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Direct control of torque and levitation force for dual-winding bearingless switched reluctance motor
Electric Power Systems Research 145 (2017) 214–222
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Direct control of torque and levitation force for dual-winding bearingless switched reluctance motor Xin Cao ∗ , Qin Sun, Chenhao Liu, Heng Zhou, Zhiquan Deng Department of Electrical Engineering, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No.169, West Shengtai Road, Nanjing 211106, Jiangsu, PR China
a r t i c l e
i n f o
Article history: Received 25 July 2016 Received in revised form 16 December 2016 Accepted 9 January 2017 Keywords: Dual-winding bearingless switched reluctance motor Direct torque control Direct force control Torque ripple Control strategy Current control algorithm
a b s t r a c t In dual-winding bearingless switched reluctance motors (BSRMs), an additional winding is placed on the stator poles in conventional switched reluctance motors (SRMs), which is mainly to achieve the function of levitation. Due to the hysteresis current control in existing control strategies for dual-winding BSRMs, the complicated derivation of winding-current expression was necessary, and some constraints were also introduced that increased the difficulties on designing the current control algorithm. In order to solve these problems, the direct control concept of torque and levitation forces is proposed and developed in this paper, named as direct torque control (DTC) and direct force control (DFC). Moreover, the torque ripple can also be reduced greatly. Firstly, the space voltage vectors of dual-winding BSRMs are defined for the main and levitation windings, respectively. After that, the rules and procedures for selecting the space voltage vectors are demonstrated in detail, and the system control block is also presented to facilitate the implementation of proposed control strategy. Finally, experimental results are provided to demonstrate the performance. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In conventional switched reluctance motors (SRMs), phase windings are mounted on the stator poles and no permanent magnets or windings on the rotor. The phase windings are energised separately by the asymmetric-half-bridge type converter. Therefore, SRMs own great advantages on the simple control strategy, fault-tolerant control, high speed capability, etc. [1–5]. In order to inherit the advantages of conventional SRMs, the bearingless motor technology was integrated with SRMs, and a reluctancetype bearingless motor was created, named as bearingless switched reluctance motors (BSRMs) [6–15]. The magnetic pulls are produced between stator and rotor poles when the phase winding is energised. The produced magnetic pull can be divided into tangential and radial components. In conventional SRMs, the radial components on each rotor poles are counteracted whereas the tangential components drive motor rotating at the expected speed. In BSRMs, the flux linkages flowing in the stator poles of the same phase are distributed asymmetrically, thus the radial components on the rotor are no longer
∗ Corresponding author. E-mail address: [email protected] (X. Cao). http://dx.doi.org/10.1016/j.epsr.2017.01.015 0378-7796/© 2017 Elsevier B.V. All rights reserved.
counteracted and the radial force is produced on the rotor accordingly. With the active control of radial forces, the rotor shaft can be levitated when the bearingless motor is rotating at a high speed, which avoids the mechanical wear between the shaft and the mechanical bearing. Therefore, for the mechanical bearing, not only the lubrication system can be removed but also the problems of the maintenance and renewal can be solved, especially in some highspeed drive applications and in harsh environments with radiation and poisonous substances [6]. In the BSRM, according to the number of windings mounted on each stator pole, it can be divided into single-winding and dualwinding BSRMs. The dual-winding BSRM was firstly implemented by A. Chiba et al. in the 1990s [7]. The operation principle and control strategies were deeply investigated to implement the rotation and levitation experimentally for BSRMs [8,9,16–20]. After that, the single-winding BSRM was proposed and developed to obtain the same winding configuration on each stator pole as that in conventional SRMs [10–12]. The difference is that the coil on each stator pole is energised separately in single-winding BSRMs, whereas the coils in the same phase are connected in series or in parallel in conventional SRMs. In the single-winding BSRM, the winding current is controlled to provide not only the torque but also the levitation force, thus the design of control strategy is difficult and the control performance on the torque and levitation interacts with each
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Nomenclature ˛ , ˇ 0 c F˛ , Fˇ h ima isa1 , isa2 l0 Lsa1 , Lsb1 Nm , Nb r T, Ta
flux linkage flux-linkage vector components in the axes of ˛ and ˇ, respectively permeability of vacuum rotor angular position constant 1.49. radial forces on the direction of ˛-axis and ˇ-axis, respectively lamination length of the iron core main-winding current of phase A levitation-winding currents of phase A at the direction of ˛- and ˇ-axes, respectively air-gap length between the stator and rotor poles and Lsc1 ˛-axis levitation-winding inductances of phases A, B and C, respectively winding turns of main winding and levitation winding, respectively radius of rotor instantaneous torque and instantaneous torque of phase A, respectively
other. Therefore, some different pole-pair structures for BSRMs were proposed and investigated to simplify the control of torque and levitation forces [13–15,21–23]. In summary, the present work for BSRMs mainly focuses on the levitation mechanism, control strategies and novel structures protect [8,9,11–13,18–20]. On the basis of 12/8 dual-winding BSRMs, this paper studies the control strategy to simplify the control of torque and levitation force compared with existing control algorithms. The existing algorithm is to directly regulate the winding currents through the hysteresis current control to produce required torque and levitation forces. Therefore, the expressions of torque and levitation forces referring to winding currents were always derived to design relevant current control algorithm. As a result, the winding currents should be derived reversely and were expressed by the torque and levitation forces. Furthermore, some constraints were introduced to facilitate the solution of equations, which also increased difficulties on designing the current control algorithm more or less [8,12,20]. This paper proposes a different control concept of torque and levitation forces for dual-winding BSRMs. The current hysteresis control is removed from the power controller, and is replaced by the hysteresis control of torque and levitation forces. Therefore, the complicated current control algorithm is then not needed yet, and the constraints for the solution can be avoided as well. Moreover, not only the radial displacements but also the levitation forces are under the closed-loop control, which was not achieved in hysteresis current control methods for dual-winding BSRMs. In addition, because the direct torque control is implemented, the torque ripple can be greatly reduced. The rest of paper is organised as follows. The operation principle of dual-winding BSRMs are demonstrated in Section 2. In order to facilitate the understanding of the DTC concept for SRMs, the principle of DTC for conventional SRMs is illustrated in Section 3. In Section 4, the space voltage vectors for main and levitation windings are defined according to the control of torque and levitation forces, respectively. After that, the rules and procedure for selecting the space voltage vectors are developed accordingly. Experimental results are presented to verify the proposed control method in Section 5 with conclusions made in Section 6.
Fig. 1. Configuration of the studied dual-winding BSRMs: (a) Definition of positive winding currents. (b) Coil connections taking phase A as an example.
2. Operation principle of dual-winding BSRM 2.1. Levitation principle In 12/8 dual-winding BSRMs, there are three phases and two sets of windings mounted on each stator pole, i.e. the main winding and the levitation winding, as shown in Fig. 1(a). Taking phase A as an example in Fig. 1(b), the four coils mounted on four phase-A poles are connected in series to form phase-A main winding. The two coils mounted on the two poles at the direction of ˛-axis, are connected in series to form ˛-axis levitation winding of phase A. Similarly, the other two coils form ˇ-axis levitation winding of phase A. The main winding conducts unipolar current to generate main magnetic flux, and the levitation winding conducts bipolar current to generate bias magnetic flux. When ˛-axis levitation winding conducts current as illustrated in Fig. 1(b), the magnetic density of air gap 1 is enhanced whereas that of air gap 3 is weakened. Therefore, a radial force on the rotor is produced at the positive direction of ˛-axis. Similarly, a positive radial force can also be produced along ˇ-axis. Through regulating the winding currents reasonably, the levitation force at arbitrary direction can be provided to satisfy the requirement of levitation. Therefore, the continuous levitation forces can be secured via the alternate conduction of three phases. 2.2. Mathematical model of torque and levitation force In dual-winding BSRMs, the torque and levitation force are relevant to motor mechanical parameters, the rotor angle position and the two winding currents. In order to obtain the expected torque and levitation force, the winding currents should be regulated reasonably according to different rotor angle positions. In hysteresis
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+
l0
(l0 + 4cr)
when ∈ ,
when ∈
Fˇ
= ima
Kf ()
0
0
Kf ()
when ∈ − 12 ,
Kf () = Nm Nb
12
isa1
−
12
0,
,0 ,
. 12
isa2
2
M25 msa1
1.5
M50 msa1
1 0.5 0 −0.5
−20 −15 −10
−5
0 θ (°)
5
10
0.2
Similarly, the radial forces can be written as F˛
M0msa1
15
20
(1)
where
−
L0ma
2.5
Fig. 2. Inductance curves with different rotor displacements.
2 2 2 2 Ta = Jt () 2Nm ima + Nb2 isa1 + Nb2 isa2 ,
⎧ hr 160 chr 0 ⎪ ⎨ l0 − (l − 4cr) , 0 Jt () = ⎪ 0 hr 160 chr ⎩
3
Inductances (mH)
current control strategies, mathematical model of the torque and levitation force were derived by using analytic method. Accordingly, the given winding currents were calculated in real time with respect to different rotor angle positions, and then were tracked by power converters in the current hysteresis controller. In the proposed control method, the torque and levitation force are then the direct control objectives, and the switching signals of power converters are directly determined by space voltage vectors. Therefore, the winding currents do not need to be pre-calculated and tracked by current hysteresis controller. However, the real torque and levitation forces should be estimated to be compared with their reference values, thus the mathematical model is still needed here. In fact, the torque and levitation force can also be observed by specially designed state observers, which requires more attention to be paid on. As derived in [8], the torque can be expressed as
,
(2)
Inductance rate of change (mH/°)
216
L0ma
0.15
M0msa1
0.1
M25 msa1
0.05
M50 msa1
0 −0.05 −0.1 −0.15
,
0 hr( − 12 | |) 6l02
+
320 hrc | | (l02 + 4crl0 | |)
−0.2
.
In proposed control method, the region out of − 12 , 12 is also used to produce torque, thus the torque coefficient Jt () in those regions are derived as (3) [24].
2.3. Influence of coupling effect between the main and levitation windings According to the levitation principle demonstrated in Section 2.1, the main winding produces the main magnetic field to
−20 −15 −10
−5
0 θ (°)
5
10
15
20
Fig. 3. Comparison of inductance rate of change with different rotor displacements.
mutual inductance between the main and levitation windings is zero. With the increase of the radial displacement, the mutual inductance is increased accordingly, but it is still small enough to be ignored compared with the self-inductance of phase-A main winding. Considering that radial displacements and levitation-winding currents are both small enough in experiments, the mutual fluxlinkage between the main and levitation windings can be ignored in the below analysis.
⎧ 2 20 hr ( − 4)(2 + /4) 2 + /4 ⎪ ⎪ + , when ∈ − − , , ⎪ ⎪ 8 12 ⎨ − 2 [r( + /6) + 2l0 ][−r( + /12) + 2l0 ] [r( + /6) + l0 ][−r( + /12) + l0 ] Jt () = (3) 2 ⎪ 2 − /4 ( − 4)(2 − /4) 20 hr ⎪ ⎪
+ , when ∈ . , ⎪ ⎩ −2 12 8 [r(/6 − ) + l0 ][r( − /12) + l0 ] [r(/6 − ) + 2l0 ][r − /12) + 2l0 generate the torque, and provide the bias magnetic field for the generation of radial forces. As illustrated in Fig. 1(b), when the main and ˛-axis levitation windings are both excited, the resultant air-gap flux density is increased in air gap1 and decreased in air gap3, which means the levitation-winding flux direction is the same as the main-winding flux direction in air gap1, but it is the opposite to the main-winding flux direction in air gap3. Therefore, the coupling influence on the main winding caused by the levitation winding is counteracted when the rotor is located in its geometric centre. Fig. 2 shows the finite element analysis (FEA) results of inductance curves with different rotor displacements, 0 is the self-inductance of phase A when ˛ = 0 m; M 0 , where Lma msa1 25 50 are the mutual inductances between the main and Mmsa1 and Mmsa1 ˛-axis levitation windings of phase A when ˛ = 0 m, ˛ = 25 m and ˛ = 50 m, respectively. When the displacement is zero, the
Fig. 3 shows the comparison of inductance rate of change between self and mutual inductances. The mutual-inductance rate of change is still very small, which means the torque generated by the mutual-inductance flux linkage can also be ignored to facilitate the modelling and control of BSRMs. If the mutual inductance was considered in the modelling of torque and radial forces, the expression can be more accurate but it becomes more complex. Therefore, it is more difficult to design the current algorithm in hysteresis current control strategies, because the winding current commands should be calculated from the torque and radial-force expressions. In the speed control of BSRMs, the speed error is regulated via a PI controller to obtain the reference torque, which is usually used together with the reference levitation forces to calculate the winding current commands. When the mutual coupling effect is not
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considered, the calculated current command is slightly higher than the real desired values. Accordingly, the real generated torque in the motor is a little larger than the real desired torque. However, the negative feedback speed control loop and the PI controller can stabilise this small difference. 3. DTC of conventional SRM In the DTC method of conventional three-phase SRMs, the three individual flux-linkage vectors are projected to the two axes ˛ and ˇ, and then the composed flux-linkage vector of the SRM can be obtained as [25] s =
Fig. 4. Power converter for levitation windings.
˛2 + ˇ2 ,
ı = arctan
˛ ˇ
(4)
.
(5)
where s and ı are the magnitude and angle of the composed fluxlinkage vector, respectively. As illustrated in [25], six switching vectors are defined and chosen to maintain the torque and flux linkage in the reference hysteresis bands, i.e. v1 (+, 0, −), v2 (0, +, −), v3 (−, +, 0), v4 (−, 0, +), v5 (0, −, +) and v6 (+, −, 0). Each switching vector is based on the operating states of three phases, where +, − or 0 represents that the terminal voltage of the phase winding is positive, negative or zero voltage. These six switching vectors locate in the centre of six zone sectors which are equally divided from the voltage diagram. The magnitude of the composed flux-linkage vector s is kept constant and the angle ı is used to detect the real-time zone sector of the flux-linkage vector. If the composed flux-linkage vector lies in the kth zone (k is one of six zones N = 1, . . ., 6), then four switching vectors vk−2 , vk−1 , vk+1 or vk+2 can be determined under four different conditions. vk−2 is to decrease the torque and the flux linkage simultaneously; vk−1 is to decrease the torque and increase the flux linkage; vk+1 is to increase the torque and the flux linkage simultaneously; vk+2 is to increase the torque and decrease the flux linkage. Therefore, once the command to decrease or increase the torque or flux linkage is determined, the switching vector can be selected accordingly.
where represents the flux; represents the flux linkage; N represents the winding turns. Accordingly, (6) can be rewritten as Fr =
2 . 2N 2 0 S
(7)
The magnetic pull Fr can be divided into tangential and radial components. In dual-winding BSRMs illustrated in Fig. 1, if only main winding is energised and the levitation windings are not energised, the flux linkages are distributed symmetrically around the energised air gaps between stator and rotor poles. As a result, the resultant of radial forces are zero, and the resultant of tangential forces is provided to drive the rotor rotating at a certain speed. In order to generate the radial force for the levitation function, the levitation windings should be energised to distribute the flux linkage asymmetrically. From this perspective, the phase legs of the power converters for levitation windings can be directly controlled by the closed-loop control of radial forces. 4.2. Calculation of synthetic flux linkage As demonstrated above, the flux linkage is the critical control target to implement the DTC and DFC method. In Fig. 1, assuming that the mutual inductances between different windings as well as that between different phases can be ignored, the flux linkage in air gap1 1 can be expressed as 1 = m + s . Similarly, the flux linkages in other three air gaps can be written as
4. Proposed DTC and DFC method for dual-winding BSRM In dual-winding BSRMs, the magnetic field existing in the air gap is energised by both of the main and levitation windings. Therefore, the space voltage vectors of the two windings should be defined and selected cooperatively to achieve DTC and DFC at the same time. The principle of DTC in BSRMs can be analysed similarly as conventional SRMs. In the same way, the principle of DTC can also be investigated from the generating mechanism of radial forces. 4.1. Principle of DFC In BSRMs, the magnetic pull Fr generated between the stator and rotor poles can be expressed as [14,15] Fr =
B2 S , 20
(6)
where B represents the air-gap flux density between stator and rotor poles; S represents the overlap area of stator and rotor poles. The flux density B can be written as B=
= , S NS
2 = m + s , 3 = m − s , 4 = m − s . Therefore, the total flux linkage of phase A can be obtained as A = 1 + 2 + 3 + 4 = 4m . It can be found that the phase-A flux linkage is only relevant to the flux linkage of phase-A main winding. Accordingly, the calculation of synthetic flux linkage by three phases only needs to calculate the flux linkages produced by the main windings. 4.3. Definition of space voltage vectors for main winding and levitation winding In dual-winding BSRMs, there are two sets of windings embedded in the stator, thus two sets of space voltage vectors should be established for two different windings. In this paper, the power converter for main winding adopts the asymmetric halfbridge topology circuit, and that for levitation windings adopts the three-phase four-leg topology circuit as shown in Fig. 4. For
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(a)
(b)
(c)
Fig. 5. Definition of switching states for levitation winding when flowing positive current: (a) “+1”. (b) “−1”. (c) “0”.
the main winding, the three switching states are defined as those in conventional SRM [25]. For the levitation winding, since the winding current flows in two directions including positive and negative directions, the three switching states are defined differently with the consideration of current flowing directions. In addition, two converters shown in Fig. 4 are required for the two different levitation windings, i.e. ˛-axis and ˇ-axis levitation windings. Fig. 5 illustrates the definition of switching states when flowing positive current, taking the levitation winding of phase A in ˛-axis as an example. The switching state “+1” represents that the positive voltage is applied to the winding via conducting Q3 and Q2 . The switching state “−1” represents that the negative voltage is applied to the winding via freewheeling through the two diodes D1 and D4 . The switching state “0” represents that the winding is freewheeling naturally through one power switch and one diode, i.e. Q3 and D1 in Fig. 5(c). Similarly, the definition of switching states can be similarly obtained when negative current flows in the winding, as illustrated in Fig. 6. For the levitation winding, since the two coils at the same axis are connected in subtractive series, this levitation winding can be seen as two equivalent coils in opposite positions. Accordingly, the two corresponding space voltage vectors are always of opposite directions. Fig. 7 shows the equivalent space voltage vectors considering the levitation function of dual-winding BSRMs. Taking positive direction of phase A as an example, the space voltage vector of main winding is “+1”, and the space voltage vectors of two coils along ˛-axis are “+˛A ” and “−˛A ”, respectively. Similarly, the two space voltage vectors along ˇ-axis are “+ˇA ” and “−ˇA ”, respectively. Therefore, the space voltage vector of phase A at this direction can be simplified as “+1”. The space voltage vector of phase A at the negative direction can be obtained similarly. As a result, the total space voltage vector of each phase is only relevant to the main-winding voltage of that phase. Accordingly, the equivalent space voltage vectors of dual-winding BSRMs can be selected as demonstrated in Section 3.
Fig. 7. Configuration of the studied dual-winding BSRMs taking phase A as an example.
4.4. Rules of selecting space voltage vectors for levitation windings For the simply control of levitation, the levitation windings are energised alternately to produce radial forces acted on the rotor, i.e. the firing angle for the levitation winding of each phase is 15◦ . Moreover, in order to provide the radial forces more efficiently, the region of ∈ [ −7.5◦ , 7.5◦ ] for each phase is selected to produce radial forces. In the proposed control method, the radial force is under closed-loop control. By the hysteresis control of radial forces, the power switches of levitation-winding converters are controlled directly based on the selected space voltage vectors. When the currents conduct in the windings as illustrated in Fig. 1, positive radial forces would be produced in ˛-axis and ˇaxis, respectively. In other words, when the positive radial force is required, the positive current should be conducted in the levitation winding. In this case, when the real radial force is less than the lower value of the hysteresis loop for levitation forces, the voltage symbol of levitation winding is “+”, and the relevant switching state is “+1”, as illustrated in Fig. 5(a). When the real radial force is greater than upper value of the hysteresis loop, the voltage symbol is “−”, and the relevant switching state is “−1”, as illustrated in Fig. 5(b). When the real radial force is located in the hysteresis band, the voltage symbol is “0”, and the relevant switching state is “0”, as illustrated in Fig. 5(c). Similarly, if the negative radial force is required, the negative current should be conducted in the levitation winding. In that case, the voltage symbols can be deduced like that in the case of conducting positive current. Accordingly, the relevant switching states can also be obtained as illustrated in Fig. 6.
4.5. Procedure for selecting the space voltage vectors
(a)
(b)
(c)
Fig. 6. Definition of switching states for levitation winding when flowing negative current: (a) “+1”. (b) “−1”. (c) “0”.
The mathematical model of torque and levitation forces demonstrated in Section 2.2 has been verified and used in experiments, which shows that the model can work well in the implementation of control algorithm. In this paper, the derived model is adopted to estimate the real torque and radial forces to compare with their reference values. In order to facilitate the implementation of proposed control method, the procedure for selecting the space voltage vectors for main-winding and levitation-winding converters are demonstrated as follows.
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Fig. 8. Control block of proposed control method.
(1) After the real torque and flux linkage are estimated, the equivalent space voltage vectors for the main winding are determined according to the demonstration in Section 3. (2) Checking the direction of required levitation forces, and then calculating the real radial forces from (2). Based on the analysis in Section 4.4, the space voltage vectors for levitation windings are then determined by the comparison between the real radial forces and their required values. (3) Finally, the selected space voltage vectors are transmitted to main-winding and levitation-winding converters, respectively. Therefore, the power switches are controlled directly by the hysteresis control of torque and levitation forces. 4.6. Control block Fig. 8 shows the system control block of proposed control method. The rotor angle position is detected by photocouplers and the real motor speed is calculated and then speed error is regulated by the PI controller to output the required torque. The real torque and flux linkage are estimated based on the measured winding currents and rotor angle position. After that, the estimated torque and flux linkage are compared with their own reference values, respectively. As a result, the hysteresis control of torque and flux linkage determines the equivalent space voltage vectors for the main-winding converter. For the levitation control, the real radial displacements in ˛and ˇ-axes are detected by eddy-current type transducers firstly. The radial-displacement errors are regulated by PID controllers to output required radial forces F˛∗ and Fˇ∗ , respectively. The real radial forces F˛ and Fˇ are calculated from (2) and then compared with their required values. Finally, the hysteresis control of radial forces is to determine the space voltage vectors for levitation-winding converters. 5. Experimental validation In order to verify the proposed control method, some experiments were carried out on a dual-winding BSRM prototype which parameters are listed in Table 1. The prototype was placed horizontally and the rotor’s weight is 5 kg. Therefore, it can be assumed that an additional radial force was applied on the rotor at the negative direction of ˇ-axis. Fig. 9 shows the test rig used in this paper.
All control algorithms including switching-signal generation and boolean calculation were programmed by TI TMS320F2812 and Altera EPM1270 based digital controller. The radial displacements were detected by eddy-current type transducers with the sensitivity of 16 V/mm, and the rotor angle position was detected by three photocouplers. In Fig. 8, the two PID controllers for the regulation of required radial forces were implemented by analogue devices whereas the PI controller for the speed regulation was programmed in the digital controller. The main-winding converter adopted the three-phase asymmetric half-bridge topology circuit and the levitation-winding converters adopted three-phase four-leg topology circuit. In order to simplify the estimation of torque, flux linkage and radial forces, the general approaches were adopted in this work. The flux linkage was estimated from the phase-winding voltage and current [25], and the torque and radial forces were calculated based on (1) and (2). In order to reduce the computing time, the coefficients used in (1) and (2) were pre-stored as look-up tables in the digital controller. Fig. 10 shows the flux-linkage locus at the speed of 1000 r/min with different control methods. It can be seen that the flux-linkage locus is of circular shape when using proposed control method whereas that is of triangular shape when using hysteresis current control method. It is worth pointing out that the flux-linkage locus cannot be regulated as the circular shape either, which forms another type of direct torque control called direct instantaneous torque control in SRMs [26].
Table 1 Parameters of the prototype Parameters
Values
Number of turns of the main winding Number of turns of the levitation winding Arc angle of rotor and stator teeth Outside diameter of stator core Inside diameter of stator pole Radius of rotor pole, r Stack length lamination, h Average air-gap length between rotor and stator, l0 Clearance length between shaft and backup bearing Sensitivity of the eddy-current type transducer Weight of rotor
14 turns 17 turns 15◦ 120 mm 60.5 mm 30 mm 75 mm 0.25 mm 0.2 mm 16 V/mm 5 kg
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Fig. 9. Pictures of the test rig: (a) Overview of the test rig. (b) Enlarged view of the motor shaft and the backup bearing.
Fig. 10. Experimental waveforms of flux-linkage locus: (a) Proposed control method. (b) Hysteresis current control method.
Fig. 11 shows experimental waveforms of the flux-linkage vector components, instantaneous torque and main-winding current of phase A. The flux-linkage vector components and instantaneous torque were estimated in real time in the digital controller and output to the oscilloscope via DA converter MAX547. Without
consideration of the sampling error and oscilloscope interference, the torque ripple is 0.037 N m with proposed control method whereas 0.146 N m with hysteresis current control method, i.e. the torque ripple can be reduced by around 75%. In the proposed control method, the instantaneous torque is directly regulated as the control target. Therefore, the torque was estimated in real time, and then was compared with the reference value to generate
Fig. 11. Experimental waveforms of torque and flux linkage: (a) Proposed control method. (b) Hysteresis current control method.
Fig. 12. Experimental waveforms of winding currents and radial displacements: (a) Proposed control method. (b) Hysteresis current control method.
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Fig. 14. Experimental results when speed increased from 1500 r/min to 2000 r/min first and then decreased to 1500 r/min.
6. Conclusions
Fig. 13. Experimental results when the shaft was knocked suddenly.
the space voltage vector for the control of power converters. Accordingly, the torque ripple can be greatly reduced. Fig. 12 shows experimental waveforms of winding currents and radial displacements with different control methods. The mainwinding current in proposed control method is greater than that in hysteresis current control method, because more negative-torque region is employed in proposed control method. However, the levitation-winding current is slightly smaller in proposed method. Furthermore, the radial displacements in proposed method is less than that in hysteresis current method, which mainly benefits from the generation region of levitation forces. The levitation force is generated in the region of ∈ [ −7.5◦ , 7.5◦ ] for each phase. In this region, with the same winding current, the generated levitation force is larger, because the inductance is relatively larger than that out of this region. In other words, the winding current can be smaller in this region for the same desired levitation force. Moreover, the main-winding current has reached the desired value when providing the levitation force in proposed control method, but it has to rise from zero when using the hysteresis current control method. Therefore, considering the rising and falling time of winding currents, the levitation-force commands can be tracked more quickly with proposed control method. In order to test the levitation performance, the shaft was knocked suddenly with a wooden hammer at the directions of ˛and ˇ-axes, respectively. Fig. 13 shows the corresponding experimental waveforms of radial displacements and winding currents at the speed of 1500 r/min. It can be seen that the shaft deviated to one side from the geometric centre once knocked by the hammer, and then was pulled back quickly because of the active control of radial forces. It is worth noting that the radial displacement at one direction is not affected when the shaft is knocked at the perpendicular direction. Therefore, the good levitation performance can be secured with proposed control method. In order to mimic the torque changing process, the experiment on the change of speed was carried out.Fig. 14 shows the relevant experimental results. The motor speed firstly increased from 1500 r/min to 2000 r/min and then decreased to 1500 r/min. Along with the change of speed, the torque changes accordingly. Because of the rotor’s weight, the radial displacement at the direction of ˇaxis was slightly increased in the speed rising and falling processes.
This paper proposes a control strategy to directly regulate the torque and levitation forces at the same time for dual-winding BSRMs. The complicated derivation of winding currents are not required as well as the design of current control algorithm, which is necessary in hysteresis current control strategies. The torque ripple can be greatly reduced because the concept of DTC is adopted, which can be reduced by 75% with the proposed control method at the speed of 1000 r/min. Moreover, the radial forces can be effectively generated with the proposed control method, and the shaft’s radial displacement can also be slightly decreased in experiments. Through the hysteresis control of torque and radial forces, the rotation and levitation of BSRMs can be regulated separately, which simplifies the control algorithms of bearingless motors. Acknowledgements The financial supports from the National Natural Science Foundation of China under grant No. 51107056, 51477074 and 51577087 are greatly appreciated. References [1] J.-J. Wang, A common sharing method for current and flux-linkage control of switched reluctance motor, Electr. Power Syst. Res. 131 (2016) 19–30. [2] X. Rain, M. Hilairet, A. Arias, Switched reluctance machines control with a minimized sampling frequency, Energy Convers. Manag. 86 (2014) 701–708. [3] W. Hua, H. Hua, N. Dai, G. Zhao, M. Cheng, Comparative study of switched reluctance machines with half-and full-teeth-wound windings, IEEE Trans. Ind. Electron. 63 (3) (2016) 1414–1424. [4] J. Marques, J. Estima, N. Gameiro, A. Cardoso, A new diagnostic technique for real-time diagnosis of power converter faults in switched reluctance motor drives, IEEE Trans. Ind. Appl. 50 (3) (2014) 1854–1860. [5] S. Song, M. Zhang, L. Ge, A new fast method for obtaining flux-linkage characteristics of SRM, IEEE Trans. Ind. Electron. 62 (7) (2015) 4105–4117. [6] A. Chiba, T. Fukao, O. Ichikawa, M. Oshima, M. Takemoto, D. Dorrell, Magnetic Bearings and Bearingless Drives, Elsevier Science, 2005. [7] M. Takemoto, K. Shimada, A. Chiba, T. Fukao, A design and characteristics of switched reluctance type bearingless motors, in: Proc. Int. Sym. Magn. Suspension Technol., vol. NASA/CP-1998-207654, 1998, pp. 49–63. [8] M. Takemoto, A. Chiba, T. Fukao, A method of determining the advanced angle of square-wave currents in a bearingless switched reluctance motor, IEEE Trans. Ind. Appl. 37 (6) (2001) 1702–1709. [9] X. Cao, Z. Deng, G. Yang, X. Wang, Independent control of average torque and radial force in bearingless switched-reluctance motors with hybrid excitations, IEEE Trans. Power Electron. 24 (5) (2009) 1376–1385. [10] B.B. Choi, M. Siebert, A bearingless switched reluctance motor for high specific power application, in: AIAA/ASME/SAE/ASEE Joint Propulsion Conf. Exhib., Sacramento, CA, 2006. [11] F.-C. Lin, S.-M. Yang, Self-bearing control of a switched reluctance motor using sinusoidal currents, IEEE Trans. Power Electron. 22 (6) (2007) 2518–2526. [12] L. Chen, W. Hofmann, Speed regulation technique of one bearingless 8/6 switched reluctance motor with simpler single winding structure, IEEE Trans. Ind. Electron. 59 (6) (2012) 2592–2600. [13] C.R. Morrison, M.W. Siebert, E.J. Ho, Electromagnetic forces in a hybrid magnetic-bearing switched-reluctance motor, IEEE Trans. Magn. 44 (12) (2008) 4626–4638. [14] H. Wang, J. Bao, B. Xue, J. Liu, Control of suspending force in novel permanent-magnet-biased bearingless switched reluctance motor, IEEE Trans. Ind. Electron. 62 (7) (2015) 4298–4306.
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[15] H. Wang, D.-H. Lee, T.-H. Park, J.-W. Ahn, Hybrid stator-pole switched reluctance motor to improve radial force for bearingless application, Energy Convers. Manag. 52 (2011) 1371–1376. [16] M. Takemoto, H. Suzuki, A. Chiba, T. Fukao, M.A. Rahman, Improved analysis of a bearingless switched reluctance motor, IEEE Trans. Ind. Appl. 37 (1) (2001) 26–34. [17] M. Takemoto, A. Chiba, H. Akagi, T. Fukao, Radial force and torque of a bearingless switched reluctance motor operating in a region of magnetic saturation, IEEE Trans. Ind. Appl. 40 (1) (2004) 103–112. [18] J. Sun, Q. Zhan, L. Liu, Modelling and control of bearingless switched reluctance motor based on artificial neural network, in: Proc. 31st Annual Conf. of IEEE Industrial Electronics Society IECON 2005, 2005. [19] G. Yang, Z. Deng, X. Cao, X. Wang, Optimal winding arrangements of a bearingless switched reluctance motor, IEEE Trans. Power Electron. 23 (6) (2008) 3056–3066. [20] Y. Yang, Z. Deng, G. Yang, X. Cao, Q. Zhang, A control strategy for bearingless switched-reluctance motors, IEEE Trans. Power Electron. 25 (11) (2010) 2807–2819.
[21] D.-H. Lee, H. Wang, J.-W. Ahn, Modeling and control of novel bearingless switched reluctance motor, in: Proc IEEE Energy Conversion Congress and Exposition ECCE 2009, 2009, pp. 276–281. [22] Z. Xu, D.H. Lee, J.W. Ahn, Comparative analysis of bearingless switched reluctance motors with decoupled suspending force control, IEEE Trans. Ind. Appl. 51 (1) (2015) 733–743. [23] M.N. Recheis, B. Schweighofer, P. Fulmek, H. Wegleiter, Selection of magnetic materials for bearingless high-speed mobile flywheel energy storage systems, IEEE Trans. Magn. 50 (4) (2014) 1–4. [24] Y. Sun, X. Liu, D. Wang, J. Mao, Extension of mathematical model to full angle for bearingless switched reluctance motors based on finite-element analysis, Trans. China Electrotech. Soc. 22 (9) (2007) 34–39 (in Chinese). [25] A. Cheok, Y. Fukuda, A new torque and flux control method for switched reluctance motor drives, IEEE Trans. Power Electron. 17 (4) (2002) 543–557. [26] H. Zeng, H. Chen, J. Shi, Direct instantaneous torque control with wide operating range for switched reluctance motors, IET Electr. Power Appl. 9 (9) (2015) 578–585.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Direct control of torque and levitation force for dual-winding bearingless switched reluctance motor Xin Cao ∗ , Qin Sun, Chenhao Liu, Heng Zhou, Zhiquan Deng Department of Electrical Engineering, College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No.169, West Shengtai Road, Nanjing 211106, Jiangsu, PR China
a r t i c l e
i n f o
Article history: Received 25 July 2016 Received in revised form 16 December 2016 Accepted 9 January 2017 Keywords: Dual-winding bearingless switched reluctance motor Direct torque control Direct force control Torque ripple Control strategy Current control algorithm
a b s t r a c t In dual-winding bearingless switched reluctance motors (BSRMs), an additional winding is placed on the stator poles in conventional switched reluctance motors (SRMs), which is mainly to achieve the function of levitation. Due to the hysteresis current control in existing control strategies for dual-winding BSRMs, the complicated derivation of winding-current expression was necessary, and some constraints were also introduced that increased the difficulties on designing the current control algorithm. In order to solve these problems, the direct control concept of torque and levitation forces is proposed and developed in this paper, named as direct torque control (DTC) and direct force control (DFC). Moreover, the torque ripple can also be reduced greatly. Firstly, the space voltage vectors of dual-winding BSRMs are defined for the main and levitation windings, respectively. After that, the rules and procedures for selecting the space voltage vectors are demonstrated in detail, and the system control block is also presented to facilitate the implementation of proposed control strategy. Finally, experimental results are provided to demonstrate the performance. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In conventional switched reluctance motors (SRMs), phase windings are mounted on the stator poles and no permanent magnets or windings on the rotor. The phase windings are energised separately by the asymmetric-half-bridge type converter. Therefore, SRMs own great advantages on the simple control strategy, fault-tolerant control, high speed capability, etc. [1–5]. In order to inherit the advantages of conventional SRMs, the bearingless motor technology was integrated with SRMs, and a reluctancetype bearingless motor was created, named as bearingless switched reluctance motors (BSRMs) [6–15]. The magnetic pulls are produced between stator and rotor poles when the phase winding is energised. The produced magnetic pull can be divided into tangential and radial components. In conventional SRMs, the radial components on each rotor poles are counteracted whereas the tangential components drive motor rotating at the expected speed. In BSRMs, the flux linkages flowing in the stator poles of the same phase are distributed asymmetrically, thus the radial components on the rotor are no longer
∗ Corresponding author. E-mail address: [email protected] (X. Cao). http://dx.doi.org/10.1016/j.epsr.2017.01.015 0378-7796/© 2017 Elsevier B.V. All rights reserved.
counteracted and the radial force is produced on the rotor accordingly. With the active control of radial forces, the rotor shaft can be levitated when the bearingless motor is rotating at a high speed, which avoids the mechanical wear between the shaft and the mechanical bearing. Therefore, for the mechanical bearing, not only the lubrication system can be removed but also the problems of the maintenance and renewal can be solved, especially in some highspeed drive applications and in harsh environments with radiation and poisonous substances [6]. In the BSRM, according to the number of windings mounted on each stator pole, it can be divided into single-winding and dualwinding BSRMs. The dual-winding BSRM was firstly implemented by A. Chiba et al. in the 1990s [7]. The operation principle and control strategies were deeply investigated to implement the rotation and levitation experimentally for BSRMs [8,9,16–20]. After that, the single-winding BSRM was proposed and developed to obtain the same winding configuration on each stator pole as that in conventional SRMs [10–12]. The difference is that the coil on each stator pole is energised separately in single-winding BSRMs, whereas the coils in the same phase are connected in series or in parallel in conventional SRMs. In the single-winding BSRM, the winding current is controlled to provide not only the torque but also the levitation force, thus the design of control strategy is difficult and the control performance on the torque and levitation interacts with each
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Nomenclature ˛ , ˇ 0 c F˛ , Fˇ h ima isa1 , isa2 l0 Lsa1 , Lsb1 Nm , Nb r T, Ta
flux linkage flux-linkage vector components in the axes of ˛ and ˇ, respectively permeability of vacuum rotor angular position constant 1.49. radial forces on the direction of ˛-axis and ˇ-axis, respectively lamination length of the iron core main-winding current of phase A levitation-winding currents of phase A at the direction of ˛- and ˇ-axes, respectively air-gap length between the stator and rotor poles and Lsc1 ˛-axis levitation-winding inductances of phases A, B and C, respectively winding turns of main winding and levitation winding, respectively radius of rotor instantaneous torque and instantaneous torque of phase A, respectively
other. Therefore, some different pole-pair structures for BSRMs were proposed and investigated to simplify the control of torque and levitation forces [13–15,21–23]. In summary, the present work for BSRMs mainly focuses on the levitation mechanism, control strategies and novel structures protect [8,9,11–13,18–20]. On the basis of 12/8 dual-winding BSRMs, this paper studies the control strategy to simplify the control of torque and levitation force compared with existing control algorithms. The existing algorithm is to directly regulate the winding currents through the hysteresis current control to produce required torque and levitation forces. Therefore, the expressions of torque and levitation forces referring to winding currents were always derived to design relevant current control algorithm. As a result, the winding currents should be derived reversely and were expressed by the torque and levitation forces. Furthermore, some constraints were introduced to facilitate the solution of equations, which also increased difficulties on designing the current control algorithm more or less [8,12,20]. This paper proposes a different control concept of torque and levitation forces for dual-winding BSRMs. The current hysteresis control is removed from the power controller, and is replaced by the hysteresis control of torque and levitation forces. Therefore, the complicated current control algorithm is then not needed yet, and the constraints for the solution can be avoided as well. Moreover, not only the radial displacements but also the levitation forces are under the closed-loop control, which was not achieved in hysteresis current control methods for dual-winding BSRMs. In addition, because the direct torque control is implemented, the torque ripple can be greatly reduced. The rest of paper is organised as follows. The operation principle of dual-winding BSRMs are demonstrated in Section 2. In order to facilitate the understanding of the DTC concept for SRMs, the principle of DTC for conventional SRMs is illustrated in Section 3. In Section 4, the space voltage vectors for main and levitation windings are defined according to the control of torque and levitation forces, respectively. After that, the rules and procedure for selecting the space voltage vectors are developed accordingly. Experimental results are presented to verify the proposed control method in Section 5 with conclusions made in Section 6.
Fig. 1. Configuration of the studied dual-winding BSRMs: (a) Definition of positive winding currents. (b) Coil connections taking phase A as an example.
2. Operation principle of dual-winding BSRM 2.1. Levitation principle In 12/8 dual-winding BSRMs, there are three phases and two sets of windings mounted on each stator pole, i.e. the main winding and the levitation winding, as shown in Fig. 1(a). Taking phase A as an example in Fig. 1(b), the four coils mounted on four phase-A poles are connected in series to form phase-A main winding. The two coils mounted on the two poles at the direction of ˛-axis, are connected in series to form ˛-axis levitation winding of phase A. Similarly, the other two coils form ˇ-axis levitation winding of phase A. The main winding conducts unipolar current to generate main magnetic flux, and the levitation winding conducts bipolar current to generate bias magnetic flux. When ˛-axis levitation winding conducts current as illustrated in Fig. 1(b), the magnetic density of air gap 1 is enhanced whereas that of air gap 3 is weakened. Therefore, a radial force on the rotor is produced at the positive direction of ˛-axis. Similarly, a positive radial force can also be produced along ˇ-axis. Through regulating the winding currents reasonably, the levitation force at arbitrary direction can be provided to satisfy the requirement of levitation. Therefore, the continuous levitation forces can be secured via the alternate conduction of three phases. 2.2. Mathematical model of torque and levitation force In dual-winding BSRMs, the torque and levitation force are relevant to motor mechanical parameters, the rotor angle position and the two winding currents. In order to obtain the expected torque and levitation force, the winding currents should be regulated reasonably according to different rotor angle positions. In hysteresis
X. Cao et al. / Electric Power Systems Research 145 (2017) 214–222
+
l0
(l0 + 4cr)
when ∈ ,
when ∈
Fˇ
= ima
Kf ()
0
0
Kf ()
when ∈ − 12 ,
Kf () = Nm Nb
12
isa1
−
12
0,
,0 ,
. 12
isa2
2
M25 msa1
1.5
M50 msa1
1 0.5 0 −0.5
−20 −15 −10
−5
0 θ (°)
5
10
0.2
Similarly, the radial forces can be written as F˛
M0msa1
15
20
(1)
where
−
L0ma
2.5
Fig. 2. Inductance curves with different rotor displacements.
2 2 2 2 Ta = Jt () 2Nm ima + Nb2 isa1 + Nb2 isa2 ,
⎧ hr 160 chr 0 ⎪ ⎨ l0 − (l − 4cr) , 0 Jt () = ⎪ 0 hr 160 chr ⎩
3
Inductances (mH)
current control strategies, mathematical model of the torque and levitation force were derived by using analytic method. Accordingly, the given winding currents were calculated in real time with respect to different rotor angle positions, and then were tracked by power converters in the current hysteresis controller. In the proposed control method, the torque and levitation force are then the direct control objectives, and the switching signals of power converters are directly determined by space voltage vectors. Therefore, the winding currents do not need to be pre-calculated and tracked by current hysteresis controller. However, the real torque and levitation forces should be estimated to be compared with their reference values, thus the mathematical model is still needed here. In fact, the torque and levitation force can also be observed by specially designed state observers, which requires more attention to be paid on. As derived in [8], the torque can be expressed as
,
(2)
Inductance rate of change (mH/°)
216
L0ma
0.15
M0msa1
0.1
M25 msa1
0.05
M50 msa1
0 −0.05 −0.1 −0.15
,
0 hr( − 12 | |) 6l02
+
320 hrc | | (l02 + 4crl0 | |)
−0.2
.
In proposed control method, the region out of − 12 , 12 is also used to produce torque, thus the torque coefficient Jt () in those regions are derived as (3) [24].
2.3. Influence of coupling effect between the main and levitation windings According to the levitation principle demonstrated in Section 2.1, the main winding produces the main magnetic field to
−20 −15 −10
−5
0 θ (°)
5
10
15
20
Fig. 3. Comparison of inductance rate of change with different rotor displacements.
mutual inductance between the main and levitation windings is zero. With the increase of the radial displacement, the mutual inductance is increased accordingly, but it is still small enough to be ignored compared with the self-inductance of phase-A main winding. Considering that radial displacements and levitation-winding currents are both small enough in experiments, the mutual fluxlinkage between the main and levitation windings can be ignored in the below analysis.
⎧ 2 20 hr ( − 4)(2 + /4) 2 + /4 ⎪ ⎪ + , when ∈ − − , , ⎪ ⎪ 8 12 ⎨ − 2 [r( + /6) + 2l0 ][−r( + /12) + 2l0 ] [r( + /6) + l0 ][−r( + /12) + l0 ] Jt () = (3) 2 ⎪ 2 − /4 ( − 4)(2 − /4) 20 hr ⎪ ⎪
+ , when ∈ . , ⎪ ⎩ −2 12 8 [r(/6 − ) + l0 ][r( − /12) + l0 ] [r(/6 − ) + 2l0 ][r − /12) + 2l0 generate the torque, and provide the bias magnetic field for the generation of radial forces. As illustrated in Fig. 1(b), when the main and ˛-axis levitation windings are both excited, the resultant air-gap flux density is increased in air gap1 and decreased in air gap3, which means the levitation-winding flux direction is the same as the main-winding flux direction in air gap1, but it is the opposite to the main-winding flux direction in air gap3. Therefore, the coupling influence on the main winding caused by the levitation winding is counteracted when the rotor is located in its geometric centre. Fig. 2 shows the finite element analysis (FEA) results of inductance curves with different rotor displacements, 0 is the self-inductance of phase A when ˛ = 0 m; M 0 , where Lma msa1 25 50 are the mutual inductances between the main and Mmsa1 and Mmsa1 ˛-axis levitation windings of phase A when ˛ = 0 m, ˛ = 25 m and ˛ = 50 m, respectively. When the displacement is zero, the
Fig. 3 shows the comparison of inductance rate of change between self and mutual inductances. The mutual-inductance rate of change is still very small, which means the torque generated by the mutual-inductance flux linkage can also be ignored to facilitate the modelling and control of BSRMs. If the mutual inductance was considered in the modelling of torque and radial forces, the expression can be more accurate but it becomes more complex. Therefore, it is more difficult to design the current algorithm in hysteresis current control strategies, because the winding current commands should be calculated from the torque and radial-force expressions. In the speed control of BSRMs, the speed error is regulated via a PI controller to obtain the reference torque, which is usually used together with the reference levitation forces to calculate the winding current commands. When the mutual coupling effect is not
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considered, the calculated current command is slightly higher than the real desired values. Accordingly, the real generated torque in the motor is a little larger than the real desired torque. However, the negative feedback speed control loop and the PI controller can stabilise this small difference. 3. DTC of conventional SRM In the DTC method of conventional three-phase SRMs, the three individual flux-linkage vectors are projected to the two axes ˛ and ˇ, and then the composed flux-linkage vector of the SRM can be obtained as [25] s =
Fig. 4. Power converter for levitation windings.
˛2 + ˇ2 ,
ı = arctan
˛ ˇ
(4)
.
(5)
where s and ı are the magnitude and angle of the composed fluxlinkage vector, respectively. As illustrated in [25], six switching vectors are defined and chosen to maintain the torque and flux linkage in the reference hysteresis bands, i.e. v1 (+, 0, −), v2 (0, +, −), v3 (−, +, 0), v4 (−, 0, +), v5 (0, −, +) and v6 (+, −, 0). Each switching vector is based on the operating states of three phases, where +, − or 0 represents that the terminal voltage of the phase winding is positive, negative or zero voltage. These six switching vectors locate in the centre of six zone sectors which are equally divided from the voltage diagram. The magnitude of the composed flux-linkage vector s is kept constant and the angle ı is used to detect the real-time zone sector of the flux-linkage vector. If the composed flux-linkage vector lies in the kth zone (k is one of six zones N = 1, . . ., 6), then four switching vectors vk−2 , vk−1 , vk+1 or vk+2 can be determined under four different conditions. vk−2 is to decrease the torque and the flux linkage simultaneously; vk−1 is to decrease the torque and increase the flux linkage; vk+1 is to increase the torque and the flux linkage simultaneously; vk+2 is to increase the torque and decrease the flux linkage. Therefore, once the command to decrease or increase the torque or flux linkage is determined, the switching vector can be selected accordingly.
where represents the flux; represents the flux linkage; N represents the winding turns. Accordingly, (6) can be rewritten as Fr =
2 . 2N 2 0 S
(7)
The magnetic pull Fr can be divided into tangential and radial components. In dual-winding BSRMs illustrated in Fig. 1, if only main winding is energised and the levitation windings are not energised, the flux linkages are distributed symmetrically around the energised air gaps between stator and rotor poles. As a result, the resultant of radial forces are zero, and the resultant of tangential forces is provided to drive the rotor rotating at a certain speed. In order to generate the radial force for the levitation function, the levitation windings should be energised to distribute the flux linkage asymmetrically. From this perspective, the phase legs of the power converters for levitation windings can be directly controlled by the closed-loop control of radial forces. 4.2. Calculation of synthetic flux linkage As demonstrated above, the flux linkage is the critical control target to implement the DTC and DFC method. In Fig. 1, assuming that the mutual inductances between different windings as well as that between different phases can be ignored, the flux linkage in air gap1 1 can be expressed as 1 = m + s . Similarly, the flux linkages in other three air gaps can be written as
4. Proposed DTC and DFC method for dual-winding BSRM In dual-winding BSRMs, the magnetic field existing in the air gap is energised by both of the main and levitation windings. Therefore, the space voltage vectors of the two windings should be defined and selected cooperatively to achieve DTC and DFC at the same time. The principle of DTC in BSRMs can be analysed similarly as conventional SRMs. In the same way, the principle of DTC can also be investigated from the generating mechanism of radial forces. 4.1. Principle of DFC In BSRMs, the magnetic pull Fr generated between the stator and rotor poles can be expressed as [14,15] Fr =
B2 S , 20
(6)
where B represents the air-gap flux density between stator and rotor poles; S represents the overlap area of stator and rotor poles. The flux density B can be written as B=
= , S NS
2 = m + s , 3 = m − s , 4 = m − s . Therefore, the total flux linkage of phase A can be obtained as A = 1 + 2 + 3 + 4 = 4m . It can be found that the phase-A flux linkage is only relevant to the flux linkage of phase-A main winding. Accordingly, the calculation of synthetic flux linkage by three phases only needs to calculate the flux linkages produced by the main windings. 4.3. Definition of space voltage vectors for main winding and levitation winding In dual-winding BSRMs, there are two sets of windings embedded in the stator, thus two sets of space voltage vectors should be established for two different windings. In this paper, the power converter for main winding adopts the asymmetric halfbridge topology circuit, and that for levitation windings adopts the three-phase four-leg topology circuit as shown in Fig. 4. For
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(a)
(b)
(c)
Fig. 5. Definition of switching states for levitation winding when flowing positive current: (a) “+1”. (b) “−1”. (c) “0”.
the main winding, the three switching states are defined as those in conventional SRM [25]. For the levitation winding, since the winding current flows in two directions including positive and negative directions, the three switching states are defined differently with the consideration of current flowing directions. In addition, two converters shown in Fig. 4 are required for the two different levitation windings, i.e. ˛-axis and ˇ-axis levitation windings. Fig. 5 illustrates the definition of switching states when flowing positive current, taking the levitation winding of phase A in ˛-axis as an example. The switching state “+1” represents that the positive voltage is applied to the winding via conducting Q3 and Q2 . The switching state “−1” represents that the negative voltage is applied to the winding via freewheeling through the two diodes D1 and D4 . The switching state “0” represents that the winding is freewheeling naturally through one power switch and one diode, i.e. Q3 and D1 in Fig. 5(c). Similarly, the definition of switching states can be similarly obtained when negative current flows in the winding, as illustrated in Fig. 6. For the levitation winding, since the two coils at the same axis are connected in subtractive series, this levitation winding can be seen as two equivalent coils in opposite positions. Accordingly, the two corresponding space voltage vectors are always of opposite directions. Fig. 7 shows the equivalent space voltage vectors considering the levitation function of dual-winding BSRMs. Taking positive direction of phase A as an example, the space voltage vector of main winding is “+1”, and the space voltage vectors of two coils along ˛-axis are “+˛A ” and “−˛A ”, respectively. Similarly, the two space voltage vectors along ˇ-axis are “+ˇA ” and “−ˇA ”, respectively. Therefore, the space voltage vector of phase A at this direction can be simplified as “+1”. The space voltage vector of phase A at the negative direction can be obtained similarly. As a result, the total space voltage vector of each phase is only relevant to the main-winding voltage of that phase. Accordingly, the equivalent space voltage vectors of dual-winding BSRMs can be selected as demonstrated in Section 3.
Fig. 7. Configuration of the studied dual-winding BSRMs taking phase A as an example.
4.4. Rules of selecting space voltage vectors for levitation windings For the simply control of levitation, the levitation windings are energised alternately to produce radial forces acted on the rotor, i.e. the firing angle for the levitation winding of each phase is 15◦ . Moreover, in order to provide the radial forces more efficiently, the region of ∈ [ −7.5◦ , 7.5◦ ] for each phase is selected to produce radial forces. In the proposed control method, the radial force is under closed-loop control. By the hysteresis control of radial forces, the power switches of levitation-winding converters are controlled directly based on the selected space voltage vectors. When the currents conduct in the windings as illustrated in Fig. 1, positive radial forces would be produced in ˛-axis and ˇaxis, respectively. In other words, when the positive radial force is required, the positive current should be conducted in the levitation winding. In this case, when the real radial force is less than the lower value of the hysteresis loop for levitation forces, the voltage symbol of levitation winding is “+”, and the relevant switching state is “+1”, as illustrated in Fig. 5(a). When the real radial force is greater than upper value of the hysteresis loop, the voltage symbol is “−”, and the relevant switching state is “−1”, as illustrated in Fig. 5(b). When the real radial force is located in the hysteresis band, the voltage symbol is “0”, and the relevant switching state is “0”, as illustrated in Fig. 5(c). Similarly, if the negative radial force is required, the negative current should be conducted in the levitation winding. In that case, the voltage symbols can be deduced like that in the case of conducting positive current. Accordingly, the relevant switching states can also be obtained as illustrated in Fig. 6.
4.5. Procedure for selecting the space voltage vectors
(a)
(b)
(c)
Fig. 6. Definition of switching states for levitation winding when flowing negative current: (a) “+1”. (b) “−1”. (c) “0”.
The mathematical model of torque and levitation forces demonstrated in Section 2.2 has been verified and used in experiments, which shows that the model can work well in the implementation of control algorithm. In this paper, the derived model is adopted to estimate the real torque and radial forces to compare with their reference values. In order to facilitate the implementation of proposed control method, the procedure for selecting the space voltage vectors for main-winding and levitation-winding converters are demonstrated as follows.
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Fig. 8. Control block of proposed control method.
(1) After the real torque and flux linkage are estimated, the equivalent space voltage vectors for the main winding are determined according to the demonstration in Section 3. (2) Checking the direction of required levitation forces, and then calculating the real radial forces from (2). Based on the analysis in Section 4.4, the space voltage vectors for levitation windings are then determined by the comparison between the real radial forces and their required values. (3) Finally, the selected space voltage vectors are transmitted to main-winding and levitation-winding converters, respectively. Therefore, the power switches are controlled directly by the hysteresis control of torque and levitation forces. 4.6. Control block Fig. 8 shows the system control block of proposed control method. The rotor angle position is detected by photocouplers and the real motor speed is calculated and then speed error is regulated by the PI controller to output the required torque. The real torque and flux linkage are estimated based on the measured winding currents and rotor angle position. After that, the estimated torque and flux linkage are compared with their own reference values, respectively. As a result, the hysteresis control of torque and flux linkage determines the equivalent space voltage vectors for the main-winding converter. For the levitation control, the real radial displacements in ˛and ˇ-axes are detected by eddy-current type transducers firstly. The radial-displacement errors are regulated by PID controllers to output required radial forces F˛∗ and Fˇ∗ , respectively. The real radial forces F˛ and Fˇ are calculated from (2) and then compared with their required values. Finally, the hysteresis control of radial forces is to determine the space voltage vectors for levitation-winding converters. 5. Experimental validation In order to verify the proposed control method, some experiments were carried out on a dual-winding BSRM prototype which parameters are listed in Table 1. The prototype was placed horizontally and the rotor’s weight is 5 kg. Therefore, it can be assumed that an additional radial force was applied on the rotor at the negative direction of ˇ-axis. Fig. 9 shows the test rig used in this paper.
All control algorithms including switching-signal generation and boolean calculation were programmed by TI TMS320F2812 and Altera EPM1270 based digital controller. The radial displacements were detected by eddy-current type transducers with the sensitivity of 16 V/mm, and the rotor angle position was detected by three photocouplers. In Fig. 8, the two PID controllers for the regulation of required radial forces were implemented by analogue devices whereas the PI controller for the speed regulation was programmed in the digital controller. The main-winding converter adopted the three-phase asymmetric half-bridge topology circuit and the levitation-winding converters adopted three-phase four-leg topology circuit. In order to simplify the estimation of torque, flux linkage and radial forces, the general approaches were adopted in this work. The flux linkage was estimated from the phase-winding voltage and current [25], and the torque and radial forces were calculated based on (1) and (2). In order to reduce the computing time, the coefficients used in (1) and (2) were pre-stored as look-up tables in the digital controller. Fig. 10 shows the flux-linkage locus at the speed of 1000 r/min with different control methods. It can be seen that the flux-linkage locus is of circular shape when using proposed control method whereas that is of triangular shape when using hysteresis current control method. It is worth pointing out that the flux-linkage locus cannot be regulated as the circular shape either, which forms another type of direct torque control called direct instantaneous torque control in SRMs [26].
Table 1 Parameters of the prototype Parameters
Values
Number of turns of the main winding Number of turns of the levitation winding Arc angle of rotor and stator teeth Outside diameter of stator core Inside diameter of stator pole Radius of rotor pole, r Stack length lamination, h Average air-gap length between rotor and stator, l0 Clearance length between shaft and backup bearing Sensitivity of the eddy-current type transducer Weight of rotor
14 turns 17 turns 15◦ 120 mm 60.5 mm 30 mm 75 mm 0.25 mm 0.2 mm 16 V/mm 5 kg
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Fig. 9. Pictures of the test rig: (a) Overview of the test rig. (b) Enlarged view of the motor shaft and the backup bearing.
Fig. 10. Experimental waveforms of flux-linkage locus: (a) Proposed control method. (b) Hysteresis current control method.
Fig. 11 shows experimental waveforms of the flux-linkage vector components, instantaneous torque and main-winding current of phase A. The flux-linkage vector components and instantaneous torque were estimated in real time in the digital controller and output to the oscilloscope via DA converter MAX547. Without
consideration of the sampling error and oscilloscope interference, the torque ripple is 0.037 N m with proposed control method whereas 0.146 N m with hysteresis current control method, i.e. the torque ripple can be reduced by around 75%. In the proposed control method, the instantaneous torque is directly regulated as the control target. Therefore, the torque was estimated in real time, and then was compared with the reference value to generate
Fig. 11. Experimental waveforms of torque and flux linkage: (a) Proposed control method. (b) Hysteresis current control method.
Fig. 12. Experimental waveforms of winding currents and radial displacements: (a) Proposed control method. (b) Hysteresis current control method.
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Fig. 14. Experimental results when speed increased from 1500 r/min to 2000 r/min first and then decreased to 1500 r/min.
6. Conclusions
Fig. 13. Experimental results when the shaft was knocked suddenly.
the space voltage vector for the control of power converters. Accordingly, the torque ripple can be greatly reduced. Fig. 12 shows experimental waveforms of winding currents and radial displacements with different control methods. The mainwinding current in proposed control method is greater than that in hysteresis current control method, because more negative-torque region is employed in proposed control method. However, the levitation-winding current is slightly smaller in proposed method. Furthermore, the radial displacements in proposed method is less than that in hysteresis current method, which mainly benefits from the generation region of levitation forces. The levitation force is generated in the region of ∈ [ −7.5◦ , 7.5◦ ] for each phase. In this region, with the same winding current, the generated levitation force is larger, because the inductance is relatively larger than that out of this region. In other words, the winding current can be smaller in this region for the same desired levitation force. Moreover, the main-winding current has reached the desired value when providing the levitation force in proposed control method, but it has to rise from zero when using the hysteresis current control method. Therefore, considering the rising and falling time of winding currents, the levitation-force commands can be tracked more quickly with proposed control method. In order to test the levitation performance, the shaft was knocked suddenly with a wooden hammer at the directions of ˛and ˇ-axes, respectively. Fig. 13 shows the corresponding experimental waveforms of radial displacements and winding currents at the speed of 1500 r/min. It can be seen that the shaft deviated to one side from the geometric centre once knocked by the hammer, and then was pulled back quickly because of the active control of radial forces. It is worth noting that the radial displacement at one direction is not affected when the shaft is knocked at the perpendicular direction. Therefore, the good levitation performance can be secured with proposed control method. In order to mimic the torque changing process, the experiment on the change of speed was carried out.Fig. 14 shows the relevant experimental results. The motor speed firstly increased from 1500 r/min to 2000 r/min and then decreased to 1500 r/min. Along with the change of speed, the torque changes accordingly. Because of the rotor’s weight, the radial displacement at the direction of ˇaxis was slightly increased in the speed rising and falling processes.
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