Networks : Instrumentationn Engineering, THE GATE ACADEMY by THE GATE ACADEMY

NETWORKS for

EC / EE / IN By

www.thegateacademy.com

Syllabus

Networks

Syllabus for Networks Network graphs: matrices associated with graphs; incidence, fundamental cut set and fundamental circuit matrices. Solution methods: nodal and mesh analysis. Network theorems: superposition, Thevenin and Norton’s, maximum power transfer, Wye-Delta transformation. Steady state sinusoidal analysis using phasors. Linear constant coefficient differential equations; time domain analysis of simple RLC circuits, Solution of network equations using Laplace transform: frequency domain analysis of RLC circuits. 2-port network parameters: driving point and transfer functions. State equations for networks.

Analysis of GATE Papers (Network Theory) Year

ECE

2013

15.00

11.00

13.00

2012

13.00

16.00

2011

9.00

6.00

2010

8.00

10.00

6.00

Over All Percentage

11.25%

10.75%

10.25%

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Contents

Networks

CONTENTS

#1.

#2.

#3.

Chapters

Page No.

Network Solution Methodology

1-43

           

1-5 5 5-6 6-7 7-8 8-9 9-12 13-22 23-30 31-35 36 36-43

Introduction Solution of Simultaneous equations Nodal Analysis Voltage/Current Source Dependent Sources Power and Energy Thevenin’s and Norton’s Theorems Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Transient/Steady State Analysis of RLC Circuits to DC Input

44-77

      

44-45 45-51 52-60 61-67 68-71 72 72-77

Transient Response Analysis Transient Response of capacitor Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Sinusoidal Steady State Analysis

78-119

          

78-82 82 82-85 85-88 88-91 91-94 95-101 102-108 109-112 113 113-119

Sinusoidal Excitation Average Power Supplied by A.C. Source Resonance Single Phase Circuit Analysis Polyphase Circuit Analysis Magnetically Coupled Circuit Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Contents

#4.

#5.

Networks

Laplace Transform

120-149

           

120 120-121 121 121-123 123 123-124 125-127 128-133 134-140 140-142 143 143-149

Definition Laplace Transform of Standard Function Initial Value Theorems Transfer Function of LTI System Circuit Analysis at a Generalized Frequency Standard Structures Properties of Driving Point Functions Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

Two Port Net works

150-172

       

150-153 153-154 154-157 158-161 162-164 165-167 168 168-172

Definitions Some Standard Networks Inter Connection of Two Port Networks Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations

#6. Network Topology           

173-197

Definitions Connected and Non-Connected Graph Formula Nodal Incidence Matrix Applications in Network Theory Loop Incidence Matrix Fundamental of Cut Set Matrix Solved Examples Assignment 1 Assignment 2 Answer Keys

173 173-174 174-175 175 175-176 176-177 178-179 180-186 187-191 191-193 194

Contents 

Networks

Explanations

194-197

Module Test

198-218



Test Questions

198-211



Answer Keys

212



Explanations

212-218

Reference Books

219

Chapter 1

Networks

CHAPTER 1 Network Solution Methodology

Introduction The passive circuit elements resistance R, inductance L and capacitance C are defined by the manner in which voltage and current are related for the individual element. The table below summarizes the voltage-current(V-I) relation, instantaneous power (P) consumption and energy stored in the period [t ,t ] for each of above elements . SL. No

Circuit element

Symbol in Units electric circuit

Resistance, R i

Inductance, L

Voltage – current relation

Instantaneous power , P

Energy stored / dissipated in [ ]

V= i R ( ohm’s law)

Vi= i

V=L

Henry (H)

(

)

Farad (F)

i=C

(

)

Ohm V ()

t )

Capacitance, C i

Table 1.1.Voltage –Current relation of network elements In the above table, if i is the current at instant m and is the voltage at instant m, total energy dissipated in a resistor (R) in [ t t ] =∫ v i dt=∫ i dt

Series and Parallel Connection of Circuit Elements Figure below summarizes equivalent resistance /inductance /capacitance for different combinations of network elements.

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Chapter 1

Networks …………

…………

………… …………

Fig1.1. Series and parallel connection of circuit elements Voltage / Current relation in series / parallel connection of resistor Figures below summarize voltage/current relations in series and parallel connection of resistors.

> +

Fig1.2. Series connection of resistor For series connection of resistors, i V= v

(∑

∀i

)

i =....... i =i

……… n

Chapter 1

Networks

Fig 1.3 Parallel combination of resistors For parallel connection of resistors, i = ii , i = i ×

(

)

∀i

……… n

……………… Kirchoff’s urrent aw ( K

=V )

The algebraic sum of current at a node in a electrical circuit is equal to zero.

Node O

Fig 1.4 Demonstration of KCL Assuming that current approaching node ‘O’ bears positive sign, and vice versa, i

Kirchoff ’s oltage aw In any closed loop electrical circuit, the algebraic sum of voltage drops across all the circuit elements is equal to emf rise in the same. Figure below demonstrates KVL and KCL equation can be written as, i

Chapter 1

Networks

Fig 1.5.Demonstration of KVL

Fig 1.6.Demonstration of KCL & KVL From Kirchoff’s current law ,i Also from KVL, (i i i ) (i i i i )

Mesh Analysis

In the mesh analysis, a current is assigned to each window of the network such that the currents complete a closed loop. They are also referred to as loop currents. Each element and branch therefore will have an independent current. When a branch has two of the mesh currents, the actual current is given by their algebraic sum. Once the currents are assigned, Kirchhoff’s voltage law is written for each of the loops to obtain the necessary simultaneous equations.

Fig 1.7.Demonstration of mesh analysis Use mesh analysis to find and , (

)

Chapter 1 (

)

Networks

The above set of simultaneous equations should be solved for

and .

Solution of Simultaneous Equations Matrix inversion method The above set of equations can be written as below in matrix from: ][ ] = *

[

This is of form, AX = B

Where

[

]

Cramer’s rule To find either of and , use Cramer’s rule as below =|

| | |

Mesh Analysis (using super mesh) When two of the loops have a common element as a current source, mesh analysis is not applied to both loops separately. Instead both the loops are merged and a super mesh is formed. Now KVL is applied to super mesh. For the circuit in figure shown below,

= E

Fig 1.8.Demonstration of mesh analysis using super mesh

Nodal Analysis Typically, electrical networks contain several nodes, where some are simple nodes and some are principal nodes. In the node voltage method, one of the principal nodes is selected as the reference and equations based on KCL are written at the other principal nodes. At each of these other principal nodes, a voltage is assigned, where it is understood that this voltage is with respect to the reference node. These voltage are the unknowns and are determined by nodal Analysis. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,  info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 5