Torque Ripple Minimization Scheme Using Torque Sharing Function Based Fuzzy Logic Control for a Switched Reluctance Motor

Abstract

This paper presents an advanced torque ripple minimization method of a switched reluctance motor (SRM) using torque sharing function (TSF). Generally, TSF is applied into the torque control. However, the conventional TSF cannot follow the expected torque well because of the nonlinear characteristics of the SRM. Moreover, the tail current that is generated at a high speed motor drive makes unexpected torque ripples. The proposed method combined TSF with fuzzy logic control (FLC). The advantage of this method is that the torque can be controlled unity at any conditions. In addition, the controller can track the torque under the condition of the wrong TSF. The effectiveness of the proposed algorithm is verified by the simulations and experiments.

1. Introduction

Recently, the switched reluctance machines (SRMs) have drawn great attention from industry and researchers. SRMs are regarded as an alternative among other electrical machines along the development of the motors which do not include rare earth resources. Due to its simple construction without magnets and brushes, SRMs offer low production and operation costs. Furthermore, SRMs have high power output and system efficiency over a wide speed range. These advantages have caused interest in SRMs for several decades [1-3].

However, SRM have a problem of torque ripple because stator and rotor have salient poles. Therefore, the SRM has very high torque ripples during the phase commutation period when the torque production is being transferred from the outgoing phase to the incoming phase. This higher torque ripple induces acoustic noise and oscillations. Therefore, many studies on reducing torque ripples have been carried out extensively [4-7]. One of the effective approaches to reducing the torque ripple is a torque sharing function (TSF) method [8-11]. Because each phase of the SRM is controlled independently, expected torque is divided into individual phase torque references. Torque references of the each phase are changed into the current command according to the rotor position using look-up table. TSF does not have a unique shape and there are many functions to regulate the torque ripple minimization. TSFs can be classified as linear or nonlinear. The former methods in which the reference torque is produced by a phase winding change with the rotor position linearly. This is introduced by Schramm et al [12]. Other methods which include the cubic, sinusoidal, exponential TSF and such make TSF nonlinear with the rotor position. The cubic TSF makes the torque reference cubic polynomial in the commutation region. Husain and Ehsani developed a sinusoidal TSF [13]. Recently, a new asymmetric TSF [14] which considers the optimal shape at the operating condition is induced. However, the asymmetric TSF utilizes a piecewise linear function and is only suitable for low speed operation. Although TSF can partially improve the performance of SRMs, they should have torque ripples in motor drive because SRMs have highly nonlinear characteristics. Moreover, because of the internal inductance of the motor, the delay of the current rising and falling time weaken the performance of the current controller about current tracking. This phase current error, in turn, induces additional torque ripples. To solve these problems, another controller is demanded.

In this paper, a torque ripple minimization method using TSF is proposed, including fuzzy logic control (FLC). When the torque ripples are generated, a sliding mode controller makes the compensating value instantaneously [15-17]. The corrected TSF makes corrected current references, which should reduce the torque ripples using a current controller. The proposed method combining TSF with FLC has robustness and good controllability. The simulation and experimental results show that the proposed methods are effective and feasible for the SRM drive system.

 

2. Torque Sharing Function for Switched Reluctance Motor

2.1 Theoretical background of SRM

An SRM is characterized by its doubly salient poles. Concentrated windings on the stator poles, along the same magnetic path are connected in series to form phases. The electromagnetic torque in the SRM is produced by exploiting the changing reluctance of the magnetic path associated with each phase. A three phase SRM with 12 stator poles and 8 rotor poles is outlined in Fig. 1.

Fig. 1.General switched reluctance motor and power converter

From the basic electromagnetic theory, the voltage equation of kth phase is shown as

where λk(θr,i)=L(θr,i)i is the phase flux linkage for the given angle variation.

The voltage equation can be rewritten as

where ωr is an angular speed of the rotor. These terms express resistor voltage drop, inductance voltage drop and back-electromotive force.

In a motor with no magnetic saturation the magnetization curves would be in a linear relationship like Fig. 2. At any position, the coenergy is defined as

Fig. 2.Principle of the torque generation

The electromagnetic torque produced by each phase is given from the partial derivative of the phase coenergy and is defined by

Eq. (4) can redefined as

Eq. (5) shows that instantaneous torque per phase is determined by the relationship of the injected current and the variation of the inductance upon the rotor position. Fig. 3(a) shows the principle of the torque generation. Because the injected current is always positive, the positive torque is generated as the positive inductance slope. The torque is zero where the phase inductance variation is constant and negative where the phase inductance variation is negative. The total instantaneous output torque is given by the sum of the individual phase torques

Fig. 3.General principle of SRM: (a) Torque generation with one phase; (b) Cause of the torque ripple

We only need positive inductance slope to generate positive uniform torque. At the transition section of the present current and next phase, excessive overlapped current and non-overlapped current make torque ripple. This is shown in Fig. 3(b). The former torque ripple occurs on account of overlapped current phase A and B. In case of the latter ripple, the current of phase B is turned off early. Therefore, lacking torque is shown. There are special strategy is needed at the commutation region.

2.2 Torque sharing function

The TSF method is one of the torque control methods to drive SRM so that overall torque becomes a constant torque. In order to make unity torque, TSF distribute the torque reference to two neighboring phase at the commutation region shown in Fig. 4.

Fig. 4.Inductance slope and the TSF for the SRM: (a) Inductance profile; (b) Linear TSF; (c) Sinusoidal TSF

Fig. 4(b) shows the linear TSF which changes linearly with the rotor position during the commutation section. Then the linear TSF can be defined by

where

θonturn-on angle θovoverlap angle θoffturn-off angle linear TSF for the rising section linear TSF for the declining section Tereference torque of the SRM θrrotor position θpthe angle of the rotor period.

Then, θov should meet control scheme like

The relationship about the rotor position and TSF is arranged as

At commutation, fup l(θ) is applied into next phase and fdn l(θ) is applied into the current phase as a reference torque. The sum of the TSF shows always constant.

Fig. 4(c) shows the sinusoidal TSF, and its type is nonlinear. The reference torque is shown to be sinusoidal with the rotor rotation. The sinusoidal TSF is determined by

where fup s(θ) and fdn s(θ) are rising and declining TSF respectively.

The relationship about rotor position and TSF is shown to be

At the overlap, the linear TSF makes the torque reference linear; on the other hand, sinusoidal TSF makes the torque reference nonlinear considering the characteristics of the SRM. The divided torque references are converted into the current command through the look-up table. This table should be made in advance. The block diagram of the TSF control scheme is shown in Fig. 5.

Fig. 5.Block diagram of the TSF control scheme

TSFs must consider the turn-on and turn-off angle for each phase. And the shapes are selected considering each motor characteristic. If TSF does not match with motor model, the torque control using TSF rather reduce the motor performance making the torque ripple. Moreover, delay time of the current rising and falling to follow the reference current makes additional problem. For example, motor inductance disturbs the current variations. In addition, short commutation time in high speed motor drive also makes a problem that the practical current cannot be kept the reference value. The tail current is extended to the negative inductance slope, than produces the negative torque. Therefore, appropriate TSF is important as considering the speed and the characteristics of the motor to avoid the torque ripple. In this paper, we presents the method that changes the TSF.

 

3. Proposed Torque Sharing Function with Fuzzy Logic Control

As mentioned above, the conventional TSF may make the unexpected torque ripple when the TSF is does not match with the motor model. In order to compensate the torque ripple, the fuzzy logic is applied. Fig. 6 shows the principle of the proposed method. This controller takes advantages of the dynamics and robustness to perturbations.

Fig. 6.Control principle of the proposed method

The proposed FLC has rule bases, fuzzification, and defuzzification processes. The FLC uses two factors as input; the error between torque reference and the real torque and the change of the error. Both the error and the change of error are normalized by multiplying with the corresponding scaling factors ge and gce. Using input values, the fuzzification is conducted. There are seven triangular membership functions which are positive big (PB), positive middle (PM), positive small (PS), zero, negative small (NS), negative middle (NM), and negative big (NB). Normally PB and NB are used to map the input variable into the normalized domain to improve the performance of a fuzzy controller when the input variable has values greatly larger than the normalized domain.

Fig. 7(a) shows the input fuzzy membership function according to the five basic linguistic values for input and output value. The input variables are combined with logical operator AND. Table 1 shows the fuzzy rule. There are 25 possible results from the combination of the error and variation of the error. The output signal is the input of the defuzzication functions.

Fig. 7.Membership functions for fuzzy logic: (a) Input; (b) Output

Table 1.Rule Table of the Fuzzy Controller

The defuzzification is performed with the output membership functions in Fig. 7(b). The output signal is multiplied by scaling gain gu. Then, this is added with the conventional TSF. The corrected TSF reduces the torque ripple instantaneously.

Fig. 8(a) shows the principle of the proposed method. In region I, the reference torque is divided by the TSF. Then the present and the next phase currents are changed to follow the TSF. In region II, even though the exciting current of the previous phase does not reach zero, there is no torque variation because there is no inductance variation. However, the exciting current is extended to the negative torque section at region III. This becomes a cause for the undesirable torque.

Fig. 8.The principle of the proposed method: (a) Conventional TSF; (b) TSF with FLC control

Fig. 8(b) represents the proposed TSF with FLC. At region ІІІ, when the torque ripple is generated by the tail current of the previous phase, the fuzzy controller makes the compensation component that is reflected on the TSF. The next phase current is increased by the modified TSF. The corrected TSF makes a corrected current reference. Then the output torque can be unity torque by the current control. It is possible to control the torque as we want through the designed system without ripple. The proposed method guarantees the robustness of the controller.

Fig. 9 shows the block diagram of the system that proposed control scheme as mentioned above. The blocked section shows the TSF with FLC. The output is transformed into reference current. The current controller regulates the currents of the each phase. The power converter is asymmetric converter which is popular. The asymmetric converter is composed with two switches and two power transistors in each phase. The asymmetric converter has several advantages such as the diversity of the control, independent current control of the each phase.

Fig. 9.Block diagram of the SRM drive with the proposed control scheme

 

4. Simulation

Simulations were performed using PSIM software to confirm the validity of the proposed method. The simulation studies were operated for a 1.5 kW SRM system.

The simulation parameters related to the system are presented in Table 2. Fig. 10 presents the simulation results for the torque control with a linear TSF and proposed method with FLC.

Table 2.Simulation Parameters

Fig. 10.Compared simulation result at 300 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

The simulation is conducted under the torque reference of the 0.1 Nm with 50 μs of control period. Actually, SRM in low speed, has rare torque ripple. However, we assume the situation that TSF is does not match with the real model to shows the performance of the proposed method. As shown in Fig. 10(a), the conventional TSF makes a torque ripple. The TSF is dislocated and overlapped. The ripple is induced by the current on the negative slope because the current follows the shape with TSF. Then this current makes a negative torque. Fig. 10(b) shows the compared simulation results about the Fig. 10(a). The torque ripple is greatly reduced compared to the conventional linear TSF. Although the current of the previous phase is flowing into the negative slope, the proposed method makes a compensation term. Then this is added into the conventional TSF instantaneously. The next phase current is increased by the corrected TSF at the overlap region. As a result, the output torque becomes unity consequently.

Fig. 11 presents the simulation result at 1,300 rpm. As same with the previous result in 300 rpm, conventional TSF makes torque ripple. However, proposed TSF modifies the TSF and reduce the torque ripple.

Fig. 11.Compared simulation result at 1300 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

Fig. 12 shows the torque control simulation result with middle speed at 2,500 rpm. As motor drives high speed, inner inductance of the motor is increased. It disturbs the current variation. In Fig. 12(a), the previous phase current falls down to zero slowly because it cannot follow the TSF; then, it reaches to the negative inductance slope. This current induces the much torque ripple making negative torque. As shown in Fig. 12(b), the torque in the proposed method is greatly reduced compare to the conventional method. Using the same method in low speed operation, when the torque ripple is generated, FLC makes the compensating component. This is applied into the TSF instantaneously, the torque ripples is advanced.

Fig. 12.Compared simulation result at 2500 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

Fig. 13 shows the simulation results in the high speed range at 4500 rpm. Fig. 13(a) shows the simulation result of the general TSF, and it can be verified that the current reduction time becomes longer since the speed back-emf becomes larger than the input voltage. Hence, the current becomes zero in the negative torque region, and the torque ripple becomes larger due to the inductance that has a negative slope. Fig. 13(b) is the simulation result of the proposed TSF and the current becomes zero in the negative torque region as shown in Fig. 13(a). However, the current compensation for the negative torque can be verified from the proposed TSF and also shows that the torque ripple is partially reduced. Nevertheless, in the case of applying the proposed algorithm in the high speed region, it can be seen that the simulation result is worse than the low- and middle-speed region. For SRMs, as the speed increases, the back-emf caused by the speed becomes larger than the input voltage. And it causes the lacking potential difference. Hence, the negative torque cannot be compensated enough within the control period by the proposed TSF method.

Fig. 13.Compared simulation result at 4500 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

Therefore, in Section IV, the simulation shows the results in the low-, middle-, and high-speed region, and in section V, the experimental shows the results in the lowand middle-speed region.

 

5. Experimental Results

The performance of the proposed scheme was experimentally verified using the 12/8 SRM in Fig. 14. The experimental conditions are the same as the rated conditions like the simulation studies described in the previous sections. The control algorithm was implemented in a digital signal processor. The part of the power conversion has 20 kHz switching frequency, and the asymmetric bridge converter consists of two IGBT modules per phase.

Fig. 14.Experimental set

Fig. 15 shows speed, output torque, current A and current B which are the results of the motor torque control with TSF at low speed. The Fig. 15(a) shows the conventional torque control based on TSF at 300 rpm. Simulated condition is that TSF is does not match with the real model. Therefore, torque ripple is generated. As it can be seen from Fig. 15(b), the torque ripple is reduced comparing to that of the conventional method. To overcome the torque ripple, next phase current contains a compensation value using proposed method. The Proposed method instantaneously corrected the TSF reflecting the torque ripple.

Fig. 15.Compared experimental results at 300 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

As can be seen from Fig. 16, the middle speed motor drive induced more torque ripples than low speed operation. At the beginning of the current conduction in each phase, the rate-of-change of the current reference is still much higher than that of the actual current due to the high phase inductance. During this period, the difference between actual current and reference value makes a dip in the output torque. On the other hands, the tail current of the previous phase, also exerts a strong influence on the motor performance. The proposed method has an ability to compensate the torque ripple caused by errors in the phase currents. Therefore, the output torque is greatly reduced than the conventional method.

Fig. 16.Compared experimental results at 1300 rpm: (a) Conventional TSF; (b) Proposed TSF control scheme

Fig. 17 presents the experimental results at 2500 rpm. The tail current makes big torque dip, because SRM operates in more middle speed. As the current follows the TSF shape correctly, the ripple should not be generated. The reasons that the current do not follow the TSF are inner inductance of the motor, incorrect TSF and others as mentioned above. Then the proposed method tries to compensate the torque ripple. Compensation term corrects the TSF and regulated current is shown in Fig. 17(b). The output torque can be reduced about 76% than the conventional TSF in real time.

Fig. 17.Compared experimental results at 2500 rpm : (a) Conventional TSF; (b) Proposed TSF control scheme

 

6. Conclusion

This paper proposes a torque ripple minimization method of SRM with TSF. In the proposed method, TSF is combined with FLC. It gives the robustness at the torque error and the uncorrected data of the TSF. Therefore, when the torque ripple is generated, FLC compensates the error instantaneously. In applying the proposed TSF, the short control period cannot guarantee the torque ripple minimization performance at the high speed range, which was shown in simulation results. In this paper, simulation and experiment were performed on a 1.5 kW SRM system model and the performance of proposed TSF is verified at the low and middle speed range.