Open access peer-reviewed chapter

# A Review of Classic Torque Control Techniques for Switched Reluctance Motors

By Marcelo Vinícius de Paula, Tárcio André dos Santos Barros and Pedro José Dos Santos Neto

Submitted: March 12th 2019Reviewed: August 30th 2019Published: September 9th 2020

DOI: 10.5772/intechopen.89450

## Abstract

In this chapter, the most relevant electronic torque control methods established in the current literature are addressed. The electronic torque control methods are divided into two technical topologies. The first one aims to control the average torque produced by the switched reluctance machine and is ideal for applications that require wide speed range of operation. The second topology enrolls the instantaneous torque production and is required in low torque ripple applications. Different torque control methods are addressed, for instance, direct average torque control, current profiling through torque sharing functions, direct torque control, and direct instantaneous torque control. Detailed information regarding the working principle and implementation of the methods is presented. Mathematical simulations are conducted in Matlab/Simulink® environment to elucidate the methods. Finally, a comparison of all addressed methods regarding the torque ripple minimization capabilities is presented.

### Keywords

• instantaneous torque control
• average torque control
• torque ripple minimization
• direct average torque control
• current profiling
• torque sharing function
• direct torque control
• direct instantaneous torque control

## 1. Introduction

Switched reluctance machines (SRMs) main advantages are their simple structure, without coils, and rare materials in the rotor, low cost, ruggedness, brushless operation, operation in high temperature environments, high reliability, fault tolerance, wide speed range of operation, high efficiency and high torque, and power density. These advantages enable SRMs as competitive to more traditional electric machines, such as induction motors, permanent magnet synchronous machines (PMSM), and DC machines in many applications [1, 2].

Besides the advantages, SRMs are strongly dependent in electronic converters for its drive. Also, automatic control of these machines is complex due to nonlinear characteristics. Furthermore, the efficiency and power factor of SRM are generally lower than those of PMSM as a result of higher losses in SRM. The switched reluctance machines have doubly salient structure and nonlinear characteristics, thus, these machines present elevated torque ripple and acoustic noise. These are the predominant disadvantages encountered in SRMs [3, 4].

The scientific community focuses on mitigating these disadvantages through electronic control or through structural changes in the machine. For instance, in [5], the authors analyze the effect of different frames and ribs structure topologies over the acoustic noise production. The methods that target the structural changes consist in manipulating the magnetic flux linkage in such way that the reluctance variation between cycles is smaller, thus resulting in shallower torque ripple. On the other hand, the speed range of operation is diminished because the torque per ampere ratio is also diminished.

In contrast, the electronic control methods aim to minimize the torque ripple by manipulating a set of control parameters, such as electric phase current, turn on angle, turn off angle, and excitation voltage, which regulate the machine operation. These parameters may be controlled directly or indirectly. The electronic control methods will be focused in this chapter.

In order to understand the working principle of the control methods, some other concepts must be addressed initially. In SRMs, the phase inductance (L) varies according to the rotor position (θ) because of the reluctance variation caused mostly by the air-gap changes as the rotor revolves. The inductance profile for a generic four-phase ideal SRM phase is highlighted in Figure 1(a). The pointed angles in the abscissa axle refer to the relative position between a point in the rotor and a point in the stator, where θasis the start of alignment position, θais the aligned position, θmsis the start of misalignment position, and θmis the misaligned position. These positions and their respective inductance are also pointed in the SRM physical structure in Figure 1(b). The aligned position occurs when the entirety of the rotor pole is bounded by the stator pole, and in this position, the inductance is at its maximum and the reluctance at its minimum. The misaligned position occurs when the entirety of the rotor pole is off the stator pole bounds [6].

Because of its working principle, SRMs require an electronic converter as a driver. The classic and most used topology is the asymmetric half bridge (AHB) [6, 7], which demand two power switches and two power diodes for its construction. There are several other proposed topologies in the literature for specific applications. In [8], the SRM phases are used as components of a charging station. In [9], the authors embed the SRM and AHB components as part of a converter with voltage boosting and power factor correction capabilities. The research presented in [10, 11] apply the SRM windings and AHB components for flexible energy conversion and multi-operation mode of the SRM. The authors in [12] propose an integrated SRM and drive system designed for hybrid energy storage systems composed of batteries and supercapacitors.

The schematic circuit for a three phase AHB is presented in Figure 2. This topology may be applied for any number of phases and enables operation in all four quadrant of the torque-speed plane. The AHB support independent voltage control for all the SRM phases. Three voltage levels are possible according to the power switches states.

Figure 3 shows all three possible voltage states that may be applied to the SRM phases using an AHB converter. In case both switches are conducting, positive voltage is applied to the phase terminals, and this state is known as positive state or magnetization (1) as depicted in Figure 3(a). If only one switch is conducting, null voltage will be applied to phase terminals and the electric current will flow through the circuit formed by the conducting switch, power diode, and SRM phase winding. This state is named free-wheel or null state (0) (Figure 3(b)). In case both switches are not conducting, negative voltage is applied to the SRM phase, characterizing the demagnetization or negative state (−1) as depicted in Figure 3(c).

The choice of the driving angles (θonand θoff) has great influence over the electronic control method applied. Some researches focus on the influence of these angles. The research presented in [13, 14] is focused on optimal angles for applications in current hysteresis control, while in [15], the authors focus the analysis on the voltage single pulse operation. The angle selection may have different objectives, for example, maximizing the efficiency, minimizing torque ripple or minimizing acoustic noise. The majority of the studies present optimizations aiming to minimize torque ripple [16, 17] and/or minimize the losses [18]. For this chapter, the focus will be on minimizing the torque ripple.

When the SRM is working at low speeds and current hysteresis control may be applied, at the beginning of the conduction period, the inductance value is almost constant, and consequently, a small amount of torque is produced. When the stator and rotor poles start to overlap, the torque production rises, thus when the rotor position overcomes the turn-off angle, a negative voltage must be applied to the phase terminals to guarantee that the electric current flowing through the phase reaches zero before the inductance starts to decrease. The magnetic flux linkage rise is constant because of the quazi-linear inductance variation. At the end of the commutation, the magnetic flux returns to zero because of the negative voltage applied. After that, zero voltage is applied to the phase.

On the other hand, when the SRM is working at high speeds, the back electromotive force (back EMF) is elevated, and the excitation voltage is not enough to guarantee appropriate hysteresis current control. Thus, voltage single pulse control is applied. At the beginning of the conduction interval, the electric current rises because there is voltage being applied to the phase terminals after the rotor position overcome the turn on angle θon. At certain point, the back EMF rises and the electric current can no longer increase, thus current hysteresis is inapplicable. The back EMF is directly proportional to the electric current and phase inductance. At certain speed, known as the base speed, the back EMF becomes the predominant term in the voltage equation (Eq. (1)), and then, the produced torque must be controlled by adjusting the θonand θoffangles. In Eq. (1), VDC, R, i, ω, and Lθiare the source voltage, the phase resistance, the phase electric current, the rotor angular speed and the angular position, and electric current dependent phase inductance, respectively.

VDC=Ri+Lθididt+dLθiE1

## 2. Torque production in switched reluctance machines

The SRM motion is directly linked to the minimum reluctance principle, that is, when the stator pole winding is energized, the closest rotor pole tends to move toward the energized stator pole, and during this process, the reluctance is minimized and the inductance is maximized. The coordination of this process leads to rotational acceleration. The electromagnetic torque produced by an SRM phase may be expressed as in Eq. (2), where Teis the electromagnetic torque.

Te=12i2LiθθE2

From Eq. (2), one can infer that: (1) the torque production does not depend on the direction of the electric current because of their quadratic relation; (2) the direction of the torque produced depends exclusively on the inductance derivative according to the rotor position. The inductance depends on the electric current and on the rotor position; and (3) the SRM is able to operate in all four torque-speed quadrants [19].

The total instantaneous torque produced by the SRM can be expressed as the sum of the individual torque produced by the phases as in Eq. (3), where Nis the number of phases and Te,kis the electromagnetic torque produced by the phase k. Each phase torque can be calculated as Eq. (2).

Ttot=k=1NTe,kE3

Thus, the average developed torque Ttot¯in a period is given by the time integral of the total torque produced in the same period as in Eq. (4), where τis one revolution period.

Ttot¯=1τ0τTtotdtE4

As aforementioned, due to the SRM working principle and driving methods, strong torque undulations occur. These undulations are known as torque ripples and one way of measuring this phenomenon is through the ratio between peak to peak torque and the average torque produced in one cycle as in Eq. (5), where maxTeand minTeare the maximum and minimum torque produced in one electric period, respectively.

Tripple=maxTeminTeTtot¯E5

## 3. Average torque control

The main aspect of average torque control (ATC) techniques is to keep a constant value of reference current throughout an electric cycle. Therefore, the major task of the controller is to adjust the control variables, for instance driving angles (θonand θoff) and electric current reference (i), for each operation point Tωin the torque-speed plane. The basic structure for ATC is presented in Figure 4.

There are two main methods for implementing the ATC [20]. The first consists of maintaining the driving angles while varying the electric current reference in each cycle, and the realization of this method is straightforward, but the operation region is limited and no efficiency optimization may be achieved. In the second method, both the electric current and the driving angles are controlled. Thus, although the control system becomes more complex and expensive, a good performance can be achieved, and the machine can operate in the entire speed range.

Studies focus on methods that enable real time variation of all control variables. This can be achieved online, through formulation and controllers or through genetic algorithms, or it can be achieved offline, using lookup tables [20, 21].

There are different combinations of control variables that attain the same control requirements for each operation point, such as minimizing the torque. It is possible, through cumbersome simulations, to create lookup tables containing the optimum values for each of these parameters according to the control objective.

Different optimization objectives may be conflicting, for example, minimizing torque ripple and maximizing efficiency. Thus, for a multi-objective optimization, a weighting process must be added to the optimization [22]. In this chapter, only torque ripple minimization is considered as objective.

Therefore, for the tables, creation diverse simulations were realized with the SRM modeling process developed in [23]. An open loop simulation of the system is implemented, while the control parameters and the load torque are varied covering all the operation region of the SRM (One Factor at a Time – OFAT optimization). After each simulation, the data are stored. The stored data incorporate the steady state rotational speed, the control variables used, and the torque ripple. By comparing the data, a combination of control parameters is selected, which enable the SRM to operate in the entire torque × speed plane Tωwith the minimum torque ripple. The selected parameters are stored in a lookup table.

Graphic representation of the lookup tables with the optimum values for the turn-on angle, turn-off angle, and reference current from the simulations are presented in Figure 5. As expected, both the turn-on and the turn-off angles must be advanced according to the rotational speed and load torque rise. On the other hand, the optimal electric current rises according to the increase in speed and load torque.

In some applications, the excitation voltage must also be considered as control parameter. The addition of one control parameter increases exponentially the number of simulations. To avoid this problem, Inderka [20] proposed a closed-loop average torque control method known as direct average torque control (DATC). The method is based on comparing the reference torque to the average torque produced. However, measuring the average torque with a torque transducer is expensive. To overcome this downside, the average torque may be estimated through the machine terminal quantities.

### 3.1 Average torque estimation

Considering SRMs with two or more phases, two phases may contribute to torque generation during commutation. Thus, the average torque estimation becomes dependent on the flux linkage derivative estimation, which may be calculated with terminal quantities, that is, phase current and phase voltage.

The torque production principle in SRM is similar to the torque production principle of solenoids. Assuming that the mechanical work incremented in a period is proportional to the rate of change in coenergy [6], the mechanical energy (Wmec) can be expressed as in Eq. (6), where ϕθiis the angular position and phase current dependent flux linkage. With this equation, the average torque (T¯) can be calculated as a function of the number of phases (Nk) and number of rotor poles (Nr), as in Eq. (7). Also, Eq. (1) can be rearranged to express the flux derivative (Eq. (8)).

Wmec=idϕθidtdtE6
T¯=NkNr2πWmecE7
dΦθidt=VDCRiE8

The average value of torque produced by the phases does not differ much from one another in each cycle due to phase symmetry, thus enabling to use only one average torque estimator. Note that in practice, the use of only one torque estimator reduces the cost of the project, but also can decrease the speed and quality of the controller. Figure 6(a) depicts the structure used to estimate the average torque of one SRM phase according to Eqs. (6)(8). Only the current and voltage quantities are required. For implementation, it is not necessary to calculate the derivative of flux, since it is given by the difference between the excitation voltage and the resistance voltage drop. Note that the integrator must be reset every time the current returns to zero in order to avoid the drifting problem during practical implementation. Also, the sample and hold logic must be used to hold the mechanical energy value at the instant the current reaches zero. Figure 6(b) presents the waveform of the quantities used to estimate the average torque produced in one cycle.

### 3.2 Direct average torque control

In total, the DATC method relies on three lookup tables, one for reference electric current of the phase ikTω, one for the turn-on angle θonTω, and one for the turn-off angle θoffTω. The block diagram structure for the DATC is presented in Figure 7. An external speed loop is responsible to generate the reference torque through a PI controller. A feedforward is added to the control system and generates a new reference torque. The lookup tables output the control parameter values. A hysteresis current regulator drives the AHB converter according to the control parameters.

Since the average torque is estimated only once per electric cycle, the DATC method requires low sampling ratio. This imply in lower cost for the controller mass production. On the other hand, the controller response to reference torque variations tend to be slow. To cope with that, one can use a torque estimator per SRM phase and recalculate the average torque at the beginning of each phase electric cycle. This solution increases the computational effort and cost of the system.

In order to verify the method, a computational model was developed in Matlab/Simulink® environment. The mathematical model of the SRM was developed as explained in [23]. The parameters for the SRM used for the simulation tests are presented in Appendix A. The base speed of the SRM is 954.93rpmor 96rad/s. The excitation voltage for the tests was maintained in 80V. A constant load torque of 2N.mwas used for the tests.

For this test and for the following tests, two types of simulation are performed. The first simulation (S1) is performed for a constant load of 2N.mand a reference speed of 30rad/s(below base speed). For the second simulation (S2), the load is maintained in 2N.mand the reference speed is increased to 130rad/s(above base speed). The same simulations setup will be used in future subsections for the other torque control methods presented, where they will be referred to as S1and S2, respectively.

The simulation results for the S1and S2setups using the DATC method are presented in Figures 8 and 9, respectively. The torque ripple in S1conditions is 69.4%, while in S2conditions, the torque ripple is 58.64%.Note that in S1, when the rotational speed is 30rad/s, the electric current is limited by the hysteresis controller, while in S2, when the rotational speed is 130rad/s, there is almost no chopping in the current waveform because the back EMF is high and preclude the current rise. The results show a high torque ripple in both simulation setups. This occurs because the DATC main objective is to control the average torque produced, and this way, the instantaneous torque is not controlled, leading to a high torque ripple response.

## 4. Instantaneous torque control

The biggest difference between ATC and instantaneous torque control (ITC) is that for the ITC methods, the electric current is regulated at every sampling period, while for ATC methods, the electric current is maintained constant through the entire excitation period. Therefore, the torque must be measured/estimated at every sampling period for ITC [24].

Another relevant factor is the operation region for each type of controller. The ATC methods may be applied through the entire speed range of operation. In contrast, the ITC methods have a shortened operation region and can only be applied while the controller is capable of controlling the current with accuracy. This becomes impractical when the speed grows above base speed because the back EMF forces the electric current in the opposite direction of the desired current.

Regarding the ITC methods, there are three main techniques present in the literature, which are the direct torque control (DTC), current profiling through torque sharing functions (TSF), and direct instantaneous torque control (DITC). Each of these techniques has its own derivations. Nonetheless, all of them depend on the instantaneous torque value. However, as in the case of the ATC, measuring the instantaneous torque may be expensive. Thus, to lower the price of the drive system, an instantaneous torque estimation technique may be employed.

### 4.1 Instantaneous torque estimation

Measuring the instantaneous torque is often expensive. Thus, sensorless torque estimation is preferable over torque transducer provided that the estimated torque accuracy is satisfactory.

The SRM model used in the simulations requires two lookup tables, one that returns the electric current according to flux linkage and electric current (iΦθ) and the other that returns the electromagnetic torque according to the electric current and rotor position (Tiθ).

If the model is accurate, the torque lookup table may be used to estimate the electromagnetic torque produced by the SRM given the rotor position and the electric current in the phase winding. The electromagnetic torque estimation may also be performed using other lookup tables. For example, the electromagnetic torque may be estimated through the magnetic flux linkage and the rotor position with a TΦθtable. This method also depends on a precise position sensor. If no position sensor may be used, the electromagnetic torque can also be estimated through the electric current and the voltage in the phase winding. With these two quantities, the flux linkage may be calculated online and used as input to a TiΦlookup table.

### 4.2 Current profiling technique through torque sharing functions

Given the relation between electromagnetic torque and electric current, it is possible to indirectly control the instantaneous torque produced through current control. On the other hand, the nonlinearities of the inductance profile must be accounted. Thus, to maintain the produced torque constant, the current profile presents a nonlinear waveform that varies according to the SRM characteristics, which is hard to interpolate.

The current profiling technique consists of controlling the electric current around a desired reference. The reference current is generated aiming to minimize the torque ripple or maximize the system efficiency. Some current profiling techniques are based on finite element analysis (FEM) [3, 4], and other methods are based on generating a reference electric current from desired torque profiles named torque sharing functions (TSF). The TSF may be generated and/or compensated online or offline. The TSF are responsible for dividing the produced torque equally through the phases and may present different formats. The most common formats are linear, sinusoidal, cubic, and exponential. In some cases, modifications are applied to these TSF by performing optimization techniques with different objectives.

Considering that the controller is fully capable of regulating the electric current, the torque production may be distributed among the phases in such a way that the sum of the torque produced by the phases is equal to the reference torque (T).

#### 4.2.1 Torque sharing functions

The reference torque and rotor position are used as inputs of the TSF. The TSF outputs the reference torque of each phase (Tk). The torque division follows the profile defined in Eq. (9), where θon, θoff, θal, and θovare the turn-on, turn-off, alignment, and overlapping angles, respectively. The functions that rule the profile while the phase is entering and leaving the commutation period are defined as fupθand fdownθ, respectively. The TSF is divided into five sectors according to angular position.

Tkθ=0,&0θθonsector1Tfupθ,&θonθθon+θovsector2T,&θon+θovθθoffθovsector3Tfdownθ,&θoffθovθθoffsector40,&θoffθθalsector5E9

In the literature, there are many functions to describe the rise and descend of the reference torque during commutation. The equations referring to the most commonly found TSF are presented hereafter along with the expected waveform for these functions. In the equations, the indexes lin, sin, cub, and expindicate the linear TSF, the sinusoidal TSF, the cubic TSF, and the exponential TSF, respectively.

The up and down functions for the linear TSF are presented in Eqs. (10) and (11). The reference torque waveform for this TSF is presented in Figure 10(a).

fuplinθ=1θovθθonE10
fdownlinθ=11θovθθoffE11

The equations for the sinusoidal TSF are presented following. The reference torque waveform is depicted in Figure 10(b).

fupsinθ=1212cosπθovθθonE12
fdownsinθ=12+12cosπθovθθoffE13

For the exponential TSF, the up and down functions are described by Eqs. (14) and (15), respectively, and the waveform of the expected reference torque is shown in Figure 10(c).

fupexpθ=1expθθon2θovE14
fdownexpθ=1fdownexpθθoff+θonE15

The last TSF is the cubic. The up and down functions are represented as followed in Eqs. (16) and (17), while the waveform of the reference torque is illustrated in Figure 10(d).

fuplinθ=1θovθθonE16
fdownlinθ=11θovθθoffE17

Observing Eqs. (10)(17), one can perceive that the firing angles play an important role on the TSF creation. Normally, these angles are kept constant throughout the operation range. Varying the firing angles enables better performance in different operation points. Thus, for this book chapter, an OFAT optimization [22] was performed, and the optimum angles for each operation point (T,ω) were stored in lookup tables. Note that the optimum firing angles for each TSF are different. Thus, two lookup tables are necessary for each TSF, totalizing eight lookup tables.

#### 4.2.2 Current profile creation

After obtaining the reference torque signals for each phase with the TSF, the reference current signals per phase (ik) must be determined. This procedure may be accomplished by inverting the Tiθlookup table into an iTθlookup table. In [25], the authors use the inverted TiθLUT to estimate electromagnetic torque and compare to a non-unity torque reference. The inverted TiθLUT is also applied in [26, 27], and in these cases, the contribution of the authors is to propose an alternative to commonly used TSFs. Alternatively, a nonlinear function of current as function of torque and rotor position can be used. The development of such function is cumbersome and requires a mathematical approach that models the inductance variation as function of electric current and rotor position. The analytical inversion function is expressed in Eq. (18), where L0pθis the inductance derivative for one SRM phase before saturation, that is, with low electric current, fθis the function that describe the ratio change between the produced torque and the electric current according to the saturation level. More detailed explanation about the derivation of Eq. (18) may be found in [28].

The iTθlookup table obtained after inverting the Tiθlookup table from the SRM model is presented in Figure 11. The parameters of the SRM used for the development of Eq. (18) and of the lookup table are presented in Appendix A.

ikTkθ=2TkL0pθfθ2+f2θ4+L0pθ2Tk1pE18

#### 4.2.3 Simulation results

The block diagram implemented in the Matlab/Simulink® environment is represented in Figure 12. The speed error is processed in a PI controller to create the reference torque (T), and the reference torque and the rotational speed are used as input of the optimum firing angles lookup tables. The optimum angles and the reference torque are used to create the phase reference torque (Tk) through the TSF. The phase reference torque is then transformed in a phase reference current. Now, a hysteresis current controller drives the AHB in order to maintain the electric current following the reference.

To verify the TSF with the best torque ripple reduction capability, simulations throughout the entire speed range of operation were done, in steps of 10rad/s.The results are presented in Figure 13. In the figure, a vertical line demarks the base speed (ωb=96rad/s). The presented TSF returns similar torque ripple. The higher disparity occurs at 170rad/s, where the exponential TSF have 19.59%less torque ripple than the cubic TSF. However, since the current profiling is an ITC method, it is expected that above base speed results are not acceptable and the interest speed range lies below the base speed. Below base speed, the sinusoidal TSF and the cubic TSF convey minor torque ripple. Thus, for future comparisons, the sinusoidal TSF will be employed because of its simple implementation.

The simulation results for sinusoidal TSF are presented in the following figures. The setups used in the simulations were the Simulation 1 (S1) and Simulation 2 (S2) setups. The results for S1and S2setups are presented in Figures 14 and 15, respectively. With S1setup, the current is accurately profiled, and the torque ripple is diminished, however, in S2, the back EMF is high, and the controller is not capable of profiling the current, thus the torque ripple is considerably higher. The torque ripple, turn-on angle, and turn-off angle for each TSF using both setups are presented in Table 1.

Turn-on angle (o)Turn-off angle (o)Torque ripple (%)
TSFS1S2S1S2S1S2
Linear25.623.043.442.513.2160.26
Sinusoidal25.323.543.542.57.9255.87
Exponential27.323.643.543.010.758.12
Cubic26.023.943.543.814.5253.24

### Table 1.

Control parameters for the different TSF using S1and S2setups.

When the speed rises, the error between reference torque and instantaneous torque produced by each phase rises as well. This phenomenon occurs for every TSF because there is a small error between the instantaneous current and the current reference at the phase energization and at the phase demagnetization. This current error is a consequence of the reduced maximum increase and decrease ratio of the electric current. When the speed rises, the back EMF also rises, diminishing the maximum current change ratio. Then, the current error is reflected in the electromagnetic torque produced, causing the torque ripple to rise.

### 4.3 Direct torque control

The direct control technique (DTC) for SRM is based on the DTC technique for induction machines from the 80s and present simple dynamic and robustness. The DTC may be adapted to SRMs with any number of phases. However, three or five phases’ adaptations are more recurrent in the literature than four phases’ adaptations. This technique does not control the electric current, and instead, it aims to control the torque directly by applying different voltage states to the phases. The voltage states are stored in vectors and are chosen according to torque and magnetic flux linkage errors [29, 30].

The DTC uses flux linkage and electromagnetic torque values as control variables. The torque and flux conditions are used to determine which voltage state vector should be applied to the AHB in order to control the instantaneous torque produced. Therefore, to better understand the working principle of this method, the relation between excitation voltage and flux linkage and how the produced torque can be controlled through magnetic flux linkage changes must be clarified.

The vector of magnetic flux linkage in the stator (λs) may be calculated as the time integral of the difference between the excitation voltage (VDC) and the voltage drop in the resistance (Rsis) minus the initial flux linkage value (λ0), as described in Eq. (19).

λs=0τVsRsisdtλ0E19

Generally, the voltage drop in the stator resistance is much smaller than the excitation voltage and then may be disregarded. In this case, the variation of flux linkage in the stator depends only on the voltage vector applied and can be described as in Eq. (20), indicating that a voltage vector applied to a phase causes a variation on the flux linkage of the same direction.

Δλs=VsΔtE20

To calculate the flux linkage vector in the stator, one must know the flux linkage from each SRM phase and parse them to their stationary component. For a three-phase SRM, the Clarke transform may be used to calculate the stationary components of the magnetic flux linkage vector (λαand λβ) through Eqs. (21) and (22), respectively.

λα=λaλbcos60o+λccos60oE21
λβ=λbsin60oλcsin60oE22

If the stationary components of flux linkage in the stator are know, it is possible to calculate the total flux linkage as:

λs=λα2+λβ2E23

Furthermore, the stationary components may be used to estimate the rotor position through Eq. (24), enabling a position sensorless application.

θ=arctanλβλαE24

Given the relation presented above, the DTC aims to maintain the flux linkage and torque values constrained in hysteresis bands. Variations in the produced torque are similar in direction to the variations in the flux linkage vector in relation to the rotor position, hence, for a torque increase, the flux linkage vector must be advanced from the rotor position vector, while for a torque decrease, the flux linkage vector must be delayed from the rotor position vector. These flux linkage variations are named flux acceleration and flux deceleration, respectively.

#### 4.3.1 Voltage state vectors definition

Consider a three-phase SRM 12/8poles with an AHB as the driver converter. As mentioned in the introduction, three different voltage states may be applied to each SRM phase. Thus, the voltage state vector (Ux) has three columns and one row as: Ux=uAuBuC, where uA, uB, and uCare the voltage states applied to phase A, phase B, and phase C, respectively. Now, to understand the vectors creation, first, it is important to understand the voltage sector division.

In the considered SRM, the phases are spaced in 120 electric degrees as in Figure 16. Tracing concentric axels over the phases, three axels are obtained. Each of them may be divided into a positive part, located at the axel portion that overlap the phase, and a negative part, on the other half of the axel, for example, for phase A, we obtain the A+portion and the Aportion. The axels divide the vector space in sixsectors of 60oeach, named S0, S1, S2, S3, S4,and S5as observed in Figure 16.

It is possible to define the state vectors for the DTC from the sectors. In total, since it is possible to apply 3 different voltage states per phase and there are 3 phases, there are 27 possible state combinations. However, some of the combinations must be avoided, such as keep the three phases turned on at the same time, remaining only six combinations of interest for this case. Each of the remaining vectors are located at the center of each sector and are named U0, U1, U2, U3, U4, and U5as in Figure 17. Each vector has three coordinates that represent the relation between the vector position and the axel portion of each phase, for example, the vector U0is located at the center of the sector S0. The sector S0is bounded by the positive portion of axel A(A+) and by the negative portion of axel C(C) and is not bounded by any portion of axel B, hence, U0is defined as 101.

Table 2 summarizes the voltage states applied to the SRM phase windings according to the vector. Note that in Figure 17, there are two circles, one more internal and one more external. These circumferences represent the inferior and superior hysteresis bands for the flux linkage, respectively. The flux linkage vector in the stator must be maintained within this bandwidth through the application of the state vectors.

VectorU0U1U2U3U4U5
States101011110101011110

### Table 2.

Voltage vectors to be applied in the AHB converter.

The choice of which vector must be applied depends on two conditions, one regarding the error between the reference torque and the instantaneous torque and other regarding the error between reference flux linkage and instantaneous flux linkage in the stator. Both conditions are implemented employing hysteresis controllers. The flux hysteresis controller receives the flux error and indicates if it necessary to increase or decrease the flux magnitude, while the torque hysteresis controller receives the torque error and indicates whether the flux vector must be accelerated or decelerated in order to control the torque production. The voltage vectors applied to make the stator flux linkage vector complete one clockwise cycle are detailed in Figure 18 (considering that it is necessary to raise the torque). In case it is necessary to reduce the electromagnetic torque, the stator flux linkage must be rotated counterclockwise.

Table 3 compiles the logic behind the voltage state vector choice according to the sector (stator flux linkage vector position) and to the torque and flux conditions. For better understanding, considering the case in which the flux linkage vector is located at the S4sector, it is demanded to escalate the torque (instantaneous torque reached the inferior hysteresis band) and to reduce flux linkage (stator flux linkage reached the superior hysteresis band, that is, the external circle of Figure 18). In such case, the vector U0must be applied in order to keep the acceleration of the flux vector (increase torque) and lessen the amplitude (decrease flux), as depicted in Figure 18.

T&λcondition
SectorAngular positionλTλTλTλT
S0060oU1011U5110U2110U4011
S160120oU2110U0101U3101U5110
S2120180oU3101U1011U4011U0101
S3180240oU4011U2110U5101U1011
S4240300oU5011U3101U0101U2110
S5300360oU0101U4011U1011U3101

### Table 3.

DTC switching table for an three-phase SRM.

#### 4.3.2 Variable magnetic flux linkage reference

The DTC requires a flux linkage reference for the hysteresis control realization. In the literature, diverse researches apply a constant reference throughout the entire operation speed range [29, 30]. However, observing the characteristics of the SRM, it is perceptible that for high values of speed (above base speed), where a high electric current is required, the flux linkage behavior becomes nonlinear according to the saturation level causing the torque ripple to increase according to the rotational speed increase. To overcome this drawback, a variable flux linkage reference may be used.

In certain operation speed, the electromagnetic torque developed starts to present drops, causing the torque ripple growth according to speed. This speed is defined as the critical speed (ωcrit). To estimate the critical speed, an open loop simulation with constant operation speed must be performed while observing the electromagnetic torque waveform. According to the speed rise, it is possible to note dips in the instantaneous torque waveform [29]. For the SRM used in this book chapter studies, the critical speed (ωcrit) is approximately 37.5rad/s.

The flux linkage reference used here is divided into three cases, one for speeds below the critical speed, the second for speeds between the critical and the base speed, and the third for speeds above the base speed. For the first case, the flux reference is obtained with a PI controller that processes the speed error. For the second case, a function that describes the optimum flux decaying according to the speed increase is implemented (λdecω). In the third case, the flux linkage reference is maintained constant at the nominal value.

The decaying flux linkage function is established using an OFAT optimization to find the optimum flux linkage values for each operation speed. Then, the optimum values are interpolated to create the λdecθfunction. The function is a sum of sinusoids presented in Eq. (25). The coefficients used in the equation are displayed in Table 4.

CoefficientsA1B1C1A2B2C2
Values0.1280.028260.44050.049070.065950.02546
CoefficientsA3B3C3A4B4C4
Values0.018760.13141.890.010240.016584.76

### Table 4.

Ai, Bi, Ci,and Dicoefficients for the λdecθfunction.

λdecω=i=14AisinBiω+CiE25

Thus, the flux linkage reference (λω) for the entire speed range of the DTC operation is described as following:

λω=PIcontrolleroutput,0ω<ωcritλdecω,ωcritω<ωb0.0875,ωbωE26

#### 4.3.3 Simulation results

The block diagram for the DTC implementation is shown in Figure 19. The error between reference and instantaneous speed is processed through a PI controller to generate the reference torque. The flux linkage reference is according to Eq. (26). The torque reference is compared to the instantaneous torque, and the error is processed by a hysteresis controller to determine the torque condition. The same happens for the flux linkage to return the flux condition. Clarke transform is employed to estimate position and to calculate the total flux linkage in the stator. The sector selector block informs the sector according to the estimated position. The sector and the conditions are then used to choose the voltage state vector according to Table 3.

The DTC controls the electromagnetic torque produced and the flux linkage in the stator. Considering the stationary frame (αβ-reference frame), the phase of the flux linkage vector must be accelerated or decelerated to produce torque, while the magnitude of the vector must be maintained constant. Plotting the αflux component (λα) against the βflux component (λβ) should result in a circular profile as depicted in Figure 20.

The results obtained using the DTC with the S1and S2conditions are presented in Figures 21 and 22, respectively. The figures contain the waveforms of the flux linkage of each phase, instantaneous and reference magnitude of the flux linkage in the stator (in stationary frame), and the instantaneous and reference electromagnetic torque. The torque ripple returned in both conditions is 24.41%, for the S1simulation, and 72.84%, for the S2simulation.

### 4.4 Direct instantaneous torque control

The direct instantaneous torque control (DITC) was first envisioned by Inderka [31]. The closed-loop control system uses a combination of double band current hysteresis controllers to generate the driving signals for each SRM phase. Differently from the current profiling through TSF, the DITC controls the torque directly.

The method dynamically divides the torque between two phases during commutation. The control system coordinates the AHB switch states according to an enabled conduction condition signal and to the instantaneous torque produced by each phase.

For better understanding of the DITC, first, some concepts must be clarified. In Figure 23, the electric currents for the three SRM phases are depicted. Two different intervals may be defined. One is the commutation interval, defined as the period in which two phases are simultaneously in conduction and both phases are responsible for the torque production. The other one is the single conduction interval, defined as the period in which only one phase is in conduction and is responsible for the entire torque production.

Furthermore, during the commutation interval, the two phases in conduction may be defined as incoming phase and outgoing phase. The outgoing phase is the phase that is in the end of the conduction period and is leaving commutation, while the incoming phase is the phase that is entering the conduction period.

The DITC classifies the phases, indicating whether the phase is enabled to enter conduction. The phase position must be in the interval between the turn-on angle and the turn-off angle (θonθθoff) to be considered enabled to enter conduction. If this condition is satisfied by only one phase, single conduction is characterized, and the control system changes the voltage state of this phase to control the torque produced. If two phases satisfy the condition, phase commutation is characterized, and the voltage states of both phases must be individually controlled in such way that the sum of the produced torque by the phases equals to the total reference torque. The control strategies used in both cases are minutely explained hereafter.

#### 4.4.1 State machines for direct instantaneous torque control

During commutation, there are two independent control logics simultaneously implemented. One for the incoming phase and one for the outgoing phase. The incoming phase is preferred for the torque production because the outgoing phase has already started the demagnetization process. Thus, the outgoing phase must be energized only if the incoming phase is not capable of producing the required reference torque by itself.

The following three state machines elucidate the working principles of DITC. The state machine in Figure 24(a) regulates the operation of the incoming phase, and the state machine in Figure 24(b) regulates the operation for the outgoing phase during commutation. Also, any phase that is in single conduction mode follows the logic within the state machine in Figure 24(c). The conditions for voltage state change are indicated in the state machines, where Terris the error between instantaneous phase torque and reference phase torque, thssis the superior torque limit, thsmis the superior medium torque limit, thimis the inferior medium torque limit, and thiiis the inferior torque limit.

The waveforms of the total torque produced are presented in Figure 25 along with the current of two commutating phases. In the figure, two regions are indicated during phase commutation. In region (1), the electromagnetic torque produced reaches the external inferior limit (thii), and then, the outgoing phase, which was in free-wheeling, is magnetized to augment the torque production, when the torque reaches back the thimlimit, the phase state goes back to free-wheeling. In region (2), the torque reaches the superior torque band thss, thus the outgoing phase is demagnetized to reduce the torque production, and when the torque reaches back the thsmlimit, the phase enter the free-wheeling state again.

Table 5 summarizes the working principle of the implemented CDTI. In this table, the voltage states (1, 0, and −1) are used to indicate the action. The “-” signal in the cells indicate that the phase is not accounted for torque production. The switching rules in Table 5 guarantee three conditions aiming to achieve low torque ripple:

• Avoid hard-chopping to minimize switching losses and reduce the torque ripple;

• Prioritize the incoming phase for the torque production so that the electric current flowing in the outgoing phase reaches zero before the inductance profile inclination becomes negative, thus avoiding negative torque production;

• Keep the incoming phase and the single conduction phase in the free-wheeling state for as long as possible to diminish the copper losses and the torque ripple.

Phase conditionState applied to phase
k1kk+1k1kk+1
000
10010
11010110
010110
01110110
001110

### Table 5.

Switch voltage state logic for DITC in a three-phase SRM.

#### 4.4.2 Simulation results

The simulated system with DITC is depicted in Figure 26. The speed error is processed through a PI controller to return the reference torque. The reference torque is compared to the total instantaneous torque produced. The output error is used to decide the voltage state applied to the phase according to the switching state machines. The logic used depends on a condition signal generated for each phase separately. To synthesize the phase condition, the rotor position and the firing angles are required. The firing angles used are provided by a lookup table and depend on the SRM operation point (T,ω) as for the previous methods introduced.

The results obtained for the DITC in S1and S2conditions are presented in Figures 27 and 28, respectively. The waveforms presented are the electric current in the phases, individual electromagnetic torque produced by each phase, and the instantaneous and reference total electromagnetic torque produced. The turn-on angle, turn-off angle, and torque ripple for both simulations are disclosed in Table 6. When the speed is higher than the base speed, the current enters in the negative torque generation zone and causes torque dips as observed in Figure 28.

SimulationTurn-on angle (o)Turn-off angle (o)Torque ripple (%)
S125.743.910.43
S223.240.559.43

### Table 6.

DITC simulation results for S1and S2conditions.

## 5. Results and discussion

For a better comparison of the methods behavior and performance throughout the SRM speed range of operation, simulations are performed from 10rad/sup to the maximum possible speed with each method (even above nominal speed) at every 20rad/s. The ripple factor at each speed for all methods presented in this chapter is calculated through Eq. (5), and the results are stored. The obtained values are grouped in Figure 29 along with the results for the case where only a simple speed control is employed. The base speed (ωb) is indicated as a vertical dashed line. The base speed for the studied machine is approximately 96rad/s.

Remark that the ITC methods present better performance than the ATC method only when the electric current may be accurately controlled. Above 150rad/s, the DATC presents better torque regulation capability, while below this speed, the DITC returned smaller torque ripple. Note that, below 30rad/s, the current profiling with the sinusoidal TSF presented the lowest torque ripple; however, in all other speeds, the DITC surpassed the current profiling.

When no torque ripple suppression method is applied, the ripple level reaches up to 570.1%at the low speed of 10rad/s. According to the speed growth, the ripple is naturally diminished as an effect of the inertia moment. Note that, when any torque control method is employed, the torque ripple is diminished.

The preferred TSF among the studied functions are the sinusoidal one because of its easy implementation and good performance in the operation speed range of interest. When the sinusoidal TSF is compared to the other methods, it has better performance in a narrow speed range (below 30rad/s) and then it is surpassed by the DITC. The DITC is the fittest ITC method since it naturally divides the torque production among phases and directly controls the total torque produced by the phases. However, as an ITC method, the DITC malfunctions in high speeds (above base speed), that is, torque dips appear as the speed increases until the instantaneous torque is no longer controlled. At above base speeds, the DATC method is preferred, because it considers only the average torque as target variable of the control.

## 6. Conclusions

In this chapter, different torque control methods were presented aiming to clarify which method is preferred for each operation point of the machine. One ATC method entitled DATC was presented, along with different ITC methods. The ITC methods addressed were the direct torque control (DTC), the current profiling through torque sharing function (TSF), and the direct instantaneous torque control (DITC). The most common TSF were used for the comparison, which are the linear, sinusoidal, exponential, and cubic TSF.

Simulations with each method were performed throughout the entire SRM speed range of operation. The DITC is the method that returned better torque ripple minimization capabilities bellow base speed, while above base speed, the DATC was the fittest method. With the presented simulations, the reader is able to understand how the SRM behavior changes according to the control method employed and operation point.

Some modifications and/or adaptations may be applied to the methods in order to improve the performance of the control system toward the chosen objective. For instance, the DATC method uses three LUT, which may be optimized according to the application and control objective. This feature makes this method very flexible.

The current profiling through torque sharing functions may be improved by creating torque profiles that account the difference between magnetization and demagnetization dynamics and how much each phase contributes to total torque produced. The DTC method can be improved by augmenting the number of sectors considered in the control strategy. The DITC technique may be modified by inserting variable turn-on and turn-off angles that can be optimized for different control objectives, such as minimize torque ripple, maximize efficiency or minimize acoustic noise, which also makes this method deeply adaptable.

## Acknowledgments

This book chapter was a result of researches supported by the São Paulo Research Foundation, FAPESP under Grant 17/21640-9 and by the Brazilian National Council for Scientific and Technological Development - CNPq under Grant 142597/2018-0.

## Appendix: switched reluctance motor parameters

This appendix presents the parameters of the SRM used in this chapter.

• Nominal power: 1.5kW;

• Nominal speed: 1500rpm;

• Number of poles: 12/8(stator/rotor);

• Operation voltage: 3080V;

• Winding resistance: 0.3Ω;

• Inertia moment (obtained through deceleration test): 0.002kgm2;

• Minimum phase inductance: 1.675mH;

• Maximum phase inductance: 13.88mH.

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Marcelo Vinícius de Paula, Tárcio André dos Santos Barros and Pedro José Dos Santos Neto (September 9th 2020). A Review of Classic Torque Control Techniques for Switched Reluctance Motors, Modelling and Control of Switched Reluctance Machines, Rui Esteves Araújo and José Roberto Camacho, IntechOpen, DOI: 10.5772/intechopen.89450. Available from:

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