Determination of current waveforms for torque ripple minimisation in switched reluctance motors using iterative learning: an investigation

N.C.Sahoo, J.X.Xu and S.K.Panda

Abstract: The paper deals with the investigations on an iterative learning approach to determine the desired current waveforms for switched reluctance motors, whch give rise to ripple-free torque. The current waveforms are generated by repeated corrections from iteration to iteration starting from the conventional rectangular pulse profile as the initial waveform. The scheme requires much less a priori knowledge of the magnetic characteristics of the motor. The algorithms have been formulated for both one-phase-on and two-phase-on schemes, for a four-phase switched reluctance motor, in the light of the principles behind iterative learning. Based on the observations from the siniulation results of these schemes, a modlfied scheme has been proposed by incorporating a suitable commutation process, often called torque sharing functions, in order to generate reasonably smooth current wavefonns for the ease of tracking by the stator circuit of the motor. The performances of all the proposed schemes have been verified by computer simulation.

1 Introduction

The switched reluctance motor (SRM) has a lot of advan- tages, due to its low cost, simple rugged construction, and relatively hgh torque-to-mass ratio. Contrary to the con- ventional motors, the SRM is intended to operate in deep magnetic saturation to increase the output power density. Thus, due to the saturation effect and the variation of mag- netic reluctance, all pertinent characteristics of the machine model (i.e. flux-linkage, inductance, phase torque etc.) are highly nonlinear functions of both rotor position and phase current. The ultimate outcome of all these nonlinearities is that the generated torque contains significant ripples when the motor is excited by the conventional rectangular pulse excitation scheme.

The problem of torque ripple minimisation has been addressed by a number of researchers 11-31. The techniques suggested in these works are mainly based on exhaustive measurement of magnetic characteristics of the SRM. All these methods address one fundamental issue (i.e. the deter- mination of a suitable currentlflux waveform so that the torque ripples are minimised). Recently, an analytical method [4] has been suggested for this purpose. However, as this scheme is based on the linear torque model, it yields poor results when tested on a more accurate nonlinear torque model of the SRM. A novel method based on fuzzy adaptive systems has been proposed in [5]. However, the performance of the scheme depends on the linguistic a pri- ori information and the parameters of the fuzzy system. A different approach has been adopted in [6, 71 where the

OIEE, 1999 IEE Proceedbzgs online no. 19990384 DOI: lO.l049/ipepa:19990384 Paper received 6th November 1998 The authors are with the Department of Electrical Engineering National Uni- versity of Sigapre, 10 Kent Ridge Crescent, Singapore 119260

IEE Proc.-Electr. Power Appl. , Vol. 146, No. 4, July 1999

optimal current waveform is computed by making use of optimisation algorithms. However, the potential benefit of simultaneous conduction of more than one phase is not considered here.

In the present work, we have tackled the problem from a control engineering point of view. We treat the current as input to the SRM, and the output is the torque which can be expressed as a nonlinear function of the rotor position and current. There are a couple of reasonably accurate models of the torque [8,9] taking the effect of magnetic sat- uration into account. To make our approach clearer, we note the following. In spite of the availability of good mod- els for the torque, it is not possible to derive an inverse expression characterising the phase currents required to produce a given torque at some specdied rotor position, without resorting to iterative numerical techniques.

This paper describes an efficient approach for finding the solution of the nonlinear function (i.e. current as a function of torque and rotor position) in a tabular form by iterative corrections. Thls approach is often termed ‘iterative learn- ing’ [13, 141. Because of the very nature of the present problem, there are a number of strategies one can adopt (i.e. for example, the one-phase-on scheme and the two- phase-on scheme (in this, again, there may be several varia- tions)). We have addressed both these schemes in their sim- plest forms. Further, a modlfied two-phase-on scheme, along with iterative learning, incorporating a set of novel pre-specified torque sharing functions (TSFs) has been pro- posed to take care of bandwidth limitations of the stator circuit. This scheme generates reasonably smooth current profiles. It is worth mentioning here that, although this work is similar, in a way, to the work in [7] in the sense that they as well as the present authors have used some sort of updating mechanism for the generation of current wave- form, the present algorithm is entirely different from that proposed in [7]. The authors would like to stress that this work is a part of their work concerned with the determina-

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tion of current waveforms. The effect of dynamics has been thoroughly investigated, but not included here because of the scope of this paper.

2 SRM torque model

The switched reluctance motor has a salient pole stator with concentrated windings and also a salient pole rotor with no magnets or coils. The basic principles of inductance variation and torque production are described in detail in [IO]. In our investigation, a four-phase SRM has been used. All the four phases are assumed to be symmetrical. Throughout the paper, reference to a generic phase j is indi- cated by a subscript j in the variables. Because of the dou- ble saliency of the motor and the intentional operation of the motor under magnetic saturation, the inductance of phasej (j = 1, ..., 4), L,, is a function of both rotor position 8 and the current 4. However, under the assumption of lin- ear magnetics, the inductance against rotor position profile can be approximated in a trapezoidal manner over one rotor pole pitch as shown in Fig. 1 (solid line). Without loss of generality, we assume that phase 1 is characterised by ths profile. The significance of the sinusoidal broken line curve is explained at a later stage in this Section. The various parameters of the profile are defined as: j3, = stator pole arc, j3, = rotor pole arc mS < P,.), a, = rotor pole pitch us + p, < a,), La = phase inductance at the aligned position, and L, = phase inductance at the una- ligned position. The parameters of the inductance profie are, taken from [15],: Output power = 7.5kW, number of phases = 4, number of rotor poles (N,) = 6, number of stator poles (N,) = 8, rated speed = 1900rpm, Lu = lOmH, La = IlOmH, a, = 1.05 rad, & = 0.35rad, /3, = 0.42rad. q$ = 1.2Wb.

t L,H 1

0 1 p5 P, pS+pr ar

8, rad - Fig. 1 Approximated inductance against rotor position for phase I

This inductance profde can be mathematically described by:

( L u + K O

Ps + Pr i 8 5 a, (1)

( 2 )

and that of phase j is, a Lj(0) = L1 (e - L(j - 1)) 4

where K is the slope of the profile in the zone of increasing inductance, given by eqn. 3.

r T

When any one phase of the SRM is energised, the torque developed [l 11, T j is expressed as:

(4)

where 0, and N,. are, respectively, the electrical angle and the number of rotor poles and 8, = N,e (We prefer to deal with electrical angle, from ths point on, rather than mechanical angle for the ease of design of iterative learning schemes later.) The co-energy W, is defined as [I 11:

where q(O,, 4) is the flux-linkage and, under linear magnet- ics of the SRM, is expressed as:

d j ( @ e , 1 j ) = Lj(@e)lj(@e) (6) Hence, the final expression for the torque can be derived as

The total torque is the algebraic sum of the individual phase torques.

Although the linear torque model, derived above, gives some basic understanding of the torque production mecha- nism, it is never accurate, especially, when the SRM is operating under magnetic saturation. A more accurate expression of the torque can be obtained by talung a flux- linkage model that incorporates the effect of magnetic satu- ration. Such an efficient model has been proposed in [9], whch is also used here. This nonlinear flux-linkage model is expressed by

4 j ( o e , 1 3 ) = 4 s (1 - e x p ( - I j f j ( @ e ) ) ) 1 2 L 0 ( 8 )

where 4A is the saturated flux-linkage and A{ e,) is expressed as a strictly positive Fourier series expansion. However, for the degree of accuracy for the present investigation, we include only the first two terms, as suggested in [9], of the Fourier series. Here, we have eva1uatedjJOJ by a sinusoi- dal approximation [12] of the inductance profile (a reasona- ble one for very low values of current). With this flux- linkage model, the torque produced by phase j can be obtained as

x (1 - [ 1 + 1 3 f 3 ( e e ) I e ~ ~ ( - 1 j j f , ( @ e ) ) }

(9)

3

The iterative learning control (ILC) is an iterative approach to the problem of improving the transient behaviour for processes that are repetitive in nature. It was origmally pro- posed by Arimoto and his colleagues [13] for the control of robots carrying out repetitive tasks. The basic idea is to employ a simple algorithm repetitively to the planthystem until perfect tracking is achieved. Thus it is an iterative scheme for generating the optimal system input so that the system output is as close as possible to the desired one. The algorithm operates over a fixed time interval. (In the fol- lowing t and k denote time and iteration number, respec- tively). To make the description more qualitative, the following general nonlinear multivariable system is consid- ered.

iterative learning control method: a brief review

x ( t ) = a(t , x) + Bs( t )u( t ) y(t> = c ( t , x ) + Ds( t )u( t ) t 2 0 (10)

where x(t) E R", u(t) E R p , y(t) E Rm, are state vector, input, and output of the system, respectively. The dimen- sions of other nodnear functions are: a(t, x) E R", B,(t) E

IEE Pro,.-Electr. Power Appl.. Vol. 146, No. 4, July 1999 370

RnxP, c(t, x) E Rm, and D,(t) E R m x P . The functions a(t, x) and c(t, x) are assumed to be umformly globally Lipschitz in x on the interval [0, tpD] (tpD is the time period of the desired output). The controller consists of an error correc- tion algorithm and a memory whch stores reference output data and system output data. Each time the system oper- ates, the input to the system d(t) is stored along with the resulting system output f l ( t) . The learning controller then evaluates the performance error as compared to the desired output, ydt), and computes a new input Uk+'(t), which is stored for use the next time the system operates. The new input is chosen in such a way as to guarantee that the per- formance error will be reduced from iteration to iteration. The idea is illustrated in Fig. 2. Typically, the learning law is expressed as (r is a gain, called as learning gain):

d + l ( t ) = d ( t ) + r ( Y d ( t ) - yk(t)) 0 5 t 5 t p g and r E RpXm (11)

According to the convergence criterion of the ILC, in the sense of hnorm, the following inequality must hold.

4

k + l

Many modifications to this basic form of learning have been proposed. An excellent survey of the recent works in this area can be found in [14]. The proposed modifications are chiefly concerned with the learning gain and the associ- ated convergence criteria. However, in spite of these varia- tions, the basic features of ths learning scheme remain the same. They are summarised below. (i) An explicitly prespecified desired output trajectory yAt), 0 5 t s tPD, is required. (ii) The learned control inputs are stored as a time sequence d(t), 0 5 t 5 tpD for the kth iteration. (iii) The sequence &(I) is used repetitively for output track- ing. (iv) The system must start with the same initial condition in every iteration.

t I

yd learning controller 4.

k k

system

~

4 Basic ILC schemes for SRM

In this Section we will describe the iterative learning control schemes to be used for learning the desired current wave- form for some specified desired torque. Both one- and two- phase-on strategies will be addressed. For ths investigation, we proceed with the assumptions that the SRM is rotating with a constant speed and that the desired current profde, as computed from the iterative learning process, is almost instantaneously established in the phase windings of the motor. The design of the learning system has been carried out using electrical angles. (The length of an electrical cycle is equal to one rotor pole pitch.) It is more convenient because the electrical cycle (time period = 2n radians (elec.)) is more fundamental for the analysis of torque production mechanism in the SRM.

IEE Proc.-Electr. Power Appl., Vol. 146, No. 4, July 1999

4. 'I One-phase-on scheme For the one-phase-on scheme, the learning system is designed such that, at any instant, only one phase is con- ducting. The complete span of an electrical cycle is allo- cated equally amongst all the four phases. Thus each phase gets a conduction angle of d2 radians (elec.). Because of the very nature of the torque production process in SRM, a phase is selected for conduction only when it is passing through a region of increasing inductance. In our case (with reference to Fig. l), we start with phase 1 and switch the other phases sequentially after successive intervals of d 2 radians (elec.) each. Thus, in the context of ILC, the interval of repetition of the process output is d2 radians (elec.). It should be pointed out here that, as all the four phases are assumed to be symmetrical with respect to each other characterised by an appropriate phase difference between them, the system does not change because of this particular mode of phase switching. Thus, in each conduc- tion interval, an incoming phase starts with the current pro- fie learnt during the conduction interval of the immediate outgoing phase, modlfies it, based on the torque (output) error during its own conduction and passes it to the next phase at the end of its conduction. This mechanism is iter- ated again and again. However, $, in practice, four phases are not exactly identical, learning iterations can be con- ducted for each phase identically for every 2n radians.

I '0

Fig. 3 Iterative leurnmg control scheme for SRM

Now we go on to specify the learning law. The iterative corrections to the current profde are based on the torque error. In the conventional ILC method, as there is no cor- rection done in the very first cycle, the performance of the system, in this first cycle, is strongly dependent on the ini- tial current profde chosen. For example, if we choose zero values of the current for the first learning cycle (first d2 radians), the torque will be zero for this entire interval. Further, since the initial choice affects the convergence characteristics [16], attempts should be made to circumvent these adverse effects by an initial choice based on the phys- ical circumstances of the problem. In our case, we make a fairly judicious choice of initial current from the linear torque model (eqn. 7). To be more specific, for a desired torque Td, we calculate the initial current 1, according to eqn. 13. Again, to make it as simple as possible, the deriva- tive in eqn. 13, dL/&, is taken to be the constant slope of the inductance profile in the increasing inductance region evaluated from the trapezoidal approximation of the inductance profile (Fig. 1). This particular choice is equiva- lent to the conventional rectangular pulse excitation scheme in the first iteration. Finally, the learning laws are stated formally in eqns. 14 and 15 without any reference to any particular phase because of the very nature of the proposed scheme. In these equations, i(= 1, ..., N) is the sample number within the iteration itself, k is the iteration index, AT(= Td- 7) is the torque error, and G is the learning gain. The same idea is also illustrated in Fig. 3 (for the one- phase-on scheme the subscript j cycles from 1 to 4).

371

With the above learning law, computer simulations are performed. The number of samples for each phase (N) is chosen to be 32 (number of samples per electrical cycle = 128). As to the choice of learning gain (0, at the moment, it is chosen by trial and error and later on, we have used its adaptation). The motor torque T is computed from its nonlinear expression (eqn. 9). Fig. 4 shows the iterative control of the torque profile and the correspondingly learnt current profile for a desired torque of 1ONm. It is seen that the learning process is almost over after five learning cycles. The simulation tests for different levels of the desired torque are carried out. The results indicate that the learning gains should be different for difTerent torque levels for con- vergence as well as the fastness of convergence. From these results, it is observed that the convergence (in terms of peaklaverage error) is acheved on the iteration axis (not within the iteration itself). These results indicate the efficacy of the iterative learning control. However, as mentioned, the need to make this learning gain adaptive arises.

4 ' 8 1 I I I 0 10 20 30 40 50 60

8, elec. rad 15 r

8, elec. rad

Fi .4 [aler completion of learning) for a aksiredorque of IONm [G = 0.9)

Iterative learning of torque pro de and current proJile for phase 1

4.7.7 Adaptive learning gain: To alleviate the diffi- culty of choosing different learning gains for dflerent torque levels, we look back to the inequality of eqn. 12 that must be satisfied for convergence of the learning algo- rithm. For that purpose, we now arrange the expression for torque in a way such that it becomes consistent with the expression for output in eqn. 10. By expanding the expres- sion for motor torque in a Taylor's series form and neglect- ing hgher order terms, we get (remember that the total torque is same as phase torque in a one-phase-on scheme),

In the above, it is assumed that the current is updated at an instant when the rotor position is held constant. In assum- ing so, we have neglected the other first-order term involv- ing the derivative of torque with respect to rotor position. This assumption is reasonable as far as the learning of cur- rent profile is concerned. As to the physical interpretation of To, it may be regarded to be equivalent to the torque that will be obtained from a linear model. That is the rea- son why we have opted for the evaluation of Io as per eqn. 13.

312

Now, the equivalent of D, for the SRM problem (in tlvs case, a single input-output system) is:

(17) Thus Ofq changes with the rotor position and the magni- tude of the current in the conducting phase. Now, the learning gain G can be made adaptive by the following that satisfies eqn. 12.

Thus, 1 - DZqG = 0 (18)

Here, a word of caution must be sounded (i.e. the conduc- tion interval must be selected such that 0,'" # 0). From eqn. 17, it can be observed that such an undesirable situa- tion (Dfq = 0) will arise in any one of the following situa- tions:

current is zero; dJ{Oe)/dOe is zero; current is lnfnitely high (a physically impossible case!).

The third condition is of theoretical interest only. However, its numerical possibihty is avoided by the initial choice which is a good approximation to the solution. With regard to the first two conditions, this is automatically eluninated, in our case, for the following reasons.

we always start with a nonzero current in the very first interval and the ILC always tries to improve it towards the solution.

a phase is excited only when it is passing through a posi- tive torque producing region (dJ{Oe)/dO, > 0). Thus, it is always ensured that D:q > 0. However, in any case for a numerically better approach, it may be advised to add a very small positive number (<< 1) in denomina- tor of eqn. 19.

45 I

0 - 20 15

loo do 40 $0 i o Id0 1;o

8, elec. rad Fig. 5 fuced lemmg g a k Desired torque = 35Nm, fured gain 0.6 __ adaptive gain . . . . . . . . . .

comparhon of iterative learning of torque profie f i r adaptive and

fuced gain

The above adaptive learning law is simulated for the one- phase-on scheme for a wide range of desired torque levels. Fig. 5 shows a comparison of the learning of torque profie for a desired torque of 35Nm. The superiority of the adap- tive learning gain over fsed gain is reflected. As expected, the finally learnt current profiles are same in both the cases. That is why they are not shown here. A better picture of the variations of the adaptive gain can be obtained from Fig. 6. As expected, the adaptive learning gain is periodic affer convergence. In any case, the learning may be stopped

IEE Proc-Electr. Power .4ppl., Vol. 146, No. 4, July 1999

after convergence. At the beginning of a conduction inter- val, D,"q is relatively high and it gradually falls to a low value towards the end of conduction (wbch is again due to the manner in which the inductance of the SRM varies). That explains this particular way of variation of the gain. Further, this also explains the reason behind the particular shape of the learnt current profile.

0, elec. rad Vaviatwm o f d p t i v e leurningguinfor desired toryue of 35Nm Fig.6

From the simulation results, it is quite clear that the iter- ative learning control method is an efficient and elegant scheme for determining the current waveform for instanta- neous torque control in SRM. It is also demonstrated that, by making the learning gain adaptive with the current and rotor position, the algorithm becomes automated for the whole torque range of the motor. As to the nature of the current waveform thus obtained, it has the following fea- tures. (i) It has sharply rising and falling edges. This is expected because a phase is instantly asked to increase its torque output from zero to the desired value at the rising edge. (The opposite is the case at the falling edge.) (ii) The amounts of riseifall at the respective edges are not necessanly equal. Particularly, as the desired torque level rises, the amount of fall is much more than the amount of rise at the respective edges. It can be seen in Fig. 4. The reason behind it has already been explained above.

It is worth mentioning here that such hgh values of cur- rent wiU result in very high resistive loss in the stator cir- cuit. Further, it may also be very difficult to injectl extinguish such high values of currents in a very short time. Thus, it is necessary to seek for a scheme where the magni- tudes of currents can be lessened. The immediate remedy is to go for a scheme where the total torque can be shared with another suitable phase during the period when the current has a very high value (i.e. at the beginning and the end of the conduction interval of the one-phase-on scheme). This modified strategy is addressed as two-phase- on scheme and is described below in detail.

4.2 Two-phase-on scheme In the two-phase-on scheme, at any instant, two phases are allowed for conduction at the most. In the scheme, we are using here, there are typically two different types of con- duction intervals following each other (i.e. an interval where only one phase is allowed to conduct and the other where two phases are allowed to conduct). In the interval where two phases are conducting, the phase which has been conducting, alone, immediately before goes out and a new phase is introduced. As before, a phase is selected for con- duction only when it is passing through its increasing inductance region. A pictorial layout of this scheme is shown in Fig. 7. The interval during which two phases are conducting is called as an overlap period and the other one is regarded as non-overlap period.

IEE Proc -Electr Power Appl , Vol 146, No 4, July 1999

: j+i,: I I I 1 I

1 1 I 1 I I I I 1 I 1 I 1 I

I i-i,i I i i,i+i I i+i I j+2 I i+2 {

{ I I I : I l l ; IV : V I VI ; Fig .7 j , j + 1, ... I, I1 ...

Typical tw@wse+n scheme for the SRM = conducting phase indices = conduction intervals

Here, we proceed with the adaptive learning gain. In this strategy, there are two different learning systems operating independently of each other (i.e. one for the overlap period and another for the non-overlap period). Again, in the overlap period, there are two learning subsystems (i.e. one for the incoming phase and another for the outgoing phase). Thus we define three different learning gains. They are: Go, for the non-overlap period; Gin and Gout for the incoming and the outgoing phases, respectively, in the over- lap period. We also define the currents accordingly (i.e. I,,, Zh, and lout, respectively). Go, is computed according to eqn. 19, as it is nothing but the one-phase-on scheme in this interval. For the computation of the other two gains, the following steps are derived.

For the overlap period, the torque can be defined as (talung only the first-order terms):

= TO + DFAI,, + D,"utAIou, (20)

Thus, for the convergence of the learning algorithm, the following inequality must be satisfied:

By setting the LHS of ths inequality to zero, as done before, we obtain

DrGi , + DgutGout = 1 (22) Now, Gin and G,, can be selected as (with Kl + K2 = 1):

In our study, we have chosen K, = K2 = 0.5. Although any other choice may serve the purpose, by ensuring both Kl and K2 to be positive always, the convergence is guaran- teed. The learning laws for the complete current waveform are defined as below. Non-overlap period:

AI;;~(Z) = AI&$) + G,,AT~(z) (24)

I,",+'(i) = I* + AI,",+'(i)

Overlap period (incoming phase) :

AI!$'(i) = AI,",(Z) + Gi,ATk((i) (26)

Overlap period (outgoing phase) :

AI:A'(i) = A1iut(Z) + Go,tATk(i) (28)

I:A1 (2) = Io + AI:;1 ( i ) (29)

In the computer simulations, the lengths of the overlap and non-overlap intervals are chosen to be 60" (elec.) and 30" (elec.), respectively. The midpoints of the overlap peri-

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ods (with reference to Fig. I) lie at points OD, 90°, 180°, and so on. The number of samples is same as before. The cur- rent profiles are updated during their respective intervals with no reference to any specific phase. In the non-overlap interval, the phases are selected in the same manner as cited in the one-phase-on scheme. In the overlap period, the two phases are selected in the following way. The outgoing phase is the phase whxh was conducting in the previous non-overlap period. The incoming phase is the phase which would have been selected next to the outgoing phase, had it been a one-phase-on scheme. Thus, with this strategy, the overlap angle should be chosen such that a phase should not be selected at an instant when it is due to produce neg- ative torque. The current profiles are appropriately assigned to the different phases.

7.5 I \ 7 1 " ' ~ " ' " ' 0 2 4 6 8 10 12 14 16 18 20

e, elec. rad Fig. 8 Iterative learning of torque in two-phare-on scheme Td = lONm

8, elec. rad 12r

e, elec. rad Fig.9 learnt current projiles Desired torque = lONm __ phase 1 . . . . . . . . . . phase 4

Torque sharing between two consecutive phuses and correspondmg

120

Q lool 80

40

. . . . . . . . . . . . . . i : : .

143 144 145 146 147

Cornparion of current proJles with drerent schemesfor torque of 8, elec.rad

Fig. 10 37Nm __ no ILC . . . . . . . . . . . one-phase-on ~ _ - _ two-phase-on

Fig. 8 shows the learning of torque profile for a desired torque of 10 Nm. The current profiles, shown in Fig. 9,

374

can be directly compared with that of Fig. 4 (both are for the same desired torque level). The current levels are reduced. Ths effect is more pronounced as the torque level rises. In Fig. 8, the convergence is faster because of the adaptive gain. Further, because we have started the learn- ing scheme with no specific assumption on the torque to be contributed by a phase to the total torque in the overlap period, it is really interesting to observe how a torque shar- ing is automatically achieved during the overlap interval (also shown in Fig. 9). Further, to show that the current levels are significantly reduced (with a two-phase-on scheme) for higher torque levels, a comparison of current profiles is shown in Fig. 10, where the conventional rectan- gular pulse has been computed from the linear torque model. (It is well known that such a scheme produces high torque ripple.)

To summarise, the two-phase-on scheme has a clear edge over the simple one-phase-on scheme in terms of peak value of the required current. However, the current profile still has sudden risinglfalling segments. This happens because of the very structure of the scheme. As an exten- sion, we have also tested the two-phase-on scheme with its full capability (i.e. a phase is allowed to conduct whenever it can produce positive torque). Thus, maximum possible overlap is allowed between any two phases. The results (not shown here) indicate that the current profile too has peaky structures. Because, the tractability of the desired current profile mainly rests upon its frequency content, the band- width of the stator phase circuit, and the inverter voltage, we need to examine these issues before going ahead. Bar- ring the finite inverter voltage, for a faithful traclung of the current profile in the stator circuit, one viable approach will be to modify the desired current profile, if necessary, so that most of its dominant frequency contents lie within the bandwidth of the stator circuit. Hence, we step into an examination of the frequency domain characteristics of the stator circuit.

Linear analysis of stator circuit: As the stator circuit of the SRM is highly nonlinear in nature, we resort to an approximate linear analysis. It will, at least, offer a rough picture. In any case, the worst situation is that of the unsaturated case (lowest possible bandwidth for all rotor positions). As before, we assume the trapezoidal inductance profile shown in Fig. 1 for simplitied analysis. Because, the inductance varies as the rotor rotates from the unaligned position to aligned position, the bandwidth of the system also changes. For our purpose, we make use of the LR model of the phase circuit taking back emf into considera- tion. The dynamics of flux in the stator phase is given by,

where is the applied voltage across the phase winding, 4m is the actual phase current, and R is the phase winding resistance. With the linear approximation of the flux model (eqn. 6), the following transfer function can be obtained (K is defined in eqn. 3 and speed of the motor, w, is assumed to be constant).

As we do not consider the effect of speed of the motor on the desired current profde, we have gone for the test of eqn. 31 for two dfierent speeds (i.e., Orpm and 1500rpm). The log-magnitude against frequency plots for the aligned and unaligned positions are shown in Fig. 11.

From the frequency response plots, it is apparent that the bandwidth is very low at the aligned position (= 1.5-2Hz)

IEE Proc-Electr. Power Appl. , Vol. 146, No. 4, July 1999

demanding an extremely low frequency current at this posi- tion. However, the bandwidth increases towards the una- ligned position. Moreover, the bandwidth, at most of the rotor positions (except those nearer to the aligned position) increases with the speed of the motor. For example, as the plots show, the bandwidth at the unaligned position increases from 15Hz at zero speed to 700-800Hz at a speed of 1500rpm. Thus, for a faithful tracking of the desired cur- rent waveform by a standard proportional controller, it is always better if the current wavefonn does not contain too many high frequency components. However, it is men- tioned again that all these analysis are approximate. Never- theless, it offers us some useful guidelines for our present purpose.

-- -2 1 0 1 2 3

10 10 10 10 10 10

frequency, Hz

50 aligned

-1001 I -2 -1 0 1 2 3 4 5

10 10 10 10 10 10 10 10

frequency, Hz

Fig. 11 Top: Orpm Bottom: 1500rpm

Frequency responses of stator circuit (linear analys&)

1 .o r

Q O..iI o . z ~ ‘0 2 4 6 8 10 12 14

frequency, kHz Fig. 12 component = 8.16A

AmjJlitude s p e c t m of current projile. Two-phe-on scheme DC

Now, coming back to the current profile obtained from the two-phase-on scheme, a 4096-point FFT is performed on the phase current waveform. The motor speed is assumed to be 1500rpm. Fig, 12 displays the amplitude spectrum of current waveform for a desired torque of 10 Nm. It contains many high frequency components, suggest- ing that it is not a very much suitable profile for hlgh per- formance applications.

Now, in the search for a suitable method for generating a reasonably smooth current waveform, some inferences drawn from the results shown in Fig. 9 will be very much helpful. The Figure shows the way the total torque is shared between the two conducting phases during the over- lap period. We have not specified it. Rather, it is automati- cally learnt in the process. A close look at the corresponding current waveform shows that it has jumps at the instants wherever there are jumps in the individual

IEE Proc -Electr Power Appl Vol 146, No 4, July 1999

phase torque profie. Also, in the sections where there is a smooth rise/fall of the torque profie, the corresponding current profile is also smooth. This suggests that the cur- rent profile in the overlap period is entirely influenced by the way the total toque is shared between the two phases. Thus, if we assign a smoother way of torque sharing between the phases, rather than leaving it free for the learn- ing process to assign, then we are certainly expected to get a smoother current profile. In the non-overlap period, because the entire torque is produced by one phase, the current profde has to be unique.

Hence, with these clues at hand, we proceed to the method of torque assignment for individual phases to get smoother current waveforms in the next Section.

5

The motivation behmd the assignment of torque to the individual phases has been explained in the previous Sec- tion. The objective is to co-ordinate the production of torque by the individual phases so that the total torque at any instant is equal to the desired torque. Like before, a TSF is defined for a phase only in the region where it can produce positive torque. For the rest of the electrical cycle, the TSF for that phase is zero. There are a number of ways one can define these TSFs. An arbitrary choice of the TSF will not be suitable for our purpose. Our approach will be obviously for a smoother function. The simplest trajectory that can be used for this purpose is the cubic segment (CS). The cubic segment, here, is a cubic polynomial of 6,. This function is characterised by four arbitrary parameters. With the two-phase-on scheme, each TSF will have a rising segment followed by one constant segment and a falling segment.

To develop the mathematical expressions for these TSFs, we define the following. The subscripts r andfstand for ris- ing and falling segments respectively. 0, = overlap angle, TJd (1, ... , 4) = desired phase torque. ee8(rl = angle at which a TSF has a zero value for the ris- ing segment. BeJnvi = angle at whch a TSF has the full value for the fall- ing segment. The mathematical expressions for the different segments are defined as: Rising segment:

ILC method with torque sharing functions

Constant segment:

T 3 d = T d (33) Falling segment:

T j d = Af $- Bf (@e - 0 e , j O ( f ) )

+ Cf (0, - ~ e , j o ( f ) ) ’ -+ ~f (0, - ~ e , j o ( f ) ) ~

(34) where the constants A, B, C, and D are to be chosen to sat- isfy the constraints. The locations of overlap regions are same as defined in the previous Section. The four con- straints for the rising segment are:

315

A similar set of constraints is used for the falling segment. With these constraints, the various constants can be derived as:

A f = T d ; B f = 0; C f = -CT; D f = -D, (38) As before, all the phases are allotted equal slots for conduc- tion. In our design, the overlap periods are symmetrical about 0, d 2 , n, 3d2, 2n, and so on. Thus, each phase is allowed to conduct for an interval of 0, + (d2 ) electrical radians. As ths is basically a two-phase-on scheme and negative torque producing region for any phase is not allowed for conduction, the maximum overlap interval is n/ 2 elec. radians. For the sake of clarity, the complete defini- tion of a typical TSF (for phase 2) is given below. For (JC - &9/2 s 0,s (n + OJ2:

(39) For (n + eJ/2 s 0,s n - (042):

For n - (042) s e, s JC + (042):

For the rest of the electrical cycle, it is zero. Fig. 13 shows two sets of TSFs for a desired torque of 5

Nm. In the upper plot, the overlap angle is d3 (elec. rads). Thus, there is a straight-line segment in each TSF. In the lower plot, the overlap angle is d 2 (elec. rads.) resulting in the absence of straight-line segment. The functions are rea- sonably smooth in both the cases.

E i 6 - g

E 2 ai e s

Fig. 13

+% : \ ! \

.. . . . . . . . . . . . . i \\ ... . . . . . . . . \ .... '0 1 2 3 4 5 6 7

8, elec, rad

i I \ ./ 3 ?\\ \

; \

8, elec, rad

E m p l e s oftorque sharing function with cubic segments

T2d

T4d T3d

The simulations are carried out for both sets of TSFs with adaptive learning gain. Here, unlike the previous

376

implementations of ILC, the learning rules use individual phase torque error. The learning laws are defined as fol- lows:

AI:+'(%) = AI!(%) + G,AT'(z) (42)

(43) I k + l , (2) = I* + AI;+l(%) Here, the learning gain for each individual phase must sat- isfy the convergence condition. Hence, the equation for the gain computation is same as eqn. 19 (GJD, = 1 and DJJ is same as 0,'y). However, as the learning law for each phase operates for the complete electrical cycle (D, goes through the zero and negative values), the learning gain GJ is modi- fied, as a safety measure, as ( E = 0.001):

As we are not interested in the negative torque producing region of any phase and, further, in order to avoid possible undesired numerical outcome near the point when a phase is just near the boundary of positive and negative torque producing regions, the following precaution can be incor- porated (tolerating negligible torque error). The constant r~ is a very small positive fraction (much less than 1).

The number of samples is again same as before. For an overlap angle of d 3 (elec. rads), the learning of torque and the final current profiles are shown in Fig. 14 for a desired torque of 1ONm. The learnt current waveforms have no sharp rising/falling segments.

12r

J

0 20 40 60 80 100 120 8, elec. rad

12 r

53 94 95 96 97 98 99 8, elec. rad

Fig. 1 4 two consecutive phasesfor &red torque of IONm __ phase 1

phase 2

Iterative leurnm of torque and the fm& learnt current prof les for

......... q[/ , , , , $ 2

1

0 0 2 4 6 8 10 12

frequency, kHz Amplitude spectra of current profile DCcompnent = 6.244 Fig. 15

From the above results, it is clear that the torque-sharing function approach for obtaining a smoother current profile

IEE Proc.-Electr. Power Appl.. Vol. 146. No. 4, July 1999

is fruitful. For a better appreciation, the amplitude spec- trum of the phase current profile for a motor speed of 1500 rpm is shown in Fig. 15. It is clear that there are not many very high frequency signals. Moreover, the dominant sig- nals are almost within the range of allowable bandwidth of the stator circuit for medium to high speeds (as inferred earlier from the linear analysis). As an additional illustra- tion, the simulation results for a desired torque of 37Nm are shown in Fig. 16.

40 r

L”

0 20 40 60 80 100 120 0, elec. rad

30 ...._ 25 I:

0, elec. rad Fig. 16 two consecutive phasesfor &sired torque of 37Nm ___ phase 1 . . . . . . . . . . . phase 2

Iterative leurnm of torque and the fmlk learnt current profiles for

Finally, it is of importance to have a comparison of the RMS currents/phase at the rated torque (37Nm) for the various schemes described in this paper. In the calculations, full cycle of each current waveform is taken. The compari- son is shown in Table 1.

Table 1: Scheme used RMS current/phase

Scheme used RMS current/phase

Normal rectangular pulse scheme 8.05A

One-phase-on scheme (with ILC) 16.44A

Standard two-phase-on scheme (with ILC) 12.47A

Two-phase-on scheme (with torque-sharing 12.48A functions and ILC)

Although the normal rectangular pulse scheme has the lowest RMS value, it gives rise to s imcan t amount of torque ripples. Hence, it is not recommended. Thus, the need for other schemes comes into picture. As explained earlier, the one-phase-on and two-phase-on schemes are expected to have higher RMS current because of its obvi- ous purpose. The RMS value of two-phase-on scheme will obviously be influenced by the overlap angle. Nevertheless, it results in a lower RMS current compared to one-phase- on scheme. In addition, the two-phase-on scheme with torque-sharing functions has got practical advantages over the standard two-phase-on scheme although they may have the same RMS value.

6 Conclusions

Determination of a suitable current profile for minimising the torque ripples in the switched reluctance motor is a burning problem. We have presented a sequence of investi- gative studies on the use of iterative learning control method for solving this problem. As it turns out, there are a number of possible schemes to meet this goal (i.e. starting from the simplest one-phase-on scheme to a two-phase-on

IEE Proc.-Electr. Power Appl., Vol. 146, No. 4, July 1999

scheme incorporating the specification of torque-sharing functions for the individual phases). All the three different schemes reported yield us different solutions to our goal. Each solution is the best possible that can be achieved under that specific scheme. However, from a hgh perform- ance application point of view, the two-phase-on scheme with the specifications of torque-sharing functions is a via- ble approach considering the limitations of the stator cir- cuit.

In the implementations of all the three schemes, estima- tion or measurement of electromagnetic torque of the motor is essential. Barring this, the algorithms are quite simple. The beauty of the ILC approach to this problem lies in the fact that it is able to learn the desired current waveform that is highly nonlinear by iterative corrections, which is difficult to obtain by any conventional means. Because of the very structure of the ILC method used here, the performance is normally poor in the very first cycle unless special care is taken (as done in this work).

The present work is expected to serve as a stepping stone to the applicability of ILC method to the SRM problem. A lot of improvements on the present state of the methods can be done (i.e. search for an optimal torque sharing func- tion) incorporating the effect of speed of the motor into the learning process etc.

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