**1. Introduction**

324 Induction Motors – Modelling and Control

Universidad de Santiago de Chile

No. 1, January, pp.60-79, ISSN 0019-0578.

Vol.138, No.2, pp. 69 -74, ISSN: 0143-7038

Técnico, Universidade Técnica de Lisboa, Portugal

Springer-Verlag, ISBN: 1-85233-073-2, London, GB

0198564651, N-13: 978-0198564652, New York, USA

Press, ISBN-10: 019859397X, ISBN-13: 978-0198593973, USA.

Travieso, J.C. (2002). *Passive equivalence of induction motors for control purposes by means of adaptive feedback*. (In Spanish). Ph.D. Thesis, Electrical Engineering Department,

Travieso-Torres, J.C. & Duarte-Mermoud, M.A. (2008), "Two simple and novel SISO controllers for induction motors based on adaptive passivity". *ISA Transactions*, Vol. 47,

Valério, D. (2005). *Fractional Robust System Control.* Ph.D. Dissertation, Instituto Superior

Van der Schaft, A. (2000). *L2-Gain and Passivity Techniques in Nonlinear Control*. 2nd Edition.

Vas, P. (1998). *Sensorless Vector and Direct Torque Control.* Oxford University Press, ISBN-10:

Vas, P. (1999). *Artificial-Intelligence-Based Electrical Machines and Drives*. Oxford University

Williams, B.W. & Green, T.C. (1991). Steady state control of an induction motor by estimation of stator flux magnitude. *IEE Proceedings, Part B, Electric Power Applications*,

> The dynamics of induction motor (IM) is traditionally represented by differential equations. The space-vector concept [13] is used in the mathematical representation of IM state variables such as voltage, current, and flux.

> The concept of complex transfer function derives from the application of the Laplace transform to differential equations in which the complex coefficients are in accordance with the spiral vector theory which has been presented by [24]. The complex transfer function concept is applied to the three-phase induction motor mathematical model and the induction motor root locus was presented in [10]. Other procedures for modeling and simulating the three-phase induction motor dynamics using the complex transfer function concept are also presented in [4].

> The induction machine high performance dynamics is achieved by the field orientation control (FOC) [1, 17]. The three-phase induction motor field orientation control using the complex transfer function concept to tune the PI controller by using the frequency-response function of the closed-loop complex transfer function of the controlled induction machine was presented in [2]. This strategy has satisfactory current response although stator currents had presented cross-coupling during the induction machine transients. An interesting solution was presented in [11] in which it was designed a stator-current controller using complex form. From this, the current controller structure employing single-complex zeros is synthesized with satisfactory high dynamic performance although low-speed tests had not been shown in mentioned strategies.

> An alternative for induction motor drive is the direct torque control (DTC), which consists of the direct control of the stator flux magnitude *λ*<sup>1</sup> and the electromagnetic torque *Te*. DTC controllers generate a stator voltage vector that allows quick torque response with the smallest

©2012 Azcue et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Azcue et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 2 Will-be-set-by-IN-TECH 326 Induction Motors – Modelling and Control Tuning PI Regulators for Three-Phase Induction Motor Space Vector Modulation Direct Torque Control Using Complex Transfer Function Concept <sup>3</sup>

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 speed operation [19].

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

(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

*<sup>M</sup>*/(*L*1*L*2) is the dispersion factor.

*q*

*λ*1*q* =


*<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>L</sup>*<sup>2</sup>

and it is shown in equation (9).

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

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

> *λ*1*dq*

> > *αr*

*λ*1*dq*| cos(*δ*)

 *λ*1*dq* − *R*<sup>1</sup>

 *R*<sup>2</sup> *σL*1*L*<sup>2</sup>

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

*λ*1*<sup>d</sup>* = | 

> *d λ*1*dq*

*a*<sup>4</sup> = −

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

The machine mechanical dynamics is given by

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

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

*dt* <sup>=</sup> <sup>−</sup>*jω*<sup>1</sup>

*dt* <sup>=</sup> *<sup>a</sup>*<sup>3</sup>

*a*<sup>3</sup> =

 *R*<sup>1</sup> *σL*<sup>1</sup> + *R*2 *σL*<sup>2</sup>  *λ*2*dq*

*d*

Tuning PI Regulators for Three-Phase Induction Motor Space

Vector Modulation Direct Torque Control Using Complex Transfer Function Concept

*<sup>i</sup>*1*dq* +*<sup>v</sup>*1*dq* (8)

*λ*1*dq* − *TL* (12)

(9)

(10)

(11)

*δ*

*<sup>λ</sup>*1*dq* <sup>+</sup> *<sup>a</sup>*<sup>4</sup>*<sup>i</sup>*1*dq* <sup>+</sup> *<sup>v</sup>*1*dq*

<sup>−</sup> *jPωmec σL*<sup>1</sup>

*σL*<sup>1</sup>

+ *j*(*ω*<sup>1</sup> − *Pωmec*)

*λ*1*dq* = *λ*1*<sup>d</sup>* + *jλ*1*<sup>q</sup>*

327

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 had not been shown.

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 motor operates at any speed.

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.
