Modeling, Simulation, And Control Of Switched Reluctance Motor Drives

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 52, NO. 6, DECEMBER 2005

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Modeling, Simulation, and Control of Switched Reluctance Motor Drives Iqbal Husain, Senior Member, IEEE, and Syed A. Hossain, Member, IEEE

Abstract—This paper presents the modeling, simulation, and control aspects of four-quadrant switched reluctance motor (SRM) drives. The design of SRM drive systems must be focused on application-based appropriate control and engineering solutions needed to overcome the practical issues. A complex model is described for the physical motor simulation to incorporate the important dynamics of the SRM. A simpler, but quite accurate, model is presented for the SRM controller. Various practical limitations have been incorporated in the simulation model to make it closer to the experimental setup. The SRM control parameters are chosen based on torque-per-ampere maximization requirement. Experimental results for a 1.0-kW SRM obtained on a digital platform are presented along with useful guidelines for prototype implementation. Index Terms—Four-quadrant controls, switched reluctance machines.

I. I NTRODUCTION

T

HE INHERENT simplicity, ruggedness, and low cost of a switched reluctance motor (SRM) makes it a viable candidate for various general-purpose adjustable-speed and servo-type applications. The SRM drives have the additional attractive features of fault tolerance and the absence of magnets. However, due to the doubly salient construction and the discrete commutation from one phase to another, high-performance torque control of this type of motor is a critical issue for servo-type applications. A sophisticated control technique can improve the operating performance for the entire motor drive system. The development of a servo-category drive system demands a good computer-simulation model to reduce the expensive and time-consuming experimental stage. The block diagram of an SRM drive system is given in Fig. 1. The controller has two parts: outer loop controller and inner loop controller. The outer loop generates the reference torque or reference speed from the position or speed error. The SRM drive system is in the inner loop. Dynamic modeling and simulation play critical roles in the inner loop controller design, drive system analysis, and future development. The importance of appropriate modeling

Manuscript received September 24, 2003; revised May 6, 2005. Abstract published on the Internet September 26, 2005. This work was supported in part by a research award from Delphi. I. Husain is with the Department of Electrical and Computer Engineering, University of Akron, Akron, OH 44325-3904 USA (e-mail: ihusain@ uakron.edu). S. A. Hossain was with the Department of Electrical and Computer Engineering, University of Akron, Akron, OH 44325-3904 USA. He is now with Globe Motors, Dayton, OH 45404-1249 USA. Digital Object Identifier 10.1109/TIE.2005.858710

is significant during both the computer-simulation and the realtime implementation stages of the drive system. The performance of an SRM drive system is enhanced through optimization of a desired criterion, which set the appropriate control parameters of turn-on angle, turn-off angle, and reference current [1]. The maximization of torque per ampere by an SRM is considered in this paper. This optimization may yield the use of a smaller motor for a given application or a faster response time for a given motor. Electromechanical actuators and traction-type loads require motor operation in the position-controlled mode with fast response characteristics. A four-quadrant drive is essential for such servo-type applications. In any operating quadrant, maximum torque per ampere is the desirable quantity, either for fast forward or reverse motion or for fast motion-direction reversal. This paper demonstrates the development of an SRM drive considering all the practical implementation issues. The issues arising at the hardware and software development stages have been addressed. II. SRM M ODEL The motor modeling lends itself to two distinctive approaches when considering the objective of modeling. A precise model is presented for the physical motor simulation to incorporate the important dynamics of the SRM. A simple, but quite accurate, model is presented for the SRM controller. A. Physical Motor Model In the computer-simulation stage, an accurate model of the physical motor is necessary to depict the real scenario, where the computation time is not at all critical. In the case of SRM, the machine is always operated in the magnetically saturated mode to maximize the energy transfer. The magnetic nonlinearities of an SRM can be taken into account by appropriate modeling of the nonlinear flux–current–angle (λ−i−θ) characteristics of the machine. The output electromagnetic torque of the machine is described by the nonlinear torque–current–angle (T −i−θ) data. The machine model may then be described by λ = λ(i, θ)

(1)

T = T (i, θ).

(2)

A first approach consists of look-up tables, with the predicted flux linked λ(i, θ) and the static torque T (i, θ) expressed as functions of current level i and rotor position θ. The look-up

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Fig. 1. SRM drive system.

table is typically populated using data from static tests or results from finite-element analysis (FEA). An alternative approach consists of using a geometry-based analytical model described in [2] and [3]. The model uses an analytical solution for the flux linked and static torque produced by one SRM phase. Separate analytical models for the flux linked by a phase of the SRM when its stator and rotor poles do and do not overlap are combined to provide a complete model of a given motor phase. When the poles overlap, saturation must be included, especially in the pole tips, whereas a linear representation can be used in the unaligned position. The flux linked by the SRM phase is determined from the sum of the main flux and the fringing flux that is linked by the phase. The analytical model could be used to populate the look-up tables mentioned earlier, or to directly calculate the motor parameters. The analytical model is preferably described in terms of the machine geometry and material properties. The geometrybased model allows extensive computer-simulation studies during the drive design stage. The general form of a geometrybased analytical expression for flux linkage used is λ(i, θ) = Am (θ, ξ) + Af (θ, ξ)
− Bm (θ, ξ) Cm (ξ) + Dm (i, ξ) + Em (i2 , ξ)
− Bf (θ, ξ) Cf (ξ) + Df (i, ξ) + Ef (i2 , ξ) (3) where A, B, C, D, and E are ξ- and θ-dependent constants. ξ stands for geometry and magnetic properties and θ stands for rotor position. Subscripts m and f are for main and fringing components of the flux linkage. The physical motor is simulated using either flux λ or current i as the state variables. Rotor position and speed (ω) constitute the state variables for the mechanical subsystem. The statespace format that describes the SRM dynamics is  λ˙ j = vj − ij rj or i˙ j =

∂λj ∂ij

−1   ∂λj ω + vj −ij rj − ∂θ

(for each jth phase) θ˙ = ω

  Nph  B 1 ω˙ = − ω +  Tj − TL  J J j=1

(4) (5) (6)

vj is the applied phase voltage, rj is the phase resistance, B is the viscous damping constant, TL is the load torque, and Tj is the torque of each phase. The output variable i is obtained from the machine (λ−i−θ) characteristics when flux is used as the electrical state variable. If currents are used as the electrical state variables, then (∂λj )/(∂θ) and (∂λj )/(∂ij ) need to be calculated from the (λ−i−θ) characteristics. The electromagnetic torque is derived from the flux-linkage expression of (3) using  i   ∂  λ(i, θ)di . Tj (θ, i) = ∂θ 0

The magnetic and mechanical loss models must also be incorporated in the physical motor, since the static (λ−i−θ) characteristics does not include the losses. B. Controller SRM Model The analytical-model equations are too complex to be implemented directly in the controller code for the real-time control of a machine. The controller model requires being fairly simple so that the computation cycle time is minimized. The following SRM flux model [5] expressed as a function of phase current and rotor position has been used in the implementation λ = λu + (λa − λu ) · f (θ)
=Lu i + (ai + b − b2 − ci + di2 ) · f (θ)

(7)

where a, b, c, and d are the machine-geometry- and ironmagnetic-property-dependent terms, which are given in Appendix I. The use of two different models for the controller and physical motor in the computer-simulation stage represents the real scenario. Equation (7) has been validated by comparing the modelpredicted result with the FEA result. The static plots of fluxlinkage (λ) and torque (T ) characteristics of an SRM obtained from the two models described above and the FEA method were compared with each other to ensure that the prediction from the proposed models are satisfactory. The characteristic data from the FEA method and static experiments are typically very close. Fig. 2 shows the λ−i−θ and T −i−θ characteristic plots for a four-phase 8/6 SRM.

HUSAIN AND HOSSAIN: MODELING, SIMULATION, AND CONTROL OF SRM DRIVES

Fig. 2.

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Comparison of torque and flux characteristics between analytical model, controller model, and FEA.

III. SRM C ONTROLLER D ESIGN The SRM drive controller parameters must be selected to optimize the design objective. The choice of control parameters to maximize the torque per ampere (T /A) is described in this section. For a high-performance drive, the SRM drive inner loop controller functions to deliver the reference torque commanded by the outer control loop. At high speeds, the available torque of SRM decreases due to back-electromotive force (EMF) voltage and input dc voltage saturation. The maximization of T /A is achieved by a field-weakening technique, which requires the advancement of turn-on and turn-off angles. At low speeds, the current limit restricts the available torque and a fixed set of turn-on and turn-off angles is sufficient for successful magnetization and demagnetization. Depending on the operating speed, the optimization problem can be divided into two parts under constraints of current or voltage limits. The transition from one constraint to the other is based on the motor base speed. A. Maximization With Current Constraint At low speeds, the current constraint is active and the controloutput parameters are turn-on angle θon , turn-off angle θoff , and reference current Iref . Therefore, the optimization problem can be defined as max

θon ,θoff ,iref

Tav = max θon ,θoff ,iref iref



qph Nr 2πiref



dJ(θon , θoff , iref ) =0 dθoff dJ(θon , θoff , iref ) = 0. diref

 h i=− + 2g

h2 −

4gT f
(θ)

2g

(8)

where g and h are machine-geometry-dependent terms explained in [4], and f
(θ) is a position-dependent term. Equation (8) has been validated by comparing the model-predicted result with the FEA result in Fig. 2. Defining the reference current in terms of the reference torque, a univariate-search technique has been used to determine the turn-on and turn-off angles [1]. The guiding logic behind univariate search is to change one variable at a time so that the function is maximized in each of the coordinate directions. The optimal-control parameters for operating speed below base speed can be represented by the following equations Optimal turn-on angle = a1 Optimal turn-off angle = a2

i(ψ, θ)dψ

where the performance index J(θon , θoff , Iref ) is Tav /Iref . The optimization problem can be solved by defining the following necessary conditions: dJ(θon , θoff , iref ) =0 dθon

An analytical solution of the necessary conditions is difficult due to the highly nonlinear characteristics of the SR machine. Numerical or graphical optimization techniques can be employed to accommodate models containing significant nonlinearities [5]. Therefore, the reference current is generated from the torque requirement using

where the constant a’s are determined through the optimization program. The optimization results show little or no variation of the turn-on and turn-off angles with motor speed as long as the speed is below the base speed. The torque is regulated by controlling the phase current according to (8).

B. Maximization With Voltage Constraint The SRM operates in single-pulse mode above the base speed. Torque-per-ampere maximization under voltage constraint yields a solution that is referred to as optimal field weakening. The control parameters are only the turn-on and turn-off

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Fig. 3. Energy versus speed with: (a) turn-off angle and (b) turn-on angle variations.

Fig. 4. Schematic representation of the controller.

angles, with the peak-current limit set to the rated value. The problem can now be defined as 

qph Nr ∗ i(ψ, θ)dψ . max Tav = max θon ,θoff θon ,θoff 2π Using the univariate-search technique, the overall solution is a function of speed ω given by Optimal turn-on angle = a11 + a∗12 ω + a∗13 ω 2 Optimal turn-off angle = a21 + a∗22 ω + a∗23 ω 2 where the coefficient a’s are determined through the optimization program. The influence of the turn-on angle and turn-off angles on energy conversion per stroke in the motoring mode at different speeds is shown in Fig. 3(a) and (b). The turn-on angle is kept constant in Fig. 3(a), while the turn-off angle is kept constant in Fig. 3(b). Observing the two figures, it is obvious that the energy conversion is increasing by advancing the turn-on and turn-off angles at high speed. IV. F OUR -Q UADRANT C ONTROL S TRATEGIES In many applications, the SRM operates in the torquecontrolled mode, with the command torque set by an outer loop position controller. The SR drive controller in the inner loop functions to maintain the desired command torque. Fast response is critical for highly dynamic loads, where the command torque and the motor speed may reverse quickly from positive

to negative and vice versa. The motor operates in all four quadrants of its torque-speed characteristic. The flow diagram of the controller is shown in Fig. 4. The controller switches the motor between the motoring and braking regions according to the control command. The appropriate and different turn-on and turn-off angles are used depending on the operating quadrant. The turn-on and turn-off angles determined in Section II are used for first- and fourth-quadrant operation. These angles are scheduled as a smooth function of speed, which maximizes the average torque at all operating points. Operation in the second and third quadrants is the mirror symmetry of that of the fourth and first quadrants, as shown in Fig. 5. The relationships between the firing angles are 2π − θon,I Nr 2π =− − θoff,I Nr

θon,III = − θoff,III

θon,II = − θon,IV θoff,II = − θoff,IV . The four-quadrant drive may be required to operate either in the position-controlled loop or in the speed-controlled loop. In the controller, the operating quadrant is determined from the sign of reference torque (or reference speed for speedcontrolled drive) and motor speed, as shown in Fig. 4. V. S IMULATION OF SR D RIVE The objective of the simulation model is to predict results that would match the experimental results such that lengthy experimental procedures can be eliminated during product development. The disparity between results from the computersimulation stage and the hardware implementation can be traced to two aspects, which are: 1) accuracy of the measured variables and 2) the delay in signal processing. Some of this disparity can be accounted for in the simulation by inserting delays in the controller execution cycle according to the speed of the processor used and the number of instructions of the controller algorithm. The delays in acquiring measured data

HUSAIN AND HOSSAIN: MODELING, SIMULATION, AND CONTROL OF SRM DRIVES

Fig. 5.

Operating points in different quadrants.

Fig. 6.

Simulation block diagram of the SRM drive.

through the analog-to-digital (A/D) channels can also be accounted for according to the A/D sampling and conversion time required. The SRM analytical model described in Section II, having the significant nonlinear saturation characteristics, has been used to represent the motor in the simulation setup. The simulation block diagram of the SRM drive is shown in Fig. 6. The simulation setup incorporated the practical nonidealities such as controller delay time, position-sensor resolution, and errors in current measurements. These measures provided results that closely matched the experimental results. A. Simulating the Execution-Time Delay The input signals to the controller e(t) are approximately constant within the execution-time interval, at a value equal to those of the signals at the preceding sampling instant. A

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zeroth-order hold (ZOH) is introduced to incorporate this effect into the simulation [5]. The ZOH implements a sample-andhold function operating at execution-time (Tex ) period rate. Therefore, for a ZOH ek (t) = e(kTex ),

kTex ≤ t < (k + 1)Tex .

Fig. 7 shows a comparison of measured current and simulated current when the motor was operated at 2000 r/min with a load torque of 0.4 N·m. The simulation-current waveform without any limitation and using 5 µs of simulation time step and the same execution time step deviates significantly from the measured current. However, the simulated current with consideration of the practical limitations such as the A/D quantization error and the delay due to execution loop frequency gives a good estimation of the actual current.

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system is directly related to controller execution time. The execution time is given by Execution time TE = Nins ∗ CPI ∗ Clock period

Fig. 7. Measured and simulated phase current at 2000 rpm and 0.4 N·m. TABLE I QUANTIZATION INTERVAL

where Nins is the number of instructions executed in one control cycle, and CPI is the average clock cycles required per instruction. The floating-point digital signal processor (DSP), programmed using C compiler, is advantageous in terms of accuracy and ease of implementation compared to a fixed-point DSP. The fixed-point processors require significant time for scaling and shifting operations for complex computations. The execution times required for a simple SRM drive controller with fixed turn-on and turn-off angles and hysteresis current control for a four-phase 8/6 SRM developed in TMS320C240 and TMS320C30 (clock speed = 40 MHz) were observed to be about 40 and 35 µs, respectively. A more advanced SRM controller implementation in TMS320C30 would typically take between 80–100 µs. B. Position-Sensor Resolution

B. Simulating the Quantization Effect The output of the encoder used for rotor-position sensing is quantized, which may lead to limit cycles in digital control systems. The quantization process involved in A/D conversion also cause errors in the measurements [6]. The dc-link voltage and phase currents are commonly digitized for the SRM controller. Both the dc-link voltage and the phase current are unipolar. Accordingly, the quantization errors of voltage and currents for an n-bit A/D converter are given by ∆ν =

Vmax 2n − 1

∆i =

Imax . 2n − 1

The rotor position, dc-link voltage, and phase currents used in the controller are to be quantized according to Table I in order to incorporate the quantization effect in the simulation. VI. R EAL -T IME I MPLEMENTATIONS An experimental setup was established to test the optimalcontrol algorithms of Section III and also to evaluate the developed simulation tool. The prototype setup of the SR drive is shown in Fig. 8. The test setup was designed for motioncontrol applications, which can be operated in variable-speedcontrolled or position-controlled loop. The parameters of the SRM are given in Appendix II. A. Processor-Speed Metrics The choice of the required processor depends on two important factors: the processor speed and the facility to allow fast and efficient programming. The performance of the drive

The resolution of the position is effective only when every position update information can be utilized in each controller execution period. The time for each updated position information when the motor is in steady state depends on the operating speed of the machine. If the position feedback system sends NA number of pulses for each mechanical revolution of the rotor, then the time for one pulse is Tp =

60 s N p · NA

where Np is the operating speed of the machine in revolutions per minute. The condition for the position-sensor feedback information to be utilized effectively is TE > TP . Alternatively, a critical speed Ncr can be defined as the maximum speed at which any particular program will be able to capture and utilize each and every updated position pulse. The critical speed is Critical speed, Ncr =

60 rev/m. T E · NA

C. Filtering Effects In SRM drives with a discrete position sensor, the encoder output is quantized, and the velocity is determined by differentiating the quantized signal. The position is also obtained by differentiation in some position-sensorless control algorithms. The differentiation amplifies the high-frequency noise. Moreover, the A/D converters introduce measurement noise to the signal. A digital recursive filter can be implemented to remove noise from the speed signal as follows: yk+1 = auk+1 + (1 − a)yk ,

0
(9)

where u and y are the input and output of the filter, respectively. The coefficient “a” determines the cutoff frequency of the

HUSAIN AND HOSSAIN: MODELING, SIMULATION, AND CONTROL OF SRM DRIVES

Fig. 8.

Hardware setup for the SRM drive system.

Fig. 9.

Filtering effect on speed measurement.

filter. The coefficient “a” can be chosen through simulation by analyzing the effect of the inherent delay introduced by the filter. Fig. 9 shows the motor speed derived from differentiation of the encoder output and then smoothened by a digital filter. The value of the coefficient a has been chosen to be 0.01.

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Fig. 10. Measured four-quadrant closed-loop speed response.

VII. E XPERIMENTAL R ESULTS In this section, experimental results are presented to show the performance of the SRM drive system. Four A/D channels having a sampling time of 4 µs are used to digitize the phasecurrent information. One A/D channel having a sampling time of 4 µs is also used to receive the bus-voltage information for the controller. The phase voltages are reconstructed from the bus voltage and gate signals. A 360-pulses-per-revolution encoder is used to provide the rotor-position feedback information. The current control is implemented through software to avoid the necessity of any additional hardware. The current hysteresis control inside the digital controller increases the hysteresis band, which leads to higher torque ripple. It also becomes difficult to maintain current within the band due to processor computation cycle time. However, the insidehysteresis current control is a good choice for many costeffective actuator applications where the torque ripple is not an important issue. A significantly smaller hysteresis band can be designed when the hysteresis current regulator is built outside the digital controller. A. Speed-Controlled Loop The SRM was first connected to an inertia load to verify the variable-speed operation. The closed-loop response for a toggling speed command from 1000 to −1000 rpm is shown in Fig. 10. The motor was thus required to switch operation

Fig. 11. Measured phase voltage, current, and estimated flux at 500 rpm.

from forward motoring (first-quadrant operation) to reverse motoring (third-quadrant operation) through regenerating mode (fourth-quadrant operation), and also from reverse motoring to forward motoring through regenerating mode (second-quadrant operation). The current is regulated in the active phase by chopping at the reference level commanded by the speed regulator. Fig. 11 presents the measured phase voltage, phase current, and estimated phase flux with inside-hysteresis current control. In the speed-controlled mode, the current is regulated in the active phase by chopping at the reference level commanded by the speed regulator. Fig. 12(a) shows the measured phase currents during motor acceleration. Fig. 12(b) shows the measured phase-A current along with the wrapped rotor position during the deceleration mode. In the wrapped rotor position, 0◦ is the aligned position and ±30◦ are the unaligned positions. The figure also shows the correct positioning of the phase current with respect to the rotor position in the regeneration mode. The optimized turn-on and turn-off angles produce a small amount

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Fig. 12. Measured phase currents during: (a) acceleration and (b) deceleration.

Fig. 13. Hydraulic drive system.

of opposite polarity torque. The opposite polarity torque produced by the rising or decaying phase current may appear to be undesirable, since it will adversely affect the machine efficiency and torque ripple. However, the small opposite polarity torque helps to maximize the average torque in each phase cycle of operation, which is required for fast actuation.

Fig. 14.

Measured and simulated closed-loop position-control response.

B. Position-Controlled Loop In this test, the SRM was used to drive a linear actuator coupled through a ballscrew arrangement. Any position change of the motor is converted into force on the piston that has a restoring spring, as shown in Fig. 13. The experimental results of the closed-loop position-controlled system are shown in Fig. 14. The rotor-position information is used to measure the translational displacement of the piston. Figs. 14 and 15 show the linear displacement and speed responses, respectively, under load-torque disturbances. The performance of the drive system is also compared with the simulation results to evaluate the prediction performance of the simulation tool. These figures represent four-quadrant response of the SRM for the linear displacement control of the actuator system. The optimized turn-on and turn-off angles obtained in Section III are used in the inner loop of the SRM drive. Table II shows the effect of the variations of turn-on and turn- off angles

Fig. 15. Measured and simulated motor speed during closed-loop positioncontrol response.

on response time. The response time (rise time) is considered as the time required for translational movement of 10%–90% of the position command. The position-controlled drive system was operated in a test bench by varying the turn-on and turn-off

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TABLE II RESPONSE-TIME COMPARISON

Fig. 17. Motor speed versus time with sudden command change.

VIII. C ONCLUSION

Fig. 16. Comparison of normal and optimal-control strategies for position control with sudden command change.

angles one mechanical degree around the optimal angles. The test results prove that the optimal turn-on and turn-off angles give the fastest response. C. Response in Highly Dynamic Operation The linear-actuator load was used to evaluate the system response in a highly dynamic mode. Fig. 16 shows the linear position versus time for a ramp command. The position command is reversed when a linear position of 0.2 mm is reached in the forward direction. The target is to minimize the subsequent overshoot and the reverse time. The figure shows the comparison of position overshoot (ξ) and reversal time (T ) for two different control strategies: one for optimal-control strategy, as described in Section III-B, and the other is the control strategy suggested by Kjaer et al. [7]. The latter control strategy used a turn-off angle at lower speeds, then switched to another turn-off angle at higher speeds. The a, b, and c reference points in Fig. 16 are for the optimalcontrol response plot. In the figure, the position command changes to 0 at point a, which changes the motor operation from forward motoring to the forward braking region. The motor operates in the forward braking region between points a and b. The forward movement after the command change leads to the overshoot of the system. From points b to c, the motor first operates in the reverse motoring mode and then in the reverse braking mode. Fig. 17 shows the motor-speed dynamics for optimal-control position response in Fig. 16. The optimally controlled four-quadrant operation of the SRM helps minimize the overshoot and reversal time of the mechanical drive system.

The importance of a good modeling and simulation tool for digital implementation of an SRM motor drive system has been emphasized in this paper. Modeling and simulation, incorporating the practical nonidealities as accurately as possible, drastically reduces the time and cost associated with extensive experimentation. Practical problems, such as measurement error, processor delay time, and quantization factors can be easily incorporated in the simulation model. Once satisfied with the performance obtained from simulation, one can start a prototype system development on a test bench. The practical problems experienced on the test bench can be used to improve the simulation model. The iterative process in developing a very useful simulation tool has been shown to be very effective in the development of a servo-type four-quadrant switched reluctance motor (SRM) drive system presented in this paper. The results obtained from the final simulation model and the experiments are extremely close. Such a model could be reliably used for performance evaluation and future development. A PPENDIX I The coefficients used in (7) and (8) are dependent on machine geometry as follows: a = λm am − Lu

b = λ m bm

c = λ2m cm

d = λ2m

and 

√  a d g= − 2 2   b c h= − √ 2 4 d   1 f (θ) = ∗ (1 − cos θ) 2 f
(θ) = 0.5 ∗ Nr ∗ (sin θe ), = 0.5 ∗ Nr ∗ tanh(π − θ),

θe < Nr (π − βr ) θe ≥ Nr (π − βr )

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where λm =

nser Np Rg · lstk · stf . · βr · µ0 · npar 2 g

am = 1 +

2g lp

bm =

npar · Bsat · [lp + (µr + 1)g] µNp

cm =

2npar · Bsat · [lp − (µr − 1)g]. µNp

Here, Np is the number of turns per pole, nser is the number of series paths, npar is the number of parallel paths, lstk is the stack length, stf is the stacking factor, Rg is the radius to rotor pole tips, g is the air-gap length, lp is the total length of rotor and stator poles, βr is the rotor-pole width, µ is the iron permeability, µr is the relative permeability, and Bsat is the saturation flux density. A PPENDIX II SRM Ratings and Parameters Inverter dc bus voltage 30 V Power 1000 W Peak current 40 A Number of stator poles 8; 22◦ pole arc Number of rotor poles 6; 22◦ pole arc Stator winding resistance 0.179 Ω

[4] I. Husain, A. Radun, and J. Nairus, “Unbalanced force calculation in switched reluctance machines,” IEEE Trans. Magn., vol. 36, no. 1, pp. 330– 338, Jan. 2000. [5] S. Hossain and I. Husain, “A geometry based simplified analytical model of switched reluctance machines for real-time controller implementation,” in Proc. IEEE Power Electronics Specialists Conf. (PESC), Cairns, Australia, Jun. 2002, pp. 839–844. [6] T. Hartley, G. Beale, and S. Chicatelli, Digital Simulation of Dynamic Systems: A Control Theory Approach. Englewood Cliffs, NJ: PrenticeHall, 1994. [7] P. C. Kjaer, J. J. Gribble, and T. J. E. Miller, “High-grade control of switched reluctance machines,” in Proc. IEEE Conf. Industry Applications, San Diego, CA, Oct. 1996, pp. 92–100.

Iqbal Husain (S’89–M’89–SM’99) received the B.Sc. degree from the Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 1987, and the M.S. and Ph.D. degrees from Texas A&M University, College Station, in 1989 and 1993, respectively, all in electrical engineering. Previously, he was a Lecturer at Texas A&M University and a Consulting Engineer at Delco Chassis, Dayton, OH. In 1996 and 1997, he was a Summer Researcher at Wright Patterson AFB Laboratories. He is currently a Professor at the Department of Electrical and Computer Engineering, University of Akron, Akron, OH, engaged in teaching and research. His research interests are in the areas of control and modeling of electrical drives, design of electric machines, and development of power conditioning circuits. He has worked extensively in the development of switched reluctance motor drives including sensorless controllers. Dr. Husain received the 1998 IEEE Industry Applications Society (IAS) Outstanding Young Member Award, the 2000 IEEE Third Millenium Medal, and the 2004 College of Engineering Outstanding Researcher Award. He was also the recipient of three IEEE IAS Committee Prize Paper Awards.

ACKNOWLEDGMENT The authors are grateful to Delphi for providing an experimental hardware for this research. R EFERENCES [1] S. Hossain, I. Husain, H. Klode, B. Lequesne, and A. Omekanda, “Four quadrant control of a switched reluctance motor for a highly dynamic actuator load,” in Applied Power Electronics Conf. and Expo. (APEC), Dallas, TX, Mar. 2002, pp. 41–47. [2] A. Radun, “Analytically computing the flux linked by a switched reluctance motor phase when the stator and rotor poles overlap,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1996–2003, Jul. 2000. [3] ——, “Analytically calculation of switched reluctance motor’s unaligned inductance,” IEEE Trans. Magn., vol. 35, no. 6, pp. 4473–4481, Nov. 1999.

Syed A. Hossain (S’01–M’02) received the B.Sc. and M.Sc. degrees in electrical and electronic engineering from Bangladesh University of Engineering and Technology, Dhaka, Bangladesh, in 1994 and 1997, and the Ph.D. degree in electrical engineering from the University of Akron, Akron, OH, in 2002. From 1994 to 1998, he was a Lecturer and then an Assistant Professor at Bangladesh University of Engineering and Technology. In Summer 2000 and 2001, he was at Delphi Research Laboratories, Shelby Township, MI. He is currently a Senior Project Engineer at Globe Motors, Dayton, OH, where he is engaged in the design and development of controls for brushless motors. His technical interests include the development of high-performance brushless motor servo drives for automotive applications.