Bond Graph Modeling And Simulation Of Three Phase PM
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009-ASMD2Bond Graph Modeling and Simulation of Three phase PM BLDCMotor1Anand Vaz, 2S.S.Dhami and 3Sandesh Trivedi*1Professor, Department of Mechanical Engineering, B.R. Ambedkar National Institute ofTechnology, Jalandhar, India.2Assistant Professor, Department of Mechanical Engineering, National Institute ofTechnical Teachers Training and Research, Chandigarh, India.3Lecturer, Department of Mechanical Engineering, GovernmentEngineering College, Ajmer, India.PPPPPPPPPPP*PSandesh Trivedi (sandeshtrivedi@yahoo.in)Pmodeling and simulation of brush less DC motor hasbeen widely studied because of their increasing usage inindustrial automation. It is therefore necessary to analyze the dynamic characteristics of a brush less dc permanent magnet motor in order to control it, simulate it,and to evaluate its performance. Many methods for obtaining the governing differential equations of dynamicsystem are well known. To make dynamic modelingsimpler and understandable, a unified approach basedupon energy transaction called Bond Graph had beeninvented by Henry Paynter in 1950s. The same approachis being used in this paper for modeling of brush less dcmotor dynamics. The code forAbstractThe purpose of this paper is to build a simple, accurate and fast running real time bond graph model of athree phase star-connected Permanent Magnet Brushless Direct Current (PMBLDC) Motor and implementation of the same in MATLAB environment. BrushlessDC motors (BLDCM) have important advantages overbrushed DC motors and induction motors. They havebetter speed/torque characteristics, high efficiency, andhigh dynamic response, are compact, need lesser maintenance.The complete dynamic model for the BLDC motorhas been systematically developed using Bond Graphs.The cause-effect relations have been analyzed and discussed. A simulation program written in MATLAB isused to verify the basic operation (performance) of theproposed topology. It is believed that the proposed model offers a reliable and low-cost solution for the emerging market of BLDC motor drives.the system equations obtained from bond graph modelof BLDC motor are written in MATLAB and then simulated by using ODE45 solver.2TBLDC Motor TheoryAs compared to a conventional DC brush motor, brushless DC (BLDC) motors are DC brush motors turnedinside out, so that the field is on the rotor and the armature is on the stator. In BLDC motor, field excitation isprovided by a permanent magnet and commutation isachieved electronically instead of using mechanicalcommutators and brushes. In BLDC motor, themechanical ‘rotating switch’ or commutator/brush gearassembly is replaced by an external electronic switchsynchronised to the rotor's position.There are two main types of BLDC motors; trapezoidal type and sinusoidal type. In the trapezoidalmotor the back-emf induced in the stator windingshas a trapezoidal shape and its phases must be supplied with quasi-square currents for ripple-free torque operation. The sinusoidal motor on the otherhand has a sinusoidal shaped back-emf and requiressinusoidal phase currents for ripple-free torque operation. The shape of the back-emf is determined by theKey words: Brushless DC motor, bond graph modeling,MATLAB1PIntroductionConventional DC permanent magnet motor is the mostwidely used actuator, but it is associated with theproblem of frequent wear out of commutator and brush.Commutation also tends to cause a great deal ofelectrical and RF noise. Without a commutator orbrushes , a brushless motor may be used in electricallysensitive devices like audio equipments or computers.BLDC motors offer several advantages over brushed DCmotors, including higher efficiency and reliability,reduced noise, longer lifetime (no brush erosion),elimination of ionizing sparks from the commutator.BLDC PM Motors are preferred actuators in automationsystems owing to their many advantages. The dynamic1
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009 Paper ID ASMD2shape of the rotor magnets and the stator windingdistribution.The sinusoidal motor needs a high resolution position sensor because the rotor position must be knownat every time instant for optimal operation. It alsorequires more complex software and hardware. Thetrapezoidal motor is a more attractive alternative formost applications due to its simplicity, lower priceand higher efficiency [7].3Operation of BLDC MotorThe three phase BLDC motor is operated in a twophases-on fashion, i.e. the two phases that producethe highest torque are energized while the thirdphase is off. Which two phases are energized dependson the rotor position.Fig. 2: Ideal back-emfs, phase currents, and positionsensor signalsFig. 1 shows a cross section of a three-phase starconnected motor along with its phase energizing sequence. Each interval starts with the rotor and statorfield lines 120 apart and ends when they are 60 apart.Maximum torque is reached when the field lines areperpendicular. The signals from the position sensorsproduce a three digit number that changes every 60 (electrical degrees) as shown in fig. 2 (HI, H2, H3).The figure also shows ideal current and back-emfwaveforms.Fig. 3: Simplified BLDC drive scheme4Bond Graph ModelingBond graphs are a domain-independent graphical description of dynamic behavior of physical systems. Thismeans that systems from different domains (electrical,mechanical, hydraulic, and thermodynamic) are described in the same way. The basis is that bond graphsare based on energy and energy exchange.Bond graph is an explicit graphical tool for capturingthe common energy structure of systems. It increases theinsight into systems behavior. Moreover, the notationsof causality provide a tool not only for formulation ofsystem equations, but also for intuition based discussionof system behavior, viz. controllability, fault diagnosisand observability [2]. In 1959, Prof. H. M. Paynter gavethe revolutionary idea of portraying systems in terms ofpower bonds, connecting elements of the physical system to the so called junction structures which were manifestations of the constraints. This power exchangeportray of a system is called Bond Graph (or Bondgraph), which can be both power and informationoriented. Later on, Bond Graph theory has been furtherdeveloped by many researchers like Karnopp, Rosenberg, Thoma and Breedveld, who have worked on extending this modeling technique to power hydraulics,mechatronic, general thermodynamic systems and toelectronics and non-energetic systems like economicsFig. 1: BLDC motor cross section and phase energizing sequence. [5]The most common method of sensing the rotor position in a BLDC motor is using hall-effect position sensors. For a BLDC motor with a trapezoidal back-EMF, itis sufficient to get position information that is updated atevery 60 degree electrical interval, called six stepscommutation. The position information is then used todecide the triggering of inverter switches. Generally,three hall-effect sensors are used for a three phase motor. Current commutation is done by a six-step inverter as shown in a simplified form in fig. 3. Theswitches are shown as bipolar junction transistors butMOSFET switches are more common.2
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009 Paper ID ASMD2and queuing theory. By this approach, a physical systemcan be represented by symbols and lines, identifying thepower flow paths. The lumped parameter elements ofresistance, capacitance and inductance are interconnected in an energy conserving way by bonds and junctions resulting in a network structure. From the pictorialrepresentation of the bond graph, the derivation of system equations is so systematic that it can be algorithmized.4Ga (θ )1&&MTFVs /2*Ga(θ)Van0fa (θ ) *KTia1ia11fa (θ ) *KEGY&&1225TaGY&&107ea13ωI : LaR : RfR: Rb5Gb (θ )&& 2MTFVs /2*Gb(θ)Vbn80ib141ibTb15eb221ωfb ( θ) *KEGY&&162623fb (θ ) *KTGY&&I :J17ωωI : LbR : Rc65 Bond Graph Model of BLDCMotorVs /2*Gc (θ)Gc (θ )&& 3MTFVcn 9ic181ic0fc ( θ) *KTGY&& dtTc19fc ( θ) *KE&&ec GY20272421θ(t)ωI : LcTypically, a Brushless dc motor is driven by a threephase inverter with, what is called as a six-step commutation as shown in fig. 4. The conducting interval foreach phase is 1200 by electrical angle.PFig. 6: Bond graph model of BLDC motorThe steps to obtain the differential equations frombond graph follow the answers to two questions:(1) What do the elements give to the system?(2) What do the integrally causaled storage elements receive from the system?The differential equations for the electrical momentums of three phases, angular momentum and angular displacement from the bond graph (fig. 6) ofBLDC motor can be obtained after it is numbered, power directed, and causaled, as these are the basic inputsnecessary to obtain the differential equations. We get thefirst order differential equations, by integrally causallingeach storage elements of bond graph by answering theabove two questions.PFig. 4: Simplified equivalent circuit of the BLDC drivesThe commutation phase sequence is AB-AC-BC-BACA-CB. Each conducting stage is called one step.Therefore, only two phases conduct current at any time,leaving the third phase floating as shown in fig. 5. Thecommutation timing is determined by the rotor position,which can be detected by Hall sensors. This is why aBLDC motor is also commonly known as an electronically commutated motor (ECM).Step I: What do the elements give to the system?f25 p1/La(1)f26 p2/Lb(2)f27 p3/Lc(3)f22 p4/J(4)e4 Ra. f4 Ra. f7 Ra. f25(5)(6)e5 Rb. f5 Rb. f8 Rb. f26e6 Rc. f6 Rc. f9 Rc. f27(7)e23 Rf. f23 Rf. f22(8)Where,f’s and e’s are the flows and efforts in respective bondsof bond graph.p1, p2 and p3 are electrical momentums in three phases;p4 is the angular momentum of rotor.Ra, Rb and Rc are resistances in three phases.La, Lb and Lc are inductance of three phases.Rf is frictional resistance.Thus the first question is answered for all the elements.Step II : What do the integrally causaled storage elements receive from the system?dp1/dt e25 -e4 e7 – e12(9 )dp2/dt e26 -e5 e8 – e16(10)dp3/dt e27 -e6 e9 – e20(11)dp4/dt e22 -e23 e11 e15 e19 (12)dq24 f24(13)Thus the second question is answered for all the elements.BBBBBBBBBBBBBBBBBBBFig. 5: Typical three phase switching pattern in theBLDC motorThe complete bond graph model of three phaseBLDC motor is shown in fig. 6.3BBBBBBBBBBBBBBBBBBBBBBBB
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009 Paper ID ASMD2at the commutation points. The notches at the commutation points occur because the rise of current in the phasethat is being turned on is slower than the decay of thecurrent in the phase that is being turned off. The notchesare the cause of the torque ripples of BLDC motor.Modeling equations for electro-mechanical part ofBLDC motor.(14)e11 fa(θ) . KT. f10Since, f10 f25; (common flow junction i.e.1- Junction)e11 fa(θ) . KT. f25(15)(16)e11 fa(θ) . KT. p1/LaSimilarly,e12 fa(θ) . KE . f13(17)But f13 f22 p4/J; (common flow junction i.e.1Junction)e12 fa(θ) . KE. p4/J(18)Where, fa (θ) . KT modulus of Gyrator element for torque.fa (θ) trapezoidal function for phase A, andKT torque constant.fa (θ). KE modulus of Gyrator element for back emf ofphase AKE back emf constant.The state variable (θ) i.e. the rotor position is required to have the function fa (θ), which is given astrapezoidal function:fa(θ) 1;if (0 θ 2.pi/3)fa(θ) 1-(θ-2pi/3)6/pi;if (2.pi/3 θ pi)fa(θ) -1;if (pi θ 5.pi/3)fa(θ) -1 (θ-5pi/3)6/pi; if (5.pi/3 θ 2pi)Similarly, trapezoidal functions for other two phasesare:fb(θ) fa(θ – 2 . pi/3)(19)(20)fc(θ) fa(θ – 4 . pi/3)Similarly,e16 fb(θ) . KE. f17(21)(22)e20 fc(θ) . KE. f21e15 fb(θ) . KT. f14(23)e19 fc(θ) . KT. f18(24)BBBBBBBBBBBBBBBBBBCurrent in phase A v/s time20BBBBphase ABBBBBBBBBBBBBB0.040.060.080.10.120.14time (s)Current in phase B v/s 060.160.180.20.080.10.120.14time (s)Current in phase C v/s time100-100.080.10.12time (s)0.14BFig. 7: Current in three phases with respect to timeBBB0.020-20B020BBB0-20phase BBphase CBBBBBBBBB100-10-20BBBB6 Simulation of Bond Graph ModelTable 1: Parameters for ltsA0.040.060.080.10.12time (s)0.140.160.180.2Angular displacement v/s 055436.16248.60.02It is clear from the fig. 8 that the current in phase Ais saturated after some time as explained in fig. 7. Afterthat sudden rise and sudden drop in current is due toinverter switching. Also the shape of back EMF wave istrapezoidal. This shape is essential in BLDC motor inorder to obtain linear relationship between torque produced by the rotor and the current. The angular displacement is continuously increasing, this confirms thattorque is continuously produced and the concept ofmodulated transformer (“MTF”), as used in bond graphfor extracting information about switching sequence iscorrect.The plot of total torque with respect to angular displacement is shown in Fig. 9.We observe from the Fig.that, initially the torque is higher. This is due to highercurrent at the beginning. As soon as the motor attainsconstant speed, the torque is stabilized.BThe simulations are done in MATLAB using the defaultsolver ode45. The simulation time has been taken as 0.2seconds, because beyond 0.2 seconds the characteristicsof motor will not be affected.ParameterMoment of InertiaInductance of statorResistance of statorDamping resistanceTorque constantBack EMF constantSource VoltageNo load current0Fig. 8: Current in phase A, Back EMF of phase A andangular displacement of rotor.BBB20BBB30BBCurrent in phase ABack EMF phase AAngular displacementBBBBCurrent in phase A, Back EMF and Angular Displacement v/s time400.40.20-0.2Fig. 7 shows the phase currents. The current startswith a high value and decays as the motor speeds upuntil it reaches no load current value, which is about 8 A,which agrees with the value given in parameters above.The current has a nearly perfect quasi-square shape. Theonly deviation from the quasi-square wave shape occurs-0.40510152025Angular displacement(rad)3035Fig. 9: Plot of torque v/s angular displacement.440
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009 Paper ID ASMD21.Dragan Antic, Biljana Vidojkovic and MiljanaMladenovic, “An Introduction to Bond Graph Modellingof Dynamic Systems”, TELSIKS 99, 13-1 5 October1999, IEEE, NIS, Yugoslavia, pp. 661-664.2.Jan F. Broenink, “Introduction to Physical Systems Modelling with Bond Graphs”, Control Laboratory, University of Twente, Netherland, 1999, pp. 1-31.3.R.G. Longoria, “Modeling of a PermanentMagnet DC Motor”, ME 244L, Dynamic Systems andControls Lab., The University of Texas at Austin, Fall2000, pp. 1-12.4.P. Pillay and R. Krishnan, "Modeling of permanent magnet motor drives", Industrial Electronics,IEEE Transactions, Vol. 35, 1988, pp. 537-541.5.T.J.E. Miller, “ Brushless Permanent-Magnetand Reluctant Motor Drives,” Oxford, 1989.6.P.C.K. Luk and C.K.Lee, “Efficient modelling for a brushless dc motor Drive”, IE, Control andInstrumentation, 1994, pp. 1-4.7.P.C.Sen. “ Principles of Electric Machines andPower Electronics”. John Wiley & Sons, 1997.8.Texas Instruments Incorporated. “DSPSolutions for BLDC Motors”, 1997.9.C.M. Ong, “Dynamic Simulation of ElectricMachinery using Matlab/Simulink”, Prentice Hall, 1998.10.Dal Y. Ohm and Jae H. Park, “About commutation and current control methods for brushless motors”,29th annual IMCSD Symposium, San Jose, July 26-29,1999, pp. 1-11.11.Uwe, Schaible, and Barna,“Dynamic MotorParameter Identification for High Speed Flux Weakening Operation of Brushless Permanent Magnet Synchronous Machines”, IEEE Transactions on EnergyConversion, Vol.14, No. 3, September 1999, pp. 486492.12.Gui-Jia Su and John W. McKeever, “Design ofa PM Brush less motor drive for hybrid electrical vehicle application”, PCIM 2000, Boston, MA Oct.2000,pp. 1-6.13.J.Jancsurak, “Motoring into DSPs”, ApplianceManufacture, Sept.2000, pp. 57- 60.14.Thomas Kaporch, “Driving the future“, Appliance Manufacture, Sept.2001, pp. 43-46.15.Joe Mattingly, “More Efficiency Standards onHorizon”, Appliance Manufacture, Oct.2001, Publishedon Internet.16.Ward Brown. “Brushless DC Motor ControlMade Easy”, Microchip Tech- nology Inc., 2002.17.Juan W. Dixon and Ivan Leal, “Currentcontrol strategy for brushless dc motors based on acommon dc signal”, IEEE Transactions on PowerElectronics, 17(2), March 2002, pp. 1-6.18.J. Filla, “ECMs Move into HVAC”, ApplianceManufacture, Mar.2002, pp. 25-27.19.Amuliu Bogdan and Proca, “Analytical modelfor Permanent Magnet motors with surface mountedmagnets”, IEEE transactions on energy conversion,Vol. 18, No.3, Sept. 2003, pp. 386-391.20.Padmaraja Yedamale, “Brushless DC(BLDC) Motor Fundamentals”, Application note 885,Microchip Technology Inc., Chandler, AZ, 2003.Angular displacement4035Angular ime (s)0.120.140.160.180.2Fig. 10: Plot of Angular displacement v/s timeFig. 10 shows that angular displacement is linearlyincreasing with time except in the beginning, wherecurve is non-linear up to 0.05second. This is due to theacceleration of the motor during this interval.Angular speed of rotor v/s time200018001600Angular speed 0.12time (s)0.140.160.180.2Fig. 11: Plot of Angular speed v/s timeThe appropriateness of bond graph model of motor isevident from the various plots as obtained during simulation.7ConclusionsThe applications of BLDC motors and drives havegrown significantly in recent years in the appliance industry and the automotive industry. The fast growth ofthe BLDC motor in applications requiring positional andspeed accuracy has increased the need for optimum andprecise modeling of their dynamics and drive circuit.Modeling of armature circuit and dynamics of BLDCmotor have been the focus of this paper.As a matter of fact the conventional modeling methods involve more computational work. The aim of thispaper was to make a model that would be simple, accurate, easy to modify and suitable for real time implementation. The simulation results of this work have shownthat these goals have been achieved. In this paper, a unified approach of bond graph for modeling of three phaseBLDC motor and its drive circuit was used, analyzed,and extended, for overcoming the drawbacks of the conventional modeling approaches.REFERENCES :5
14th National Conference on Machines and Mechanisms (NaCoMM09),NIT, Durgapur, India, December 17-18, 2009NaCoMM-2009 Paper ID ASMD221.M.A. Malik and A. Khurshid, “Bond graphModeling and Simulation of mechatronic systems”,IEEE proceeding INMIC, 2003, pp. 309-315.22.Min Dai, Ali Keyhani, and Tomy Sebastian,“Torque Ripple Analysis of a PM Brush less DC motorusing Finite Element Method”, IEEE transactions onenergy conversion, Vol. 19, No.1, March 2004, pp. 4045.23.M.A.Jabbar, “Modeling and Numerical simulation of brushless PM DC motor in dynamic conditionsby Time-stepping technique”, IEEE transactions on Industry Applications, Vol. 40, No.3, May/June2004, pp.763-770.24.Leonard N. Elevich, “Three phase BLDC motorcontrol with Hall sensors using 56800/E digital signalcontroller”, Application note AN1916, Freescale Semiconductor, Rev.2.0, 11/2005.25.M. Kumar, B. Singh and B.P. Singh, “ Singlecurrent based speed control of PM BLDC motor usingDSP ”, IE(I) Journal, EL 2005, Vol.86, pp. 17-21.26.Salih Baris Ozturk, “modelling, simulation andanalysis of low-cost direct torque control of PMSM using hall-effect sensors”, Master of Science thesis December, 2005, Texas A&M University.27.C. Gencer and M. Gedikpinar, “ Modeling andSimulation of BLDCM using MATLAB/SIMULINK”,Journal of Applied Science 6(3), 2006, pp. 688-691.28.Ronald De Four and Emily Ramoutar, “Operational characteristics of brushless dc motors”, The University of the West Indies St. Augustine, Trinidad, 2007.29.Bolton, “Mechatronics” , 2nd edition.30.Mukherjee, A., Karmakar, R. & Samantaray,Arun Kumar, “Bond graph in modeling simulation andfault identification”, I.K. international, N.Delhi.31.Karnopp, Dean, Margolis, C. Donald, L. andRosenberg, Ronald C., “System Dynamic Modeling andsimulation of Mechatronic System”, Fourth Edition,John Wiley & Sons,Inc.32.Marc Vila Mani, “A quick overview on rotatoryBrush and Brushless DC Motors”, Ingenia-Cat - MotionControl Department, Barcelona – Spain.PP6
4 Bond Graph Modeling Bond graphs are a domain-independent graphical de-scription of dynamic behavior of physical systems. This mechanical, hydraulic, and thermodynamic) are de-scribed in the same way. The basis is that bond graphs are based on energy and energy exchange. Bond g
Bond Graph Theory Bond graph, short for the power bond graph, is a graphical approach to deal with multiple energy categories of engineering system, based on the law of conservation of energy. The bond graph method was proposed by Professor H.M.Paynter of MIT in 1959 and developed in
bond-graph is the added integration of q& to produce q. This increases the order of the system from three to four. Figure 3 shows the Dymola implementation of this bond graph using the Dymola Bond-Graph Library. Figure 3. Dymola Implementation of the Gyroscope Bond-Graph Although the
The totality of these behaviors is the graph schema. Drawing a graph schema . The best way to represent a graph schema is, of course, a graph. This is how the graph schema looks for the classic Tinkerpop graph. Figure 2: Example graph schema shown as a property graph . The graph schema is pretty much a property-graph.
As bond order increases, bond length decreases, and bond energy increases H 2 bond order 1 A bond order of 1 corresponds to a single bond Bond order (number of-bonding e ) - (number of antibonding e ) 2 electrons/bond 38 MO Energy Diagram for He 2 Four electrons, so both and *
Fig. 3. The general structure of an acausal bond graph. 2.3 The bond graph formalism Created by Paynter [1], the bond graph formalism allows to model continuous systems and by extension, the continuous part of hybrid dynamical systems. The bond graph is a combination of bonds. A bond is a
Oracle Database Spatial and Graph In-memory parallel graph analytics server (PGX) Load graph into memory for analysis . Command-line submission of graph queries Graph visualization tool APIs to update graph store : Graph Store In-Memory Graph Graph Analytics : Oracle Database Application : Shell, Zeppelin : Viz .
and derived bond graph, a rational path to pseudo bond graph. In Chapter – 11 we establish the power variables and junction structure in hydraulic circuits followed by modeling of a hydraulic servomotor. Chapter – 12 deals with the conversion of bond graph
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