3.1 Mathematical Models of Induction Machines
3.1.3.1 Types of Models of Induction Machines
Mathematical modeling plays a very important role in the design, exploitation and control of electric drives. Modeling and computer simulation, whether with regard to electric drive or in other branches of engineering, that is adequate and effective reduces the time needed and the cost of gaining an optimum design of a drive and its control system. Thus, new opportunities are offered in terms of reducing lead times in the prototype testing phase of the design. The modeling of an induction motor is complex to the degree that we have to do with an electromechanical de- vice with a large number of degrees of electrical freedom, represented by charges and electric currents in phase windings and, additionally, that can account for magnetic linkages. The latter are delivered by the magnetic field in the ferromag- netic material in which the windings of the stator operate and the ferromagnetic core is often in the condition of magnetic saturation. The simultaneous and com- prehensive accounting for electromagnetic and electromechanical processes in an induction motor that involves saturation of the active iron in the stator and rotor, energy losses during alternate magnetization, precise mapping of linkages between the windings, the non-steady working regime of the rotor and the potential effect of the heat generated on the properties of the system is in fact too complex and too costly and, hence, even in the most advanced models of induction machines these processes tend to be simplified. The basic and most common simplification con- sists in the distinction made between the magnetic and electric field due to the small frequencies of the alternation of the field. For that reason, the field is con- sidered to be magnetostatic. Moreover, there is a tendency to simplify the issues associated with energy losses during the alternate magnetization of the iron, and sometimes it is disregarded. Phase windings in a machine are most commonly considered as electric circuits with lumped parameters and their connection with the magnetic field is expressed by flux linkage ψk, where subscript k denotes the number of the adequate winding. Overall, the problem is associated with the de- termination of the flux linkage as the function of electric currents in the particular phase windings of a machine [90]. The issue of the mechanical motion of a rotor is not a complex phenomenon since a typical induction motor has only a single de- gree of mechanical freedom – angle of rotation of the rotor θr. In mathematical modeling of a an induction machine drive we take into consideration two cases:
non-homogenous motion of the rotor in the dynamic states – for example during
start-up or braking of the motor and motion under a constant angular speed, i.e. in a steady condition of the drive in operation. As a consequence of not accounting for the parasitic torques with synchronic characteristics we do not take into con- sideration small oscillations of the speed around the balance state; this comes as a consequence of their marginal role in a designed drive. The basic and the common foundation during the development of a mathematical model of an induction ma- chine is the assumption of its geometrical and material symmetry. This allows very largely to simplify the model and it is most often followed in the issues asso- ciated with the electric drive. Abandoning of the assumptions of symmetry during the modeling of an induction machine is necessary only in special circumstances, such as modeling of emergency conditions for a drive and for example in the stud- ies devoted to the tolerance of the engineering structure of the machine to its char- acteristics and potential emergencies. Such an example encountered during the analysis of an induction machine is the study of the effect of the asymmetry of the air gap between the stator and rotor to the resulting forces of magnetic pull and bearing’s wear. The assumption of the symmetry also enables one to limit the area of calculation undertaken with an aim of developing field models and determina- tion of boundary conditions for such calculations. Due to the presented impedi- ments and complications the models of induction motors usually account for a number of simplifications which form an adaptation of the examined question and can lead to the statement of an answer. In this respect we can identify three gen- eral categories of mathematical modeling of a drive. The categories include: mod- els serving for the optimization of the construction characteristics of a motor, sec- ondly, models used for the determination of electromechanical characteristics and, thirdly, models whose object is to apply an induction motor drive control.
The presented three categories of models can be described as follows: a mathe- matical model of an induction motor aimed at the optimization of its construction with regard to the structure of a magnetic circuit is, as a rule, a field based model whose solution is presented in 3D or 2D space, with a particular emphasis on the shape of a ferromagnetic core along with the design of the stator’s and rotor’s slots as well as spatial distribution of the windings. The ferromagnetic material is con- sidered as non-linear taking into account its characteristics of magnetization. The considerations tend to more frequently involve a magnetic hysteresis loop and less often the occurrence of eddy currents [17,49]. Hence, calculations are performed for fixed positions of the stator in relation to the rotor or a constant speed of the motion, while the current density in the windings is as a rule constant over the en- tire cross-section of the winding in the slot. For the case of winding bars with large dimensions we have to account for the non-homogenous distribution of the current density in the radial direction. The construction of a typical induction mo- tor due to the plane-parallel field representation enables one to perform field cal- culations in 2D space without affecting the precision of the results. The calcula- tions apply professional software suites using Finite Elements Method (FEM) or Edge Elements Method (EEM). Such software contains procedures making it pos- sible to gain various data and images regarding field characteristics in a particular subject, to obtain a number of integrated parameters such as the value of energy and co-energy of the magnetic field, electromagnetic torque, forces calculated by
means of various methods and inductance of the windings in the area of calcula- tion [24,48]. As one can conclude from this description, field models are applica- ble not only with an aim of improving the engineering and considering details of material parameters but can also provide valuable data in the form of lumped pa- rameters for the calculation of the problems encountered in the drive. In particular, relevant insight is offered by the data regarding the inductance of the windings and its relation to the magnetic saturation. The mathematical models serving for the determination of electromechanical characteristics of a drive, both in static and dynamic states, as a rule are formed as models with lumped parameters. The reason is that in this case the engineering details are related to in an indirect way using a small number of parameters, which subsequently combine a number of physical properties of a machine. During the determination of characteristics, in particular the mechanical ones, the parasitic phenomena are frequently accounted for in the form of additional elements of electromagnetic torque derived from higher harmonics of the magnetic flux and harmonics associated with variable terms expressed by other elements in the permeneance of the air gap. The models which are applicable for stating the characteristics in many cases have to be precise in terms of energy balance since one of their application is in the determi- nation of the losses of energy and efficiency of the drive. The analysis of lumped parameters is performed by a number of specialized calculation methods. This is based on field calculations in the electrical machines for the specific conditions of operation [37,48,91]. The mathematical models applied in the issues associated with drive control tend to be the most simplified models. As a principle, they dis- regard the losses in the iron, the phenomena of magnetic saturation and nuances in the form of multi-harmonic spectrum of the magnetic field in the air gap. Such models take the form of a system of ordinary differential equations. The models are transformed using the properties of the machine’s symmetry into systems of equations, in which the form of the equations is relatively simple in the sense of the assumption of constant parameters of a system, the number and structure of expressions. Thus, the models correspond to the requirements of the control sys- tem due to its interaction with the transformed measurement signals derived from feedback in the system. The rationale for using the possibly most uncomplicated (in terms of calculations) models in the questions of control is associated with the fact that they are later used for the calculation of the vector of state variables of the drive in real time. The mathematical models of the induction motor find an in- creasingly wider application in the modern methods of control concerned with lin- earization through non-linear feedback of the dynamic model of a drive, which, in reality, has a non-linear structure. A type of this kind of control is also named con- trol with inverse dynamics. An arising question is concerned with the practical ap- plication of models that do not account for a number of phenomena in induction machines including magnetic saturation. The solution proposed involves the con- temporary control methods, also applied in electric drive, which are more resistant to the uncertainty of the parameters of the model and disturbances along the measurement paths. Such models include robust and adaptive control [22,45,53,57,65,75,105], in which case the mathematical model is combined with estimation of the parameters in real time. The currently solved tasks in drive
control apply the following procedure: simple and functional control models in terms of calculations are accompanied with the correction of discrepancies result- ing from parameter estimation using signals that are easily accessible by way of measurement. From the point of view of the current book the principal interest focuses on the mathematical models designed for determination of the characteris- tics of the drives and the ones applied for the purposes of control.