STUDY PACKAGE 1e GATE INSTRUMENTATION ENGINEERING Vol 5 of 5 Signals and systems Communication systems Control systems and process control R. K. Kanodia Ashish Murolia NODIA & COMPANY
GATE Instrumentation Engineering Vol 5 of 5 RK Kanodia and Ashish Murolia Copyright By NODIA & COMPANY Information contained in this book has been obtained by author, from sources believes to be reliable. However, neither NODIA & COMPANY nor its author guarantee the accuracy or completeness of any information herein, and NODIA & COMPANY nor its author shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that NODIA & COMPANY and its author are supplying information but are not attempting to render engineering or other professional services. MRP 540.00 NODIA & COMPANY B - 8, Dhanshree Ist, Central Spine, Vidyadhar Nagar, Jaipur - 30039 Ph : +91-141 - 101150, email : enquiry@nodia.co.in Printed by Nodia and Company, Jaipur
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Preface The objective of this study package is to develop in the GATE aspirants the ability to solve GATE level problems of Instrumentation Engineering Paper. The highly increased competition in GATE exam from last few years necessitate an in-depth knowledge of the concepts for the GATE aspirants. There are lots of study packages available for GATE Instrumentation Engineering, which includes the theory and problem sets. But through this package our notion is to develop the problem solving approach rather than just introducing the theory and problem set. This study package fulfills all the requirements of a GATE aspirant to prepare for the exam. There is no special pre-requisite before starting this study package. Although it is always recommended to refer other standard text books to clear doubts in a typical problem. The study package is published in 5 different volumes that cover the different subjects of GATE Instrumentation Engineering Paper. As the weightage of General Aptitude and Engineering Mathematics in the Instrumentation Engineering paper are 15 % each, and the subjects are very much wide in the syllabus; these subjects are published in separate volumes to provide practice problem set on all the important topics of the subjects. Rest three volumes cover the core subjects of GATE Instrumentation Engineering. In the very first volume of this study package, General Aptitude is introduced. General aptitude is divided into two sections: verbal ability and numerical ability. Some important rules of grammar is introduced at the starting of verbal ability section, and then different types of verbal ability problems are given in separate chapters. At the end of each chapter answers of the problems are described with detailed theory and grammatical rule. The numerical ability part does not include theory as it is expected from an engineering students that they are very well known to the basic mathematical formulas of under 10th class. In numerical ability section, the chapters are organized such as to cover all types of problems asked in previous GATE papers. There is the detailed solutions available for each of the numerical ability problems such that even an average student can clear his/her doubts easily. In volume of the study package, Engineering Mathematics is introduced. Each chapter of Engineering Mathematics introduces a brief theory with problem solving methodology and important formulas at the starting and then the problems are given in a graded manner from basic to advance level. At last, the solutions are given with a detailed description of formulas and concepts used to solve it. Volumes 3, 4 and 5 include the core subjects of instrumentation. The subjects with interrelated topics are taken in the same volume. Volume 3 includes the subjects:
Basics of Measurement Systems; Electrical & Electronic Measurement; Transducers, Mechanical Measurement and Industrial Instrumentation; Analytical, Optical & Biomedical Instrumentation. Volume 4 includes the subjects: Basics of Circuits, Analog Electronics, Digital Electronics. Volume 5 includes the subjects: Signals & Systems; Communication Systems; Control Systems and Process Control. For each of the subjects, the chapters are organized in a manner to cover the complete syllabus with a balanced number of problems on each topic. In starting of each chapter, a brief theory is given that includes formula, problem solving methodology and some important points to remember. There are enough number of problems to cover all the varieties, and the problems are graded from basic to advance level such that a GATE aspirant can easily understand concepts while solving problems. Each and every problems are solved with a good description to avoid any confusion or doubt. There are two types of problems being asked in GATE exam: MCQ (Multiple Choice Questions) and NAT (Numerical Answer Type questions). Both type of problems are given in this study package. Solutions are presented in a descriptive and step-bystep manner. The diagrams in the book are clearly illustrated. Overall, a very simple language is used throughout this study package to facilitate easy understanding of the concepts. We believe that each volume of GATE Study Package helps a student to learn fundamental concepts and develop problem solving skills for a subject, which are key essentials to crack GATE. Although we have put a vigorous effort in preparing this book, some errors may have crept in. We shall appreciate and greatly acknowledge all constructive comments, criticisms, and suggestions from the users of this book at rajkumar.kanodia@gmail.com We wish you good luck! Authors Acknowledgements We would like to express our sincere thanks to all the co-authors, editors, and reviewers for their efforts in making this project successful. We would also like to thank Team NODIA for providing professional support for this project through all phases of its development. At last, we express our gratitude to God and our Family for providing moral support and motivation. Authors
Syllabus General Aptitude (GA): Verbal Ability : English grammar, sentence completion, verbal analogies, word groups, instructions, critical reasoning and verbal deduction. Numerical Ability : Numerical computation, numerical estimation, numerical reasoning and data interpretation. Section 1 : Engineering Mathematics Linear Algebra: Matrix algebra, systems of linear equations, Eigen values and Eigen vectors. Calculus: Mean value theorems, theorems of integral calculus, partial derivatives, maxima and minima, multiple integrals, Fourier series, vector identities, line, surface and volume integrals, Stokes, Gauss and Green s theorems. Differential equations: First order equation (linear and nonlinear), higher order linear differential equations with constant coefficients, method of variation of parameters, Cauchy s and Euler s equations, initial and boundary value problems, solution of partial differential equations: variable separable method. Analysis of complex variables: Analytic functions, Cauchy s integral theorem and integral formula, Taylor s and Laurent s series, residue theorem, solution of integrals. Probability and Statistics: Sampling theorems, conditional probability, mean, median, mode and standard deviation, random variables, discrete and continuous distributions: normal, Poisson and binomial distributions. Numerical Methods: Matrix inversion, solutions of non-linear algebraic equations, iterative methods forsolving differential equations, numerical integration, regression and correlation analysis. Instrumentation Engineering Section : Electrical Circuits: Voltage and current sources: independent, dependent, ideal and practical; v - i relationships of resistor, inductor, mutual inductor and capacitor; transient analysis of RLC circuits with dc excitation. Kirchoff s laws, mesh and nodal analysis, superposition, Thevenin, Norton, maximum power transfer and reciprocity theorems. Peak-, average- and rms values of ac quantities; apparent- active- nd reactive powers; phasor analysis, impedance and admittance; series and parallel resonance, locus diagrams, realization of basic filters with R, L and C elements. One-port and two-port networks, driving point impedance and admittance, open-, and short circuit parameters. Section 3: Signals and Systems Periodic, aperiodic and impulse signals; Laplace, Fourier and z-transforms; transfer function, frequency response of first and second order linear time invariant systems, impulse response of systems; convolution, correlation. Discrete time system: impulse response, frequency response, pulse transfer function; DFT and FFT; basics of IIR and FIR filters. Section 4: Control Systems Feedback principles, signal flowgraphs, transient response, steady-state-errors, Bode plot, phase and
gain margins, Routh and Nyquist criteria, root loci, design of lead, lag and lead-lag compensators, state-space representation of systems; time-delay systems; mechanical, hydraulic and pneumatic system components, synchro pair, servo and stepper motors, servo valves; on-off, P, P-I, P-I-D, cascade, feedforward, and ratio controllers. Section 5: Analog Electronics Characteristics and applications of diode, Zener diode, BJT and MOSFET; small signal analysis of transistor circuits, feedback amplifiers. Characteristics of operational amplifiers; applications of opamps: difference amplifier, adder, subtractor, integrator, differentiator, instrumentation amplifier, precision rectifier, active filters and other circuits. Oscillators, signal generators, voltage controlled oscillators and phase locked loop. Section 6: Digital Electronics Combinational logic circuits, minimization of Boolean functions. IC families: TTL and CMOS. Arithmetic circuits, comparators, Schmitt trigger, multi-vibrators, sequential circuits, flip-flops, shift registers, timers and counters; sample-and-hold circuit, multiplexer, analog-to-digital (successive approximation, integrating, flash and sigma- delta) and digital-to-analog converters (weighted R, R-R ladder and current steering logic). Characteristics of ADC and DAC (resolution, quantization, significant bits, conversion/settling time); basics of number systems, 8-bit microprocessor and microcontroller: applications, memory and input-output interfacing; basics of data acquisition systems. Section 7: Measurements SI units, systematic and random errors in measurement, expression of uncertainty -accuracy and precision index, propagation of errors. PMMC, MI and dynamometer type instruments; dc potentiometer; bridges for measurement of R, L and C, Q-meter. Measurement of voltage, current and power in single and three phase circuits; ac and dc current probes; true rms meters, voltage and current scaling, instrument transformers, timer/counter, time, phase and frequency measurements, digital voltmeter, digital multimeter; oscilloscope, shielding and grounding. Section 8: Sensors and Industrial Instrumentation Resistive-, capacitive-, inductive-, piezoelectric-, Hall effect sensors and associated signal conditioning circuits; transducers for industrial instrumentation: displacement (linear and angular), velocity, acceleration, force, torque, vibration, shock, pressure (including low pressure), flow (differential pressure, variable area, electromagnetic, ultrasonic, turbine and open channel flow meters) temperature (thermocouple, bolometer, RTD (3/4 wire), thermistor, pyrometer and semiconductor); liquid level, ph, conductivity and viscosity measurement. Section 9: Communication and Optical Instrumentation Amplitude-and frequency modulation and demodulation; Shannon s sampling theorem, pulse code modulation; frequency and time division multiplexing, amplitude-, phase-, frequency-, pulse shift keying for digital modulation; optical sources and detectors: LED, laser, photo-diode, light dependent resistor and their characteristics; interferometer: applications in metrology; basics of fiber optic sensing. **********
Contents SIGNALS AND SYSTEMS 1 Continuous Time Signals 1.1 Introduction 3 1. Signal-classification 3 1..1 Analog and Discrete Signals 3 1.. Deterministic and Random Signal 3 1..3 Periodic and Aperiodic Signal 4 1..4 Even and Odd Signal 4 1..5 Energy and Power Signal 4 1.3 Basic operations on signals 5 1.3.1 Addition of Signals 6 1.3. Multiplication of Signals 6 1.3.3 Amplitude Scaling of Signals 6 1.3.4 Time-Scaling 6 1.3.5 Time-Shifting 7 1.3.6 Time-Reversal or Folding 7 1.4 Basic Continuous Time Signals 7 1.4.1 The Unit-Impulse Function 7 1.4. The Unit-Step Function 8 1.4.3 The Unit-Ramp Function 9 1.4.4 Unit Rectangular Pulse Function 9 1.4.5 Unit Triangular Function 10 1.4.6 Unit Signum Function 10 1.4.7 The Sinc Function 11 Continuous Time Systems.1 Introduction 5. Continuous Time System & Classification 5.3 Linear Time Invariant System 6.3.1 Impulse Response of LTI System 6.3. Convolution of LTI System 6.4 Impulse Response of Interconnected Systems 8.4.1 Systems in Parallel Configuration 8.4. System in Cascade 8.5 Correlation 9.5.1 Cross-Correlation 9.5. Auto-Correlation 30.5.3 Relationship between Correlation and Convolution 30.6 Block Diagram Representation 30 3 DISCRETE TIME SIGNALS 3.1 Introduction 47 3. Representation of Discrete Time signals 47 3..1 Graphical Representation 47 3.. Functional Representation 47 3..3 Sequence Representation 48 3.3 Classification of discrete time signals 48 3.3.1 Periodic and Aperiodic DT Signals 48 3.3. Even and Odd DT Signals 49 3.3.3 Energy and Power Signals 49 3.4 Basic Operations on Discrete time SignalS 50 3.5 Basic Discrete Time Signals 51 3.5.1 Discrete Unit Impulse Function 51
3.5. Discrete Unit Step Function 5 3.5.3 Discrete Unit Ramp Function 53 3.5.4 Unit-Rectangular Function 53 3.5.5 Unit-Triangular Function 53 3.5.6 Unit-Signum Function 54 4 DISCRETE TIME SYSTEMS 4.1 Introduction 69 4. Classification of Discrete Time Systems 69 4.3 Linear-Time Invariant Discrete System 70 4.3.1 Impulse Response of LTI System 70 4.3. Convolution of LTI System 70 4.4 Properties of Discrete LTI system In Terms of Impulse Response 71 4.5 Impulse Response of Interconnected Systems 7 4.5.1 Systems in Parallel 7 4.5. System in Cascade 7 4.6 Correlation 7 4.6.1 Cross-Correlation 7 4.6. Auto-Correlation 73 4.7 Block Diagram Representation 74 5 LAPLACE TRANSFORM 5.1 Introduction 87 5. Bilateral and unilateral Laplace transform 87 5..1 Existence of Laplace Transform 88 5.. Region of Convergence 88 5.3 Inverse Laplace Transform 91 5.4 Properties of The Laplace Transform 9 5.5 Stability and Causality of Continuous LTI System Using Laplace Transform 93 5.5.1 Causality 93 5.5. Stability 93 5.5.3 Stability and Causality 93 5.6 System Function For Interconnected LTI Systems 93 5.6.1 Parallel Connection 93 5.6. Cascaded Connection 94 5.6.3 Feedback Connection 94 5.7 Block Diagram Representation of Continuous LTI System 94 6 Z-TRANSFORM 6.1 Introduction 107 6. Bilateral and unilateral z -transform 107 6..1 Existence of z-transform 107 6.. Region of Convergence of z-transform 108 6.3 Inverse Z-Transform 110 6.3.1 Partial Fraction Method 111 6.3. Power Series Expansion Method 111 6.4 Properties of Z-Transform 111 6.5 Stability and Causality of LTI Discrete Systems Using z-transform 113 6.5.1 Causality 113 6.5. Stability 113 6.5.3 Stability and Causality 113 6.6 Block Diagram Representation 113 7 CONTINUOUS TIME FOURIER TRANSFORM 9.1 Introduction157
9. Different forms of continuous time Fourier series 157 9..1 Trigonometric Fourier Series 157 9.. Exponential Fourier Series 158 9..3 Polar Fourier Series 159 9.3 Existence of Fourier Series 159 9.4 Properties of Exponential CTFS 159 9.5 Relation Between CTFT and CTFS 160 9.5.1 CTFT using CTFS Coefficients 160 9.5. CTFS Coefficients as Samples of CTFT 161 8 DiSCRETE TIME FOURIER TRANSFORM 8.1 Introduction145 8. Definition of discrete time Fourier transform 145 8..1 Existence of DTFT 145 8.. Inverse DTFT 146 8.3 Properties of Discrete-Time Fourier Transform 146 8.4 Relation Between The DTFT and The z-transform 148 8.5 Discrete Fourier Transform (DFT) 148 8.6 Fast Fourier Transform (FFT) 150 8.6.1 Radix-r FFT 150 8.6. Number of Calculations 150 9 CONTINUOUS TIME FOURIER SERIES 9.1 Introduction157 9. Different forms of continuous time Fourier series 157 9..1 Trigonometric Fourier Series 157 9.. Exponential Fourier Series 158 9..3 Polar Fourier Series 159 9.3 Existence of Fourier Series 159 9.4 Properties of Exponential CTFS 159 9.5 Relation Between CTFT and CTFS 160 9.5.1 CTFT using CTFS Coefficients 160 9.5. CTFS Coefficients as Samples of CTFT 161 10 DISCRETE TIME FOURIER SERIES 10.1 Introduction 173 10. Definition of Discrete time Fourier series 173 10.3 Properties of Discrete Time Fourier SerieS 173 11 DIGITAL FILTERS 11.1 introduction181 11. Classification of digital filters 181 11..1 Classification Based on Shape of Magnitude Spectrum 181 11.. Classification Based on Length of Impulse Response 18 11.3 filter realization 183 11.4 FIR filters 183 11.4.1 Direct Form 184 11.4. Cascaded Form 184 11.4.3 Linear-phase FIR Filters 184 11.5 iir filters 185 11.5.1 Direct form I 186 11.5. Direct form II 186 11.5.3 Cascaded Form 187 11.5.4 Parallel Form 188
COMMUNICATION SYSTEMS 1 AMPLITUDE MODULATION 1.1 Introduction 3 1. Amplitude Modulation 3 1.3 DSB-SC AM Signal 7 1.4 SSB-SC AM Signal 9 1.5 Vestigial-Sideband AM Signal 10 ANGLE MODULATION.1 Introduction 31. Angle Modulation 31.3 Types of Angle modulation 3.3.1 Phase Modulation System 3.3. Frequency Modulation System 3.4 Modulation Index 33.5 Transmission Bandwidth of Angle modulated Signal 33.5.1 Deviation Ratio 34.5. Expression of Transmission Bandwidth in Terms of Deviation Ratio 34.6 Power in Angle Modulated Signal 34 3 DIGITAL TRANSMISSION 3.1 Introduction 54 3. Sampling Process 54 3.3 Sampling Theorem 55 3.3.1 Nyquist Rate 55 3.3. Nyquist Interval55 3.4. Digital Pulse Modulation 56 3.5 Delta Modulation 59 3.6 Multiplexing 59 3.6.1 Frequency-Division Multiplexing 60 3.6. Time Division Multiplexing 60 4 DIGITAL MODULATION SCHEME 4.1 Introduction 79 4. Digital Bandpass Modulation 79 4.3 Coherent Binary systems 80 4.3.1 Amplitude Shift Keying 81 4.3. Binary Phase Shift Keying 8 4.3.3 Coherent Binary Frequency Shift Keying 83 4.4 Noncoherent Binary Systems 83 4.4.1 Differential Phase Shift Keying 84 4.4. Noncoherent Frequency Shift Keying 85 4.5 Multilevel modulated bandpass Signaling 85 4.5.1 Relations between Bit and Symbol Characteristics for Multilevel Signaling 85 4.5. M-ary Phase Shift Keying (MPSK) 86 4.5.3 Quadrature Phase Shift Keying (QPSK) 87 4.5.4 Quadrature Amplitude Modulation 88 4.5.5 M-ary Frequency Shift Keying (MFSK) 89 4.6 Comparison between Various Digital Modulation Scheme 89 4.7 Constellation Diagram 90 3.4 Pulse Modulation 55 3.4.1 Analog Pulse Modulation 55
CONTROL SYSTEMS AND PROCESS CONTROL 1 TRANSFER FUNCTIONS 1.1 Introduction 3 1. Control System 3 1.3 Transfer function 5 1.4 Feedback system 5 1.5 Block diagrams 6 1.6 Signal flow graph 9 1.6.1 Basic Terminologies of SFG 9 1.6. Gain Formula for SFG (Mason s Rule) 10 STABILITY.1 Introduction 31. Stability 31.3 Dependence of stability on location of poles 31.4 Methods of determining stability 34.5 Routh-hurwitz criterion 34.5.1 Routh s Tabulation 34.5. Location of Roots of Characteristic Equation using Routh s Table 35.5.3 Limitations of Routh-Hurwitz Criterion 37 3 TIME RESPONSE 3.1 Introduction 53 3. Time response 53 3.3 First Order Systems 53 3.3.1 Unit Impulse Response of First Order System 54 3.3. Unit Step Response of First Order System 54 3.4 Second Order System 55 3.4.1 Unit Step Response of Second Order System 56 3.5 Steady state errors 58 3.5.1 Steady State Error for Unity Feedback System 58 3.5. Steady State Error for Non-unity Feedback 60 3.6 Sensitivity 61 4 ROOT LOCUS TECHNIQUE 4.1 Introduction 79 4. The Root-Locus Concept 79 4.3 Properties of Root Locus 80 4.4 Rules for sketching root locus 81 5 FREQUENCY DOMAIN ANALYSIS 5.1 Introduction 101 5. Frequency response 101 5.3 Polar plot 103 5.4 Nyquist criterion 103 5.4.1 Principle of Argument 103 5.4. Nyquist Stability Criterion 104 5.5 Bode plots 104 5.6 Gain margin and phase margin 105 5.6.1 Determination of Gain Margin and Phase Margin using Nyquist Plot 106 5.6. Determination of Gain Margin and Phase Margin using Bode Plot 107
5.6.3 Stability of a System 107 5.7 Constant m-circles and constant n-circles 107 5.7.1 M-Circles 108 5.7. N-Circles 108 5.8 Nichols charts109 6 STATE VARIABLE ANALYSIS 6.1 Introduction 133 6. State variable system 133 6..1 State Differential Equations 134 6.. Block Diagram of State Space 134 6..3 Comparison between Transfer Function Approach and State Variable Approach 135 6.3 Solution of state equation 135 6.4 Transfer function from the state model 136 6.4.1 Characteristic Equation 137 6.4. Eigen Values 137 6.4.3 Eigen Vectors 137 6.4.4 Determination of Stability Using Eigen Values 137 6.5 Controllability and observability 138 6.5.1 Controllability 138 6.5. Output Controllability 138 6.5.3 Observability 138 6.6 Steady state error in state space 139 6.6.1 Analysis Using Final Value Theorem 139 6.6. Analysis Using Input Substitution 139 7 MODELLING OF PHYSICAL SYSTEM 7.1 Introduction165 7. Mechanical system 165 7..1 Translational Mechanical Systems 165 7.. Rotational Mechanical System 166 7.3 Hydraulic system 167 7.3.1 Model of Hydraulic Linear Actuators 167 7.3. Electrohydraulic Servovalves 168 7.4 Pneumatic system 168 7.4.1 Basic Elements of Pneumatic System 168 7.4. Construction and Working of Pneumatic System 169 7.5 Synchros 170 7.6 Servomotors 17 7.6.1 DC Servomotors 173 7.6. AC Servomotors 174 7.7 Stepper Motor 175 7.7.1 Classification of Stepper Motors 175 7.7. Characteristics of Stepper Motors 177 8 DESIGN OF CONTROL SYSTEMS 8.1 Introduction 195 8. ControllerS 195 8..1 Proportional Controller 195 8.. Proportional-Derivative (PD) Controller 196 8..3 Proportional-Integral (PI) Controller 197 8..4 Derivative Feedback Control 197 8..5 Proportional-Integral-Derivative (PID) Controller 198 8.3 CompensatorS 199 8.3.1 Lead Compensator 199 8.3. Lag Compensator 01 8.3.3 Lag-Lead compensator 0
8.4 Fuzzy controller 03 8.4.1 Fuzzy Logic 03 8.4. Fuzzy Set 04 8.4.3 Fuzzy Relations 05 8.4.4 Fuzzification 05 8.4.5 Defuzzification 06 8.4.6 Function Analysis of Fuzzy Control System 07 **********
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING 4.1 Introduction CHAPTER 4 Root Locus Technique A graph showing how the roots of the characteristic equation move around the s-plane as a single parameter varies is known as a root locus plot. Following topics related to root locus technique are included in the chapter: Basic concept of the root locus method Properties of root locus Useful rules for constructing the root loci 4. The Root-Locus Concept The roots of the characteristic equation of a system provide a valuable concerning the response of the system. To understand the root locus concept, consider the characteristics equation qs ^ h = 1+ GsHs ^ h ^ h= 0 Now, we rearrange the equation so that the parameter of interest, K, appears as the multiplying factor in the form, 1 + KP^s h = 0 For determining the locus of roots as K varies from 0 to 3, consider the polynomial in the form of poles and zeros as m % K ^s+ Zih i 1 + n = 0 ^s+ Pih % n m or % ^s+ Pjh+ K% ^s+ Zih = 0 j j i when K = 0, the roots of the characteristic equation are the poles of P^sh. n i.e. % ^s+ P j h = 0 j when K = 3, the roots of the characteristic equation are the zeros of P_ si. m i.e. % ^s+ Z i h = 0 i Hence, we noted that the locus of the roots of the characteristic equation 1+ KP^s h = 0 begins at the poles of P^sh and ends at the zeros of P^sh as K increases from zero to infinity.
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 17 4.3 Properties of Root Locus To examine the properties of root locus, we consider the characteristic equation as 1 + GsHs ^ h ^ h = 0 or 1 + KG1^shH1^s h = 0 where G1^shH1^sh does not contain the variable parameter K. So, we get G1^shH1^sh = K 1 From above equation, we conclude the following result: 1. For any value of s on the root locus, we have the magnitude G1^shH1^sh = 1 ; 3 < K < 3 K G1^shH1^sh m % i = 1 n j = 1 ^s+ Zih = ^s+ Pjh = % 1 K ; 3 < K < 3. For any value of s on the root locus, we have G1^shH1^sh = ^k + 1h π ; where k = 0,! 1,!,... = odd multiple of 180c for 0 # K < 3 G1^shH1^sh = k π ; where k = 0,! 1,!,... = even multiple of 180c for 3 < K # 0 3. Once the root locus are constructed, the values of K along the loci can be determined by K K = n % j = 1 m % i = 1 ^s+ Pjh ^s+ Zih The value of K at any point s 1 on the root locus is obtained from above equation by substituting value of s 1. Graphically, we write Productofvector lenghtsdrawn from the poles of GsHs ^ h ^ hto s = Productofvector lengths drawn from the zeros ofgshs ^ h ^ hto s Points to remember 1. Root loci are trajectories of roots of characteristic equation when a system parameter varies.. In general, this method can be applied to study the behaviour of roots of any algebraic equation with one or more variable parameters. 3. Root loci of multiple variable parameters can be treated by varying one parameter at a time. The resultant loci are called the root contours. 1 1
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 18 Root Locus Technique Chap 4 4. Root-Loci refers to the entire root loci for 3 < K < 3, 5. In general, the values of K are positive ^0 < K < 3h. Under unusual conditions when a system has positive feedback or the loop gain is negative, then we have K as negative. 4.4 Rules for sketching root locus Some important rules are given in the following texts that are useful for sketching the root loci. Rule 1: Symmetry of Root Locus Root locus are symmetrical with respect to the real axis of the s-plane. In general, the root locus are symmetrical with respect to the axes of symmetry of the pole-zero configuration of G^sHs h ^ h. Rule : Poles and Zeros on the Root Locus To locate the poles and zeros on root locus, we note the following points. 1. The K = 0 points on the root loci are at the poles of G^sHs h ^ h.. The K =!3 points on the root loci are at zeros of G^sHs h ^ h. Rule 3: Number of Branches of Root Locus The number of branches of the root locus equals to the order of the characteristic polynomial. Rule 4: Root Loci on the Real axis While sketching the root locus on real axis, we must note following points: 1. The entire real axis of the s-plane is occupied by the root locus for all values of K.. Root locus for K $ 0 are found in the section only if the total number of poles and zeros of G^sHs h ^ h to the right of the section is odd. The remaining sections of the real axis are occupied by the root locus for K # 0. 3. Complex poles and zeros of G^sHs h ^ h do not affect the type of root locus found on the real axis. Rule 5: Angle of Asymptotes of the Root Locus When n is the number of finite poles and m is the number of finite zeros of G^sHs h ^ h, respectively. Then n- m branches of the root locus approaches the infinity along straight line asymptotes whose angles are given by
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 19 ^q + 1hp q a =! ; for K $ 0 n m and θ a! qhp = n ^ m ; for K # 0 where q = 0, 1,,... ^n-m-1h Rule 6: Determination of Centroid The asymptotes cross the real axis at a point known as centroid, which is given by s A = / Rule 7: Angle of Departure Real partsofpoles ofgshs ^ h ^ h / Real partsofzeros of GsHs ^ h ^ h n m The angle of departure from an open loop pole is given by (for K $ 0) f D =! 6 ^q + 1hπ+ φ@; q = 0, 1,,... where, f is the net angle contribution at this pole, of all other open loop poles and zeros. For example, consider the plot shown in figure below. Figure 4.1: Illustration of Angle of Departure From the figure, we obtain f = q3+ q5 ^q1+ q+ q4h and f D =! 6 ^q + 1hπ+ φ@; q = 0, 1,,... Rule 8: Angle of Arrival The angle of arrival at an open loop zero is given by (for K $ 0) f a =! 6 ^q + 1hπ φ@; q = 0, 1,,... where f = net angle contribution at this zero, of all other open loop poles and zeros. For example, consider the plot shown in figure below.
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 0 Root Locus Technique Chap 4 Figure 4.: Illustration of Angle of Arrival From the figure, we obtain and ^ h =! 6 ^q + 1hπ φ@; q = 0, 1,,... f = q q1+ q3 NOTE For K # 0, departure and arrival angles are given by φ D = " 7_ q + 1iπ+ φa and φ a = " 7_ q + 1iπ φa Rule 9: Break-away and Break-in Points f a To determine the break-away and break-in points on the root locus, we consider the following points: 1. A root locus plot may have more than one breakaway points.. Break away points may be complex conjugates in the s-plane. 3. At the break away or break-in point, the branches of the root 180c locus form an angle of n with the real axis, where n is the number of closed loop poles arriving at or departing from the single breakaway or break-in point on the real axis. 4. The breakaway and break-in points of the root locus are the solution of dk = 0 ds i.e. breakaway and break in points are determined by finding maximum and minimum points of the gain K as a function of s with s restricted to real values. Rule 10: Intersects of Root Locus on Imaginary Axis Routh-Hurwitz criterion may be used to find the intersects of the root locus on the imaginary axis. **********
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 1 MCQ 4.1 NAT 4. MCQ 4.3 EEXERCIS Consider the sketch shown below. The root locus can be (A) (1) and (3) (B) () and (3) (C) () and (4) (D) (1) and (4) The forward-path transfer function of a ufb system is Ks ( + ) Gs () = ( s + 3)( s + s + ) The angle of departure from the complex poles is!φ D; where φ D = An open-loop pole-zero plot is shown below degree
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page Root Locus Technique Chap 4 The general shape of the root locus is MCQ 4.4 An open-loop pole-zero plot is shown below. The general shape of the root locus is
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 3 MCQ 4.5 The root locus is a (A) time-domain approach (B) frequency domain approach (C) combination of both (D) none of these NAT 4.6 MCQ 4.7 MCQ 4.8 A unity feedback control system has an open-loop transfer function Gs () = K ss ( + 7s + 1) The gain K for which s = 1+ j1 will lie on the root locus of this system is The root locus can be used to determine (A) the absolute stability of a system of a system (B) the relative stability of a system (C) both absolute and relative stabilities of a system (D) none of these The forward-path open-loop transfer function of a ufb system is Ks ( + 1) Gs () = s The root locus of this system is
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 4 Root Locus Technique Chap 4 MCQ 4.9 Consider the feedback system shown below. NAT 4.10 MCQ 4.11 NAT 4.1 For this system, the root locus is The open loop transfer function of a system is Ks ^ + 4h GsHs ^ h ^ h = ss ^ + h The value of K at breakaway point is The root locus of the system having the loop transfer function, GsHs () () = K ss ( + 4)( s + 4s+ 5) has (A) 3 break-away points (B) 3 break-in points (C) break-in and 1 break-away point (D) break-away and 1 break-in point The open loop transfer function of a system is GsHs ^ h ^ h = K ^ s + 1 h^ s + 5 h What is the value of K, so that the point s = 3+ j5 lies on the root locus?
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 5 MCQ 4.13 Consider the following statements : Gopal/450/9.6 (P) The effect of compensating pole is to pull the root locus towards left. (Q) The effect of compensating zero is to press the locus towards right. (A) None of the above statements is true (B) Statement (P) is true but statement (Q) is false (C) Statement (P) is false but statement (Q) is true (D) Both the statements are true Ks ( + 6) MCQ 4.14 For the system GsHs () () =, consider the following ( s + )( s + 4) characteristic of the root locus : 1. It has one asymptote.. It has intersection with jw-axis. 3. It has two real axis intersections. 4. It has two zeros at infinity. The root locus have characteristics (A) 1 and (B) 1 and 3 (C) 3 and 4 (D) and 4 MCQ 4.15 GK/98-17/90 MCQ 4.16 If the characteristic equation of a closed-loop system is 1 + K = 0 ss ^ + 1h^s + h the centroid of the asymptotes in root-locus will be (A) zero (B) (C) - 1 (D) - The characteristic equation of a closed-loop system is ss ( + 1)( s+ 3) + K( s+ ) = 0, K > 0 Which of the following statements is true? (A) Its roots are always real (B) It cannot have a breakaway point in the range 1 < Re[] s < 0 (C) Two of its roots tend to infinity along the asymptotes Re[] s = 1 (D) It may have complex roots in the right half plane.
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 6 Root Locus Technique Chap 4 MCQ 4.17 The open loop transfer function of a control system is s Anand k./373 GsHs Ke /6.1 ^ h ^ h = ss ^ + h For low frequencies, consider the following statements regarding the system. 1. s =.73 is break-away point.. s = 073. is break-away point. 3. s = 073. is break-in point. 4. s = 73. is break-in point. Which of the following is correct? (A) 1 and (B) 3 and 4 (C) 1 and 3 (D) and 4 MCQ 4.18 Consider the system with delay time ^tdh shown below. Suppose delay time t D = 1 sec. In root locus plot of the system, the break-away and break-in points are respectively (A) 0, 4.83 (B) 4.83, 0 (C) - 083., 4.83 (D) 4.83, -0. 83 **********
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 7 SOL 4.1 SolUTionS Correct option is (D). Here, option () and option (3) both are not symmetric about real axis. So, both can not be root locus. SOL 4. Correct answer is 108.4. Forward path transfer function of given ufb system is Ks ^ + h Gs ^ h = ^s + 3h^s + s + h So, we have the open loop poles and zeros as zero: s = poles: s = 3 and s = 1! j1 Therefore, we get the pole-zero plot as Angle of departure at pole P 1 is given by f D =! 6180c + f@ where f is net angle contribution at pole P 1 due to all other poles and zeros. f = fz fp = fz1 6 fp+ fp3@ where f = tan 1 1 1; f = 90c; f = tan 1 Z1 1 1 So, f = tan 1 90c + tan 1 : D Therefore, we obtain the departure angle as P P3
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 8 Root Locus Technique Chap 4 f D =! 6180c + f@ 1 1 =! 180c+ tan 1 90c tan 1 : D SOL 4.3 SOL 4.4 =! 6180 + 45 90 6. 56@ f D =! 108. 4c Hence, departure angle for pole P 1 is + 108.4c and departure angle for pole P is- 108.4c because P 1 and P are complex conjugate. Correct option is (A). Given open loop pole-zero plot is From the given plot, we have Number of poles, P = Number of zeros, Z = 1 Since, the number of branches of root locus is equal to number of poles, so we have Number of branches = Thus (B) and (D) are not correct. Again, the branch of root locus always starts from open loop pole and ends either at an open loop zero (or) infinite. Thus, (C) is incorrect and remaining Correct option is (A). Correct option is (A). Root locus always starts from open loop pole, and ends at open loop zero (or) infinite. Only option (A) satisfies this condition. We can find the root locus of given plot as follows Number of poles, P = Number of zeros, Z = 0 So, we have number of asymptotes P- Z = Also, the angle of asymptotes is obtained as f a ^q + 1h180c = P Z ; q = 0, 1
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 9 ^0 1 180c f a = + h = 90c; q = 0 SOL 4.5 and f a ^+ 1h180c = = 3 # 180c = 70c; q = 1 P Z Hence, we sketch the root locus plot as Correct option is (A). SOL 4.6 Correct answer is 10. Given the open loop transfer function, Gs _ i = K ss ^ + 7s + 1h So, we have the characteristic equation 1 + GsHs ^ h ^ h = 0 or 1 + K = 0 ss ^ + 7s + 1h or s + 7s + 1s+ K = 0...(1) If point s = 1+ j1 lies on root locus, then it satisfies characteristic equation. Substituting s = 1+ j1 in equation (1), we get 3 ^ 1+ jh + 7^ 1+ jh + 1^ 1+ jh + K = 0 or 10 + K = 0 So, K =+ 10 SOL 4.7 SOL 4.8 Correct option is (C). Correct option is (A). Forward path open loop transfer function of given ufb system is Ks ^ + 1h Gs ^ h = s So, we obtain the zeros of the system as s + 1 = 0 or s =! j1
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 30 Root Locus Technique Chap 4 Also, the poles of the system are s = 0; s = 0 So, we have the pole-zero plot for the system as SOL 4.9 Hence, option (B) and (D) may not be correct option. A point on the real axis lies on the root locus if the total number of poles and zeros to the right of this point is odd. This is not satisfied by (C) because at origin there are double pole. Thus, remaining Correct option is (A). Correct option is (A). The given system is shown below. We redraw the block diagram after moving take off point as shown below. So, the forward path transfer function is Ks ^ + h Gs ^ h = ^s + 1h^s + 3h Root locus is plotted for K = 0 to K = 3. But, here the gain K is negative. So, we will plot for K = 3 to K = 0. This is called complementary root locus. For this case, the root locus on the real axis is found to the left of an even count of real poles and real zeros of G^sh. Also, the plot will start from pole and ends on zero. Only option (A) satisfies the condition for given system.
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 31 SOL 4.10 Correct answer is 0.. For the given system, we have the characteristic equation 1 + GsHs ^ h ^ h = 0 or Ks ^ + 4h 1 + = 0 ss ^ + h or ^s + sh K = s + 4 The break points of the root locus are given by solution of dk ds = 0 or 8^s + 4h^s+ h ^s + sh^shb = 0 ^s + 4h or s -8s- 8 = 0 or s -4s- 4 = 0 So, s = 48., -0. 8 The root locus will be on real axis at any point, if total number of poles and zeros are odd to the right of that point. SOL 4.11 Hence, breakaway point should lie between s = 0 and -. So, breakaway point is s = 08.. Now, the open loop transfer function is Ks ^ + 4h GsHs ^ h ^ h = ss ^ + h So, we obtain the value of K at break-away point as GsHs ^ h ^ h = 1 at s = 08. K ^ 08. h + 4 or = 1 ^ 08. h ^ 0. 8 + h or K = 08. # 1. 18 = 0. 467. Correct option is (D). Open loop transfer function for given system is GsHs ^ h ^ h = K ss ^ + 4h^s + 4s+ 5h So, we have the characteristic equation 1 + K = 0 ss ^ + 4h^s + 4s+ 5h
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 3 Root Locus Technique Chap 4 or K = ss ^ + 4h^s + 4s+ 5h...(1) Differentiating the above equation with respect to s and equating it to zero, we have dk = s 4 s 4s 5 s 4s s 4 ds 8^ + h^ + + h+ ^ + h^ + hb = 0 or ^s+ 4h^s + 4s+ 5+ s + 4sh = 0 or ^s+ h^s + 8s + 5h = 0...() Solving the above equation, we get s = and s = 0. 775, -3. 5 Now, we check for maxima and minima value of gain K at above point. If gain is maximum, then that point will be break away point. If gain is minimum, then that point will be break in point. Again, differentiating equation () with respect to s, we get d K = 6 ^s + 8s+ 5h+ ^s+ h^4s+ 8h@ ds = ^6s + 4s+ 1h For s = 0. 775 and s = 3. 5, we have d K = 60<. 0 ds So, the points s = 0. 775 and - 3. 5 are maxima points. Hence, s = 0. 775 and s = 3. 5 are break away points. Again, for s =, we have d K =+ 3 > 0 ds So, the point s = is minima points. Hence, s = is break in point. Thus, there is two break away points ^s =. 0775, 3. 5h and one break in point ( s = ). SOL 4.1 Correct answer is 9. First we check if point lies on root locus. For this, we use angle criterion GsHs ^ h ^ h =! 180 s= s 0 Since, we have GsHs ^ h ^ h s= 3+ j5 = K ^ 3 + j5 + 1 h^ 3 + j5 + 5 h = K ^ + j5 h^ + j5 h 1 So, GsHs ^ h ^ h tan 5 1 = tan 5 s= 3+ j5 b l b l 1 180 tan 5 1 = c + tan 5 180c = i.e. the given point satisfies angle criterion. Now, using magnitude condition, we have
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 33 GsHs ^ h ^ h s= 3+ j5 = 1 SOL 4.13 SOL 4.14 or or K ^ + j5h^+ j5h = 1 K 4+ 5 4+ 5 = 1 Thus, K = 9 Correct option is (A). The effect of compensating pole is to pull the root locus towards right half of s-plane. The effect of compensating zero is to pull the root locus towards left half of s-plane. Correct option is (B). Given the open loop transfer function, Ks ^ + 6h GsHs ^ h ^ h = ^s + h^s + 4h So, we have the open loop poles and zeros as Poles : s = and s = 4 Zeros : s = 6 Therefore, the number of asymptotes is P- Z = 1 = 1 So, the characteristic (1) is correct. Now, we have the characteristic equation for the system ^s+ h^s+ 4h+ Ks ^ + 6h = 0 or s + ^6 + Kh s + 8 + 6K = 0 For the characteristic equation, we form the Routh s array as s 1 8+ 6K s 1 s 0 6 + K 8+ 6K Root locus is plotted for K = 0 to 3. i.e. K > 0. Here, for K > 0 root locus does not intersect jw axis because s 1 row will not be zero. Thus, characteristic () is incorrect. For the given system, we have two poles and one zero. So, one imaginary zero lies on infinite. Therefore, the characteristic (4) is incorrect. Hence, (B) must be correct option. But, we check further for characteristic (3) as follows. We sketch the root locus for given system as
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 34 Root Locus Technique Chap 4 SOL 4.15 SOL 4.16 It has two real axis intersections. So, characteristic (3) is correct. Correct option is (C). The given characteristic equation is 1 + K = 0 ss ^ + 1h^s + h or 1 + GsHs ^ h ^ h = 0 So, the open loop transfer function is GsHs ^ h ^ h = K ss ^ + 1h^s + h The centroid s A is, ^0 1 h ^0h s A = = 1 3 0 Correct option is (C). Characteristic equation of the given closed loop system is ss ^ + 1h^s+ 3h+ K^s + h = 0; K > 0 Ks ^ + h or 1 + = 0 ss ^ + 1h^s + 3h So, the open loop transfer function is given as Ks ^ + h GsHs ^ h ^ h = ss ^ + 1h^s + 3h Therefore, we have the open loop poles and zeros as poles: s = 0, s = 1, s = 3 zero: s = So, we obtain the pole zero plot for the system as For the pole-zero location, we obtain the following characteristic of root locus
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 35 Number of asymptotes: P- Z = 3 1 = Angles of asymptotes: ^q + 1h180c f a = ; P Z =, q = 0, 1 P Z f a = 90c and 70c Centroid: Sum of Re6P@ Sum of Re6Z@ s A = P Z ^0 1 3h ^ h = = 3 1 SOL 4.17 = 1 Thus, from above analysis, we sketch the root locus as For the root locus, we conclude the following points 1. The break away point lies in the range, 1 < Re6@ s < 0. Two of its roots tends to infinite along the asymptotes Re6@ s = 1. 3. Root locus lies only in left half of s-plane. Correct option is (D). For low frequencies, we have e -s. 1 - s So, the open loop transfer function is K^1 sh Ks ^ 1h GsHs ^ h ^ h = = ss ^ + h ss ^ + h K^1 sh or GsHs ^ h ^ h = = 1 ss ^ + h ss ^ + h or K = 1 s The break points are given by solution of dk = 0 ds
GATE STUDY PACKAGE INSTRUMENTATION ENGINEERING Set of 5 Books by NODIA Publication Page 36 Root Locus Technique Chap 4 or d ss ^ + h ds ; 1 s E = 0 SOL 4.18 or ^1 sh^s+ h ss ^ + h^ 1h = 0 or s -s- = 0 Therefore, the break points are s = +! 4 8 1! 3 + = =.73, 073. or -0.73 Since, G^sHs h ^ h is negative, so the root locus will be complementary root locus and will exist at any point on the real axis, if the total number of poles and zeros to the right of that point is even. Root locus will exist on real axis between s = and 0 and also for s >+ 1. Hence, break away point will be s = 0.73 and break in point will be s =+.73 Correct option is (C). For delay time t D = 1 sec, the characteristic equation of the system is s 1 + Ke = 0; K $ 0 s Now, we have the approximation e - s 1 s/, = s 1 + s/ + s So, the characteristic equation becomes Ks ^ h 1 = 0...(1) ss ^ + h Therefore, the open loop transfer function is
SALIENT FEATURES * Brief Theory * Methodology * Important Points * *MCQ * Numerical Answer Type Questions * Memory Based Questions * Detailed Solution for Each and Every Problem* Chap 4 Root Locus Technique Page 37 Ks ^ h GsHs ^ h ^ h = ss+ Since, G^sHs h ^ h is negative, so the root locus will be complementary root locus and will exist at any point on the real axis, if the total number of poles and zeros to the right of that point is even. ^ So, the root locus (complementary) for the given system will exist on real axis in the region < s < 0 and s > The break points of the system are given by solution of dk ds = 0...() From equation (1), we have ss ^ + h K = ^s h Substituting it in equation (), we get d ss ^ + h ds > H = 0 ^s h or ^s h^s+ h ss ^ + h = 0 or s -s-4-s - s = 0 or s -4s- 4 = 0 So, s = 4! 5. 657 = 483., 0. 83 Hence, break away point is s = 083. and beak in point is s = 483.. ********** h