PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII  SPETO pod patronatem. Summary


 Heather Hodges
 3 years ago
 Views:
Transcription
1 PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8  TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC XIII  SPETO pod patonatem PN STBILITY TESTING OF DC CICUITS USING ITIONL METHODS Summay The pape compises consideations concening some thought souces of cicuit theoy. It is shown that some etun to the categoies of analytical mechanics and nonequilibium pocess themodynamics can give a hand with the solution of such poblems as the stability of DC cicuits with nonecipocal elements. t fist sight, analysis of cicuits with contolled souces seems to be the closed pat of cicuit theoy. Howeve, putting the accustoming pocedues into pactice can lead to the paadoxical situations as shown fo example in "school" poblem with the independence of tansfe function on the input polaity of opeational amplifie. In ou pape, the oots of some paadoxical situations in cicuit theoy ae investigated in moe details. We investigate two estictions of Newton's mechanics and analytical dynamics that have also influenced cicuit theoy.
2 . INTODUCTION nalysis of cicuits with contolled souces belongs to the wellknown pats of cicuit theoy. Howeve, putting the accustoming pocedues into pactice can lead to the paadoxical situations as shown in following "school" example. Detemine the voltage gain F = 2 / accoding to Fig.. Conside infinite input amplifie esistance and zeo output esistance. Gain is abitay nonzeo eal numbe. Utilization of abitay method based on Kichhoff's laws and Ohm's law leads to the esult K 2 = F., F =, 0, (.) K2 2 whee K = and K2 =. 2 2 It is emakable that lim F = lim F = 2. (.2) Equation (.2) imposes the idea that the function of amplifie in Fig. does not depend on the polaity of input banch of Opmp. Howeve, this conclusion does not coespond to the pactical expeiences. Paadox oigin inside the thought system can point at cetain estictions of this system. Fo example, the wellknown antique paadox about the chilles and totoise has aisen in consequence of the fact that the late knowledge of the aea of infinitesimal calculus could not be included to the consideations. Next paadox  when the pile of stones stops to be the pile taking away stone by stone  stops be the paadox consideing some pinciples of fuzzy sets. Ou "Opmp" paadox is caused due to peculia position of stability in the cicuit theoy. It is conditioned histoically because this doctine has been unde the influence of the Newton's mechanics and analytical dynamics. Equation (.) namely does not compise infomation if the equilibium point is stable o unstable. This equation only ensues that Kichhoff's laws and coupling condition 2 = will be complied. Thee is no sense in definition of voltage gain F in case of unstable equilibium point. In this way, the paadox (.2) is explained. It emains to investigate the stability conditions of ou DC solution. s usual in cicuit theoy, the simple model in Fig.2 can be compiled using equations those have led to the esult (.). The sign of open loop gain detemines the stability, e.g. 2 K. 2 < 0 o <. (.3) Let us exploe the oots of the paadox (.2) in moe details. We investigate two estictions of Newton's mechanics and analytical dynamics those have also influenced cicuit theoy. We concentate ou attention on the contolled souces fom this athe unusual point of view. 2. NEWTON'S SPECTS IN THE CICUIT THEOY It has been poved duing 50th yeas that Kichhoff's voltage law (KL) in the cicuit theoy K 2 coesponds to the 2nd Newton's postulate of the classical physics and consequently to the basic equation Fig.2 of vaiational calculus, i.e. Eule equation []. Choice of electical chage q as genealized Fig. K
3 coodinates has been the assumption of this analogy. Then cuents i = q& have coesponded to genealized speeds, electical voltages to genealized foces and magnetic fluxes to momentum. In this way, the ingenious tools of analytical dynamics could be used mainly fo the analysis methods based on the KL, e.g. fo the littleknown method of loop chages and fo the method of loop cuents. It is less known that this pinciple is also utilized to the cicuit analysis using nodal analysis method (NM). Utilizing the dual popeties of functions and cofunctions those descibe electical, magnetic and dissipative fields of used elements, the cicuit Lagange function can be constucted as a function of nodal voltages and thei deivatives L( v, v& ) = T, whee T () v & and () v ae time deivatives of electical field coenegy of chaged capacitos and the magnetic field enegy of inductos, v is the vecto of nodal voltages. In the aea of NM, the Hamilton vaiational pinciple can be fomulated as follows: The tajectoy connecting points v( t0 ) and v( t) that system chooses fo its motion, minimizes the time integal of system Lagange function t (2.) t 0 t δ L( v, v & )dt thus L dt = 0, (2.2) t0 whee δ means the symbol of vaiation of the whole tajectoy between points v( t ) and v( t ) aound the actual tajectoy. Eule equation is then equivalent to equation (2.2) d L L 0 & =. (2.3) dt v v The eal tajectoy v () t is the solution of Eule equation fo given initial conditions. The eal tajectoy is the extemale of vaiational task (2.2). The cicuit compising also esistos and excited by cuent souces can be descibed using extended Eule equation d L L d & I = dt v& v dt v (2.4) whee I is the vecto of cuent souces connected to cicuit nodes, is the ayleigh dissipative cofunction that maps the enegy dissipation on esistos. If the cicuit is chaacteized by conductance matix G, then T T v = I dv~ = G v~ dv~ (2.5) () ( ) Γ is the linie integal along the contou Γ fom the oigin of coodinates to the point v. In case of cicuits compising linea ecipocal esistos, the ayleigh function is as follows: () v v T = G v. 2 If the cicuit contains the elements those set coupling of type f ( v, v &) = 0 (i.e. contolled souces, contolled switches etc.), this condition will be included to the Lagange system function using the method of indefinite coefficients λ: L = Lλ. f. Γ 0
4 The physical intepetation of coefficient λ is the system esponse to the coupling f. This is the genealized foce that the system has to expend to keep condition f. Moe infomation s can be found in the classical wok [2]. If the cicuit consists only of the esistive elements those ae coupled by condition f, the simplified fom of extended Eule equation may be used: ( λf ) = I. (2.6) v In case of linea ecipocal esistos we can wite ( λ f ) G. v = I. (2.7) v Neglecting coupling element, the equation (2.7) epesents the basic equation of NM. We have seen that algoithms commonly used in the cicuit theoy ae stongly influenced by the methods developed duing past centuies in the aea of classical physics. We have paticulaly followed the influence of fist two Newton's postulates. Howeve, the cicuit theoy has own peculiaities concening 3d Newton's postulate of eaction. 3. POSTULTE OF ECTION s mentioned in the opening example, the infomation about the cicuit stability does not follow automatically the analysis method. This infomation must be obtained independently of the method additionally using special test. It looks to be something infeio not included to the main theoy. This sepaation of the stability test and the est analysis is typical not only fo the cicuit theoy but also fo many othe banches like contol theoy. This is connected with the fact that these banches have assumed the conception of fist two Newton's laws of classical mechanics but they have modified the thid Newton's postulate of eaction accoding to own needs. Contol theoy has modified this postulate to the fom "eaction is equal to zeo". This fomulation has enabled to decompose oiginally twoway couplings in the system to the unilateal ones and to intoduce the system as oientated "block" diagam (the sample of such diagam consisting exclusively of the elements with the unilateal fowad coupling is in Fig.2). Moe infomation can be found in [3]. Cicuit theoy let the eaction chaacte to the natue of individual elements that can be eithe ecipocal (action = eaction) o nonecipocal (action eaction). Due to modification of the postulate of eaction, vaious theoies aise. Some thei conclusions seem paadoxically fom the classical physics point of view. Fo instance, the pepeetum mobile can exist in the cicuit and contol theoy (oscillato etc.). It is inteesting fo ou suggestion that invalidity of the law of enegy consevation is one of the implications of the 3d Newton's postulate modification. This fact is pactically pojected to the possibility of system instability. This possibility then follows natually the axiomatic of given theoy. Hee thee is necessay to seek explanation of why the stability test seems to be alien element in the aea of such theoy. Let us then ty to analyze ou cicuit in Fig. fom the positions mentioned in chapte 2. Fig.3 shows the same cicuit pepaed fo the NM desciption. The input voltage souce has been eplaced by the cuent souce I = G. pplying equation (2.7) yields G G G 2 2 G G I =.λ 0 λ (3.) Equilibium in the esistive netwok with the coupling condition f = = 2 0 (3.2)
5 is eached by means of compensation cuent souces affecting 2 2 both nodes and!. On the othe hand, it is known that ideal voltage amplifie ensues condition (3.2) only using its output, I i.e. affecting only node!. This contadiction has aose because of the egulation (2.7) comes out the validity of the Newton's  postulate of the equality of action and eaction. Howeve, the voltage amplifie as nonecipocal element violates this Fig.3 egulaity by its own law that "eaction is equal to zeo". Taking into account afoementioned facts, equation (2.7) can be genealized as follows ( λ f ) G. v = I K (3.3) v whee K is the matix of weight coefficients that adapts elation 2 2 between the action and eaction accoding to eality. Let us then inset infomation that ou voltage amplifie has infinite input esistance to equation (3.3). We obtain I λ G v = I λ. (3.4) Fig.4 Meaning of the undetemined coefficient λ is then the cuent supplied to the netwok by the output of amplifie. Equation (3.4) along with the condition (3.2) give diections fo computation of both nodal voltages and unknown cuent λ. Now let us investigate how to test stability of cicuits without enegy stoage elements using vaiational pinciples. 4. POBLEM OF ENEGY DISSIPTION If the system has no stoage elements, then the most wellknown citeion cannot be used to stability test because the time facto is not entiely pesented. Fo instance, this poblem is solved in [4] in espect of the algoithms fo computation of DC opeating points in the simulation pogam SPICE. The tem "potential stability" is intoduced in this wok. The pesented algoithms opeate in this way that paasitic inductances o capacitances ae added to the cicuit with the aim to each cetain dynamics. s shown in afoementioned chapte, the alone possibility of state instability is paadox as a matte of fact which is caused by the modification of the 3d Newton's postulate of eaction. s egads cicuit without inetia, this poblem is still moe complicated because alone Newton's dynamics is oiginally the doctine about consevative systems. esisto as the element dissipating enegy into heat is the alien element in the theoy based on the Newton's conception. This fact is well evident in the extended Eule's equation (2.4): the ayleigh's function is violently placed hee to expess eal esistos. esulting tajectoy of such system is not extemale of vaiational pinciple (2.2). Hamilton's pinciple is in its essence valid only in case of consevative systems. This is the sign of the peiod in which the poblems of celestial mechanics and othe phenomena wee solved. In these phenomena the system dynamics played dominant ole in compaison with the compaatively negligible dissipation of kinetic enegy. Howeve, ideal esisto as an element geneating heat has became the cental point of elatively young doctine  nonequilibium themodynamics. This doctine also epots to the tems as capacito and inducto, gives diections fo the compilation of system motion equations and it also has its own vaiational pinciples those mostly coespond to the pinciples of analytical dynamics. The nonequilibium themodynamics eveals that the integal Hamilton's pinciple cannot be efomed fo the systems with dissipative elements as esistos those themodynamic essence baffles Newton's conception.
6 It is possible to seach answes to the stability poblems of pue dissipative systems in the themodynamical vaiational pinciples. mong them, the diffeential pinciples dominate, i.e. pinciples efeing to the instantaneous system behavio in the concete tajectoy point. One of the fom of least enegy dissipation pinciple given by Glansdoff and Pigogine [5] is suitable fo the cicuit desciption using NM. This pinciple can be loosely fomulated as follows: Duing motion, system minimizes the expession v v I T Φ =. (4.) () () v in view of the possible instantaneous vaiations of nodal voltages. Function is the ayleigh's system cofunction defined accoding to (2.5). This pinciple can be fomulated in moe pactical way: Stable DC opeating point occus in the local minimum of function (4.). emak: s egads nonecipocal systems, the choice of the contou Γ of integation is impotant fo the ayleigh cofunction definition (2.5). Fig.5 Let us attempt to analyze stability of ou voltage amplifie cicuit using the function Φ. fte disconnection of outside excitation to node " accoding to Fig.3, we can only constuct ayleigh cofunction using (4.). We disconnect output of amplifie. Ou task is to seach function of cicuit with open feedback loop (see Fig.5): ( ) I( v ~ ) dv ~ =. It can be easily poved that I = whee = 2. Thus ( ) 2 2 =. If > 0 then ayleigh cofunction has local minimum in the opeating point = 0. Then < 2. Howeve, this is the elation (.3) deived in the fist chapte. 5. CONCLUSION Equation (2.3) shows that KCL and Eule's equation of vaiational calculus ae equivalent if the suitably chosen cicuit voltages ae selected as system coodinates. This fact enables to use vaiational methods fo the cicuit analysis using NM. Equation (3.3) is univesal. Fo instance, DC cicuits with nonideal Opmps those models ae given using conditions f ( v, v & i ) = 0 can be analyzed using this equation. Themodynamical stability citeion (4.) can be also used fo the stability testing of lage cicuits with nonecipocal elements as Opmps. The suitable selection of integation contou fo the constuction of ayleigh cofunction is impotant key to success. EFEENCES  2 I [] MYSLÍK,J.: nalysis of electical cicuits using vaiational methods. Electical Engineeing Jounal, No.4, 972. Czechoslovakia. [2] LNCZOS,C.: The vaiational pinciples of mechanics. Mathematical expositions, No.4, Toonto Pess, 962. [3] GLPEIN,I.I.: utomatics as onesided mechanics. Moskva 964. [4] GEEN,M.M.WILLSON,.N.: How to Identify Unstable DC Opeating Points. IEEE Tans. on CS I, vol. 39, No.0, 992, pp [5] GYMTI,I.: Nonequilibium themodynamics. Spingeelag, New Yok
7 SPŠE OŽNOĚ P.. Ing. Zdeněk Biolek Školní 60 alašská OŽNO P.., CZECH EPUBLIC OŽNO P.., CZECH EPUBLIC MILITY CDEMY BNO Depatment of Electical Engineeing K30 ssoc.pof. Ing. Dalibo Biolek, CSc. Kounicova 65, PS 3 Hoácké nám. 9/ BNO, CZECH EPUBLIC BNO, CZECH EPUBLIC
2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationSELFINDUCTANCE AND INDUCTORS
MISN0144 SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. SelfInductance L.........................................
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More informationThe transport performance evaluation system building of logistics enterprises
Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194114 Online ISSN: 213953 Pint ISSN: 2138423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationSTUDENT RESPONSE TO ANNUITY FORMULA DERIVATION
Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts
More informationIlona V. Tregub, ScD., Professor
Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation
More informationResearch on Risk Assessment of the Transformer Based on Life Cycle Cost
ntenational Jounal of Smat Gid and lean Enegy eseach on isk Assessment of the Tansfome Based on Life ycle ost Hui Zhou a, Guowei Wu a, Weiwei Pan a, Yunhe Hou b, hong Wang b * a Zhejiang Electic Powe opoation,
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationContinuous Compounding and Annualization
Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More information1.4 Phase Line and Bifurcation Diag
Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes
More informationConcept and Experiences on using a Wikibased System for Softwarerelated Seminar Papers
Concept and Expeiences on using a Wikibased System fo Softwaeelated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wthaachen.de,
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationMechanics 1: Work, Power and Kinetic Energy
Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationSoftware Engineering and Development
I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining
More informationON THE (Q, R) POLICY IN PRODUCTIONINVENTORY SYSTEMS
ON THE R POLICY IN PRODUCTIONINVENTORY SYSTEMS Saifallah Benjaafa and JoonSeok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poductioninventoy
More informationest using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.
9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationINITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS
INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in
More informationThe Supply of Loanable Funds: A Comment on the Misconception and Its Implications
JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently FieldsHat
More information2. SCALARS, VECTORS, TENSORS, AND DYADS
2. SCALARS, VECTORS, TENSORS, AND DYADS This section is a eview of the popeties of scalas, vectos, and tensos. We also intoduce the concept of a dyad, which is useful in MHD. A scala is a quantity that
More informationVISCOSITY OF BIODIESEL FUELS
VISCOSITY OF BIODIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use
More informationThe LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.
Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the
More informationAN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM
AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationExplicit, analytical solution of scaling quantum graphs. Abstract
Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 064590155, USA Email: ydabaghian@wesleyan.edu (Januay 6, 2003)
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationChapte 3 Is Gavitation A Results Of Asymmetic Coulomb Chage Inteactions? Jounal of Undegaduate Reseach èjurè Univesity of Utah è1992è, Vol. 3, No. 1, pp. 56í61. Jeæey F. Gold Depatment of Physics, Depatment
More informationDYNAMICS AND STRUCTURAL LOADING IN WIND TURBINES
DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationRelativistic Quantum Mechanics
Chapte Relativistic Quantum Mechanics In this Chapte we will addess the issue that the laws of physics must be fomulated in a fom which is Loentz invaiant, i.e., the desciption should not allow one to
More informationChapter 4: Fluid Kinematics
Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian
More informationGravitational Mechanics of the MarsPhobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning
Gavitational Mechanics of the MasPhobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationQuestions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing
M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow
More informationRisk Sensitive Portfolio Management With CoxIngersollRoss Interest Rates: the HJB Equation
Risk Sensitive Potfolio Management With CoxIngesollRoss Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More informationCloud Service Reliability: Modeling and Analysis
Cloud Sevice eliability: Modeling and Analysis YuanShun Dai * a c, Bo Yang b, Jack Dongaa a, Gewei Zhang c a Innovative Computing Laboatoy, Depatment of Electical Engineeing & Compute Science, Univesity
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More informationAn application of stochastic programming in solving capacity allocation and migration planning problem under uncertainty
An application of stochastic pogamming in solving capacity allocation and migation planning poblem unde uncetainty YinYann Chen * and HsiaoYao Fan Depatment of Industial Management, National Fomosa Univesity,
More informationPhysics 505 Homework No. 5 Solutions S51. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.
Physics 55 Homewok No. 5 s S5. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance the availability
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationSamples of conceptual and analytical/numerical questions from chap 21, C&J, 7E
CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known
More information2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationQuestions for Review. By buying bonds This period you save s, next period you get s(1+r)
MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the twopeiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume
More informationConverting knowledge Into Practice
Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading
More informationSUPPORT VECTOR MACHINE FOR BANDWIDTH ANALYSIS OF SLOTTED MICROSTRIP ANTENNA
Intenational Jounal of Compute Science, Systems Engineeing and Infomation Technology, 4(), 20, pp. 677 SUPPORT VECTOR MACHIE FOR BADWIDTH AALYSIS OF SLOTTED MICROSTRIP ATEA Venmathi A.R. & Vanitha L.
More informationApproximation Algorithms for Data Management in Networks
Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationDesign of Wind Energy System on the Building Tower Applications
ISSN(Online): 398753 ISSN (Pint) :34767 (An ISO 397: 7 Cetified Oganization) Vol. 4, Issue, Febuay 5 Design of Wind Enegy System on the Building owe Applications D.Anusha, L V Suesh Kuma, G.V. Nagesh
More informationAn Introduction to Omega
An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei iskewad chaacteistics? The Finance Development Cente 2002 1 Fom
More informationAdvanced Control of Active Filters. in a Battery Charger Application. Martin Bojrup
Advanced Contol of Active Filtes in a Battey Chage Application Matin Bojup Lund 999 ii Cove pictue Measuement on the dynamic esponse of the MRI hamonic filte contolle: load cuent (top), esulting line cuent
More informationComplex Envelope Vectorization for the solution of midhigh frequency acoustic problems. A. Sestieri
Complex Envelope Vectoization fo the solution of midhigh fequency acoustic poblems A. Sestiei Depatment of Mechanical and Aeospace Engineeing Univesity of Rome la Sapienza Pesentation layout  Low fequency
More informationNontrivial lower bounds for the least common multiple of some finite sequences of integers
J. Numbe Theoy, 15 (007), p. 393411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationChannel selection in ecommerce age: A strategic analysis of coop advertising models
Jounal of Industial Engineeing and Management JIEM, 013 6(1):89103 Online ISSN: 0130953 Pint ISSN: 013843 http://dx.doi.og/10.396/jiem.664 Channel selection in ecommece age: A stategic analysis of
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationMULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION
MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe and subsolution method with
More informationModal Characteristics study of CEM1 SingleLayer Printed Circuit Board Using Experimental Modal Analysis
Available online at www.sciencediect.com Pocedia Engineeing 41 (2012 ) 1360 1366 Intenational Symposium on Robotics and Intelligent Sensos 2012 (IRIS 2012) Modal Chaacteistics study of CEM1 SingleLaye
More informationMATHEMATICAL SIMULATION OF MASS SPECTRUM
MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationData Center Demand Response: Avoiding the Coincident Peak via Workload Shifting and Local Generation
(213) 1 28 Data Cente Demand Response: Avoiding the Coincident Peak via Wokload Shifting and Local Geneation Zhenhua Liu 1, Adam Wieman 1, Yuan Chen 2, Benjamin Razon 1, Niangjun Chen 1 1 Califonia Institute
More informationSeshadri constants and surfaces of minimal degree
Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth
More information2. Orbital dynamics and tides
2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationProc. Int. Joint Conf. On Neural Networks, (1):587{592, San Diego, June Nonlinear Prediction with Selforganizing Maps
Poc. Int. Joint Conf. On Neual Netwoks, (1):587{592, San Diego, June 1990 Nonlinea Pediction with Selfoganizing Maps Jog Walte, Helge Ritte and Klaus Schulten BeckmanInstitute and Depatment of Physics
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationThe impact of migration on the provision. of UK public services (SRG.10.039.4) Final Report. December 2011
The impact of migation on the povision of UK public sevices (SRG.10.039.4) Final Repot Decembe 2011 The obustness The obustness of the analysis of the is analysis the esponsibility is the esponsibility
More informationSemipartial (Part) and Partial Correlation
Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated
More informationChris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment
Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability
More information12.1. FÖRSTER RESONANCE ENERGY TRANSFER
ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 11 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More information867 Product Transfer and Resale Report
867 Poduct Tansfe and Resale Repot Functional Goup ID=PT Intoduction: This X12 Tansaction Set contains the fomat and establishes the data contents of the Poduct Tansfe and Resale Repot Tansaction Set (867)
More informationDatabase Management Systems
Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012
More informationComparing Availability of Various Rack Power Redundancy Configurations
Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dualpath powe distibution to IT equipment ae used to enhance
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationAn Epidemic Model of Mobile Phone Virus
An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More informationCIRCUITS LABORATORY EXPERIMENT 7
CIRCUITS LABORATORY EXPERIMENT 7 Design of a Single Tansisto Amplifie 7. OBJECTIVES The objectives of this laboatoy ae to: (a) Gain expeience in the analysis and design of an elementay, single tansisto
More informationOverencryption: Management of Access Control Evolution on Outsourced Data
Oveencyption: Management of Access Contol Evolution on Outsouced Data Sabina De Capitani di Vimecati DTI  Univesità di Milano 26013 Cema  Italy decapita@dti.unimi.it Stefano Paaboschi DIIMM  Univesità
More informationProblem Set # 9 Solutions
Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new highspeed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease
More informationThe Predictive Power of Dividend Yields for Stock Returns: Risk Pricing or Mispricing?
The Pedictive Powe of Dividend Yields fo Stock Retuns: Risk Picing o Mispicing? Glenn Boyle Depatment of Economics and Finance Univesity of Cantebuy Yanhui Li Depatment of Economics and Finance Univesity
More informationFirstmark Credit Union Commercial Loan Department
Fistmak Cedit Union Commecial Loan Depatment Thank you fo consideing Fistmak Cedit Union as a tusted souce to meet the needs of you business. Fistmak Cedit Union offes a wide aay of business loans and
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationYARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH
nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More informationOffice of Family Assistance. Evaluation Resource Guide for Responsible Fatherhood Programs
Office of Family Assistance Evaluation Resouce Guide fo Responsible Fathehood Pogams Contents Intoduction........................................................ 4 Backgound..........................................................
More informationCONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS
CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.
More informationHow Much Should a Firm Borrow. Effect of tax shields. Capital Structure Theory. Capital Structure & Corporate Taxes
How Much Should a Fim Boow Chapte 19 Capital Stuctue & Copoate Taxes Financial Risk  Risk to shaeholdes esulting fom the use of debt. Financial Leveage  Incease in the vaiability of shaeholde etuns that
More information