Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models

Size: px
Start display at page:

Download "Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models"

Transcription

1 Curret Progred Cotrol.e. Pek Curret-Mode Cotrol eture lde prt More Aurte Model ECEN 5807 Drg Mkovć

2 Sple Frt-Order CPM Model: Sury Aupto: CPM otroller operte delly, Ueful reult t low frequee, well uted for oervtve reltvely low bdwdth deg of voltge feedbk loop roud CPM otrolled overter tto: The ple odel doe ot dte poble tblty of the urret otroller, or the eed for opeto rtfl rp Doe ot lude hgh-frequey dy, whh relevt for wdebdwdth voltge loop deg Doe ot orretly odel le-to-output repoe CPM buk or buk-derved overter eve t low frequee t orretly predt oplete rejeto of le dturbe

3 More Aurte CPM Model: Outle Spled-dt odelg of dutor dy urret progred ode Itblty of the urret loop d the eed for opeto rtfl rp Iproved odelg of hgh-frequey dy to eble deg of wde-bdwdth voltge loop More urte verged odel rge-gl d ll-gl verged odultor odel Aurte verged ll-gl odel, ludg hgh-frequey dy Aurte odelg of le-to-output repoe Duo of reult for b overter CPM odel for ulto Deg exple 3

4 4

5 5

6 Idutor urret tret t t t dt d T dt d T 6

7 7 Hgh-frequey ll-gl dutor-urret dy Aue tht voltge perturbto re eglgbly ll t hgh frequee: the lope d be odered ott Apply pled-dt odelg: * t t z z jk T k ω

8 Sll-gl perturbto - t dt t 8

9 9 Drete-te dy T d T d

10 0 Drete-te dy T d T d ' D D M M

11 Itblty for D > 0.5 M M D D'

12

13 3

14 4

15 Sll-gl perturbto wth opeto rp t - t dt t 5

16 6 Drete-te dy wth opeto rp: T d T d

17 7 T d ' D D Drete-te dy wth opeto rp: T d

18 8

19 9

20 0

21 Drete-te dy: z z Z-trfor: z z z z z z z Drete-te z-do otrol-todutor urret trfer futo: T j T e e ω Dfferee equto: Pole t z Stblty odto: pole de the ut rle, < Frequey repoe ote tht z orrepod to dely of T te do:

22 Equvlet hold: t, z t - t dt t t T

23 Equvlet hold The repoe fro the ple of the dutor urret to the dutor urret perturbto t pule of pltude d legth T Hee, frequey do, the equvlet hold h the trfer futo prevouly derved for the zeroorder hold: e T 3

24 4 Coplete pled-dt trfer futo T T T e e ' D D Cotrol-to-dutor urret ll-gl repoe:

25 5 Exple CPM buk overter: V g 0V, 5 µh, C 75 µf, D 0.5, V 5 V, I 0 A, R V/I 0.5 Ω, f 00 khz Idutor urret lope: V g V/ A/µ V/ A/µ ' D D D 0.5: CPM otroller tble for y opeto rp, / > 0 T T T e e

26 Cotrol-to-dutor urret repoe for everl opeto rp / preter gtude db MATAB fle: CPMfr. / gtude d phe repoe / 0. / 0.5 / / phe deg frequey Hz 6

27 7 Frt-order pproxto hf T T T e e ω π ω / / / π ω π ω T e π π hf f D D f f Cotrol-to-dutor urret repoe behve pproxtely gle-pole trfer futo wth hgh-frequey pole t

28 Cotrol-to-dutor urret repoe for everl opeto rp / 0., 0.5,, 5 0 / gtude d phe repoe 0 gtude db t -order trfer-futo pproxto 0 phe deg frequey Hz 8

29 9 Seod-order pproxto / / T T T e e ω ω π / / / / T e ω ω π ω ω π D D Q π π Cotrol-to-dutor urret repoe behve pproxtely eodorder trfer futo wth orer frequey f / d Q-ftor gve by

30 Cotrol-to-dutor urret repoe for everl opeto rp / 0., 0.5,, 5 0 / gtude d phe repoe 0 gtude db d -order trfer-futo pproxto 0 phe deg frequey Hz 30

31 Coluo I CPM overter, hgh-frequey dutor dy deped trogly o the opeto rtfl rp lope Wthout opeto rp 0, CPM otroller utble for D > 0.5, reultg perod-doublg or other ub-hro or eve hot ollto For 0.5, CPM otroller tble for ll D Reltvely lrge opeto rp > 0.5 prtl hoe ot jut to eure tblty of the CPM otroller, but lo to redue etvty to oe For reltvely lrge vlue of, hgh-frequey dutor urret dy be well pproxted by gle hgh-frequey pole Seod-order pproxto very urte for y Next: ore urte verged odel, ludg hgh-frequey dy 3

ECEN 5807 Lecture 26

ECEN 5807 Lecture 26 ECEN 5807 eture 6 HW 8 due v D Frdy, rh, 0 S eture 8 on Wed rh 0 wll be leture reorded n 0 he week of rh 5-9 Sprng brek, no le ody: Conlude pled-dt odelng of hghfrequeny ndutor dyn n pek urrentode ontrolled

More information

2. Elementary Linear Algebra Problems

2. Elementary Linear Algebra Problems . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )

More information

The Z-Transform in DSP Lecture Andreas Spanias

The Z-Transform in DSP Lecture Andreas Spanias The Z-Trsform DSP eture - Adres Ss ss@su.edu 6 Coyrght 6 Adres Ss -- Poles d Zeros of I geerl the trsfer futo s rtol; t hs umertor d deomtor olyoml. The roots of the umertor d deomtor olyomls re lled the

More information

Chapter Gauss-Seidel Method

Chapter Gauss-Seidel Method Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos

More information

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4 // Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples

More information

Position and Speed Control. Industrial Electrical Engineering and Automation Lund University, Sweden

Position and Speed Control. Industrial Electrical Engineering and Automation Lund University, Sweden Poton nd Speed Control Lund Unverty, Seden Generc Structure R poer Reference Sh tte Voltge Current Control ytem M Speed Poton Ccde Control * θ Poton * Speed * control control - - he ytem contn to ntegrton.

More information

Module B3 3.1 Sinusoidal steady-state analysis (single-phase), a review 3.2 Three-phase analysis. Kirtley

Module B3 3.1 Sinusoidal steady-state analysis (single-phase), a review 3.2 Three-phase analysis. Kirtley Module B.1 Siusoidl stedy-stte lysis (sigle-phse), review.2 Three-phse lysis Kirtley Chpter 2: AC Voltge, Curret d Power 2.1 Soures d Power 2.2 Resistors, Idutors, d Cpitors Chpter 4: Polyphse systems

More information

Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2)

Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2) A ew Algorth for er Progrg Dhy P. ehedle Deprtet of Eletro See, Sr Prhurhu College, lk Rod, Pue-00, d dhy.p.ehedle@gl.o Atrt- th pper we propoe ew lgorth for ler progrg. h ew lgorth ed o tretg the oetve

More information

Numerical Analysis Topic 4: Least Squares Curve Fitting

Numerical Analysis Topic 4: Least Squares Curve Fitting Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto

More information

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as. Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o

More information

Linear Open Loop Systems

Linear Open Loop Systems Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce

More information

Chapter #2 EEE Subsea Control and Communication Systems

Chapter #2 EEE Subsea Control and Communication Systems EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio

More information

Collapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder

Collapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder Collapg to Saple ad Reader Mea Ed Staek Collapg to Saple ad Reader Average order to collape the expaded rado varable to weghted aple ad reader average, we pre-ultpled by ( M C C ( ( M C ( M M M ( M M M,

More information

ROUTH-HURWITZ CRITERION

ROUTH-HURWITZ CRITERION Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 ROUTH-HURWITZ CRITERION The Routh-Hurwitz riterio i lytil proedure for determiig

More information

Chapter Simpson s 1/3 Rule of Integration. ( x)

Chapter Simpson s 1/3 Rule of Integration. ( x) Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule

More information

Numerical Differentiation and Integration

Numerical Differentiation and Integration Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te

More information

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n 0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke

More information

Introduction to Modern Control Theory

Introduction to Modern Control Theory Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4

More information

CURVE FITTING LEAST SQUARES METHOD

CURVE FITTING LEAST SQUARES METHOD Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued

More information

Note 7 Root-Locus Techniques

Note 7 Root-Locus Techniques Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd Lecture Note

More information

Analyzing Control Structures

Analyzing Control Structures Aalyzg Cotrol Strutures sequeg P, P : two fragmets of a algo. t, t : the tme they tae the tme requred to ompute P ;P s t t Θmaxt,t For loops for to m do P t: the tme requred to ompute P total tme requred

More information

Mathematically, integration is just finding the area under a curve from one point to another. It is b

Mathematically, integration is just finding the area under a curve from one point to another. It is b Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom

More information

ELEG 3143 Probability & Stochastic Process Ch. 5 Elements of Statistics

ELEG 3143 Probability & Stochastic Process Ch. 5 Elements of Statistics Deprtet of Electricl Egieerig Uiversity of Arkss ELEG 3143 Probbility & Stochstic Process Ch. 5 Eleets of Sttistics Dr. Jigxi Wu wuj@urk.edu OUTLINE Itroductio: wht is sttistics? Sple e d sple vrice Cofidece

More information

CHAPTER 1: PEREVIEW. 1. Nature of Process Control Problem CLIENT HARDWARE. LEVEL-6 Planning. LEVEL-5 Scheduling. LEVEL-4 Real-Time Optimization

CHAPTER 1: PEREVIEW. 1. Nature of Process Control Problem CLIENT HARDWARE. LEVEL-6 Planning. LEVEL-5 Scheduling. LEVEL-4 Real-Time Optimization CHPER : PEREVIEW. Nature o Proe Cotrol Proble CLIEN IMERME CIVIY HRDWRE Upper level aageet Week-Mot LEVEL-6 Plaig Corporate oplex etwork Plat ager Da-Week LEVEL-5 Sedulig Platwide Ioratio te Proe Egieer

More information

2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission

2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission /0/0 Topcs Power Flow Part Text: 0-0. Power Trassso Revsted Power Flow Equatos Power Flow Proble Stateet ECEGR 45 Power Systes Power Trassso Power Trassso Recall that for a short trassso le, the power

More information

Chap8 - Freq 1. Frequency Response

Chap8 - Freq 1. Frequency Response Chp8 - Freq Frequecy Repoe Chp8 - Freq Aged Prelimirie Firt order ytem Frequecy repoe Low-p filter Secod order ytem Clicl olutio Frequecy repoe Higher order ytem Chp8 - Freq 3 Frequecy repoe Stedy-tte

More information

LECTURE 23 SYNCHRONOUS MACHINES (3)

LECTURE 23 SYNCHRONOUS MACHINES (3) ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 3 SYNCHRONOUS MACHINES (3) Acknowledgent-Thee hndout nd lecture note given in cl re bed on teril fro Prof. Peter Suer ECE 330 lecture note. Soe lide

More information

Math 10 Discrete Mathematics

Math 10 Discrete Mathematics Math 0 Dsrete Mathemats T. Heso REVIEW EXERCISES FOR EXM II Whle these problems are represetatve of the types of problems that I mght put o a exam, they are ot lusve. You should be prepared to work ay

More information

The z-transform. LTI System description. Prof. Siripong Potisuk

The z-transform. LTI System description. Prof. Siripong Potisuk The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put

More information

Linear predictive coding

Linear predictive coding Liner predictive coding Thi ethod cobine liner proceing with clr quntiztion. The in ide of the ethod i to predict the vlue of the current ple by liner cobintion of previou lredy recontructed ple nd then

More information

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1

Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1 Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule

More information

Analysis of error propagation in profile measurement by using stitching

Analysis of error propagation in profile measurement by using stitching Ay o error propgto proe eureet y ug ttchg Ttuy KUME, Kzuhro ENAMI, Yuo HIGASHI, Kej UENO - Oho, Tuu, Ir, 35-8, JAPAN Atrct Sttchg techque whch ee oger eureet rge o proe ro eer eure proe hg prty oerppe

More information

2 SKEE/SKEU v R(t) - Figure Q.1(a) Evaluate the transfer function of the network as

2 SKEE/SKEU v R(t) - Figure Q.1(a) Evaluate the transfer function of the network as SKEE/SKEU 073 PART A Q. ) A trfer futio i ued to deribe the reltiohi betwee the iut d outut igl of ytem. Figure Q.) how RC etwork ued to form filter futio. V it) R + v Rt) - C + v t) - Figure Q.) i) ii)

More information

The Auto-Tuning PID Controller for Interacting Water Level Process

The Auto-Tuning PID Controller for Interacting Water Level Process World Aademy of See, Egeerg ad Tehology Iteratoal Joural of Eletral ad Iformato Egeerg Vol:, No:, 7 The Auto-Tug PID Cotroller for Iteratg Water Level Proe Satea Tuyarrut, Taha Sukr, Arj Numomra, Supa

More information

Chapter #5 EEE Control Systems

Chapter #5 EEE Control Systems Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,

More information

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr

More information

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is

In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I

More information

The linear system. The problem: solve

The linear system. The problem: solve The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly

More information

11. Ideal Gas Mixture

11. Ideal Gas Mixture . Ideal Ga xture. Geeral oderato ad xture of Ideal Gae For a geeral xture of N opoet, ea a pure ubtae [kg ] te a for ea opoet. [kol ] te uber of ole for ea opoet. e al a ( ) [kg ] N e al uber of ole (

More information

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1. SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,

More information

Laplace Examples, Inverse, Rational Form

Laplace Examples, Inverse, Rational Form Lecure 3 Ouline: Lplce Exple, Invere, Rionl For Announceen: Rein: 6: Lplce Trnfor pp. 3-33, 55.5-56.5, 7 HW 8 poe, ue nex We. Free -y exenion OcenOne Roo Tour will e fer cl y 7 (:3-:) Lunch provie ferwr.

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule

More information

When current flows through the armature, the magnetic fields create a torque. Torque = T =. K T i a

When current flows through the armature, the magnetic fields create a torque. Torque = T =. K T i a D Motor Bic he D pernent-gnet otor i odeled reitor ( ) in erie with n inductnce ( ) nd voltge ource tht depend on the ngulr velocity of the otor oltge generted inide the rture K ω (ω i ngulr velocity)

More information

SOLUTIONS TO ASSIGNMENT NO The given nonrecursive signal processing structure is shown as

SOLUTIONS TO ASSIGNMENT NO The given nonrecursive signal processing structure is shown as SOLUTIONS TO ASSIGNMENT NO.1 3. The given nonreursive signl proessing struture is shown s X 1 1 2 3 4 5 Y 1 2 3 4 5 X 2 There re two ritil pths, one from X 1 to Y nd the other from X 2 to Y. The itertion

More information

Discrete random walk with barriers on a locally infinite graph

Discrete random walk with barriers on a locally infinite graph Drete rdo wl wth rrer o loll fte grh Theo Ue Aterd Shool of Teholog Weeerde 9 97 DZ Aterd The etherld El: te@hl Atrt We ot eeted er of rrl orto rolte d eeted te efore orto for etr drete rdo wl o loll fte

More information

To Determine the Characteristic Polynomial Coefficients Based On the Transient Response

To Determine the Characteristic Polynomial Coefficients Based On the Transient Response ICCAS Jue -, KINTEX, Gyeogg-Do, Kore To Determe the Chrctertc Polyoml Coeffcet Bed O the Tret Repoe Mohmmd Her d Mohmmd Sleh Tvzoe Advced Cotrol Sytem Lb., Electrcl Egeerg Deprtmet, Shrf Uverty of Techology,

More information

Signal Recovery - Prof. S. Cova - Exam 2016/02/16 - P1 pag.1

Signal Recovery - Prof. S. Cova - Exam 2016/02/16 - P1 pag.1 gal Recovery - Pro.. Cova - Exam 06/0/6 - P pag. PROBEM Data ad Note Appled orce F rt cae: tep ple ecod cae: rectaglar ple wth drato p = 5m Pezoelectrc orce eor A q =0pC/N orce-to-charge covero C = 500pF

More information

PROBLEM SET #4 SOLUTIONS by Robert A. DiStasio Jr.

PROBLEM SET #4 SOLUTIONS by Robert A. DiStasio Jr. PROBLM ST # SOLUTIONS y Roert. DStso Jr. Q. Prove tht the MP eergy s sze-osstet for two ftely seprted losed shell frgmets. The MP orrelto eergy s gve the sp-ortl ss s: vrt vrt MP orr Δ. or two moleulr

More information

DC-DC Converters - Dynamic Model Design and Experimental Verification

DC-DC Converters - Dynamic Model Design and Experimental Verification D-D erter - Dym Mdel De d Expermetl Verft Jh, Bet 005 k t publt tt fr publhed er APA: Jh, B. 005. D-D erter - Dym Mdel De d Expermetl Verft Deprtmet f Idutrl Eletrl Eeer d Autmt, ud Ittute f Tehly Geerl

More information

CS 4758 Robot Kinematics. Ashutosh Saxena

CS 4758 Robot Kinematics. Ashutosh Saxena CS 4758 Rt Kemt Ahuth Se Kemt tude the mt f de e re tereted tw emt tp Frwrd Kemt (ge t pt ht u re gve: he egth f eh he ge f eh t ht u fd: he pt f pt (.e. t (,, rdte Ivere Kemt (pt t ge ht u re gve: he

More information

8. INVERSE Z-TRANSFORM

8. INVERSE Z-TRANSFORM 8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere

More information

Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1

Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1 Addedum Addedum Vetor Review Deprtmet of Computer Siee d Egieerig - Coordite Systems Right hded oordite system Addedum y z Deprtmet of Computer Siee d Egieerig - -3 Deprtmet of Computer Siee d Egieerig

More information

Chapter 1 Counting Methods

Chapter 1 Counting Methods AlbertLudwgs Uversty Freburg Isttute of Empral Researh ad Eoometrs Dr. Sevtap Kestel Mathematal Statsts - Wter 2008 Chapter Coutg Methods Am s to determe how may dfferet possbltes there are a gve stuato.

More information

INTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek

INTERPOLATION(2) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek ELM Numerl Alss Dr Murrem Merme INTEROLATION ELM Numerl Alss Some of te otets re dopted from Luree V. Fusett Appled Numerl Alss usg MATLAB. rete Hll I. 999 ELM Numerl Alss Dr Murrem Merme Tod s leture

More information

ALGEBRA II CHAPTER 7 NOTES. Name

ALGEBRA II CHAPTER 7 NOTES. Name ALGEBRA II CHAPTER 7 NOTES Ne Algebr II 7. th Roots d Rtiol Expoets Tody I evlutig th roots of rel ubers usig both rdicl d rtiol expoet ottio. I successful tody whe I c evlute th roots. It is iportt for

More information

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University Revised by Prof. Jang, CAU

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University Revised by Prof. Jang, CAU Part 4 Capter 6 Sple ad Peewe Iterpolato PowerPot orgazed y Dr. Mael R. Gutao II Duke Uverty Reved y Pro. Jag CAU All mage opyrgt Te MGraw-Hll Compae I. Permo requred or reproduto or dplay. Capter Ojetve

More information

INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS) Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)

More information

Recent Progresses on the Simplex Method

Recent Progresses on the Simplex Method Reet Progresses o the Smple Method www.stford.edu/~yyye K.T. L Professor of Egeerg Stford Uversty d Itertol Ceter of Mgemet See d Egeerg Ng Uversty Outles Ler Progrmmg (LP) d the Smple Method Mrkov Deso

More information

z line a) Draw the single phase equivalent circuit. b) Calculate I BC.

z line a) Draw the single phase equivalent circuit. b) Calculate I BC. ECE 2260 F 08 HW 7 prob 4 solutio EX: V gyb' b' b B V gyc' c' c C = 101 0 V = 1 + j0.2 Ω V gyb' = 101 120 V = 6 + j0. Ω V gyc' = 101 +120 V z LΔ = 9 j1.5 Ω ) Drw the sigle phse equivlet circuit. b) Clculte

More information

UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II

UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II Awer Key Nme: Dte: UNIT # SEQUENCES AND SERIES COMMON CORE ALGEBRA II Prt I Quetio. For equece defied by f? () () 08 6 6 f d f f, which of the followig i the vlue of f f f f f f 0 6 6 08 (). I the viul

More information

G x, x E x E x E x E x. a a a a. is some matrix element. For a general single photon state. ), applying the operators.

G x, x E x E x E x E x. a a a a. is some matrix element. For a general single photon state. ), applying the operators. Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Proble Set # Quetio With G x, x E x E x E x E x G pqr p q r where G pqr i oe trix eleet For geerl igle

More information

2. The Laplace Transform

2. The Laplace Transform . The Lplce Trnform. Review of Lplce Trnform Theory Pierre Simon Mrqui de Lplce (749-87 French tronomer, mthemticin nd politicin, Miniter of Interior for 6 wee under Npoleon, Preident of Acdemie Frncie

More information

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

More information

Chapter #3 EEE Subsea Control and Communication Systems

Chapter #3 EEE Subsea Control and Communication Systems EEE 87 Chter #3 EEE 87 Sube Cotrol d Commuictio Sytem Cloed loo ytem Stedy tte error PID cotrol Other cotroller Chter 3 /3 EEE 87 Itroductio The geerl form for CL ytem: C R ', where ' c ' H or Oe Loo (OL)

More information

5 - Determinants. r r. r r. r r. r s r = + det det det

5 - Determinants. r r. r r. r r. r s r = + det det det 5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow

More information

The Algebraic Least Squares Fitting Of Ellipses

The Algebraic Least Squares Fitting Of Ellipses IOSR Jourl of Mthets (IOSR-JM) e-issn: 78-578 -ISSN: 39-765 Volue 4 Issue Ver II (Mr - Ar 8) PP 74-83 wwwosrjourlsorg he Algebr Lest Squres Fttg Of Ellses Abdelltf Betteb Dertet of Geerl Studes Jubl Idustrl

More information

On the energy of complement of regular line graphs

On the energy of complement of regular line graphs MATCH Coucato Matheatcal ad Coputer Chetry MATCH Cou Math Coput Che 60 008) 47-434 ISSN 0340-653 O the eergy of copleet of regular le graph Fateeh Alaghpour a, Baha Ahad b a Uverty of Tehra, Tehra, Ira

More information

Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.

Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n. Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx

More information

GENERALIZED OPERATIONAL RELATIONS AND PROPERTIES OF FRACTIONAL HANKEL TRANSFORM

GENERALIZED OPERATIONAL RELATIONS AND PROPERTIES OF FRACTIONAL HANKEL TRANSFORM S. Res. Chem. Commu.: (3 8-88 ISSN 77-669 GENERLIZED OPERTIONL RELTIONS ND PROPERTIES OF FRCTIONL NKEL TRNSFORM R. D. TYWDE *. S. GUDDE d V. N. MLLE b Pro. Rm Meghe Isttute o Teholog & Reserh Bder MRVTI

More information

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B]

Concept of Activity. Concept of Activity. Thermodynamic Equilibrium Constants [ C] [ D] [ A] [ B] Conept of Atvty Equlbrum onstnt s thermodynm property of n equlbrum system. For heml reton t equlbrum; Conept of Atvty Thermodynm Equlbrum Constnts A + bb = C + dd d [C] [D] [A] [B] b Conentrton equlbrum

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt

More information

Algebra 2 Readiness Summer Packet El Segundo High School

Algebra 2 Readiness Summer Packet El Segundo High School Algebr Rediess Suer Pcket El Segudo High School This pcket is desiged for those who hve copleted Geoetry d will be erolled i Algebr (CP or H) i the upcoig fll seester. Suer Pcket Algebr II Welcoe to Algebr

More information

ELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University

ELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University ELEC 37 LECTURE NOTES, WEE 6 Dr mir G ghdm Cocordi Uiverity Prt of thee ote re dpted from the mteril i the followig referece: Moder Cotrol Sytem by Richrd C Dorf d Robert H Bihop, Pretice Hll Feedbck Cotrol

More information

xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +.

xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +. Ler Trsfortos d Group Represettos Hoework #3 (06-07, Aswers Q-Q re further exerses oer dots, self-dot trsfortos, d utry trsfortos Q3-6 volve roup represettos Of these, Q3 d Q4 should e quk Q5 s espelly

More information

Chapter 2. LOGARITHMS

Chapter 2. LOGARITHMS Chpter. LOGARITHMS Dte: - 009 A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog

More information

6 Random Errors in Chemical Analysis

6 Random Errors in Chemical Analysis 6 Rndom Error n Cheml Anl 6A The ture of Rndom Error 6A- Rndom Error Soure? Fg. 6- Three-dmenonl plot howng olute error n Kjeldhl ntrogen determnton for four dfferent nlt. Anlt Pree Aurte 4 Tle 6- Pole

More information

Diagonally Implicit Runge-Kutta Nystrom General Method Order Five for Solving Second Order IVPs

Diagonally Implicit Runge-Kutta Nystrom General Method Order Five for Solving Second Order IVPs WSEAS TRANSACTIONS o MATHEMATICS Fudzh Isml Dgoll Implt Ruge-Kutt Nstrom Geerl Method Order Fve for Solvg Seod Order IVPs FUDZIAH ISMAIL Deprtmet of Mthemts Uverst Putr Mls Serdg Selgor MALAYSIA fudzh@mth.upm.edu.m

More information

R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder Sampled-data response: i L /i c Sampled-data transfer function : î L (s) î c (s) = (1 ) 1 e st

More information

PAIR OF STRAIGHT LINES. will satisfy L1 L2 0, and thus L1 L. 0 represent? It is obvious that any point lying on L 1

PAIR OF STRAIGHT LINES. will satisfy L1 L2 0, and thus L1 L. 0 represent? It is obvious that any point lying on L 1 LOCUS 33 Seto - 3 PAIR OF STRAIGHT LINES Cosder two les L L Wht do ou thk wll L L represet? It s ovous tht pot lg o L d L wll stsf L L, d thus L L represets the set of pots osttutg oth the les,.e., L L

More information

Chapter #2 EEE State Space Analysis and Controller Design

Chapter #2 EEE State Space Analysis and Controller Design Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td

More information

v v at 1 2 d vit at v v 2a d

v v at 1 2 d vit at v v 2a d SPH3UW Unt. Accelerton n One Denon Pge o 9 Note Phyc Inventory Accelerton the rte o chnge o velocty. Averge ccelerton, ve the chnge n velocty dvded by the te ntervl, v v v ve. t t v dv Intntneou ccelerton

More information

2.Decision Theory of Dependence

2.Decision Theory of Dependence .Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give

More information

Spectral Characteristics of Digitally Modulated Signals

Spectral Characteristics of Digitally Modulated Signals Strl Chrtrt of Dgtlly odultd Sgl 6:33:56 Wrl Couto holog Srg 5 Ltur7&8 Drtt of Eltrl Egrg Rutgr Uvrty Ptwy J 89 ught y Dr. ry dy ry@wl.rutgr.du Doutd y Bozh Yu ozh@d.rutgr.du trt: h ltur frt trodu th tdrd

More information

7-1: Zero and Negative Exponents

7-1: Zero and Negative Exponents 7-: Zero nd Negtive Exponents Objective: To siplify expressions involving zero nd negtive exponents Wr Up:.. ( ).. 7.. Investigting Zero nd Negtive Exponents: Coplete the tble. Write non-integers s frctions

More information

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root

Summer MA Lesson 4 Section P.3. such that =, denoted by =, is the principal square root Suer MA 00 Lesso Sectio P. I Squre Roots If b, the b is squre root of. If is oegtive rel uber, the oegtive uber b b b such tht, deoted by, is the pricipl squre root of. rdicl sig rdicl expressio rdicd

More information

Three-Phase Voltage-Source Converters

Three-Phase Voltage-Source Converters CURET Fll Three-Phe olge-soure Coerer Oule B Oero & Alo Pule-Wh oulo AC-Se Curre Corol DC-k olge Regulo Su C 85, ju@r.eu Three-Phe SC Three-Phe SC Cru / / S S S S S S A erle erfe ewee DC Three-Phe AC le

More information

Ideal Gas behaviour: summary

Ideal Gas behaviour: summary Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive)

More information

The state space model needs 5 parameters, so it is not as convenient to use in this control study.

The state space model needs 5 parameters, so it is not as convenient to use in this control study. Trasfer fuctio for of the odel G θ K ω 2 θ / v θ / v ( s) = = 2 2 vi s + 2ζωs + ω The followig slides detail a derivatio of this aalog eter odel both as state space odel ad trasfer fuctio (TF) as show

More information

CHAPTER 5 Vectors and Vector Space

CHAPTER 5 Vectors and Vector Space HAPTE 5 Vetors d Vetor Spe 5. Alger d eometry of Vetors. Vetor A ordered trple,,, where,, re rel umers. Symol:, B,, A mgtude d dreto.. Norm of vetor,, Norm =,, = = mgtude. Slr multplto Produt of slr d

More information

Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University

Advanced Algorithmic Problem Solving Le 3 Arithmetic. Fredrik Heintz Dept of Computer and Information Science Linköping University Advced Algorthmc Prolem Solvg Le Arthmetc Fredrk Hetz Dept of Computer d Iformto Scece Lköpg Uversty Overvew Arthmetc Iteger multplcto Krtsu s lgorthm Multplcto of polyomls Fst Fourer Trsform Systems of

More information

Preliminary Examinations: Upper V Mathematics Paper 1

Preliminary Examinations: Upper V Mathematics Paper 1 relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0

More information

A New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming

A New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research

More information

Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation

Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777

More information

Moments of Generalized Order Statistics from a General Class of Distributions

Moments of Generalized Order Statistics from a General Class of Distributions ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt

More information

DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini

DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg

More information

Dopant Compensation. Lecture 2. Carrier Drift. Types of Charge in a Semiconductor

Dopant Compensation. Lecture 2. Carrier Drift. Types of Charge in a Semiconductor Lecture OUTLIE Bc Semcoductor Phycs (cot d) rrer d uo P ucto odes Electrosttcs ctce ot omesto tye semcoductor c be coverted to P tye mterl by couter dog t wth ccetors such tht >. comested semcoductor mterl

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg

More information

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol

More information

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:

Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11: Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1

More information

Problem Set 4 Solutions

Problem Set 4 Solutions 4 Eoom Altos of Gme Theory TA: Youg wg /08/0 - Ato se: A A { B, } S Prolem Set 4 Solutos - Tye Se: T { α }, T { β, β} Se Plyer hs o rte formto, we model ths so tht her tye tke oly oe lue Plyer kows tht

More information