Maximum Power Point Tracking Algorithm with Turn Round Measurement and Curve Fitting Method for Solar Generation System
|
|
- Bennett Francis
- 6 years ago
- Views:
Transcription
1 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin Mximum Power Point Trking Algorithm with Turn Round Mesurement nd Curve Fitting Method for Solr Genertion System S Pn, Ynd Li, Dept of Eletril Engineering, Anyng Norml niversity PRChin pnsn@ynuedun, liyd@ynuedun Sn-Bo Pn, Chendong Lin Dept of Eletril Engineering, Shnghi Dinji niversity PRChin pns@sdjuedun Astrt: - A vel mximum power point trking (T lgorithm is proposed in this pper The vel lgorithm dopted round k mesurement whih mkes it esy to void misjudge when the irrdition vries rpidly, espeilly omprison to trditionl T strtegy Wht s more, round k mesurement mkes it possile to imitte PV rry s P- urve with mthemtil method whih mkes it possile to redue the osilltion ner PV s mximum power point ( Lgrnge Interpoltion Formul is used in this pper to fit PV s P- urve nd simultion with Mtl/Simulink tool to verify the proposed hs een done The result revels proposed T lgorithm is more ury in irrdition vrition sitution nd less osilltion ner the These fetures mke the lgorithm good ide in high-power solr genertion system Key-Words: T, solr genertion system, irrdition vrition, osilltion, urve fitting, mtl/simulink 1 Introdution Photovolti (PV power genertion system is widely studied reently for energy risis nd the hevily polluted environment As the hrteristi of PV rry is nliner nd pt to e influened y mient temperture nd irrdition ondition A mximum power point trking (T tehlogy is essentil in PV system to mximize PV s output power PV system s performne is mostly ffeted y T ontroller s stility, effiieny nd dynmi hrteristi Widely used T lgorithms n e onluded s perturtion nd oservtion (P&O nd inrement ondutne (InCond These two lgorithms filed to trk PV s mximum power point ( when the irrdition vries rpidly The min reson is tht trditionl T lgorithm doesn t tke the reltionship etween power nd voltge into ount in detil To solve this prolem, vel T lgorithm using turn round mesurement method hs een proposed in this pper Wht s more, turn round mesurement mkes it possile to lulte the erene voltge of PV rry t whih redues the osilltion ner the nd the effiieny is prominent improved The operting priniple nd simultion result with Mtl/Simulink tool hs een given nd disussed in this pper in detil Fig1 shows the hrteristis of PV Cell P P V _ or _ I Fig1 Power urve of PV ell V or I The prolem fixed y the T tehnique is to find the Vmpp or Impp of PV ell Mximiztion output power n e otined when the PV ell opertes t V mpp or I mpp udner speifi irrdition nd temperture ondition It is ted tht prtil shding sitution resulted multiple of PV ell However, there is still single true overll Different T tehniques n respond to oth irrdine nd temperture But some re pt to del with irrdine or temperture situtions speifilly Some T tehniques n e filed when irrdition vries fst 3 T Tehniques 31 T priniple Aording to optil eletroni theory, PV rry s mthemtil module is s Fig shown Prolem Overview E-ISSN: 4-66X 101 Volume 16, 017
2 Iph WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin Is Rs I r ID Ish Rsh VJ Cj o DC/DC R L Fig Equivlent iruit model for photovolti ell The reltionship etween PV s output urrent nd voltge n e desried s eqution (1 q( V IRS V IRS I IL I0 exp[ ] 1 (1 AKT RSh PV s I- urve nd P- urve re shown s Fig3 nd Fig4 respetively Fig3 I- urve of PV ells Fig4 P- urve of PV ells PV ell s model n e simplified s Fig5 o represents the open iruit voltge nd resister r n e onsidered s the output impedne of PV ells When PV ell s lod resister R equls to output resister r, PV ell s output power n e mximized Fig6 shows the T topology with Boost onverter Equivlent impedne of oost onverter nd the lod R L n e desried s R eq : Req (1 D RL ( Among whih, D represents oost onverter s duty yle PV ell output mximum power when R equls to r o r Fig5 PV ell s simplified model RL eq Fig6 T with DCDC topology As irrdition vries, P&O nd InCond lgorithm filed to response fst Thus, the T lgorithm need to trk the gin whih mkes PV s effiieny redue signifintly under irrdition rpid hnge irumstne Most ommonly used T tehniques re desried in this hpter nd the filure priniple re nlyzed in detil 3 Hill-Climing/Perturtion nd Oservtion Hill-Climing/Perturtion nd Oservtion re most fvored T tehniques Hill-liming method perturs the duty rtio of the power onverter nd tht results the output urrent hnge of PV ell Perturtion nd oservtion method perturs the erene of PV ell When the output voltge of PV ell hnges, the output urrent vries onsequenetly Hill-Climing nd P&O method re different wy to envision the sme fundmentl method In Fig1, it n e seen tht inresing(deresing the voltge inreses(dereses the power when PV ell operting on the left of the nd deresing (inresing the voltge dereses (inreses the power when on the right of the Theore, next step s perturtion diretion is determined y the output power s vrition If the power inreses, the prior perturtion diretion is orret, otherwise reversed the perturtion diretion The reltions etween hnging in power nd next perturtion diretion re s Tle1 shown Tle1 Summry of hill-liming nd P&O method Perturtion Chnge in Power Next Perturtion Positive Positive Positive Positive Negtive Negtive Negtive Positive Negtive Negtive Negtive Positive The perturtion nd oservtion proess is repeted periodilly until the is rehed The system then osilltes round the The osilltion wstes PV s power if the step size is too lrge In order to minimize the osilltion ner the, the step size should e deresed However, tht will slow down T speed In some thesis, two stge step size is dopted to trk the A E-ISSN: 4-66X 10 Volume 16, 017
3 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin igger step size is used in first stge to offer fst trking speed nd finer step size is dopted in the seond stge to derese the osilltion Hill-liming nd P&O methods fil under rpidly hnging irrdition onditions s shown in Fig7 P P P1 A C B Inputs:V(t,I(t I I( t I( t t V V ( t V ( t t V 0 I / V I / V I 0 O V V V Fig7 Hill liming method invlid shemti digrm Assume the PV ells opertes t point A under irrdition P1 The output power will derese from A to B if the perturtion V is positive The next step should derese the voltge of PV ells The output power will hnge from A to C if the irrdition hnges from P1 to P The ontroller will keep on inresing the operting voltge for the output power inreses It is ovious tht the perturtion diretion is wrong for the reson of irrdition rpidly hnging 33 Inrementl Condutne The inrementl ondutne method is sed on the ft tht the slope of the PV rry power urve is zero t the, positive on the left of the, nd negtive on the right, s given y formul (3 dp / dv 0, t _ dp / dv 0, left _ of _ (3 dp / dv 0, right _ of _ Sine dp / dv d( IV / dv I VdI / dv I VI / V (4 (n e rewritten s I / V I / V, t _ I / V I / V, left _ of _ (5 I / V I / V, right _ of _ The n thus e trked y ompring the instneous ondutne(i/v to the inrementl ondutne ( I / V s shown in Fig8 V is the erene voltge t whih the PV rry is fored to operte At the, V equls to V One the is rehed, the opertion of the PV rry is mintined t this point unless hnge in I is ted, inditing hnge in tmospheri onditions nd the The lgorithm derements or inrements V to trk the new Inrement V I / V I / V Derement V I( t t I( t V ( t t V ( t return I 0 Derement V Fig8 InCond lgorithm flowht Inrement V The inrement size determines how fst the is trked Fst trking n e hieved with igger inrements ut the system might t operte extly t the nd osillte out it insted; so there is trdeoff Referenes [] nd [3] propose method tht rings the operting point of the PV rry lose to the in first stge nd then uses InCond to extly trk the in seond stge By proper ontrol of the power onverter, the initil operting point is set to mth lod resistne proportionl to the rtio of the open-iruit voltge Vo to the short-iruit urrent I s of the PV rry This two-stge lterntive lso ensures tht the rel is trked in se of multiple lol mxim A less ovious, ut effetive wy of performing the InCond tehnique is to use the instntneous ondutne nd the inrementl ondutne to generte n error signl e I / V di / dv (6 s suggested in [5,6] From (6, we kw tht e goes to zero t the A simple proportionl integrl (PI ontrol n then e used to drive e to zero Mesurements of the instntneous PV rry voltge nd urrent require two sensors InCond method lends itself well to DSP nd miroontroller ontrol, whih n esily keep trk of previous vlues of voltge nd urrent E-ISSN: 4-66X 103 Volume 16, 017
4 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin For ll other pplitions t mentioned here, we put together Tle, ontining the mjor hrteristis of ll the T tehniques Tle should help in hoosing n pproprite T method T Tehnique Tle T tehniques omprison Converg Sensed e Prmeters speed Irrdition Chnge? Hill-liming/ Vri es Voltge, P&O Current InCond Vries Voltge, Current Frtionl Vo Medium Voltge Frtionl Is Medium Current Fuzzy Logi Fst Vries Control Neurl Network Fst Vries Current Sweep Slow Voltge, Current DC Cpitor Medium Voltge Droop Control Liner Current Control Fst Irrdition 4 Proposed T lgorithm with turn round mesurement nd urve fitting lultion 41 Turn round mesurement operting priniple Control system detets the open-iruit voltge o of photovolti (PV ells The initil erene voltge is set s 08 o ording to ppers relted Reord PV s urrent voltge nd power s (, P A perturtion voltge 001 is given to hieve two different o opertion point nd The dt is reorded s (, P nd (, P respetively The next perturtion diretion is determined y the reltion etween P, P, P nd,, Given P P P nd it mens is the proximl point to the Grdient e1 n e P P e otined s 1 nd fter PI ontroller we get the fresh perturtion voltge PI ontroller is designed to mke the perturtion voltge self-dption vlue whih mkes the trking proess fster when it is fr wy from the trget vlue Wht s more, PI ontroller derese the perturtion step ner the nd tht redue the osilltion ner the After onfirmtion the new perturtion step, is set s the new voltge strting point The new trking proess is the sme s efore A turn round mesurement retes more operting steps for the ontrol system However, it is essentil to tke this step to void misjudge when the irrdition vries rpidly Trditionl perturtion nd oservtion lgorithm filed under this sitution It is explined s Fig9 P O 400W/m 00W/m P P P P Fig9 Photovolti rry s output power nd voltge The origin point of PV is (, P nd re otined with the perturtion voltge Suppose the irrdition inreses t point nd the P- urve of PV trnsfer from A to B The output power of PV t point is P If trdition P&O lgorithm is used here,, P P, nd tht mens the is on the right side of nd the ontroller should derese PV s output voltge It is ovious tht the trking diretion is wrong The turn round method proposed in this pper n otin three operting points t the sme moment A riterion is used in this pper to judge the irrdition sitution T lgorithm dopts the ltest PV output point if irrdition vries A new trking proess is initited fter the irrdition The riteri proposed in this pper re shown s Tle3 Tle 3 T trking diretion riteri PV voltge PV power Trking diretion << P<P<P Inrese voltge << P<P<P Curve fitting << P<P<P Irrdition vried << P<P<P Irrdition vried << P<P<P Curve fitting << P<P<P Derese voltge As the tle showed, P P P or P P P revels the is loted t the setion, When this ondition is stisfied, turn-round method is stopped nd Lgrnge Interpoltion Formul is used to find the Curve fitting hs een done to simulte PV s P- urve with Lgrnge Interpoltion Formul Aording to the derived P- A B o E-ISSN: 4-66X 104 Volume 16, 017
5 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin urve eqution, is lulted nd the erene voltge for PV is given 4 Curve fitting ner the When the PV s operting point is ner the, for the reson of turn round mesurement method, three points of PV ells re otined t the sme moment Tht mkes it possile to restore PV s P- urve when it is ner the nd the error will e deresed signifintly With Lgrnge Interpoltion Formul, PV s P- urve re fitted y three operting point ner the The urve fitting of PV s P- urve mkes it possile to trk the with the derived urve whih is muh esier nd fster thn trditionl T lgorithm The osilltion round the will e prominently deresed with the lultion method rther thn perturtion method Whih mkes it suit for high-power photovolti modules pplition The operting priniple of urve fitting is desried s follow: Assume the interpolted point re (, P, (, P nd (, P The interpoltion polymil of PV s P- urve is s follow: L ( ( ( ( P l P l P l (7 Among whih: ( ( l ( P (8 ( ( ( ( l ( P ( ( ( ( l ( P (10 ( ( Mrix Hishmn-Sigmr model of qudrti funtion n e derived s follow: L ( x x x ( P P x1 ( ( ( ( P ( ( P ( P ( x ( ( ( ( P ( ( ( P P x3 ( ( ( ( (9 (1 (13 P (14 ( ( The fitted urve s mximum point x 4x1 x3 x is(, Only three points (, P, (, P x1 4x1 nd (, P re needed to fit PV s P- urve With the sustitution into the formul with three points, the oeffiient x 1, x nd x 3 re derived Aording to the derived P- prol urve of PV ells, the erened voltge t point n e desried x s Set the PV s output voltge to m x1 m y the ontroller PV s output power m P t operting point m need to e reorded Three mximum power points re seleted from (, P, (, P, (, P nd ( m, P m The seleted points re mrked s ( 1, P 1, (, P nd ( 3, P 3 If urve fitting is implemented sequentilly with the fresh three points, they must meet the onditions P1 P P3 nd 1 3 Itertion with the fresh operting points y Lgrnge Interpoltion Formul s efore until the is trked PV s urrent output power P w is ompred with the previous yle s power P efore When the reltionship etween P w nd P efore stisfies Pw Pefore, the is trked nd urve fitting proess need to e hlted The lgorithm flow ht is s Fig10: Detet PV s open iruit voltge o Set PV s erene voltge =08o Reord PV s voltge nd power s(,p PV s erene voltge s:=- ;=+ P P P P P e1 PI 001o Reord PV s urrent voltge nd power s:(,p,(,p,(,p Reord PV s voltge nd power s:(,p,(,p P P P P P e PI Reord PV s urrent voltge nd power s(,p,(,p,(,p P P P P P P Curve fitting for PV s P- urve with (,P,(,P,(,P nd lulte s erene voltge m m Reord PV s voltge nd power (m,pm Selet three mximum power points nd mrked s(1,p1,(,p,(3,p3 respetively whih stisfy P1<P<P3 mx P P, P P 3>1nd 3> Fig10 Flow ht of proposed lgorithm T stopped P P P P P P =,P=P 001o Reord PV s voltge nd power s:(,p,(,p,(,p E-ISSN: 4-66X 105 Volume 16, 017
6 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin 5 Simultion nd Experiment Results The simultion nd experiment results show the differene etween InCond T ontrol lgorithm nd proposed T ontrol lgorithm Fig14 P- urve with Proposed T lgorithm when irrdition hnges Fig11 Fig6 Proposed T model Irrdition vries t point A from P1 to P Aording to the simultion model, the irrdition vrition proess is ontinuous proess T lgorithm s trking proess re showed s A to B B point re PV ell s fresh under P irrdition Fig13 nd Fig14 revel tht proposed T lgorithm osilltes lesser ner the nd it n respond to the fresh point B fster Tht mens the proposed lgorithm is more effiient thn InCond lgorithm when irrdition vries rpidly Fig1 Proposed T lgorithm The simultion onditions re s follow: Temperture: 5 C Irrdition P1: 600W / m Irrdition P: 800W / m PV ell s key prmeters under different irrdition re s Tle4 shows Tle4 PV ell s key prmeter Irrdition Condition P1 P Open iruit voltge(v 4 46 Short iruit Current(A voltge(v urrent(a Power(W Fig13 P- urve with InCond when irrdition hnges Fig15 PV s output power with proposed lgorithm when irrdition hnges Fig15 shows the output power of PV ell Aording to the result, proposed T lgorithm onvergene speed is exellent when irrdition vries from time(ms 15 to time 0 Tht mens the proposed T lgorithm n trk to the fresh fstly in 5ms whih mkes it proper to irrdition vries fst irumstne Tle5 Theoretil V, I nd P of PV ell Cse No G (W/m^ T ( V (V I (A P (W E-ISSN: 4-66X 106 Volume 16, 017
7 WSEAS TRANSACTIONS on CIRCITS nd SYSTEMS S Pn, Ynd Li, Sn-Bo Pn, Chendong Lin Tle6 V, I nd P lgorithm Cse G T No (W/m^ ( of PV ell with InCond V I P (W (V (A Tle7 V, I nd P lgorithm Cse G T No (W/m^ ( (V of PV ell with proposed V I P (W (A Tle5-7 re dt reorded of PV ells respetively under different irrdition nd temperture irumstne These three tles show InCond method nd proposed T method s effiieny nd lso the dt re ompred to the theoretil vlue of PV ell The dt tells tht PV ells with proposed lgorithm outputs more power thn InCond lgorithm 6 Conlusion The proposed T lgorithm flututes less ner PV s when the irrdition vries rpidly Simultion result in Fig15 shows the proposed T lgorithm respond to irrdition hnge fst When irrdition vries, proposed T lgorithm tkes less time to trk the new For the ondition of rpidly hnging solr rdition, the proposed T lgorithm will e very useful Shnghi Muniipl Edution Commission (14YZ160 Referenes: [1] LZhng,B Yunfei nd A Al-Amoudi, GA-RBF neutrl network sed mximum power point trking in photovolti systems, Int Power Eletronis Mhines nd Drives Conf, pp18-3,00 [] K Irisw, T Sito, I Tk, nd Y Swd, "Mximum power point trking ontrol of photovolti genertion system under nuniform insoltion y mens of monitoring ells," in Conf Reord of the Twenty-Eighth IEEE Photovolti Speilists Conf, 000, pp [3] K Koyshi, I Tk, nd Y Swd, "A study on two stge mximum power point trking ontrol of photovolti system under prtilly shded insoltion onditions," in IEEE Power Eng Soiety Generl Meeting, 003, pp [4] H Koizumi nd K Kurokw, "A vel mximum power point trking method for PV module integrted onverter," in 36th Annul IEEE Power Eletron Speilists Conf, 005, pp [5] E N Costogue nd S Linden, "Comprison of ndidte solr rry mximum power utiliztion pprohes," in Intersoiety Energy Conversion Eng Conf, 1976, pp [6] J Hrd nd G Zho, "Controlled powerinterfe etwee n solr ells nd soures," in IEEE Teleommun Power Conf, 1989, pp 1/1-1/7 [7] Bidrm, A, Dvoudi, A, Blog, RS: Control nd iruit tehniques to mitigte prtil shding effets in photovolti rrys,ieee J photovolt, 01,,(4,pp [8] Kerekes, T, Koutroulis,E, Ser,D, Teodoresu, R, Ktsnevkis, M, An optimiztion method for designing lrge PV plnts, IEEE J Photovolt, 013,3,(,pp814-8 Akwledgments This work ws supported in prt y the Ntionl Nturl Siene Foundtion of Chin (104515, in prt y the Supported y Invtion Progrm of E-ISSN: 4-66X 107 Volume 16, 017
Dorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More informationUniversity of Sioux Falls. MAT204/205 Calculus I/II
University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationReview Topic 14: Relationships between two numerical variables
Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph
More informationTable of Content. c 1 / 5
Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More informationa) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.
Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You
More informationANALYSIS AND MODELLING OF RAINFALL EVENTS
Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.
More informationTHE ANALYSIS AND CALCULATION OF ELECTROMAGNETIC FIELD AROUND OVERHEAD POWER LINE HongWang Yang
5th Interntionl Conferene on Advned Mterils nd Computer Siene (ICAMCS 6) THE ANALYSIS AN CALCULATION OF ELECTROMAGNETIC FIEL AROUN OVERHEA POWER LINE HongWng Yng eprtment of eletril engineering, North
More informationFirst compression (0-6.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35.
0.9 First ompression (0-6.3 GP) First deompression (6.3-2.7 GP) Seond ompression (2.7-35.5 GP) Seond deompression (35.5-0 GP) V/V 0 0.7 0.5 0 5 10 15 20 25 30 35 P (GP) Supplementry Figure 1 Compression
More informationEE 330/330L Energy Systems (Spring 2012) Laboratory 1 Three-Phase Loads
ee330_spring2012_l_01_3phse_lods.do 1/5 EE 330/330L Energy Systems (Spring 2012) Lortory 1 ThreePhse Lods Introdution/Bkground In this lortory, you will mesure nd study the voltges, urrents, impednes,
More informationResearch Article Comparative Studies of Different Switching Patterns for Direct and Indirect Space Vector Modulated Matrix Converter
dvnes in Power Eletronis Volume, rtile ID 854, 8 pges doi:.55//854 Reserh rtile omprtive Studies of Different Swithing Ptterns for Diret nd Indiret Spe Vetor Modulted Mtrix onverter min Shnpour, Ssn Gholmi,
More informationTHE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL
THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More informationSECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS
SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS IN-SITU PROBING OF DOMAIN POLING IN Bi 4 Ti 3 O 12 THIN FILMS BY OPTICAL SECOND HARMONIC GENERATION YANIV BARAD, VENKATRAMAN GOPALAN Mterils Reserh Lortory
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationNON-DETERMINISTIC FSA
Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationAlgorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 5 Supplement Greedy Algorithms Cont d Minimizing lteness Ching (NOT overed in leture) Adm Smith 9/8/10 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov,
More informationFor a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then
Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities:
More informationUnit 4. Combinational Circuits
Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute
More informationNEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE
NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE V.S. Gordeev, G.A. Myskov Russin Federl Nuler Center All-Russi Sientifi Reserh Institute of Experimentl Physis (RFNC-VNIIEF)
More informationA Mathematical Model for Unemployment-Taking an Action without Delay
Advnes in Dynmil Systems nd Applitions. ISSN 973-53 Volume Number (7) pp. -8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte
More informationSpacetime and the Quantum World Questions Fall 2010
Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1
More informationPolyphase Systems. Objectives 23.1 INTRODUCTION
Polyphse Systems 23 Ojetives eome fmilir with the opertion of threephse genertor nd the mgnitude nd phse reltionship etween the three phse voltges. e le to lulte the voltges nd urrents for three-phse Y-onneted
More informationAlgorithm Design and Analysis
Algorithm Design nd Anlysis LECTURE 8 Mx. lteness ont d Optiml Ching Adm Smith 9/12/2008 A. Smith; sed on slides y E. Demine, C. Leiserson, S. Rskhodnikov, K. Wyne Sheduling to Minimizing Lteness Minimizing
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationNow we must transform the original model so we can use the new parameters. = S max. Recruits
MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationSolutions to Assignment 1
MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove
More informationSymmetrical Components 1
Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationMATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE
U.P.B. Si. Bull., Series C, Vol. 75, Iss. 2, ISSN 86 354 MATRIX INVERSE ON CONNEX PARALLEL ARCHITECTURE An-Mri CALFA, Gheorghe ŞTEFAN 2 Designed for emedded omputtion in system on hip design, the Connex
More informationGeneralization of 2-Corner Frequency Source Models Used in SMSIM
Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More informationExercise 3 Logic Control
Exerise 3 Logi Control OBJECTIVE The ojetive of this exerise is giving n introdution to pplition of Logi Control System (LCS). Tody, LCS is implemented through Progrmmle Logi Controller (PLC) whih is lled
More informationPolyphase Systems 22.1 INTRODUCTION
22 Polyphse Systems 22.1 INTRODUTION n genertor designed to develop single sinusoidl voltge for eh rottion of the shft (rotor) is referred to s single-phse genertor. If the numer of oils on the rotor is
More informationAppendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities
Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the
More informationLecture 6. CMOS Static & Dynamic Logic Gates. Static CMOS Circuit. PMOS Transistors in Series/Parallel Connection
NMOS Trnsistors in Series/Prllel onnetion Leture 6 MOS Stti & ynmi Logi Gtes Trnsistors n e thought s swith ontrolled y its gte signl NMOS swith loses when swith ontrol input is high Peter heung eprtment
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More informationINTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable
INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd
More informationGM1 Consolidation Worksheet
Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up
More informationMatrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix
tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri
More informationArrow s Impossibility Theorem
Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep
More informationLecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and
Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly
More informationCS 491G Combinatorial Optimization Lecture Notes
CS 491G Comintoril Optimiztion Leture Notes Dvi Owen July 30, August 1 1 Mthings Figure 1: two possile mthings in simple grph. Definition 1 Given grph G = V, E, mthing is olletion of eges M suh tht e i,
More informationQUADRATIC EQUATION. Contents
QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,
More informationIntegration. antidifferentiation
9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further
More informationAdaptive Controllers for Permanent Magnet Brushless DC Motor Drive System using Adaptive-Network-based Fuzzy Interference System
Amerin Journl of Applied Sienes 8 (8): 810-815, 2011 ISSN 1546-9239 2011 Siene Pulitions Adptive Controllers for Permnent Mgnet Brushless DC Motor Drive System using Adptive-Network-sed Fuzzy Interferene
More informationEngr354: Digital Logic Circuits
Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost
More informationDistributed Generation Placement in Unbalanced Distribution System with Seasonal Load Variation
Distriuted Genertion Plement in Unlned Distriution System with Sesonl Lod Vrition Rvi Tej Bhimrsetti Dept. of Eletril Engg., NT Kurukshetr Kurukshetr, ndi svrtej@gmil.om Ashwni Kumr, Memer, EEE Dept. of
More informationCounting Paths Between Vertices. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs. Isomorphism of Graphs
Isomorphism of Grphs Definition The simple grphs G 1 = (V 1, E 1 ) n G = (V, E ) re isomorphi if there is ijetion (n oneto-one n onto funtion) f from V 1 to V with the property tht n re jent in G 1 if
More informationThermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam
April 08 Therml energy Soures of het Trnsport of het How to use het The First Lw of Thermoynmis U W Or, if we wnt to otin externl work: U W 009 vrije Universiteit msterm Close yle stem power plnt The trik
More informationNovel Fiber-Optical Refractometric Sensor Employing Hemispherically-Shaped Detection Element
Novel Fier-Optil Refrtometri Sensor Employing Hemispherilly-Shped Detetion Element SERGEI KHOTIAINTSEV, VLADIMIR SVIRID Deprtment of Eletril Engineering, Fulty of Engineering Ntionl Autonomous University
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More informationI 3 2 = I I 4 = 2A
ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents
More informationAP CALCULUS Test #6: Unit #6 Basic Integration and Applications
AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret
More informationArrow s Impossibility Theorem
Rep Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationTHE ASYMMETRY OF COASTAL WATER LEVEL RESPONSE TO LANDFALLING HURRICANES SIMULATED BY A THREE-DIMENSIONAL STORM SURGE MODEL
THE ASYMMETRY OF COASTAL WATER LEVEL RESPONSE TO LANDFALLING HURRICANES SIMULATED BY A THREE-DIMENSIONAL STORM SURGE MODEL Mhun Peng *, Lin Xie nd Leonrd J. Pietrfes Deprtment of Mrine, Erth nd Atmospheri
More informationLearning Partially Observable Markov Models from First Passage Times
Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).
More informationMATH Final Review
MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationElectromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors
LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly
More information( ) as a fraction. Determine location of the highest
AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to
More informationAC/DC/AC Converters: Two-Level and Multilevel VSI
Sortes Ersmus Visit A/D/A onerters: Two-Leel nd Multileel VSI Josep Pou Antoni Aris Pge 1 Sortes Ersmus Visit Outline 1. Two-Leel Inerter 2. Multileel Inerters - sde H-Bridge Inerter - Flying-pitor Inerter
More informationAvailable online at ScienceDirect. Procedia Engineering 120 (2015 ) EUROSENSORS 2015
Aville online t www.sienediret.om SieneDiret Proedi Engineering 10 (015 ) 887 891 EUROSENSORS 015 A Fesiility Study for Self-Osillting Loop for Three Degreeof-Freedom Coupled MEMS Resontor Fore Sensor
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationChem Homework 11 due Monday, Apr. 28, 2014, 2 PM
Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationMagnetically Coupled Coil
Mgnetilly Coupled Ciruits Overview Mutul Indutne Energy in Coupled Coils Liner Trnsformers Idel Trnsformers Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Mgnetilly Coupled Coil i v L
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationFormula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx
Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint
More informationChapter 4 State-Space Planning
Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different
More informationIndustrial Electrical Engineering and Automation
CODEN:LUTEDX/(TEIE-719)/1-7/(7) Industril Electricl Engineering nd Automtion Estimtion of the Zero Sequence oltge on the D- side of Dy Trnsformer y Using One oltge Trnsformer on the D-side Frncesco Sull
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationParabola and Catenary Equations for Conductor Height Calculation
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, 6 (), nr. 3 9 Prbol nd Ctenr Equtions for Condutor Height Clultion Alen HATIBOVIC Abstrt This pper presents new equtions for ondutor height lultion bsed on the
More information332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006
2:221 Principles of Electricl Engineering I Fll 2006 Nme of the student nd ID numer: Hourly Exm 2 Novemer 6, 2006 This is closed-ook closed-notes exm. Do ll your work on these sheets. If more spce is required,
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationBehavior Composition in the Presence of Failure
Behvior Composition in the Presene of Filure Sestin Srdin RMIT University, Melourne, Austrli Fio Ptrizi & Giuseppe De Giomo Spienz Univ. Rom, Itly KR 08, Sept. 2008, Sydney Austrli Introdution There re
More informationEECS 141 Due 04/19/02, 5pm, in 558 Cory
UIVERSITY OF CALIFORIA College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Lst modified on April 8, 2002 y Tufn Krlr (tufn@eecs.erkeley.edu) Jn M. Rey, Andrei Vldemirescu Homework
More information3/8" Square (10 mm) Multi-Turn Cermet Trimmer
3/8" Squre ( mm) Multi-Turn Cermet FEATURES Industril grde W t 70 C Tests ording to CECC 400 or IEC 60393-1 Contt resistne vrition < 1 % typil Complint to RoHS Diretive 2002/95/EC The Model is smll size
More informationTOPIC: LINEAR ALGEBRA MATRICES
Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED
More informationNetwork Analysis and Synthesis. Chapter 5 Two port networks
Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More informationAlgorithms & Data Structures Homework 8 HS 18 Exercise Class (Room & TA): Submitted by: Peer Feedback by: Points:
Eidgenössishe Tehnishe Hohshule Zürih Eole polytehnique fédérle de Zurih Politenio federle di Zurigo Federl Institute of Tehnology t Zurih Deprtement of Computer Siene. Novemer 0 Mrkus Püshel, Dvid Steurer
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationPolynomials. Polynomials. Curriculum Ready ACMNA:
Polynomils Polynomils Curriulum Redy ACMNA: 66 www.mthletis.om Polynomils POLYNOMIALS A polynomil is mthemtil expression with one vrile whose powers re neither negtive nor frtions. The power in eh expression
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More informationCS 2204 DIGITAL LOGIC & STATE MACHINE DESIGN SPRING 2014
S 224 DIGITAL LOGI & STATE MAHINE DESIGN SPRING 214 DUE : Mrh 27, 214 HOMEWORK III READ : Relte portions of hpters VII n VIII ASSIGNMENT : There re three questions. Solve ll homework n exm prolems s shown
More information22: Union Find. CS 473u - Algorithms - Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u - Algorithms - Spring 2005 April 14, 2005 1 Union-Fin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x) - rete set tht ontins the single element x. 2. Fin(x)
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationHyers-Ulam stability of Pielou logistic difference equation
vilble online t wwwisr-publitionsom/jns J Nonliner Si ppl, 0 (207, 35 322 Reserh rtile Journl Homepge: wwwtjnsom - wwwisr-publitionsom/jns Hyers-Ulm stbility of Pielou logisti differene eqution Soon-Mo
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationLOSS COMPARISON OF TWO AND THREE-LEVEL INVERTER TOPOLOGIES
LOSS COPARSON OF TWO AND THREE-LEVEL NVERTER TOPOLOGES G.. Orfnoudis*, S.. Shrh*,. A. Yurtih nd. A. Ausr* * University of Southmpton, UK, TSL Tehnology, UK G..Orfnoudis@soton..u Keywords: nverter, DC-lin
More information