1. Introduction
Although the switched reluctance motor (SRM) has many merits, such as robustness, a simple construction, low cost, and no permanent magnets, hence higher operating temperature limits [
1,
2], its deployment in servo applications is restrained due to the following: firstly, acoustic noise caused by radial vibration [
3]; secondly, severe torque ripple (TR) during commutation (the transfer of torque production from an outgoing phase to an incoming phase) due to the discrete and non-linear nature of SRM torque production [
4]; and thirdly, non-standard bridge converter configuration [
5,
6]. The high torque ripple may cause mechanical vibration stresses (possibly resulting in mechanical resonance effects) and speed oscillation at low speeds, which are undesirable in applications such as an electric vehicle (EV) [
7]. Many solutions have been proposed to alleviate the first two undesirable SRM features [
8]. Generally, two main approaches are available to reduce TR, namely the motor design approach and the control approach [
9].
Increasing the number of SRM stator and rotor poles results in an overlap of more than two phases during commutation, which can minimise torque ripple. However, the increased switching frequency results in more converter losses and machine losses. In addition, extra converter switches are required (if the number of phases increases), increasing the converter cost [
10].
A new SRM approach explores an SRM with a higher number of rotor poles than stator poles, as presented in [
11]. The new motor concept has a lower mass and copper loss than the conventional SRM. Due to the extra space available in the stator slot area, windings with a higher number of turns and thicker cross-sectional area can be deployed [
12]. Also, the increased rotor pole number minimises TR, which is vital for EV application [
13]. However, since the interpolar rotor air gaps are narrower in the new motor design, the unaligned inductance is significantly higher than that of the conventional SRM. The increase in unaligned inductance reduces the energy conversion area and thus decreases the developed torque [
14]. Also, the current rise time at phase turn-on is prolonged as a consequence of higher unaligned inductance, which implies using higher dc-link voltages to increase the current rate of rise [
15]. Introducing a notch in the rotor pole was presented in [
16] for TR reduction. This approach was extended in [
17] for a mutually coupled SRM. The rotor pole shoe is discussed in [
18]. On the other hand, a stator pole slant is demonstrated in [
19], and a non-uniform air gap for a two-phase SRM is presented in [
20]. All these variations attempt to minimise TR. However, the machine torque density is reduced. Increasing the average torque and reducing torque ripple was proposed in [
21] by optimising the stator and rotor poles for a four-phase 8/6 SRM. However, the approach of modifying the machine design is limited to a narrow speed range and for rated load conditions. Moreover, changes to the basic rotor/stator design result in a reduced power output, which is an unacceptable SRM limitation in EVs.
As opposed to the machine design approach, the control approach is less expensive, more effective and flexible, and can cover a wide range of speeds [
22]. Ripple-free torque could be obtained by current profiling [
23], where optimal current contours, producing constant output torque, are generated using static SRM data. These current profiles are stored using look-up tables (LUTs), and current controllers are used to track these profiles [
24]. A high bandwidth current controller is used to accurately track the stored current profiles [
25,
26,
27]. Using conventional pulse-width modulation (PWM) may result in current tracking errors, especially at high speeds [
28,
29,
30,
31,
32]. In [
33,
34], current profiling along with a new SRM design with a flat torque profile are combined to reduce the torque ripple. Yet, the new machine design has a narrow speed range compared with conventional SRM designs.
Generally, optimal current profiles are generated as discrete points, and the intermediate points are found using linear interpolation. Increasing the number of points will improve the accuracy at the expense of time and storage memory in the form of LUTs. In [
35], harmonic coefficients are used to produce the required current profiles, where interpolation is not required. However, this method is based on the accuracy of the analytical model. Fuzzy logic and neural networks presented in [
36,
37] can generate the required magnetisation data without the need for a huge memory. However, the computational complexity increases significantly. In [
38,
39], a control scheme is developed based on iterative learning control. The advantage of this method is that it does not depend on model accuracy and does not need a large memory to store the magnetisation data. Yet, it has slow dynamics with a poor response during transients, which is not tolerable in EV applications.
Average torque control (ATC) was presented in [
40,
41], where torque is calculated using co-energy that is derived based on terminal quantities (voltage and current) [
42]. The method is extended in [
43] for four-quadrant operation at low speeds. However, this method relies on high-precision measurements of voltage and current. In addition, the effect of varying the stator resistance (due to temperature change) is not considered, which affects the co-energy estimation.
The principle of direct torque control (DTC), which is similar to that used for induction motors, was proposed in [
44] and optimised in [
45]. The concept is based on controlling the flux, which, in turn, controls the developed torque [
46]. Improving the torque per ampere ratio and, hence, improving the efficiency, was considered in [
47,
48]. A model predictive controller was presented in [
49] to choose the suitable voltage vector. However, the method relies on the SRM mathematical model, which does not reflect machine non-linearity.
Direct instantaneous torque control (DITC) can continuously control the instantaneous value of developed torque [
50], so it significantly minimises TR. In [
51], a four-quadrant DITC was presented, while [
52] presented an extension of the method for SRMs with more than four phases. However, this method requires a complicated algorithm [
53]. In addition, there is no current control, which poses a limitation for SRM operation with one failure phase [
54].
Among different control approaches, torque-sharing functions (TSFs) reliably reduce the TR generated during commutation. In [
55], the objective was to optimise linear and sinusoidal TSFs in order to reduce the TR along with the phase rms current. However, saturation was neglected, and analytical expressions were required within the optimisation process. In [
56], the concept of the maximum rate of change of flux linkage was introduced to assess conventional TSFs (linear, sinusoidal, cubic, and exponential). It was shown that a low rate of change of flux linkage increases the ripple-free speed range. Additionally, optimal values for turn-on and overlap angles were advised. A logical non-linear TSF was proposed in [
57], with no attention to the maximum rate of change of flux linkage. In [
58], an online TSF was proposed, where the selection between commutating phases is based on the rate of change of flux linkage. Nevertheless, this method requires a PI controller to compensate for the error between the reference and estimated torque. Also, the conversion of torque expression to current expression requires analytical formulation. An offline TSF was presented in [
59], which offers a trade-off between a low rms current and low rate of change of flux linkage. The model requires accurate analytical expressions to derive the current profiles. A DITC-based TSF was proposed in [
60], which requires fewer current sensors. However, the phase current detection accuracy is low. Also, circuit modification along with two high-frequency PWM signals are required for phase current extraction. A hybrid switching mode (a combination between hard and soft switching) was proposed in [
61] instead of the hard switching mode usually adopted in conventional TSFs. However, there was no illustration of the maximum ripple-free speed range of this method. In [
62], an off-line TSF based on a SRM’s magnetic characteristics is proposed, where a weighting parameter is adjusted to minimise the rms current while ensuring that the rate of change of flux linkage is below the dc-link voltage for accurate current tracking. However, the dc-link voltage is not fully utilised. Hence, the zero TR speed ranges are limited.
In the literature, all TSFs have some salient shortcomings, which can be summarised as follows:
All TSFs are limited to one or two phases conducting simultaneously. There has been no attempt to investigate TSF performance for more than two phases conducting simultaneously.
The dc-link voltage was not fully exploited, which limits the ripple-free speed range.
Either hard or hybrid switching is required for accurate current tracking, which increases switching losses.
The turn-on angle cannot be advanced beyond the unaligned position for rapid current buildup.
Reference [
63] proposed a new torque control function for TR minimisation, extending the zero TR speed range by fully utilising the dc-link voltage. A ripple-free operation was possible up to 71% of the base speed, which is still far below the SRM base speed. In addition, there was no attention to the rms current and, hence, the SRM efficiency. Moreover, the method only allows for two-phase overlap.
This paper presents a torque control function (TCF, as opposed to TSF, since torque will be controlled by controlling one phase only) based on an SRM’s magnetic characteristics. The new TR analysis concept separates TR into two independent components, namely phase commutation torque ripple and switching (PWM/hysteresis) torque ripple.
The merits of the proposed TCF are as follows:
The method is suitable for an SRM with any number of phases and stator/rotor poles.
The concept of TSF is generalised to include more than two phases conducting simultaneously.
Maximum utilisation of available dc-link voltage is achieved at turn-on and turn-off, with no switching, hence reducing switching losses and increasing the ripple-free speed range.
The SRM maximum speed range with theoretically zero TR (commutation ripple elimination, leaving ripple due to PWM switching only) is determined.
Soft switching control is deployed (as opposed to hard or hybrid switching as usually adopted for traditional TSFs), which reduces switching losses.
The proposed TCF requires switching for a period equal to the stroke angle only instead of the full conduction period, which reduces switching losses.
At any instant (regardless of the number of phases conducting simultaneously), only one phase current is controlled, hence termed torque control function, TCF. Effects due to phase coupling, short/long flux paths, saturation, Cu temperature, etc., can be compensated by adjusting the single controlling current.
The paper is organised as follows:
Section 2 presents the new TCF with the proposed control technique.
Section 3 investigates the maximum torque ripple-free speed range achieved at different torque demands using the proposed TCF. The proposed method is validated by simulations in
Section 4.
Section 5 discusses the effect of varying the turn-on angle for rms/peak current reduction and compares the proposed TCF against other TR-minimisation methods. Conclusions form
Section 6.
2. Proposed Torque Control Function
The proposed TCF for commutation ripple reduction generates a flux linkage profile that fully utilises the dc-link voltage at both turn-on and turn-off. This flux profile is transformed to a current profile stored in a (3D finite element-based) look-up table (LUT).
Figure 1 shows the block diagram of the conventional TSF and the proposed TCF. In the former, the total torque demand is used to generate the reference phase torques. Actual phase currents are transformed into phase torques using LUTs. A hysteresis band controller (HBC) compares the reference and actual phase torques to generate the gating signals. On the other hand, the proposed TCF generates the required reference current profiles depending on the motor speed and the torque demand. Actual phase currents are compared with reference currents, and an HBC generates the required driving signals.
A 4φ, an 8/6 SRM, with specifications given in
Table 1 and finite element analysis data in
Appendix A, is selected to demonstrate the proposed TCF, noting that the proposed method is suitable to any SRM. The rotor pole pitch and the phase shift for an 8/6 SRM are
and
, respectively. For the SRM under study, the unaligned position is at
, while the aligned position is at
(negative torque
to
; positive torque
to
). All the angles are mechanical degrees.
The SRM is driven by a conventional asymmetric half-bridge converter (ASHB) with two switches and two diodes per phase. Three voltage states are available for the operation of the converter, namely magnetisation, freewheeling, and de-magnetisation states.
Magnetisation, , is where the full dc-link voltage is applied on the phase; thus, the current builds up quickly, reaching the reference value.
Freewheeling,, is where zero voltage is applied across the phase winding to regulate the current. Hence, the current is maintained within the reference value.
De-magnetisation,, is where the full negative dc-link voltage is applied on the phase. The de-magnetisation energy via the motor winding is pumped back to the dc link, reducing the winding current to zero before the SRM enters the negative torque-production region.
Figure 2 shows the typical voltage and current waveforms for the three states, illustrating the switching and conduction periods.
For the proposed TCF, two modes of SRM operation exist:
These two modes of operation are investigated in the following subsections.
2.1. Mode #1 (Phase Current Conduction ≤ 30°)
In this mode of operation, either one or two phases (simultaneously) conduct to generate the required torque. The maximum conduction period for each phase is limited, in this mode, to
.
Figure 3 shows the phase torques and total torque for an arbitrary conduction period,
.
In regions I and III, two phases overlap and additively produce the demanded torque, while in region II, only one phase produces the required torque. If the conduction period is (that is, and ), the second region (II) vanishes, and only regions I and III exist.
The goal is to generate a phase torque profile, which fully utilises the available dc-link voltage and, in so doing, will offer the widest zero TR speed range. The phase torque profile will be composed of five parts. To illustrate the concept, a numerical example is given with and .
When the incoming phase is at 35°, a full positive dc-link voltage, +, is applied continuously (no switching) to this phase for rapid current and, hence, torque buildup, producing portion 1 (in this case, the incoming phase is termed the master phase). Simultaneously, the outgoing phase is at 50° (due to a 15° phase shift). The outgoing phase generates the rest of the torque so that the torque demand level is reached. Hence, portion 2 is determined (in this case, the outgoing phase is termed the control phase).
When angle is reached, the outgoing phase takes precedence. Hence, the outgoing phase becomes the master phase, and a full negative dc-link voltage, −, is applied continuously (no switching) for rapid current extinction, resulting in portion 3. Simultaneously, the incoming phase becomes the control phase supplying the torque deficiency to maintain the torque demand. Hence, portion 4 is specified. Finally, portion 5, region II, is when only one phase conducts to generate the demanded torque. Concatenating the five portions, the phase torque profile is obtained.
Table 2 demonstrates this process in the overlap region (the single-phase conduction region, portion 5, is excluded).
The final step is to calculate the value of which maximises the zero TR speed, ().
Equation (1) defines the voltage equation at turn-on/off for one SRM phase, neglecting phase-winding resistance.
Here, λ is the flux linkage. + is applied continuously at phase turn-on (portion 1), while − is applied continuously at phase turn-off (portion 3).
Integrating (1):
where
,
are the final and initial values of flux linkage, respectively, while t is time.
At phase turn-on, the initial flux linkage of the phase winding is zero. Hence, the final flux linkage is calculated by:
At phase turn-off, the flux linkage must decay to zero before/at the aligned position (to avoid entering the negative torque-production region). Hence, the final flux linkage is zero, and the initial flux linkage is calculated from:
Equations (3) and (4) are solved to calculate
and
ensuring that (5) is satisfied.
For the numerical example with a torque demand of 25 Nm, the angle is 36.65°, is 355 rpm, is 0.32 Wb-t, and is 0.65 Wb-t. The developed phase torques for the incoming and outgoing phases are 10.35 Nm and 14.65 Nm, respectively, and the rms phase current is 11.63 A.
Figure 4 shows the flowchart, which describes the operation steps in model #1.
2.2. Mode #2 (Phase Current Conduction > 30°)
Figure 5 shows the operation in mode #2, where two/three phases conduct simultaneously to produce the required torque.
In the three-phase conduction region, there are two master phases and one control phase. A full dc-link voltage, +Vdc, is applied continuously on the incoming phase (Master #1), giving portion 1, and a full negative dc voltage, −Vdc, is applied continuously on the outgoing phase (Master #2), giving portion 2. Meanwhile, the middle phase is the control phase, which supplies the rest of the demand torque, producing portion 3.
The two-phase conduction period provides the remaining portions (4, 5, 6, and 7), as specified in mode #1.
For the numerical example with a torque demand of 25 Nm, the angle is , is 1765 rpm, is 0.39 Wb-t, and is 0.58 Wb-t. The developed phase torques for the incoming and outgoing phases are 11.1 Nm and 13.9 Nm, respectively, and the rms phase current is 14.1 A.
Figure 6 shows the flowchart, which describes the operation steps in model #2.