**Abstract**

The direct torque control (DTC) was proposed as an alternative to the vector control in the middle of 1980s for AC machine control. This strategy bases on the direct determination of inverter switching states and offers a simpler scheme and less sensitivity to machine parameters. However, the variable switching frequency of DTC causes high flux and torque ripples which lead to an acoustical noise and degrade the performance of the control technique, especially at low-speed regions. In the objective of improving the performance of DTC for the induction motor, this work addresses the most important points concerning this issue. The reduction of high ripples, which are the major drawbacks, by applying a constant switching frequency using the space vector modulation (SVM) has been done firstly. Then, fuzzy DTC-SVM strategy with adaptive fuzzy-PI speed controller has been proposed. The results of all the discussed aspects of this chapter have been obtained by numerical simulation using MATLAB/Simulink software.

**Keywords:** induction motor, conventional direct torque control (DTC), space vector modulation, DTC-SVM, fuzzy-PI speed controller

#### **1. Introduction**

Advanced control of electrical machines requires an independent control of magnetic flux and torque. For that reason it was not surprising that the DC machine played an important role in the early days of high-performance electrical drive systems, since the magnetic flux and torque are easily controlled by the stator and rotor current, respectively. The introduction of field oriented control meant a huge turn in the field of electrical drives, since with this type of control the robust induction machine can be controlled with a high performance. Later in the 1980s, a new control method for induction machines was introduced: The direct torque control (DTC) method. It was proposed by Takahashi and Depenbrock [1, 2]. It bases on the direct selecting of the switching states to control the voltage source inverter (VSI) through a switching look-up table. Due to the limits of the conventional DTC strategy, especially the high torque and flux ripples problem, various control structures are presented to improve the performances of control, [3, 4]. The constant switching frequency DTC using the space vector modulation (DTC-SVM) is a well discussed solution; in order to improve the DTC-SVM performances, hysteresis comparators of electromagnetic torque and stator flux have been

replaced by PI controllers, [5, 6]. The main drawbacks of DTC-SVM using PI controllers are the sensitivity of the performances to the system-parameter variations and the inadequate rejection of external disturbances and load changes [7–11]. To cope with this disadvantage, it is suggested to replace the conventional regulators used for the speed control, flux, and electromagnetic torque by intelligent controllers by fuzzy logic to make the controls more robust against the disturbances of the parameters of the machine. The aim of this chapter is to design and compare three strategies for the direct torque control (DTC) of induction motor (IM). The first method is a conventional direct torque control (C-DTC) where the torque and the flux are regulated by the hysteresis controllers. The second one is direct torque control by space vector modulation strategy (SVM-DTC) where the torque and flux are regulated by PI controllers. The third one is fuzzy SVM-DTC with adaptive fuzzy-PI speed controller where the torque and flux are regulated by fuzzy logic controllers. The main feature of the proposed (SVM-DTC) strategy is the reduction of torque and flux ripples.

#### **2. Model of induction motor dedicated for direct torque control**

The mathematical model of induction motor can be described in the stator fixed reference frame (*α,β*) (stationary frame) by assuming the rotor and the stator flux as state variables:

$$\frac{dX}{dt} = AX + BU\tag{1}$$

**3. Two-level voltage source inverter (VSI) model**

*DOI: http://dx.doi.org/10.5772/intechopen.90199*

*Direct Torque Control Strategies of Induction Machine: Comparative Studies*

frequency.

diodes are so helpful.

three-phase load.

a logical value of either 0 or 1.

**4. Principles of direct torque control**

*Three-phase VSI fed star-connected induction machine.*

commutation period.

**19**

**Figure 1.**

Two-level three-phase voltage source inverter (VSI) is considered as a mature technology and becoming an industrial standard for the demand for energy saving. The output phase voltages are produced by the rectifier (Vdc) is delivered to the inverter input, which, thanks to controlled transistor switches, converts this voltage to three-phase AC voltage signal with wide range variable voltage amplitude and

The type of the used switches depends on the power of the inverter and switching frequency. In the most applications, IGBT transistors with antiparallel

**Figure 1** shows the two-level three-phase voltage source inverter (VSI) with six transistor switches, S1–S6, and a dc constant voltage source Vdc connecting a

*Vdc Sa* þ *Sbe*

Direct torque control principle was introduced in the late 1980s by [1, 2]. It achieves a decoupled control of the stator flux and the electromagnetic torque in the stationary frame (α, β), and it allows induction machines to have an accurate and fast electromagnetic torque response. It uses a switching table for the selection of an appropriate voltage vector. The selection of the switching states is related directly to the variation of the stator flux and the torque of the machine. Hence, the selection is made by restricting the flux and torque magnitudes within two hysteresis bands. Those controllers ensure a separated regulation of both of these quantities [12–14]. The inputs of hysteresis controllers are the flux and the torque errors as well as their outputs determine the appropriate voltage vector for each

where *Sa*, *Sb*, and *Sc* are three-phase inverter switching functions, which can take

*j* 2*π* <sup>3</sup> þ *Sce j* 4*π* 3 h i (3)

The model of two-level voltage inverter is shown in **Figure 1**.

The voltage vector is generated by the following equation:

ffiffiffi 2 3 r

*V* ! *<sup>s</sup>* ¼

with

$$X = \begin{bmatrix} \phi\_{\infty} \\ \phi\_{\beta\delta} \\ \phi\_{r\alpha} \\ \phi\_{r\beta} \end{bmatrix}, A = \begin{bmatrix} -\frac{R\_{s}}{\sigma L\_{s}} & 0 & \frac{MR\_{s}}{\sigma L\_{r}L\_{s}} & 0 \\ 0 & -\frac{R\_{s}}{\sigma L\_{s}} & 0 & \frac{MR\_{s}}{\sigma L\_{r}L\_{s}} \\ \frac{MR\_{r}}{\sigma L\_{r}L\_{s}} & 0 & -\frac{R\_{r}}{\sigma L\_{r}} & -\alpha \\ 0 & \frac{MR\_{r}}{\sigma L\_{r}L\_{s}} & \alpha & -\frac{R\_{r}}{\sigma L\_{r}} \end{bmatrix}, B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{bmatrix} \text{ and }$$

$$U = \begin{bmatrix} v\_{\alpha\alpha} \\ v\_{\beta\delta} \end{bmatrix} \text{ where}$$

$$\begin{aligned} \phi\_{\infty}, \phi\_{\delta\beta}, \phi\_{\alpha\gamma}, \phi\_{r\delta} \text{ are stator and rotor flux components.}\\ R\_{s}, R\_{r} \text{ are stator and rotor resistance.}\\ \Delta\_{s}, L\_{r} \text{ are star and rotor inductance.}\\ \Delta\_{s}, L\_{r} \text{ are star and rotor inductance.} \end{aligned}$$

*M* is the mutual stator-rotor inductance.

*<sup>σ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>M</sup>*<sup>2</sup> *LrLs* is the Blondel's coefficient.

*<sup>ω</sup>* is the machine speed (*<sup>ω</sup>* <sup>¼</sup> <sup>p</sup><sup>Ω</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>s</sup>‐*ω*<sup>r</sup> and *<sup>p</sup>* is the pole pair number).

The rotor motion can be described by:

$$J\frac{d\Omega}{dt} = T\_{em} - T\_L - f\,\Omega\tag{2}$$

where *J* is the motor inertia,*Tem* is the electromagnetic torque,*TL* is the load torque, and *f* is the friction coefficient.

*Direct Torque Control Strategies of Induction Machine: Comparative Studies DOI: http://dx.doi.org/10.5772/intechopen.90199*

#### **3. Two-level voltage source inverter (VSI) model**

replaced by PI controllers, [5, 6]. The main drawbacks of DTC-SVM using PI controllers are the sensitivity of the performances to the system-parameter variations and the inadequate rejection of external disturbances and load changes [7–11]. To cope with this disadvantage, it is suggested to replace the conventional regulators used for the speed control, flux, and electromagnetic torque by intelligent controllers by fuzzy logic to make the controls more robust against the disturbances of the parameters of the machine. The aim of this chapter is to design and compare three strategies for the direct torque control (DTC) of induction motor (IM). The first method is a conventional direct torque control (C-DTC) where the torque and the flux are regulated by the hysteresis controllers. The second one is direct torque control by space vector modulation strategy (SVM-DTC) where the torque and flux are regulated by PI controllers. The third one is fuzzy SVM-DTC with adaptive fuzzy-PI speed controller where the torque and flux are regulated by fuzzy logic controllers. The main feature of the proposed (SVM-DTC) strategy is the reduction

*Direct Torque Control Strategies of Electrical Machines*

**2. Model of induction motor dedicated for direct torque control**

*dX*

<sup>0</sup> *MRs οLrLs*

<sup>0</sup> � *Rr*

*σLr*

*<sup>ω</sup>* is the machine speed (*<sup>ω</sup>* <sup>¼</sup> <sup>p</sup><sup>Ω</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>s</sup>‐*ω*<sup>r</sup> and *<sup>p</sup>* is the pole pair number).

where *J* is the motor inertia,*Tem* is the electromagnetic torque,*TL* is the load

The mathematical model of induction motor can be described in the stator fixed reference frame (*α,β*) (stationary frame) by assuming the rotor and the stator flux

0

�*ω*

*σLr*

<sup>0</sup> *MRs οLrLs*

*<sup>ω</sup>* � *Rr*

*d t* <sup>¼</sup> *AX* <sup>þ</sup> *BU* (1)

, *B* ¼

*dt* <sup>¼</sup> *Tem* � *TL* � *<sup>f</sup>* <sup>Ω</sup> (2)

of torque and flux ripples.

as state variables:

*ϕs<sup>α</sup> ϕsβ ϕr<sup>α</sup> ϕr<sup>β</sup>* 3 7 7 7 5, *<sup>A</sup>* <sup>¼</sup>

where

� *Rs σLs*

*Rs, Rr* are stator and rotor resistance. *Ls*, *Lr* are stator and rotor inductance. *M* is the mutual stator-rotor inductance.

The rotor motion can be described by:

torque, and *f* is the friction coefficient.

*MRr οLrLs*

<sup>0</sup> � *Rs*

<sup>0</sup> *MRr οLrLs*

*ϕs<sup>α</sup>*, *ϕs<sup>β</sup>*, *ϕr<sup>α</sup>*, *ϕr<sup>β</sup>* are stator and rotor flux components.

*<sup>J</sup> <sup>d</sup>*<sup>Ω</sup>

*LrLs* is the Blondel's coefficient.

*σLs*

with

*X* ¼

*<sup>U</sup>* <sup>¼</sup> *vs<sup>α</sup> vsβ* � �

**18**

*<sup>σ</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>M</sup>*<sup>2</sup>

Two-level three-phase voltage source inverter (VSI) is considered as a mature technology and becoming an industrial standard for the demand for energy saving. The output phase voltages are produced by the rectifier (Vdc) is delivered to the inverter input, which, thanks to controlled transistor switches, converts this voltage to three-phase AC voltage signal with wide range variable voltage amplitude and frequency.

The type of the used switches depends on the power of the inverter and switching frequency. In the most applications, IGBT transistors with antiparallel diodes are so helpful.

The model of two-level voltage inverter is shown in **Figure 1**.

**Figure 1** shows the two-level three-phase voltage source inverter (VSI) with six transistor switches, S1–S6, and a dc constant voltage source Vdc connecting a three-phase load.

The voltage vector is generated by the following equation:

$$\overrightarrow{\mathbf{V}}\_{s} = \sqrt{\frac{2}{3}} \mathbf{V}\_{dc} \left[ \mathbf{S}\_{a} + \mathbf{S}\_{b} \mathbf{e}^{\beta \frac{\mathbf{r}}{\beta}} + \mathbf{S}\_{c} \mathbf{e}^{\beta \frac{\mathbf{r}}{\beta}} \right] \tag{3}$$

where *Sa*, *Sb*, and *Sc* are three-phase inverter switching functions, which can take a logical value of either 0 or 1.

**Figure 1.** *Three-phase VSI fed star-connected induction machine.*

#### **4. Principles of direct torque control**

Direct torque control principle was introduced in the late 1980s by [1, 2]. It achieves a decoupled control of the stator flux and the electromagnetic torque in the stationary frame (α, β), and it allows induction machines to have an accurate and fast electromagnetic torque response. It uses a switching table for the selection of an appropriate voltage vector. The selection of the switching states is related directly to the variation of the stator flux and the torque of the machine. Hence, the selection is made by restricting the flux and torque magnitudes within two hysteresis bands. Those controllers ensure a separated regulation of both of these quantities [12–14]. The inputs of hysteresis controllers are the flux and the torque errors as well as their outputs determine the appropriate voltage vector for each commutation period.

#### **5. Estimation of stator flux and electromagnetic torque**

#### **5.1 Control of stator flux**

Basing on the induction motor model in stationary frame, the stator flux equation can be expressed as follows [15–19]:

$$\begin{cases} \phi\_{sa} = \int (v\_{sa} - R\_i i\_{sa}) dt \\\\ \phi\_{s\beta} = \int (v\_{s\beta} - R\_i i\_{s\beta}) dt \end{cases} \tag{4}$$

The logical outputs of the flux controller are defined as:

*Direct Torque Control Strategies of Induction Machine: Comparative Studies*

where *hϕ*<sup>s</sup> is hysteresis band of stator flux.

*Two-level hysteresis comparator for stator flux control.*

*DOI: http://dx.doi.org/10.5772/intechopen.90199*

flux and the actual estimated value:

**5.2 Control of electromagnetic torque**

relation:

**Figure 3.**

where:

**21**

*p* is the number of poles pairs.

Cflx ¼ 1 if Δ*ϕ*s>*hϕ*<sup>s</sup>

<sup>Δ</sup>*ϕ*<sup>s</sup> <sup>¼</sup> *<sup>ϕ</sup>*<sup>∗</sup> *s* � � �

During one sampling period, the rotor flux vector is supposed invariant.

*:* <sup>1</sup> 1 þ *jσωTr*

Finally, between the modules of the two flux vectors, we have the following

*M σLsLr*

*M σLsLr* *ϕs:ϕ*<sup>∗</sup> *r*

*:* <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ð Þ *σωTr* <sup>2</sup>

The rotor and stator flux vectors are linked by the following relation:

*<sup>ϕ</sup><sup>r</sup>* <sup>¼</sup> *<sup>M</sup> Ls*

*<sup>ϕ</sup><sup>r</sup>* j j <sup>¼</sup> *<sup>M</sup> Ls*

The general expression of electromagnetic torque is given by:

*Tem* ¼ *p*

*Tem* ¼ *p*

The angle between these two vectors is given by:

Cflx ¼ 0 if Δ*ϕ*<sup>s</sup> ≤ � *hϕ*<sup>s</sup>

The stator flux error is defined by the difference between the reference value of

� � *ϕ*<sup>s</sup> j j (9)

*ϕ<sup>s</sup>* (10)

*δ* ¼ Arctan *g*ð Þ *σωTr* (11)

<sup>q</sup> *<sup>ϕ</sup><sup>s</sup>* j j (12)

� � (13)

*ϕs:ϕr*Sinð Þ*δ* (14)

(8)

Considering that the control of the switches of the inverter is done by control period (or sampling) *Te* and that at each of these periods the states *Sa*, *Sb*, and *Sc* are kept constant, the method of numerical integration of the rectangles makes it possible to obtain an expression of the sample *k+1* of the stator flux in the following form:

$$\begin{cases} \phi\_{sa}(k+\mathbf{1}) = \phi\_{sa}(k) + \left(v\_{sa}(k) - R\_{s}\dot{\imath}\_{sa}(k)\right)T\_{\epsilon} \\\\ \phi\_{s\boldsymbol{\beta}}(k+\mathbf{1}) = \phi\_{s\boldsymbol{\beta}}(k) + \left(v\_{s\boldsymbol{\beta}}(k) - R\_{s}\dot{\imath}\_{s\boldsymbol{\beta}}(k)\right)T\_{\epsilon} \end{cases} \tag{5}$$

A vector inscription of this expression can be given by:

$$
\overrightarrow{\phi}\_s(\mathbf{k}+\mathbf{1}) = \overrightarrow{\phi}\_s(\mathbf{k}) + \left(\overrightarrow{V}\_s(\mathbf{k}) - R\_s \overrightarrow{I}\_s(\mathbf{k})\right) T\_e \tag{6}
$$

We can neglect the stator resistance voltage drop compared to *Vs* for high speed regions. Then Eq. (6) can be written as:

$$
\overrightarrow{\phi}\_s(\mathbf{k}+\mathbf{1}) = \overrightarrow{\phi}\_s(\mathbf{k}) + \overrightarrow{V}\_s(\mathbf{k})T\_\varepsilon \tag{7}
$$

Eq. (7) means that the stator flux can be changed by the application of stator voltage during a time *k*. The stator flux vector's extremity moves in direction given by the voltage vector and making a circular trajectory.

A two-level hysteresis comparator is used for flux regulation. It allows to drop easily the flux vector extremity within the limits of the two concentric circles with close radius. The choice of the hysteresis bandwidth depends on the switching frequency of the inverter **Figures 2** and **3**.

**Figure 2.** *Evolution of stator flux vector in the complex plan.*

*Direct Torque Control Strategies of Induction Machine: Comparative Studies DOI: http://dx.doi.org/10.5772/intechopen.90199*

**Figure 3.** *Two-level hysteresis comparator for stator flux control.*

**5. Estimation of stator flux and electromagnetic torque**

(

A vector inscription of this expression can be given by:

!

*<sup>s</sup>*ð Þ¼ *k* þ 1 *ϕ*

*<sup>s</sup>*ð Þþ *k V* !

!

Eq. (7) means that the stator flux can be changed by the application of stator voltage during a time *k*. The stator flux vector's extremity moves in direction given

A two-level hysteresis comparator is used for flux regulation. It allows to drop easily the flux vector extremity within the limits of the two concentric circles with close radius. The choice of the hysteresis bandwidth depends on the switching

We can neglect the stator resistance voltage drop compared to *Vs* for high speed

*<sup>s</sup>*ð Þþ *k V* !

*<sup>s</sup>*ð Þ� *k Rs I*

� �

! *<sup>s</sup>*ð Þ*k*

*<sup>s</sup>*ð Þ¼ *k* þ 1 *ϕ*

*ϕ* !

by the voltage vector and making a circular trajectory.

Basing on the induction motor model in stationary frame, the stator flux equa-

*<sup>ϕ</sup>s<sup>α</sup>* <sup>¼</sup> <sup>Ð</sup> *vs<sup>α</sup>* � *Rsi* ð Þ *<sup>s</sup><sup>α</sup> dt*

� �*dt*

(4)

(5)

*Te* (6)

*<sup>s</sup>*ð Þ*k Te* (7)

*<sup>ϕ</sup>s<sup>β</sup>* <sup>¼</sup> <sup>Ð</sup> *vs<sup>β</sup>* � *Rsis<sup>β</sup>*

Considering that the control of the switches of the inverter is done by control period (or sampling) *Te* and that at each of these periods the states *Sa*, *Sb*, and *Sc* are kept constant, the method of numerical integration of the rectangles makes it possible to obtain an expression of the sample *k+1* of the stator flux in the following

> *ϕs<sup>α</sup>*ð Þ¼ *k* þ 1 *ϕs<sup>α</sup>*ð Þþ *k vsα*ð Þ� *k Rsi* ð Þ *<sup>s</sup><sup>α</sup>*ð Þ*k Te <sup>ϕ</sup>s<sup>β</sup>*ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>ϕ</sup>s<sup>β</sup>*ð Þþ *<sup>k</sup> vsβ*ð Þ� *<sup>k</sup> Rsisβ*ð Þ*<sup>k</sup>* � � *Te*

**5.1 Control of stator flux**

form:

**Figure 2.**

**20**

tion can be expressed as follows [15–19]:

*Direct Torque Control Strategies of Electrical Machines*

(

*ϕ* !

regions. Then Eq. (6) can be written as:

frequency of the inverter **Figures 2** and **3**.

*Evolution of stator flux vector in the complex plan.*

The logical outputs of the flux controller are defined as:

$$\begin{aligned} \text{Cflx} &= \mathbf{1} \quad \text{if } \Delta \phi\_s \mathbf{>} h\_{\phi\_s} \\ \text{Cflx} &= \mathbf{0} \quad \text{if } \Delta \phi\_s \le -h\_{\phi\_s} \end{aligned} \tag{8}$$

where *hϕ*<sup>s</sup> is hysteresis band of stator flux.

The stator flux error is defined by the difference between the reference value of flux and the actual estimated value:

$$
\Delta \phi\_{\mathbf{s}} = \left| \phi\_{\mathbf{s}}^{\*} \right| - \left| \phi\_{\mathbf{s}} \right| \tag{9}
$$

#### **5.2 Control of electromagnetic torque**

During one sampling period, the rotor flux vector is supposed invariant. The rotor and stator flux vectors are linked by the following relation:

$$\phi\_r = \frac{M}{L\_s} \cdot \frac{1}{1 + j\sigma a T\_r} \phi\_s \tag{10}$$

The angle between these two vectors is given by:

$$\delta = \text{Actual } \lg(\sigma a T\_r) \tag{11}$$

Finally, between the modules of the two flux vectors, we have the following relation:

$$|\phi\_r| = \frac{M}{L\_s} \frac{1}{\sqrt{1 + \left(\sigma \alpha T\_r\right)^2}} |\phi\_s|\tag{12}$$

The general expression of electromagnetic torque is given by:

$$T\_{cm} = p \frac{M}{\sigma L\_s L\_r} \left[ \phi\_s.\phi\_r^\* \right] \tag{13}$$

$$T\_{cm} = p \frac{M}{\sigma L\_s L\_r} \phi\_s.\phi\_r \text{Sin}(\delta) \tag{14}$$

where:

*p* is the number of poles pairs.

*ϕs*, *ϕ<sup>r</sup>* are stator and rotor flux vectors.

*δ* angle between the stator and rotor flux vectors.

From expression (14), it is clear that the electromagnetic torque is controlled by the stator and rotor flux amplitudes. If those quantities are maintaining constant, the torque can be controlled by adjusting the load angle *δ*.

The stator currents components *is<sup>α</sup>* and *is<sup>β</sup>* can be obtained also by applying

ffiffiffi 3 2 r *isa*

The produced electromagnetic torque of the induction motor can be determined using the cross product of the stator quantities (i.e., stator flux and stator currents).

*Tem* ¼ *p ϕs<sup>α</sup>is<sup>β</sup>* � *ϕs<sup>β</sup>is<sup>α</sup>*

To maintain a decoupled control, a pair of hysteresis comparators receives the stator flux and torque errors as inputs. Then, the comparators outputs determine the appropriate voltage vector selection. However, the choice of voltage vector is not only depending on the output of hysteresis controllers but on the position of stator flux vector also. Thus, the circular stator flux vector trajectory will be divided

For each sector, the vectors (*Vi and V3+i*) are not considered because both of them can increase or decrease the torque in the same sector according to the position of flux vector on the first or the second sector. If the zero vectors V0 and V7 are selected, the stator flux will stop moving, its magnitude will not change, and the electromagnetic torque will decrease, but not as much as when the active voltage vectors are selected. The resulting look-up table for DTC which was proposed by

*ϕ<sup>s</sup> Vi*�*<sup>1</sup> and Vi+1 Vi+2 and Vi*�*<sup>2</sup> Tem Vi+1 and Vi+2 Vi*�*<sup>1</sup> and Vi*�*<sup>2</sup>*

**Flux Torque 1 2 3 4 5 6 Comparator** Cflx = 1 Ctrq = 1 V2 V3 V4 V5 V6 V1 Two-level Ctrq = 0 V7 V0 V7 V0 V7 V0

Cflx = 0 Ctrq = 1 V3 V4 V5 V6 V1 V2 Two-level Ctrq = 0 V0 V7 V0 V7 V0 V7

Ctrq = �1 V6 V1 V2 V3 V4 V5 Three-level

Ctrq = �1 V5 V6 V1 V2 V3 V4 Three-level

**7. Switching table construction and control algorithm design**

� � (20)

**Increases Decreases**

(19)

*is<sup>α</sup>* ¼

8 >>>><

*Direct Torque Control Strategies of Induction Machine: Comparative Studies*

>>>>:

*is<sup>β</sup>* <sup>¼</sup> <sup>1</sup> ffiffi 2 p *isb* � *i* ð Þ*sc*

Concordia transformation on the measured currents:

**6.2 Electromagnetic torque estimation**

*DOI: http://dx.doi.org/10.5772/intechopen.90199*

into six symmetrical sectors (**Table 1**).

Takahashi is presented in **Table 2**.

*Look-up table for basic direct torque control.*

**Table 1.**

**Table 2.**

**23**

*Generalized switching table.*

The torque formula is expressed as the following:

The torque regulation can be realized using three-level hysteresis comparator. (**Figure 4**).

#### **Figure 4.**

*Three-level hysteresis comparator for electromagnetic torque control.*

The logical outputs of the torque controller are defined as:

$$\begin{aligned} \textbf{Ctrq} &= \textbf{1} & \textbf{if } \Delta T\_{\text{em}} &> h\_{T\_{\text{ou}}}\\ \textbf{Ctrq} &= \textbf{0} & \textbf{if } -h\_{T\_{\text{ou}}} \le \Delta T\_{\text{em}} &\le h\_{T\_{\text{ou}}}\\ \textbf{Ctrq} &= -\textbf{1} & \textbf{if } \Delta T\_{\text{em}} &< -h\_{T\_{\text{ou}}} \end{aligned} \tag{15}$$

$$\text{where } h\_{T\_{am}} \text{ is hysteresis band of torque.}$$

The torque error is defined by the difference between the references values of the torque and the actual estimated values:

$$
\Delta T\_{em} = T\_{em}^\* - T\_{em} \tag{16}
$$

#### **6. Estimation of stator flux and electromagnetic torque**

#### **6.1 Stator flux estimation**

The amplitude of the stator flux is estimated from its two-phase components *ϕs<sup>α</sup>* and *ϕs<sup>β</sup>*:

$$
\phi\_s = \sqrt{\phi\_{sa}^2 + \phi\_{s\beta}^2} \tag{17}
$$

Or *ϕs<sup>α</sup>* and *ϕs<sup>β</sup>* are estimated using Eq. (4) which requires knowledge of the components of the stator current vector *is<sup>α</sup>* and *is<sup>β</sup>* and that of the vector stator voltage *vs<sup>α</sup>* and *vsβ*.

The stator voltage components *vs<sup>α</sup>* and *vs<sup>β</sup>* are obtained by applying Concordia transformation on the output voltage of the three-phase VSI which are given by:

$$\begin{cases} \boldsymbol{\upsilon}\_{sa} = \sqrt{\frac{2}{3}} \boldsymbol{V}\_{dc} \left[ \mathbf{S}\_{a} - \frac{1}{2} (\mathbf{S}\_{b} + \mathbf{S}\_{c}) \right] \\\\ \boldsymbol{\upsilon}\_{s\boldsymbol{\rho}} = \sqrt{\frac{1}{2}} \boldsymbol{V}\_{dc} (\mathbf{S}\_{b} - \mathbf{S}\_{c}) \end{cases} \tag{18}$$

The stator currents components *is<sup>α</sup>* and *is<sup>β</sup>* can be obtained also by applying Concordia transformation on the measured currents:

$$\begin{cases} \dot{i}\_{sa} = \sqrt{\frac{3}{2}} \dot{i}\_{sa} \\\\ \dot{i}\_{s\beta} = \frac{1}{\sqrt{2}} (\dot{i}\_{sb} - \dot{i}\_{sc}) \end{cases} \tag{19}$$

#### **6.2 Electromagnetic torque estimation**

*ϕs*, *ϕ<sup>r</sup>* are stator and rotor flux vectors.

*Direct Torque Control Strategies of Electrical Machines*

(**Figure 4**).

**Figure 4.**

*δ* angle between the stator and rotor flux vectors.

the torque can be controlled by adjusting the load angle *δ*.

The logical outputs of the torque controller are defined as:

**6. Estimation of stator flux and electromagnetic torque**

where *hT*e*<sup>m</sup>* is hysteresis band of torque.

*Three-level hysteresis comparator for electromagnetic torque control.*

the torque and the actual estimated values:

**6.1 Stator flux estimation**

*ϕs<sup>α</sup>* and *ϕs<sup>β</sup>*:

**22**

voltage *vs<sup>α</sup>* and *vsβ*.

Ctrq ¼ 1 if Δ*T*em>*hT*e*<sup>m</sup>*

<sup>Δ</sup>*Tem* <sup>¼</sup> *<sup>T</sup>* <sup>∗</sup>

Ctrq ¼ 0 if � *hT*e*<sup>m</sup>* ≤ Δ*T*em ≤ *hT*e*<sup>m</sup>* Ctrq ¼ �1 if Δ*T*em < � *hT*e*<sup>m</sup>*

The torque error is defined by the difference between the references values of

The amplitude of the stator flux is estimated from its two-phase components

q

Or *ϕs<sup>α</sup>* and *ϕs<sup>β</sup>* are estimated using Eq. (4) which requires knowledge of the components of the stator current vector *is<sup>α</sup>* and *is<sup>β</sup>* and that of the vector stator

The stator voltage components *vs<sup>α</sup>* and *vs<sup>β</sup>* are obtained by applying Concordia transformation on the output voltage of the three-phase VSI which are given by:

*Vdc Sa* � <sup>1</sup>

*Vdc*ð Þ *Sb* � *Sc*

2

ð Þ *Sb* þ *Sc* � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ϕ*2 *<sup>s</sup><sup>α</sup>* <sup>þ</sup> *<sup>ϕ</sup>*<sup>2</sup> *sβ*

*ϕ<sup>s</sup>* ¼

ffiffiffi 2 3 r

ffiffi 1 2 r

*vs<sup>α</sup>* ¼

8 >>><

>>>:

*vs<sup>β</sup>* ¼

(15)

(17)

(18)

*em* � *Tem* (16)

From expression (14), it is clear that the electromagnetic torque is controlled by the stator and rotor flux amplitudes. If those quantities are maintaining constant,

The torque regulation can be realized using three-level hysteresis comparator.

The produced electromagnetic torque of the induction motor can be determined using the cross product of the stator quantities (i.e., stator flux and stator currents). The torque formula is expressed as the following:

$$T\_{em} = p\left(\phi\_{sa}i\_{s\beta} - \phi\_{s\beta}i\_{sa}\right) \tag{20}$$
