3.1 Mathematical Models of Induction Machines
3.1.2 Construction and Types of Induction Motors
A typical induction machine is a cylinder shaped machine whose ratio of the di- ameter to length is in the range of 1.2-0.8. Induction motors are built to meet the requirements of various numbers of phases; however, most commonly they are three-phase machines. The air gap between the stator and the rotor is as small as it is achievable and windings are located in the slots (Fig 3.1) of the stator and rotor.
The ferromagnetic circuit is made of a laminated elastic steel magnetic sheets in
order to limit energy losses associated with alternating magnetization of the iron during the operation of the machine. Very important role is played by the wind- ings, which are engineered in several basic types. In the stators of high voltage machines they are often made in the form of bars from isolated rectangular shaped conductors formed into coils inserted into the slots of the stator. In this case the slots are open, rectangular and it is possible to insert the ready made rigid coils into them in contrast to the semi-closed slots applied in windings made of coil wire. The material which conducts the current is the high conductivity copper.
Another important classification is associated with the single and two layer wind- ings. For the case of single layer winding the side of the coil occupies the entire space available in the slot, while in two layer windings inside the slot there are two sides belonging to two different coils, one above the other, while the sides could belong to the same or different phases of the winding. In machines with higher capacity we usually have to do with windings in two layers. Still another classification of windings in induction motors is associated with integral and frac- tional slot windings [101,102,103,104]. Integral slot winding is the one in which the number of slots per pole and phase is an integer number. Most induction ma- chines apply integral slot windings since they offer better characteristics of mag- netic field in the air gap. Fractional slot windings are used in the cases when the machine is designed in a way that has a large number of poles but it is not justified to apply too large a number of slots in a small cross section. Another reason for the application of fractional slot windings is associated with economic factors when the same ferromagnetic sheets are used for motors with various numbers of pole pairs. In this case for a given number of slots and certain number of pole pairs we have to do with fractional slot windings.
However, the most important role of an engineer in charge of the design of an induction machine is to focus on the development of such a winding whose mag- netic field in the air gap resulting from the flow of current through a winding fol- lows as closely as possible a sine curve (Fig 3.2). The windings in the rotors of induction motors are encountered in two various models whose names are adopted by the types of induction motors: slip-ring motor and squirrel-cage motor. The winding in a slip-ring motor is made of coils just as for a stator in the form of a three phase winding with the same number of pole pairs as a winding in a stator and the terminals of phases are connected to slip rings.
With these rings and by adequate butting contact using brushes slipping over the rings it is possible to connect an external element to the windings in a rotor.
This possibility is used in order to facilitate the start-up of a motor and in many cases also to control its rotational speed. The squirrel cage forms the other variety of an induction motor rotor’s winding that is more common. It is most often made of cast bars made of aluminum or, more rarely of bars made from welded copper alloys placed in the slots. Such bars are clamped using rings on both sides of the rotor. In this way a cage is formed (Fig 3.3); hence, the name squirrel cage was coned. The cage formed in this way does not enable any external elements or sup- ply sources to be connected. It does not have any definite number of phases, or more strictly speaking: each mesh in the network formed by two adjacent bars and connecting ring segments form a separate phase of the winding. Hence, a squirrel
Fig. 3.1 Cross section of an induction machine with semi-closed slots in the stator
Fig. 3.2 Shape of a magnetic field produced in the air gap of a three-phase induction ma- chine and its fundamental harmonic for a 24 slot stator with the number of pole-pairs p = 2 and for a 36 slot stator and p =1
cage winding with m bars in a detailed analysis could be considered as a winding with m phases. Moreover, a squirrel cage winding does not have a defined number of pole pairs. In the most basic analysis of an induction motor one can assume that a squirrel cage winding is a secondary winding that passively adapts in response to the magnetic field as a result of induced voltages and consequently currents. It is possible to further assume that the magnetic field in an air gap with p pole pairs induces in the bars of a cage a system of voltages and currents with p pole pairs as well. Since the number of phases in the rotor is basically arbitrary as the winding is not supplied from an external source this is also a three-phase winding similar to the winding in a stator. Hence, in its basic engineering drawing along the circum- ference of the stator the magnetic field in the air gap of the induction motor is
described by sine curve with p times recurrence during the round of the gap’s circumference (Fig 3.2). The difference between the actual shape of the magnetic field in the air gap and the fundamental harmonic of the order ρ = p is approxi- mated by a set of sine curves, forming the higher harmonics of the field, whose spectra and amplitudes can be calculated by accounting for all construction details of the stator and rotor of a machine. The basic reason for the occurrence of higher harmonics of the field in the air gap is associated with the discreetly located con- ductors in the slots and their accumulation in a small space, the particular span of the coils carrying currents and non-homogenous magnetic permeance in the air gap [80]. This air gap despite having its constant engineering width δ is in the sense of the magnetic permeance relative to the dimensions of the slots in the sta- tor and rotor. The higher harmonics of the magnetic field in the air gap account for a number of undesirable phenomena in induction machines called parasitic phe- nomena. They involve asynchronous and synchronous parasitic torques that de- form the basic characteristics of the electromagnetic torque [80], as well as addi- tional losses resulting from higher harmonics and specific frequencies present in the acoustic signal emitted by the machine.
Fig. 3.3 A frequent shape of a squirrel-cage winding of a rotor of induction motor
In the currently manufactured induction motors parasitic phenomena are en- countered on a relatively low level and do not disturb the operation of the drive.
Hence, in the discussion of the driving characteristics the induction motor is repre- sented by a mathematical model whose magnetic field displays monoharmonic properties. The only harmonic is the fundamental one with the number ρ = p, which is equal to the number of pole pairs. The limitation of parasitic phenomena and construction of a machine that is virtually monoharmonic comes as a result of a number of engineering procedures, of which the most basic one involves an ap- propriate selection of a number of slots in the stator and rotor. The numbers in question are Ns and Nr, respectively and they are never equal to each other and
their selection depends largely on the designed number of pole pairs p. Consider- able progress has been made in the design and engineering of induction motors over their more than 100 year old history. The measure of this progress not only involves the limitation of parasitic phenomena but also an increase the effective- ness of the structures in terms of the torque rating per kilogram of the machine’s mass, long service life, energy efficiency, ecological characteristics, progress in the use of insulation materials, which makes it possible to supply from converters with high frequency and amplitude of voltage harmonics.