Physical Modeling of Switched Reluctance Motors Using Modelica

Physical Modeling of Switched Reluctance Motors 6 2 Fundamental of Switched Reluctance Drives

beginning beginning of beginning using Modelicaof overlap full overlap of overlap Y. Ji J. Bals

L t t t active flat active Abstract—This paper presents a novel Modelica library for L physical modeling of switched reluctance machines. In order max to deal with the nonlinear characteristics of switched reluctance drives, an analytical approximation function is applied when building a motor model. Furthermore, the look-up table L based approach derived by the finite element method for min computing the nonlinear model is also considered. Compared with other modeling approaches, physical modeling using the 0 180 360 q (°elec.) object-oriented modeling language-Modelica often exhibits a significant progress regarding model reusability, flexibility and range of modeling. One application example with a 16 stator 12 rotor poles 4 phase witched reluctance machine is presented to illustrate the use of the developed Modelica switched reluctance unalignedaligned unaligned machine library. Fig. 1.T Ideal inductance function of rotor position Index Terms—switched reluctance machine, modeling and simulation, motion control. q II.BASICCHARACTERISTICSOF SRM I.INTRODUCTION A. OperationFigure 2.4: principlePhase inductance and torque as a function of rotor position Switched reluctance machine is a doubly salient pole In contrast to conventional DC-machine, synchronous and A machine fed by an unipolar power converter. Concern- On the electrical side, switched reluctance machines are described by the generalized inductionmachine equation, machine whose output torque can be presented ing its simplicity, robustness, low cost and high efficiency, the by the vector product between current and magnetic field, dΨ(t) switched reluctance machine become a good competitor to v(t)=Ri(t)+ (2.2) switched reluctance machines workdt on the principle of re- conventional ac machines in aircraft and automobile industry luctanceEquation 2.2 force. is generally The valid torque for both linear is developedand nonlinear operations from theof switched physical reluc- [1]. Some new research works have presented the possibility naturetance machines. of magnetic For the linear paths operation, which (2.2) is always developed attemptfurther by the to assumption minize to use the switched reluctance machine as generator in more that the flux-linkage Ψ is a product of inductance L and current i and the inductance the magnetic resistance in the magnetic loop [3]. electric aircraft [2] and as major actuator in the upcoming characteristic L(θ) remains unchanged for all current levels (no magnetic saturation). The voltage equation with a link to the mechanical side via rotor speed is expressed by (2.3). electric vehicle. Computer modeling and simulation can For the basic understanding, switched reluctance machines ease the whole product development process and controller are often analyzed assuming linear operation without mag- design. Model based design is getting more and more at- netic saturation in the iron lamination. This greatly simplifies tention in the last years. However, current modeling tools the mathematical machine model since without the magnetic supporting switched reluctance machines e.g. Simpower by saturation the machine characteristic, i.e. phase inductance, Mathworks does not provide users full flexibility to build can be considered as a function of rotor position only. In the models. In this work, a switched reluctance drive library linear consideration, the phase torque product of the switched has been made using Modelica- an object oriented modeling reluctance machine is described as language to achieve the maximum flexibility, motor type 1 dL(θ) converge and model reusability. In this library, all needed T = i2 (1) fundamental components for modeling switched reluctance 2 dθ machine are addressed. Modelica is a non-proprietary, object- Fig.1 shows the phase inductance curve as a function of oriented, equation based language to conveniently model rotor position for one electrical period. It starts at the rotor complex physical systems containing, e.g., mechanical, elec- position with the minimum inductance defined as ”unaligned trical, electronic, hydraulic, thermal, control, electric power. position”. Modelica Simulation Environments are available commer- According to Eq. 1, torque can be produced in the position cially and free of charge, such as CATIA Systems, Dymola, where the inductance gradient dL(θ)/dθ is nonzero. Torque LMS AMESim, JModelica.org, MapleSim, MathModelica, can be produced for both negative and positive directions. OpenModelica, SCICOS, SimulationX. Modelica models can This also implies that switched reluctance machines are be imported conveniently into Simulink using export features basically capable of breaking or generating operation. Note of Dymola, MapleSim and SimulationX. or process-oriented that the input current influences only the amplitude of the subcomponents. Considering the good support of the newest output torque due to the quadratic term i2. Hence, the Modelica standard, the Dymola system is selected to imple- only possibility to command the direction of the output ment the switched reluctance machine library. torque is controlling the phase excitation period according to the rotor position. The electrical characteristic of switched Y. Ji and J. Bals are with the Department of System Dynamics and Control, Institute of Robotics and Mechatronics, German Aerospace Center, reluctance machines can be described by the generalized Muenchnerstr. 20, Wessling, Germany [email protected] machine equation 2.1 Operation Principle and Characteristics 9

By applying diﬀerentiation, the instantaneous torque T is expressed as

dW i ∂Ψ(i, θ) T (i, θ)= = di (2.5) dθ ∂θ 0

As a conclusion, a mathematical model of a switched reluctance machine with nonlinear operation can be derived from the nonlinear flux-linkage characteristic Ψ(i, θ). By applying (2.5) to the flux-linkage characteristic, the torque characteristic as a function of rotor position for different currents can be determined, as depicted on the right of Fig. 2.6. Note that the representation using the flux-linkage-current diagram is widely employed in literature to illustrate the nonlinear torque production of switched reluctance machines [Sta95][Kri01][Mil01].

By considering a switched reluctance machine as a system with state variables, there are totally four variables representing the behavior of the machine, i.e. current, flux-linkage, rotor position and torque as summarized in Fig. 2.7. Since the direction of output torque depends only on the rotor position, an unambiguous relationship between the electrical physical dimensions and characteristics of the SRM. The BL= d – Ldsat m – LdsatIm (6) and mechanical side can be drawn only by using rotor position as an input variable. details usually required for this calculation are: laminations where L is the d-axis non-saturated inductance and is the In addition, both current and rotor position can be accurately measured using physical dimensions and magnetic characteristics, air gap length, stator d m and rotor poles arcs, size and number of turns of the stator flux linkage at i = Im. sensors, e.g. hall sensors for current measurement and resolvers for rotor position. As windings. These construction details are available only if you The intermediate magnetizations curves corresponding to a result, two fundamental characteristics for describing switched reluctance machines are are designing your own SRM. Otherwise, they are not avail- rotor positions between the aligned position (d-axis) and the able from the manufacturer. unaligned position (q-axis) can be deduced from the two preferably formulated as a function of current and rotor position, Ψ(i, θ)andT (i, θ)as Finite-element computation of the SRM magnetization extreme curves d and q by using a nonlinear function f( ) shown in Fig. 2.7(right). curves is also a time-consuming process. that approximately represents the variation of magnetic flux 3) Analytical expression of the magnetization curves: For linkage in function of the rotor position in a standard structure These characteristics can be determined by analytical calculations, finite element sim- converter and control system design purpose, it would be con- SRM. This nonlinear function can be written for a general case ulations (FEM) [Moa90][Ome97] or experimental measurements [Kri89a] [Ras98][Che01] venient and desirable to calculate the magnetization curves as: from main parameters normally available or easily measurable. 3 3 3 2 2 2 [Fue06]. During the design stage and optimization, analytical calculations or finite element f = 2Nr –13Nr + (7) The details on the machine geometry will be supposed to be where N is the number of rotor poles. simulations are generally used to study the performance and characteristics of the designed those of a “standard construction” SRM. In this way, the r machines. For the design verification and control purposes, experimental measurements generic model can represent a category of SRM of standard The magnetization characteristic of the SRM can be thus expressed as a function of stator current and rotor position: are preferred since the measured data incorporate imperfections in manufacturing and construction having common main parameters. In the proposed generic model, the extreme magnetization –Bi i = L iL+ iA1e+ – – L i f (8) physical effects often neglected in finite element calculations. curves, corresponding to aligned and unaligned rotor positions, q dsat q are approximated by analytical functions as shown in Fig. 5. D. Torque characteristic The SRM electromagnetic torque is equal to the sum of the Y i q Tiq Slope = Ldsat individual torques developed in each phase. Due to the nonlin- (, ) (, ) flux-linkageY m Aligned earity of the magnetization curves, the developed torque is a Slope = Ld nonlinear function of the stator current and the rotor position. i Y 300 0.25 current d 200 0.2 T The electromagnetic torque developed by one phase of the elec. 100 0.15 0 SRM can be calculated as the derivative of the machine co- 0.1 -100 energy: 0.05 -200

torque

0 -300 Unaligned 300 of SRMs 300 flux-linkage 360 360 T i = W i (9) 200 200 e current100 180 q current T 100 180 q State variables torque 0 0 q i i 0 0 where W’ is calculated as

mech. rotor position rotor position rotor position q i W i = O i di (10) 0 Slope = L Figure 2.7: State variables andFig. characteristics2. Magnetic ﬂux and of a torque switched functions reluctance of current and machine rotor position q In case where the magnetization curves are calculated from i experimental data or from finite-element results, the torque 0Im characteristic Te(i, ) can be numerically calculated using Eq. 9. Fig. 5 Aligned and unaligned magnetization curves used to build the SRM ana- Figure 6 shows the SRM torque characteristic calculated dΨ(t) Fig. 3. Aligned and unalignedlytical magnetization model curves for building SRM v(t) = Ri(t) + (2) analytical model from the magnetization curves shown in Fig. 4. dt The magnetization curve at unaligned rotor position (q axis) Machine SRM 6/4 (60 kW) is represented as a straight line with slope equal to the mini- 300 Substituting the Ψ(t) by L(θ)i(t) Eq. 2 can be developed to i = 450 A presentedmum inductance in this paper Lq: can deal with all three modeling meth- i = 400 A di(t) dL(θ) dθ ods. For fast prototyping the = converterL i and control strategy,(3) 250 v(t) = Ri(t) + L(θ) + i (3) q q i = 350 A dt dθ dt analytical approximation of the magnetization curves and The magnetization curve at aligned rotor position (d axis) is i = 300 A torquea nonlinear characteristic function of is the very stator convenient current i: and desirable. By 200 where dθ/dt = ω The last term can be considered as i = 250 A –Bi induced back-emf. However, it should be noted that the the analytical approximation = L approachiA1e+ – [9], the magnetization(4) ﬂux performance cand be presenteddsat by 150 i = 200 A

product of the phase current and the induced back-emf (N.m) Couple where Ldsat is the d-axis saturated inductance, and A and B are in switched reluctance machines does not represent the i = 150 A constantsψq = determinedLqi by conditions at i = 0 and i = Im (Im is the 100 mechanical power conversion in the same way like in the −Bi i = 100 A maximumψd = currentLdsat ini stator+ A (1windings).− e ) conventional machines, since only the half of the power flow- In supposing that3 the3 machine3 is really2 2 saturated2 at i = Im, 50 f(θ) = (2Nr /ψ )θ − (3Nr /ψ )θ + 1 i = 50 A ing into the equivalent back-emf voltage source is converted –BI one have e m 0 and the constants A and−Bi B can be deduced into mechanical energy while the other half is stored in a ψ = Lqi + [Ldsati + A(1 − e ) − Lqi]f(θ) (6) 0 0 5 10 15 20 25 30 35 40 45 as: Position du rotor (degre) form of magnetic energy. When the machine runs with high where A = – L I (5) velocity, back-emf will be dominant in the Eq. 3. As result, m dsat m Fig. 6 Torque characteristic calculated from the magnetization the maximum speed of a switched reluctance machine can A = ψm − LdsatIm curves shown in Fig. 4. be approximately considered as proportional to the phase B = (Ld − Ldsat)/(ψm − LdsatIm) (7) voltage. The description in Eq. 2 is generally valid for 1558 Furthermore, the electromagnetic torque results from the both linear and nonlinear operations of switched reluctance sum of the individual torques developed in each phase. The machines. analytical expression for the torque of each phase can be written as B. Modeling approaches 2 −Bi 0 τe(i, θ) = [0.5(Ldsat − Lq)i + Ai + A(1 − e ]f (θ) To model switched reluctance machines in practical work- 0 3 3 2 2 2 f (θ) = (6Nr /φ )θ − (6Nr /φ )θ (8) ing conditions, the magnetic saturation has to be treated. Some main factors, which affect the performance of a The parameters used in Eq. 6-Eq. 8 are explained in Tab. I switched reluctance machine, are machine structure, rotor and graphically interpreted in Fig. 3. and stator poles, magnetization characteristic of the lamina- TABLE I tions and control strategy. The highly complex characteristic SYMBOLSUSEDFORANALYTICALEXPRESSIONOFTHE is presented by two nonlinear relations. As first one, the MAGNETIZATION CURVES machine flux linkage is dependent on the stator current and rotor angular position such as Parameter Description Lq Non-saturated inductance at unaligned position ψ = ψ(i, θ) (4) Ld Non-saturated inductance at aligned position Ldsat Saturated inductance at aligned position The second nonlinear functions reveals that the electro- Im Maximum current in stator windings magnetic torque produced by each phase of the switched ψq Magnetization curve at aligned position ψd Magnetization curve at unaligned position reluctance machine is determined by ψm Flux linkage with im Nr Rotor pole number τ = τ(i, θ) (5) The motor performance is fixed once these two nonlinear relationships are known. Such kind of state variable de- III.IMPLEMENTATION OF MODELICASWITCHED pendencies can be presented by Fig.2. These characteristics RELUCTANCE DRIVE LIBRARY can be typically determined by analytical calculations, finite The components needed for modeling a switched reluc- element methods [4], [5] or experimental measurements [6], tance drive mainly contain power converter, air gap, mechan- [7], [8]. The Modelica switched reluctance machine library ical parts, control unit, communication bus system, human currentSensor

Mechanical

A onePhaseStator1 Imeasure1 L=L(theta,i)

pow erDeMultiplex3P controlBus

ModAnglephase1

currentSensor1 Phimeasure onePhaseStator2 phi Electrical A L=L(theta,i) Imeasure2 ModAnglephase2 flange_b1 angleSensor

currentSensor2 onePhaseStator3

A L=L(theta,i) inertia Imeasure3 ModAnglephase3 J=1e-6

positiveP… currentSensor3

pow erDeMultiplex… onePhaseStator4 tau A L=L(theta,i) Taumeasure Imeasure4 ModAnglephase4 torqueSensor Fig. 4. Graphical interface to model an air gap negative… torque tau machine interface and power supply. Using the Modelica standard library, which covers a wide range of infrastructural Fig. 5. Rotor model from a 12 rotor poles switched reluctance machine model in numerous physical domains, the major parts of

Inductance function verification switched reluctance machine library can be easily modularly 0.46

0.44 established. 2.4 Control of Switched Reluctance Drives 13 0.42

0.40 states.0.38 Note that the sign of the current slope in freewheeling states is decisive for selecting A. Model of rotor the0.36 proper excitation scheme. Applying a wrong excitation scheme may lead to damages due0.34 to overcurrent. Hence, the controller must be designed to be able to assign proper The rotor of switched reluctance machine primarily com- 0.32 excitations corresponding to the current operating mode. Examples of control solutions 0.30 prises a number of air gaps. To model the air gap of a capable of both motoring and generating modes are presented in [Fue05b] and [Car04]. [Wb] 0.28 switched reluctance machine, the analytical representation 0.26 described in the last section is applied. Typically, the aligned 0.24 2.40.22 Control of Switched Reluctance Drives and unaligned magnetization curves in Fig. 3 are derived 0.20 by finite element method or measurement. The result tables Output0.18 torque of switched reluctance machines is commanded by two variables, i.e. excita- tion0.16 level represented by current or flux-linkage and rotor position. Therefore, a high-grade can be imported directly into the Modelica library. After- 0.14 torque controller must possess two feedback signals, for either current or flux-linkage and 0.12 0 5 10 15 20 25 30 wards all the parameters needed in the analytical model are for rotor position, to obtain a full access on outputRotorPosition torque. Fig. 2.10 shows two generalized automatically computed and put into the machine model. forms of torque controllers for switched reluctance machines [Vas98]. The main difference The rotor of a switched reluctance machine is modeled by Fig.between 6. both Testing controllers of magnetization is the secondary curve control variable, current or flux-linkage. object oriented approach, which allows to easily build a - Current-based torque controller v desired rotor model by putting air gap models together. The dc (optional) setup window to model an air gap is depicted in Fig. 4. SRM Current reference i Current s Converter T refph ph A rotor model with 12 rotor poles is depicted in Fig. 5. ref and controller Commutation (hysteresis, The rotor model in Fig. 5 primarily consists of 4 air gaps control delta mod. or PWM) i besides some current sensor, data bus system and machine (look-up table) ph inertial. Additionally, some test blocks for verifying the analytical representation of magnetization curves are offered q in the library. For example, the magnetization performance by analytical modeling approach is depicted in Fig. 6, where a) Current-based torque control Fig. 7. Schematic current based torque control the phase current i = 200A and the rotor position varies v dc (optional) from 0 to 30 degree. SRM Flux-linkage Y Flux-linkage s Converter T refph ph ref reference controller loop any more.and In this paper, only the current based torque B. Model of control unit control unitCommutation is considered. The(PWM) schematic diagram of current control i Y ph Output torque of switched reluctance machines is com- based torque(look-up control table) is depicted inph Fig. 7. Besides analytical Flux-linkage manded by two variables, i.e. excitation level represented by modeling approach, the result of theestimator finite element method current or flux-linkage and rotor position. Therefore, a high- for modeling the magnetic and mechanic characteristicsq in grade torque controller must possess two feedback signals, Eq. 4 and 5 can also be imported into the library by lookup for either current or flux-linkage and for rotor position, to tables. Based on thoseb) lookup Flux-linkage-based tables, the torque model control of a switched obtain a full access on output torque. There are two major reluctanceFigure 2.10: machineFundamental can torque also controllers be done. for switched However, reluctance machines those kinds[Vas98] control strategies e.g current based torque control and direct of models are not quite suitable for control design due to high torque control [10], [12]. The major difference between two computation cost. The major components in a current based control strategies is that output torque is directly managed torque control unit are power converter, current controller as a control variable and there is no inner current control and some signal sensors for current and rotor angle. u

PAmat…

dc idealClosingSwitch idealDiode1 idealClosingSwit… idealDiode

NVdc

NAm Fig. 8. Determination of phase related angle Fig. 9. Asymmetric bridge power converter Asymmetric bridge type power converter is addressed in the Modelica switched reluctance machine library. Using

Modelica standard library, a power converter depicted in Fig. hysteresis feedback 9 can be easily modeled. Even more detail model converging Id

Imeasure Imeasure1 thermal effects and power loss could also be treated. Since - uL… uH… the turn on/off angles of each phase in rotor play a crucial controlBus role for torque control. The current angle in relation to each hysteresis1 feedback1 phase/rotor pair has to be conducted by the measured rotor Id

Imeasure2 rotation angle. Taking the example of a 16 stators 12 rotor - uL… uH… poles switched reluctance machine, the angle between rotor and phase i , θmi with the assumption that the rotor at the hysteresis2 y Idesired feedback2 starting point shown in Fig. 8 can be described as Id

Imeasure3 - uL… uH…

θmi = Θ mod 30 − (i − 1) × 7.5 (9)

hysteresis3 where Θ is the measured rotor rotation angle. feedback3 Id

Besides major components to build a switched reluctance Imeasure4 - uL… uH… machine, the infrastructure models of current controller provide user a convenient way to implement own control unit. The hysteresis current regulator offers a very fast dynamic Fig. 10. Implementation of current controller response by keeping the output current in a given hysteresis band. Its switching frequency varies considerably with torque speed operation points and rotor position [11]. The hysteresis based approach for current control is incorporated in the currentController CurrentIs Modelica library. The hysteresis band of current control is Anglephase1 uLow uHigh made by the Hysteresis block from Modelica standard library. Angleon

AngleoffPh… powerMultiplex3P Furthermore, some algorithms of direct torque control will CurrentIs also be included in the library. In order present the physical model in reality, all data of Anglephase2 Angles control commands and sensor measurements are collected Angleon in a special bus system, which is implemented by the AngleoffPh… expandable connector in Modelica. A structural overview of

the Modelica switched reluctance machine library is shown Anglephase3 in Fig. 12. Additionally, the Modeling templates for three Angleon Angleoff mostly used motor types (6/4, 8/6 16/12) are already included Ph… powerMultiplex3N in the Modelica switched reluctance machine library.

positiveP Anglephase4 PPLICATION EXAMPLE ODELINGOFA Angleon IV. A :M 16/12 4 AngleoffPh… PHASESSWITCHEDRELUCTANCEMACHINE As demonstration, the Modelica switched reluctance ma- negative chine library has been applied to build a test rig of 16 pin_p pin_n stators/12 rotors 4 phases switched reluctance machine. The test rig in Fig. 13 comprises a human machine interface for deﬁning desired current and turn on/off angle, a power Fig. 11. Control unit for a 4 phase switched reluctance machine supply, a motor control unit including current controller Virtual Switched Reluctance Motor (16/12) Test Rig

HMI

controlBus torqueStep inertia

time Pow er Sup… Motion Control SRM 16… J=0.5

Fig. 13. Testrig of a 16/12 switched reluctance machine

Phase current

200

100 [A]

0.00 0.05 0.10 0.15 0.20

Output Torque 4000

Fig. 12. Overview of the switched reluctance machine library 3000

2000

[N.m] 1000 and the 16/12 switched reluctance machine as well as a 0 -1000 mechanical load. The test rig is depicted in the Fig. 13. 0.00 0.05 0.10 0.15 0.20 The parameters used for the test rig are listed in Tab. II. Two test cases have been performed. With the first test case, Machine speed the switched reluctance machine is simulated without any load to indicate the idealized operation boundary. Simulation 200 results of phase current, output torque and motor velocity 100 are depicted in Fig. 14. With the second test case, a negative [rad/s] torque has been placed as load at 0.1s. The performance of 0 the motor is shown by the phase current, output torque and 0.00 0.05 0.10 0.15 0.20 velocity simulation results in Fig.15. Fig. 14. Simulation result without load TABLE II PARAMETERSINTHEDEMONSTRATION and their corresponding intelligent controllers. In the near Parameter Value future, the direct torque control unit will also be implemented Lq 0.9e-3 Henry in the Modelica switched reluctance machine library. Ld 40e-3 Henry Ldsat 0.2e-3 Henry Im 450 Ampere ACKNOWLEDGMENT ψm 0.5 Weber Nr 12 The research leading to these results has received funding Rs 0.02 Ohm from the European Union’s Seventh Framework Programme Current Command 200 A (FP7/2007-2013) for the Clean Sky Joint Technology Initia- Turnon angle 18 Deg Turnoff angle 25 Deg tive under grant agreement Nr. CSJU-GAM-SGO-2008-001. Load -1000 Nm REFERENCES [1] R. Krishnan, Switched Reluctance Motor Drives. CRC Press, 2001. [2] L. Han, J. Wang, and D. Howe, “Small-signal stability studies of V. CONCLUSION 270v dc power system for more electric aircraft employing switched In this paper, a novel Modelica library for modeling reluctance generator technology,” 25th Internatinal Congress of the Aeronautical Sciences, 2006. switched reluctance machines are presented. The basic com- [3] T. J. E. Miller, Switched reluctance motos and their control. Magna ponents to build a arbitrary type of switched reluctance ma- Phsics Publishing, 1993. chine have been implemented using object oriented physical [4] M. Moallem and C. Ong, “Predicting the torque of a switched reluctance machine from its finite element field solution,” IEEE modeling approach. Additionally, control units for switched Transactions on Energy Conversion, vol. 5, 1990. reluctance machines are also partly incorporated e.g. hystere- [5] A. Omekanda, C. Broche, and M. Renglet, “Calculation of the electro- sis current based torque controller. The work addressed here magnetic parameters of a switched reluctance motor using an improved fembiem application to different models for the torque calculation,” is a part from an ongoing project of a Modelica Power Drive IEEE Transactions on Industry Applications, vol. 33, pp. 914–918, library, which will contain mostly up-to-date machine types 1997. Phase current

200

100 [A]

0.00 0.05 0.10 0.15 0.20

Output Torque 4000

3000

2000

[N.m] 1000

-1000 0.00 0.05 0.10 0.15 0.20

Machine speed 200

150

100

[rad/s] 50

-50 0.00 0.05 0.10 0.15 0.20

Fig. 15. Simulation result with predeﬁned load proﬁle

[6] R. Krishnan and P. Materu, “Measurement and instrumentation of a switched reluctance motor,” IEEE Industry Application Society Annual Meeting, 1989. [7] P. O. Rasmussen, G. Andersen, L. Helle, F. Blaabjerg, and J. K. Peder- sen, “Fully automatic characterization system for switched reluctance motors,” ICEM, 1998. [8] N. H. Fuengwarodsakul, S. E. Bauer, and R. W. D. Doncker, “Charac- teristic measurement system for automotive class switched reluctance machines,” European Power Electronics and Drives Journal, vol. 16, 2006. [9] H. Le-Huy and B. P., “Versatile nonlinear switched reluctance motor model in simulink using realisic and analytical magnetization characteristics,” 31st Annual Industrial Electronics Society Conference, 2005. [10] A. D. Cheok and Y. Fukuda, “A new torque and ﬂux control method for switched reluctance motor drives,” IEEE TRANSACTIONS ON POWER ELECTRONICS, vol. 17, 2002. [11] F. Blaabjerg, P. C. Kjaer, P. O. Rasmussen, and C. Cossar, “Improved digital current control methods in switched reluctance motor drives,” IEEE Transactions on Power Electronics, 1999. [12] R. Inderka and R. Doncker, “Ditc-direct instantaneous torque control of switched reluctance drives,” IEEE Transactions on Industrial Ap- plications, vol. 39, 2003.