Harmonic Modelling of Thyristor Bridges using a Simplified Time Domain Method
|
|
- Phoebe Sherman
- 6 years ago
- Views:
Transcription
1 1 Harmonic Moelling of Thyristor Briges using a Simplifie Time Domain Metho P. W. Lehn, Senior Member IEEE, an G. Ebner Abstract The paper presents time omain methos for harmonic analysis of a 6-pulse thyristor brige connecte to an ac network. Balance firing of the converter an a balance interface transformer inuctance are assume. For the special case when the ac network is linear, a close form solution for the harmonic inection of the converter is evelope. For the more general case of a nonlinear ac network, a moular converter moel is evelope that relies on iterative methos. The converter moel moule takes as input the ac voltage harmonics at the point of common coupling an outputs the corresponing harmonic current inection into the network. The moels are first valiate against time omain simulation results. The results from the evelope moels are then compare with those obtaine from approximate analytical techniques which assume zero c ripple current. For a system with typical parameter values, it is shown that the zero ripple assumption may yiel acceptable results for some operating points, but highly erroneous results for others. Inex Terms Thyristor brige, HVDC, converter, harmonics, steay state analysis I. INTRODUCTION Power electronics converters contribute significantly to the harmonic pollution of istribution an transmission networks. In particular, converters that switch at line frequency, such as thyristor briges, inect large harmonic currents into the network. The amplitue an phase of these harmonic current inections may be significantly influence by the presence of resonance conitions on either the ac or c sie of the converter. Accurate preiction of the harmonic current inection of a converter into the ac network must therefore take into account ynamics of both the ac an c sie networks. Harmonic interactions in thyristor briges may be calculate either in the frequency omain 1, 2 or in the time omain 3, 4, 5. The focus of this paper will be on the evelopment of simplifie, computationally efficient time omain methos. In orer to solve for the steay state of a thyrsitor brige Grötzbach employe a state space formulation of the converter 6. For a simplifie ac system, he analytically etermine initial conitions of the states that woul lea to a steay state solution. Base on this founation he later etermine the influence of ac an c sie reactances on the ac sie current harmonics 3. For the special case of a linear, balance ac network Herol an Weinl emonstrate the existence of a completely analytical solution for the steay state of the thyristor brige 4. Pivotal to this analytical solution was the nee to moel the P.W. Lehn is with the Dept. of Electrical an Computer Engineering, University of Toronto, Toronto, Canaa ( lehn@ecf.utoronto.ca). G. Ebner is with the Institute of Electrical Power Systems, University of Erlangen-Nuremberg, Erlangen, Germany, ( ebner@eev.e-technik.unierlangen.e). ac system all the way back to an ieal, harmonic free, infinite bus. After fining the initial conition associate with the steay state, a 1-perio time omain simulation was employe, followe by a Fourier Analysis, to etermine converter current harmonics. Perkins 5 employe a more general iterative metho, similar to the one propose by Dobson for ioe circuits, that allows inclusion of arbitrary linear ac networks. Base again on time omain concepts, iteration is employe to fin (i) the commutation angle of the converter an (ii) the initial conition associate with the steay state. Base on the steay state initial conition a 1-perio time omain simulation coul be employe, followe by a Fourier Analysis, to etermine converter current harmonics. All the above works employ ieal switching evices (zero or infinite impeance) to avoi singularity an/or convergence problems associate with high/low impeance switch moels. This leas to a converter representation with 2 state equations uring the commutation interval an only one state equation uring the non-commutation interval (see commutation an non-commutation circuit iagrams of Fig. 3). This time varying imension of the system significantly complicates the analysis. In this paper it is shown that an assumption of ieal switching evices oes not preclue the evelopment of a fixe imension formulation of the thyristor brige. It is then emonstrate that this fixe imension moel of the converter may be easily augmente to the state matrices of an arbitrary linear time invariant, balance ac network. For analysis of ac networks that contain multiple converters, a moular converter moel is also evelope. While solution of the moular moel requires Newton iteration, convergence is extremely fast since: (i) imension of the system Jacobian is 1x1, (ii) an accurate initial estimate for the solution is available, (iii) the equation being iterate is nearly linear in the vicinity of the solution. II. CONVERTER WITH IDEAL SOURCE Steay state analysis of a 6-pulse or 12-pulse converter is most easily accomplishe if: converter interface inuctance an gating are balance the ac network parameters are balance the ac network is linear the infinite bus is balance an free of harmonics. Uner these conitions a complete analytic moel of the unregulate converter may be evelope. Key to this evelopment is the observation that the state transition matrices of the system epen only on the commutation interval length,
2 2 Y D Y E L D L E 5 / / G 5 G Y F Y GF D Y D Y E L D L E 5 / / G 5 G Y F Y GF E Fig. 3. The equivalent circuits of the converter: (a) uring commutation an (b) after commutation. Fig. 1. Fig. 2. Flow chart for fining the steay state solution of a thyristor brige. Y D Y E Y F L D L E L F 5 / The simplifie schematic iagram a thyristor brige. L GF / G 5 G Y GF β 8. Consequently, taking β as an input variable allows a solution algorithm to procee, as per Fig. 1, that outputs the associate firing angle, α, an the associate initial conition x(0) that leas to a steay state solution. The following subsections etail the solution algorithm. For simplicity, the ac an c networks are first replace by ieal voltage sources, as shown in Fig. 2. It will later be shown how this moel may be extene to consier more general ac network configurations. A. State Transition Map Uner balance operation, the 6-pulse converter isplays 6 th perio symmetry that may be exploite to simplify harmonic analysis. Assuming continuous conuction, the two circuits shown in Fig. 3 characterize the behavior of the converter. The circuit of Fig. 3(a) hols uring the commutation interval, while the circuit of Fig. 3(b) hols for the remainer of the 6 th perio. In general, the state transition map will epen on the choice of state variables. In this work, a new state assignment is selecte to simplify analysis. States are selecte as phase currents i a an i b subect to the constraint that i a = 0 for the entire interval after commutation. The propose selection of state is in contrast to the common approach associating two state variables with the circuit of Fig. 3(a), an only one state variable with the circuit of Fig. 3(b). By maintaining the same number of state variables uring commutation an non-commutation intervals, the propose approach avois the nee for proection an inection matrices (as use in 5, ), or the nee for partial matrix inverses (as use in 4), thereby simplifying implementation. During commutation the state moel of the system is given by: ia ia va L 1 = R t i 1 + B b i 1 + D b v 1 v c (1) b 2L L L L L 1 = L L 2L L 2R + R R + R R 1 = R + R 2R + R 1 D 1 = B 1 = 1 2 After commutation phase a becomes open circuite an i a = 0 for the remainer of the sixth perio. To avoi elimination of state variable i a we employ an auxiliary ifferential equation of the form: i a t = 0 t ɛ β, π/3. Given that i a (β) = 0 by efinition, this simple state moel yiels the esire solution: i a (t) = 0 t ɛ β, π/3. The state equations after commutation are therefore given by: ia ia va L 2 = R t i 2 + B b i 2 + D b v 2 v c, (2) b
3 3 1 0 L 2 = 0 2L L 0 0 R 2 = 0 2R + R 0 D 2 = B 2 = 1 2 Assuming the converter connection allows no zero sequence currents to flow, the αβ frame equations of the system uring an after commutation are obtaine by transformation of (1) an (2): iα t i β = CL 1 R C 1 iα i β. + CL 1 B C 1 vα v β + CL 1 D v c (3) setting subscript = 1 gives the equation uring commutation an = 2 gives the equations after the commutation. The matrix C is a 2 2 variant of the Clarke Transform, as given in the Appenix. The ieal source voltage vector T v α v β is represente by a set of ieal oscillator equations, while the c source is represente by the equation z c /t = 0 as per 9. This allows formation of an equivalent autonomous system equation: with A = t CL 1 x z ac z c = A x z ac z c (4) R C 1 CL 1 B C 1 CL 1 0 Ω ac Ω c Ω ac = Ω c = 0 an the frequency of the ac source has been normalize. Amplitue an phase information of the ac voltage source, as well as amplitue information of the c voltage source is containe only in the initial conitions z(0) = T. Over a sixth of a perio the state transitions may be foun accoring to: x(π/3) z ac (π/3) z c (π/3) = Φ x(0) z ac (0) D (5) Φ is the state transition matrix of the system over a sixth of a perio given by: Φ = e A2(π/3 β) e A1β. (6) It is important to note that the above state transition matrix is only a function of the commutation angle β an oes not explicitly epen on the the firing angle of the converter. B. Perioicity Constraint on State Traectories For a time omain formulation, the conitions that must be satisfie in the steay state are 10: (i) perio excitation, (ii) perioicity of the state traectories (iii) perioicity of the switching events. Perioicity of the excitation is assume for any harmonic analysis an will not be iscusse further. Perioicity of state variables will be aresse in this section, while perioicity of the switching times will be aresse in the subsequent section. Perioicity of the state traectories requires that x(t+2π) = x(t). For a balance system this constraint may be converte to an equivalent constraint on the state traectories over a sixth of a perio 10. Over a sixth of a perio ac current an source space vectors unergo a rotation of π/3 raians: x(π/3) = Θ ac x(0) () Θ ac is the π/3 rotation matrix: cos π/3 sin π/3 Θ ac = sin π/3 cos π/3. (8) Dc quantities repeat ever sixth perio, thus their state rotation matrix is simply the unity matrix: Θ c = 1. (9) To apply the state perioicity constraint () to the state traectory equation (5), the state transition matrix (6) is first evaluate for the specifie commutation angle. It has the form 1 : x(π/3) z ac (π/3) z c (π/3) = Ap N p ac N p c 0 Ω p ac Ω p c x(0) z ac (0) (10) Applying the state perioicity constraint to (10) yiels a constraints on the initial conitions associate with the steay state solution: x(0) = (Θ ac A p ) 1 N p ac N p c z ac (0).. (11) Contrary to the analysis of VSC circuits 9, in analysis of the thyristor brige (11) provies only one of two constraints necessary to solve for the steay state. A secon constraint must be impose to ensure perioicity of the switching times. C. Perioicity Constraint on Switching Times Analysis leaing up to (11) is base on the assumption that the commutation interval length β is known a priori. In fact, the commutation interval length is equal to β if an only if the current i a in Fig. 3(a) reaches zero precisely at time t = β. Mapping this constraint into the αβ-frame yiels: i α (β) = 0. (12) It is assume that a unique firing angle α is associate with each commutation interval length β. Thus we may aust the phase of the ac voltage vector until constraint (12) is satisfie. 1 It may be easily proven that Ω p ac = Θ ac an that Ω p c = Θ c.
4 4 Solution procees as follows. First x(β) is expresse in a from similar to (10) by evaluating the state transition matrix e A1β : x(β) A β N β ac N β c z ac (β) = 0 Ω β ac 0 x(0) z ac (0). (13) z c (β) 0 0 Ω β z c c (0) Introucing the constraint (11) an solving for x(β) yiels: x(β) = F (14) F = A β (Θ ac A p ) 1 N p ac N p c + N β ac N β c. (15) Both the c an ac voltage information as well as the firing angle information is containe in the initial conition vector z(0). Assuming a c voltage of V c, an ac voltage of V an a firing angle of α, the associate initial conition vector is given by (16): z(0) = = V cos(α + π/3) V sin(α + π/3) V c An expression for i α (β) is extracte from (14):. (16) i α (β) = F 1,1 F 1,2 F 1,3 z(0). (1) Finally the constraint (12) is applie to etermine the firing angle α. α = cos 1 F 1,3 F 2 1,1 + F 2 1,2 V c V + tan 1 { F1,2 F 1,1 } π 3. (18) In other wors, a converter operate at the above calculate firing angle will settle into a steay state with the stipulate commutation interval length β. The initial conitions of the states, x(0), associate with this solution may be etermine by applying initial conition (16) to (11). This yiels a fully analytic solution of the converter. III. CONVERTER WITH ARBITRARY AC NETWORK It is often necessary to stuy the harmonic interaction of a converter with its filters an an existing AC network. In this case, ac network equations must be ae to the basic converter equations. This may be one either by augmenting the abc-frame equations of of (1) an (2) augmenting the αβ-frame equations of of (3). Generally, it is easier to employ the latter metho, as it leas to a more moular moel of the system. Network equations are therefore expresse in the form: x ac = A ac x ac + B ac u ac (19) t y ac = C ac x ac (20) input excitation u ac comes from an ieal oscillator then is use to represent the infinite bus within the ac network: u ac = z ac, an output of the network equations gives the ac input voltage neee by the converter moel of (3): vα = y ac. (21) v β This approach allows the influence of network parameters on the operation of the converter to be etermine uner balance operation. IV. CONVERTER WITH AC SOURCE DISTORTION In the previous section, a set of linear network equations were simply augmente to the converter moel. A maor limitation of this classical approach is that no other switching circuits may exist within the moele ac network. Possible interactions between neighboring converters cannot be stuie. To overcome this limitation, a fully moular harmonic moel of the converter can be evelope subect to the following reuce set of limitations: converter interface inuctance an gating are balance only characteristic harmonics exist in the ac network, i.e. negative sequence 5, 11, 1, etc. an positive sequence 1,, 13, etc. The moel buils irectly on the results of Section II. Ac excitation is now provie not by merely a funamental frequency ac source, but by a set of harmonic sources all summe together. The harmonic oscillator matrix Ω ac is now given by: Ω ac = iag(ω 1, 5Ω 1, +Ω 1, 11Ω 1, +13Ω 1,...) (22) Ω 1 = The phase angles of the bus voltage harmonics are efine with respect to the phase angle of the funamental: v α + v β = V +1 0e t + V 5 φ 5 e 5t + V + φ + e t +.. Assuming the funamental of the bus voltage to have zero phase, the necessary initial conitions for the oscillator states are: V +1 cos(0) V +1 sin(0) V 5 cos(φ 5 ) z ac (0) = V 5 sin(φ 5 ). (23) V + cos(φ + ) V + sin(φ + ) : The solution algorithm time shifts the ac excitation voltage to meet the commutation constraint i α (β) = 0. Introucing harmonics on the bus voltage has one critical implication on this solution algorithm. Since harmonics have their phase angles efine relative to the funamental, time shifting of the funamental results in an associate phase shift of all.
5 5 Fig. 4. Test system for valiation of the analytical moel. harmonics. The require initial conition vector therefore has the form: V +1 cos(α + π/3) V +1 sin(α + π/3) V 5 cos(φ 5 5(α + π/3)) z(0)= = V 5 sin(φ 5 5(α + π/3)) V + cos(φ + + (α + π/3)). (24) V + sin(φ + + (α + π/3)) : V c Matrix F may be evaluate ust as in (15), albeit with larger matrix imensions for A β, N p ac an N β ac. Assuming a total of n ac sie harmonics (incluing the funamental) (1) becomes: i α (β) = 0 = F 1,1 F 1,2 : F 1,2n 1 F 1,2n F 1,2n+1 T. (25) All terms in the vector z ac (0) have trigonometric epenance on firing angle α, hence a close form solution to constraint equation (25) oes not exist. Instea the firing angle is solve through iteration. V. MODEL VALIDATION A test system, as epicte in Fig. 4, is employe to valiate the propose moel against time omain simulation results. Parameters for the test system are given in Table I. A complete analytical representation of the system is evelope, as per Sections II an III, an converter current harmonic inections an bus voltage harmonics at the point of common coupling are valiate. The operating point is specifie by the selection of β = o. The resulting firing angle of the converter is solve using the analytical moel to be o. This firing angle is use in the time omain simulation. Table II compares the α an β values from simulation with those from the analytical moel. Table III compares the resulting voltage harmonics at the point of common coupling (PCC) an the converter current harmonics. The results of Tables II an III clearly valiate the accuracy of the analytical moel propose in Sections II an III. TABLE I TEST SYSTEM PARAMETERS. Ac System Converter System Quantity Value Quantity Value V s kv ln, peak V c 9.00 kv R s Ω R Ω X s Ω X Ω R l Ω R Ω X l 0.10 Ω X Ω X Cl 3858 Ω X Cf Ω R f Ω X f Ω X Cf Ω R f Ω X f Ω TABLE II VALIDATION OF ANALYTICAL MODEL - ANALYTICALLY OBTAINED FIRING AND COMMUTATION ANGLES COMPARED WITH SIMULATION Quantity Analysis Simulation β o o α o o TABLE III VALIDATION OF THE ANALYTICAL MODEL - ANALYTICALLY OBTAINED PCC VOLTAGE AND CURRENT HARMONICS COMPARED WITH SIMULATION Harmonic Analysis Simulation Analysis Simulation Number Vpcc h Vpcc h I h I h h (V ln, peak ) (V ln, peak ) (A peak ) (A peak ) Next the moular converter moel of Section IV is valiate. The moular moel takes as input voltage harmonics at the PCC. For a given β value, it outputs the resulting converter current harmonics. Again we assume β = 18.0 o. Table IV shows the PCC voltage harmonics applie to the converter an the resulting converter current harmonics obtaine from simulation an from the moular converter moel. In steay state, the firing angle associate with these PCC harmonics an the specifie β value is foun to be α = o. Excellent agreement is again seen between the results from time simulation an those obtaine from the propose metho.
6 6 TABLE IV VALIDATION OF THE MODULAR MODEL FOR AN ARBITRARY SET OF V pcc 350 HARMONICS Output Output Harmonic Input Input I h I h Number Vpcc h Vpcc h (A peak ) (A peak ) h (V ln, peak ) (egrees) Analysis Simulation I 5, I (Amps) I 5 with c ripple no c ripple I 100 In contrast to the previous moel, no assumption on the linearity of the ac network is mae in the evelopment of the moular moel. Thus the moular moel may be interface to an arbitrary network, provie the network is balance. VI. COMPARISON WITH APPROXIMATE MODELS Many classical texts analyze the thyristor brige base on an assumption of zero c ripple current 11. In other wors, they assume the c smoothing reactor (X in Fig. 4) is infinitely large. In this section, the accuracy of this zero ripple assumption is investigate. The system of Fig. 4 is first simulate with the nominal smoothing reactance of the test system: X = 3.982Ω. It is then simulate with a near infinite smoothing reactance. Fig. 5 shows the resulting ac sie 5 th an th harmonic currents as a function of the commutation angle β. As may be seen from Fig. 5, the zero ripple assumption yiels fairly accurate results for some operating points (e.g. for β > 30 o ), however for other operating points (e.g. for β < 30 o ) large errors result. While the zero ripple assumption sometimes yiels sufficiently accurate results, there is no obvious means of etermining whether its results shoul be truste. VII. CONCLUSION A time omain metho for harmonic analysis of thyristor briges was presente. The complexity of the time omain formulation was reuce through the introuction of an auxiliary ifferential equation. The auxiliary ifferential equation makes the equations uring commutation equal to the orer of the equations after commutation. This eliminates the nee for inection/proection matrices or the nee for non-unique partial matrix inverses, significantly simplifying the solution. The propose metho leas to a fully analytic solution of the converter equations provie the ac system is linear, balance an contains no other harmonic sources. A more general moular converter moel is also evelope. The moular moel may be interface to ac networks containing nonlinearities, an other harmonic sources, however, the solution of the moular moel relies on iterative methos. The propose moel is employe for a simple emonstrative stuy investigating the accuracy of the common zero c ripple assumption. The stuy shows that significant errors BETA (eg) Fig. 5. Errors resulting from a zero c current ripple assumption. 5 th an th ac current harmonics with an without inclusion of c ripple. may occur when using simplifie analytical moels base on a zero c ripple assumption. VIII. ACKNOWLEDGEMENTS The authors woul like to thank Prof. Dr. G. Herol of the University of Erlangen-Nuremberg for hosting this research proect. Without his support this collaborative work woul not have been possible. REFERENCES 1 J. Rittiger an B. Kulicke, Calculation of HVDC converter harmonics in frequency omain with regar to asymmetries an comparison with time omain simulations, IEEE Transactions on Power Delivery, Vol. 10, No. 4, Oct. 1995, pp B.C. Smith, N.R. Watson, A.R. Woo an J. Arrillaga, Steay state moel of AC/DC converter in the harmonic omain, IEE Proc.- Generation, Transmission an Distribution, Vol. 142, No. 2, March 1995, pp M. Grötzbach, an W. Frankenberg, Inecte currents of controlle AC/DC converters for harmonic analysis in iustrial power plants, IEEE Transactions on Power Delivery, Vol. 8, No. 2, April 1993, pp G. Herol an C. Weinl, Calculation of 6-pulse current converters in steay state an optimise esign of AC series wave traps, PEMC Proceeings, 1996, vol. 3, pp B. K. Perkins, an M. R. Iravani, Novel calculation of HVDC converter harmonics by linearization in the time-omain, IEEE Transactions on Power Delivery, Vol. 12, No. 2, April 199, pp M. Grötzbach, an R. von Lutz, Unifie moelling of rectifer-controlle DC-power supplies, IEEE Transactions on Power Electronics, Vol. PE-1, No. 2, April 1986, pp I. Dobson, an S. G. Jalali, Surprising simplification of the acobian of a ioe switching circuit, IEEE Intl. Symposium on Circuits an Systems, May 1993, pp G. Herol, Resonanzbeingungen sechspulsiger Stromrichtersysteme, Elektrie, vol. 40, 1986, pp P. W. Lehn, an K. L. Lian Frequency coupling matrix of a voltage source converter erive from piecewise linear ifferential equations, submitte to the IEEE Transactions on Power Delivery, Jan P. W. Lehn, Exact moeling of the voltage source converter, IEEE Transactions on Power Delivery, Vol. 1, No. 1, January 2002, pp S.B. Dewan, A. Straughen, Power semiconuctor circuits, Wiley, New York, 195.
Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory
Design A Robust Power System Stabilizer on SMIB Using Lyapunov Theory Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper proposes a robust power system stabilizer (PSS)
More information'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21
Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting
More informationArm Voltage Estimation Method for Compensated Modulation of Modular Multilevel Converters
Arm Voltage Estimation Metho for Compensate Moulation of Moular Multilevel Converters Ael A. Taffese Elisaetta Teeschi Dept. of Electric Power Engineering Norwegian University of Science an Technology
More informationQuasi optimal feedforward control of a very low frequency high-voltage test system
Preprints of the 9th Worl Congress The International Feeration of Automatic Control Quasi optimal feeforwar control of a very low frequency high-voltage test system W. Kemmetmüller S. Eberharter A. Kugi
More informationAn inductance lookup table application for analysis of reluctance stepper motor model
ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR
More informationDetermine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Validation
Determine Power Transfer Limits of An SMIB System through Linear System Analysis with Nonlinear Simulation Valiation Yin Li, Stuent Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract This paper extens
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationA Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation
A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationSwitching Time Optimization in Discretized Hybrid Dynamical Systems
Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More information4. CONTROL OF ZERO-SEQUENCE CURRENT IN PARALLEL THREE-PHASE CURRENT-BIDIRECTIONAL CONVERTERS
4. CONRO OF ZERO-SEQUENCE CURREN IN PARAE HREE-PHASE CURREN-BIDIRECIONA CONVERERS 4. A NOVE ZERO-SEQUENCE CURREN CONRO 4.. Zero-Sequence Dynamics he parallel boost rectifier moel in Figure.4 an the parallel
More informationLeast-Squares Regression on Sparse Spaces
Least-Squares Regression on Sparse Spaces Yuri Grinberg, Mahi Milani Far, Joelle Pineau School of Computer Science McGill University Montreal, Canaa {ygrinb,mmilan1,jpineau}@cs.mcgill.ca 1 Introuction
More informationDynamics of the synchronous machine
ELEC0047 - Power system ynamics, control an stability Dynamics of the synchronous machine Thierry Van Cutsem t.vancutsem@ulg.ac.be www.montefiore.ulg.ac.be/~vct These slies follow those presente in course
More informationA Comparison between a Conventional Power System Stabilizer (PSS) and Novel PSS Based on Feedback Linearization Technique
J. Basic. Appl. Sci. Res., ()9-99,, TextRoa Publication ISSN 9-434 Journal of Basic an Applie Scientific Research www.textroa.com A Comparison between a Conventional Power System Stabilizer (PSS) an Novel
More information6.003 Homework #7 Solutions
6.003 Homework #7 Solutions Problems. Secon-orer systems The impulse response of a secon-orer CT system has the form h(t) = e σt cos(ω t + φ)u(t) where the parameters σ, ω, an φ are relate to the parameters
More informationDesign and Application of Fault Current Limiter in Iran Power System Utility
Australian Journal of Basic an Applie Sciences, 7(): 76-8, 13 ISSN 1991-8178 Design an Application of Fault Current Limiter in Iran Power System Utility M. Najafi, M. Hoseynpoor Department of Electrical
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationTHE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE
Journal of Soun an Vibration (1996) 191(3), 397 414 THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationThe canonical controllers and regular interconnection
Systems & Control Letters ( www.elsevier.com/locate/sysconle The canonical controllers an regular interconnection A.A. Julius a,, J.C. Willems b, M.N. Belur c, H.L. Trentelman a Department of Applie Mathematics,
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationPredictive control of synchronous generator: a multiciterial optimization approach
Preictive control of synchronous generator: a multiciterial optimization approach Marián Mrosko, Eva Miklovičová, Ján Murgaš Abstract The paper eals with the preictive control esign for nonlinear systems.
More informationHomework 7 Due 18 November at 6:00 pm
Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationDifferentiability, Computing Derivatives, Trig Review
Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationTransmission Line Matrix (TLM) network analogues of reversible trapping processes Part B: scaling and consistency
Transmission Line Matrix (TLM network analogues of reversible trapping processes Part B: scaling an consistency Donar e Cogan * ANC Eucation, 308-310.A. De Mel Mawatha, Colombo 3, Sri Lanka * onarecogan@gmail.com
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationNonlinear Adaptive Ship Course Tracking Control Based on Backstepping and Nussbaum Gain
Nonlinear Aaptive Ship Course Tracking Control Base on Backstepping an Nussbaum Gain Jialu Du, Chen Guo Abstract A nonlinear aaptive controller combining aaptive Backstepping algorithm with Nussbaum gain
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationInvestigation of local load effect on damping characteristics of synchronous generator using transfer-function block-diagram model
ORIGINAL ARTICLE Investigation of local loa effect on amping characteristics of synchronous generator using transfer-function block-iagram moel Pichai Aree Abstract of synchronous generator using transfer-function
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationDifferentiability, Computing Derivatives, Trig Review. Goals:
Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an
More informationChapter 31: RLC Circuits. PHY2049: Chapter 31 1
Chapter 31: RLC Circuits PHY049: Chapter 31 1 LC Oscillations Conservation of energy Topics Dampe oscillations in RLC circuits Energy loss AC current RMS quantities Force oscillations Resistance, reactance,
More informationLecture 6: Control of Three-Phase Inverters
Yoash Levron The Anrew an Erna Viterbi Faculty of Electrical Engineering, Technion Israel Institute of Technology, Haifa 323, Israel yoashl@ee.technion.ac.il Juri Belikov Department of Computer Systems,
More informationState-Space Model for a Multi-Machine System
State-Space Moel for a Multi-Machine System These notes parallel section.4 in the text. We are ealing with classically moele machines (IEEE Type.), constant impeance loas, an a network reuce to its internal
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationECE 422 Power System Operations & Planning 7 Transient Stability
ECE 4 Power System Operations & Planning 7 Transient Stability Spring 5 Instructor: Kai Sun References Saaat s Chapter.5 ~. EPRI Tutorial s Chapter 7 Kunur s Chapter 3 Transient Stability The ability of
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationDeriving ARX Models for Synchronous Generators
Deriving AR Moels for Synchronous Generators Yangkun u, Stuent Member, IEEE, Zhixin Miao, Senior Member, IEEE, Lingling Fan, Senior Member, IEEE Abstract Parameter ientification of a synchronous generator
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More informationSTUDENT S COMPANIONS IN BASIC MATH: THE FOURTH. Trigonometric Functions
STUDENT S COMPANIONS IN BASIC MATH: THE FOURTH Trigonometric Functions Let me quote a few sentences at the beginning of the preface to a book by Davi Kammler entitle A First Course in Fourier Analysis
More informationPower Generation and Distribution via Distributed Coordination Control
Power Generation an Distribution via Distribute Coorination Control Byeong-Yeon Kim, Kwang-Kyo Oh, an Hyo-Sung Ahn arxiv:407.4870v [math.oc] 8 Jul 204 Abstract This paper presents power coorination, power
More informationNOTES ON EULER-BOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULER-BOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between Euler-MacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationFinal Exam: Sat 12 Dec 2009, 09:00-12:00
MATH 1013 SECTIONS A: Professor Szeptycki APPLIED CALCULUS I, FALL 009 B: Professor Toms C: Professor Szeto NAME: STUDENT #: SECTION: No ai (e.g. calculator, written notes) is allowe. Final Exam: Sat 1
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationx f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.
Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?
More informationDesigning Information Devices and Systems II Spring 2019 A. Sahai, J. Roychowdhury, K. Pister Midterm 1: Practice
EES 16B Designing Information Devices an Systems II Spring 019 A. Sahai, J. Roychowhury, K. Pister Miterm 1: Practice 1. Speaker System Your job is to construct a speaker system that operates in the range
More informationState observers and recursive filters in classical feedback control theory
State observers an recursive filters in classical feeback control theory State-feeback control example: secon-orer system Consier the riven secon-orer system q q q u x q x q x x x x Here u coul represent
More informationA Control Scheme for Utilizing Energy Storage of the Modular Multilevel Converter for Power Oscillation Damping
A Control Scheme for Utilizing Energy Storage of the Moular Multilevel Converter for Power Oscillation Damping Ael A. Taffese, Elisaetta Teeschi Dept. of Electric Power Engineering Norwegian University
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationTIME-DELAY ESTIMATION USING FARROW-BASED FRACTIONAL-DELAY FIR FILTERS: FILTER APPROXIMATION VS. ESTIMATION ERRORS
TIME-DEAY ESTIMATION USING FARROW-BASED FRACTIONA-DEAY FIR FITERS: FITER APPROXIMATION VS. ESTIMATION ERRORS Mattias Olsson, Håkan Johansson, an Per öwenborg Div. of Electronic Systems, Dept. of Electrical
More informationLaplacian Cooperative Attitude Control of Multiple Rigid Bodies
Laplacian Cooperative Attitue Control of Multiple Rigi Boies Dimos V. Dimarogonas, Panagiotis Tsiotras an Kostas J. Kyriakopoulos Abstract Motivate by the fact that linear controllers can stabilize the
More informationSensors & Transducers 2015 by IFSA Publishing, S. L.
Sensors & Transucers, Vol. 184, Issue 1, January 15, pp. 53-59 Sensors & Transucers 15 by IFSA Publishing, S. L. http://www.sensorsportal.com Non-invasive an Locally Resolve Measurement of Soun Velocity
More informationOn Characterizing the Delay-Performance of Wireless Scheduling Algorithms
On Characterizing the Delay-Performance of Wireless Scheuling Algorithms Xiaojun Lin Center for Wireless Systems an Applications School of Electrical an Computer Engineering, Purue University West Lafayette,
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section-2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationImpact of DFIG based Wind Energy Conversion System on Fault Studies and Power Swings
Impact of DFIG base Win Energy Conversion System on Fault Stuies an Power Swings Likin Simon Electrical Engineering Department Inian Institute of Technology, Maras Email: ee133@ee.iitm.ac.in K Shanti Swarup
More informationA SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS
International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationPMU-Based System Identification for a Modified Classic Generator Model
PMU-Base System Ientification for a Moifie Classic Generator Moel Yasser Wehbe, Lingling Fan, Senior Member, IEEE Abstract The paper proposes to use PMU measurements (voltage phasor, real an reactive powers)
More informationModule FP2. Further Pure 2. Cambridge University Press Further Pure 2 and 3 Hugh Neill and Douglas Quadling Excerpt More information
5548993 - Further Pure an 3 Moule FP Further Pure 5548993 - Further Pure an 3 Differentiating inverse trigonometric functions Throughout the course you have graually been increasing the number of functions
More informationSome Remarks on the Boundedness and Convergence Properties of Smooth Sliding Mode Controllers
International Journal of Automation an Computing 6(2, May 2009, 154-158 DOI: 10.1007/s11633-009-0154-z Some Remarks on the Bouneness an Convergence Properties of Smooth Sliing Moe Controllers Wallace Moreira
More informationTAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS
MISN-0-4 TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS f(x ± ) = f(x) ± f ' (x) + f '' (x) 2 ±... 1! 2! = 1.000 ± 0.100 + 0.005 ±... TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS by Peter Signell 1.
More informationOptimized Schwarz Methods with the Yin-Yang Grid for Shallow Water Equations
Optimize Schwarz Methos with the Yin-Yang Gri for Shallow Water Equations Abessama Qaouri Recherche en prévision numérique, Atmospheric Science an Technology Directorate, Environment Canaa, Dorval, Québec,
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationAdjoint Transient Sensitivity Analysis in Circuit Simulation
Ajoint Transient Sensitivity Analysis in Circuit Simulation Z. Ilievski 1, H. Xu 1, A. Verhoeven 1, E.J.W. ter Maten 1,2, W.H.A. Schilers 1,2 an R.M.M. Mattheij 1 1 Technische Universiteit Einhoven; e-mail:
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationA Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges
Plasma Science an Technology, Vol.16, No.1, Oct. 214 A Simple Moel for the Calculation of Plasma Impeance in Atmospheric Raio Frequency Discharges GE Lei ( ) an ZHANG Yuantao ( ) Shanong Provincial Key
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationx f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.
Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?
More informationPolynomial Inclusion Functions
Polynomial Inclusion Functions E. e Weert, E. van Kampen, Q. P. Chu, an J. A. Muler Delft University of Technology, Faculty of Aerospace Engineering, Control an Simulation Division E.eWeert@TUDelft.nl
More informationII. First variation of functionals
II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent
More informationSimulink model for examining dynamic interactions involving electro-mechanical oscillations in distribution systems
University of Wollongong Research Online Faculty of Engineering an Information Sciences - Papers: Part A Faculty of Engineering an Information Sciences 205 Simulink moel for examining ynamic interactions
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationModeling time-varying storage components in PSpice
Moeling time-varying storage components in PSpice Dalibor Biolek, Zenek Kolka, Viera Biolkova Dept. of EE, FMT, University of Defence Brno, Czech Republic Dept. of Microelectronics/Raioelectronics, FEEC,
More informationQuantum Search on the Spatial Grid
Quantum Search on the Spatial Gri Matthew D. Falk MIT 2012, 550 Memorial Drive, Cambrige, MA 02139 (Date: December 11, 2012) This paper explores Quantum Search on the two imensional spatial gri. Recent
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationGLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED
More informationOptimum design of tuned mass damper systems for seismic structures
Earthquake Resistant Engineering Structures VII 175 Optimum esign of tune mass amper systems for seismic structures I. Abulsalam, M. Al-Janabi & M. G. Al-Taweel Department of Civil Engineering, Faculty
More informationComputing Derivatives
Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations
More informationCalculus of Variations
16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationPlacement and tuning of resonance dampers on footbridges
Downloae from orbit.tu.k on: Jan 17, 19 Placement an tuning of resonance ampers on footbriges Krenk, Steen; Brønen, Aners; Kristensen, Aners Publishe in: Footbrige 5 Publication ate: 5 Document Version
More information16.30/31, Fall 2010 Recitation # 1
6./, Fall Recitation # September, In this recitation we consiere the following problem. Given a plant with open-loop transfer function.569s +.5 G p (s) = s +.7s +.97, esign a feeback control system such
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More information