**1. Introduction**

A proper study of the induction machine operation, especially when it comes to transients and unbalanced duties, requires effective mathematical models above all. The mathematical model of an electric machine represents all the equations that describe the relationships between electromagnetic torque and the main electrical and mechanical quantities.

The theory of electrical machines, and particularly of induction machine, has mathematical models with *distributed* parameters and with *concentrated* parameters respectively. The first mentioned models start with the cognition of the magnetic field of the machine components. Their most important advantages consist in the high generality degree and accuracy. However, two major disadvantages have to be mentioned. On one hand, the computing time is rather high, which somehow discountenance their use for the real-time control. On the other hand, the distributed parameters models do not take into consideration the influence of the temperature variation or mechanical processing upon the material properties, which can vary up to 25% in comparison to the initial state. Moreover, particular constructive details (for example slots or air-gap dimensions), which essentially affects the parameters evaluation, cannot be always realized from technological point of view.

The mathematical models with concentrated parameters are the most popular and consequently employed both in scientific literature and practice. The equations stand on resistances and inductances, which can be used further for defining magnetic fluxes, electromagnetic torque, and et.al. These models offer results, which are globally acceptable but cannot detect important information concerning local effects (Ahmad, 2010; Chiasson, 2005; Krause et al., 2002; Ong, 1998; Sul, 2011).

© 2012 Livadaru 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 Livadaru 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.

The family of mathematical models with concentrated parameters comprises different approaches but two of them are more popular: *the phase coordinate* model and the *orthogonal (dq)* model (Ahmad, 2010; Bose, 2006; Chiasson, 2005; De Doncker et al., 2011; Krause et al., 2002; Marino et al., 2010; Ong, 1998; Sul, 2011; Wach, 2011).

Mathematical Model of the Three-Phase Induction Machine

for the Study of Steady-State and Transient Duty Under Balanced and Unbalanced States 5

*abcs*

*ABCR*

(4)

*dt* 

(3)

*dt* 

*d*

*d*

*abcs s abcs*

*u Ri*

*ABCR R ABCR*

**Figure 1.** Schematic model of three-phase induction machine: a. real; b. reduced rotor

component: Mjk=Ljk=Lhj=Lhk. The expressions in matrix form are:

The quantities in brackets represent the matrices of voltages, currents, resistances and total flux linkages for the stator and rotor. Obviously, the total fluxes include both main and mutual components. Further, we define the self-phase inductances, which have a leakage and a main component: Ljj=Lσs+Lhs for stator and LJJ=LΣR+LHR for rotor. The mutual inductances of two phases placed on the same part (stator or rotor) have negative values, which are equal to half of the maximum mutual inductances and with the main self-phase

(a) (b)

(1 / 2) (1 / 2)

*hs hs s hs*

(1 / 2) (1 / 2)

*HR HR R HR*

*RR R*

cos cos cos 2

*RR R*

*u u*

 

> 

(5-1)

(5-2)

(5-3)

*L L LL*

*R HR HR HR*

*L L LL*

cos 2 cos cos cos cos 2 cos

*u u*

(1 / 2) (1 / 2)

(1 / 2) (1 / 2)

*LL L L*

*RR HR R HR HR*

*L L LL L*

*sR Rs sR R t R R*

*LLL u u*

*LL L L*

*ss hs s hs hs*

*L L LL L*

*s hs hs hs*

(1 / 2) (1 / 2)

(1 / 2) (1 / 2)

*u Ri*

The first category works with the real machine. The equations include, among other parameters, the mutual stator-rotor inductances with variable values according to the rotor position. As consequence, the model becomes non-linear and complicates the study of dynamic processes (Bose, 2006; Marino et al., 2010; Wach, 2011).

The orthogonal (dq) model has begun with Park's theory nine decades ago. These models use parameters that are often independent to rotor position. The result is a significant simplification of the calculus, which became more convenient with the defining of the *space phasor* concept (Boldea & Tutelea, 2010; Marino et al., 2010; Sul, 2011).

Starting with the ″classic″ theory we deduce in this contribution a mathematical model that exclude the presence of the currents and angular velocity in voltage equations and uses total fluxes alone. Based on this approach, we take into discussion two control strategies of induction motor by principle of constant total flux of the stator and rotor, respectively.

The most consistent part of this work is dedicated to the study of unbalanced duties generated by supply asymmetries. It is presented a comparative analysis, which confronts a balanced duty with two unbalanced duties of different unbalance degrees. The study uses as working tool the Matlab-Simulink environment and provides variation characteristics of the electric, magnetic and mechanical quantities under transient operation.
