**2. The complex model of the induction motor**

The three-phase induction motor mathematical model in synchronous reference frame (dq) is given by [16]

$$
\vec{\upsilon}\_{1dq} = R\_1 \vec{i}\_{1dq} + \frac{d\vec{\lambda}\_{1dq}}{dt} + j\omega\_1 \vec{\lambda}\_{1dq} \tag{1}
$$

$$0 = R\_2 \vec{i}\_{2dq} + \frac{d\vec{\lambda}\_{2dq}}{dt} + j\left(\omega\_1 - P\omega\_{mec}\right)\vec{\lambda}\_{2dq} \tag{2}$$

the relationship between fluxes and currents

$$
\vec{\lambda}\_{1dq} = L\_1 \vec{i}\_{1dq} + L\_M \vec{i}\_{2dq} \tag{3}
$$

$$
\vec{\lambda}\_{2dq} = L\_M \vec{i}\_{1dq} + L\_2 \vec{i}\_{2dq} \tag{4}
$$

The electromagnetic torque is expressed in terms of the cross-vectorial product of the stator flux and the stator current space vectors.

$$T\_{\varepsilon} = \frac{3}{2} P \frac{L\_M}{L\_2 L\_1 \sigma} \vec{\lambda}\_{2dq} \times \vec{\lambda}\_{1dq} \tag{5}$$

$$T\_{\varepsilon} = \frac{3}{2} P \frac{L\_M}{L\_2 L\_1 \sigma} \left| \vec{\lambda}\_{2dq} \right| \left| \vec{\lambda}\_{1dq} \right| \sin(\alpha\_r - \delta) \tag{6}$$

$$T\_{\varepsilon} = \frac{3}{2} P \frac{L\_M}{L\_2 L\_1 \sigma} \left| \vec{\lambda}\_{2dq} \right| \left| \vec{\lambda}\_{1dq} \right| \sin(\mathfrak{a}) \tag{7}$$

Equation (7) shows that variations in stator flux will reflect variations on rotor flux.

2 Will-be-set-by-IN-TECH

variation of the stator flux. The principles of the DTC using hysteresis controllers and variable switching frequency have been presented by [22] and [6]. It has disadvantages such as low

The PI-PID controllers are widely used in control process in industry [18]. The PI controller was applied to the IM direct torque control has been presented by [23]. Some investigations to tune the PI gains of speed controller have been presented using genetic-fuzzy [20] and neural networks [21]. These strategies have satisfactory torque and flux response although a method to tune the PI controllers for stator flux and electromagnetic torque loop and low-speed tests

To overcome low speed operation shortcomings, various approaches for DTC applying flux vector acceleration method [9, 14] and deadbeat controller [5, 12, 15] have been reported. These strategies aim the induction motor control at low speed. In this case, the complex transfer function was not used to tune PI controllers for such strategy when the induction

The aim of this book chapter is to provide the designing and tuning method for PI regulators, based on the three-phase induction motor mathematical model complex transfer function to be used in induction motor direct torque control when the machine operates at low speed which is a problem so far. This methods is in accordance with the present state of the art. The PI controller was designed and tuned by frequency-response function of the closed loop system. The controller also presents a minor complexity to induction motor direct torque control implementation. Experimental results are carried out to validate the controller design.

The three-phase induction motor mathematical model in synchronous reference frame (dq) is

*d λ*1*dq dt* <sup>+</sup> *<sup>j</sup>ω*<sup>1</sup>

*dt* <sup>+</sup> *<sup>j</sup>*(*ω*<sup>1</sup> <sup>−</sup> *<sup>P</sup>ωmec*)

*λ*1*dq* (1)

*<sup>i</sup>*1*dq* + *LM<sup>i</sup>*2*dq* (3)

*i*2*dq* (4)

*λ*1*dq* (5)

sin(*α<sup>r</sup>* − *δ*) (6)

sin(*α*) (7)

*λ*2*dq* (2)

*<sup>i</sup>*1*dq* +

*λ*2*dq* = *LMi*1*dq* + *L*<sup>2</sup>

 *λ*2*dq λ*1*dq* 

 *λ*2*dq λ*1*dq* 

The electromagnetic torque is expressed in terms of the cross-vectorial product of the stator

*d λ*2*dq*

speed operation [19].

had not been shown.

given by [16]

motor operates at any speed.

**2. The complex model of the induction motor**

0 = *R*<sup>2</sup>

*Te* <sup>=</sup> <sup>3</sup> 2 *<sup>P</sup> LM L*2*L*1*σ <sup>λ</sup>*2*dq* <sup>×</sup>

*Te* <sup>=</sup> <sup>3</sup> 2 *<sup>P</sup> LM L*2*L*1*σ*

*Te* <sup>=</sup> <sup>3</sup> 2 *<sup>P</sup> LM L*2*L*1*σ*

the relationship between fluxes and currents

flux and the stator current space vectors.

*v*1*dq* = *R*<sup>1</sup>

*<sup>i</sup>*2*dq* +

 *λ*1*dq* = *L*<sup>1</sup>

Where *δ* and *αr* are the angle of the stator flux and rotor flux space vector with respect to the direct-axis of the synchronous reference frame respectively as is shown in Fig. 1, *α* = *α<sup>r</sup>* − *δ* is the angle between the stator and rotor flux space vectors, *P* is a number of pole pairs and *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup> *<sup>M</sup>*/(*L*1*L*2) is the dispersion factor.

Combining equations (1), (2), (3) and (4), after some manipulations, the induction machine model can be written as a complex space state equation in the synchronous reference frame (dq) and the state variables are stator current*<sup>i</sup>*1*dq* = *<sup>i</sup>*1*<sup>d</sup>* + *ji*1*<sup>q</sup>* and stator flux *λ*1*dq* = *λ*1*<sup>d</sup>* + *jλ*1*<sup>q</sup>* and it is shown in equation (9).

**Figure 1.** stator and rotor fluxes space vectors in synchronous reference frame.

$$\frac{d\vec{\lambda}\_{1dq}}{dt} = -j\omega\_1 \vec{\lambda}\_{1dq} - \mathcal{R}\_1 \vec{i}\_{1dq} + \vec{v}\_{1dq} \tag{8}$$

$$\frac{d\vec{i}\_{1dq}}{dt} = a\_3 \vec{\lambda}\_{1dq} + a\_4 \vec{i}\_{1dq} + \frac{\vec{v}\_{1dq}}{\sigma L\_1} \tag{9}$$

$$a\_3 = \left(\frac{R\_2}{\sigma L\_1 L\_2} - \frac{jP\omega\_{mec}}{\sigma L\_1}\right) \tag{10}$$

$$a\_4 = -\left[\frac{R\_1}{\sigma L\_1} + \frac{R\_2}{\sigma L\_2} + j\left(\omega\_1 - P\omega\_{\text{mec}}\right)\right] \tag{11}$$

The machine mechanical dynamics is given by

$$J\frac{d\omega\_{mec}}{dt} = \frac{3}{2} P \frac{L\_M}{L\_2 L\_1 \sigma} \vec{\lambda}\_{2dq} \times \vec{\lambda}\_{1dq} - T\_{\rm L} \tag{12}$$

The *ω*<sup>1</sup> is the synchronous speed, *ωmec* is the machine speed, *R*<sup>1</sup> and *R*<sup>2</sup> are the estator and rotor windings per phase electrical resistance, *L*<sup>1</sup> , *L*<sup>2</sup> and *Lm* are the proper and mutual inductances of the stator and rotor windings, *v* is the voltage vector , *P* is the machine number of pair of poles, *J* is the load and rotor inertia moment, the symbol "\*" represents the conjugate of the complex number and *TL* is the load torque.

IM complex transfer function, *<sup>v</sup>*1*dq* is considered as the input and the *<sup>i</sup>*1*dq* is considered as the output. For this purpose it is assumed that the mechanical time constant of the motor is much larger than the transient electromagnetic time constants and the saturation effects is neglected. Thus, *ωmec* = constant is a valid approximation [24], [11]. Therefore the induction machine complex transfer function *H*(*s*) is derived from application of the Laplace transform in equations (8) and (9) and it is the closed loop system of Figure 4 without machine

If the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become

The electromagnetic torque (14) can be quickly changed by changing the angle *α* in the desired direction. The angle *α* can be easily changed when choosing the appropriate stator voltage

> *λ*2*αβ λ*1*αβ*

For simplicity, let us assume that the stator phase ohmic drop could be neglected in *v*1*αβ* =

vector at a speed which is proportional to the magnitude of the stator voltage space vector. By selecting step-by-step the appropriate stator voltage vector, it is possible to change the stator

The stator flux oriented direct torque control (SFO-DTC) have two PI regulators. The outputs of the PI flux and torque controllers can be interpreted as the stator voltage components in the stator flux oriented coordinates as shown in Fig. 5 [23], [3]. The control strategy relies on a simplified description of the stator voltage components, expressed in stator-flux-oriented

*v*1*<sup>d</sup>* = *R*1*i*1*<sup>d</sup>* +

*dλ*<sup>1</sup>

Δ

 *s* + *jω*<sup>1</sup> *σL*<sup>1</sup>

 + *a*<sup>3</sup>

Vector Modulation Direct Torque Control Using Complex Transfer Function Concept

Tuning PI Regulators for Three-Phase Induction Motor Space

*λ*1*αβ*/*dt* = *v*1*αβ*. During a short time Δ*t*, when the voltage space

*λ*1*αβ* ≈ *v*1*αβ* · Δ*t* (15)

*λ*1*αβ* in the direction of the stator voltage space

*dt* (16)

*v*1*<sup>q</sup>* = *R*1*i*1*<sup>q</sup>* + *ω*1*λ*<sup>1</sup> (17)

(13)

329

sin(*α*) (14)

(*s* + *jω*1) (*s* + *a*4) + *R*<sup>1</sup> *a*<sup>3</sup>

mechanical dynamics. Thus, it has the form given in 13.

*H*(*s*) =

and *V*1*dq* = L

constant because it changes slower than the stator flux.

*Te* <sup>=</sup> <sup>3</sup> 2 *<sup>P</sup> LM L*2*L*1*σ*

where *I*1*dq* = L

space vector.

*<sup>i</sup>*1*αβ* <sup>+</sup> *<sup>d</sup>*

coordinates as:

*λ*1*αβ*

vector is applied it has:

flux in the required direction.

*dt* . Therefore *<sup>d</sup>*

Thus, the stator flux space vector moves by Δ

**3.1. Stator flux oriented direct torque control**

*R*1

 *<sup>i</sup>*1*dq*

**3. Direct torque control**

*I*1*dq V*1*dq* =

> *<sup>v</sup>*1*dq* .

In order to obtain the induction motor complex transfer function the Laplace transform is applied to the equations (8) and (9) in accordance with the complex transfer function concept [24], [10]. Thus, the equation (8) complex transfer function is shown in Figure 2.

**Figure 2.** Equation (8) complex transfer function.

And the equation (9) complex transfer function complex transfer function is shown in Figure 3. Thus, the induction motor block diagram originated by use of the equations (8)

**Figure 3.** Equation (9) complex transfer function.

and (9) complex transfer functions shown in Figures 2 and 3 and the machine mechanical dynamics (12) is shown in Figure 4. When designing the DTC control system through the

**Figure 4.** Induction motor block diagram.

IM complex transfer function, *<sup>v</sup>*1*dq* is considered as the input and the *<sup>i</sup>*1*dq* is considered as the output. For this purpose it is assumed that the mechanical time constant of the motor is much larger than the transient electromagnetic time constants and the saturation effects is neglected. Thus, *ωmec* = constant is a valid approximation [24], [11]. Therefore the induction machine complex transfer function *H*(*s*) is derived from application of the Laplace transform in equations (8) and (9) and it is the closed loop system of Figure 4 without machine mechanical dynamics. Thus, it has the form given in 13.

$$H(s) = \frac{I\_{1dq}}{V\_{1dq}} = \frac{\left(\frac{s + j\omega\_1}{\sigma L\_1}\right) + a\_3}{(s + j\omega\_1)\left(s + a\_4\right) + R\_1 a\_3} \tag{13}$$

where *I*1*dq* = L *<sup>i</sup>*1*dq* and *V*1*dq* = L *<sup>v</sup>*1*dq* .

## **3. Direct torque control**

4 Will-be-set-by-IN-TECH

of pair of poles, *J* is the load and rotor inertia moment, the symbol "\*" represents the conjugate

In order to obtain the induction motor complex transfer function the Laplace transform is applied to the equations (8) and (9) in accordance with the complex transfer function concept

And the equation (9) complex transfer function complex transfer function is shown in Figure 3. Thus, the induction motor block diagram originated by use of the equations (8)

and (9) complex transfer functions shown in Figures 2 and 3 and the machine mechanical dynamics (12) is shown in Figure 4. When designing the DTC control system through the

[24], [10]. Thus, the equation (8) complex transfer function is shown in Figure 2.

of the complex number and *TL* is the load torque.

**Figure 2.** Equation (8) complex transfer function.

**Figure 3.** Equation (9) complex transfer function.

**Figure 4.** Induction motor block diagram.

If the sample time is short enough, such that the stator voltage space vector is imposed to the motor keeping the stator flux constant at the reference value. The rotor flux will become constant because it changes slower than the stator flux.

The electromagnetic torque (14) can be quickly changed by changing the angle *α* in the desired direction. The angle *α* can be easily changed when choosing the appropriate stator voltage space vector.

$$T\_{\varepsilon} = \frac{3}{2} P \frac{L\_M}{L\_2 L\_1 \sigma} \left| \vec{\lambda}\_{2a\beta} \right| \left| \vec{\lambda}\_{1a\beta} \right| \sin(a) \tag{14}$$

For simplicity, let us assume that the stator phase ohmic drop could be neglected in *v*1*αβ* = *R*1 *<sup>i</sup>*1*αβ* <sup>+</sup> *<sup>d</sup> λ*1*αβ dt* . Therefore *<sup>d</sup> λ*1*αβ*/*dt* = *v*1*αβ*. During a short time Δ*t*, when the voltage space vector is applied it has:

$$
\Delta \vec{\lambda}\_{1a\beta} \approx \vec{v}\_{1a\beta} \cdot \Delta t \tag{15}
$$

Thus, the stator flux space vector moves by Δ *λ*1*αβ* in the direction of the stator voltage space vector at a speed which is proportional to the magnitude of the stator voltage space vector. By selecting step-by-step the appropriate stator voltage vector, it is possible to change the stator flux in the required direction.

### **3.1. Stator flux oriented direct torque control**

The stator flux oriented direct torque control (SFO-DTC) have two PI regulators. The outputs of the PI flux and torque controllers can be interpreted as the stator voltage components in the stator flux oriented coordinates as shown in Fig. 5 [23], [3]. The control strategy relies on a simplified description of the stator voltage components, expressed in stator-flux-oriented coordinates as:

$$v\_{1d} = R\_1 i\_{1d} + \frac{d\lambda\_1}{dt} \tag{16}$$

$$
\omega\_{1\emptyset} = R\_1 i\_{1\emptyset} + \omega\_1 \lambda\_1 \tag{17}
$$

Where *ω*<sup>1</sup> is the angular speed of the stator flux vector. The above equations show that the component *v*1*<sup>d</sup>* has influence only on the change of stator flux magnitude, and the component *v*1*q*, if the term *ω*1*λ*<sup>1</sup> is decoupled, can be used for torque adjustment. Therefore, after coordinate transformation *dq*/*αβ* into the stationary frame, the command values *v*1*dref* , *v*1*qref* , are delivered to SVM module.

The reference stator voltage vector *v*1*dqref* is transformed by using stator flux angle *δ<sup>s</sup>* to obtain

the stator voltage at stationary reference frame *αβ*.

**Figure 6.** DTC strategy with PI regulators and complex signals.

 *λ*1*αβ* = The stator flux angle is estimated by using the trigonometric transfer function

(*v*1*αβ* − *R*<sup>1</sup>

A satisfactory flux estimation for induction motor at low speed using Equation (21) is obtained by using the integration method presented in [8] and the block diagram for the flux stimation

*<sup>δ</sup><sup>s</sup>* <sup>=</sup> arctan *<sup>λ</sup>*1*<sup>β</sup>*

In order to tune the PI regulator it is necessary the closed-loop complex transfer function of the controlled induction motor. The complex transfer function of the controlled induction motor

In accordance with the DTC control strategy the induction motor output has to be the stator flux magnitude *λ*<sup>1</sup> and the torque *Te*. Therefore the H(s) (13) outputs have to become the stator flux magnitude *λ*<sup>1</sup> and the torque *Te*. The expression to obtain the stator flux by using

was also used to tune a complex gain controller in which has been presented in [7].

*λ*1*<sup>α</sup>*

*λ*<sup>1</sup> = *λ*1*<sup>d</sup>* ∼= *G σ L*<sup>1</sup> *i*1*<sup>d</sup>* (23)

*<sup>i</sup>*1*αβ*)*dt* (21)

Tuning PI Regulators for Three-Phase Induction Motor Space

Vector Modulation Direct Torque Control Using Complex Transfer Function Concept

(22)

331

**4.1. Stator flux estimation**

is presented in Figure 7

The stator flux estimation is done by

**4.2. Design of the PI regulator gains**

the stator current *i*1*<sup>d</sup>* is given by

This SFO-DTC scheme requires the flux and the torque estimators, which can be performed as it is proposed in Fig. 5. Therefore, the control signals are fed to the power electronics drive.

**Figure 5.** Stator flux oriented direct torque control scheme.

## **4. Design and tuning PI gains**

By using stator field orientation, the torque and stator flux must become parts of a complex number, where the magnitude of the stator flux *λ*<sup>1</sup> is the real component and the torque *Te* is the imaginary component. Hence, the reference signals and the error become a complex number. Thus, the PI regulators presented in the before section [23] has the function to generate a voltage reference space vector using the stator flux-torque error vector (*ελ* + *jεT*). This way the stator-voltage vector in this control strategy is given by

$$
\vec{\sigma}\_{1dq\_{ref}} = (\varepsilon\_\lambda + j\varepsilon\_T) \left( Kp + \frac{Ki}{s} \right) \tag{18}
$$

Which means that the direct and quadrature axis of the voltage vector are

$$v\_{1d\_{ref}} = (e\_\lambda) \left( Kp + \frac{Ki}{s} \right) \tag{19}$$

$$w\_{1q\_{ref}} = (\varepsilon\_T) \left( Kp + \frac{Ki}{s} \right) \tag{20}$$

Where *kp* is the proportional gain, *ki* is the integral gain, *ελ* is the flux error signal and *ε<sup>T</sup>* is the torque error signal.

The block diagram of the strategy with the PI regulators is shown in Figure 6.

The reference stator voltage vector *v*1*dqref* is transformed by using stator flux angle *δ<sup>s</sup>* to obtain the stator voltage at stationary reference frame *αβ*.

**Figure 6.** DTC strategy with PI regulators and complex signals.

## **4.1. Stator flux estimation**

6 Will-be-set-by-IN-TECH

Where *ω*<sup>1</sup> is the angular speed of the stator flux vector. The above equations show that the component *v*1*<sup>d</sup>* has influence only on the change of stator flux magnitude, and the component *v*1*q*, if the term *ω*1*λ*<sup>1</sup> is decoupled, can be used for torque adjustment. Therefore, after coordinate transformation *dq*/*αβ* into the stationary frame, the command values *v*1*dref* , *v*1*qref* ,

This SFO-DTC scheme requires the flux and the torque estimators, which can be performed as it is proposed in Fig. 5. Therefore, the control signals are fed to the power electronics drive.

*δs*

*αβ*

*dq*

SVM

Voltage

*αβ*

*i*<sup>1</sup> *ia*

*ABC*

*Vdc*

PWM

PWM signals

*ib*

(18)

(19)

(20)

calculation Flux angle

*v*1

*λ*1*<sup>α</sup> λ*1*<sup>β</sup>*

Flux and torque estimator

By using stator field orientation, the torque and stator flux must become parts of a complex number, where the magnitude of the stator flux *λ*<sup>1</sup> is the real component and the torque *Te* is the imaginary component. Hence, the reference signals and the error become a complex number. Thus, the PI regulators presented in the before section [23] has the function to generate a voltage reference space vector using the stator flux-torque error vector (*ελ* + *jεT*).

> *K p* +

 *K p* +

 *K p* +

Where *kp* is the proportional gain, *ki* is the integral gain, *ελ* is the flux error signal and *ε<sup>T</sup>* is

*Ki s* 

*Ki s* 

*Ki s* 

*λ*1

This way the stator-voltage vector in this control strategy is given by

*v*1*dqref* = (*ελ* + *jεT*)

*v*1*dref* = (*ελ*)

*v*1*qref* = (*εT*)

The block diagram of the strategy with the PI regulators is shown in Figure 6.

Which means that the direct and quadrature axis of the voltage vector are

*Te*

PI

*v*1*qref*

*v*1*dref*

are delivered to SVM module.

*λ*1*ref*

+

*Teref*

−

PI

+

**Figure 5.** Stator flux oriented direct torque control scheme.

**4. Design and tuning PI gains**

the torque error signal.

−

The stator flux estimation is done by

$$
\vec{\lambda}\_{1a\beta} = \int (\vec{v}\_{1a\beta} - R\_1 \vec{i}\_{1a\beta}) dt \tag{21}
$$

A satisfactory flux estimation for induction motor at low speed using Equation (21) is obtained by using the integration method presented in [8] and the block diagram for the flux stimation is presented in Figure 7

The stator flux angle is estimated by using the trigonometric transfer function

$$\delta\_{\rm s} = \arctan\left(\frac{\lambda\_{1\beta}}{\lambda\_{1a}}\right) \tag{22}$$

### **4.2. Design of the PI regulator gains**

In order to tune the PI regulator it is necessary the closed-loop complex transfer function of the controlled induction motor. The complex transfer function of the controlled induction motor was also used to tune a complex gain controller in which has been presented in [7].

In accordance with the DTC control strategy the induction motor output has to be the stator flux magnitude *λ*<sup>1</sup> and the torque *Te*. Therefore the H(s) (13) outputs have to become the stator flux magnitude *λ*<sup>1</sup> and the torque *Te*. The expression to obtain the stator flux by using the stator current *i*1*<sup>d</sup>* is given by

$$
\lambda\_1 = \lambda\_{1d} \cong G \sigma \, L\_1 \, i\_{1d} \tag{23}
$$

**Figure 7.** Block diagram for the stator flux stimation.

and to obtain the electromagnetic torque in the *dq* reference frame one may use the expression:

$$T\_{\varepsilon} = \frac{3}{2} P \lambda\_1 \ i\_{1q} \tag{24}$$

**Bode Diagram**

**2 Hz 4.15 Hz 5 Hz 6 Hz 10Hz**

Vector Modulation Direct Torque Control Using Complex Transfer Function Concept

Tuning PI Regulators for Three-Phase Induction Motor Space

333

**2 HzI 4.15 Hz 5 Hz 6 Hz 10 Hz**

**Frequency (Hz) 10−2 10−1 <sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> −30**

As the variables at synchronous referential are constants the angle of output *Xλ<sup>T</sup>* is neglected. At the frequency of 2Hz, 4.16Hz and 6Hz the *kp* and *ki* gains are chosen by using simulations, considering slip approximately null and the 0 dB magnitude. Them values are *kp* = 155 and *ki* = 15. The frequency-response function of Equation (26) is shown in Figure 10 and its

**Bode diagram**

**Frequency (Hz)**

**100 101 102**

**−25**

**Figure 8.** Equation (25) frequency-response function.

**−2 −1.5 −1 −0.5 0 0.5 1 1.5 2**

**Figure 10.** Frequency-response function of the equation (26)

**Magnitude (dB)**

**Figure 9.** System to design the PI regulator.

magnitude is near 0 dB.

**−20**

**−15**

**Magnitude (dB)**

**−10**

**−5**

**0**

As the stator flux magnitude *λ*<sup>1</sup> is assumed to be essentially constant through of the equations (13), (23) and (24) the new transfer function is achieved with torque and flux as output and it is given by

$$\frac{X\_{\lambda T}}{V\_{1dq}} = H(s) \left( G\sigma L\_1 + jP\frac{3}{2}\lambda\_1 \right) \tag{25}$$

where *Xλ<sup>T</sup>* = L {*λ*<sup>1</sup> + *jTe*}.

The low speeds utilized in this book chapter are 60 rpm (6.25*rad*/*s*), 125 rpm (13*rad*/*s*), 150 rpm (16*rad*/*s*) , 180 rpm (17*rad*/*s*) that corresponds to 2Hz, 4.16Hz, 5Hz and 6Hz respectively. The frequency-response function of Equation (25) is presented in Figure 8 at frequencies 2Hz, 4.16Hz, 5Hz and 6Hz in accordance with the induction motor desired speed.

Then, from Equations (18) and (25) one obtain the control system block diagram and it is shown in Figure 9.

The expression of the closed loop transfer function of the system to design the PI regulators showed in Figure 9 is given by

$$\frac{X\_{\lambda T}}{X\_{\lambda T\_{nf}}} = \frac{\left(Kp + \frac{Ki}{s}\right)H(s)\left(G\sigma L\_1 + jP\frac{3}{2}\lambda\_1\right)}{1 + \left(Kp + \frac{Ki}{s}\right)H(s)\left(G\sigma L\_1 + jP\frac{3}{2}\lambda\_1\right)}\tag{26}$$

where *XλTref* = L *<sup>λ</sup>*1*ref* <sup>+</sup> *jTeref* .

**Figure 8.** Equation (25) frequency-response function.

**Figure 9.** System to design the PI regulator.

8 Will-be-set-by-IN-TECH

and to obtain the electromagnetic torque in the *dq* reference frame one may use the expression:

As the stator flux magnitude *λ*<sup>1</sup> is assumed to be essentially constant through of the equations (13), (23) and (24) the new transfer function is achieved with torque and flux as

*<sup>G</sup>σL*<sup>1</sup> <sup>+</sup> *jP*<sup>3</sup>

2 *λ*1 

*<sup>G</sup>σL*<sup>1</sup> <sup>+</sup> *jP*<sup>3</sup>

*<sup>G</sup>σL*<sup>1</sup> <sup>+</sup> *jP*<sup>3</sup>

2 *λ*1 

> 2 *λ*1

(26)

The low speeds utilized in this book chapter are 60 rpm (6.25*rad*/*s*), 125 rpm (13*rad*/*s*), 150 rpm (16*rad*/*s*) , 180 rpm (17*rad*/*s*) that corresponds to 2Hz, 4.16Hz, 5Hz and 6Hz respectively. The frequency-response function of Equation (25) is presented in Figure 8 at frequencies 2Hz,

Then, from Equations (18) and (25) one obtain the control system block diagram and it is

The expression of the closed loop transfer function of the system to design the PI regulators

*Pλ*<sup>1</sup> *i*1*<sup>q</sup>* (24)

(25)

*Te* <sup>=</sup> <sup>3</sup> 2

= *H*(*s*)

4.16Hz, 5Hz and 6Hz in accordance with the induction motor desired speed.

*Xλ<sup>T</sup> V*1*dq*

**Figure 7.** Block diagram for the stator flux stimation.

output and it is given by

where *Xλ<sup>T</sup>* = L {*λ*<sup>1</sup> + *jTe*}.

showed in Figure 9 is given by

*Xλ<sup>T</sup> XλTref*

*λ*1*ref* + *jTeref*

=

 *K p* + *Ki s H*(*s*) 

1 + *K p* + *Ki s H*(*s*) 

 .

shown in Figure 9.

where *XλTref* = L

As the variables at synchronous referential are constants the angle of output *Xλ<sup>T</sup>* is neglected. At the frequency of 2Hz, 4.16Hz and 6Hz the *kp* and *ki* gains are chosen by using simulations, considering slip approximately null and the 0 dB magnitude. Them values are *kp* = 155 and *ki* = 15. The frequency-response function of Equation (26) is shown in Figure 10 and its magnitude is near 0 dB.

**Figure 10.** Frequency-response function of the equation (26)
