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  1. 1. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571050 | P a g eMathematical Modelling And Position Control Of Brushless Dc(Bldc) MotorHemchand ImmaneniGITAM UNIVERSITY, VISAKHAPATNAM, INDIA.ABSTRACTThe aim of the paper is to design asimulation model of Permanent MagnetBrushless DC (PMBLDC) motor and to controlits position. In the developed model, thecharacteristics of the speed, torque, back EMF,voltages as well as currents are effectivelymonitored and analysed. The PID controller isused to control the position of a Permanentmagnet brushless DC motor by changing thecurrent flow to control the average voltage andthereby the average current. Most usefulapplication is in controlling of CNC machine.KEY WORDS:-Brushless dc (BLDC) motor,position control, mathematical modelling, PIDcontrollerINTRODUCTIONThe economic constraints and newstandards legislated by governments placeincreasingly higher requirements on electricalsystems. New generations of equipment must havehigher performance parameters such as betterefficiency and reduced electromagneticinterference .System flexibility must be high tofacilitate market modifications and to reducedevelopment time. All these improvements must beachieved while, at the same time, decreasing systemcost. Brushless motor technology makes it possibleto achieve these specifications. Such motorscombine high reliability with high efficiency, andfor a lower cost in comparison with brush motors.The Brushless DC Motor (BLDC) motor isconventionally defined as a permanent magnetsynchronous motor with a trapezoidal back ElectroMotive Force (EMF) waveform shape.A system based on the Direct Current (DC)motor provides a good, simple and efficient solutionto satisfy the requirements of a variable speed drive.Although DC motors possess good controlcharacteristics and ruggedness, their performanceand applications in wider areas is inhibited due tosparking and commutation problems. Inductionmotor do not possess the above mentionedproblems, they have their own limitations such aslow Power factor and non-linear speed torquecharacteristics. With the advancement of technologyand development of modern control techniques, thePermanent Magnet Brushless DC (PMBLDC)motor is able to overcome thelimitations mentioned above and satisfy therequirements of a variable speed drive.Electric motors influence almost everyaspect of modern living. Refrigerators, vacuumcleaners, air conditioners, fans, computer harddrives, automatic car windows, and multitudes ofother appliances and devices use electric motors toconvert electrical energy into useful mechanicalenergy. In addition to running the common placeappliances that we use every day, electric motors arealso responsible for a very large portion of industrialprocesses.FIGURE (1): PERMANENT MAGNET BLDCMOTORPOSITIONING APPLICATIONSMost of the industrial and automation typesof application come under this category. Theapplications in this category have some kind ofpower transmission, which could be mechanicalgears or timer belts, or a simple belt driven system.In these applications,The dynamic response of speed and torqueare important. Also, these applications may havefrequent reversal of rotation direction. The load onthe motor may vary during all of these phases,causing the controller to be complex.These systems mostly operate in closedloop. There could be three control loops functioningsimultaneously: Torque Control Loop, SpeedControl Loop and Position Control Loop. Opticalencoder or synchronous resolvers are used formeasuring the actual speed of the motor. In somecases, the same sensors are used to get relativeposition information. Otherwise, separate positionsensors may be used to get absolute positions.
  2. 2. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571051 | P a g eComputer Numeric Controlled (CNC) machines area good example of this. Process controls, machinerycontrols and conveyer controls have plenty ofapplications in this category.MATHEMATICAL MODELLINGBrushless DC Motors are permanentmagnet motors where the function ofcommutatorand brushes were implemented by solidstate switches. BLDC motors come in single-phase,2-phase and 3-phase configurations. Correspondingto its type, the stator has the same number ofwindings. Out of these, 3-phase motors are the mostpopular and widely used. Because of the specialstructure of the motor, it produces a trapezoidal backelectromotive force (EMF) and motor currentgenerate a pulsating torque.Three phase BLDC motor equations:-Va=iaRa+La𝑑𝑖𝑎𝑑𝑡+ 𝑀𝑎𝑏𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑎𝑐𝑑𝑖𝑐𝑑𝑡+ 𝑒𝑎Vb=ibRb+Lb𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑏𝑎𝑑𝑖𝑎𝑑𝑡+ 𝑀𝑏𝑐𝑑𝑖𝑐𝑑𝑡+ 𝑒𝑏Vc=icRc+Lc𝑑𝑖𝑐𝑑𝑡+ 𝑀𝑐𝑏𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑐𝑎𝑑𝑖𝑎𝑑𝑡+ 𝑒𝑐R: Stator resistance per phase, assumed to be equalfor all phasesL: Stator inductance per phase, assumed to be equalfor all phases.M: Mutual inductance between the phases.ia,ib,ic: Stator current/phase.Va,Vb,Vc: are the respective phase voltage of thewindingThe stator self-inductances are independent of therotor position, hence:La=Lb=Lc=LAnd the mutual inductances will have the form:Mab=Mac=Mbc=Mba=Mca=Mcb=MAssuming three phase balanced system, all thephase resistances are equal:Ra=Rb=Rc=RRearranging the above equationsVa=iaR+L𝑑𝑖𝑎𝑑𝑡+ 𝑀𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑑𝑖𝑐𝑑𝑡+ 𝑒𝑎Vb=ibR+L𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑑𝑖𝑎𝑑𝑡+ 𝑀𝑑𝑖𝑐𝑑𝑡+ 𝑒𝑏Vc=icR+L𝑑𝑖𝑐𝑑𝑡+ 𝑀𝑑𝑖𝑏𝑑𝑡+ 𝑀𝑑𝑖𝑎𝑑𝑡+ 𝑒𝑐Neglecting mutual inductanceVa=iaR+L𝑑𝑖𝑎𝑑𝑡+ 𝑒𝑎Vb=ibR+L𝑑𝑖𝑏𝑑𝑡+ 𝑒𝑏Vc=icR+L𝑑𝑖𝑐𝑑𝑡+ 𝑒𝑐TRAPEZOIDAL BACK EMFWhen a BLDC motor rotates, eachwinding generates a voltage known as backElectromotive Force or back EMF, which opposesthe main voltage supplied to the windings accordingto Lenz‟s Law. The polarity of this back EMF is inopposite direction of the energized voltage. BackEMF depends mainly on three factors: Angular velocity of the rotor Magnetic field generated by rotor magnets The number of turns in the stator windingsOnce the motor is designed, the rotormagnetic field and the number of turns in the statorwindings remain constant. The only factor thatgoverns back EMF is the angular velocity or speedof the rotor and as the speed increases, back EMFalso increases. The potential difference across awinding can be calculated by subtracting the backEMF value from the supply voltage. The motors aredesigned with a back EMF constant in such a waythat when the motor is running at the rated speed,the potential difference between the back EMF andthe supply voltage will be sufficient for the motor todraw the rated current and deliver the rated torque.If the motor is driven beyond the rated speed, backEMF may increase substantially, thus decreasing thepotential difference across the winding, reducing thecurrent drawn which results in a drooping torquecurve.In general, Permanent Magnet Alternatingcurrent (PMAC) motors are categorized into twotypes. The first type of motor is referred to as PMsynchronous motor (PMSM). These producesinusoidal back EMF and should be supplied withsinusoidal current / voltage. The second type ofPMAC has trapezoidal back EMF and is referred toas the Brushless DC (BLDC) motor. The BLDCmotor requires that quasi-rectangular shapedcurrents are to be fed to the machine.When a brushless dc motor rotates,each winding generates a voltage known aselectromotive force or back EMF, which opposesthe main voltage supplied to the windings. Thepolarity of the back EMF is opposite to theenergized voltage. The stator has three phasewindings, and each winding is displaced by 120degree. The windings are distributed so as toproduce trapezoidal back EMF. The principle of thePMBLDC motor is to energize the phase pairs thatproduce constant torque. The three phase currentsare controlled to take a quasi-square waveform inorder to synchronize with the trapezoidal back EMFto produce the constant torque. The back EMF is afunction of rotor position (θ) and hasthe amplitudeE= Ke* ω (Ke is the back EMF constant).The instantaneous back EMF in BLDC iswritten as:Ea= fa(θ)*Ka*ωEb = fb(θ)*Kb*ω
  3. 3. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571052 | P a g eEc= fc(θ)*Kc*ωWhere, “ω” is the rotor mechanical speed and “θ” isthe rotor electrical position.Themodelling of the back EMF isperformed under the assumption that all threephaseshave identical back EMF waveforms. Basedon the rotor position, the numerical expression ofthe back EMF can be obtained Therefore, with thespeed command and rotor position, the symmetricthree-phase back EMF waveforms can be generatedat every operating speed.The respective back EMF in the windingsis represented by the equations:𝑒𝑎 =6𝐸𝜋𝜃 0 < 𝜃 <𝜋6𝐸𝜋6< 𝜃 <5𝜋6−6𝐸𝜋𝜃 + 6𝐸5𝜋6< 𝜃 <7𝜋6−𝐸7𝜋6< 𝜃 <11𝜋66𝐸𝜋𝜃 − 12𝐸11𝜋6< 𝜃 < 2𝜋𝑒𝑏 =−𝐸 0 < 𝜃 <𝜋26𝐸𝜋𝜃 − 4𝐸𝜋2< 𝜃 <5𝜋6𝐸5𝜋6< 𝜃 <9𝜋6−6𝐸𝜋𝜃 + 10𝐸9𝜋6< 𝜃 <11𝜋6−𝐸 (11𝜋6< 𝜃 < 2𝜋)𝑒𝑐 =𝐸 0 < 𝜃 <𝜋6−6𝐸𝜋𝜃 + 2𝐸𝜋6< 𝜃 <𝜋2−𝐸𝜋2< 𝜃 <7𝜋66𝐸𝜋𝜃 − 8𝐸7𝜋6< 𝜃 <9𝜋6𝐸 (9𝜋6< 𝜃 < 2𝜋)By putting E=1 in the above back EMFequations a back EMF function is obtained. Theback EMF function is a function of the rotorposition which is represented as fa(θ), fb(θ)& fc(θ)with limit values between -1 & 1 is defined as:𝑓𝑎(𝜃) =6𝜋𝜃 0 < 𝜃 <𝜋61𝜋6< 𝜃 <5𝜋6−6𝜋𝜃 + 65𝜋6< 𝜃 <7𝜋6−17𝜋6< 𝜃 <11𝜋66𝜋𝜃 − 12 (11𝜋6< 𝜃 < 2𝜋)𝑓𝑏(𝜃)=−1 0 < 𝜃 <𝜋26𝜋𝜃 − 4𝜋2< 𝜃 <5𝜋615𝜋6< 𝜃 <9𝜋6−6𝜋𝜃 + 109𝜋6< 𝜃 <11𝜋6−1 (11𝜋6< 𝜃 < 2𝜋)𝑓𝑐(𝜃)=1 0 < 𝜃 <𝜋6−6𝜋𝜃 + 2𝜋6< 𝜃 <𝜋2−1𝜋2< 𝜃 <7𝜋66𝜋𝜃 − 87𝜋6< 𝜃 <9𝜋61 (9𝜋6< 𝜃 < 2𝜋)The induced EMFs do not have sharpcorners, but rounded edges.The quasi-square trapezoidal back EMFwaveform and the phase current of the PMBLDCmotor with respect to the rotor position is shown inthe figure The graph is presented for one completecycle rotation of 360 degreesFIGURE (2): BACK EMF AND PHASECURRENTS WAVEFORMS OF BLDC
  4. 4. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571053 | P a g eTORQUE GENERATIONThe Torque is the product of the theoreticalmotor constant „Kt„the supplied current‟ I‟. In asingle pole system, usable torque is only producedfor 1/3 of the rotation. To produce useful torquethroughout the rotation of the stator, additional coils,or “phases” are added to the fixed stator. Thedeveloped torque by each phase is the product of themotor constant „kt„and the current „I‟.The sum of the torques isTa + Tb + TcAssumption made is all the phases are perfectsymmetryKt(motor)=Kt(a)=Kt(b)=Kt(c)imotor = ia= ib = icAt any given angle θ, the applied torque asmeasured on the rotor shaft isTmotor = 2* Kt(motor) * imotorThe key to effective torque and speedcontrol of a BLDC motor is based on relativelysimple torque and back EMF equations, which aresimilar to those of the DC motor. The generatedelectromagnetic torque is given byTe = [ eaia + ebib + ecic] / ω (in N.m)The electromagnetic torque is also relatedwith motor constant and the product of the currentwith the electrical rotor position which is given asTe = Kt{ fa(θ) ia+ fb(θ) ib + fc (θ) ic}The equation of motion for simple system is,J(dω/dt) +Bω = Te -TlWhere,Tl is the load torque, J is motor inertia, B is dampingconstant.The relation between angular velocity and angularposition (electrical) is given bydθ/dt = (P/2) * ωWhere, P is numbers of Poles,The Simulink diagram based on themathematical equations as described above isdesigned in MATLABSIMULINK as shown in thefigure The mat lab function block in the figure isdescribed the back EMF function. The equations ofback EMF functionis to be fed into “S-FunctionBlock” in Mat lab Simulink which passes theprogram written in M-file to the Mat lab workspace.FIGURE (3): MATHEMATICAL MODELREPRESENTATION OF BLDC MOTORPOSITION CONTROLIn most of the industrial processes likeelectrical, mechanical, construction, petroleumindustry, iron & steel industry, power sectors,development sites, paper industry, beveragesindustry the need for higher productivity isplacing new demands on mechanisms connectedwith electrical motors. They lead to differentproblems in work operation due to fast dynamicsand instability. That is why control is needed by thesystem to achieve stability and to work at desired settargets. The position control of electrical motors ismost important due to various non-linear effects likeload anddisturbance that affects the motor to deviate from itsnormal operation. The position control of the motoris to be widely implemented in machine automation.The position of the motor is the rotation ofthe motor shaft or the degree of the rotation which isto be controlled by giving the feedback to thecontroller which rectifies the controlled output toachieve the desired position. The applicationincludes robots (each joint in a robot requires aposition servo), computer numeric control (CNC)machines, and laser printers. The commoncharacteristics of all such systems is that thevariable to be controlled (usually position orvelocity) is fed back to modify the command signal.The BLDC motor employs a dc power supplyswitched to the stator phase windings of the motor
  5. 5. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571054 | P a g eby power devices, the switching sequence beingdetermined from rotor position. The phase current ofBLDC motor, in typically rectangular shape, issynchronized with the back EMF to produceconstant torque at a constant speed. The mechanicalcommutator of the brush dc motor is replaced byelectronic switches, which supply current to themotor windings as a function of the rotor position.To control the position of motor shaft, thesimplest strategy is to use a proportional controllerwith gain K. Figure shows the position control ofPMBLDC motor in which the motor output angularvelocity is integrated to obtain the actual position ofthe motor. The output is feedback to the input andthe error signal which is the differencebetween setpoint and actual motor position acts as the commandsignal for the PID controller.FIGURE (4): POSITION CONTROL OF PMBLDCMOTORLOOP TUNINGTuning a control loop is the adjustment ofits control parameters (gain/proportionalband,integral gain/reset, derivative gain/rate) to theoptimum values for the desired control response.Stability (bounded oscillation) is a basicrequirement, but beyond that,different systems have different behaviour,Differentapplications have differentrequirements,and requirements may conflict with one another.Some processes have a degree of non-linearity andso parameters that work well at full-load conditionsdont work when the process is starting up from no-load; this can be corrected by gain scheduling (usingdifferent parameters in different operating regions).PID controllers often provide acceptable controlusing default tunings, but performance can generallybe improved by careful tuning, and performancemay be unacceptable with poor tuning. PID tuning isa difficult problem, even though there are only threeparameters and in principle is simple to describe,because it must satisfy complex criteria within thelimitations ofPID control. There are accordinglyvarious methods for loop tuning, and moresophisticated techniques are the subject of patents;this section describes some traditional manualmethods for loop tuning. If the PIDcontrollerparameters (the gains of the proportional,integral and derivative terms) are chosen incorrectly,the controlled process input can be unstable, i.e. itsoutput diverges, with or without oscillation, and islimited only by saturation or mechanical breakage.Instability is caused by excess gain, particularly inthe presence of significant lag. Generally, stabilityof response is required and the process must notoscillate for any combination of process conditionsand set points, though sometimes marginal stability(bounded oscillation) is acceptable ordesired.The optimum behaviour on a processchange or set point change varies depending on theapplication. Two basic requirements are regulation(disturbance rejection - staying at a given set point)and command tracking (implementing set pointchanges) - these refer to how well the controlledvariable tracks the desired value. Specific criteriafor command tracking include rise time and settlingtime. Some processes must not allow an overshootof the process variable beyond the set point if, forexample, this would be unsafe. Other processesmust minimize the energy expended in reaching anew set point. There are several methods for tuninga PID loop. The most effective methods generallyinvolve the development of some form of processmodel, then choosing P, I, and D based on thedynamic model parameters. Manual tuning methodscan be relatively inefficient, particularly if the loopshave response times on the order of minutes orlonger. The choice of method will depend largely onwhether or not the loop can be taken "offline" fortuning, and the response time of the system. If thesystem can be taken offline, thebest tuning method often involves subjecting thesystem to a step change in input, measuring theoutput as a function of time, and using this responseto determine the control parameters.MANUAL TUNINGIf the system must remain online, onetuning method is to first set Ki and Kd values tozero. Increase the Kp until the output of the looposcillates, then the Kp should be set toapproximately half of that value for a "quarteramplitude decay" type response. Then increase Kiuntil any offset is correct in sufficient time for theprocess. However, too much Ki will causeinstability. Finally, increase Kd, if required, untilthe loop isacceptably quick to reach its referenceafter a load disturbance. However, too much Kdwill cause excessive response and overshoot. A fastPID loop tuning usually overshoots slightly to reachthe set point more quickly; however, somesystems cannot accept overshoot, in which case anover-damped closed-loop system is required, whichwill require a Kp setting significantly less than halfthat of the Kp setting causing oscillation.SIMULATION: - MATLAB Simulink model ofPermanent Magnet Brushless DC Motor andits position control using PID controller.
  6. 6. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571055 | P a g eREFERENCE CURRENTSRotor position(Degrees)Reference currentsIa Ib Ic0-30 0 -I I30-90 I -I 090-150 I 0 -I150-210 0 I -I210-270 -I I 0270-330 -I 0 I330-360 0 -I IFIGURE (5): TABLE SHOWING REFFERENCECURRENTSSIMULINK MODEL OF PMBLDCMOTORFIGURE (6): SIMULINK MODEL OFMATHAMATICAL MODEL OF PMBLDCMATHEMATICAL MODELLINGRESULTSFIGURE (7): REFERENCE CURRENTS (X-AXIS:TIME, Y-AXIS: CURRENT)FIGURE (8): PHASE CURRENTS (X-AXIS:TIME, Y-AXIS: CURRENT)FIGURE (9): BACK-EMF FUNCTION (X-AXIS:TIME, Y-AXIS: BACK-EMF)FIGURE (10): PHASE BACK-EMF‟S (X-AXIS:TIME, Y-AXIS: BACK-EMF)FIGURE (11): PHASE TORQUES (X-AXIS:TIME, Y-AXIS: TORQUE)
  7. 7. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571056 | P a g eFIGURE (12): TOTAL TORQUE (X-AXIS: TIME,Y-AXIS: TORQUE)FIGURE (13): SPEED (X-AXIS: TIME, Y-AXIS:SPEED)FIGURE (14): ROTOR POSITION (X-AXIS:TIME, Y-AXIS: DEGREES)SIMULINK MODEL OF POSITIONCONTROL OF PMBLDC MOTORFigure (15) below shows the positioncontrol of PMBLDC motor with PID controllerwhich is manually tuned to obtain the desired rotorposition. The PID values used is to average thecurrent which is fed to the inverter. The value atwhich the position is obtained at Kp=0.4, Ki=0.05and Kd=0.01. Subsequently graphical rotor positionwith time is shown below for the various angles.FIGURE (15): SIMULINK MODEL OF POSITIONCONTROL OF PMBLDCPOSITION CONTROL AT DIFFERENTVALUES OF THETAFIGURE (16): ROTOR POSITION FOR θ=350 (X-AXIS: TIME, Y-AXIS: DEGREES)FIGURE (17): ROTOR POSITION FOR θ=300 (X-AXIS: TIME, Y-AXIS: DEGREES)
  8. 8. Hemchand Immaneni / International Journal of Engineering Research and Applications(IJERA) ISSN: 2248-9622 www.ijera.comVol. 3, Issue 3, May-Jun 2013, pp.1050-10571057 | P a g eFIGURE (18): ROTOR POSITION FOR θ=250 (X-AXIS: TIME, Y-AXIS: DEGREES)FIGURE (19): ROTOR POSITION FOR θ=200 (X-AXIS: TIME, Y-AXIS: DEGREES)CONCLUSIONElectric machines are used to generateelectrical power in power plants and providemechanical work in industries. The DC machine isconsidered to be a basic electric machine. ThePermanent Magnet BrushlessDC (PMBLDC)motors are one of the electrical drives that arerapidly gaining popularity, due to their highefficiency, good dynamic response and lowmaintenance. The brushless DC (BLDC) motors anddrives have grown significantly in recent years inthe appliance industry and the automotive industry.BLDC drives are very preferable for compact, lowcost, low maintenance, and high reliability system.In this paper, a mathematical model ofbrushless DC motor is developed. The mathematicalmodel is presented in block diagram representationform. The simulation of the Permanent MagnetBrushless DC motor is done using the softwarepackage MATLAB/SIMULINK and its phasevoltage, phase current, back emf and torquewaveform are analysed. A PID controller has beenemployed for position control of PMBLDC motor.FUTURE SCOPE Tuning of PID controller for positioncontrol using Artificial Intelligencetechniques. Implementation of real time hardwaresystem for PMBLDC motor positioncontrol.REFERENCES1. P. S. Bimbhra, “Generalized Theory ofElectrical Machines”, Khanna Publishers,Delhi, India, 2001, pp. 93-98.2. G. K. Dubey, “Power Semiconductorcontrolled Drives”, Englewood, Cliffs,N.J.Prentice Hall, 19893. P. C. Sen, “Electric Motor Drives andControl: Past, Present and Future”, IEEETransaction on Industrial Electronics, Vol.IE37, No. 6, 1990, pp. 562-5754. J. J. D‟Azzo and C. H. Houpis, “Linearcontrol system analysis and design”,McGraw Hill, New York, 1995.5. Sim power Systems for use with Simulink,user‟s guide, Math Works Inc., Natick,MA, 2002. Math Works, 2001,Introduction to MATLAB, the MathWorks, Inc.6. T.J.E. Miller, “Brushless PermanentMagnet & Reluctance Motor Drives”Clarendon Press, Oxford, Vol.2, pp: 192-199, 1989