement = 01100011)2

2’s complement = 01100100)2

Conversion of this number 01100100)2

01100100)2=(0*27)+(1*26)+(1*25)+(0*24)+(0*23)+(1*22)+(0*21)+(0*20)

                    =(0*128)+(1*64)+(1*32)+(0*16)+(0*8)+(1*4)+(0*2)+(0*1)

                    =64+32+4

                    =100

10011100)2 = -100)10

01100100)2 = 100)10

Recommended Books:

My favorite books by my favorite authors. You can get all digital electronics concepts in these book. loads of examples, illustrations, exercises, applications. This book is very easy to understand for beginner to intermediate level.   

Hexadecimal Division Examples

This is my fourth and last article on hexadecimal arithmetic. In this post I will solve some examples on hexadecimal number system division. Before starting this topic you should know how to multiply hexadecimal numbers. Here is my article on hexadecimal numbers multiplication.

Key Questions:

• How to perform division base 16 or hexadecimal number system?
• How to perform division on fractional hexadecimal numbers?

Hexadecimal Division Examples:

The hexadecimal numbers division is same as the division performs in other number systems. All the rules and principles are same. You can verify your results by using this online calculator.

5÷6  in this 5 is dividend and 6 is divisor

I will discuss three cases with the help of three examples. In first case dividend and divisor both are integers. In second case dividend has a fractional part while divisor is an integer. In third cas both dividend and divisor have fractional parts.

Example#01:AE887)16÷3A)16

First make multiplication table for 3A and it's multiples

3A*1	3A
3A*2	74
3A*3	AE
3A*4	E8
3A*5	122
3A*6	15C
3A*7	196
3A*8	1D0
3A*9	20A
3A*A	244
3A*B	27E

                  3025.A3

         3A⟌AE887  

                 AE      

                   X88

                      74

                      147

                      122

                        250

                        244

                           B0

                           AE

                             2

Answer:3025.A3)16

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Example#02:ECE.46)16÷12)16

First make multiplication table for 12

12*1	12
12*2	24
12*3	36
12*4	48
12*5	5A
12*6	6C
12*7	7E
12*8	90
12*9	A2
12*A	B4
12*B	C6
12*C	D8
12*D	EA

 

                       D2.92

         12⟌ECE.46

                 EA

                    2E

                    24

                      A4

                      A2

                         26

                         24

                           2

Answer:D2.92)16

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Example#03:257.8A)16÷5.1)16

Shifting hexadecimal point makes the problem easy. According to mathematical rule if you shift hexadecimal point of the dividend upto one place then you have to shift the hexadecimal point of the divisor.

After shifting

2578.A÷51

51*1	51
51*2	A2
51*3	F3
51*4	144
51*5	195
51*6	1E6
51*7	237
51*8	288
51*9	2D9
51*A	32A
51*B	37B
51*C	3CC
51*D	41D

                         76.6D

         51⟌2578.A

                 237

                   208

                  1E6

                     22A

                     1E6

                       440

                       41D

                          33

                                   

Answer: 76.6D)16