ELEC 372 LECTURE NOTES, WEEK 11 Dr. Amir G. Aghdam Concordia University
|
|
- June Allison
- 5 years ago
- Views:
Transcription
1 ELEC 37 LECTURE NOTES, WEE Dr Amir Aghdam Cncrdia Univrity Part f th nt ar adaptd frm th matrial in th fllwing rfrnc: Mdrn Cntrl Sytm by Richard C Drf and Rbrt H Bihp, Prntic Hall Fdback Cntrl f Dynamic Sytm by n F Franklin, J David Pwll and Abba Emami-Naini, Prntic Hall Autmatic Cntrl Sytm by Farid lnaraghi and Bnamin C u, Jhn Wily & Sn, Inc, Stability analyi in frquncy dmain cnt d - Fr tability analyi it i vry imprtant t nt that th cld-lp pl ar th zr f and th pn-lp pl ar th pl f r - T u th principl f th argumnt, w nd t dfin a cld path that cntain all pint in th RHP W dfin th fllwing cld path a th Nyquit path r Nyquit cntur: Im{} -plan Radiur r R{} - Nw, upp that Z i th numbr f zr f inid untabl cld-lp pl and P i th numbr f pl f inid inc th pl f and ar qual, thi i in fact numbr f untabl pn-lp pl In tability analyi P i knwn frm th pn-lp tranfr functin Lctur Nt Prpard by Amir Aghdam
2 and it i dird t find Z Thi can b accmplihd by finding N and uing th principl f th argumnt - W dfin N,, t b th numbr f ncirclmnt f th rigin by th imag f undr - Uing a impl hift in th -plan, it can b aily n that N,, N,, In thr wrd, th numbr f ncirclmnt f th rigin in th -plan i qual t th numbr f ncirclmnt f th pint -, in th -plan Th pint -, i uually rfrrd t a th critical pint - Fr th tability f th cld-lp ytm w mut hav Z r N P - If th pn-lp tranfr functin i tabl, w hav P and thn fr th tability f th cld-lp ytm w mut hav N - It i t b ntd that th imag f th imaginary axi in th Nyquit path undr i th Nyquit diagram f - Th Nyquit tability critrin tat that a cld-lp ytm with ngativ fdback i tabl if and nly if th numbr f cuntrclckwi ncirclmnt f th pint -, in th -plan i qual t th numbr f pl f in th RHP, whr rprnt th pn-lp tranfr functin - Exampl : U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: r t yt - - Slutin: W hav: Lctur Nt Prpard by Amir Aghdam
3 3 Im{} -plan S R{} Frm th abv figur w knw that P n pn-lp pl in th RHP Nw, w hav t find th map f th Nyquit path undr Fr th prtin f th Nyquit path which i n th imaginary axi frm t, th map i in fact th Nyquit diagram f W hav: 5 9 R{ } <, 4 Im{ } <, Th prtin f th Nyquit path that i a circl > > φ r r, 9 φ 9, will b mappd int th rigin f th -plan thi i alway th ca fr trictly prpr tranfr functin Thrfr, th Nyquit map will b a fllw: Lctur Nt Prpard by Amir Aghdam
4 4 A it can b n frm th diagram, th numbr f ncirclmnt f th critical pint -, by th imag f undr i qual t zr r N,, S, frm th principl f th argumnt w will hav: Z N P Thi impli that th cld-lp ytm ha n pl in th RHP and, it i untabl Fr thi impl xampl, withut uing th Nyquit critrin, w knw that th charactritic quatin f th cld-lp ytm i S, th charactritic quatin ha a rt at and th cld-lp ytm i untabl Nyquit critrin can b vry uful fr mr cmplx ytm in gnral - If thr i a cntant gain in th pn-lp tranfr functin, th charactritic quatin will b In thi ca, fr Nyquit tability analyi, n can find th map f th Nyquit path undr imilar t th prviu ca but th critical pint will b intad f Thi can b n frm th fllwing quatin: Lctur Nt Prpard by Amir Aghdam
5 5 Thi impli that th numbr f ncirclmnt f th rigin by th imag f undr i qual t th numbr f ncirclmnt f th pint, by th imag f undr In thr wrd: N,, N,, N,, S, in gnral w will u, a th critical pint - It i t b ntd that th tability analyi uing th Nyquit critrin can b gnralizd t th nn-unity fdback ytm If th frward path tranfr functin i dntd by and th tranfr functin in th fdback path i dntd by H, th charactritic quatin will b H Thrfr, th nly diffrnc i that fr th tability analyi w mut find th imag f undr H intad f - Exampl : U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: 6 r t y t Slutin: W hav: Im{} -plan R{} Lctur Nt Prpard by Amir Aghdam
6 6 Sinc thr i n pn-lp pl inid th Nyquit path, w hav P Frm th rult f Exampl 95, w hav th fllwing Nyquit diagram map f th imaginary axi undr : Th numbr f ncirclmnt f th critical pint, in thi xampl dpnd n th valu f W will hav th fllwing ca nt that P : Fr < < r quivalntly < <, w will hav:,, Z N P Th cld - lp ytm i tabl N Fr < < r quivalntly >, w will hav: N,, Z N P Th cld - lp ytm ha untabl pl 3 Fr < < r quivalntly <, w will hav: N,, Z N P Th cld - lp ytm ha untabl pl Lctur Nt Prpard by Amir Aghdam
7 7 4 Fr < < r quivalntly < <, w will hav:,, Z N P Th cld - lp ytm i tabl N - Opn-lp pl n th imaginary axi: Sinc th Nyquit path mut nt pa thrugh any pl and zr f th charactritic quatin, if th pn-lp tranfr functin ha pl n th imaginary axi, w will hav t u mall micircl a hwn in th fllwing figur t g arund th pl Im{} -plan R{} - Exampl 3: U th Nyquit critrin t acrtain th tability f th fllwing cld-lp ytm: r t y t - - Slutin: W will u th fllwing Nyquit path: Lctur Nt Prpard by Amir Aghdam
8 Lctur Nt Prpard by Amir Aghdam 8 Th Nyquit diagram can b btaind a fllw: 9 << 8 >> > < > <, } Im{, } R{ 4 Sinc th micircl i vry cl t th pl, it will b mappd int infinity but w nd t find th hap f th map Fr thi purp, w can ch thr pint A, B and C a hwn in th figur and find th angl f th imag f th pint Aum that th radiu f th mall micircl i qual t Dfin A, B and C a th imag f A, B and C, rpctivly W will hav: 9 : : 9 : M C C M B B M A A Im{} R{} -plan - A B C
9 9 W knw that th imag f a curv in th cmplx plan undr a ratinal functin i a cnfrmal map Thi man that th angl f th -plan cntur will b rtaind in th -plan On can u thi rult t implify th prc f finding th imag f th crnr f th Nyquit path Fr xampl, aum that uing th tchniqu givn fr drawing th Nyquit diagram, aum that th imag f th imaginary axi undr th functin i btaind If w mv frm tward n th imaginary axi, w will hav t turn 9 dgr t th lft at th C pint W hav xactly th am thing in th imag f th Nyquit path undr at th C pint W will nw u th Nyquit critrin fr th tability analyi f Exampl 3 Sinc thr ar n pl f th pn-lp tranfr functin inid th Nyquit path, w hav P W will hav th fllwing tw ca: Fr < < r quivalntly >, w will hav:,, Z N P Th cld - lp ytm i tabl N Fr < < r quivalntly <, w will hav: Lctur Nt Prpard by Amir Aghdam
10 N,, Z N P Th cld - lp ytm ha untabl pl Cnditinal tability - Cnidr th fllwing cld-lp ytm: φ C r t y t - whr C rprnt th nminal pn-lp tranfr functin fr and φ - Th trm φ can rprnt th mdling rrr f th prc Aum that th nminal cld-lp ytm crrpnding t and φ i tabl - Fr φ, th largt and mallt valu f which rult in a tabl cldlp ytm ar calld upward gain margin and dwnward gain margin, rpctivly - Fr, th largt valu f φ which rult in a tabl cld-lp ytm i calld pha margin - Th pha margin and gain margin ar indicatd in th fllwing Nyquit diagram: R C plan φ - d d Im Lctur Nt Prpard by Amir Aghdam
11 Aum that th cld-lp ytm i tabl Frm thi figur w that rtating th Nyquit plt by any angl l than φ will nt chang th numbr f ncirclmnt f th critical pint Thi impli that th pha margin i qual t φ On th thr hand, if th pn-lp tranfr functin C i multiplid by any valu gratr than d and l than, th numbr f d ncirclmnt f th critical pint will nt chang Thi man that th upward gain margin i qual t gain margin i qual t d < < d and, d r lg d in th db cal and th dwnward d < < d d r lg d in th db cal Nt that - A a dignr, w want th pha margin and upward gain margin t b a larg a pibl and dwnward gain margin t b a mall a pibl t mak ur that th cld-lp ytm will rmain tabl in th prnc f uncrtainty in th plant mdl - A ytm with a nnzr dwnward gain margin and finit upward gain margin i calld a cnditinally tabl ytm Uually fr many practical ytm th dwnward gain margin i qual t and th trm gain margin i rfrrd t th upward gain margin - Pha margin and upward gain margin ar dntd by PM and M, rpctivly - PM and M rprnt th rlativ pitin f th Nyquit plt with rpct t th critical pint and ar ud a quantitativ maur fr rlativ tability In thr wrd, PM rprnt th angl by which th Nyquit diagram huld rtat, uch that th Nyquit plt pa thrugh th critical pint M rprnt th gain by which th Nyquit diagram huld b multiplid uch that th Nyquit plt pa thrugh th critical pint Lctur Nt Prpard by Amir Aghdam
6. Negative Feedback in Single- Transistor Circuits
Lctur 8: Intrductin t lctrnic analg circuit 36--366 6. Ngativ Fdback in Singl- Tranitr ircuit ugn Paprn, 2008 Our aim i t tudy t ffct f ngativ fdback n t mall-ignal gain and t mall-ignal input and utput
More information[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then
SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd
More informationLECTURE 5 Guassian Wave Packet
LECTURE 5 Guassian Wav Pact 1.5 Eampl f a guassian shap fr dscribing a wav pact Elctrn Pact ψ Guassian Assumptin Apprimatin ψ As w hav sn in QM th wav functin is ftn rprsntd as a Furir transfrm r sris.
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationTopic 5: Discrete-Time Fourier Transform (DTFT)
ELEC36: Signals And Systms Tpic 5: Discrt-Tim Furir Transfrm (DTFT) Dr. Aishy Amr Cncrdia Univrsity Elctrical and Cmputr Enginring DT Furir Transfrm Ovrviw f Furir mthds DT Furir Transfrm f Pridic Signals
More informationEven/Odd Mode Analysis of the Wilkinson Divider
//9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Evn/Odd Md Analyi f th Wilkinn Dividr Cnidr a matchd Wilkinn pwr dividr, with a urc at prt : Prt Prt Prt T implify thi chmatic, w rmv th grund plan, which
More informationChapter 8. Root Locus Techniques
Chapter 8 Rt Lcu Technique Intrductin Sytem perfrmance and tability dt determined dby cled-lp l ple Typical cled-lp feedback cntrl ytem G Open-lp TF KG H Zer -, - Ple 0, -, -4 K 4 Lcatin f ple eaily fund
More informationModern Physics. Unit 5: Schrödinger s Equation and the Hydrogen Atom Lecture 5.6: Energy Eigenvalues of Schrödinger s Equation for the Hydrogen Atom
Mdrn Physics Unit 5: Schrödingr s Equatin and th Hydrgn Atm Lctur 5.6: Enrgy Eignvalus f Schrödingr s Equatin fr th Hydrgn Atm Rn Rifnbrgr Prfssr f Physics Purdu Univrsity 1 Th allwd nrgis E cm frm th
More informationAnother Explanation of the Cosmological Redshift. April 6, 2010.
Anthr Explanatin f th Csmlgical Rdshift April 6, 010. Jsé Francisc García Juliá C/ Dr. Marc Mrncian, 65, 5. 4605 Valncia (Spain) E-mail: js.garcia@dival.s h lss f nrgy f th phtn with th tim by missin f
More informationENGR 7181 LECTURE NOTES WEEK 5 Dr. Amir G. Aghdam Concordia University
ENGR 78 LETURE NOTES WEEK 5 r. mir G. dam onordia Univrity ilinar Tranformation - W will now introdu anotr mtod of tranformation from -plan to t - plan and vi vra. - Ti tranformation i bad on t trapoidal
More informationEE 119 Homework 6 Solution
EE 9 Hmwrk 6 Slutin Prr: J Bkr TA: Xi Lu Slutin: (a) Th angular magniicatin a tlcp i m / th cal lngth th bjctiv ln i m 4 45 80cm (b) Th clar aprtur th xit pupil i 35 mm Th ditanc btwn th bjctiv ln and
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationChapter 2 Linear Waveshaping: High-pass Circuits
Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Chaptr 2 Linar Wavshaping: High-pass Circuits. A ramp shwn in Fig.2p. is applid t a high-pass circuit. Draw t scal th utput wavfrm
More informationChapter 9 Compressible Flow 667
Chapter 9 Cmpreible Flw 667 9.57 Air flw frm a tank thrugh a nzzle int the tandard atmphere, a in Fig. P9.57. A nrmal hck tand in the exit f the nzzle, a hwn. Etimate (a) the tank preure; and (b) the ma
More informationLecture 26: Quadrature (90º) Hybrid.
Whits, EE 48/58 Lctur 26 Pag f Lctur 26: Quadratur (9º) Hybrid. Back in Lctur 23, w bgan ur discussin f dividrs and cuplrs by cnsidring imprtant gnral prprtis f thrand fur-prt ntwrks. This was fllwd by
More informationChapter 10 Time-Domain Analysis and Design of Control Systems
ME 43 Sytm Dynamic & Control Sction 0-5: Stady Stat Error and Sytm Typ Chaptr 0 Tim-Domain Analyi and Dign of Control Sytm 0.5 STEADY STATE ERRORS AND SYSTEM TYPES A. Bazoun Stady-tat rror contitut an
More informationLecture 13 - Boost DC-DC Converters. Step-Up or Boost converters deliver DC power from a lower voltage DC level (V d ) to a higher load voltage V o.
ecture 13 - Bt C-C Cnverter Pwer Electrnic Step-Up r Bt cnverter eliver C pwer frm a lwer vltage C level ( ) t a higher la vltage. i i i + v i c T C (a) + R (a) v 0 0 i 0 R1 t n t ff + t T i n T t ff =
More informationUser s Guide. Electronic Crossover Network. XM66 Variable Frequency. XM9 24 db/octave. XM16 48 db/octave. XM44 24/48 db/octave. XM26 24 db/octave Tube
U Guid Elctnic Cv Ntwk XM66 Vaiabl Fquncy XM9 24 db/ctav XM16 48 db/ctav XM44 24/48 db/ctav XM26 24 db/ctav Tub XM46 24 db/ctav Paiv Lin Lvl XM126 24 db/ctav Tub Machand Elctnic Inc. Rcht, NY (585) 423
More informationRevision: August 21, E Main Suite D Pullman, WA (509) Voice and Fax
2.7.1: Sinusidal signals, cmplx xpnntials, and phasrs Rvisin: ugust 21, 2010 215 E Main Suit D ullman, W 99163 (509 334 6306 ic and Fax Ovrviw In this mdul, w will rviw prprtis f sinusidal functins and
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationOutline. Heat Exchangers. Heat Exchangers. Compact Heat Exchangers. Compact Heat Exchangers II. Heat Exchangers April 18, ME 375 Heat Transfer 1
Hat Exangr April 8, 007 Hat Exangr Larry artt Manial Engrg 375 Hat ranfr April 8, 007 Outl Bai ida f at xangr Ovrall at tranfr ffiint Lg-man tmpratur diffrn mtd Efftivn NU mtd ratial nidratin Hat Exangr
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationAcid Base Reactions. Acid Base Reactions. Acid Base Reactions. Chemical Reactions and Equations. Chemical Reactions and Equations
Chmial Ratins and Equatins Hwitt/Lyns/Suhki/Yh Cnptual Intgratd Sin During a hmial ratin, n r mr nw mpunds ar frmd as a rsult f th rarrangmnt f atms. Chaptr 13 CHEMICAL REACTIONS Ratants Prduts Chmial
More informationCalculation of electromotive force induced by the slot harmonics and parameters of the linear generator
Calculation of lctromotiv forc inducd by th lot harmonic and paramtr of th linar gnrator (*)Hui-juan IU (**)Yi-huang ZHANG (*)School of Elctrical Enginring, Bijing Jiaotong Univrity, Bijing,China 8++58483,
More informationLectur 22. RF and Microwave Circuit Design Γ-Plane and Smith Chart Analysis. ECE 303 Fall 2005 Farhan Rana Cornell University
ctur RF ad Micrwav Circuit Dig -Pla ad Smith Chart Aalyi I thi lctur yu will lar: -pla ad Smith Chart Stub tuig Quartr-Wav trafrmr ECE 33 Fall 5 Farha Raa Crll Uivrity V V Impdac Trafrmati i Tramii i ω
More informationMATHEMATICS FOR MANAGEMENT BBMP1103
Objctivs: TOPIC : EXPONENTIAL AND LOGARITHM FUNCTIONS. Idntif pnntils nd lgrithmic functins. Idntif th grph f n pnntil nd lgrithmic functins. Clcult qutins using prprtis f pnntils. Clcult qutins using
More informationName Student ID. A student uses a voltmeter to measure the electric potential difference across the three boxes.
Name Student ID II. [25 pt] Thi quetin cnit f tw unrelated part. Part 1. In the circuit belw, bulb 1-5 are identical, and the batterie are identical and ideal. Bxe,, and cntain unknwn arrangement f linear
More informationLecture 2a. Crystal Growth (cont d) ECE723
Lctur 2a rystal Grwth (cnt d) 1 Distributin f Dpants As a crystal is pulld frm th mlt, th dping cncntratin incrpratd int th crystal (slid) is usually diffrnt frm th dping cncntratin f th mlt (liquid) at
More informationA Quadratic Serendipity Plane Stress Rectangular Element
MAE 323: Chaptr 5 Putting It All Togthr A Quadratic Srndipity Plan Str Rctangular Elmnt In Chaptr 2, w larnd two diffrnt nrgy-bad mthod of: 1. Turning diffrntial quation into intgral (or nrgy) quation
More informationChapter 9. Design via Root Locus
Chapter 9 Deign via Rt Lcu Intrductin Sytem perfrmance pecificatin requirement imped n the cntrl ytem Stability Tranient repne requirement: maximum verht, ettling time Steady-tate requirement :.. errr
More informationFrequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser
Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99 Idal Filrs W wan sudy Hω funcins which prvid frquncy slciviy such as: Lw Pass High Pass Band Pass Hwvr, w will lk a idal filring, ha is,
More informationLecture 4: Parsing. Administrivia
Adminitrivia Lctur 4: Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationRoot locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s
The given TFs are: 1 1() s = s s + 1 s + G p, () s ( )( ) >> Gp1=tf(1,ply([0-1 -])) Transfer functin: 1 ----------------- s^ + s^ + s Rt lcus G 1 = p ( s + 0.8 + j)( s + 0.8 j) >> Gp=tf(1,ply([-0.8-*i
More informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationELEG 413 Lecture #6. Mark Mirotznik, Ph.D. Professor The University of Delaware
LG 43 Lctur #6 Mrk Mirtnik, Ph.D. Prfssr Th Univrsity f Dlwr mil: mirtni@c.udl.du Wv Prpgtin nd Plritin TM: Trnsvrs lctrmgntic Wvs A md is prticulr fild cnfigurtin. Fr givn lctrmgntic bundry vlu prblm,
More informationIntroduction to Smith Charts
Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary
More informationLesson #15. Section BME 373 Electronics II J.Schesser
Feedack and Ocillatr Len # Tranient and Frequency Repne Sectin 9.6- BME 373 Electrnic II J.Scheer 78 Cled-Lp Gain in the Frequency Dmain ume that th the pen-lp gain, and the eedack, β are unctin requency
More informationA Brief and Elementary Note on Redshift. May 26, 2010.
A Brif and Elmntary Nt n Rdshift May 26, 2010. Jsé Francisc García Juliá C/ Dr. Marc Mrncian, 65, 5. 46025 Valncia (Spain) E-mail: js.garcia@dival.s Abstract A rasnabl xplanatin f bth rdshifts: csmlgical
More informationSER/BER in a Fading Channel
SER/BER in a Fading Channl Major points for a fading channl: * SNR is a R.V. or R.P. * SER(BER) dpnds on th SNR conditional SER(BER). * Two prformanc masurs: outag probability and avrag SER(BER). * Ovrall,
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationSource code. where each α ij is a terminal or nonterminal symbol. We say that. α 1 α m 1 Bα m+1 α n α 1 α m 1 β 1 β p α m+1 α n
Adminitrivia Lctur : Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationCHAPTER 3 INEQUALITIES. Copyright -The Institute of Chartered Accountants of India
CHAPTER 3 INEQUALITIES Cpyright -The Institute f Chartered Accuntants f India INEQUALITIES LEARNING OBJECTIVES One f the widely used decisin making prblems, nwadays, is t decide n the ptimal mix f scarce
More information5 Curl-free fields and electrostatic potential
5 Curl-fr filds and lctrstatic tntial Mathmaticall, w can gnrat a curl-fr vctr fild E(,, ) as E = ( V, V, V ), b taking th gradint f an scalar functin V (r) =V (,, ). Th gradint f V (,, ) is dfind t b
More informationModule 6. Actuators. Version 2 EE IIT, Kharagpur 1
Mul 6 ctuatr Vrin EE T, haragpur 1 Ln 30 Pnumatic Cntrl Sytm Vrin EE T, haragpur ntructinal Objctiv t th n thi ln, th tunt hul b abl t Sktch th chmatic iagram a pnumatic prprtinal cntrllr. pply linariatin
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More informationChapter 7 Control Systems Design by the Root Locus Method
haptr 7 ntrl Sytm Dgn by th t Lu Mthd 7. Intrdutn! Prfrman Spfatn: h rqurmnt mpd n th ntrl ytm ar plld ut a prfrman pfatn, whh gnrally rlat t auray, rlatv tablty, and pd f rpn.! Sytm mpnatn: Sttng th gan
More informationLecture 27: The 180º Hybrid.
Whits, EE 48/58 Lctur 7 Pag f 0 Lctur 7: Th 80º Hybrid. Th scnd rciprcal dirctinal cuplr w will discuss is th 80º hybrid. As th nam implis, th utputs frm such a dvic can b 80º ut f phas. Thr ar tw primary
More information1. Introduction: A Mixing Problem
CHAPTER 7 Laplace Tranfrm. Intrductin: A Mixing Prblem Example. Initially, kg f alt are dilved in L f water in a tank. The tank ha tw input valve, A and B, and ne exit valve C. At time t =, valve A i pened,
More informationPre-Calculus Individual Test 2017 February Regional
The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted
More informationLHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationChapter 33 Gauss s Law
Chaptr 33 Gauss s Law 33 Gauss s Law Whn askd t find th lctric flux thrugh a clsd surfac du t a spcifid nn-trivial charg distributin, flks all t ftn try th immnsly cmplicatd apprach f finding th lctric
More informationCoulomb s Law Worksheet Solutions
PHLYZIS ulb Law Wrkht Slutin. w charg phr 0 c apart attract ach thr with a frc f 3.0 0 6 N. What frc rult fr ach f th fllwing chang, cnir paratly? a Bth charg ar ubl an th itanc rain th a. b An uncharg,
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More information. This is made to keep the kinetic energy at outlet a minimum.
Runnr Francis Turbin Th shap th blads a Francis runnr is cmplx. Th xact shap dpnds n its spciic spd. It is bvius rm th quatin spciic spd (Eq.5.8) that highr spciic spd mans lwr had. This rquirs that th
More informationPhysics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1
Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationSupport-Vector Machines
Supprt-Vectr Machines Intrductin Supprt vectr machine is a linear machine with sme very nice prperties. Haykin chapter 6. See Alpaydin chapter 13 fr similar cntent. Nte: Part f this lecture drew material
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationWhen a substance heats up (absorbs heat) it is an endothermic reaction with a (+)q
Chemistry Ntes Lecture 15 [st] 3/6/09 IMPORTANT NOTES: -( We finished using the lecture slides frm lecture 14) -In class the challenge prblem was passed ut, it is due Tuesday at :00 P.M. SHARP, :01 is
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationLyapunov Stability Stability of Equilibrium Points
Lyapunv Stability Stability f Equilibrium Pints 1. Stability f Equilibrium Pints - Definitins In this sectin we cnsider n-th rder nnlinear time varying cntinuus time (C) systems f the frm x = f ( t, x),
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationExercises for lectures 7 Steady state, tracking and disturbance rejection
Exrc for lctur 7 Stady tat, tracng and dturbanc rjcton Martn Hromčí Automatc control 06-3-7 Frquncy rpon drvaton Automatcé řízní - Kybrnta a robota W lad a nuodal nput gnal to th nput of th ytm, gvn by
More information3. Classify the following Numbers (Counting (natural), Whole, Integers, Rational, Irrational)
After yu cmplete each cncept give yurself a rating 1. 15 5 2 (5 3) 2. 2 4-8 (2 5) 3. Classify the fllwing Numbers (Cunting (natural), Whle, Integers, Ratinal, Irratinal) a. 7 b. 2 3 c. 2 4. Are negative
More informationFundamental Concepts in Structural Plasticity
Lecture Fundamental Cncepts in Structural Plasticit Prblem -: Stress ield cnditin Cnsider the plane stress ield cnditin in the principal crdinate sstem, a) Calculate the maximum difference between the
More informationECE 5318/6352 Antenna Engineering. Spring 2006 Dr. Stuart Long. Chapter 6. Part 7 Schelkunoff s Polynomial
ECE 538/635 Antenna Engineering Spring 006 Dr. Stuart Lng Chapter 6 Part 7 Schelkunff s Plynmial 7 Schelkunff s Plynmial Representatin (fr discrete arrays) AF( ψ ) N n 0 A n e jnψ N number f elements in
More informationHomology groups of disks with holes
Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.
More informationTHE ALIGNMENT OF A SPHERICAL NEAR-FIELD ROTATOR USING ELECTRICAL MEASUREMENTS
THE ALIGNMENT OF A SPHERICAL NEAR-FIELD ROTATOR USING ELECTRICAL MEASUREMENTS ABSTRACT Th mchanical rotator mut b corrctly alignd and th prob placd in th propr location whn prforming phrical nar-fild maurmnt.
More informationChapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms
Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical
More informationChapter 3. Electric Flux Density, Gauss s Law and Divergence
Chapter 3. Electric Flu Denity, Gau aw and Diergence Hayt; 9/7/009; 3-1 3.1 Electric Flu Denity Faraday Eperiment Cncentric phere filled with dielectric material. + i gien t the inner phere. - i induced
More informationUniversity Chemistry Quiz /04/21 1. (10%) Consider the oxidation of ammonia:
University Chemistry Quiz 3 2015/04/21 1. (10%) Cnsider the xidatin f ammnia: 4NH 3 (g) + 3O 2 (g) 2N 2 (g) + 6H 2 O(l) (a) Calculate the ΔG fr the reactin. (b) If this reactin were used in a fuel cell,
More informationN J of oscillators in the three lowest quantum
. a) Calculat th fractinal numbr f scillatrs in th thr lwst quantum stats (j,,,) fr fr and Sl: ( ) ( ) ( ) ( ) ( ).6.98. fr usth sam apprach fr fr j fr frm q. b) .) a) Fr a systm f lcalizd distinguishabl
More informationDINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2009/2010 Paper 1 Assessing Units 1 & 2. Time allowed - 1 hour 30
INGWLL EMY Mathematics Higher Prelim Eaminatin 009/00 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationSolutions. Digital Control Systems ( ) 120 minutes examination time + 15 minutes reading time at the beginning of the exam
BSc - Sample Examination Digital Control Sytem (5-588-) Prof. L. Guzzella Solution Exam Duration: Number of Quetion: Rating: Permitted aid: minute examination time + 5 minute reading time at the beginning
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationChapter 8: Electron Configurations and Periodicity
Elctron Spin & th Pauli Exclusion Principl Chaptr 8: Elctron Configurations and Priodicity 3 quantum numbrs (n, l, ml) dfin th nrgy, siz, shap, and spatial orintation of ach atomic orbital. To xplain how
More informationDINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2.
INGWLL EMY Mathematics Higher Prelim Eaminatin 00/0 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0
More informationPlan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations
STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.
More informationENGR 7181 LECTURE NOTES WEEK 3 Dr. Amir G. Aghdam Concordia University
ENGR 78 LECTURE NOTES WEEK Dr. ir G. ga Concoria Univrity DT Equivalnt Tranfr Function for SSO Syt - So far w av tui DT quivalnt tat pac ol uing tp-invariant tranforation. n t ca of SSO yt on can u t following
More informationAppendices on the Accompanying CD
APPENDIX 4B Andis n th Amanyg CD TANSFE FUNCTIONS IN CONTINUOUS CONDUCTION MODE (CCM In this st, w will driv th transfr funt v / d fr th thr nvrtrs ratg CCM 4B- Buk Cnvrtrs Frm Fig. 4-7, th small signal
More informationDead-beat controller design
J. Hetthéssy, A. Barta, R. Bars: Dead beat cntrller design Nvember, 4 Dead-beat cntrller design In sampled data cntrl systems the cntrller is realised by an intelligent device, typically by a PLC (Prgrammable
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationMultiplierless Modules for Forward and Backward Integer Wavelet Transform
Multiplirl Mdul fr Frward and Backward Intgr Wavlt Tranfrm Vail Klv Abtract: Thi articl i abut nw architctur f a intgr DWT with rprgrammabl lgic. It i bad n cnd gnratin f wavlt with a rducd f numbr f pratin.
More informationSolution to HW14 Fall-2002
Slutin t HW14 Fall-2002 CJ5 10.CQ.003. REASONING AND SOLUTION Figures 10.11 and 10.14 shw the velcity and the acceleratin, respectively, the shadw a ball that underges unirm circular mtin. The shadw underges
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationSolutions to Supplementary Problems
Solution to Supplmntary Problm Chaptr 5 Solution 5. Failur of th tiff clay occur, hn th ffctiv prur at th bottom of th layr bcom ro. Initially Total ovrburn prur at X : = 9 5 + = 7 kn/m Por atr prur at
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 4: Mdel checing fr ODE mdels In Petre Department f IT, Åb Aademi http://www.users.ab.fi/ipetre/cmpmd/ Cntent Stichimetric matrix Calculating the mass cnservatin relatins
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationCHEM 2400/2480. Lecture 19
Lecture 19 Metal In Indicatr - a cmpund whse clur changes when it binds t a metal in - t be useful, it must bind the metal less strngly than EDTA e.g. titratin f Mg 2+ with EDTA using erichrme black T
More informationCharacteristic Equations and Boundary Conditions
Charatriti Equation and Boundary Condition Øytin Li-Svndn, Viggo H. Hantn, & Andrw MMurry Novmbr 4, Introdution On of th mot diffiult problm on i onfrontd with In numrial modlling oftn li in tting th boundary
More information