instrumentation engineering : networks, the gate academy

NETWORKS

for

EC / EE / IN

By

www.thegateacademy.com

http://www.thegateacademy.com/

Syllabus Networks

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th

Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com

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 EE IN

2013 15.00 11.00 13.00

2012 13.00 13.00 16.00

2011 9.00 9.00 6.00

2010 8.00 10.00 6.00

Over All Percentage 11.25% 10.75% 10.25%

Contents Networks


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page I

C O N T E N T S

Chapters Page No.

#1. Network Solution Methodology 1-43

Introduction 1-5 Solution of Simultaneous equations 5 Nodal Analysis 5-6 Voltage/Current Source 6-7 Dependent Sources 7-8 Power and Energy 8-9 Thevenin’s and Norton’s Theorems 9-12 Solved Examples 13-22 Assignment 1 23-30 Assignment 2 31-35 Answer Keys 36 Explanations 36-43

#2. Transient/Steady State Analysis of RLC Circuits to DC Input 44-77 Transient Response Analysis 44-45 Transient Response of capacitor 45-51 Solved Examples 52-60 Assignment 1 61-67 Assignment 2 68-71 Answer Keys 72 Explanations 72-77

#3. Sinusoidal Steady State Analysis 78-119 Sinusoidal Excitation 78-82 Average Power Supplied by A.C. Source 82 Resonance 82-85 Single Phase Circuit Analysis 85-88 Polyphase Circuit Analysis 88-91 Magnetically Coupled Circuit 91-94 Solved Examples 95-101 Assignment 1 102-108 Assignment 2 109-112 Answer Keys 113 Explanations 113-119

Contents Networks


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page II

#4. Laplace Transform 120-149

Definition 120

Laplace Transform of Standard Function 120-121

Initial Value Theorems 121

Transfer Function of LTI System 121-123

Circuit Analysis at a Generalized Frequency 123

Standard Structures 123-124

Properties of Driving Point Functions 125-127

Solved Examples 128-133

Assignment 1 134-140


Answer Keys 143

Explanations 143-149

#5. Two Port Net works 150-172

Definitions 150-153

Some Standard Networks 153-154

Inter Connection of Two Port Networks 154-157




Answer Keys 168


#6. Network Topology 173-197

Definitions 173

Connected and Non-Connected Graph 173-174

Formula 174-175

Nodal Incidence Matrix 175

Applications in Network Theory 175-176

Loop Incidence Matrix 176-177

Fundamental of Cut Set Matrix 178-179




Answer Keys 194

Contents Networks


Cross, 10th

Main, Jayanagar 4th

Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page III


Module Test 198-218

Test Questions 198-211

Answer Keys 212


Reference Books 219

Chapter 1 Networks

THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 1

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 .

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.

SL. No Circuit element

Symbol in electric circuit

Units Voltage – current relation

Instantaneous power , P

Energy stored / dissipated in [ ]

1 Resistance, R

Ohm ( )

V= i R ( ohm’s law)

Vi= i

(t t )

2 Inductance, L

Henry ( H )

V = L

Li

(

)

3 Capacitance, C

Farad ( F )

i = C

C

(

)

i

V

i

V

V

i

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.

For series connection of resistors, i i = . . . . . . . i = i

V = v v

(∑ )

∀ i ……… n

i i

> >

+ - >

Fig1.2. Series connection of resistor

R i

…………

=

…………

=

…………

= …………

= …………

=

…………

Chapter 1 Networks


For parallel connection of resistors,

i = ii , i = i ×

(

)

∀ i ……… n

……………… = V

Kirchoff’s urrent aw ( K )

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

Assuming that current approaching node ‘O’ bears positive sign, and vice versa,

i i i i 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 i

Fig 1.4 Demonstration of KCL

Node O

i

i

i

i

Fig 1.3 Parallel combination of resistors

R i

Chapter 1 Networks


From Kirchoff’s current law ,i i 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.

Use mesh analysis to find and ,

( )

Fig 1.7.Demonstration of mesh analysis

E

Fig 1.6.Demonstration of KCL & KVL

E

Fig 1.5.Demonstration of KVL

i

Chapter 1 Networks


( ) = 0

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

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.

Fig 1.8.Demonstration of mesh analysis using super mesh

I

instrumentation engineering : networks, the gate academy

Technology

th cross

th main

th block

networks network graphs

nodal analysis

polyphase circuit analysis

network theorems

answer keys