Tất cả những kiến thức cần và đủ về điện ba pha, từ nguồn, truyền tải cho đến phân phối, nguyên lý hoạt động, thông số của các thiết bị điện, rơle, tính toán thiết kế phục vụ cho chuyên nghành hệ thống điện, các bản vẽ chi tiết dễ hiểu.
Fleckenstein Engineering – Electrical Three-Phase Electrical Power Three-Phase Electrical Power addresses all aspects of three-phase power circuits The book treats the transmission of electrical power from the common sources where it is generated to locations where it is consumed At typical facilities where electrical power is used, the book covers the important topics of grounding, currents, power, demand, metering, circuit protection, motors, motor protection, power factor correction, tariffs, electrical drawings, and relays Included in the text are the necessary methods of computing currents and power in all possible types of circuit applications as those that are balanced, unbalanced, leading, lagging, three-wire, and four-wire Focusing on electrical gear, programs, and issues related to the generation and use of three-phase electrical power, this contemporary educational guide: • Uses simple, straightforward language to explain key concepts and their underlying theory • Introduces numerous examples, illustrations, and photographs to aid in comprehension • Employs phasor concepts throughout the text to aid in the analysis of three-phase circuits • Encourages applied learning by supplying practical problems at the end of each chapter • Provides extensive references and a glossary of symbols, acronyms, and equations Three-Phase Electrical Power delivers a much-needed modern-day treatment of three-phase electrical power for electrical engineering students and practitioners alike Three-Phase Electrical Power K26554 ISBN: 978-1-4987-3777-7 90000 781498 737777 Joseph E Fleckenstein www.EngineeringEBooksPdf.com Three-Phase Electrical Power www.EngineeringEBooksPdf.com www.EngineeringEBooksPdf.com Three-Phase Electrical Power Joseph E Fleckenstein Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business www.EngineeringEBooksPdf.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20150626 International Standard Book Number-13: 978-1-4987-3778-4 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.EngineeringEBooksPdf.com Contents Preface xiii Author xix Alternating Current 1.1 Single-Phase Alternating Current .2 1.1.1 Instantaneous Voltage and Instantaneous Current 1.1.2 RMS Voltage and RMS Current 1.1.3 Single-Phase Circuits 1.1.3.1 Resistive Loads 1.1.3.2 Leading and Lagging Power Factor 1.1.4 Phasor Diagrams of Single-Phase Circuits 11 1.1.5 Parallel Single-Phase Loads 14 1.1.6 Polar Notation 17 1.2 Three-Phase Alternating Current: Basic Concepts 18 1.2.1 Three-Phase Alternating Current Source 18 1.2.2 Delta and Wye Circuits Defined 19 1.2.3 Common Service Voltages 20 1.2.3.1 240/120 VAC Three-Phase Four-Wire Delta 20 1.2.3.2 480 VAC Three-Phase Two-Wire Delta or Three-Phase Three-Wire Delta 21 1.2.3.3 208/120 VAC Wye (Three-Phase Four-Wire) .22 1.2.3.4 480 VAC Wye (Three-Phase Four-Wire) 23 1.2.3.5 480 VAC Wye (Three-Phase Three-Wire) 24 1.2.4 Phases and Lines 24 1.2.5 Balanced and Unbalanced Three-Phase Circuits 25 1.2.6 Lead, Lag, and Power Factor in a Balanced Three-Phase Circuit 25 1.2.7 Phasor Diagrams of Three-Phase Circuits 27 1.2.8 Advantage of Three-Phase Electricity over Single-Phase Electricity 30 1.3 ac Impedance and Admittance 33 1.3.1 Impedance 33 1.3.2 Admittance 49 Problems 53 v www.EngineeringEBooksPdf.com vi Contents Generation, Transmission, and Distribution 57 2.1 Generation 57 2.1.1 Fossil-Fired Steam Power Plants 58 2.1.2 Biofuel-Fired Steam Power Plant 61 2.1.3 Nuclear Power Plant 61 2.1.4 Gas Turbine Generators 61 2.1.5 Wind Turbines 62 2.1.6 Solar 62 2.1.7 Hydro 63 2.1.8 Geothermal 63 2.2 Transmission .63 2.3 Distribution 65 2.4 Electrical Gear 66 2.4.1 Generators 66 2.4.2 Transformers 72 2.4.2.1 Power Transformers 73 2.4.2.2 Instrument Transformers 79 2.5 Per Unit Calculations 81 2.5.1 Systems with Transformers 88 2.6 Cybersecurity .90 Problems 92 Grounding 95 3.1 Definitions 95 3.1.1 Ground 96 3.1.2 Equipment Grounding Conductor 96 3.1.3 Neutral Conductor 96 3.1.4 Bonding 96 3.1.5 Ground Loops 97 3.2 Reasons for Grounding 97 3.2.1 Non-Current-Carrying Ground 99 3.2.2 Current-Carrying Ground 99 3.2.3 Lightning Grounds 99 3.2.4 Grounding of LV, MV, and HV Cable Screens 100 3.3 Single-Phase Grounding 101 3.4 Three-Phase Grounding 103 3.4.1 Utility Three-Phase Grounding Practices 103 3.4.2 User Three-Phase Grounding Practices 103 3.4.3 Grounding Resistors 104 3.4.3.1 Low-Resistance NGR 105 3.4.3.2 High-Resistance NGR 106 3.4.4 Ground Fault Detector 106 3.4.5 Neutral Grounding Reactor 107 Problems 108 www.EngineeringEBooksPdf.com vii Contents Calculating Currents in Three-Phase Circuits 109 4.1 Calculating Currents in Balanced Three-Phase Circuits 109 4.1.1 Calculating Currents in a Balanced Three-Phase Delta Circuit: General 109 4.1.1.1 Resistive Loads 110 4.1.1.2 Capacitive Loads 115 4.1.1.3 Inductive Loads 116 4.1.1.4 Two or More Loads 119 4.1.2 Calculating Currents in a Balanced Three-Phase Wye Circuit: General 127 4.1.2.1 Resistive Loads 128 4.1.2.2 Capacitive Loads 130 4.1.2.3 Inductive Loads 130 4.1.2.4 Two or More Loads 130 4.2 Calculating Currents in Unbalanced Three-Phase Circuits: General 130 4.2.1 Unbalanced Three-Phase Delta Circuits 131 4.2.1.1 Unbalanced Three-Phase Delta Circuits with Resistive, Inductive, or Capacitive Loads 131 4.2.1.2 Unbalanced Three-Phase Delta Circuit with Only Resistive Loads 142 4.2.2 Unbalanced Three-Phase Wye Circuit 145 4.3 Combined Balanced or Unbalanced Three-Phase Circuits 147 Problems 158 Calculating Three-Phase Power 161 5.1 Units 161 5.2 Calculating Power of a Single-Phase Circuit 162 5.3 Calculating Power in Balanced Three-Phase Circuits 166 5.3.1 Calculating Power in Balanced Three-Phase Wye Circuits 169 5.3.1.1 Calculating Power in Balanced Three-Phase Wye Circuits: Resistive Loads 170 5.3.1.2 Calculating Power in Balanced Three-Phase Wye Circuits: Inductive or Capacitive Loads 172 5.3.2 Calculating Power in Balanced Delta Three-Phase Circuits 172 5.3.2.1 Calculating Power in Balanced Three-Phase Delta Circuits: Resistive Loads 173 5.3.2.2 Calculating Power in Balanced Three-Phase Delta Circuits: Inductive or Capacitive 174 5.4 Calculating Power in Unbalanced Three-Phase Circuits 174 5.4.1 Calculating Power in Unbalanced Three-Phase Wye Circuits 174 www.EngineeringEBooksPdf.com viii Contents 5.4.1.1 Calculating Power in Unbalanced Three-Phase Wye Circuits: Resistive Loads 175 5.4.1.2 Calculating Power in Unbalanced Three-Phase Wye Circuits: Inductive or Capacitive Loads 178 5.4.2 Calculating Power in Unbalanced Three-Phase Delta Circuits 180 5.4.2.1 Calculating Power in Unbalanced Three-Phase Delta Circuits: Resistive Loads 181 5.4.2.2 Calculating Power in Unbalanced Three-Phase Delta Circuits: Inductive or Capacitive 182 5.5 Calculating Power in a Three-Phase Circuit with Mixed Wye and Delta Loads 188 5.5.1 Using Phase Parameters 188 5.5.2 Using Line Parameters 189 5.6 Nonlinear Electrical Circuits 189 5.6.1 Linear Circuits and Power Factor 191 5.6.2 Root Mean Square 193 5.6.3 Nonlinear Circuits and Power Factor 194 5.7 Conclusion 196 Problems 197 Demand and Demand Response 201 6.1 Demand Defined 201 6.1.1 Why Demand? 202 6.1.2 Decreasing Demand 202 6.2 Demand Response 203 6.2.1 Definitions 203 6.2.2 Implementation Examples of Demand Response 205 6.2.2.1 NYISO (New York–Based Utility) 205 6.2.2.2 TXU Energy (Texas-Based Utility) 206 6.2.2.3 EnerNOC (Boston-Based Utility) 206 6.3 Smart Grid 207 6.3.1 Definition 207 6.3.2 Implementation Examples of Smart Grid 207 6.3.2.1 Florida Power and Light 208 Problems 208 Instruments and Meters 209 7.1 Power Measurements 209 7.1.1 Single-Phase Power Measurements 210 7.1.2 Three-Phase Power Measurements 211 www.EngineeringEBooksPdf.com ix Contents 7.2 Measuring Power in Three-Phase, Three-Wire Circuits 212 7.2.1 Using One Single-Phase Wattmeter to Measure Power in a Three-Phase, Three-Wire Circuit 213 7.2.2 Using Two Single-Phase Wattmeters to Measure Power in a Three-Phase, Three-Wire Circuit 214 7.2.3 Using Three Single-Phase Wattmeters to Measure Power in a Three-Phase, Three-Wire Circuit 216 7.2.4 Using a Three-Phase Wattmeter to Measure Power in a Three-Wire, Three-Phase Circuit 219 7.3 Measuring Power in a Three-Phase, Four-Wire Circuit 220 7.3.1 Using One Single-Phase Wattmeter to Measure Power in a Three-Phase, Four-Wire Circuit 220 7.3.2 Using Three Single-Phase Wattmeters to Measure Power in a Three-Phase, Four-Wire Circuit 220 7.3.3 Using a Three-Phase Wattmeter to Measure Power in a Three-Phase, Four-Wire Circuit 220 7.4 Power Analyzers and Power Quality Meters .222 7.5 Watt-Hour Meters 222 7.6 Demand Meters 224 7.7 Smart Meters 226 7.7.1 Definitions 226 7.7.2 Operation 227 7.7.3 Implementation Examples 228 7.7.3.1 Ontario Power Authority, Canada 229 7.7.3.2 State of Pennsylvania 229 7.8 Submeters 229 7.9 Phase Sequence Meters 230 7.10 Earth Resistance 231 7.11 Megohmmeter 232 7.12 High-Potential Tester 233 7.13 Multimeters .234 Problems 235 Circuit Protection 237 8.1 General Requirements 237 8.2 Ungrounded Systems 240 8.3 Overcurrent 240 8.4 Faults 241 8.5 Circuit Breakers 243 8.5.1 Molded Case Circuit Breakers 243 8.5.1.1 Thermal-Magnetic Circuit Breakers 244 8.5.1.2 Magnetic-Only Circuit Breaker 246 8.5.1.3 Electronic Circuit Breakers 246 8.5.1.4 Sizing MCCBs 248 www.EngineeringEBooksPdf.com Symbols, Acronyms, and Equations 389 where I(t) is the current expressed as a function of time (rms amps) I is the numerical value of current (rms) θ SP is the angle of lead or angle of lag (radians) (current with respect to voltage in a single-phase circuit) for a lagging power factor, θ SP < for a leading power factor, θ SP > For single-phase circuits (Figure 1.12): The magnitude and lead/lag of line current Ia resulting from the addition of line current I1 at lag/lead angle θ1 (to line voltage) and line current I2 at lead/lag angle θ2 (to line voltage) and … line current In at lead/lag angle θn (to line voltage): |I a |= {(I a - x )2 + (I a - y )2 }½ (1.5) where I a - x = I1 cos q1 + I cos q2 + I n cos qn , and I a - y = I1 sin q1 + I sin q2 + I n sin qn qa = sin -1(I a - y ¸ I a ) Power consumption of a balanced, linear three-phase wye or balanced, linear three-phase delta load: (1.6) P = ( ) VL I L cos qP Equation for the relationship of phase current and line current lead/lag angle in a balanced delta and a balanced or unbalanced wye circuit: qP = qL – 30° (1.7) where θL is the line lead/lag angle between line current and line voltage (degrees or radians) (for lagging current, θL < 0; for leading current, θL > 0) θ P is the phase lead/lag angle between phase current and phase voltage (degrees or radians) (for lagging current, θ P < 0; for leading current, θ P > 0) www.EngineeringEBooksPdf.com 390 Symbols, Acronyms, and Equations Specifically, For balanced delta circuits: qP-CA = qL -A/CA - 30°, where θP–CA is the lead/lag of current in phase C–A with respect to voltage C–A θL–A/CA is the lead/lag of current in conductor A with respect to voltage C–A qP - AB = qL - B/AB - 30°, where θP–AB is the lead/lag of current in phase A–B with respect to voltage A–B θL–B/AB is the lead/lag of current in conductor B with respect to voltage A–B qP - BC = qL - C/BC - 30°, where θP–BC is the lead/lag of current in phase B–C with respect to voltage B–C θL–C/BC is the lead/lag of current in conductor C with respect to voltage B–C For balanced or unbalanced wye circuits: qP-A/AD = qL -A/CA - 30°, where θP–A/AD is the lead/lag of current in phase A–D with respect to voltage A–D θL–A/CA is the lead/lag of current in conductor A with respect to voltage C–A qP-B/BD = qL -B/AB - 30°, where θP–B/BD is the lead/lag of current in phase B–D with respect to voltage B–D θL–B/AB is the lead/lag of current in conductor B with respect to voltage A–B qP-C/CD = qL -C/BC - 30°, where θP–C/CD is the lead/lag of current in phase C–D with respect to voltage C–D θL–C/BC is the lead/lag of current in conductor C with respect to voltage B–C www.EngineeringEBooksPdf.com 391 Symbols, Acronyms, and Equations Impedance: Z = V/I (1.8) Admittance: Y = I/Z (1.9) Line current of a balanced three-phase delta load: IL = ( ) IP (4.1) For balanced three-phase circuits, delta, or wye The magnitude and lead/ lag of line current IB resulting from the addition of line current I1 at lag/ lead angle θ1 (to line voltage) and line current I2 at lead/lag angle θ2 (to line voltage) and up to …line current In at lead/lag angle θn (to line voltage) (Figure 4.9) Although Figure 4.9 shows a delta load the equation is applicable to any combination of balanced loads I B = {(I B - x )2 + (I B - y )2 }½ (4.2) where I B - x = I1 cos q1 + I cos q2 + I n cos qn , and I B - y = I1 sin q1 + I sin q2 + I n sin qn qB = sin -1(I B - y ¸ I B ) In a balanced three-phase circuit, the three-line currents, IA, IB, and IC, are equal in absolute value Therefore, IB = IA = IC Phase voltage in a balanced three-phase wye load or an unbalanced threephase load with a grounded neutral: VL = ( ) VP www.EngineeringEBooksPdf.com (4.3) 392 Symbols, Acronyms, and Equations Equations 4.4 through 4.9 Equations for calculating line currents when the phase currents and respective leads/lags in an unbalanced delta circuit are known With reference to Figure 1.14—where Iab is the current in phase a–b; I bc is the current in phase b–c; Iac is the current in phase a–c; θ P–AB is the lead/lag of c urrent in phase A–B; θ P–BC is the lead/lag of current in phase B–C; θP–CA is the lead/lag of current in phase C–A; IA, IB, and IC are the line currents in, respectively, conductors A, B, and C; θL–A, is the lead/lag of the line current in phase A; θL–B is the lead/lag of the line current in phase B; and θL–C is the lead/lag of the line current in phase C { |I A | = (I A- x )2 + (I A- y )2 } ½ qL - A = (l - 120°) (4.4) (4.5) where I A is the current in line A qL - A is the lead (lag ) of current I A with respect to line voltage Vca I ba - x = – I ab cos qP - AB I ca - x = – I ca (1/2) é( ) sin qP - CA + cos qP - CA ù ë û I A - x = I ba - x + I ca - x I ba - y = – I ab sin qP - AB ( ) I ca - y = I ca (1/2) é cos qP - CA – sin qP - CA ù ë û I A - y = I ba - y + I ca - y l = sin –1(I A - y ¸ I A ) Valid range of θP-AB and θP-CA: ±90°; valid range of θL-A: +120° to –60° { |I B | = (I B - x )2 + (I B - y )2 } qL - B = sin -1(I B - y I B ) ẵ www.EngineeringEBooksPdf.com (4.6) (4.7) 393 Symbols, Acronyms, and Equations where I B is the current in line B qL -B is the lead (lag) of current I B with respect to line voltage Vab I ab- x = I ab cos qP-AB I cb- x = -I bc (1/2) é( ) sin qP-BC - cos qP-BC ù ë û I B- x = I ab- x + I cb- x I ab- y = I ab sin qP-AB I cb- y = I bc (1/2) é( ) cos qP-BC + sin qP-BC ù ë û I B- y = I ab- y + I cb- y Valid range of θP-AB and θP-CB: ±90°; valid range of θL-B: +120° to –60° { } I C = ( I C - x )2 + ( I C - y ) ½ qL - C = (j - 240° ) (4.8) (4.9) where I C is the current in line C qL - A is the lead (lag ) of current I C with respect to line voltage Vbc ( ) ( ) I bc - x = I bc (1/2) é sin qP - BC - cos qP - BC ù ë û é I ac - x = I ca (1/2) sin qP - CA - cos qP - CA ù ë û I C = I bc - x + I ac - x I bc - y = -I bc (1/2) é( ) cos qP - BC + sin qP - BC ù ë û é I ac - y = -I ca ( 1/2 ) ( ) cos qP - CA - sin qP - CA ù ë û I C - y = I bc - y + I ac - y j = sin -1(I C - y ¸ I C ) Valid range of θP-BC and θP-AC: ±90°; valid range of θL-C: +120° to –60° Equations 4.10 through 4.15 Equations for determining line currents in an unbalanced delta circuit when the loads on all three phases are resistive www.EngineeringEBooksPdf.com 394 Symbols, Acronyms, and Equations With reference to Figure 1.14—where Iab is the current in phase a–b; Ibc is the current in phase b–c; Iac is the current in phase a–c; θP−AB is the lead/lag of current in phase A–B; θP−BC is the lead/lag of current in phase B–C; θP−CA is the lead/lag of current in phase C–A; IA, IB, and IC are the line currents in, respectively, conductors A, B, and C; θL−A, is the lead/lag of the line current in phase A; θL−B is the lead/lag of the line current in phase B; and θL−C is the lead/lag of the line current in phase C { } ½ (4.10) I A = (I A– x )2 + (I A– y )2 qL – A = (l – 120° ) (4.11) where I A is the current in line A qL -A is the lead (lag ) of current I A with respect to line voltage Vca I A- x = – I ab – (1/2) I ca I A- y = I ca l = sin –1 ( 3/2 ) ( I A- y ¸ I A ) { } ½ (4.12) I B = ( I B - x )2 + ( I B - y ) qL – B = sin –1(I B– y ¸ I B ) (4.13) where I B is the current in line B qL -B is the lead (lag) of current I B with respect to line voltage Vab I B = – I ab – (1/2) I bc I B- y = I bc ( 3/2 ) { |I C | = (I C - x )2 + (I C - y )2 } ½ (4.14) www.EngineeringEBooksPdf.com 395 Symbols, Acronyms, and Equations qL – C = (j – 240° ) (4.15) where I C is the current in line C qL -C is the lead (lag ) of current I C with respect to line voltage Vbc I C- x = (1/2)I ca - (1/2) I bc I C- y = I bc ( ) 3/2 - I ca ( 3/2 ) -1 (j) = sin (I C-y ¸ I C ) Equations 4.16 through 4.21 Equations for determining the currents in a feeder that delivers power to two or more three-phase loads any of which could be balanced or unbalanced Reference is made to Figure 4.27 Conductor A-1 is the conductor connecting common conductor A to phase A of load #1; A-2 is the conductor connecting conductor A to phase A of load #2 I A, I B, and IC are the line currents in, respectively, feeder conductors A, B, and C, θ L−A is the lead/lag of the line current in phase A, θ L−B is the lead/lag of the line current in phase B, and θ L−C is the lead/lag of the line current in phase C { |I A | = (I A - x )2 + (I A - y )2 } ½ (4.16), and qL – A = ( k – 120°) (4.17) where k = sin -1(I A- y ¸ I A ) I A- x = å(I A1- x + I A 2- x + I A 3- x + I AN - x ) I A- y = å(I A1- y + I A 2- y + I A 3- y + I AN - y ) I A1- x = -I A1(1/2) é( ) sin qA1 + cos qA1 ù ë û é ù I A 2- x = – I A (1/2) ( ) sin qA + cos qA ¼ to ë û www.EngineeringEBooksPdf.com 396 Symbols, Acronyms, and Equations I AN - x = – I AN (1/2) é( ) sin qAN + cos qAN ù ë û I A1- y = I A1(1/2) é( ) cos qA1 – sin qA1 ù ¼to ë û I AN -y = I AN (1/2) é( ) cos qAN – sin qAN ù ë û I A1 is the line current in branch of conductor A to load # I A2 is the line current in branch of conductor A to load # ¼ to I AN is the line current in branch of conductor A to load # N qA1 is the lead/lag of line current I A-1 with respect to line voltage Vca qA is the lead/lag of line current I A-2 with respect to line voltage Vca ¼ to qAN is the lead/lag of line current I A-N with respect to line voltage Vca { |I B | = (I B - x )2 + (I B - y )2 } ½ , and (4.18) qL – B = sin –1(I B – y ¸ I B ) (4.19) where I B- x = å(I B1- x + I B 2- x + I B 3- x ¼+ I BN - x ) I B- y = å(I B1- y + I B 2- y + I B 3- y ¼+ I BN - y ) I B1- x = I B1 cos qB1 ¼ to I BN - x = I BN cos qBN I B1- y = I B1 sin qB1 ¼ to I BN - y = I BN sin qLN I B1 is the line current in branch of conductor B to load #1 I B is the line current in branch of conductor B to load # 2¼ to I BN is the line current in branch of conductor B to load # N qB1 is the lead/lag of line current I B1 with respect to line voltage Vca qB is the lead/lag of line current I A with respect to line voltage Vca ¼ to qBN is the lead / lag of line current I AN with respect to line voltage Vca I B1 is the line current in branch of conductor B to load #1 I B is the line current in branch of conductor B to load # ¼ to I BN is the line current in branch of conductor B to load # N qB1 is the lead/lag of line current I B-1 with respect to line voltage Vab qB is the lead/lag of line current I B-2 with respect to line voltage Vab ¼ to qBN is the lead/lag of line current I B-N with respect to line voltageVab www.EngineeringEBooksPdf.com 397 Symbols, Acronyms, and Equations { |I C | = (I C - x )2 + (I C - y )2 } ½ and (4.20) qL -C = z – 240° (4.21) z = sin -1(I C- y ¸ I C ) I C- x = å(I C1- x + I C 2- x + I C 3- x + I CN - x ) I C- y = å(I C1- y + I C 2- y + I C 3- y + I CN - y ) I C1 is the current in branch of conductor C to load #1 I C is the current in branch of conductor C to load # 2¼ to I CN is the current in branch of conductor C to load # N I C1- x = I C1(1/2) é( ) sin qC1 - cos qC1 ù ë û é I C 2- x = I C (1/2) ( ) sin qC - cos qC ù ¼to ë û I CN - x = I CN (1/2) é( ) sin qCN - cos qCN ù ë û I C1- y = -I C1(1/2) é( ) cos qC1 + sin qC1 ù ë û I C 2- y = -I C (1/2) é( ) cos qC + sin qC ù ¼to ë û é I CN - y = -I CN (1/2) ( ) cos qCN + sin qCN ù ë û qC1 is the lead / lag of line current I C1 with respect to line voltage Vbc qC is the lead/lag of line current I C with respect to line voltage Vbc ¼to qCN is the lead/lag of line current I CN with respect to line voltage Vbc Equation 5.22 Equation that defines root mean square of a known function: T2 ì ü2 ï ï f (t) rms = í1/(T2 - T1 ) [ f (t)] dt ý T1 ỵï þï (5.22) ò Equation 5.23 Equation for determining root mean square from known values of a function sampled at uniformly spaced distances: { } f (X ) rms = éë1/n ùû éë X12 + X 22 + X n2 ùû www.EngineeringEBooksPdf.com 1/2 (5.23) 398 Symbols, Acronyms, and Equations Equation B.1: Guidelines for readily determining the arc sin (sin−1) of angles λ, φ, κ, or ζ Some of the computations of this textbook require the calculation of the arc sin (sin–1) of angles λ, φ, κ, and ζ in order to determine the displacement angle of current vectors in the CCW direction from the positive abscissa Unless a person is well experienced and well practiced in calculating the sin–1 of angles, it is easy to make errors in the performance of this exercise Presented here are simple guidelines that may be used as an assist to correctly and simply determine the value of the sin–1 of an angle in any quadrant Reference is made to Figure B.1 that depicts a vector “I” in quadrant I Vector I is also the hypotenuse of a triangle with vertical component “y” and horizontal component “x,” and vector I is at an angle “a” from the positive abscissa By definition, the sin of angle a is y/I, and the arc sin of y/I (sin–1 y/I) is the angle a While dealing with angles and values pertinent to quadrant I, little confusion usually ensues Confusion usually comes about when computing values in quadrant II, quadrant III, and quadrant IV In quadrant II, the sine of I is the sin of angle “b.” Similarly in quadrant III, the sin of vector I is the sine of angle “c,” and in quadrant IV, the sin of I is the sine of angle “d.” If, say, angle b in quadrant II is 120° and a handheld calculator is used to compute the sin of b, the calculator would correctly indicate the true sin of b to be 0.866 However, if the same calculator is used to compute the sin–1 of 0.866, the calculator indicates the angle to be 60° since the calculator does not “know” that the angle under consideration is in quadrant II Therein is the potential for mistaken calculations Following are some simple rules of thumb that will prove helpful to accurately determine the values of the sin–1 of any angle in any quadrant Given: The value of sin–1 of vector I, which is in quadrant I, II, III, or IV Determine: The angle of I measured CCW from the positive abscissa (i.e., λ, φ, κ, or ζ) QI QII x I b a c d QIII QIV FIGURE B.1 Aid in determining arc sin of an angle www.EngineeringEBooksPdf.com y 399 Symbols, Acronyms, and Equations Let M = │sin–1 λ│, │sin–1 φ│, │sin–1 κ│, or │sin–1 ζ│ Compute M by calculator or table (The calculated value of M will be between 0° and 90°.) From Table B.1, determine the quadrant in which I is located Calculate the value of λ, φ, κ, or ζ per fourth row of Table B.1 Summary of three-phase power equations: Three-phase delta, wye, or mixed circuit power using line parameters is shown in Tables B.2 through B.4 TABLE B.1 Determination of Quadrant Value of x ≥0 ≤0 ≤0 ≥0 Value of y Quadrant ≥0 ≥0 ≤0 ≥0 I II III IV Value of λ, φ, κ, or ζ =M = 180° − M = 180° + M = 360° – M TABLE B.2 Summary of Three-Phase Power Equations: Three-Phase Wye Circuit Equation Numbera Type Circuit Balanced resistive Balanced inductive or capacitive Unbalanced resistive Phase Parametersb Line Parameters 5.6 5.7 P = VL I L P = 3VPIP 5.8 P = 3VPIPcos θP 5.9 P = VL I L cos(qL - 30°) 5.14 P = VP[IA + IB + IC] 5.15 P = (1/ ) VL [I A cos(qL-A/CA - 30°) + I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] Unbalanced inductive or capacitive 5.16 P = VP[IAcos θP−AD + IBcos θP−BD + ICcos θP−CD] 5.17 P = (1/ ) VL [I A cos(qL-A/CA - 30°) +I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] a b The equation number is the same as the associated paragraph number In a wye circuit, VP = 1/ VL ( ) www.EngineeringEBooksPdf.com 400 TABLE B.3 Summary of Three-Phase Power Equations: Three-Phase Delta Circuit Type Circuit Balanced resistive Equation Numbera Phase Parametersb Line Parameters 5.10 5.11 P = 3VPIP Balanced inductive or capacitive 5.12 P = 3VPIPcos θP 5.13 P = VL I L cos(qL - 30°) Unbalanced resistive 5.18 P = VP[IP−AB + IP−BC + IP−CA] 5.19 P = 1/ VL [ I A cos(qL-A/CA - 30°) P = VL I L ( ) + I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] Unbalanced inductive or capacitive 5.20 P = VP [ I A-B cos qP-AB + I B-C cos qP-BC + I C-A cos qP-CA ] 5.21 ( ) P = 1/ VL [ I A cos(qL-A/CA - 30°) + I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] a b The equation number is the same as the associated paragraph number In a delta circuit VP = VL TABLE B.4 Summary of Three-Phase Power Equations: Three-Phase Mixed Circuit Type of Circuit Equation Balanced resistive P = VL I L Balanced inductive or capacitive P = VL I L cos(qL - 30°) Unbalanced resistive P = 1/ VL [ I A cos(qL-A/CA - 30°) ( ) + I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] Unbalanced inductive or capacitive ( ) P = 1/ VL [ I A cos(qL-A/CA - 30°) + I B cos(qL-B/AB - 30°) + I C cos(qL-C/BC - 30°)] www.EngineeringEBooksPdf.com References 1.1 J.W Hammond, Charles Proteus Steinmetz—A Biography, The Century Co New York/London, 1924, pp 195–227 1.2 G.C Blalock, Principles of Electrical Engineering, 3rd edn., McGraw-Hill Book Company, New York, 1950, p 270 1.3 T.L Floyd, Principles of Electric Circuits, 6th edn., Prentice Hall, Upper Saddle River, NJ, 2000, p 627 1.4 T.L Floyd, Principles of Electric Circuits, 6th edn., Prentice Hall, Upper Saddle River, NJ, 2000, pp 430–455 1.5 T.L Floyd, Principles of Electric Circuits, 6th edn., Prentice Hall, Upper Saddle River, NJ, 2000, p 892 1.6 V Del Toro, Electrical Engineering Fundamentals, 2nd edn., Prentice Hall, Upper Saddle River, NJ, 1986, pp 305–314 2.1 U.S Energy Information Administration, Coal Overview, January 2013 2.2 L Ward, Going with the flow, Wall Street Journal, November 11, 2013 2.3 Global Energy Network Institute, National Energy Grid Canada, January 2014 3.1 Merriam-Webster, Inc., Webster’s Ninth New Collegiate Dictionary, MerriamWebster, Inc., Springfield, MA, 1990 3.2 National Institute of Safety and Health, Publication 98-131, Worker Deaths by Electrocution, May 1998 3.3 IEEE Std 141-1993, Recommended Practice for Electrical Power Distribution for Industrial Plants (Red Book), Paragraph 4.4.1.2 4.1 J.W Nilsson, S.A Riedel, Electric Circuits, 6th edn., Prentice Hall, Upper Saddle River, NJ, 2000, p 553 4.2 J.W Nilsson, S.A Riedel, Electric Circuits, 6th edn., Prentice-Hall, Upper Saddle River, NJ, 2000, p 548 7.1 New York Public Services Commission Bulletin, A Primer on Smart Metering, Fall 2013 7.2 Maryland Public Service Commission, Approved Electric Submeters, COMAR 20.25.01.04A.(3), 2010 8.1 IEEE Std 242-2001, Recommended Practices for Protection and Coordination of Industrial and Commercial Power Systems (Buff Book) 8.2 Eaton Corporation, Short Circuit Current Rating and Available Fault Currents, August 27, 2013 8.3 GE Publication GET3550F, Short Circuit Current Calculations for Industrial and Commercial Power Systems, 1989 8.4 UL Standard 489, Molded Case Circuit Breakers and Circuit Breaker Enclosures 8.5 IEEE Std C37.13, Standard for Low Voltage AC Power Circuit Breakers Used in Enclosures, 2008 9.1 C Wester, Motor Protection Principles, GE Multilin Publication, 2014 401 www.EngineeringEBooksPdf.com 402 References 9.2 USA National Electric Code, 2002, National Fire Protection Association, Article 430 9.3 NEMA Standard MG 1-2003, Motors and Generators A.1 R.S Burlington, Handbook of Mathematical Tables and Formulas, 3rd edn., Handbook Publishers, Inc., Sandusky, OH, 1954, p 18 www.EngineeringEBooksPdf.com Fleckenstein Engineering – Electrical Three-Phase Electrical Power Three-Phase Electrical Power addresses all aspects of three-phase power circuits The book treats the transmission of electrical power from the common sources where it is generated to locations where it is consumed At typical facilities where electrical power is used, the book covers the important topics of grounding, currents, power, demand, metering, circuit protection, motors, motor protection, power factor correction, tariffs, electrical drawings, and relays Included in the text are the necessary methods of computing currents and power in all possible types of circuit applications as those that are balanced, unbalanced, leading, lagging, three-wire, and four-wire Focusing on electrical gear, programs, and issues related to the generation and use of three-phase electrical power, this contemporary educational guide: • Uses simple, straightforward language to explain key concepts and their underlying theory • Introduces numerous examples, illustrations, and photographs to aid in comprehension • Employs phasor concepts throughout the text to aid in the analysis of three-phase circuits • Encourages applied learning by supplying practical problems at the end of each chapter • Provides extensive references and a glossary of symbols, acronyms, and equations Three-Phase Electrical Power delivers a much-needed modern-day treatment of three-phase electrical power for electrical engineering students and practitioners alike Three-Phase Electrical Power K26554 ISBN: 978-1-4987-3777-7 90000 781498 737777 www.EngineeringEBooksPdf.com Joseph E Fleckenstein