Chapter 2: Principles of steadystate converter analysis


 Hilda Thompson
 2 years ago
 Views:
Transcription
1 Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer example 2.5. Esimaing he ripple in converers conaining wopole lowpass filers 2.6. Summary of key poins 1
2 2.1 Inroducion Buck converer SPDwich changes dc componen 1 2 v s v Swich oupu volage waveform Duy cycle D: 0 D 1 complemen D : D = 1  D v s D D' 0 D swich posiion:
3 Dc componen of swich oupu volage v s area = D <v s > = D 0 D 0 Fourier analysis: Dc componen = average value v s = T 1 v s d s 0 v s = 1 (D )=D 3
4 Inserion of lowpass filer o remove swiching harmonics and pass only dc componen 1 2 v s C v V v v s = D D 4
5 Three basic dcdc converers Buck a) 1 2 i C v M(D) M(D)=D D Boos b) i 1 2 C v M(D) M(D)= 1 1D D Buckboos c) 1 2 i C v M(D) D M(D)= D 1D 5
6 Objecives of his chaper Develop echniques for easily deermining oupu volage of an arbirary converer circui Derive he principles of inducor volsecond balance and capacior charge (ampsecond) balance Inroduce he key small ripple approximaion Develop simple mehods for selecing filer elemen values Illusrae via examples 6
7 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion Acual oupu volage waveform, buck converer Buck converer conaining pracical lowpass filer 1 2 i v C i C v Acual oupu volage waveform v=vv ripple v V Dc componen V acual waveform v = V v ripple 0 7
8 The small ripple approximaion v=vv ripple v V Dc componen V acual waveform v = V v ripple 0 In a welldesigned converer, he oupu volage ripple is small. Hence, he waveforms can be easily deermined by ignoring he ripple: v ripple << V v V 8
9 Buck converer analysis: inducor curren waveform original converer 1 2 i v C i C v swich in posiion 1 swich in posiion 2 i v i C v i C C v i C v 9
10 Inducor volage and curren Subinerval 1: swich in posiion 1 Inducor volage v = v i v i C Small ripple approximaion: C v v V Knowing he inducor volage, we can now find he inducor curren via v = di d Solve for he slope: di d = v V The inducor curren changes wih an essenially consan slope 10
11 Inducor volage and curren Subinerval 2: swich in posiion 2 Inducor volage v =v Small ripple approximaion: v V v i C i C v Knowing he inducor volage, we can again find he inducor curren via v = di d Solve for he slope: di d V The inducor curren changes wih an essenially consan slope 11
12 Inducor volage and curren waveforms v V D D' i I i (0) V i (D ) V swich posiion: V i v = di d 0 D 12
13 Deerminaion of inducor curren ripple magniude i i (D ) I i (0) V V i 0 D (change in i )=(slope)(lengh of subinerval) 2 i = V DT s i = V 2 D = V 2 i D 13
14 Inducor curren waveform during urnon ransien i i ( ) i (0)=0 0 D v v i (n ) 2 n (n1) i ((n1) ) When he converer operaes in equilibrium: i ((n 1) )=i (n ) 14
15 The principle of inducor volsecond balance: Derivaion Inducor defining relaion: v = di d Inegrae over one complee swiching period: i ( )i (0) = 1 0 v d In periodic seady sae, he ne change in inducor curren is zero: 0= v d 0 Hence, he oal area (or volseconds) under he inducor volage waveform is zero whenever he converer operaes in seady sae. An equivalen form: 0= 1 v T d = v s 0 The average inducor volage is zero in seady sae. 15
16 Inducor volsecond balance: Buck converer example Inducor volage waveform, previously derived: v V oal area λ D Inegral of volage waveform is area of recangles: λ = v d =( V)(D )(V)(D' ) Average volage is v 0 = λ = D( V)D'(V) V Equae o zero and solve for V: 0=D (DD')V=D V V = D 16
17 The principle of capacior charge balance: Derivaion Capacior defining relaion: i C =C dv C d Inegrae over one complee swiching period: v C ( )v C (0) = 1 C 0 i C d In periodic seady sae, he ne change in capacior volage is zero: 0= T 1 i C d s 0 = i C Hence, he oal area (or charge) under he capacior curren waveform is zero whenever he converer operaes in seady sae. The average capacior curren is hen zero. 17
18 2.3 Boos converer example 2 Boos converer wih ideal swich i v 1 i C C v D 1 ealizaion using power MOSFET and diode i v D Q 1 i C C v 18
19 Boos converer analysis 2 original converer i v 1 i C C v swich in posiion 1 swich in posiion 2 i v i C i v i C C v C v 19
20 Subinerval 1: swich in posiion 1 Inducor volage and capacior curren v = i C =v/ i v i C C v Small ripple approximaion: v = i C =V/ 20
21 Subinerval 2: swich in posiion 2 Inducor volage and capacior curren v = v i C = i v / i v i C C v Small ripple approximaion: v = V i C = I V / 21
22 Inducor volage and capacior curren waveforms v D D' V i C I V/ D V/ D' 22
23 Inducor volsecond balance Ne volseconds applied o inducor over one swiching period: v D D' 0 v d =( )D ( V)D' V Equae o zero and collec erms: (D D')VD'=0 Solve for V: V = D' The volage conversion raio is herefore M(D)= V = 1 D' = 1 1D 23
24 Conversion raio M(D) of he boos converer 5 4 M(D) = 1 D' = 1 1D M(D) D 24
25 Deerminaion of inducor curren dc componen Capacior charge balance: i C I V/ D D' 0 i C d =( V )D (I V )D' V/ Collec erms and equae o zero: V Solve for I: I = (D D')ID'=0 V D' Eliminae V o express in erms of : I = D' 2 I ( / ) D 25
26 Deerminaion of inducor curren ripple Inducor curren slope during subinerval 1: di d di d = v = v = Inducor curren slope during subinerval 2: = V i I V 0 D Change in inducor curren during subinerval 1 is (slope) (lengh of subinerval): i 2 i = D Solve for peak ripple: i = 2 D Choose such ha desired ripple magniude is obained 26
27 Deerminaion of capacior volage ripple Capacior volage slope during subinerval 1: dv C d = i C C = V C v V V C I C V C v Capacior volage slope during subinerval 2: 0 D dv C d = i C C = I C V C Change in capacior volage during subinerval 1 is (slope) (lengh of subinerval): 2 v= V C D Solve for peak ripple: v = V 2C D Choose C such ha desired volage ripple magniude is obained In pracice, capacior equivalen series resisance (esr) leads o increased volage ripple 27
28 2.4 Cuk converer example Cuk converer, wih ideal swich 1 C 1 2 i 1 v i C 2 v 2 Cuk converer: pracical realizaion using MOSFET and diode 1 C 1 2 i 1 v i 2 1 Q 1 D 1 C 2 v 2 28
29 Cuk converer circui wih swich in posiions 1 and 2 Swich in posiion 1: MOSFET conducs Capacior C 1 releases energy o oupu 1 i 2 i 1 v i v 1 C1 2 2 v 1 C 1 C 2 i C2 v 2 Swich in posiion 2: diode conducs Capacior C 1 is charged from inpu 1 i 1 i 2 v 1 i C1 2 C 1 v 1 C 2 v 2 i C2 v 2 29
30 Waveforms during subinerval 1 MOSFET conducion inerval Inducor volages and capacior currens: v 1 = 1 i 2 i 1 v i v 1 C1 2 2 v 1 C 1 C 2 i C2 v 2 v 2 =v 1 v 2 i C1 =i 2 i C2 =i 2 v 2 Small ripple approximaion for subinerval 1: v 1 = v 2 =V 1 V 2 i C1 =I 2 i C2 =I 2 V 2 30
31 Waveforms during subinerval 2 Diode conducion inerval Inducor volages and capacior currens: v 1 = v 1 v 2 =v 2 i C1 =i 1 1 i 1 i 2 v 1 i C1 C 1 v 1 C 2 2 v 2 i C2 v 2 i C2 =i 2 v 2 Small ripple approximaion for subinerval 2: v 1 = V 1 v 2 =V 2 i C1 =I 1 i C2 =I 2 V 2 31
32 Equae average values o zero The principles of inducor volsecond and capacior charge balance sae ha he average values of he periodic inducor volage and capacior curren waveforms are zero, when he converer operaes in seady sae. Hence, o deermine he seadysae condiions in he converer, le us skech he inducor volage and capacior curren waveforms, and equae heir average values o zero. Waveforms: Inducor volage v 1 v 1 Volsecond balance on 1 : D D' v 1 = D D'( V 1 )=0 V 1 32
33 Equae average values o zero Inducor 2 volage v 2 V 2 D D' V 1 V 2 Average he waveforms: Capacior C 1 curren i C1 I 1 v 2 = D(V 1 V 2 )D'(V 2 )=0 i C1 =DI 2 D'I 1 =0 D I 2 D' 33
34 Equae average values o zero Capacior curren i C2 waveform i C2 D I 2 V 2 / (= 0) D' i C2 =I 2 V 2 =0 Noe: during boh subinervals, he capacior curren i C2 is equal o he difference beween he inducor curren i 2 and he load curren V 2 /. When ripple is negleced, i C2 is consan and equal o zero. 34
35 Cuk converer conversion raio M = V/ 0 D M(D) M(D)= V 2 = D 1D 5 35
36 Inducor curren waveforms Inerval 1 slopes, using small ripple approximaion: di 1 d di 2 d = v 1 1 Inerval 2 slopes: di 1 d di 2 d = 1 = v 2 2 = V 1 V 2 2 = v 1 1 = V 1 1 = v 2 2 = V 2 2 i 1 i 2 I 1 i 1 1 V 1 1 I 2 V 1 V 2 2 D D V 2 2 i2 36
37 Capacior C 1 waveform Subinerval 1: dv 1 d Subinerval 2: dv 1 d = i C1 C 1 = I 2 C 1 = i C1 C 1 = I 1 C 1 v 1 v 1 V 1 I 2 I 1 C 1 D C 1 37
38 ipple magniudes Analysis resuls i 1 = D 2 1 i 2 = V 1 V v 1 = I 2D 2C 1 D Use dc converer soluion o simplify: i 1 = D 2 1 i 2 = D 2 2 v 1 = D 2 2D'C 1 Q: How large is he oupu volage ripple? 38
39 2.5 Esimaing ripple in converers conaining wopole lowpass filers Buck converer example: Deermine oupu volage ripple 1 2 i i C C v C i Inducor curren waveform. Wha is he capacior curren? i I i (0) V i (D ) V 0 D i 39
40 Capacior curren and volage, buck example Mus no neglec inducor curren ripple! i C oal charge q / 2 i D D' If he capacior volage ripple is small, hen essenially all of he ac componen of inducor curren flows hrough he capacior. v C V v v 40
41 Esimaing capacior volage ripple v i C v C V oal charge q D / 2 v D' i v Curren i C is posiive for half of he swiching period. This posiive curren causes he capacior volage v C o increase beween is minimum and maximum exrema. During his ime, he oal charge q is deposied on he capacior plaes, where q = C (2 v) (change in charge)= C(change in volage) 41
42 Esimaing capacior volage ripple v i C oal charge q D / 2 D' i The oal charge q is he area of he riangle, as shown: q = 1 2 i 2 Eliminae q and solve for v: v C v = i 8 C V v v Noe: in pracice, capacior equivalen series resisance (esr) furher increases v. 42
43 Inducor curren ripple in wopole filers Example: problem i T Q1 i 1 i 2 2 V g C 1 v C1 D1 C 2 v v oal flux linkage λ v / 2 D D' can use similar argumens, wih λ = i i λ = inducor flux linkages I i i = inducor volseconds 43
44 2.6 Summary of Key Poins 1. The dc componen of a converer waveform is given by is average value, or he inegral over one swiching period, divided by he swiching period. Soluion of a dcdc converer o find is dc, or seadysae, volages and currens herefore involves averaging he waveforms. 2. The linear ripple approximaion grealy simplifies he analysis. In a welldesigned converer, he swiching ripples in he inducor currens and capacior volages are small compared o he respecive dc componens, and can be negleced. 3. The principle of inducor volsecond balance allows deerminaion of he dc volage componens in any swiching converer. In seadysae, he average volage applied o an inducor mus be zero. 44
45 Summary of Chaper 2 4. The principle of capacior charge balance allows deerminaion of he dc componens of he inducor currens in a swiching converer. In seadysae, he average curren applied o a capacior mus be zero. 5. By knowledge of he slopes of he inducor curren and capacior volage waveforms, he ac swiching ripple magniudes may be compued. Inducance and capaciance values can hen be chosen o obain desired ripple magniudes. 6. In converers conaining muliplepole filers, coninuous (nonpulsaing) volages and currens are applied o one or more of he inducors or capaciors. Compuaion of he ac swiching ripple in hese elemens can be done using capacior charge and/or inducor fluxlinkage argumens, wihou use of he smallripple approximaion. 7. Converers capable of increasing (boos), decreasing (buck), and invering he volage polariy (buckboos and Cuk) have been described. Converer circuis are explored more fully in a laer chaper. 45
6.334 Power Electronics Spring 2007
MI OpenCourseWare hp://ocw.mi.edu 6.334 Power Elecronics Spring 2007 For informaion abou ciing hese maerials or our erms of Use, visi: hp://ocw.mi.edu/erms. Chaper 5 Inroducion o DC/DC Converers Analysis
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationPart II Converter Dynamics and Control
Par II onverer Dynamics and onrol 7. A equivalen circui modeling 8. onverer ransfer funcions 9. onroller design 1. Inpu filer design 11. A and D equivalen circui modeling of he disconinuous conducion mode
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67  FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1  TRANSIENTS
EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67  FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1  RANIEN Uni conen 2 Undersand he ransien behaviour of resisorcapacior (R) and resisorinducor (RL) D circuis
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationDCDC Boost Converter with Constant Output Voltage for Grid Connected Photovoltaic Application System
DCDC Boos Converer wih Consan Oupu Volage for Grid Conneced Phoovolaic Applicaion Sysem PuiWeng Chan, Syafrudin Masri Universii Sains Malaysia Email: edmond_chan85@homail.com, syaf@eng.usm.my Absrac
More informationUsing RCtime to Measure Resistance
Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a builin ADC (Analog o Digial Converer)
More informationConverter Topologies
High Sepup Raio DCDC Converer Topologies Par I Speaker: G. Spiazzi P. Teni,, L. Rosseo,, G. Spiazzi,, S. Buso,, P. Maavelli, L. Corradini Dep. of Informaion Engineering DEI Universiy of Padova Seminar
More informationELECTRICAL CIRCUITS 7. NONLINEAR COMPARATOR OSCILLATORS
87 ELETIAL IUITS 7. NONLEA OMPAATO OSILLATOS Inroducion A linear oscillaor is a basic feedback conrol sysem ha has been made deliberaely unsable a he frequency of oscillaion. The linear oscillaor sysem
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationLECTURE 9. C. Appendix
LECTURE 9 A. BuckBoos Converer Design 1. VolSec Balance: f(d), seadysae ransfer funcion 2. DC Operaing Poin via Charge Balance: I(D) in seadysae 3. Ripple Volage / C Spec 4. Ripple Curren / L Spec 5.
More informationI) EQUATION 1: SHUNTCONTROLLED
Swich Mode Power Supply (SMPS) Topologies (Par I) Auhor: Mohammad Kamil Microchip Technology Inc. EQUATION 1: SHUNTCONTROLLED REGULATOR POWER LOSS INTRODUCTION The indusry drive oward smaller, ligher
More informationSwitched Mode Converters (1 Quadrant)
(1 Quadran) Philippe Barrade Laboraoire d Elecronique Indusrielle, LEI STI ISE Ecole Polyechnique Fédérale de Lausanne, EPFL Ch1015 Lausanne Tél: +41 21 693 2651 Fax: +41 21 693 2600 Philippe.barrade@epfl.ch
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationPHYS245 Lab: RC circuits
PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of
More informationPower Electronics Introduction
Power Elecronics nroducion Y. Baghzouz EE 442642 11 Power Elecronics: an Overview Power elecronics is an inerdisciplinary subjec wihin elecrical engineering. 12 Power Elecronic Sysem A power elecronic
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationµ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ
Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationLaboratory #3 Diode Basics and Applications (I)
Laboraory #3 iode asics and pplicaions (I) I. Objecives 1. Undersand he basic properies of diodes. 2. Undersand he basic properies and operaional principles of some dioderecifier circuis. II. omponens
More informationORDER INFORMATION TO pin 300 ~ 360mV AMC7150DLF 300 ~ 320mV AMC7150ADLF 320 ~ 340mV AMC7150BDLF 340 ~ 360mV AMC7150CDLF
www.addmek.com DESCRIPTI is a PWM power ED driver IC. The driving curren from few milliamps up o 1.5A. I allows high brighness power ED operaing a high efficiency from 4Vdc o 40Vdc. Up o 200KHz exernal
More informationModule 3. RL & RC Transients. Version 2 EE IIT, Kharagpur
Module 3  & C Transiens esson 0 Sudy of DC ransiens in  and C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series
More information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
More informationAN1207. Switch Mode Power Supply (SMPS) Topologies (Part II) REQUIREMENTS AND RULES INTRODUCTION CONTENTS. Microchip Technology Inc.
Swich Mode Power Supply (SMPS) opologies (Par II) Auhor: INRODUCION his applicai noe is he secd of a wopar series Swich Mode Power Supply (SMPS) opologies. he firs applicai noe in his series AN1114 
More informationLLC Resonant Converter Reference Design using the dspic DSC
LLC Resonan Converer Reference Design using he dspic DSC 2010 Microchip Technology Incorporaed. All Righs Reserved. LLC Resonan Converer Webinar Slide 1 Hello, and welcome o his web seminar on Microchip
More information11. Properties of alternating currents of LCRelectric circuits
WS. Properies of alernaing currens of Lelecric circuis. Inroducion Socalled passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationVoltage level shifting
rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bipolar
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationSwitching Regulator IC series Capacitor Calculation for Buck converter IC
Swiching Regulaor IC series Capacior Calculaion for Buck converer IC No.14027ECY02 This applicaion noe explains he calculaion of exernal capacior value for buck converer IC circui. Buck converer IIN IDD
More informationMaking Use of Gate Charge Information in MOSFET and IGBT Data Sheets
Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
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 informationDesigning A HighVoltage NonIsolated BuckBoost Converter with the Si9121DY
Designing A HighVolage NonIsolaed BuckBoos Converer wih he Si9DY AN Niin Kalje The Si9DY is a nonisolaed buckboos converer IC, operaing from a wide inpu volage range of 0 o 0 V wih minimal exernal
More informationPulseWidth Modulation Inverters
SECTION 3.6 INVERTERS 189 PulseWidh Modulaion Inverers Pulsewidh modulaion is he process of modifying he widh of he pulses in a pulse rain in direc proporion o a small conrol signal; he greaer he conrol
More informationLecture10 BJT Switching Characteristics, Small Signal Model
1 Lecure1 BJT Swiching Characerisics, Small Signal Model BJT Swiching Characerisics: The circui in Fig.1(b) is a simple CE swich. The inpu volage waveform v s shown in he Fig.1(a) is used o conrol he
More informationRC Circuit and Time Constant
ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisorcapacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationIntroduction to Power Electronics
Inroducion o Power Elecronics Fang Z. Peng Dep. of Elecrical and Compuer Engineering Michigan Sae Universiy Phone: 5174323331, Fax: 5173531980 Email: fzpeng@egr.msu.edu F. Z. Peng: Slide 1 Conens Chaper
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More information2 Electric Circuits Concepts Durham
Chaper 3  Mehods Chaper 3  Mehods... 3. nroducion... 2 3.2 Elecrical laws... 2 3.2. Definiions... 2 3.2.2 Kirchhoff... 2 3.2.3 Faraday... 3 3.2.4 Conservaion... 3 3.2.5 Power... 3 3.2.6 Complee... 4
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Wholerange Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More information6.003 Homework #4 Solutions
6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationENE 104 Electric Circuit Theory
Elecric Circui heory Lecure 0: AC Power Circui Analysis (ENE) Mon, 9 Mar 0 / (EE) Wed, 8 Mar 0 : Dejwoo KHAWPARSUH hp://websaff.ku.ac.h/~dejwoo.kha/ Objecives : Ch Page he insananeous power he average
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationBasic Circuit Elements  Prof J R Lucas
Basic Circui Elemens  Prof J ucas An elecrical circui is an inerconnecion of elecrical circui elemens. These circui elemens can be caegorized ino wo ypes, namely acive elemens and passive elemens. Some
More informationPhotovoltaic Power Control Using MPPT and Boost Converter
23 Phoovolaic Power Conrol Using MPP and Boos Converer A.Aou, A.Massoum and M.Saidi Absrac he sudies on he phoovolaic sysem are exensively increasing because of a large, secure, essenially exhausible and
More information6.003: Signals and Systems
6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20 2 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers. 3 Fourier Series
More informationTLE 472x Family Stepper Motor Drivers. Current Control Method and Accuracy
Applicaion Noe, V 1.0, Augus 2001 ANPS063E TLE 472x Family Sepper Moor Drivers Curren Conrol Mehod and Accuracy by Frank Heinrichs Auomoive Power N e v e r s o p h i n k i n g.  1  TLE 472x sepper moor
More informationPhysics 111 Fall 2007 Electric Currents and DC Circuits
Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels
More informationSEMICONDUCTOR APPLICATION NOTE
SEMICONDUCTOR APPLICATION NOTE Order his documen by AN1542/D Prepared by: C. S. Mier Moorola Inc. Inpu filer design has been an inegral par of power supply designs. Wih he adven of inpu filers, he designer
More informationECEN4618: Experiment #1 Timing circuits with the 555 timer
ECEN4618: Experimen #1 Timing circuis wih he 555 imer cæ 1998 Dragan Maksimović Deparmen of Elecrical and Compuer Engineering Universiy of Colorado, Boulder The purpose of his lab assignmen is o examine
More informationSolidstate Marx generator design with an energy recovery reset circuit for output transformer association
Solidsae Marx generaor design wih an energy recovery rese circui for oupu ransformer associaion L M Redondo J Fernando Silva P Tavares E Margao Insiuo Superior de Engenharia Lisboa, CEEI, CFNUL Insiuo
More informationTransient Analysis of First Order RC and RL circuits
Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationCAPACITANCE AND INDUCTANCE
CHAPTER 6 CAPACITANCE AND INDUCTANCE THE LEARNING GOALS FOR THIS CHAPTER ARE: Know how o use circui models for inducors and capaciors o calculae volage, curren, and power Be able o calculae sored energy
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationBrown University PHYS 0060 INDUCTANCE
Brown Universiy PHYS 6 Physics Deparmen Sudy Guide Inducance Sudy Guide INTODUCTION INDUCTANCE Anyone who has ever grabbed an auomobile sparkplug wire a he wrong place, wih he engine running, has an appreciaion
More informationUMR EMC Laboratory UMR EMC Laboratory Technical Report: TR
UMR EMC Laboraory UMR EMC Laboraory Dep. of Elecrical & Compuer Engineering 870 Miner Circle Universiy of Missouri Rolla Rolla, MO 654090040 UMR EMC Laboraory Technical Repor: TR0800 Effec of Delay
More informationPRM and VTM Parallel Array Operation
APPLICATION NOTE AN:002 M and V Parallel Array Operaion Joe Aguilar VI Chip Applicaions Engineering February 2014 Conens Page Inroducion 1 HighLevel Guidelines 1 Sizing he Resisor 4 Arrays of Six or 5
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
More informationTorsion of Closed Thin Wall (CTW) Sections
9 orsion of losed hin Wall (W) Secions 9 1 Lecure 9: ORSION OF LOSED HIN WALL (W) SEIONS ALE OF ONENS Page 9.1 Inroducion..................... 9 3 9.2 losed W Secions.................. 9 3 9.3 Examples......................
More informationModule 3 Design for Strength. Version 2 ME, IIT Kharagpur
Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationFE Review Basic Circuits. William Hageman
FE eview Basic Circuis William Hageman 804 FE opics General FE 4. Elecriciy, Power, and Magneism 7 A. Elecrical fundamenals (e.g., charge, curren, volage, resisance, power, energy) B. Curren and volage
More informationA Family of Zero Current Switching Switched Capacitor DCDC Converters
A Family of Zero Curren Swiching Swiched Capacior DCDC Converers Dong Cao Deparmen of Elecrical & Compuer Engineering Michigan Sae Universiy Eas Lansing, MI 48824, USA caodong@msu.edu Fang Zheng Peng
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More information6.003: Signals and Systems
6.003: Signals and Sysems Signals and Sysems Sepember 8, 2011 1 6.003: Signals and Sysems Today s handous: Single package conaining Slides for Lecure 1 Subjec Informaion & Calendar Lecurer: Insrucors:
More informationSignal Rectification
9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, halfwae and fullwae. Le s firs consider he ideal
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationThyristor Based Speed Control Techniques of DC Motor: A Comparative Analysis
Inernaional Journal of Scienific and Research Publicaions, Volume 2, Issue 6, June 2012 1 Thyrisor Based Speed Conrol Techniques of DC Moor: A Comparaive Analysis Rohi Gupa, Ruchika Lamba, Subhransu Padhee
More informationKinematics in 1D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.
Chaper 2 Kinemaics in 1D From Problems and Soluions in Inroducory Mechanics (Draf ersion, Augus 2014) Daid Morin, morin@physics.harard.edu As menioned in he preface, his book should no be hough of as
More informationModule 4. Singlephase AC Circuits. Version 2 EE IIT, Kharagpur
Module Singlephase AC Circuis Version EE, Kharagpur Lesson Generaion of Sinusoidal Volage Wavefor (AC) and Soe Fundaenal Conceps Version EE, Kharagpur n his lesson, firsly, how a sinusoidal wavefor (ac)
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationt t t Numerically, this is an extension of the basic definition of the average for a discrete
Average and alues of a Periodic Waveform: (Nofziger, 8) Begin by defining he average value of any imevarying funcion over a ime inerval as he inegral of he funcion over his ime inerval, divided by : f
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationSection 5.1 The Unit Circle
Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P
More information