More on Magnetically C Coupled Coils and Ideal Transformers
|
|
- Katherine Warren
- 5 years ago
- Views:
Transcription
1 Appenix ore on gneiclly C Couple Coils Iel Trnsformers C. Equivlen Circuis for gneiclly Couple Coils A imes, i is convenien o moel mgneiclly couple coils wih n equivlen circui h oes no involve mgneic coupling. Consier he wo mgneiclly couple coils shown in Fig. C.. The resisnces R R represen he wining resisnce of ech coil. The gol is o replce he mgneiclly couple coils insie he she re wih se of inucors h re no mgneiclly couple. Before eriving he equivlen circuis, we mus poin ou n imporn resricion: The volge eween erminls mus e zero. In oher wors, if erminls cn e shore ogeher wihou isuring he volges currens in he originl circui, he equivlen circuis erive in he meril h follows cn e use o moel he coils. This resricion is impose ecuse, while he equivlen circuis we evelop oh hve four erminls, wo of hose four erminls re shore ogeher. Thus, he sme requiremen is plce on he originl circuis. We egin eveloping he circui moels y wriing he wo equions h rele he erminl volges o he erminl currens. For he given references polriy os, R L c R Figure C. The circui use o evelop n equivlen circui for mgneiclly couple coils. L + (C.) +. (C.) The T-Equivlen Circui To rrive n equivlen circui for hese wo mgneiclly couple coils, we seek n rrngemen of inucors h cn e escrie y se of equions equivlen o Eqs. C. C.. The key o fining he rrngemen is o regr Eqs. C. C. s mesh-curren equions wih s he mesh vriles. Then we nee one mesh wih ol inucnce of L H secon mesh wih ol inucnce of H. Furhermore, he wo meshes mus hve common inucnce of H. The T-rrngemen of coils shown in Fig. C. sisfies hese requiremens. R L c R Figure C. The T-equivlen circui for he mgneiclly couple coils of Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. 787 This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
2 788 ore on gneiclly Couple Coils Iel Trnsformers You shoul verify h he equions reling o reuce o Eqs. C. C.. Noe he sence of mgneic coupling eween he inucors he zero volge eween. The p-equivlen Circui We cn erive p-equivlen circui for he mgneiclly couple coils shown in Fig. C.. This erivion is se on solving Eqs. C. C. for he erivives > > hen regring he resuling expressions s pir of noe-volge equions. Using Crmer s meho for solving simulneous equions, we oin expressions for > >: L L - - L - ; (C.3) L L - - L - + L L -. (C.4) Now we solve for y muliplying oh sies of Eqs. C.3 C.4 y hen inegring: (0) + L - - L0 L - L0 (C.5) (0) - L - + L0 L L -. L0 (C.6) If we regr s noe volges, Eqs. C.5 C.6 escrie circui of he form shown in Fig. C.3. L B c (0) L A L C (0) Figure C.3 The circui use o erive he p-equivlen circui for mgneiclly couple coils. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
3 C. Equivlen Circuis for gneiclly Couple Coils 789 All h remins o e one in eriving he p-equivlen circui is o fin L A, L B, L C s funcions of L,,. We esily o so y wriing he equions for in Fig. C.3 hen compring hem wih Eqs. C.5 C.6. Thus (0) + v L + (v A L0 L - ) B L0 (0) + + v L A L - B L 0 L B L0 (C.7) (0) + v + (v C L0 - ) B L0 (0) + L B L0 + + L B L C L 0. (C.8) Then L B L -, (C.9) L A - L -, (C.0) L C L - L -. (C.) When we incorpore Eqs. C.9 C. ino he circui shown in Fig. C.3, he p-equivlen circui for he mgneiclly couple coils shown in Fig. C. is s shown in Fig. C.4. R (0) L L L L (0) c R Figure C.4 The p-equivlen circui for he mgneiclly couple coils of Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
4 790 ore on gneiclly Couple Coils Iel Trnsformers L R c R L L L Figure C.5 The p-equivlen circui use for sinusoil sey-se nlysis. Noe h he iniil vlues of re explici in he p-equivlen circui u implici in he T-equivlen circui. We re focusing on he sinusoil sey-se ehvior of circuis conining muul inucnce, so we cn ssume h he iniil vlues of re zero. We cn hus elimine he curren sources in he p-equivlen circui, he circui shown in Fig. C.4 simplifies o he one shown in Fig. C.5. The muul inucnce crries is own lgeric sign in he T- p-equivlen circuis. In oher wors, if he mgneic polriy of he couple coils is reverse from h given in Fig. C., he lgeric sign of reverses. A reversl in mgneic polriy requires moving one polriy o wihou chnging he reference polriies of he erminl currens volges. Exmple C. illusres he pplicion of he T-equivlen circui. Exmple C. ) Use he T-equivlen circui for he mgneiclly couple coils shown in Fig. C.6 o fin he phsor currens I I. The source frequency is 400 r>s. ) Repe (), u wih he polriy o on he seconry wining move o he lower erminl. Soluion ) For he polriy os shown in Fig. C.6, crries vlue of +3 H in he T-equivlen circui. Therefore he hree inucnces in he equivlen circui re L H ; H ; 3 H. Figure C.7 shows he T-equivlen circui, Fig. C.8 shows he frequency-omin equivlen circui frequency of 400 r>s. Figure C.9 shows he frequency-omin circui for he originl sysem. 500 j00 00 I V j V j3600 j600 j500 V 800 I 6 H H Figure C.6 The frequency-omin equivlen circui for Exmple C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge 3 H Figure C.7 The T-equivlen circui for he mgneiclly couple coils in Exmple C.. j400 j400 j00 Figure C.8 The frequency-omin moel of he equivlen circui. 400 r>s 500 j j 400 j V I I 800 j00 j500 Figure C.9 The circui of Fig. C.6, wih he mgneiclly couple coils replce y heir T-equivlen circui.
5 Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge C. Equivlen Circuis for gneiclly Couple Coils 79 I Here he mgneiclly couple coils re moele y he circui shown in Fig. C.8. To fin he phsor currens I I, we firs fin he noe volge cross he 00 Æ inucive recnce. If we use he lower noe s he reference, he single noe-volge equion is V j500 + V j00 + Solving for V yiels V 36 - j l V (rms). Then I (36 - j8) j500 V j l ma (rms) 36 - j j l63.43 ma (rms). ) When he polriy o is move o he lower erminl of he seconry coil, crries vlue of -3 H in he T-equivlen circui. Before crrying ou he soluion wih he new T-equivlen circui, we noe h reversing he lgeric sign of hs no effec on he soluion for I shifs I y 80. Therefore we nicipe h As efore, we firs fin he noe volge cross he cener rnch, which in his cse is cpciive recnce of -j00 Æ. If we use he lower noe s reference, he noe-volge equion is V j V -j00 + V j Solving for V gives Then V -8 - j56 I I l V (rms) (-8 - j56) j l ma (rms) -8 - j j l ma (rms). I 63.5 l ma (rms) j4800 j 800 I l ma (rms). We now procee o fin hese soluions y using he new T-equivlen circui. Wih -3 H, he hree inucnces in he equivlen circui re j00 Figure C.0 The frequency-omin equivlen circui for -3 H v 400 r>s. L (-3) H ; (-3) 7 H ; -3 H. A n opering frequency of 400 r>s, he frequency-omin equivlen circui requires wo inucors cpcior, s shown in Fig. C.0. The resuling frequency-omin circui for he originl sysem ppers in Fig. C j00 00 j4800 j V I I 800 j00 j500 Figure C. The frequency-omin equivlen circui for Exmple C.().
6 79 ore on gneiclly Couple Coils Iel Trnsformers C. The Nee for Iel Trnsformers in he Equivlen Circuis The inucors in he T- p-equivlen circuis of mgneiclly couple coils cn hve negive vlues. For exmple, if L 3 mh, mh, 5 mh, he T-equivlen circui requires n inucor of - mh, he p-equivlen circui requires n inucor of -5.5 mh. These negive inucnce vlues re no roulesome when you re using he equivlen circuis in compuions. However, if you re o uil he equivlen circuis wih circui componens, he negive inucors cn e ohersome. The reson is h whenever he frequency of he sinusoil source chnges, you mus chnge he cpcior use o simule he negive recnce. For exmple, frequency of 50 kr>s, - mh inucor hs n impence of -j00 Æ. This impence cn e moele wih cpcior hving cpcince of 0. mf. If he frequency chnges o 5 kr>s, he - mh inucor impence chnges o -j50 Æ. A 5 kr>s, his requires cpcior wih cpcince of 0.8 mf. Oviously, in siuion where he frequency is vrie coninuously, he use of cpcior o simule negive inucnce is prciclly worhless. You cn circumven he prolem of eling wih negive inucnces y inroucing n iel rnsformer ino he equivlen circui. This oesn compleely solve he moeling prolem, ecuse iel rnsformers cn only e pproxime. However, in some siuions he pproximion is goo enough o wrrn iscussion of using n iel rnsformer in he T- p-equivlen circuis of mgneiclly couple coils. L L Iel Iel () () L L L L Iel Iel (L ) (L ) (L ) L (c) () Figure C. The four wys of using n iel rnsformer in he T- p-equivlen circui for mgneiclly couple coils. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
7 C. The Nee for Iel Trnsformers in he Equivlen Circuis 793 An iel rnsformer cn e use in wo ifferen wys in eiher he T-equivlen or he p-equivlen circui. Figure C. shows he wo rrngemens for ech ype of equivlen circui. Verifying ny of he equivlen circuis in Fig. C. requires showing only h, for ny circui, he equions reling o > > re ienicl o Eqs. C. C.. Here, we vlie he circui shown in Fig. C.(); we leve i o you o verify he circuis in Figs. C.(), (c), (). To i he iscussion, we rerew he circui shown in Fig. C.() s Fig. C.3, ing he vriles i 0 v 0. From his circui, L i 0 N N v 0 Iel () Figure C.3 The circui of Fig. C.() wih efine. i 0 v 0 L - + ( + i 0 ) (C.) v 0 - i 0 +. (C.3) (i 0 + ) The iel rnsformer imposes consrins on : v 0 i 0 v 0 ; i 0. (C.4) (C.5) Susiuing Eqs. C.4 C.5 ino Eqs. C. C.3 gives L + () (C.6) () +. (C.7) From Eqs. C.6 C.7, L + (C.8) +. (C.9) Equions C.8 C.9 re ienicl o Eqs. C. C.; hus, insofr s erminl ehvior is concerne, he circui shown in Fig. C.3 is equivlen o he mgneiclly couple coils shown insie he ox in Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
8 Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge 794 ore on gneiclly Couple Coils Iel Trnsformers i L k In showing h he circui in Fig. C.3 is equivlen o he mgneiclly couple coils in Fig. C., we plce no resricions on he urns rio. Therefore, n infinie numer of equivlen circuis re possile. Furhermore, we cn lwys fin urns rio o mke ll he inucnces posiive. Three vlues of re of priculr ineres: L, (C.0) L () Iel, (C.) ( k ) Iel () k Figure C.4 Two equivlen circuis when >L. L ( k ) k L () Iel i Iel () k Figure C.5 Two equivlen circuis when >. (C.) The vlue of given y Eq. C.0 elimines he inucnces L - > L - from he T-equivlen circuis he inucnces (L (L - )>( - )>( L - ) L - ) from he p-equivlen circuis. The vlue of given y Eq. C. elimines he inucnces ( > ) - (>) - from he T-equivlen circuis he inucnces (L - )>( - ) (L - )>( - ) from he p-equivlen circuis. Also noe h when >L, he circuis in Figs. C.() (c) ecome ienicl, when >, he circuis in Figs. C.() () ecome ienicl. Figures C.4 C.5 summrize hese oservions. In eriving he expressions for he inucnces here, we use he relionship kl. Expressing he inucnces s funcions of he self-inucnces L he coefficien of coupling k llows he vlues of given y Eqs. C.0 C. no only o reuce he numer of inucnces neee in he equivlen circui, u lso o gurnee h ll he inucnces will e posiive. We leve o you o invesige he consequences of choosing he vlue of given y Eq. C.. The vlues of given y Eqs. C.0 C. cn e eermine experimenlly. The rio >L is oine y riving he coil esigne s hving N urns y sinusoil volge source. The source frequency is se high enough h vl W R, he N coil is lef open. Figure C.6 shows his rrngemen. Wih he coil open, N C L. I V jvl jv N N V V jvi. Now, s jvl W R, he curren I is (C.3) Figure C.6 Experimenl eerminion of he rio >L. I V. jvl (C.4)
9 Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge C. The Nee for Iel Trnsformers in he Equivlen Circuis 795 Susiuing Eq. C.4 ino Eq. C.3 yiels V V I 0 L, (C.5) in which he rio >L is he erminl volge rio corresponing o coil eing open; h is, I 0. We oin he rio > y reversing he proceure; h is, coil is energize coil is lef open. Then V. V I 0 (C.6) Finlly, we oserve h he vlue of given y Eq. C. is he geomeric men of hese wo volge rios; hus C V V V I 0 V I 0 C L. CL (C.7) For coils woun on nonmgneic cores, he volge rio is no he sme s he urns rio, s i very nerly is for coils woun on ferromgneic cores. Becuse he self-inucnces vry s he squre of he numer of urns, Eq. C.7 revels h he urns rio is pproximely equl o he geomeric men of he wo volge rios, or N V V. CL N C V I 0 V I 0 (C.8)
10 Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationSOME USEFUL MATHEMATICS
SOME USEFU MAHEMAICS SOME USEFU MAHEMAICS I is esy o mesure n preic he behvior of n elecricl circui h conins only c volges n currens. However, mos useful elecricl signls h crry informion vry wih ime. Since
More informationThe solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.
[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More informationECE Microwave Engineering
EE 537-635 Microwve Engineering Adped from noes y Prof. Jeffery T. Willims Fll 8 Prof. Dvid R. Jcson Dep. of EE Noes Wveguiding Srucures Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationHW #1 Solutions. Lewis Structures: Using the above rules, determine the molecular structure for Cl2CO. Hint: C is at the center.
HW # Soluions Cron Mss Prolem: ssuming n erge surfce pressure of m, n erge ropospheric emperure of 55 K, n glol CO mixing rio of 385 ppm, wh is he curren mospheric Cron reseroir (in unis of g m -? Compre
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples
More informationECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)
EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for
More information14. The fundamental theorem of the calculus
4. The funmenl heorem of he clculus V 20 00 80 60 40 20 0 0 0.2 0.4 0.6 0.8 v 400 200 0 0 0.2 0.5 0.8 200 400 Figure : () Venriculr volume for subjecs wih cpciies C = 24 ml, C = 20 ml, C = 2 ml n (b) he
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More informationCharacteristic Function for the Truncated Triangular Distribution., Myron Katzoff and Rahul A. Parsa
Secion on Survey Reserch Mehos JSM 009 Chrcerisic Funcion for he Trunce Tringulr Disriuion Jy J. Kim 1 1, Myron Kzoff n Rhul A. Prs 1 Nionl Cener for Helh Sisics, 11Toleo Ro, Hysville, MD. 078 College
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationSome basic notation and terminology. Deterministic Finite Automata. COMP218: Decision, Computation and Language Note 1
COMP28: Decision, Compuion nd Lnguge Noe These noes re inended minly s supplemen o he lecures nd exooks; hey will e useful for reminders ou noion nd erminology. Some sic noion nd erminology An lphe is
More informationMagnetostatics Bar Magnet. Magnetostatics Oersted s Experiment
Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were
More informationHonours Introductory Maths Course 2011 Integration, Differential and Difference Equations
Honours Inroducory Mhs Course 0 Inegrion, Differenil nd Difference Equions Reding: Ching Chper 4 Noe: These noes do no fully cover he meril in Ching, u re men o supplemen your reding in Ching. Thus fr
More informationProperties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)
Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =
More informationMATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
More informationPHYSICS 1210 Exam 1 University of Wyoming 14 February points
PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationP441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba
Lecure 3 Mondy - Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics - I oupled Oscillors c Alex R. Dzierb oupled oscillors - rix echnique In Figure we show n exple of wo coupled oscillors,
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationMathematical Modeling
ME pplie Engineering nlsis Chper Mhemicl Moeling Professor Ti-Rn Hsu, Ph.D. Deprmen of Mechnicl n erospce Engineering Sn Jose Se Universi Sn Jose, Cliforni, US Jnur Chper Lerning Ojecives Mhemicl moeling
More informationA Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION
Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN 99-878 A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi,
More informationA LOG IS AN EXPONENT.
Ojeives: n nlze nd inerpre he ehvior of rihmi funions, inluding end ehvior nd smpoes. n solve rihmi equions nlill nd grphill. n grph rihmi funions. n deermine he domin nd rnge of rihmi funions. n deermine
More informationFuji Power MOSFET Power calculation method
Fuji Power MOSFE Power clculi mehod Design ool Cher. Overview is necessry o check wheher he ower loss hs no exceeded he Asolue Mximum Rings for using MOSFE. Since he MOSFE loss cnno e mesured using ower
More informationRESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π
RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... () nd b τ F( ) ω d, τ,,... (3)
More information3D Transformations. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd 1/26/07 1
D Trnsformions Compuer Grphics COMP 770 (6) Spring 007 Insrucor: Brndon Lloyd /6/07 Geomery Geomeric eniies, such s poins in spce, exis wihou numers. Coordines re nming scheme. The sme poin cn e descried
More informationSolutions to assignment 3
D Sruure n Algorihm FR 6. Informik Sner, Telikeplli WS 03/04 hp://www.mpi-.mpg.e/~ner/oure/lg03/inex.hml Soluion o ignmen 3 Exerie Arirge i he ue of irepnie in urreny exhnge re o rnform one uni of urreny
More informationCollision Detection and Bouncing
Collision Deecion nd Bouncing Collisions re Hndled in Two Prs. Deecing he collision Mike Biley mj@cs.oregonse.edu. Hndling he physics of he collision collision-ouncing.ppx If You re Lucky, You Cn Deec
More informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationT-Match: Matching Techniques For Driving Yagi-Uda Antennas: T-Match. 2a s. Z in. (Sections 9.5 & 9.7 of Balanis)
3/0/018 _mch.doc Pge 1 of 6 T-Mch: Mching Techniques For Driving Ygi-Ud Anenns: T-Mch (Secions 9.5 & 9.7 of Blnis) l s l / l / in The T-Mch is shun-mching echnique h cn be used o feed he driven elemen
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
More informationAdvanced Integration Techniques: Integration by Parts We may differentiate the product of two functions by using the product rule:
Avance Inegraion Techniques: Inegraion by Pars We may iffereniae he prouc of wo funcions by using he prouc rule: x f(x)g(x) = f (x)g(x) + f(x)g (x). Unforunaely, fining an anierivaive of a prouc is no
More informationChapter Five - Eigenvalues, Eigenfunctions, and All That
Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl
More informationAn integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.
Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl
More informationThe Natural Logarithm
The Naural Logarihm 5-4-007 The Power Rule says n = n + n+ + C provie ha n. The formula oes no apply o. An anierivaive F( of woul have o saisfy F( =. Bu he Funamenal Theorem implies ha if > 0, hen Thus,
More informationThink of the Relationship Between Time and Space Again
Repor nd Opinion, 1(3),009 hp://wwwsciencepubne sciencepub@gmilcom Think of he Relionship Beween Time nd Spce Agin Yng F-cheng Compny of Ruid Cenre in Xinjing 15 Hongxing Sree, Klmyi, Xingjing 834000,
More informationSection 5: Chain Rule
Chaper The Derivaive Applie Calculus 11 Secion 5: Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More informationNMR Spectroscopy: Principles and Applications. Nagarajan Murali Advanced Tools Lecture 4
NMR Specroscop: Principles nd Applicions Ngrjn Murli Advnced Tools Lecure 4 Advnced Tools Qunum Approch We know now h NMR is rnch of Specroscop nd he MNR specrum is n oucome of nucler spin inercion wih
More information( ) ( ) ( ) ( u) ( u) = are shown in Figure =, it is reasonable to speculate that. = cos u ) and the inside function ( ( t) du
Porlan Communiy College MTH 51 Lab Manual The Chain Rule Aciviy 38 The funcions f ( = sin ( an k( sin( 3 38.1. Since f ( cos( k ( = cos( 3. Bu his woul imply ha k ( f ( = are shown in Figure =, i is reasonable
More information3 Motion with constant acceleration: Linear and projectile motion
3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2
ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More informationA 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE
More informationEXERCISE - 01 CHECK YOUR GRASP
UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus
More informationCSC 373: Algorithm Design and Analysis Lecture 9
CSC 373: Algorihm Deign n Anlyi Leure 9 Alln Boroin Jnury 28, 2013 1 / 16 Leure 9: Announemen n Ouline Announemen Prolem e 1 ue hi Friy. Term Te 1 will e hel nex Mony, Fe in he uoril. Two nnounemen o follow
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationChapter Introduction. 2. Linear Combinations [4.1]
Chper 4 Inrouion Thi hper i ou generlizing he onep you lerne in hper o pe oher n hn R Mny opi in hi hper re heoreil n MATLAB will no e le o help you ou You will ee where MATLAB i ueful in hper 4 n how
More informationIf we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as
Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x
More informationA LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly
More informationLecture 2: Network Flow. c 14
Comp 260: Avne Algorihms Tufs Universiy, Spring 2016 Prof. Lenore Cowen Srie: Alexner LeNil Leure 2: Nework Flow 1 Flow Neworks s 16 12 13 10 4 20 14 4 Imgine some nework of pipes whih rry wer, represene
More informationHomework 5 for BST 631: Statistical Theory I Solutions, 09/21/2006
Homewok 5 fo BST 63: Sisicl Theoy I Soluions, 9//6 Due Time: 5:PM Thusy, on 9/8/6. Polem ( oins). Book olem.8. Soluion: E = x f ( x) = ( x) f ( x) + ( x ) f ( x) = xf ( x) + xf ( x) + f ( x) f ( x) Accoing
More information1. Find a basis for the row space of each of the following matrices. Your basis should consist of rows of the original matrix.
Mh 7 Exm - Prcice Prolem Solions. Find sis for he row spce of ech of he following mrices. Yor sis shold consis of rows of he originl mrix. 4 () 7 7 8 () Since we wn sis for he row spce consising of rows
More informationFURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)
Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationNecessary and sufficient conditions for some two variable orthogonal designs in order 44
University of Wollongong Reserch Online Fculty of Informtics - Ppers (Archive) Fculty of Engineering n Informtion Sciences 1998 Necessry n sufficient conitions for some two vrile orthogonl esigns in orer
More information1. Consider a PSA initially at rest in the beginning of the left-hand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.
In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lef-hnd
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction
Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the
More informationAn object moving with speed v around a point at distance r, has an angular velocity. m/s m
Roion The mosphere roes wih he erh n moions wihin he mosphere clerly follow cure phs (cyclones, nicyclones, hurricnes, ornoes ec.) We nee o epress roion quniiely. For soli objec or ny mss h oes no isor
More informationSeminar 5 Sustainability
Seminar 5 Susainabiliy Soluions Quesion : Hyperbolic Discouning -. Suppose a faher inheris a family forune of 0 million NOK an he wans o use some of i for himself (o be precise, he share ) bu also o beques
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationFaraday s Law. To be able to find. motional emf transformer and motional emf. Motional emf
Objecie F s w Tnsfome Moionl To be ble o fin nsfome. moionl nsfome n moionl. 331 1 331 Mwell s quion: ic Fiel D: Guss lw :KV : Guss lw H: Ampee s w Poin Fom Inegl Fom D D Q sufce loop H sufce H I enclose
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationA new model for solving fuzzy linear fractional programming problem with ranking function
J. ppl. Res. Ind. Eng. Vol. 4 No. 07 89 96 Journl of pplied Reserch on Indusril Engineering www.journl-prie.com new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion Spn Kumr Ds
More informationThe order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.
www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or
More informationCS3510 Design & Analysis of Algorithms Fall 2017 Section A. Test 3 Solutions. Instructor: Richard Peng In class, Wednesday, Nov 15, 2017
Uer ID (NOT he 9 igi numer): gurell4 CS351 Deign & Anlyi of Algorihm Fll 17 Seion A Te 3 Soluion Inruor: Rihr Peng In l, Weney, Nov 15, 17 Do no open hi quiz ookle unil you re iree o o o. Re ll he inruion
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationReleased Assessment Questions, 2017 QUESTIONS
Relese Assessmen Quesions, 17 QUESTIONS Gre 9 Assessmen of Mhemis Aemi Re he insruions elow. Along wih his ookle, mke sure ou hve he Answer Bookle n he Formul Shee. You m use n spe in his ook for rough
More information1 jordan.mcd Eigenvalue-eigenvector approach to solving first order ODEs. -- Jordan normal (canonical) form. Instructor: Nam Sun Wang
jordnmcd Eigenvlue-eigenvecor pproch o solving firs order ODEs -- ordn norml (cnonicl) form Insrucor: Nm Sun Wng Consider he following se of coupled firs order ODEs d d x x 5 x x d d x d d x x x 5 x x
More informationf(x) dx with An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples dx x x 2
Impope Inegls To his poin we hve only consideed inegls f() wih he is of inegion nd b finie nd he inegnd f() bounded (nd in fc coninuous ecep possibly fo finiely mny jump disconinuiies) An inegl hving eihe
More informationM r. d 2. R t a M. Structural Mechanics Section. Exam CT5141 Theory of Elasticity Friday 31 October 2003, 9:00 12:00 hours. Problem 1 (3 points)
Delf Universiy of Technology Fculy of Civil Engineering nd Geosciences Srucurl echnics Secion Wrie your nme nd sudy numer he op righ-hnd of your work. Exm CT5 Theory of Elsiciy Fridy Ocoer 00, 9:00 :00
More informationChapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
More informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationExact Minimization of # of Joins
A Quer Rewriing Algorihm: Ec Minimizion of # of Joins Emple (movie bse) selec.irecor from movie, movie, movie m3, scheule, scheule s2 where.irecor =.irecor n.cor = m3.cor n.ile =.ile n m3.ile = s2.ile
More informationConservation Law. Chapter Goal. 6.2 Theory
Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationCHAPTER 2: Describing Motion: Kinematics in One Dimension
CHAPTER : Describing Moion: Kinemics in One Dimension Answers o Quesions A cr speeomeer mesures only spee I oes no gie ny informion bou he irecion, n so oes no mesure elociy By efiniion, if n objec hs
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationSimulation of Tie Lines in Interconnected Power Systems Akhilesh A. Nimje 1, Rajendra M. Bhome 2, Chinmoy Kumar Panigrahi 3
nernionl Journl of pplicion or nnovion in Engineering & Mngemen (JEM) We Sie: www.ijiem.org Emil: edior@ijiem.org ediorijiem@gmil.com SSN 9-8 Specil ssue for Nionl Conference On Recen dvnces in Technology
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationgraph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationC H A P T E R 5. Integrals. PDF Created with deskpdf PDF Writer - Trial :: A r e a s a n d D i s t a n c e s
License o: jsmuels@mcc.cun.eu PDF Cree wih eskpdf PDF Wrier - Tril :: hp://www.ocuesk.com C H A P T E R 5 To compue n re we pproime region recngles n le he numer of recngles ecome lrge. The precise re
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More information