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Network Theory NETWORK THEORY For ELECTRICAL ENGINEERING INSTRUMENTATION ENGINEERING ELECTRONICS & COMMUNICATION ENGINEERING NETWORK THEORY SYLLABUS Network graph, KCL, KVL, Node and Mesh analysis, Transient response of dc and ac networks, Sinusoidal steady‐state analysis, Resonance, Passive filters, Ideal current and voltage sources, Thevenin’s theorem, Norton’s theorem, Superposition theorem, Maximum power transfer theorem, Two‐port networks, Three phase circuits, Power and power factor in ac circuits. ANALYSIS OF GATE PAPERS ELECTRONICS ELECTRICAL INSTRUMENTATION 1 Mark 2 Mark 1 Mark 2 Mark 1 Mark 2 Mark Exam Year Ques. Ques. Total Ques. Ques. Total Ques. Ques. Total 2003 4 7 18 3 6 15 ‐ 1 2 2004 5 5 15 1 7 15 ‐ ‐ ‐ 2005 5 6 17 4 7 18 ‐ 2 4 2006 6 ‐ 6 2 6 14 1 4 9 2007 2 4 10 ‐ 7 14 2 4 10 2008 2 7 16 2 6 14 2 7 16 2009 3 4 11 2 6 14 ‐ 2 4 2010 2 3 8 3 4 11 2 2 6 2011 3 3 9 3 5 13 2 3 8 2012 4 4 12 5 6 17 4 6 16 2013 3 6 15 2 3 8 3 5 13 2014 Set‐1 2 4 10 2 2 6 3 4 11 2014 Set‐2 2 4 10 3 2 7 ‐ ‐ ‐ 2014 Set‐3 2 4 10 3 3 9 ‐ ‐ ‐ 2014 Set‐4 2 4 10 ‐ ‐ ‐ ‐ ‐ ‐ 2015 Set‐1 4 3 10 4 3 10 3 5 13 2015 Set‐2 3 3 9 3 3 9 ‐ ‐ ‐ 2015 Set‐3 3 2 7 ‐ ‐ ‐ ‐ ‐ ‐ 2016 Set‐1 1 2 5 4 5 14 3 3 9 2016 Set‐2 4 2 8 5 4 13 ‐ ‐ ‐ 2016 Set‐3 1 3 7 ‐ ‐ ‐ ‐ ‐ ‐ 2017 Set‐1 2 3 8 1 2 5 4 4 12 2017 Set‐1 0 4 8 2 2 6 ‐ ‐ ‐ 2018 2 3 8 2 3 8 3 3 9 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission CONTENTS Topics Page No 1. NETWORK BASICS 1.1 Introduction 1.2 Classifications of Network Elements 1.3 Circuit Components 1 1.4 Kirchoff's Laws 2 1.5 Mesh & Nodal Analysis 3 1.6 Equivalent Circuits 6 Gate Questions 137 9 2. NETWORK THEOREMS 2.1 Introduction 2.2 Superposition Theorem 2.3 Thevenin's & Norton's Theorem 49 2.4 Maximum Power Transfer Theorem 49 2.5 Tellegen's Theorem 50 2.6 Reciprocity Theorem 51 2.7 Substitution Theorem 53 2.8 Millman's Theorem 53 2.9 Duality Principle 54 Gate Questions 54 54 3. TRANSIENTS 58 3.1 Introduction 3.2 Steady State & Transient Response 3.3 DC Transients 89 Gate Questions 89 91 4. AC ANALYSIS 99 4.1 Introduction 4.2 Sinusoidal Steady state analysis 4.3 Series Circuits 129 4.4 Parallel Circuits 129 Gate Questions 13013 132 5. RESONANCE 6 5.1 Introduction 149 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission 5.2 Series Resonance 5.3 Parallel Resonance Gate Questions 149 152 6. GRAPH THEORY 155 6.1 Graph of a Network 6.2 Incidence Matrix 6.3 Tie‐Set Matrix 157 6.4 F‐Cut Set Matrix 159 Gate Questions 159 163 7. COUPLED CIRCUITS 164 7.1 Introduction 7.2 Mutual Inductance 7.3 The Coupling Coefficient 174 7.4 Series Connection of Coupled Inductors 174 7.5 Parallel Connection of Coupled Coils 176 7.6 Ideal Transformer 176 177 8. TWO PORT NETWORKS 177 8.1 Introduction 9 8.2 Open Circuit Impedance (Z) Parameters 9 8.3 Short Circuit Admittance (Y) Parameters 17 8.4 Transmission (A B C D) Parameters 170 8.5 Inverse Transmission (A' B' C' D') Parameters 180 8.6 Hybrid (H) Parameters 18 8.7 Inverse Hybrid (G) Parameters 181 8.8 Conditions for Reciprocity and Symmetry 181 8.9 Inter Relationships of Different Parameters 182 8.10 Interconnection of Two‐Port Network 182 Gate Questions 1182 184 9. NETWORK SYNTHESIS 89 9.1 LC, RC, RL Impedance & Admittance Functions Gate Questions 211 10. ASSIGNMENT 212 217 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission 1 NETWORK BASICS 1.1 INTRODUCTION Q I = (ampere) t In terms of the atomic theory concept, an Where, I is the current electric current in an element is the time Q is the charge rate of flow of free electrons in the element. T is the time The material may be classified as The current through a circuit element is the • Conductors, where availability of free time derivative of the electric charge i.e. electrons is very large, as in the case dq i = (c/s) or (Ampere) metals. dt • Insulators, where the availability of Where, dq is small change in charge. free electrons is rare, as in case of glass, dt is small change in time. mica, plastics etc. • Other materials, such as germanium 1.1.3 VOLTAGE and silicon called semiconductors, may play a significant role in All opposite charges possess a certain electronics. Thermally generated amount of potential energy because of the electrons are available as free electrons separation between them. The difference is at room temperature, and act as potential energy of the charges is called the conductors, but at 0 Kelvin they act as potential difference. The potential insulator. difference in electrical terminology is known as voltage, and is denoted either by 1.1.1 CHARGE ‘V’ or ‘v’. Voltage is expressed in terms of energy (W) per unit charge (Q) i.e. According to basic physics, we know that W dw there are two types of charges: Positive V = Or ν = Q dq (corresponding to proton) and Negative (corresponding to electron). The (J/c) or (volt) fundamental unit of charge is the Where, dw is the small change in energy coulomb[c]. A single electron has a charge dq is the small change in charge. − One Volt, is the potential difference of −×1.602 10 19 C and a single proton −19 between two points when one joule of has a charge of +×1.602 10 C where one energy is used to pass one coulomb of coulomb is defined as one ampere second. charge from one point to the other Charge in coulomb Q = It where, I is current in ampere and t is time in second. 1.1.4 ENERGY 1.1.2 CURRENT Energy is capacity for doing work. Energy may exist in many forms such as The phenomenon of transferring charge mechanical, chemical; electrical is called from one point is a circuit to another is ‘Joule’. Energy is denoted by ‘W’. The termed as “electric current”. The eclectic energy delivered to a circuit element over current, denoted by either ‘I’ or ‘i’ .The the time interval (to, t) is given by unit of electric current is ‘ampere’ which t is denoted by ‘A’. Electric current W=pxdx∫ () mathematically expressed as t0 © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission 1.1.5 POWER time interval. For example, passive elements like inductors and capacitor Power is the rate of change of energy. It is capable of storing a finite amount of denoted by ‘P’ or ‘p’, unit of power is energy. Resistor is also a passive ‘Watts’. element. Energy W Power() p = = time t 2) BILATERAL AND UNILATERAL dw In Bilateral elements, the voltage– or P = dt current relation is same for the current Where, dw is the change in energy flowing in either direction. For the dt is the change in time entire time‘t’ element offers the same dw impedance for the different directions we can also write, P = of the same current flow and hence the dt resistor is said to bilateral. dw dq = × dt dt P = v × iwatts So, the instantaneous power p(t) delivered to a circuit element is the product of the instantaneous value of voltage v(t) and current i(t) of the element has different P = v() t × i(t) A unilateral element relations between voltage and current for the two possible directions of 1.2 CLASSIFICATION OF NETWORK current. Vacuum diodes, silicon diodes ELEMENTS and metal rectifiers are examples of unilateral elements. 3) LINEAR AND NONLINEAR ELEMENTS An element is said to be linear (device is linear if it is characterized by an equation of the form y=mx, where m is constant) if its voltage-current characteristics is at all times a straight line through the origin. For example, 1) ACTIVE AND PASSIVE the current passive through a resistor is Active network elements are those proportional to the voltage applied which are capable of delivering power through it, and relation is expressed as to some external device. Specifically an V∝= IORV IR active element (energy sources like A linear element or network is one voltage and current sources) is capable which satisfies the principle of of delivering an average power greater superposition, i.e. the principle of than zero to some external device over homogeneity and additivity. An element an infinite time interval. For example, which does not satisfy the above ideal sources are active elements. principle is called a non linear element. Passive network elements are those which are capable only of receiving 4) LUMPED AND DISTRIBUTED power. A passive element is defined as Lumped elements are those elements one that cannot supply average power which are very small in size and in that is greater than zero over an infinite which simultaneous actions takes place © Copyright Reserved by Gateflix.in No part of this material should be copied or reproduced without permission for any given cause at the same instant dq ∴=vR of time.
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