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141767414 Switched Reluctance Motor

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    Chapter 2

    Principle of Operation of

    the Switched Reluctance

    Motor

    2.1 Introduction

    Switched reluctance motor (SRM) drives are simpler in construction comparedto induction and synchronous machines. Their combination with power elec-tronic controllers may yield an economical solution [Bos 04]. The structure ofthe motor is simple with concentrated coils on the stator and neither windingsnor brushes on the rotor. This apparent simplicity of its construction is decep-tive [Ste 95]. The Switched Reluctance Motor drives present several advantagesas high efficiency, maximum operating speed, good performance of the motorin terms of torque/inertia ratio together with four-quadrant operation, mak-ing it an attractive solution for variable speed applications [Giu 91]. The very

    wide size, power and speed range together with the economical aspects of itsconstruction, will give the SRM place in the drives family.The performances of switched reluctance motor strongly depend on the

    applied control. Figure 2.1 shows the principal parts of a switched reluctancedrive. Three main parts can be identified: the motor itself, which can have var-ious topologies as explained in the next section, the power electronic converterand the controller. The drive system, comprising signal processing, power con-verter and motor must be designed as a whole for a specific application.

    There is one converter unit per phase. A battery or a rectifier supplies the dcpower. The basic principle is simple: each phase is supplied with dc voltage byits power-electronic converter unit as dictated by the control unit, developinga torque, which tends to move the rotor poles in line with the energized statorpoles in order to maximize the inductance of the excited coils. An important

    fact is that the torque production is independent of the direction of current,

    13

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    14 Principle of Operation

    UDC Converter

    Controller Phase current

    SRM

    Rotor position

    Figure 2.1: Switched reluctance drive system.

    which contributes to the reduction of the number of switches per phase.This chapter presents the main topologies of switched reluctance motors,

    the energy flows and control variables. The electromagnetic principles aredescribed along classical lines. The machine operations in all of its four quad-rants, the torque versus speed characteristics, and the mathematical model ofthe equivalent circuit are formulated. The magnetically linear model is used

    to provide a structure for understanding the SRM control. The chapter pro-vides the description of a four-phase 8/6 SRM motor and its control scheme.The simulated annealing method is proposed to find the optimal speed con-troller gains. The simulations carried out and their most important results arediscussed.

    2.2 Machine Topologies

    As any other motor, the structure of the switched reluctance motor consistsof a stator and a rotor. Both stator and rotor are laminated. Stacking thelaminations punched from steel lamination with high magnetic quality yieldsthe rotor cores. The stator is formed from punched laminations too bonded

    into a core, and the coils are placed on each of the stator poles.Each stator pole carries an excitation coil, and opposite coils are connected

    to form one phase. There are no windings on the rotor. The number of statorand rotor poles are chosen using a series of criteria developed in Chapter IV.In this chapter it is supposed that the number of rotor poles Nr, and statorpoles Ns are known without discussing the criteria for their choice.

    Switched reluctance machines can offer a wide variety of aspect ratios andsalient pole topologies without affecting performance too much. This meansthat each application is likely to be better suited for a specific SR topology.

    Single Phase Motor

    These are the simplest SR motors having the advantage of fewest connections

    between machine and power electronics. However, the very high torque ripple

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    2.3. Basic SRM Principles 15

    and inability to start at all angular positions represents a drawback. They canpresent interest only for very high-speed applications.

    Two Phase Motor

    The use of a stepping the air-gap can avoid the starting problems. For a two

    phase SRM the high torque ripple is an important drawback.

    Three Phase Motor

    The most popular topology of a three-phase SRM is the 6/4 form (Ns = 6and Nr = 4). It represents a good compromise between starting and torqueripple problems and number of phases. Alternative three-phase machines withdoubled-up pole numbers can offer a better solution for lower speed applica-tions.

    Four Phase Motor

    The four-phase motor is known for reducing torque ripple. The large number

    of power electronic devices and connections is a major drawback, limiting four-phase motors to a specific application field. A practical limitation to considerlarger phase numbers is the increase of the converter phase units, hence of thetotal cost.

    2.3 Basic SRM Principles

    The switched reluctance motor with its passive rotor has a simple construction.However, the solution of its mathematical model is relatively difficult due to itsdominant non-linear behaviour. The SRM is characterized by its geometricallayout, the characteristic of the magnetic material and electrical parameters.The cross sectional view of a four-phase SRM is shown in Figure 2.2.

    The selection of the stator and rotor teeth number Ns and Nr is made withthe respect to several constraints as rotor deformation, capability of torqueproduction at all rotor positions and four-quadrant operation. The relation-ships among all these constraints will be presented in Chapter IV. The numberof phases is identified from the stator and rotor pole numbers:

    q=

    Ns

    |NsNr |,q integer

    2Ns|NsNr |

    ,q non-integer(2.1)

    Once the number of poles is chosen, the next parameters are stator s androtor r pole arcs in order to minimize the inductance, maximize the inductanceratio, avoid dead zones and allow four quadrant operation. The stator and rotorpole tapering angles s and r are direct functions of the number of stator and

    rotor teeth:

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    16 Principle of Operation

    1

    2

    3

    4 bs

    br

    ar

    as

    Figure 2.2: Cross sectional view of a four-phase SRM.

    s =2

    Nsrad and r =

    2

    Nrrad (2.2)

    A torque is produced when one phase is energized and the magnetic circuittends to adopt a configuration of minimum reluctance, i.e. the rotor polesaligned with the excited stator poles in order to maximize the phase inductance.As the motor is symmetric, it means that the one phase inductance cycle iscomprised between the aligned and unaligned positions or vice versa (Figure2.3).

    Figure 2.3: Inductance profile of SRM.

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    2.3. Basic SRM Principles 17

    The aligned position (La)

    Consider a pair of rotor and the stator poles to be aligned. Applying a cur-rent to phase establishes a flux through stator and rotor poles. If the currentcontinues to flow through this phase, the rotor remains in this position, therotor pole being stuck face to face to the stator pole. This position is called

    aligned position, and the phase inductance is at its maximum value (Lmax orLa) as the magnetic reluctance of the flux path is at its minimum.

    Intermediate rotor positions (Lint)

    At intermediate positions the rotor pole is between two stator poles. In thiscase the induction is intermediate between the aligned and unaligned values. Ifthere is any overlap at all, the flux is diverted entirely to the closer rotor poleand the leakage flux path starts to increase at the base of the stator pole onone side.

    The unaligned position (Lu)

    In the unaligned position, the magnetic reluctance of the flux path is at itshighest value as a result of the large air gap between stator and rotor. Theinductance is at its minimum (Lmin or Lu). There is no torque production inthis position when the current is flowing in one the adjacent phases. However,the unaligned position is one of unstable equilibrium.

    Mathematically, the inductance profile of phase j may be approximated by:

    L()j = L1()

    rq

    (j 1)

    (2.3)

    Figure 2.3 shows the idealised inductance profile of one phase as a function ofthe rotor position for a pair of stator poles. The number of cycles of inductancevariation per revolution is proportional to the number of rotor pole pairs, and

    the length of the cycle is equal to the rotor pole pitch. In reality the rotor polearc r is always larger than the stator pole s if Ns > Nr. The value of theinterval r r between the rotor teeth is larger than s in order to have theminimum value of the inductance Lmin as low as possible. For the calculation,the value of the air gap is considered to be constant in the interval where thestator and rotor teeth are face to face.

    The equation of the inductance profile can be rewritten as:

    L() =

    Lu, 1 < < 0Lu + k, 0 sLa, s rLu + k( r s), r r + s

    (2.4)

    where k is the slope of the profile in the zone of increasing inductance:

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    18 Principle of Operation

    k =La Lu

    s(2.5)

    The torque developed by a phase in which current flows tends to move therotor in such a direction as to increase the phase inductance, i.e. the alignedposition. This means that the motoring torque can be produced only in thedirection of the rising inductance. The instantaneous torque is obviously notconstant, as shown further, depending of the rotor position and the instanta-neous phase current. Note that the torque is independent of the direction ofcurrent flow, the motoring or braking torque production only depending of therotor position, suggesting the existence of the impact of switching angles of thepower electronic switches. This particularity of the switched reluctance motoris discussed further.

    The control scheme is based on the torque-speed characteristic (Figure 2.4).Lawrenson [Law 80] describes three basic modes of operation of switched reluc-tance motor based on the torque speed characteristic. Currents in the statorcircuits are switched on and off in accordance to the rotor position. With thissimplest form of control, the switched reluctance motor inherently develops the

    torque speed characteristics typical of d.c. machine.

    TChopping

    Current-limited Const. power Natural

    T = const.

    T = constw

    T = c ons tw2

    0 wb wp w

    qD increasing qD fixed

    max qD

    Figure 2.4: SRM Torque - Speed characteristics.

    This first mode is the natural one with fixed supply voltage and fixed switch-ing angles. The operating region is the constant torque region, below ratedspeed. Base speed (b) is defined as the highest speed at which maximum cur-rent can be supplied to the motor (Imax) at rated voltage, with fixed switchingangles. There is, of course, a family of characteristics for varying supply volt-ages. At given speed the flux is proportional to the voltage U, and the torquevaries with the current squared. The chopping voltage control is able to controlan SRM drive only in the mode below rated speed where the generated voltage,being larger than the back-EMF, forces the drive states on the sliding surface.

    If fixed switching angles are maintained at speeds above b, the torque falls

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    2.4. Mathematical approach 19

    as 1/. This is the second important mode of operation, when the machinespeed is above base speed (b). A control alternative for the switched reluctancemotor is to reduce the conduction angle c = off on at constant voltage.In this mode, the voltage generator is fully applied across the phase till offand the current decreases.

    There is a practical limitation of increasing the conduction angle. If it were

    increased so that the turn-off angle corresponds to the next cycle turn-on angle,then the flux level would not return to zero at the end of each pulse. In thiscase, the net flux in the phase winding would increase until the machine becamecontinuously saturated. This corresponds to a rotor speed p. Running abovethis sped implies a fall of the torque production as 1/2.

    2.4 Mathematical approach

    An accurate analysis of the motor behaviour requires a formal, and relativelycomplex, mathematical approach. The instantaneous voltage across the ter-minals of a single phase of an SRM drive winding is related to the flux linkedby the winding. Conform to Tomko [Tom 98], the flux linkage is a function of

    two variables, the current i and the rotor position (angle ). The mathematicalmodel describes the equivalent circuit for one phase (Figure 2.5).

    U

    R

    S

    UR

    L( ,i)q

    R

    Figure 2.5: Equivalent circuit for one phase.

    U = Ri +(, i)

    i

    di

    dt+

    (, i)

    d

    dt(2.6)

    where U is the supply voltage, i is the phase current, R is the phase resis-tance, is the flux-linkage , and is the rotor angular position.

    The general torque expression is:

    T(, i) =

    10 (, i) (2.7)

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    20 Principle of Operation

    In general, the dynamical model of an SRM is characterized by the rotorangular speed-angular position relationship:

    =d

    dt(2.8)

    T Tload = Jd

    dt + B (2.9)

    It is a set of four non-linear partial differential equations. Its solution,neglecting the nonlinearity due to magnetic saturation is:

    (, i) = iL(, i) (2.10)

    can be written as:

    U = Ri + L(, i)di

    dt+ i

    dL(, i)

    d(2.11)

    The average torque can be written depending on the number of phases ofthe SRM as:

    T =n

    phase=1

    Tphase (2.12)

    In this section it is assumed that the drive works in the linear region, limitedby the saturation value of the current Imax.

    2.5 SRM modelling

    The torque or force production in a switched reluctance motor may be foundfrom the variation of the stored magnetic energy as a function of the rotorposition (virtual work principle). This relationship is also used to analyseelectromagnetic relays, holding magnets, solenoid actuators, and other devices

    where force is produced between two magnetic surfaces, including all machineswith saliency.

    2.5.1 Linear analysis of the voltage equation and torque

    production

    A linear analysis assume that the inductance is unaffected by the current, thusno magnetic saturation occurs. For the sake of simplicity it is also assumedthat all the flux crosses the air gap in the radial direction, the mutual couplingbetween phases may be ignored, and the effect of fringing flux around the polecorners is also negligible. In the linear region, the equation of the magneticcharacteristics is

    = L()i (2.13)

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    2.5. SRM modelling 21

    where L() is the inductance of a phase as a function of the angle . Itsidealized profile, one rotor pole pitch in length, has been presented in Figure2.3.

    The rate of charge of the energy, i.e. the power is:

    U i =

    d 12 L(, i)i2dt +

    i2

    2

    dL(, i)

    d (2.14)This equation shows that when the rotor operates in the first quadrant,

    the input electrical power goes partially to the increase of the magnetic energy

    stored in the self-inductance

    12 Li

    2

    and the other part i2

    2dL()d is transformed

    into mechanical power. As seen in the Figure 2.5, the equivalent circuit isequipped with a switch representing the power electronic component. When theswitch S is closed, part of energy from source 1 (U) is converted into mechanicaloutput and the other part is stored magnetically. When S is open, the storedmagnetic energy is partly transferred to the second source (UR)(charging, thusrecovering energy) and partly is transformed in mechanical energy.

    For an SRM, an operating cycle consists in energising the phase followed bydemagnetising, achieved by zeroing the current. When one phase is energised,

    a torque is produced in order to minimise the reluctance of the phase by pullingthe pair of rotor poles into alignment with the corresponding stator phase. Themechanical output power is the product of the electromagnetic torque and rotorspeed.

    Pa = mTe (2.15)

    from which the torque is obtained:

    T =1

    2i2

    dL(, i)

    d(2.16)

    Equation (2.16) shows that the torque is proportional to the current square,hence the current can be unipolar to produce unidirectional torque. The slopeof the inductance versus rotor position characteristic gives the torque constant.

    Intervals (0, s) and (r, s + r) are effective torque zones. These intervalshave to be as large as possible. Interval (s, r) is a dead zone required for theflux to be reduced to zero. Interval (s+r, r) represents the interval betweenrotor and stator pole-corners in the unaligned position. The instantaneoustorque over a period can be expressed by substituting (2.4) in (2.16):

    T =

    0, 1 012 ki

    2, 0 s0, s r

    12

    ki2, r (r + s)

    (2.17)

    Graphically, this can be visualized as in Figure 2.6. In the linear analysis,the torque value is a quadratic function of the current and a linear one of the

    commutation angle. When the conduction angle is situated on the non-zero

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    22 Principle of Operation

    inductance slope zone, a non-zero torque is produced. This torque can bedeveloped until the phase voltage reaches the rated value UN, that is, up to aspeed defined as:

    b = UUN

    KIm(2.18)

    L, ii L

    tT

    t

    Figure 2.6: Torque production in SRM - idealized representation.

    Conform with the voltage equation of one phase winding (2.11), the varia-tion of the phase current can be written as

    di

    dt=

    UN idL()d

    L()(2.19)

    where the second term of the numerator is the back-EMF, which dependson the phase current, rotor position, and machine speed. As the maximumphase current is Im, the linear inductance slope is constant, it results that themaximum value of back-EMF is function of the shaft speed.

    The speed torque characteristic of the switched reluctance motor can bemathematical expressed by considering the basic modes of operation. Two basicmodes of operation, named A and B are possible, depending of the machinespeed. Mode A (bellow the base speed) occurs when supply voltage is largerthan the back-EMF, and mode B occurs in the opposite case (above the basespeed). Figure 2.7 presents the voltage, current, and inductance profiles forone phase of switched reluctance motor in mode A, and respectively mode Bof operation.

    In mode A, the applied voltage is larger then any value possible of back-EMF

    for the shaft speed range of [0 b], which is equivalent to a possible current

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    2.5. SRM modelling 23

    Figure 2.7: Basic modes of operation for SRM.

    variation control depending of the applied chopping voltage. The switchingangles (turn-on and turn-off) are fixed and depend of the machine configurationand control constraints, as developed further.

    In mode B, the level of current diminishes because even for a fully applied

    voltage the back-EMF is larger. Mode B has, therefore, the merit of enlargingthe torque speed characteristic of an SRM. Here the control variables are turn-on and turn-off angles. The turn-on angle is controlled from the inrush moment(the moment when full voltage is applied) to a maximum value. The turn-offangle is controlled from zero to a maximum value that, at high speeds andcurrents, is restricted by the extinction constraints.

    2.5.2 Nonlinear analysis of torque production

    The analysis of switched reluctance motor made till now has avoided the ques-tion of the influence of the nonlinear, saturation characteristic of real magneticsteel. However, a proper understanding and handling of saturation is essential.Such analysis is based on magnetization curves. A magnetization curve is a

    curve of flux-linkage versus current i at a particular rotor position (Figure2.8).

    The difference between these characteristics and the ideal ones is obvi-ous. The two most important magnetization curves, the aligned and theunaligned, can be easily seen on Figure 2.8. In the aligned position, the curveis similar to that of an iron-cored inductor with an air gap. At low flux density,the curve is linear. The unaligned curve is straight because of the dominatinglarge air-gap. The saturation effect is observed at current levels that are usuallytoo high for normal operation and therefore the unaligned curve is assumed tobe linear.

    There are two distinct effects of saturation. One is related to the values ofthe phase current, being similar the saturation effect in other types of machines.The second effect depends on the rotor position, and is known as local effect.

    In the switched reluctance motor both effects are present and interact, but their

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    24 Principle of Operation

    Figure 2.8: Magnetization curves of SRM.

    impacts can be isolated by observations at particular rotor angles. The firsteffect can be observed best for aligned position between stator and rotor poleswhere there is no influence of the local saturation. The second effect is evidentfor rotor positions corresponding to partial overlapping of rotor and statorpoles. The nonlinear effect of the magnetic circuit is well seen in Figure 2.8.In the linear part at any position the co-energy, represented by the area belowthe magnetization curve, is equal to the stored field energy, Wf, representedby the area above the magnetization curve as (Figure 2.9):

    Wf = W

    =1

    2L(, i)i2 (2.20)

    where L(, i) represents the inductance at a particular current value androtor position.

    Flux-linkage

    Y

    Current

    Wf

    W

    Figure 2.9: The nonlinear effect of a magnetic circuit.

    The co-energy is defined:

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    2.5. SRM modelling 25

    W

    =

    i0

    di (2.21)

    The most general expression for the torque produced by one phase at anyrotor position is given by the change in magnetic co-energy (virtual work prin-

    ciple):

    T =

    W

    i=constant

    (2.22)

    In conventional switched reluctance motors, the torque produced is deter-mined directly from the area enclosed by the flux-linkage/current (/i) of eachphase. The instantaneous torque represents the work variation Wm at con-stant current for an infinitesimal rotor displacement . This is illustrated inFigure 2.10. During the displacement there is an exchange of energy with thesupply, and there is also a change in the stored magnetic energy. The constant-current constraint ensures that during such a displacement, the mechanicalwork done is exactly equal to the change in magnetic co-energy.

    Flux-linkage

    Y

    Current

    A

    BC

    D

    iO

    DWm D DW = We mDW +f

    Figure 2.10: Determination of electromagnetic torque.

    As the rotor moves from A to B by a displacement at constant currenti, the machine exchanges energy with the supply:

    We = ABCD (2.23)

    The change in stored magnetic energy is:

    Wf = OBCOAD (2.24)

    The input electrical power goes partly to the increase of the magnetic energystored in the self-inductance. The other part is transformed into mechanical

    output power:

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    26 Principle of Operation

    We = Wf + Wm (2.25)

    and the mechanical work done is found to be equal to the area enclosed byboth flux-linkage curves:

    Wm = T = We Wf = OAB (2.26)By applying the co-energy method to each rotor position and for the whole

    range of phase currents, the instantaneous torque curves can be build. An im-portant observation is that not all the supply energy is converted into mechan-ical work, some of it being stored in the magnetic field. This has an importanteffect on the rating of the controller and the need for filter capacitors [Mil 93].

    The torque curves for a four-phase 8/6 SRM are presented in Figure 2.11.They have been obtained using the finite element analysis, as it will be devel-oped in Chapter IV.

    0 10 20 30 40 50 60-4

    -2

    0

    2

    4

    T[Nm

    ]

    q [ ]

    Imax

    Imin

    Figure 2.11: Torque curves of a four-phase 8/6 SRM.

    When the rotor pole pair is exactly aligned with the stator pole pair for anycurrent flowing in the phase, no torque is produced because the rotor is at aposition of maximum inductance. As explained earlier, the sign of the torquedepends on the sign of the inductance slope.

    Much of the classical theory of torque control in electric drives is based onthe independently excited dc machine, in which the torque is proportional tothe flux and current product. The control law of such machines is based onthe capability of independent control of flux and current. Generally speaking,in classical d.c. and a.c. machines the flux is maintained constant while thecurrent is varied in response to torque demand. The switched reluctance motoris a singly excited machine and therefore the orthogonality of the flux andcurrent is very difficult to see. In this way, the armature and field current areindistinguishable from the actual phase current. Therefore no equivalent of

    field-oriented theory is applicable in switched reluctance motor.

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    2.6. SRM Drives - Converters and Dynamic Operation 27

    The torque in a switched reluctance motor is composed of a sequence ofimpulses and the flux in each phase must usually be built-up from zero andreturned to zero during each stroke. To achieve continuous control of theinstantaneous torque, the current waveform must be modulated according to acomplex mathematical model of the machine, as shown later. For a q phase andNr rotor pole SRM, the torque averaged over one revolution and the efficiency,

    are:

    Tave =qNr2

    W (2.27)

    =T

    T + qRI2RMS(2.28)

    IRMS is the root mean-square value of the current in one phase. The torqueripple Tr is:

    Tr =Tmax Tmin

    Tave(2.29)

    where Tmax, Tmin and Tave are, respectively, the maximum, minimum andaverage torque values.

    2.6 SRM Drives - Converters and Dynamic

    Operation

    As developed till now, the basic operating principle of the SRM is quite simple:as current is passed through one of the stator windings, torque is generated bythe tendency of the rotor to align with the excited stator pole. The directionof the torque generated is a function of the rotor position with respect tothe energized phase, and is independent of the direction of current flowingthrough the phase winding. Continuous torque can be produced by intelligently

    synchronizing each phases excitation with the rotor position. The amount ofcurrent flowing through the SRM winding is controlled by switching on andoff power electronic devices, such as MOSFETs or IGBTs, which can connecteach SRM phase to the DC bus. The power electronic inverter topology isan important issue in SRM control because it largely dictates how the motorcan be controlled. During the last years, various converters configurationsused in SRM drive have been developed in the research laboratories. Theirfunctionality emerges from some basic technical and economical requirementsand constraints.

    There are numerous options available, and invariably the decision will comedown to a trade-off between the cost of the converter components against havingenough control capability (independent control of phases, current feedback,etc.) built into the drive. The dependency of the torque production cycles of

    the rotor position and the current value flowing into the phase winding suggests

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    28 Principle of Operation

    the existence ofcontrol intervals. The construction of the SRM drive convertershas to be done after the analysis of the control cycle interval. Based on this, aconfiguration of converters used in SRM drives is developed.

    2.6.1 Control interval and switching angles

    As seen previously, the torque production of the switched reluctance motorstrongly depends on rotor position and phase current. In the linear analysisthe influence of the nonlinear, saturation characteristics is neglected. This de-pendency affects the control strategy of the machine. For an SRM, an operatingcycle consists in energising the phase and demagnetising, achieved by zeroingthe current, which suggests the existence of control intervals.

    In practice a dc voltage source supplies the SRM, by applying to the motora two-level voltage (UN,UN). In order to drive an SRM, three angles areidentified dividing the period into four intervals: inrush, chopping, extinctionand rest. The names of each interval represent in fact the command. Figure2.12 shows the four intervals (first quadrant operation) as a function of therotor position. The four intervals are located among three important angles:on, off and ext. The difference between the turn-on and turn-off angle is

    called the dwell angle. The interval controller output gives four values 1, 2, 3and 4 as a function of the rotor position and the reference torque sign.

    0 20 40 60 80 100 120

    I quadrant

    0

    1

    2

    3

    4

    5

    Interva

    l

    Rotor position [ ], q

    IV quadrant

    0 20 40 60 80 100 120

    0

    1

    2

    3

    4

    5

    Rotor position [ ], q

    Interva

    l

    -120 -100 - 80 -60 -40 -20 00

    1

    2

    3

    4

    5

    Rotor position [ ], q

    Interva

    l

    II quadrant

    -120 -100 -80 -60 -40 -20 00

    1

    2

    3

    4

    5

    Rotor position [ ], q

    Interva

    l

    III quadrant

    Figure 2.12: SRM interval controller.

    The inrush (output value 4) and the chopping (output value 1) intervals

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    2.6. SRM Drives - Converters and Dynamic Operation 29

    are located between on and off. At on, the supply voltage is fullyapplied across the phase provoking an increase of the phase current. Thechopping interval begins from the moment the current reaches its presetmaximum value (established by the control or in natural way by thesaturation current Im). The angular difference (off on) is called theconduction interval of the phase and its length is from zero to s in order

    to prevent the split of the current between two adjacent phases.

    As the motor speed increases, the back-EMF becomes significant and it isnecessary to advance the switching angle in order to reach the referencecurrent in the phase winding before the start of pole overlap. An algo-rithm for calculating the correct phase angle has to be made neglectingresistive voltage drop Ri in the stator winding and the motor back-EMFin (2.11). The required value of the current is reached at the moment s(the beginning of the positive inductance slope) by calculating the valueof 1 using (2.11) where i(s) is the value of the current at s and L isthe inductance (Lmin motor or Lmax for generator operation). This isthe inrush interval. The output of interval controller is 4.

    1 = Li(s)UN

    (2.30)

    In order to avoid the ripple torque, for motoring and generating mode, theadvanced turn-on angle has a limitation imposed by the motor geometry.In the motoring mode, the maximum advanced turn-on angle is limitedto the angular dimension of unaligned position. For generating mode, thelimitation is due to the angular dimension of aligned position.

    1 r (r + s) motoring mode1 | r s | generating mode

    (2.31)

    The extinction interval (output of interval controller is 2) extends from

    turn-off angle, off, to extinction angle, ext. At off a fully invertedvoltage is applied in order to remove the current quickly. The existence ofa current in the phase during the negative inductance slope period impliesa negative torque, i.e. ext r. The system detects the instant whenthe current reaches zero and automatically passes to the next interval,even if ext has not been reached yet.

    The rest interval (output value 3) extends from ext to on + r. Theswitch S is open and no current flows in the phase circuit. A major prob-lem might be the instant when the chopping interval (positive currentin the coil) becomes rest interval, the switch S being open. This can becaused by a sudden change in rotation direction of the drive: the torquebecomes negative. In this case the slope inductance is positive, the cur-

    rent is not equal to zero and the voltage is zero provoking an uncontrolled

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    30 Principle of Operation

    increase of the current. The solution is to transform a natural rest intoextinction interval, independent of the rotor position. The process isnot easy. Meanwhile the SRM continues to rotate, which may provoke achange of the interval.

    For an SRM an operation session consists in energising the phase, constant

    current and de-energising it by zeroing the current. When phases are energizedin a clockwise sequence, an anticlockwise torque is produced minimising thereluctance by pulling the pair of rotor poles into alignment with the corre-sponding stator phase. The anticlockwise direction is considered as positive.

    L, i

    i L

    T

    0

    L, i L, i

    L Li i

    T T

    a) b) c)

    Figure 2.13: Torque of SRM (linear analysis).

    Figure 2.13a. shows an idealised operation. The current i, increased

    during the interval of constant L = Lmin, reaches maximum value duringthe interval of increasing L. It decreases and becomes zero during theinterval of maximum value ofL = Lmax;

    Figure 2.13b. shows a less good operation: the current did not reach itsmaximum value when L begins to increase. A negative torque is producedif the current is not zero while the inductance slope is negative;

    A braking operation is presented in Figure 2.13c., the switch S being openduring the period decreasing of L.

    The discussion so far has established the existence of three angles thatgovern the switched reluctance motor control: turn-on, turn-off, and extinction.Since the torque in SRM drives is independent of the excitation current polarity,

    SRM drives require only one switch per phase winding. Various converter

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    2.6. SRM Drives - Converters and Dynamic Operation 31

    topologies exist, and each one has its own merits and drawbacks. Functionof the converter topology, a fourth angle can be established called freewheelingangle, delimiting the freewheeling interval. The role of the freewheeling intervalis to return the energy to the supply via the diodes after the commutation.During a typical motoring stroke the locus of the operating point [i, ] followsa curve similar to the one shown in Figure 2.14c. (current with respect of the

    rotor position). In the same figure, the inductance curve for a constant currentis presented in order to make the whole process clearer.

    At A the power electronic switch is turn-on and the current starts to flowin the phase winding. It increases till the angle B where it reaches its referencevalue. Usually this angle coincides with the beginning of overlap of rotor andstator poles. The turn-on angle A is situated on the unaligned magnetizationcurve (Figure 2.14a.). At turn-off angle C the supply voltage is reversed andthe current freewheels through the diode.

    Figure 2.14: Analysis of energy-conversion loop.

    At C the accumulated energy from the supply is equal to the total areaW = Wm + Wsme (Figure 2.14a.). The stored magnetic energy is equal toWsme. The area enclosed between the curve ABC and the magnetization curveC represents the mechanical work Wm done during the conduction period ofthe power electronic switch.

    The freewheeling interval starts after the commutation point C and is rep-

    resented by the area enclosed between the magnetization curve C and the

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    32 Principle of Operation

    curve CD (Figure 2.14b.). The supply voltage is reversed and the energy WDis delivered. During the freewheeling interval, the current still flows into thephase winding, which obviously generates a torque. The area WmD in this caseexpresses the mechanical work done. Mathematically, this can be expressed asWmD = Wsme WD.

    A rough energy balance can be deducted graphically. The result suggests

    that at each stroke the total mechanical work done during the power electronicdevice and freewheeling diode period represents about 2/3 of the supplied en-ergy, while the remaining energy is returned to the supply. As seen, the useof a freewheeling angle is not necessary for controlling the switched reluctancemotor. Its only advantage is the energy efficiency improvement. However,this advantage has its limitations by the torque ripple produced during thefreewheeling period. Without a fully inverted voltage immediately after theturn-off angle C, the time necessary for zeroing the current is higher. For highspeeds, this can provoke negative torque generation, and the price paid is a sig-nificant degradation of the motor performances (average torque, ripple torque,etc.).

    2.6.2 Four-quadrant operationVariable speed applications require usually a four-quadrant operation. Theswitched reluctance motor allows this kind of control. The advantage of theSRM is that forward and reverse motoring/braking operations do not dependon the direction of the current flowing in the phase windings, but only on therotor position (Figure 2.15).

    Forward motoring requires a positive electromagnetic torque during the for-ward motion and is developed when the four phase windings of the motor areswitched in the sequence A, B, C, D during their rising inductance zone. Re-verse motoring, similarly, requires negative electromagnetic torque during neg-ative (read reverse) direction of rotation and is obtained by exciting the phasewindings in the reverse switching sequence B, A, D, C, again during their ris-

    ing inductance zone. A braking action requires reverse electromagnetic torquewith respect to the actual direction of rotation. Forward braking requires theswitching sequence to be A, B, C, D during the falling inductance zone whereasreverse braking is accomplished by the sequence B, A, D, C during the fallinginductance zone of the phase windings.

    2.6.3 Dynamic Operation

    Summarising the previous discussion, the control of the switched reluctancemotor can be realized with unidirectional phase currents, and the work sessionconsists of four intervals:

    Inrush interval where the full positive voltage is applied; it extends from

    advance turn-on angle to turn-on (onadv on);

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    2.6. SRM Drives - Converters and Dynamic Operation 33

    q

    T

    q

    0

    L, i

    iL

    w < 0T > 0

    w

    T

    0

    q

    q

    w > 0T > 0

    T

    L, i i

    L

    T

    L i

    q

    q0

    w > 0T < 0

    L, i

    0

    Tq

    q

    w < 0T < 0

    L, iLi

    Figure 2.15: Four-quadrant operation of SRM.

    Chopping or single-pulse interval; it extends from turn-on to turn-offangle (onoff); during the chopping interval, the phase current is con-trolled by chopping the supplied dc voltage generator; during the single-pulse interval, the full positive voltage is applied;

    Extinction interval where the dc voltage is fully inverted across the phaseto remove the current quickly; it extends from turn-off to the extinctionangle (off ext);

    Rest interval where the phase is open and no current flows;

    Depending on the chosen strategy, a fifth interval can be considered, thefreewheel interval. It does not contribute to the control of the machineand its only purpose is to maximize the efficiency, as it is used to returnthe energy to the supply via the diodes; it extends from the freewheelingangle to the turn-off angle (frwheel off).

    The flux in the switched reluctance motor is not constant, but must be built

    up from zero during every stroke. Switching the supply voltage on at turn-on

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    34 Principle of Operation

    angle and off at the commutation angle controls the process. Current control isachieved with closed-loop control by a chopping action using either a PWM orhysteresis switching control of the converter. The current controller structureis different, depending on the switching strategy chosen.

    PWM Current Control

    The selection of the Pulse Width Modulation strategy is an important issuein SRM control, as it dictates how the motor can be controlled. The PWMstrategy is also directly related to the power electronic converter topology.Assuming that each phase of the SRM can be independently controlled, thereare three PWM strategies.

    Single pulse operation

    Each phase must be energized at its turn-on angle and switched off atits turn-off angle. In single pulse operation, the power supply is keptswitched on during the dwell angle and is switched off at the phase com-mutation angle. As there is no control of the current and as there isa sharp increase in the rate of change of current, this PWM strategyis used when the amount of time available to get the desired current isshort. Typically, single pulse operation is used at high speed.

    Chopping voltage strategy

    The chopping voltage strategy is useful for controlling the current at lowspeeds. This PWM strategy works with a fixed chopping frequency. Inthis case, the supply voltage is chopped at a fixed frequency with a dutycycle depending on the current error. Thus both current and rate ofchange of current can be controlled.

    The chopping voltage strategy can be separated into two modes: hard andsoft chopping strategies. The fixed duty-cycle is defined as d = ton/T, where

    ton is the on time and T is the period of the chopping frequency. PWM widthis determined by comparing the measured phase current and the required ref-erence current (Figure 2.16).

    iref

    i+ -

    ke+

    +

    if duty_cycle Tduty_cycle = T

    end;

    if duty_cycle 0duty_cycle = 0

    end;

    z-1

    duty_cycle

    Figure 2.16: Duty cycle flow chart.

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    2.6. SRM Drives - Converters and Dynamic Operation 35

    K is the proportional gain and depends on the motor parameters and alsoon dc bus voltage. Considering S the number of steps allowed in one PWMcycle, the proportional gain K can be determined. Let i be the change inphase current for 100% change in PWM duty cycle. The parameter i dependson motor and converter types. The proportional gain K can be defined as:

    K = Si

    (2.32)

    Krishnan in [Kri 01] suggests that to obtain a high response speed, its valueis in the range of 30 to 70.

    Chopping current strategy

    The chopping current strategy is a hysteresis type current control in which thevoltage supply is chopped according to whether the current is larger or lessthan a reference value. At low speed, the current is controlled using choppingcontrol. Chopping controls the plant in a closed-loop in order to force the stateof the plant to slide along a given surface (sliding surface). To keep the currentsufficiently close to its reference value, the switching function is defined as:

    i = i i (2.33)

    The existence conditions of the sliding regime are

    ididt

    < 0 (2.34)

    and

    | Ueq | 0

    (2.36)

    Soft chopping

    U =

    UN i < 0

    0 i > 0(2.37)

    The equivalent voltage for constant current(di/ = 0) is:

    Ueq = ki (2.38)

    The hysteresis controller is used to limit the phase current within a preset

    hysteresis band. As the supply voltage is fixed, the result is that the switching

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    36 Principle of Operation

    frequency varies as the current error varies. The current chopping is not fixedfrequency. This PWM method is more commonly implemented in drives, wheremotor speed and load do not vary too much, so that the variation in switching issmall. As seen previously, here again both hard and soft chopping schemes arepossible. Hysteresis controllers realize the switching logic, one for each phase(Figure 2.17). Note that the strategy described in the figure is soft chopping,

    the output voltage having two values +Udc and 0. A similar strategy can beachieved by switching the output voltage between +Udc and Udc.

    Figure 2.17: Current control using hysteresis PWM.

    The simple hysteresis controller maintains the current between an upperand a lower limit, the hysteresis band. As the supply voltage is fixed, theresult is that the switching frequency decreases as the incremental inductanceof the phase winding increases. This control technique is valid only for anSRM drive operating in mode A, the dc voltage is larger than the back-EMF.

    Hysteresis current control allows rapid variation of the motor phase currentand thus the motor torque, because the full bus voltage is instantly availableto force the current to change quickly.

    Hard or Soft chopping

    Chopping is necessary to control the current at low speed. Till now it wasshown that chopping the supply voltage using various PWM strategies cancontrol the phase current. The voltage can be chopped between three definedvalues as:

    U = UN equivalent to INRUSH interval

    0 equivalent to FREEWHEEL interval

    UN equivalent to EXCTINCTION interval

    (2.39)

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    2.6. SRM Drives - Converters and Dynamic Operation 37

    Switching the voltage between full positive voltage and 0 is called soft chop-ping (Figure 2.18a.). Hard chopping (Figure 2.18b.) consists in switching thesupply voltage between fully positive and fully negative voltage. This controlstrategy puts more ripples into dc link capacitor, thus reducing its lifetime andincreasing the switching losses of the power switches due to frequent switchingnecessitated by the energy exchange. This can be improved by an alternate

    switching strategy [Kir 01]. The chopping interval is a synthesis among thethree other intervals.

    i

    L

    + UDC

    - UDC

    0

    + UDC

    - UDC

    0

    i

    LL, i L, i

    t t

    t t

    Figure 2.18: Hard and soft chopping current control.

    If the switching frequency remains constant, hard chopping increases thecurrent ripple by a large factor. Miller [Mil 93] suggests that the typical increaseis by a factor between 5 and 10. This is the main reason why this kind of controlstrategy is not desirable for motoring operation. However, in generating orbraking operation, it may be necessary as the only feasible means to control

    the current. Experience suggests that soft chopping produces lower acousticnoise and less EMI. It also decreases the dc ripple current in the supply andsubstantially reduces the requirement for filter capacitance.

    2.6.4 Converter Structures for SRM

    The subject of converter structures for SRM is not well covered in the literature,relatively few being published about their development. Most of the authorsfocus on the development of the control strategies, considering the converter asideal, thus practically ignoring it during their analysis. The objective of thiswork is to synthesise, from the knowledge of the specifications, the structure ofa power electronic converter capable of controlling the SRM in accordance tothe rules, intervals and switching angles developed previously. In this section,

    the purpose is to find straightforward way without using empirical data, the

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    38 Principle of Operation

    structure of the converter which answers best the imposed requirements. Ahysteresis current controller is considered here due to its simplicity in conceptand implementation.

    Therefore, it is necessary to determine:

    the structure of the converter, i.e. the number and the place of the

    switches;

    the static and dynamic characteristics of the switches.

    It is necessary to know the characteristics of the dc supply and of severaltypes of power electronics switches. A battery can supply and receive power.It is a reversible voltage source. The winding of an electrical motor phase is areversible current source. During its operation, the converter connects throughthese switches the sources ensuring and controlling the exchange of energy.In order to make these links, a certain number of rules have to be respectedimperatively:

    a voltage source should not ever be short-circuited, but it can be opened;

    the circuit of a current source should not never be opened but it can beshort-circuited;

    two sources of the same nature should never be connected;

    only a current source and a voltage source can be connected .

    Conform to Foch et al. [Foc 88], the sources of input and output of aconverter being characterized (voltage or current source, reversibility) and theinter-connection rules of the sources being known, the converter structure canbe deducted. It is called the basic configuration diagram which, without as-sumption of the characteristics of the switches, allows all possible intercon-nections between a given input and output source. In case of an SRM, the

    input and output sources have different nature, therefore a direct connectionconfiguration will be used.Resuming, the problem to solve is: the input source being a voltage source

    and the output source a current source, which are the different interconnectionpossibilities of these two sources and which is the structure that makes it pos-sible to carry out all these interconnections? Having these two sources and therules presented previously, the connections of Figure 2.19 are feasible.

    to connect in one way input and output source (state I)

    to connect in the opposite way the input and the output sources (stateII)

    to separate both by opening the voltage source and by shorting-circuiting

    the current source (state III)

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    2.6. SRM Drives - Converters and Dynamic Operation 39

    +

    -

    +

    -

    +

    -

    I II III

    Figure 2.19: Possibilities of interconnections of a voltage and a current source.

    It is noted that these three interconnections are necessary to allow all ex-changes and to control the energy flow between voltage and the current source.If all three states have to be realized without using a middle point convertertopology, the simplest solution is to use a four switches bridge topology (Figure2.20).

    K1 and K3 closed gives state I;

    K2 and K4 closed gives state II;

    K1 and K4 closed or K2 and K3 closed gives state III.

    +

    -

    K1

    K2

    K4

    K3

    Figure 2.20: Base configuration of a voltage - current converter.

    This diagram will be considered from now as the basic configuration forthe switched reluctance motor converter. As an observation, a direct converteris an electric circuit made only of switches. It is unable to store energy, theenergy transfer being carried out directly from input to output. If the losses inthe converter are neglected, input and output power are equal at each instant.The current control in switched reluctance motor is realized by chopping thesupplied voltage among three values: +Udc, 0, and Udc. Since the torque ofa switched reluctance motor is independent of the excitation current polarity,the direction of the current flowing into the phase windings will be the same inall the cases. Motoring and braking in four-quadrant operation are made in thesame way, the difference being given by the instant when the voltage is applied,

    in accordance to the intervals and switching angles developed previously. The

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    40 Principle of Operation

    dc source may be a battery but usually is a rectified ac supply with a filter toprovide a dc input source to the SRM converters [Kri 01].

    The Figure 2.21 presents the four control sequences of a switched reluctancemotor. The chopping interval is in fact a synthesis among the inrush, freewheel(soft chopping) or extinction (hard chopping) intervals. The description corre-sponds to the control sequences presented in Figure 2.21a.

    Figure 2.21: Four control sequences of a switched reluctance motor.

    By closing K1 and K3, and opening K2 and K4, the SRM phase is suppliedwith fully positive voltage provoking a rise of the current, corresponding tothe inrush interval (1). The chopping interval consists of a hard chopping,corresponding to a switch of the voltage between fully positive (2) and fullyinverted voltage supply (2). The sub-interval (2) is identical with the inrushinterval, and the (2) sub-interval is realized by closing K2 and K4, and openingK1 and K3. The freewheeling interval (3) is obtained by closing K1 and K4,and opening K2 and K3. The fully inverted voltage is obtained in a similarway as sub-interval (2), corresponding to the extinction interval (4).

    Before starting the analysis of the control sequence, one important remarkhas to be made. In the steady state, a switch can be characterized as a non-linear resistance, very small at turn-on, very high at turn-off. Considering the

    switch as a load, its static characteristic I = f(U) can be represented as in

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    2.6. SRM Drives - Converters and Dynamic Operation 41

    Figure 2.22. If the static operating points of the sequences before and aftercommutation are on two half-axes of same signs, this commutation can only beforced (A B and C D). If the static operation points of the sequencesbefore and after commutation are on two half-axes with contrary signs, thiscommutation can only be spontaneous (A D and B C) [Foc 88].

    D

    U

    A

    B

    C

    I

    U

    I

    Figure 2.22: Static characteristic of a switch.

    The control sequence of the proposed SRM control strategy is (1) (2) (2) (3) (4). Now, the operating point for each switch can be describedfor each control sequence. Examining for each switch the way to pass from oneoperating point to another, the commutation type can be deducted. Figure2.23 shows the operation points for switches K1, K2, K3, and K4.

    Figure 2.23: Characteristics of the switches for SRM control sequences.

    Summarizing, it can be seen that the switches K1 and K3 present two states,

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    42 Principle of Operation

    both being forced. Their characteristics suggest that K1 and K3 are IGBTs.The characteristics of K2 and K4 switches suggest a spontaneous turn-on andrespectively turn-off states. This corresponds to forward biased diodes. Theanalysis leads to the converter presented in Figure 2.24.

    T1 D2

    D1 T2

    +

    -

    Figure 2.24: Asymmetric converter for SRM with freewheel and regenerationcapability.

    Figure 2.24 shows an asymmetric bridge converter for one phase of theswitched reluctance motor. The rest of the phases are similarly connected.Turning-on and -off of T1 and T2 for the proposed strategy is shown in Figure2.25. Turning on T1 and T2 circulates a current in the SRM phase. If thecurrent rises above the reference value, T1 and T2 are turned off. The energystored in the motor winding recharges the dc source through the diodes D1 andD2, bringing rapidly the current below the reference value, the phase voltagebeing negative. During this interval, there is a repeated exchange of energybetween the dc source and machine winding. During turn-on and -off of T1and T2, the machine phase winding experiences twice the rate of change of dclink voltage, resulting in a higher deterioration of the insulation. This controlstrategy puts more ripples into the dc link capacitor, thus reducing its lifetimeand also increasing the switching losses of the power electronic switches.

    From Figure 2.18, it can be seen that inrush and extinction intervals auto-matically impose simultaneous turn-on and -off of T1 and T2. Soft choppingimplies the existence of the freewheel interval, thus the energy stored into phasewinding is not anymore returned to the source. In this case the time for thephase current to return under the reference value is higher than previous asseen in Figure 2.18b. The advantage of the soft chopping is that reduces theswitching frequency and hence the switching losses. This is realized by keepingT1 turned-on during the inrush, chopping, and freewheel interval. During thefreewheel interval, the current continues to flow in the phase winding, convert-ing the stored energy to useful mechanical work.

    Krishnan [Kri 01] describes a unipolar switching strategy derived from thesoft chopping strategy presented previously. The inrush, soft chopping, andfreewheel intervals are identical as presented, but instead of an extended ex-

    tinction interval, the strategy alternates freewheeling and extinction interval.

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    2.6. SRM Drives - Converters and Dynamic Operation 43

    0

    T1

    T2

    D1

    D2

    + UDC

    - UDC

    L

    i

    Figure 2.25: Hard chopping strategy.

    The result is a more judicious choice of negative or zero voltages across thewinding to obtain a fast current response.

    A lightly modified soft chopping strategy can be realized by alternatingthe freewheeling cycle through the upper and the lower diode. This is madeby alternatively turning-off T1 and T2 respectively in the chopping region(

    on

    frwheel) (Figure 2.26). The advantage of using this strategy is that it

    gives equal ratings for the power devices and diodes in each phase conductor.Each switch conducts for two on times and one off time and is turned offfor one off time. Such a switching strategy enables equal rms currents in theswitches and equal average currents in the diodes.

    During the chopping interval (on frwheel), the average duty cycle is:

    d =Ip(Rs + m)

    Udc(2.40)

    Various power converter configurations exist to control a switched reluc-tance motor: two-stage power converter, single-switch-per-phase converters,etc. Each category has advantages and drawbacks. For the present thesis, theasymmetric converter developed previously has been considered. During the

    simulations only the control strategy has been retained, the converter being

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    44 Principle of Operation

    Figure 2.26: Modified soft chopping strategy.

    considered as ideal.

    2.7 SRM system modelling

    Closed-loop motor control has the attractive properties of response and optimalperformance for varying load conditions. The switched reluctance motor hasstrong similarity to dc and synchronous reluctance machine, but in control it isvery familiar to these machines and therefore, analogous control developmentsare not possible [Kri 01]. To overcome the torque ripple of the motor and thenonlinearity of the torque characteristics, various control solutions have beendeveloped.

    This section deals with the description of the control scheme and its maincomponents. The switched reluctance drive is an electro-mechanical unit, com-posed of a SRM, a power electronic converter and a controller, all componentsbeing coupled The development starts with the motor model based on themathematical description of one phase. It continues with the description of theinterval controller, speed controller and tuning of the speed controller gainsusing simulated annealing algorithm.

    2.7.1 Motor model

    The magnetization curves allow the complete mathematical description of themotor. Considering the equivalent circuit of one phase, the voltage equation

    is:

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    2.7. SRM system modelling 45

    U = Ri +(i, )

    i

    di

    dt+

    (i, )

    d

    dt(2.41)

    The first term of the equation corresponds to the voltage due to the phaseresistance R. The second is the contribution of the inductive voltage, and thelast term corresponds to the back-EMF.

    The partial derivates of the flux with the current (i,)i and respectivelywith the rotor position (i,) are easily found from the magnetization as seenin Figure 2.27. The variation in time of the current is expressed by the voltageequation:

    di

    dt=

    1(i,)

    i

    URi

    (i, )

    (2.42)

    Figure 2.27: Partial derivatives of the flux with respect to current and rotorposition.

    This non-linear model of the switched reluctance motor (Figure 2.28) is very

    handy and uses only the partial derivates of the flux-linkage with respect ofrotor position and phase current, derivatives that are stored as look-up tables.

    Intervaland

    CurrentController

    U

    iq

    iref

    wref ( )

    --

    = wq

    qyqy

    .,

    ..,

    1 iiRu

    i

    idt

    di

    Figure 2.28: Non-linear switched reluctance motor model.

    The interval controller output gives four values as a function of the rotor

    position and the reference torque sign. The output values correspond to the

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    46 Principle of Operation

    inrush, chopping, extinction and rest intervals, and the intervals extreme lim-its are named switching angles, which can be found using different analyticalor/and simulation methods, function of the criteria chosen (maximising the av-erage torque output, maximizing the machine efficiency, reducing ripple torquelevel, etc.). This will be developed for an SRM four-phases 8/6 in the nextsection.

    Postulating a very small sample time, the following transformation fromcontinuous time to discrete time state space causes a negligible error. By thistransformation, the mechanical equation can be rewritten as explained. Theintegration in the continuous time of a variable X is transformed to:

    Int

    X=

    1

    s(2.43)

    In discrete time space the same integration is written as Figure 2.29, whichcan be mathematically written as:

    Int

    X=

    Ts1 z1

    (2.44)

    XTs

    +

    +

    Int

    1z

    Figure 2.29: Integration in discrete time space.

    Equalizing, the transformation from continuous time to discrete time spaceyields:

    s =

    1 z1

    Ts (2.45)

    The mechanical equation written in discrete time space is presented in Fig-ure 2.30. Tolerating a small discretisation error, the transformation from con-tinuous time to the discrete time state space causes a negligible error.

    Figure 2.30: Mechanical equation.

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    2.7. SRM system modelling 47

    2.7.2 General scheme

    The switched reluctance drive is an electro-mechanical unit, composed of anSRM, a power electronic converter and a controller, all components being cou-pled. A specific power electronic converter supplies the switched reluctancemotor. The converters turn-on and turn-off conditions can follow different

    schemes in order to control the motor speed.By combining these various blocks of the drive system, system equations

    are assembled containing differential, algebraic, and conditional equations. Thedifferential equations are solved for a given instant of time with a sufficientlysmall sample time using Matlab/Simulink. The general scheme of the drive isshown in Figure 2.31. Variations from this block diagram may be minor, andare usually confined to the controller section, being specific to the application.

    Figure 2.31: Block diagram of the SRM controller.

    Interval controller

    The optimal switching angle and interval control is very important. The In-terval controller module assures the choice of the right instant of each intervalfunction of rotor position, reference current, and operation quadrant. Themodule contains a series of logic functions and look-up tables, the output databeing the optimal switching angles.

    Finding the optimal switching angle for a specific SRM application, is anoptimisation problem. The interest points in controlling SRM systems is toassure the lowest torque ripple, maximum efficiency, maximum average torque,minimal acoustic noise, reducing reactive power flow, etc. These objectives canbe targeted together or individually as a function of the required application.As described above, optimal switching angles choice strongly depends on the

    reference current and machine speed. This feature is developed in Chapter IV.

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    48 Principle of Operation

    2.7.3 Speed Controller

    Various control strategies exist in the literature. In the following subsections,fuzzy logic and proportional-integral (PI) controller are presented. Finally, thePI controller is chosen for the switched reluctance motor speed loop regulation.Tuning of PI gains can be realized using various methods. In the present

    thesis the simulated annealing method is used to determine the optimum gaintuning. It is not intended in this thesis to advocate one or another speedcontroller type as best solution for Switched Reluctance Motor, neither to findthe optimal tuning method, but rather to present the effectiveness of usingsimulated annealing algorithm in tuning the PI controller.

    Fuzzy logic

    Fuzzy logic applications in power electronics and drives are relatively new. A.Zadeh has developed the theory of sets, in the middle of seventh decade [Zad96]. It is based on the mathematics of vague notions, as it imitates humanperception expressed in words as not really, almost, etc. This theory appearsto be the opposite of the more common bivalent logic. There is a very wide

    range of controllers based on fuzzy logic: fuzzy controllers of the PID type,fuzzy controllers - sliding mode, fuzzy controllers type Sugeno and Takagi,direct and indirect adaptive fuzzy controllers, etc [Bir 99]. The advantages offuzzy logic controllers are:

    There is no need of exact knowledge of the mathematical model of thecontrolled process;

    More efficient control of non-linear systems due to the non-linear natureof the controller;

    Fuzzy controllers are relatively easy to implement;

    Lower cost than other intelligent control systems.

    However, the main problem of a fuzzy controller is its stability. A compar-ison of the fuzzy-controlled system performance with that of a PID control isgiven in [Li 89] proving the superiority of the former.

    Proportional-integral (PI) controllers

    Proportional-integral (PI) controllers are widely used in industry for drives. Inmany industrial processes accurate speed control associated with good speedholding capability in the presence of load disturbance is essential to ensureproduct quality. Many non-linearities arise in drives.

    The speed controller converts the speed error in a torque reference value(or current reference value). Keeping torque and current within predeterminedboundaries is achieved by limiting the output of the speed controller. The

    most commonly used speed controller for drives contains two separate control

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    2.7. SRM system modelling 49

    loops (Figure 2.32). The inner loop is responsible for the current control andincorporates a PWM hysteresis controller, activated by the error between setand measured motor current. The current/torque reference is generated bythe outer control loop, in which the error between reference and actual speedactivates the proportional-integral (PI) speed controller.

    Figure 2.32: Cascade control of SRM.

    In order to simplify the calculations, the system can be decoupled in twomodes: quick mode (electrical mode) and slow mode (mechanical mode). Thishypothesis is mostly true as the mechanical time constant is much larger than

    the electrical. In this way the current control loop can be considered as beingunity. The model equations are non-linear, because the current and its controlenter the speed equation.

    Tuning the speed controller

    The tuning of electric drive controllers is a complex problem due to the manynon-linearities of the machine, power electronic converter and controller. Theinherent non-linearity of the SRM is difficult to handle for a proper tuning ofthe controller parameters. The power converter presents a non-linear transfercharacteristic because of imposed switching dead times. A further obstaclein finding the optimum settings for the parameters of a proportional-integral(PI) speed controller is the difficulty of characterizing the load. Many methods

    have been published. Miller suggests that specifying the damping ratio , thenatural frequency n and the maximum overshoot Mr yields the PI-controllerparameters [Mil 01]. For small-signal analysis the spring load force can beregarded as constant. The system can be described as a standard second-ordersystem with a zero:

    (s)

    ref(s)=

    KJ (s + a)

    s2 +

    f+KJ

    s + KaJ

    =2na

    (s + a)

    s2 + 2ns + 2n(2.46)

    Tuning the controller for optimal performance in a non-linear system (pos-sibly together with an anti-windup system) may become a difficult task, es-pecially when load disturbances are involved. Da Silva et al. [Sil 01] propose

    on-line optimization of a PI controller with anti-windup circuit using genetic

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    50 Principle of Operation

    algorithm. The genetic algorithm has proven to be capable of finding the op-timum or near optimum settings for proportional and integral gains, togetherwith the setting of the non-linear blocks of anti-windup circuit. An alternativeis the simulated annealing method.

    Simulated annealing method

    The simulated annealing method is an optimisation technique that can be em-ployed to find global minima or maxima. The inspiration for simulated anneal-ing comes from the physical process of cooling molten materials down to theirsolid state. When molten steel is cooled too quickly, its atoms have no timeto find the equilibrium position and the product presents a low mechanicalresistance. A slow cooled down schedule ends up with the best final product,bringing the steel to a low-energy, optimal state.

    The application of the simulated annealing method in tuning PI speed con-trollers has given very good results [Aca 02]. Stochastic search techniques avoidthe requirement for mathematical modelling of the power electronic converter,drive and load, being able to deal with non-linearities. They simulate the ran-dom evolution of a physical system and reach equilibrium as the steady-state

    distribution over states of a corresponding Markov chain. Simulated annealingcan be shown to converge to a globally optimal solution. However, it can beextremely computationally expensive.

    A physical system, as it cools down, seeks to reach to a minimum-energystate. For any discrete set of particles, minimizing the total energy is a combina-torial optimisation problem. Through random transitions generated accordingto the above probability distribution, the physics to solve arbitrary combinato-rial optimisation problems can be simulated. The simulated annealing tuningalgorithm for proportional-integral (PI) can be formulated as follows.

    Set the independent parameters to the initial values;

    Set Temperature to a relatively high value;

    Perturb the independent parameters;

    Calculate the new cost function in the new conditions;

    Compare the new and best till now cost functions:- If the new cost function is lower than or equal to the best, the new onebecomes the best till now;- If not, choose a random number r uniformly from [0,1]. Ifr < e(E/kT),accept the worst solution (the new one) as best till now;

    Repeat steps 3-5 an arbitrary number of times n;

    If an improvement has been made after n iterations, set the center point

    to be the best point.

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    2.7. SRM system modelling 51

    Reduce the temperature following the preset schedule;

    Repeat the algorithm till the stop criterion.

    Acarnley et al. [Aca 02] introduce the application of simulated anneal-ing to the problem of tuning electric drive speed controllers. In this thesis

    a similar algorithm is used for finding optimal parameters of a PI controllerof a switched reluctance drive. The algorithm consists of comparing a ran-domly generated potential solution B[KpB, KiB,IAEB ], to an existing solutionA[KpA, KiA,IAEA]. The solution space defines the maximum and minimumvalues of the PI gain, and any potential solution B is in the allowed systemsolution space. The algorithm consists of two steps.

    Proximity of the solutions

    The first step is to generate a randomly potential solution B within the allowedsystem solution space and to calculate an acceptance probability, PA, functionof the distance between A and B, and the instantaneous temperature. Thereare two acceptance probabilities, corresponding to the two gains, PAA and PABdefined as:

    PA = exp

    displacementrange

    kT

    (2.47)

    where the gain displacement (for proportional and respectively integralgains) is defined by:

    displacementP =| KpA KpB |displacementI =| KiA KiB |

    (2.48)

    Each acceptance probability is then compare to a random value, r, between[0, 1]. Ifr > PA, the potential solution gain is rejected, otherwise it is accepted.If both potential solution gains are accepted, the entire potential solution, B, is

    accepted for the second step of the algorithm, otherwise the algorithm proposesanother potential solution and the loop is restarted. This prevents the searchof a potential solution far away from the actual solution.

    The acceptance probability, PA, is a function depending on the instan-taneous temperature, T, and the ratio displacement/range. The acceptanceprobability is higher for any displacement/range ratio at the beginning of theprocess when the temperature is high than at the end of the process (Fig-ure 2.33). A proposed solution B has a higher acceptance probability to beevaluated all along the process, if it is close to the existing solution A.

    A number of 500 solutions per temperature level, B, is proposed for eval-uation in the first step for each temperature. Not all of them are accepted,

    just those situated in the proximity of the existing solution A (represented asa circle on Figure 2.34). At the beginning, the proposed solutions, B, are dis-

    tributed in the whole solution space around A. As temperature decreases, the

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    52 Principle of Operation

    0 0.2 0.4 0.6 0.8 10

    0.5

    1Acceptance probability

    PA

    Temperature

    displacement/range

    1

    0.10.3

    0.5

    0.8

    Acceptance

    pro

    ba

    bility,P

    A

    Figure 2.33: Effect of temperature on the acceptance probability PA.

    acceptance probability is more and more sensitive to the displacement of B toA, the accepted solutions, B, being chosen closer to A (Figure 2.34).

    Figure 2.34: Effect of the temperature on the accepted displacement.

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    2.7. SRM system modelling 53

    Cost function

    The second step of the algorithm consists in evaluating the proposed solutionB, and corresponds to the point (5) of the simulated annealing algorithm. Theevolution of the error signal in the transitory regime is a required parameter inthe evaluation of quality of an automatic control system. The most common

    evaluation criteria for a PI controller are the IAE (integral absolute error), ITSE(integral of time-multiplied square error), ISE (integral square error) or ITAE(integral of time-multiplied absolute-value of error) [Mar 87]. For the presentalgorithm, the IAE criterion has been chosen for evaluation. The so-called costfunction is:

    E =

    0.150

    | e(t) | dt (2.49)

    The corresponding change in cost function Eis calculated and it is decidedupon this difference whether the proposed transition is accepted. If EB < EA,B is accepted and A is replaced by B. Else, if the new result (EB) is higher thanthe actual best (EA), a random number r in the range of [0,1] is generated and

    compared to a change probability. To get out of a local minimum, an increaseof the cost function is accepted with a certain change probability. Hence, anew state with a larger cost has a high probability of being accepted. Theprobability of accepting a worse state is high at the beginning and decreasesas the temperature decreases (Figure 2.35). For each temperature, the systemmust reach equilibrium i.e., a number of new states must be tried, before thetemperature is reduced typically by 10%. The change probability is defined as:

    Pc =1

    1 + exp

    EkT

    (2.50)

    Figure 2.35: Effect of temperature on the change probability, PC.

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    54 Principle of Operation

    Temperature decrement function

    Frequently an exponential decay in temperature by a multiplication with aconstant factor is used. The value of the multiplication factor is a compromisebetween temperature decrement between stages and number of iterations perstage. A larger number of iteration per stage allows the use of a lower multipli-

    cation factor. The best values for the multiplication factor are between 0,8 and0,98. In this thesis, the multiplication factor is 0.95 with an initial temperatureof 1 (Figure 2.36).

    Tempera

    ture

    []

    Number of temperature levels

    0 10 20 30 40 50 60 700

    0.2

    0.4

    0.6

    0.8

    1

    Figure 2.36: Temperature decrement function.

    Number of iteration per stage

    In practice many criteria can be used to limit the number of iteration perstage. A common one is a constant number of iteration, or a constant num-ber of accepted transitions per stage. Experiments show that better resultsare achieved by keeping the temperature constant until the cost function isoscillating around a constant value. Another option is to impose the num-ber of iterations. Typically, 100 to 1,000 iterations might be permitted beforelowering the temperature.

    Stop criterion

    Typically, when the value of the current solution has not changed or improvedwithin the last iteration or so, the search is terminated and the current solution

    reported.

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    2.7. SRM system modelling 55

    2.7.4 Four-quadrant control of SRM drive - linear analy-

    sis

    This section deals with control of a four-phase 8/6 switched reluctance motor inall four quadrants. The scheme contains a speed loop (PI control), current loop(chopping control) and interval controller developed in the previous sections

    (Figure 2.37). A specially designed algorithm for the interval controller isproposed to control the voltage in order to minimise torque ripple. The rotorposition is calculated from the speed integration. Simulation results confirmtorque ripple reduction, illustrate performance and robustness of the controlscheme proposed and show good overall behaviour of an SRM drive modelunder various types of disturbances.

    The simulated SRM has the following data and parameters: PN = 7.5kW,N = 1900r/min, UN = 460V, IN = 32A, Im = 8A, q = 4, Ns = 8, Nr = 6,Lmin = 10mH, Lmax = 110mH, r = 1.05rad, s = 0.35rad, r = 0.42rad,f = 0.004Nms/rad, J = 0.0016kg m2.

    Reg (s)T

    *I

    *

    USRM

    UNSgn(T )*

    Interva

    l

    Contro

    ller

    w

    w* +

    -

    q

    Hysteres

    is

    Contro

    l

    Figure 2.37: Linear model - control schema.

    The controller applies the correspondent voltage at each phase as a functionof the rotor position in order to maintain the phase current close to the referencevalue given by the PI controller. A special algorithm was designed to drop the

    phase current to zero (interval 3) in order to minimise the ripple torque. Therotor angle is detected by the integration of the drive speed value. The systemwas designed in modules in order to provide a simple solution in implementingfurther research (running above the rated speed, sensorless control, differentkinds of controllers, different types of SRM drives, etc).

    In case of a linear analysis, the inductance profile of one phase (j) is ap-proximated by (2.3). Figure 2.3 shows the idealised inductance profile of onephase as a function of the rotor position for a pair of stator poles.

    In most of dc drives, torque control is a synonym of current control. Be-cause of its nonlinearity, the SRM control is significantly different in this regard.Torque is a nonlinear function of the current and rotor position. The instanta-neous torque of a single phase over a period can be expressed by (2.17). Theinstantaneous torque developed by the motor is the sum of the instantaneous

    torques developed by the individual phases.

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    56 Principle of Operation

    Interval controller and switching angles

    The four intervals are located among the three important angles: on, off andext. For each quadrant, the choice of the switching angles is made with therespect to the torque production and to the assumption that only one phase isactive at each instant.

    The control variables of an SRM drive are switching angles and currentamplitude. Function of the strategy adopted, the relations among these controlvariables are established. To produce a positive motoring torque it is expectedthat the current pulse coincide more or less exactly with the rising part of thephase self-inductance curve. Thus the turn-on angle, on, must be chosen atthe beginning of the torque zone. The turn-off, off, is controlled from 0 tos. It is essential that the phase current is brought to zero before negativeslope inductance begins in order to avoid any negative torque production. Itis evident that 0 on off ext.

    Conform to the analysis made beforehand, the switching angles of the firstphase in all four quadrants are chosen as presented in Table 2.1:

    Table 2.1: Switching angles in four quadrants.I II III IV

    on 0 s + r s roff

    rq

    (s + r) rq

    s rq

    r +rq

    The advance turn-on angle is chosen for the current to reach the desiredlevel i(0) at = 0:

    onadv = Lui(0)

    UN(2.51)

    The simulation circuit requires the detection of the motor speed and the

    phase currents. The rotor position is calculated from the integration of thedrive speed, the initial start position being the 0 angle of the first phase inthe positive rotation sense (counterclockwise) and the 0 angle of the secondphase in the opposite rotation sense (clockwise). As an observation, for thepositive rotation direction, the phase order is L1, L2, L3, L4, L1, L2,... and forthe negative rotation direction the phase order is L2, L1, L3, L4, L2, L1,...

    The phase voltage is manipulated in both the chopping and the extinctioninterval. The extinction angle is not fixed, but automatically calculated asa function of current and speed. The SRM was designed to operate in four-quadrants, the control strategies being different. For motor operations in bothdirections the start of the chopping interval is the instant when the induc-tance starts to increase. For generation operation the chopping interval beginswhen the inductance starts to decrease. The SRM performance depends on the

    current value in each phase as a function of the rotor position.

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    2.7. SRM system modelling 57

    An important aspect in speed control of a drive is the torque ripple. TheSRM drive system has the particularity of a non-linear relation between air-gap torque and the excitation current. The complexity of the control task isamplified in this case by the commutation strategy. During the commutation,one phase is demagnetised and the next one is energised. Without a propercontrol of the current flow in both adjacent phases, the sum of the air gap

    torques contributes to an increase of the torque ripple.A solution is to control the torques (currents) in two adjacent phases through

    a torque (current) distribution function, so that the summation of the two airgap torques equals at any instant the reference torque. For two adjacent phases,A and B, the air gap torque might be written as

    T = TA + TB (2.52)

    The torques of each phase can be expressed using the torque distributionfunctions (fA() and fB()) as:

    TA = T fA() and T

    B = T

    fB() (2.53)

    Note that they are function of the rotor position. Combining (2.52) with(2.53) results that at any instant:

    fA() + fB() = 1 (2.54)

    These functions can be determined based on many criteria. As presentedbeforehand, there are four intervals located among the three important an-gles: inrush, chopping, extinction and rest. However, introducing the conceptof torque distribution function implies the existence of another interval, thecommutation interval. This interval can be identified with a chopping interval,since the current is controlled after a falling (respectively rising) distributionfunction. It is obviously that in this case, the inrush and extinction intervalsare replaced by a chopping commutation interval. The extreme limits of thisinterval can be geometrical determined for each operation quadrant: initial

    (i) and final (f) commutation angles. A number of distribution functionshave been suggested in the literature having as variables the initial and finalcommutation angles [Kri 01]. In this thesis based on the observations, it willbe shown that the distribution function is dependent only of the initial (i)commutation angle. In the first quadrant, where T > 0, the commutationangles can be expressed as:

    0 < i < f < s (2.55)

    The conduction period of one phase is limited by the effective torque zone(2.17). Between two phases, the rotor has to move with an angular distance ofr/q. Before this angle, only the first phase (in this case phase A) is able toproduce torque. The maximum effective torque zone of one phase is determined

    by the stator pole-arc width, s. Thus, the limits of the commutation interval

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    58 Principle of Operation

    are [rq , s]. The initial commutation angle (i) must be situated in this inter-

    val. For higher speeds and currents (as it will be shown in Chapter IV), it ispreferably to chose i as close as possible of the minimum extre


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