EECE 370 Problems, Spring 2010

i Instructor’s Manual to accompany Chapman Electric Machinery Fundamentals Fourth Edition Stephen J. Chapman BAE SYSTEMS Australia ii Instructor’s Manual to accompany Electric Machinery Fundamentals, Fourth Edition Copyright  2004 McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. No part of this book may be used or reproduced in any manner whatsoever without written permission, with the following exception: homework solutions may be copied for classroom use. ISBN: ??? iii TABLE OF CONTENTS CHAPTER 1: INTRODUCTION TO MACHINERY PRINCIPLES 1 CHAPTER 2: TRANSFORMERS 23 CHAPTER 3: INTRODUCTION TO POWER ELECTRONICS 63 CHAPTER 4: AC MACHINERY FUNDAMENTALS 103 CHAPTER 5: SYNCHRONOUS GENERATORS 109 CHAPTER 6: SYNCHRONOUS MOTORS 149 CHAPTER 7: INDUCTION MOTORS 171 CHAPTER 8: DC MACHINERY FUNDAMENTALS 204 CHAPTER 9: DC MOTORS AND GENERATORS 214 CHAPTER 10: SINGLE-PHASE AND SPECIAL-PURPOSE MOTORS 270 APPENDIX A: REVIEW OF THREE-PHASE CIRCUITS 280 APPENDIX B: COIL PITCH AND DISTRIBUTED WINDINGS 288 APPENDIX C: SALIENT POLE THEORY OF SYNCHRONOUS MACHINES 295 APPENDIX D: ERRATA FOR ELECTRIC MACHINERY FUNDAMENTALS 4/E 301 iv PREFACE TO THE INSTRUCTOR This Instructor’s Manual is intended to accompany the fourth edition of Electric Machinery Fundamentals. To make this manual easier to use, it has been made self-contained. Both the original problem statement and the problem solution are given for each problem in the book. This structure should make it easier to copy pages from the manual for posting after problems have been assigned. Many of the problems in Chapters 2, 5, 6, and 9 require that a student read one or more values from a magnetization curve. The required curves are given within the textbook, but they are shown with relatively few vertical and horizontal lines so that they will not appear too cluttered. Electronic copies of the corresponding open- circuit characteristics, short-circuit characteristics, and magnetization curves as also supplied with the book. They are supplied in two forms, as MATLAB MAT-files and as ASCII text files. Students can use these files for electronic solutions to homework problems. The ASCII files are supplied so that the information can be used with non-MATLAB software. Please note that the file extent of the magnetization curves and open-circuit characteristics have changed in this edition. In the Third Edition, I used the file extent *.mag for magnetization curves. Unfortunately, after the book was published, Microsoft appropriated that extent for a new Access table type in Office 2000. That made it hard for users to examine and modify the data in the files. In this edition, all magnetization curves, open-circuit characteristics, short-circuit characteristics, etc. use the file extent *.dat to avoid this problem. Each curve is given in ASCII format with comments at the beginning. For example, the magnetization curve in Figure P9-1 is contained in file p91_mag.dat. Its contents are shown below: % This is the magnetization curve shown in Figure % P9-1. The first column is the field current in % amps, and the second column is the internal % generated voltage in volts at a speed of 1200 r/min. % To use this file in MATLAB, type "load p91_mag.dat". % The data will be loaded into an N x 2 array named % "p91_mag", with the first column containing If and % the second column containing the open-circuit voltage. % MATLAB function "interp1" can be used to recover % a value from this curve. 0 0 0.0132 6.67 0.03 13.33 0.033 16 0.067 31.30 0.1 45.46 0.133 60.26 0.167 75.06 0.2 89.74 v 0.233 104.4 0.267 118.86 0.3 132.86 0.333 146.46 0.367 159.78 0.4 172.18 0.433 183.98 0.467 195.04 0.5 205.18 0.533 214.52 0.567 223.06 0.6 231.2 0.633 238 0.667 244.14 0.7 249.74 0.733 255.08 0.767 259.2 0.8 263.74 0.833 267.6 0.867 270.8 0.9 273.6 0.933 276.14 0.966 278 1 279.74 1.033 281.48 1.067 282.94 1.1 284.28 1.133 285.48 1.167 286.54 1.2 287.3 1.233 287.86 1.267 288.36 1.3 288.82 1.333 289.2 1.367 289.375 1.4 289.567 1.433 289.689 1.466 289.811 1.5 289.950 To use this curve in a MATLAB program, the user would include the following statements in the program: % Get the magnetization curve. Note that this curve is % defined for a speed of 1200 r/min. load p91_mag.dat if_values = p91_mag(:,1); ea_values = p91_mag(:,2); n_0 = 1200; Unfortunately, an error occurred during the production of this book, and the values (resistances, voltages, etc.) in some end-of-chapter artwork are not the same as the values quoted in the end-of-chapter problem text. I have attached corrected pages showing each discrepancy in Appendix D of this manual. Please print these pages and distribute them to your students before assigning homework problems. (Note that this error will be corrected at the second printing, so it may not be present in your student’s books.) vi The solutions in this manual have been checked carefully, but inevitably some errors will have slipped through. If you locate errors which you would like to see corrected, please feel free to contact me at the address shown below, or at my email address schapman@tpgi.com.au. I greatly appreciate your input! My physical and email addresses may change from time to time, but my contact details will always be available at the book’s Web site, which is http://www.mhhe.com/engcs/electrical/chapman/. I am also contemplating a homework problem refresh, with additional problems added on the book’s Web site mid- way through the life of this edition. If that feature would be useful to you, please provide me with feedback about which problems that you actually use, and the areas where you would like to have additional exercises. This information can be passed to the email address given below, or alternately via you McGraw-Hill representative. Thank you. Stephen J. Chapman Melbourne, Australia January 4, 2004 Stephen J. Chapman 278 Orrong Road Caulfield North, VIC 3161 Australia Phone +61-3-9527-9372 1 Chapter 1: Introduction to Machinery Principles 1-1. A motor’s shaft is spinning at a speed of 3000 r/min. What is the shaft speed in radians per second? SOLUTION The speed in radians per second is ( ) 1 min 2 rad 3000 r/min 314.2 rad/s 60 s 1 r π ω    = =       1-2. A flywheel with a moment of inertia of 2 kg ⋅ m2 is initially at rest. If a torque of 5 N ⋅ m (counterclockwise) is suddenly applied to the flywheel, what will be the speed of the flywheel after 5 s? Express that speed in both radians per second and revolutions per minute. SOLUTION The speed in radians per second is: ( ) 2 5 N m 5 s 12.5 rad/s 2 kg m t t J τ ω α ⋅   = = = =   ⋅   The speed in revolutions per minute is: ( ) 1 r 60 s 12.5 rad/s 119.4 r/min 2 rad 1 min n π    = =       1-3. A force of 5 N is applied to a cylinder, as shown in Figure P1-1. What are the magnitude and direction of the torque produced on the cylinder? What is the angular acceleration α of the cylinder? SOLUTION The magnitude and the direction of the torque on this cylinder is: , CCW sin ind θ τ = rF ( )( ) ind 0.25 m 10 N sin 30 1.25 N m, CCW τ = ° = ⋅ The resulting angular acceleration is: 2 2 1.25 N m 0.25 rad/s 5 kg m J τ α ⋅ = = = ⋅ 1-4. A motor is supplying 60 N ⋅ m of torque to its load. If the motor’s shaft is turning at 1800 r/min, what is the mechanical power supplied to the load in watts? In horsepower? SOLUTION The mechanical power supplied to the load is ( )( ) 1 min 2 rad 60 N m 1800 r/min 11,310 W 60 s 1 r P π τω = = ⋅ = 2 ( ) 1 hp 11,310 W 15.2 hp 746 W P = = 1-5. A ferromagnetic core is shown in Figure P1-2. The depth of the core is 5 cm. The other dimensions of the core are as shown in the figure. Find the value of the current that will produce a flux of 0.005 Wb. With this current, what is the flux density at the top of the core? What is the flux density at the right side of the core? Assume that the relative permeability of the core is 1000. SOLUTION There are three regions in this core. The top and bottom form one region, the left side forms a second region, and the right side forms a third region. If we assume that the mean path length of the flux is in the center of each leg of the core, and if we ignore spreading at the corners of the core, then the path lengths are 1l = 2(27.5 cm) = 55 cm, 2l = 30 cm, and 3l = 30 cm. The reluctances of these regions are: ( )( )( )( ) 1 7 0.55 m 58.36 kA t/Wb 1000 4 10 H/m 0.05 m 0.15 m r o l l A A µ µ µ π − = = = = ⋅ × R ( )( )( )( ) 2 7 0.30 m 47.75 kA t/Wb 1000 4 10 H/m 0.05 m 0.10 m r o l l A A µ µ µ π − = = = = ⋅ × R ( )( )( )( ) 3 7 0.30 m 95.49 kA t/Wb 1000 4 10 H/m 0.05 m 0.05 m r o l l A A µ µ µ π − = = = = ⋅ × R The total reluctance is thus TOT 1 2 3 58.36 47.75 95.49 201.6 kA t/Wb = + + = + + = ⋅ R R R R and the magnetomotive force required to produce a flux of 0.003 Wb is ( )( ) 0.005 Wb 201.6 kA t/Wb 1008 A t = φ = ⋅ = ⋅ F R and the required current is 1008 A t 2.52 A 400 t i N ⋅ = = = F The flux density on the top of the core is ( )( ) 0.005 Wb 0.67 T 0.15 m 0.05 m B A = φ = = 3 The flux density on the right side of the core is ( )( ) 0.005 Wb 2.0 T 0.05 m 0.05 m B A = φ = = 1-6. A ferromagnetic core with a relative permeability of 1500 is shown in Figure P1-3. The dimensions are as shown in the diagram, and the depth of the core is 7 cm. The air gaps on the left and right sides of the core are 0.070 and 0.020 cm, respectively. Because of fringing effects, the effective area of the air gaps is 5 percent larger than their physical size. If there are 4001 turns in the coil wrapped around the center leg of the core and if the current in the coil is 1.0 A, what is the flux in each of the left, center, and right legs of the core? What is the flux density in each air gap? SOLUTION This core can be divided up into five regions. Let 1 R be the reluctance of the left-hand portion of the core, 2 R be the reluctance of the left-hand air gap, 3 R be the reluctance of the right-hand portion of the core, 4 R be the reluctance of the right-hand air gap, and 5 R be the reluctance of the center leg of the core. Then the total reluctance of the core is ( )( ) 1 2 3 4 TOT 5 1 2 3 4 + + = + + + + R R R R R R R R R R ( )( )( )( ) 1 1 7 0 1 1.11 m 90.1 kA t/Wb 2000 4 10 H/m 0.07 m 0.07 m r l µ µ A π − = = = ⋅ × R ( )( )( )( ) 2 2 7 0 2 0.0007 m 108.3 kA t/Wb 4 10 H/m 0.07 m 0.07 m 1.05 l µ A π − = = = ⋅ × R ( )( )( )( ) 3 3 7 0 3 1.11 m 90.1 kA t/Wb 2000 4 10 H/m 0.07 m 0.07 m r l µ µ A π − = = = ⋅ × R ( )( )( )( ) 4 4 7 0 4 0.0005 m 77.3 kA t/Wb 4 10 H/m 0.07 m 0.07 m 1.05 l µ A π − = = = ⋅ × R ( )( )( )( ) 5 5 7 0 5 0.37 m 30.0 kA t/Wb 2000 4 10 H/m 0.07 m 0.07 m r l µ µ A π − = = = ⋅ × R The total reluctance is 1 In the first printing, this value was given incorrectly as 300. 4 ( )( ) ( )( ) 1 2 3 4 TOT 5 1 2 3 4 90.1 108.3 90.1 77.3 30.0 120.8 kA t/Wb 90.1 108.3 90.1 77.3 + + + + = + = + = ⋅ + + + + + + R R R R R R R R R R The total flux in the core is equal to the flux in the center leg: ( )( ) center TOT TOT 400 t 1.0 A 0.0033 Wb 120.8 kA t/Wb φ = φ = = = ⋅ F R The fluxes in the left and right legs can be found by the “flux divider rule”, which is analogous to the current divider rule. ( ) ( ) ( ) 3 4 left TOT 1 2 3 4 90.1 77.3 0.0033 Wb 0.00193 Wb 90.1 108.3 90.1 77.3 φ φ + + = = = + + + + + + R R R R R R ( ) ( ) ( ) 1 2 right TOT 1 2 3 4 90.1 108.3 0.0033 Wb 0.00229 Wb 90.1 108.3 90.1 77.3 φ φ + + = = = + + + + + + R R R R R R The flux density in the air gaps can be determined from the equation φ = BA : ( )( )( ) left left eff 0.00193 Wb 0.375 T 0.07 cm 0.07 cm 1.05 B A = φ = = ( )( )( ) right right eff 0.00229 Wb 0.445 T 0.07 cm 0.07 cm 1.05 B A = φ = = 1-7. A two-legged core is shown in Figure P1-4. The winding on the left leg of the core (N1) has 400 turns, and the winding on the right (N2) has 300 turns. The coils are wound in the directions shown in the figure. If the dimensions are as shown, then what flux would be produced by currents i1 = 0.5 A and i2 = 0.75 A? Assume r µ = 1000 and constant. 5 SOLUTION The two coils on this core are would so that their magnetomotive forces are additive, so the total magnetomotive force on this core is ( )( ) ( )( ) TOT 1 1 2 2 400 t 0.5 A 300 t 0.75 A 425 A t N i N i = + = + = ⋅ F The total reluctance in the core is ( )( )( )( ) TOT 7 0 2.60 m 92.0 kA t/Wb 1000 4 10 H/m 0.15 m 0.15 m r l µ µ A π − = = = ⋅ × R and the flux in the core is: TOT TOT 425 A t 0.00462 Wb 92.0 kA t/Wb φ ⋅ = = = ⋅ F R 1-8. A core with three legs is shown in Figure P1-5. Its depth is 5 cm, and there are 200 turns on the leftmost leg. The relative permeability of the core can be assumed to be 1500 and constant. What flux exists in each of the three legs of the core? What is the flux density in each of the legs? Assume a 4% increase in the effective area of the air gap due to fringing effects. SOLUTION This core can be divided up into four regions. Let 1 R be the reluctance of the left-hand portion of the core, 2 R be the reluctance of the center leg of the core, 3 R be the reluctance of the center air gap, and 4 R be the reluctance of the right-hand portion of the core. Then the total reluctance of the core is ( ) 2 3 4 TOT 1 2 3 4 + = + + + R R R R R R R R ( )( )( )( ) 1 1 7 0 1 1.08 m 127.3 kA t/Wb 1500 4 10 H/m 0.09 m 0.05 m r l µ µ A π − = = = ⋅ × R ( )( )( )( ) 2 2 7 0 2 0.34 m 24.0 kA t/Wb 1500 4 10 H/m 0.15 m 0.05 m r l µ µ A π − = = = ⋅ × R ( )( )( )( ) 3 3 7 0 3 0.0004 m 40.8 kA t/Wb 4 10 H/m 0.15 m 0.05 m 1.04 l µ A π − = = = ⋅ × R ( )( )( )( ) 4 4 7 0 4 1.08 m 127.3 kA t/Wb 1500 4 10 H/m 0.09 m 0.05 m r l µ µ A π − = = = ⋅ × R The total reluctance is 6 ( ) ( ) 2 3 4 TOT 1 2 3 4 24.0 40.8 127.3 127.3 170.2 kA t/Wb 24.0 40.8 127.3 + + = + = + = ⋅ + + + + R R R R R R R R The total flux in the core is equal to the flux in the left leg: ( )( ) left TOT TOT 200 t 2.0 A 0.00235 Wb 170.2 kA t/Wb φ = φ = = = ⋅ F R The fluxes in the center and right legs can be found by the “flux divider rule”, which is analogous to the current divider rule. ( ) 4 center TOT 2 3 4 127.3 0.00235 Wb 0.00156 Wb 24.0 40.8 127.3 φ φ = = = + + + + R R R R ( ) 2 3 right TOT 2 3 4 24.0 40.8 0.00235 Wb 0.00079 Wb 24.0 40.8 127.3 φ φ + + = = = + + + + R R R R R The flux density in the legs can be determined from the equation φ = BA : ( )( ) left left 0.00235 Wb 0.522 T 0.09 cm 0.05 cm B A = φ = = ( )( ) center center 0.00156 Wb 0.208 T 0.15 cm 0.05 cm B A = φ = = ( )( ) left right 0.00079 Wb 0.176 T 0.09 cm 0.05 cm B A = φ = = 1-9. A wire is shown in Figure P1-6 which is carrying 5.0 A in the presence of a magnetic field. Calculate the magnitude and direction of the force induced on the wire. SOLUTION The force on this wire can be calculated from the equation ( ) ( )( )( ) 5 A 1 m 0.25 T 1.25 N, into the page i ilB = × = = = F l B 7 1-10. The wire is shown in Figure P1-7 is moving in the presence of a magnetic field. With the information given in the figure, determine the magnitude and direction of the induced voltage in the wire. SOLUTION The induced voltage on this wire can be calculated from the equation shown below. The voltage on the wire is positive downward because the vector quantity v ×B points downward. ( ) ( )( )( ) ind cos 45 5 m/s 0.25 T 0.50 m cos 45 0.442 V, positive down e vBl = × ⋅ = ° = ° = v B l 1-11. Repeat Problem 1-10 for the wire in Figure P1-8. SOLUTION The induced voltage on this wire can be calculated from the equation shown below. The total voltage is zero, because the vector quantity v ×B points into the page, while the wire runs in the plane of the page. ( ) ( )( )( ) ind cos 90 1 m/s 0.5 T 0.5 m cos 90 0 V e vBl = × ⋅ = ° = ° = v B l 1-12. The core shown in Figure P1-4 is made of a steel whose magnetization curve is shown in Figure P1-9. Repeat Problem 1-7, but this time do not assume a constant value of µr. How much flux is produced in the core by the currents specified? What is the relative permeability of this core under these conditions? Was the assumption in Problem 1-7 that the relative permeability was equal to 1000 a good assumption for these conditions? Is it a good assumption in general? 8 SOLUTION The magnetization curve for this core is shown below: The two coils on this core are wound so that their magnetomotive forces are additive, so the total magnetomotive force on this core is ( )( ) ( )( ) TOT 1 1 2 2 400 t 0.5 A 300 t 0.75 A 425 A t N i N i = + = + = ⋅ F Therefore, the magnetizing intensity H is 9 425 A t 163 A t/m 2.60 m c H l ⋅ = = = ⋅ F From the magnetization curve, B = 0.15 T and the total flux in the core is ( )( )( ) TOT 0.15 T 0.15 m 0.15 m 0.0033 Wb BA φ = = = The relative permeability of the core can be found from the reluctance as follows: A l r 0 TOT TOT µ µ φ = R = F Solving for µr yields ( )( ) ( )( )( )( ) TOT -7 TOT 0 0.0033 Wb 2.6 m 714 425 A t 4 10 H/m 0.15 m 0.15 m r l A φ µ µ π = = = ⋅ × F The assumption that r µ = 1000 is not very good here. It is not very good in general. 1-13. A core with three legs is shown in Figure P1-10. Its depth is 8 cm, and there are 400 turns on the center leg. The remaining dimensions are shown in the figure. The core is composed of a steel having the magnetization curve shown in Figure 1-10c. Answer the following questions about this core: (a) What current is required to produce a flux density of 0.5 T in the central leg of the core? (b) What current is required to produce a flux density of 1.0 T in the central leg of the core? Is it twice the current in part (a)? (c) What are the reluctances of the central and right legs of the core under the conditions in part (a)? (d) What are the reluctances of the central and right legs of the core under the conditions in part (b)? (e) What conclusion can you make about reluctances in real magnetic cores? 10 SOLUTION The magnetization curve for this core is shown below: (a) A flux density of 0.5 T in the central core corresponds to a total flux of ( )( )( ) TOT 0.5 T 0.08 m 0.08 m 0.0032 Wb BA φ = = = By symmetry, the flux in each of the two outer legs must be 1 2 0.0016 Wb φ = φ = , and the flux density in the other legs must be ( )( ) 1 2 0.0016 Wb 0.25 T 0.08 m 0.08 m B = B = = The magnetizing intensity H required to produce a flux density of 0.25 T can be found from Figure 1-10c. It is 50 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 0.50 T is 70 A·t/m. Therefore, the total MMF needed is TOT center center outer outer H l H l = + F ( )( ) ( )( ) TOT 70 A t/m 0.24 m 50 A t/m 0.72 m 52.8 A t = ⋅ + ⋅ = ⋅ F and the required current is TOT 52.8 A t 0.13 A 400 t i N ⋅ = = = F (b) A flux density of 1.0 T in the central core corresponds to a total flux of ( )( )( ) TOT 1.0 T 0.08 m 0.08 m 0.0064 Wb BA φ = = = By symmetry, the flux in each of the two outer legs must be 1 2 0.0032 Wb φ = φ = , and the flux density in the other legs must be ( )( ) 1 2 0.0032 Wb 0.50 T 0.08 m 0.08 m B = B = = 11 The magnetizing intensity H required to produce a flux density of 0.50 T can be found from Figure 1-10c. It is 70 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 1.00 T is about 160 A·t/m. Therefore, the total MMF needed is TOT center center outer outer H I H I = + F ( )( ) ( )( ) TOT 160 A t/m 0.24 m 70 A t/m 0.72 m 88.8 A t = ⋅ + ⋅ = ⋅ F and the required current is TOT 88.8 A t 0.22 A 400 t i N φ ⋅ = = = This current is less not twice the current in part (a). (c) The reluctance of the central leg of the core under the conditions of part (a) is: ( )( ) TOT cent TOT 70 A t/m 0.24 m 5.25 kA t/Wb 0.0032 Wb φ ⋅ = = = ⋅ F R The reluctance of the right leg of the core under the conditions of part (a) is: ( )( ) TOT right TOT 50 A t/m 0.72 m 22.5 kA t/Wb 0.0016 Wb φ ⋅ = = = ⋅ F R (d) The reluctance of the central leg of the core under the conditions of part (b) is: ( )( ) TOT cent TOT 160 A t/m 0.24 m 6.0 kA t/Wb 0.0064 Wb φ ⋅ = = = ⋅ F R The reluctance of the right leg of the core under the conditions of part (b) is: ( )( ) TOT right TOT 70 A t/m 0.72 m 15.75 kA t/Wb 0.0032 Wb φ ⋅ = = = ⋅ F R (e) The reluctances in real magnetic cores are not constant. 1-14. A two-legged magnetic core with an air gap is shown in Figure P1-11. The depth of the core is 5 cm, the length of the air gap in the core is 0.06 cm, and the number of turns on the coil is 1000. The magnetization curve of the core material is shown in Figure P1-9. Assume a 5 percent increase in effective air-gap area to account for fringing. How much current is required to produce an air-gap flux density of 0.5 T? What are the flux densities of the four sides of the core at that current? What is the total flux present in the air gap? 12 SOLUTION The magnetization curve for this core is shown below: An air-gap flux density of 0.5 T requires a total flux of ( )( )( )( ) eff 0.5 T 0.05 m 0.05 m 1.05 0.00131 Wb φ = BA = = This flux requires a flux density in the right-hand leg of ( )( ) right 0.00131 Wb 0.524 T 0.05 m 0.05 m B A = φ = = The flux density in the other three legs of the core is ( )( ) top left bottom 0.00131 Wb 0.262 T 0.10 m 0.05 m B B B A φ = = = = = 13 The magnetizing intensity required to produce a flux density of 0.5 T in the air gap can be found from the equation ag ag o B = µ H : ag ag 7 0 0.5 T 398 kA t/m 4 10 H/m B H µ π − = = = ⋅ × The magnetizing intensity required to produce a flux density of 0.524 T in the right-hand leg of the core can be found from Figure P1-9 to be right 410 A t/m H = ⋅ The magnetizing intensity required to produce a flux density of 0.262 T in the top, left, and bottom legs of the core can be found from Figure P1-9 to be top left bottom 240 A t/m H H H = = = ⋅ The total MMF required to produce the flux is TOT ag ag right right top top left left bottom bottom H l H l H l H l H l = + + + + F ( )( ) ( )( ) ( )( ) TOT 398 kA t/m 0.0006 m 410 A t/m 0.40 m 3 240 A t/m 0.40 m = ⋅ + ⋅ + ⋅ F TOT 278.6 164 288 691 A t = + + = ⋅ F and the required current is TOT 691 A t 0.691 A 1000 t i N ⋅ = = = F The flux densities in the four sides of the core and the total flux present in the air gap were calculated above. 1-15. A transformer core with an effective mean path length of 10 in has a 300-turn coil wrapped around one leg. Its cross-sectional area is 0.25 in2, and its magnetization curve is shown in Figure 1-10c. If current of 0.25 A is flowing in the coil, what is the total flux in the core? What is the flux density? SOLUTION The magnetizing intensity applied to this core is 14 ( )( ) ( )( ) 300 t 0.25 A 295 A t/m 10 in 0.0254 m/in c c Ni H l l = = = = ⋅ F From the magnetization curve, the flux density in the core is B = 1.27 T The total flux in the core is ( )( ) 2 2 0.0254 m 1.27 T 0.25 in 0.000205 Wb 1 in BA φ = = = 1-16. The core shown in Figure P1-2 has the flux φ shown in Figure P1-12. Sketch the voltage present at the terminals of the coil. SOLUTION By Lenz’ Law, an increasing flux in the direction shown on the core will produce a voltage that tends to oppose the increase. This voltage will be the same polarity as the direction shown on the core, so it will be positive. The induced voltage in the core is given by the equation ind d e N dt φ = so the voltage in the windings will be 15 Time dt N d φ eind 0 < t < 2 s ( 500 t) 0.010 Wb 2 s 2.50 V 2 < t < 5 s ( 500 t) 0.020 Wb 3 s − -3.33 V 5 < t < 7 s ( 500 t) 0.010 Wb 2 s 2.50 V 7 < t < 8 s ( 500 t) 0.010 Wb 1 s 5.00 V The resulting voltage is plotted below: 1-17. Figure P1-13 shows the core of a simple dc motor. The magnetization curve for the metal in this core is given by Figure 1-10c and d. Assume that the cross-sectional area of each air gap is 18 cm2 and that the width of each air gap is 0.05 cm. The effective diameter of the rotor core is 4 cm. 16 SOLUTION The magnetization curve for this core is shown below: The relative permeability of this core is shown below: Note: This is a design problem, and the answer presented here is not unique. Other values could be selected for the flux density in part (a), and other numbers of turns could be selected in part (c). These other answers are also correct if the proper steps were followed, and if the choices were reasonable. (a) From Figure 1-10c, a reasonable maximum flux density would be about 1.2 T. Notice that the saturation effects become significant for higher flux densities. (b) At a flux density of 1.2 T, the total flux in the core would be (1.2 T)(0.04 m)(0.04 m) 0.00192 Wb φ = BA = = (c) The total reluctance of the core is: 17 TOT stator air gap 1 rotor air gap 2 = + + + R R R R R At a flux density of 1.2 T, the relative permeability r µ of the stator is about 3800, so the stator reluctance is ( )( )( )( ) stator stator 7 stator stator 0.48 m 62.8 kA t/Wb 3800 4 10 H/m 0.04 m 0.04 m l A µ π − = = = ⋅ × R At a flux density of 1.2 T, the relative permeability r µ of the rotor is 3800, so the rotor reluctance is ( )( )( )( ) rotor rotor 7 stator rotor 0.04 m 5.2 kA t/Wb 3800 4 10 H/m 0.04 m 0.04 m l A µ π − = = = ⋅ × R The reluctance of both air gap 1 and air gap 2 is ( )( ) air gap air gap 1 air gap 2 7 2 air gap air gap 0.0005 m 221 kA t/Wb 4 10 H/m 0.0018 m l A µ π − = = = = ⋅ × R R Therefore, the total reluctance of the core is TOT stator air gap 1 rotor air gap 2 = + + + R R R R R TOT 62.8 221 5.2 221 510 kA t/Wb = + + + = ⋅ R The required MMF is ( )( ) TOT TOT 0.00192 Wb 510 kA t/Wb 979 A t = φ = ⋅ = ⋅ F R Since F = Ni , and the current is limited to 1 A, one possible choice for the number of turns is N = 1000. 1-18. Assume that the voltage applied to a load is 208 30 V = ∠ − ° V and the current flowing through the load is 5 15 A = ∠ ° I . (a) Calculate the complex power S consumed by this load. (b) Is this load inductive or capacitive? (c) Calculate the power factor of this load? (d) Calculate the reactive power consumed or supplied by this load. Does the load consume reactive power from the source or supply it to the source? SOLUTION (a) The complex power S consumed by this load is ( )( ) ( )( ) * 208 30 V 5 15 A 208 30 V 5 15 A = = ∠ − ° ∠ ° = ∠ − ° ∠ − ° * S VI 1040 45 VA = ∠ − ° S (b) This is a capacitive load. (c) The power factor of this load is ( ) PF cos 45 0.707 leading = − ° = (d) This load supplies reactive power to the source. The reactive power of the load is ( )( ) ( ) sin 208 V 5 A sin 45 735 var Q VI θ = = − ° = − 1-19. Figure P1-14 shows a simple single-phase ac power system with three loads. The voltage source is = 120 0 V ∠ ° V , and the three loads are 1 5 30 = ∠ ° Ω Z 2 5 45 = ∠ ° Ω Z 3 5 = ∠ − 90 ° Ω Z 18 Answer the following questions about this power system. (a) Assume that the switch shown in the figure is open, and calculate the current I, the power factor, and the real, reactive, and apparent power being supplied by the source. (b) Assume that the switch shown in the figure is closed, and calculate the current I, the power factor, and the real, reactive, and apparent power being supplied by the source. (c) What happened to the current flowing from the source when the switch closed? Why? + - I V + - + - + - Z1 Z2 3 Z = 120 0 V ∠ ° V SOLUTION (a) With the switch open, only loads 1 and 2 are connected to the source. The current 1I in Load 1 is 1 120 0 V 24 30 A 5 30 A ∠ ° = = ∠ − ° ∠ ° I The current 2I in Load 2 is 2 120 0 V 24 45 A 5 45 A ∠ ° = = ∠ − ° ∠ ° I Therefore the total current from the source is 1 2 24 30 A 24 45 A 47.59 37.5 A = + = ∠ − ° + ∠ − ° = ∠ − ° I I I The power factor supplied by the source is ( ) PF cos cos 37.5 0.793 lagging θ = = − ° = The real, reactive, and apparent power supplied by the source are ( )( ) ( ) cos 120 V 47.59 A cos 37.5 4531 W P VI θ = = − ° = ( )( ) ( ) cos 120 V 47.59 A sin 37.5 3477 var Q VI θ = = − ° = − ( )( ) 120 V 47.59 A 5711 VA S =VI = = (b) With the switch open, all three loads are connected to the source. The current in Loads 1 and 2 is the same as before. The current 3I in Load 3 is 3 120 0 V 24 90 A 5 90 A ∠ ° = = ∠ ° ∠ − ° I Therefore the total current from the source is 1 2 3 24 30 A 24 45 A 24 90 A 38.08 7.5 A = + + = ∠ − ° + ∠ − ° + ∠ ° = ∠ − ° I I I I The power factor supplied by the source is ( ) PF cos cos 7.5 0.991 lagging θ = = − ° = The real, reactive, and apparent power supplied by the source are ( )( ) ( ) cos 120 V 38.08 A cos 7.5 4531 W P VI θ = = − ° = 19 ( )( ) ( ) cos 120 V 38.08 A sin 7.5 596 var Q VI θ = = − ° = − ( )( ) 120 V 38.08 A 4570 VA S = VI = = (c) The current flowing decreased when the switch closed, because most of the reactive power being consumed by Loads 1 and 2 is being supplied by Load 3. Since less reactive power has to be supplied by the source, the total current flow decreases. 1-20. Demonstrate that Equation (1-59) can be derived from Equation (1-58) using simple trigonometric identities: ( ) ( ) ( ) ( ) 2 cos cos p t v t i t VI t t ω ω θ = = − (1-58) ( ) ( ) cos 1 cos2 sin sin 2 p t VI t VI t θ ω θ ω = + + (1-59) SOLUTION The first step is to apply the following identity: ( ) ( ) 1 cos cos cos cos 2 α β α β α β = The result is ( ) ( ) ( ) ( ) 2 cos cos p t v t i t VI t t ω ω θ = = − ) ( ) ( ) 1 ( ) 2 cos cos 2 p t VI t t t t ω ω θ ω ω θ = ( ) ( ) cos cos 2 p t VI t θ ω θ = Now we must apply the angle addition identity to the second term: ( ) cos cos cos sin sin α β α β α β − = + The result is [ ] ( ) cos cos2 cos sin 2 sin p t VI t t θ ω θ ω θ = + + Collecting terms yields the final result: ( ) ( ) cos 1 cos2 sin sin 2 p t VI t VI t θ ω θ ω = + + 1-21. A linear machine has a magnetic flux density of 0.5 T directed into the page, a resistance of 0.25 Ω, a bar length l = 1.0 m, and a battery voltage of 100 V. (a) What is the initial force on the bar at starting? What is the initial current flow? (b) What is the no-load steady-state speed of the bar? (c) If the bar is