bldc

1576 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005

A Phase Variable Model of Brushless dc MotorsBased on Finite Element Analysis and Its Coupling

With External CircuitsOsama A. Mohammed, Fellow, IEEE, S. Liu, Senior Member, IEEE, and Z. Liu

Department of Electrical and Computer Engineering, Energy Systems Laboratory, Florida International University,Miami, FL 33174 USA

This paper presents a fast and accurate brushless dc motor (BLDC) phase variable model for drive system simulations. The developedmodel was built based on nonlinear transient finite-element analysis to obtain the inductances, back electromotive force as well as thecogging torque. The model was implemented in a Simulink environment through the creation of an adjustable inductance componentto account for the dependence of inductances on rotor position. Since no model for BLDC actually exists, the significance of thiswork is that it provides an accurate equivalent circuit model of BLDC motors for utilization in simulation environments. Using thedeveloped model, the sensorless control and the torque ripple control issues were investigated and the simulation results show its practicaleffectiveness.

Index Terms—Brushless dc motor (BLDC), finite element analysis (FEA), motor control, phase variable model.

I. INTRODUCTION

ACCURATE and efficient simulation of brushless dc motor(BLDC) machines, driven by power electronic switching

devices, is important for drive system design and optimization[1]–[5]. Two key issues related to this topic are the machinemodeling and the coupling between the machine model and ex-ternal circuits.

A fast and accurate machine model is always desirable. Com-pared with an equivalent electric circuit model, the finite ele-ment (FE) description is more accurate but can be time con-suming.

Two types of circuit models are available for machines: the-model and the phase variable model. BLDC has a trape-

zoidal back electromotive force (EMF) and requires rectangularstator currents to produce constant torque. The variation of theself and mutual inductances of the stator windings is nonsinu-soidal. No particular advantage exists in transforming theequations to the frame. The commonly used model as-sumes that the self and mutual inductances are constant [1]. Dueto the physical rotation of the rotor and the nonlinear magneti-zation property of stator iron, the inductance varies with rotorposition and winding current. Since the magnetic field of perma-nent magnet (PM) machines is mainly established by permanentmagnets, the effects of current are usually ignored. The rotorposition dependence of inductances can be accurately evalu-ated through nonlinear transient FE analysis. Similarly, the rotorposition dependence of the back EMF and the cogging torquecan be calculated from nonlinear transient FE analysis as well.Using these rotor position dependent parameters, the physicalphase variable model of BLDC is developed.

For dynamic performance studies, the time-stepping FE pro-cedure strongly couples the circuit equation with the FE motorequations then solve the coupled system simultaneously [3]–[5].

Digital Object Identifier 10.1109/TMAG.2005.845042

This gives accurate results but is time consuming. Using the de-veloped phase variable model, the drive system simulationbehaves much faster with the same level of accuracy.

The equation-based Simulink phase variable model is intro-duced by the authors in the context of PM synchronous machineimplementation[6]. Inorder toconnect theequation-basedmodeland external circuits, line voltage must be measured. For BLDCmachines, each commutation sequence has one winding that isenergized to positive power (current enters into the winding), onewinding is deenergized (current exits the winding) and the thirdwinding is in a nonenergized condition. This means that there al-ways exists one phase which is open circuited. The input voltageto this phase is immeasurable. This shows that the equation basedmodel cannot be applied to BLDC. An alternative model, com-posed of circuit components, is built to implement the phasevariable model of BLDC presented here. An adjustable induc-tance component is developed to represent the inductance depen-dence on the rotor position.

II. PHYSICAL PHASE VARIABLE MODEL

The phase variable model of BLDC machines is given as

(1)

(2)

(3)

and (4)

where is the back EMF, is the cogging torque,is the flux linkage contributed by the stator winding, andis the matrix of apparent inductance. The rest variables are usedas their conventional meanings.

The , , and profiles are obtained from the non-linear transient FE solutions, in which the rotor position depen-dence as well as the saturation effect are considered.

0018-9464/$20.00 © 2005 IEEE

Authorized licensed use limited to: RMIT University. Downloaded on February 24,2010 at 09:00:07 EST from IEEE Xplore. Restrictions apply.

MOHAMMED et al.: A PHASE VARIABLE MODEL OF BDCM 1577

Fig. 1. Circuit diagram of BLDC stator phase “a” winding.

III. SIMULINK IMPLEMENTATION

A. Voltage Equation

Expanding the derivative term of (1), one has

(5)

Substituting (2) into (5) and considering that the current in-dependence of the winding inductances, one can obtain

(6)

where are incremental inductances.Substituting (6) into (1), the voltage (1) becomes

(7)

Based on (7), the circuit diagram is constructed. As an ex-ample, Fig. 1 shows the diagram of phase “a” winding. The con-trolled voltage source (CVS) component is adopted to describethe voltage drop due to the flux cutting by the moving rotor.It represents the summation of the third and forth terms of (7).The derivative of apparent inductances with respect to the ro-tation angle ( , ), are calculated in advanceand stored in look-up tables.

An adjustable inductance component is developed to describethe self-inductance voltage drop of the BLDC ( ,

), seen in the second term of (7). As the inductancecurrent is a state variable, the adjustable inductance componentis built according to the integral description of inductance

(8)

Using as an example, the circuit diagram of the devel-oped adjustable inductance is illustrated in Fig. 2. The initialvalue of the integrator is set to zero. The voltage measurementblock (VM) and the controlled current source (CCS) are used torealize the connection of the adjustable inductance componentand the external circuits.

Fig. 2. Adjustable self inductance block.

Fig. 3. Equivalent circuit of phase “a” inductive voltage drop.

The mutual inductance voltage drop in the second term of (7),( , ), are represented by CVSs.

Fig. 3 is the equivalent circuit of phase “a” inductive voltagedrop. The current passing through the weighted line is . Theinductances , , and are retrieved from the incre-mental inductance table by picking the values corresponding toa specific rotor position.

B. Torque Calculation

As the rotor speed is used as a denominator in the torquecalculation of (3), it causes problems at the initial simulationstep due to the zero rotor speed. In order to solve this problem,a very small number is assigned to when starting the simula-tion. In addition, an initial value of the electromagnetic torque

, which is larger than the load torque is assigned to .Otherwise, according to (4), one knows that the motor will notstart moving. Consequently, the back EMF, , , and , equalzero and there will not be output toque .

The cogging toque are stored in tables and retrieved ac-cording to the rotor position.

C. Phase Variable Model

The developed physical phase variable model is shown inFig. 4. Subsystem 1 is the implementation of voltage (1), as in-troduced in section A. This subsystem is built usingand in form of tables. Subsystem 2 performsthe torque calculation according to (2). A value of 3.5 Nm, max-imum torque of this BLDC motor, is used as the initial torque to


1578 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005

Fig. 4. Developed physical phase variable model for BLDC.

Fig. 5. (a) Self inductance L . (b) Per-unit speed back EMF. (c) Coggingtorque. (d) dL =d�.

start the rotor’s motion. It lasts 0.1ms, which is controlled by astep function block. The motion (4) is performed at the right por-tion of Fig. 4. The two tables in Fig. 4 are the cogging torque andback EMF. The unit speed back EMF is storedin the back EMF table to perform speed control simulation.

The table data are obtained from nonlinear transient FE anal-ysis. In order to obtain high accuracy, attentions to the rotatingair gap mesh and the time step in transient FE analysis are given.Nodes on the lateral dimension of the rotating air gap must beevenly distributed. The time step of transient analysis should bekept as the time required for moving the radial angle betweentwo contingent nodes.

IV. MODEL VERIFICATION

As an example, the phase variable model of a 4-pole 24-slot24-V BLDC motor is built. The incremental inductance ,per-unit speed back EMF, cogging torque, and the derivative of

Fig. 6. (a)–(b) From full FE model. (c)–(d) From phase variable model.

Fig. 7. Zero crossing detection of back EMF.


MOHAMMED et al.: A PHASE VARIABLE MODEL OF BDCM 1579

Fig. 8. (a), (c), and (e): without torque ripple control; (b), (d), and (f): withtorque ripple control.

mutual inductance with respect to rotor position areshown in Fig. 5.

The performance of the developed physical phase variablemodel is examined by comparing it with the full FE model inan electric commutation circuit. The obtained torque and threephase current profiles during the starting process are given inFig. 6. It shows that the developed physical phase variable modelis capable of providing the same dynamic simulation character-istics as the full FE model.

V. APPLICATIONS

Sensor-less control and torque ripple minimization of ma-chines are two topics of major interess by researchers. The con-trol strategy study is usually performed via experiments and/orcomputer simulation. The computer simulation is the only pos-sible way in the machine design stage. A fast and accurate modelfacilitates such a study.

A. Sensorless Control

The zero-crossing instance of back EMF waveform canbe utilized to perform senseless motor control. As the backEMF waveform of the motor cannot be directly measured,various methods were proposed to identify the back EMFzero-crossing. [7] introduced a method to find the zero-crossinginstance through the phase to ground PWM waveform, whichwas verified by experiments. In this paper, we provide thesimulation verification of their zero-crossing detection methodby using the developed physical phase variable model. Theback EMF waveform obtained from simulation as well as theswitching off process of PWM is shown in Fig. 7(a), whichreproduces the experiment results given in [7]. Therefore, wecan say that the developed phase variable model provides an

alternative way for the verification of control strategies. Fur-thermore, with the developed model, the accuracy of the controlstrategy can be investigated. In addition, the inverter parameterdesign can be studied. Fig. 7(b) demonstrates the consequenceof inappropriate inverter parameters to the sensorless controlstrategy of [7]. In this case, the zero crossing point cannot beidentified properly.

B. Torque Ripple Reduction

The ripple torque occurs as the results of fluctuations of thefield distribution and the armature MMF. Using the developedphysical phase variable model, the fluctuations of the field dis-tribution and the armature MMF can be captured. In this way,the torque ripple control can be studied under the real situa-tions. As an example, the phase variable model is applied toa ripple torque forward-fed compensation method. The resultsare shown in Fig. 8, which demonstrates the effects of the torqueripple control. The torque ripple is greatly reduced and the speedbecomes smooth.

VI. CONCLUSION

A physical phase variable model of BLDC machines is pro-posed. Its parameters are obtained from transient FE analysisof the machine. Simulink implementation details of the pro-posed model are presented. The proposed phase variable modelhas the accuracy of the full FE model with much faster simula-tion speed. To maintain the accuracy of the proposed model, theinductance was distinguished as apparent and incremental in-ductances. The per-unit speed back EMF was used as the backEMF table data. The rotating air gap meshing and the time step-ping FE analysis were related. The effectiveness of the proposedphase variable model was tested through implementation of sen-sorless control and torque ripple reduction examples.

REFERENCES

[1] P. Pillay and R. Krishnan, “Modeling, simulation, and analysis of perma-nent-magnet motor drives, part 2: The brushless dc motor drive,” IEEETrans. Ind. Appl., vol. 25, no. 2, pp. 274–279, Mar./Apr. 1989.

[2] M. A. Inerfield, S. L. Garverick, W. S. Newman, and Y. Zhang, “ASPICE model for a novel brushless adjustable-speed drive,” IEEE Trans.Ind. Electron., vol. 47, no. 6, pp. 1307–1318, Dec. 2000.

[3] S. L. Ho, W. N. Fu, H. L. Li, H. C. Wong, and H. Tan, “Performanceanalysis of brushless dc motors including features of the control loopin the finite element modeling,” IEEE Trans. Magn., vol. 27, no. 5, pp.3370–3374, Sep. 2001.

[4] M. A. Jabbar, H. N. Phyu, Z. Liu, and C. Bi, “Modeling and numericalsimulation of a brushless permanent-magnet dc motor in dynamic con-ditions by time-stepping technique,” IEEE Trans. Ind. Appl., vol. 40, no.3, pp. 763–770, May/Jun. 2004.

[5] G. H. Jang, J. H. Chang, D. P. Hong, and K. S. Kim, “Finite-element anal-ysis of an electromechanical field of a BLDC motor considering speedcontrol and mechanical flexibility,” IEEE Trans. Magn., vol. 38, no. 2,pp. 945–948, Mar. 2002.

[6] O. A. Mohammed, S. Liu, and Z. Liu, “Physical modeling of PM syn-chronous motors for integrated coupling with machine drives,” IEEETrans. Magn., vol. 41, no. 5, pp. 1628–1631, May 2005.

[7] T. Shao, D. Nolan, M. Teissier, and D. Swanson, “A novel micro-controller-based brushless dc(BLDC) motor drive for automotivefuel pumps,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1734–1740,Nov./Dec. 2003.

Manuscript received June 8, 2004.


bldc

Documents