More Emphsis on Complex Numbers? The i 's Hve It! Rlph Fehr, P.E. St. Petersburg Junior College Clerwter Cmpus Presented t the Twenty-Ninth Annul Meeting Florid Section Mthemticl Assocition of Americ Mrch, 996 Reprinted December 2000 Jointly Hosted By
More Emphsis on Complex Numbers? The i s Hve It! Rlph Fehr, P.E. Electricl engineers perhps hve more prcticl pplictions of complex numbers thn ny other profession. Most students entering engineering progrms hve hd exposure to complex numbers in high school lgebr, but mny do not hve thorough understnding of the concepts relted to rel nd imginry numbers. Although student my be ble to perform rithmetic opertions on complex numbers correctly, the opertions often hve no significnt mening or physicl importnce to the student. If this is the cse, engineering principles built on the theory of complex numbers will be too bstrct to be comprehended totlly, resulting in n incomplete understnding of the engineering principle. BASIC APPLICATION OF COMPLEX NUMBERS The most importnt reltionship in electricl engineering is Ohm s Lw, which sttes tht voltge cross circuit element is equl to the product of the current through the element nd the impednce of the element. In lternting current (AC) circuits, the quntities of voltge, current, nd impednce re phsors in the complex plne. The horizontl xis of the complex plne indictes rel quntities, while the verticl xis denotes imginry quntities. The significnce of rel versus imginry is best seen by nlyzing nother complex quntity power. Power is the phsor product of voltge nd the complex conjugte of the current. To differentite between the rel nd imginry components of power, consider n electric motor. The rel component of electric power input to motor provides the output horsepower required to rotte the motor shft. The imginry component of electricl power estblishes the electromgnetic fields necessry for the motor to operte. Although the imginry component of the input power mkes no contribution to the mechnicl output of the motor, it does perform vitl function. In prcticl terms, the difference between the rel nd imginry components of electricl power is 90 phse displcement. A 90 rottion in the complex plne is chieved using the i opertor, where i =. Understnding tht the i opertor forces rottion in the complex plne is extremely importnt in comprehending lternting current theory. Without tht bsic understnding, the mthemetics nd the physics tht describe electricity remin decoupled. Thoroughly understnding the fundmentls of complex numbers becomes even more criticl when studying more complex topics, such s symmetricl components. SYMMETRICAL COMPONENTS OF UNBALANCED PHASOR SYSTEMS A pper on polyphse network nlysis ws presented by C. L. Fortescue t meeting of the Americn Institute of Electricl Engineers in 98. This pper proved tht ny unblnced system of n relted phsors cn be resolved into n systems of blnced phsors clled symmetricl components of the unblnced system. The method of symmetricl components is one of the most powerful nlysis tools vilble to the electricl engineer, s mny prcticl problems cnnot be solved redily without this technique. Understnding symmetricl components requires good understnding of complex numbers. Specificlly, opertions performed on complex quntities must hve physicl significnce to the student. If the opertions re crried out simply by executing formuls, n understnding of the problem is unlikely lthough correct numeric nswers might be found. To pply the concept of symmetricl components to n unblnced phsor system, the unblnced system must be decomposed into series of blnced (symmetricl) systems, ech contining the sme number of phses s the unblnced system. The phse displcement in ech sequence of symmetricl components is equl to 360 divided by the number of relted phsors in the unblnced system. In system of two unblnced phsors, the phse displcement between the sequences is 360 2=80. The 80 rottion is chieved by pplying n opertor of to one sequence to obtin the next.
Note tht cn be written in exponentil form using Euler s Identity s = e π i where the rel prt of the exponent indictes the rottion in the complex plne mesured in rdins. Since three phse systems re the most commonly-encountered polyphse electricl systems, n opertor tht cuses 20 rottion (360 3) is required. The letter is used to represent the 20 opertor, nd is defined s follows: = e 2π 3 i = 0.5 + 0.866i Consider the unblnced system of phsors V, V b, nd V c shown below. Figure Unblnced Phsor System This phsor digrm models typicl three-phse power system, where the phse A, phse B, nd phse C voltge mgnitudes re unequl, nd the ngulr displcement between the phses is lso unequl. The entire phsor system rottes in the counterclockwise direction t the rte of 20π rdins per second (for 60 hertz power system frequency, s is used in the United Sttes). Anlyzing n unblnced phsor system is very difficult. If the system ws blnced ( V = V b = V c, nd the ngulr displcement from one phsor to the next is 20 ), then it could be nlyzed using only V ( single phse system) nd pplying principles of symmetry to determine the behvior of phses B nd C. Single phse nlysis is reltively simple, but unfortuntely it is not pplicble to unblnced systems. If the unblnced system ws trnsformed to three blnced systems, however, simpler nlysis techniques could be employed. The first symmetricl system hs phse sequence the sme s the unblnced system, nd is clled the positive sequence, denoted with subscript of. The second symmetricl system hs phse sequence opposite of the unblnced system, so it is clled the negtive sequence, nd is indicted with subscript of 2. The third symmetricl system hs no ngulr displcement between its phses, nd is termed the zero sequence, signified with subscript of 0. All three sequences rotte in the sme direction s the genertor rotor tht produced them. Figure 2 Positive, Negtive, nd Zero Sequences
In his pper, Fortescue defined bidirectionl trnsformtions between blnced nd unblnced phsor systems. The following mtrix eqution shows how n unblnced system of phsors V, V b, nd V c cn be resolved into symmetricl components V 0, V, nd V 2. V0 V = V2 3 2 V 2 Vb V c For phse A, voltge V is the phsor sum of the three symmetricl components tht mke up V, nmely, V, V 2, nd V 0. Phses B nd C re obtined similrly. Vectorlly, the reltionship between the unblnced system nd the blnced systems cn be seen esily. Figure 3 Symmetricl Components of Unblnced System Once the unblnced system of V, V b, nd V c is trnsformed to the blnced systems of V 0, V, nd V 2, bsic circuit nlysis techniques cn be used to nlyze the system s single phse system. The single phse solutions cn be trnsformed bck into unblnced phsors by pplying the following mtrix eqution. V Vb = V c 2 V0 V 2 V2 Although the mthemtics re reltively strightforwrd, to fully understnd the electricl theory behind symmetricl components, thorough understnding of the mthemticl theory of complex numbers is necessry. Fmilirity with mnipultions on the complex plne, where the horizontl xis represents rel vlues nd the verticl xis represents imginry vlues, is fundmentl to mstering the electricl principles modeled by the symmetricl components. For exmple, it must be relized tht multiplying phsor by ( 0.5 + 0.866 i) simply rottes the phsor 20 counterclockwise. If the mthemtics re blindly performed, the correct numeric nswer will be obtined, but the problem will not truly be understood. This scenrio (getting the right nswer but not relly knowing how) is common t ll levels, from grde school through the professionl world, nd poses mjor obstcle to problem solving. It is essentilly cused by prtil understnding of concept s opposed to complete mstery. Achieving thorough understnding of bsic mthemticl principles is n excellent strt to the complete understnding of complicted scientific situtions. For the electricl engineer, mstery of complex numbers provides the foundtion necessry to understnd the most complicted electricl systems.
POSSIBLE SOLUTION TO THE PROBLEM OF "PARTIAL UNDERSTANDING" Involvement nd ppliction re perhps the key ingredients for success in solving the problem of prtil understnding. Demonstrtions cn be beneficil to some students, but being involved in the exercise tends to reinforce the importnt concepts to higher degree thn simply observing demonstrtion. Selecting ctivities where the ppliction is pprent is lso importnt. Every mth techer hs herd the line, Wht will I ever use this for? Grnted, some mthemticl concepts re more esily nd more frequently pplied to everydy problems thn others, but every topic tught hs some kind of ppliction. If not, why is the topic tught? If the instructor goes out of his or her wy to illustrte the pplictions of ech topic s it is studied, severl results re likely to be seen. First, since the pplictions re mde known to the student from the strt, mny students will mke more conscientious effort to understnd the mteril thoroughly s opposed to simply lerning it just well enough to pss the next test. Also, more clss prticiption is likely when pplictions of theory re presented. It is this uthor s experience tht s pplictions re presented to the clss, ttention spns increse, discussions ensue, nd grdes improve. To convince students of the importnce of understnding theoreticl mthemticl concepts thoroughly, prcticl exercises bsed on mthemticl theory should begin t the elementry school level. Involving students in exercises tht require the solution of prgmtic problems by using mthemticl methods is n effective wy of coupling the theoreticl mthemtics with the prcticl ppliction. As students re exposed to this pproch, they often begin seeking ties between theory nd ppliction on their own. The use of interctive exercises helps to hold the ttention of some students who might otherwise lose interest. New technologies hve much to offer in the wy of providing interctive ctivities. Computer-bsed simultions, for exmple, provide n excellent medium for presenttions s well s individul or tem investigtion. As students work with softwre-bsed eductionl mterils, they re lso developing computer skills tht undoubtedly will continue to grow in importnce. Computers re lso excellent tools to use for visulizing mthemticl processes. An opertion such s convolution, for exmple, seems bstrct when viewed mthemticlly, but becomes quite understndble when viewed grphiclly. Integrtion is nother topic where visul pproch is very beneficil. Although figure in textbook my dequtely convey point, n interctive nd nimted pproch on computer screen cn communicte the sme informtion, nd will likely hve more of n influence on the typicl student thn pge from textbook. Mny students feel tht mthemtics my be importnt to some specific occuptions, but fil to see the diversity of its pplictions. As n engineer, it is tempting to show engineering-relted pplictions of the mteril covered in clss while overlooking the non-engineering pplictions. Relisticlly, however, very few of my students will become engineers. A deliberte effort must be mde to show how mny diverse pplictions of mthemtics exist. By intentionlly vrying the scope of pplictions s much s possible, the brod spectrum of voctions tht rely on mthemtics cn be seen. It is fctors such s this kind of vriety tht cn mke mthemtics fscinting subject to study. CONCLUSION The field of engineering contins mny exmples of complicted concepts tht re bsed on more fundmentl ides. Mny of these fundmentl ides re mthemticl in nture. While cursory understnding of the mthemtics my be sufficient for the student to solve some engineering problems, thorough understnding of the bsics is necessry for totl comprehension of the more involved concepts. Complex numbers s pplied in the field of electricl engineering provide one of mny exmples of topic tht requires thorough understnding of bsic mthemticl principles. Often when concept is first tught, it is presented s theory without strong ties to prcticl pplictions. When this hppens, mny students fil to see pplictions on their own. Consequently, they do not hve the motivtion necessry to invest the effort needed to fully understnd the subject. In mny cses, the subject will be lerned well enough to pss the next test, but will soon be forgotten. Even if it is not forgotten totlly, the student will
probbly not be ble to pply it to the solution of prcticl problems, since the concepts were relly never understood. When topic is presented to students, the instructor should strive to mximize the interest level. Involving students in discussions, tem projects relting course mteril to rel world situtions, nd utilizing udiovisul nd computer presenttion methods often help increse interest levels, but every clss responds differently. The instructor needs to be wre of interest level nd be willing to mke djustments to increse it s necessry. In order to truly understnd concept, student must first hve desire to understnd it. Prt of the instructor s role should be to encourge tht desire in ll of his or her students. The methods of encourgement re limited only by the imgintion of the instructor. BIBLIOGRAPHY. Chrles L. Fortescue, Method of Symmetricl Coordintes Applied to the Solution of Polyphse Networks, Trnsctions of the AIEE, vol. 37, pp. 027-40, 98. 2. Willim D. Stevenson, Jr., Elements of Power System Anlysis, 4 th edition, McGrw-Hill, pp. 275-302, 982, ISBN 0-07-06278-. ABOUT THE AUTHOR Rlph Fehr hs been n djunct mthemtics instructor t St. Petersburg Junior College, Clerwter Cmpus, since 994. He erned B.S. degree in Electricl Engineering from the Pennsylvni Stte University nd n M.E. degree in Electricl Engineering (power) from the University of Colordo t Boulder. He tught clsses on computer operting systems nd computer-ided design t the University of New Mexico in Albuquerque. He is senior engineer t Florid Power Corportion, nd is registered professionl engineer in Florid nd New Mexico.