## Network Theory

Table of Contents

## Introduction to Network Theory

Network theory is the study of graphics as the representation of symmetrical or asymmetrical relationships between discrete objects. In computer science and network science, network theory is part of graph theory: a network in graph theory can be defined as a graph in which nodes and/or edges have attributes (such as names).

Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering, biology, economics, finance, operations research, climatology, ecology, public health, sociology, and Neuroscience. The applications of network theory include logistics networks, the World Wide Web, the Internet, gene regulation networks, metabolic networks, social networks, epistemological networks, and so on.

### Overview of Network Theory

Network theory is the study and solution of problems in circuits or electrical networks. In this introductory chapter, let us first discuss the basic terminology of circuits and types of network elements.

Basic Terminology

In-Network Theory, we will frequently come across the following terms −

• Electric Circuit
• Electric Network
• Current
• Voltage
• Power

As a result, before continue, we must first have a fundamental understanding of these words. Let’s begin with the Electric Circuit.

Circuit Electric

An electric circuit has a closed channel that allows electrons to flow from a voltage or current source. An electric circuit’s constituents will be connected in series, parallel, or any mix of series and parallel connections.

Electricity Network

An electric network does not have to have a closed channel for electrons to travel from a voltage or current source. As a result, we can infer that “all-electric circuits are electric networks,” but the opposite is not required.

Current

The current “I” passing through a conductor is nothing more than the charge’s time rate of flow. It may be expressed mathematically as

I=DQ/DT

Where,

Q is the charge and its unit is Colum b.

t is the time and its unit is second.

Electric current may be compared to the flow of water via a pipe. Amperes are used to measure current.

In general, electron current flows from the source’s negative terminal to the positive terminal, whereas conventional current flows from the source’s positive terminal to the negative terminal.

Electron current results from the movement of free electrons, whereas conventional current results from the movement of free positive charges. Both of them are referred to as electric currents.

Voltage

Voltage “V” is nothing more than an electromotive force that causes charge (electrons) to flow. It may be expressed mathematically as

V=dW/dQ

Where,

W is the potential energy and its unit is Joule.

Q is the charge and its unit is Columb.

Voltage may be compared to the pressure of water that causes water to flow through a conduit. Volts are used to measure it.

Power

The power “P” is just the time rate at which electrical energy flows. It may be expressed mathematically as

P=dW/dt

Where,

W is the electrical energy and it is measured in terms of Joule.

t is the time and it is measured in seconds.

We can re-write the above equation a

P=dW/dt=dW/dQ×dQ/dt=VI

As a result, power is just the product of voltage V and current I. It is measured in Watts.

### Network Elements of Various Types

Based on specific characteristics, we may categorize Network components into several kinds. The types of network elements are as follows:

• Active Elements and Passive Elements
• Linear Elements and Non-linear Elements
• Bilateral Elements and Unilateral Elements

Active Elements and Passive Elements

Based on their capacity to supply power, network elements can be classified as either active or passive.

Active Components provide power to the other elements in an electric circuit. They may absorb power as passive components at times. That is, active components are capable of both providing and absorbing power. Voltage and current sources are two examples.

Passive Elements cannot transfer power (energy) to other elements, but they can absorb it. That is, these components either waste energy in the form of heat or store energy in the form of a magnetic or electric field. Resistors, inductors, and capacitors are some examples.

Linear Elements and Non-Linear Elements

Based on their ability to obey the property of linearity, network elements can be classified as linear or non-linear.

Linear Elements are those that exhibit a linear connection between voltage and current. Resistors, inductors, and capacitors are some examples.

Non-Linear Elements do not have a linear connection between voltage and current. Voltage and current sources are two examples.

Bilateral Elements and Unilateral Elements

Based on the direction of current flows through the network elements, network elements can also be categorized as bilateral or unilateral.

Bilateral Elements are those that enable current to flow in both directions and have the same impedance in both directions. Resistors, inductors, and capacitors are some examples.

#### The concept of Bilateral elements is illustrated in the following figures.

Bilateral Elements

In the above figure, the current (I) is flowing from terminals A to B through a passive element having an impedance of Z Ω. It is the ratio of voltage (V) across that element between terminals A & B and current (I).

Bilateral

In the above figure, the current (I) is flowing from terminals B to A through a passive element having an impedance of Z Ω. That means the current (–I) is flowing from terminals A to B. In this case, too, we will get the same impedance value, since both the current and voltage having negative signs concerning terminals A & B.

Unilateral Elements are those that allow the current in only one direction. Hence, they offer different impedances in both directions.

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