Math Models of Electric Drives as Induction Machines

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INTRO: This Section deals with electrical drives with induction machines. After short introduction concerning basic construction variants and general problems of modeling induction motor drives, mathematical models are developed on a basis of previously introduced Lagrange's method. Models transformed in a classical way are presented by use of orthogonal transformation along with the options governing free parameters of these transformations.



Subsequently, other models are developed in which only one side electrical variables are transformed - stator's or rotor's, while the other side remains untransformed, with natural variables. These models are applied in simulation and presenting various problems of drive systems with electronic power converters and for other external asymmetry cases. Some of them are classical drive problems like DC breaking, operation of Bi-directional drive or soft-start systems. Other concern modern systems with electronic power converters, more precisely and successfully modeled and computed in this way. They are drives with two- and three-level voltage source inverters used in PWM control (SVM, DPWM) as well as current source inverters - CSI. These kinds of models are also applied to present vector control (VC) or field oriented control (FOC) and direct torque control (DTC) of induction motor drives. Abovementioned problems are widely illustrated by examples computed for various dynamic states, with and without automatic control, and for that purpose four induction motors of different rated power are presented. Finally a problem of structural linearization of induction motor drive is covered, beside a number of useful state observer systems that are discussed.

Introduction

The history of construction and application of induction motors in electric drive dates back as far as over 100 years and the induction machine constitutes the basic unit energized from alternating current in a symmetrical three-phase power distribution system. Beside the three-phase layout some small size machines can be supplied from a single phase for household applications and two-phase machines can be used in the drives of servomechanisms. The practical meaning of three phase induction machines is emphasized by the fact that they consume nearly 70% of the generated electric power. The name given to induction motor originates from the single sided power supply to such machines most frequently occurring from the direction of the stator. Inside the rotor there is a fixed winding which generates electric power as a result of induced voltages resulting from magnetic flux that changes (rotates) in relation to the rotor's windings. The mutual interaction of magnetic field in the air gap with the currents in rotor's windings leads to the origin of electromagnetic torque Te, which sets the rotor in motion. In the induction motor the air gap between the stator and rotor is as narrow as it’s technically possible for the requirements of the mechanical structure, since the energy powering the rotor by the magnetic field has a considerably high value. This field should have a high value of flux density in the gap which requires an adequate magnetizing current, that is approximately proportional to the width of the air gaped. Three-phase induction motors find an application in all branches of industry and in municipal utility management as well as in farming and service workshops.

This group of machines involves devices produced within a wide range of power ratings from under 100 [W] and can reach as much as 20 [MW]. The traditional induction machines were applied in drives that don’t require the control of rotational speeds. This was due to the cost and problems associated with the use of such control devices while securing the maintenance of high efficiency of trans forming electric energy into mechanical one. The course of events has changed considerably over the past 20 years. The increase of accessibility and relative fall in the prices of power electronic switches such as SCRs, GTOs, MOSFETs and IGBTs was followed by continuous development of diverse electronic converters. Their application makes it possible to transform electric power with the parameters of the supply network into variable parameters required at the input of induction motors to meet the needs of the effective control of rotational speed. Such control sometimes known as scalar regulation is discussed in detail. The following stage in the development of the control systems of induction machines focused on the improvement of the power electronic devices involved in the execution of commands and, in particular, with the development of processors for transferring information, including signal processors adaptable for industrial applications. As a result, it was possible not only to design and implement dynamic control of drives containing induction motors but also develop control that tracks the trajectory of the position and rotor speed. This type of control is encountered mostly in two varieties.

Field Oriented Control (FOC) and Direct Torque Control (DTC). The methods applied in this respect are discussed.

Construction and Types of Induction Motors

A typical induction machine is a cylinder shaped machine whose ratio of the diameter 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’s achievable and windings are located in the slots () 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 windings, 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’s 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 windings. 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 fractional slot windings. Integral slot winding is the one in which the number of slots per pole and phase is an integer number. Most induction machines apply integral slot windings since they offer better characteristics of magnetic 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’s 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 magnetic field in the air gap resulting from the flow of current through a winding follows as closely as possible a sine curve. 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.

+-+-+-1 Cross section of an induction machine with semi-closed slots in the stator

+-+-+-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

With these rings and by adequate butting contact using brushes slipping over the rings it’s 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’s 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; 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 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’s 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 circumference 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. The difference between the actual shape of the magnetic field in the air gap and the fundamental harmonic of the order ? = p is approximated 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 conductors 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. This air gap despite having its constant engineering width d is in the sense of the magnetic permeance relative to the dimensions of the slots in the stator 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 phenomena. They involve asynchronous and synchronous parasitic torques that de form the basic characteristics of the electromagnetic torque, as well as additional losses resulting from higher harmonics and specific frequencies present in the acoustic signal emitted by the machine.

+-+-+-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 don’t disturb the operation of the drive.

Hence, in the discussion of the driving characteristics the induction motor is represented by a mathematical model whose magnetic field displays mono-harmonic properties. The only harmonic is the fundamental one with the number ? = 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 appropriate 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 effectiveness 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.

Fundamentals of Mathematical Modeling

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 ferromagnetic material in which the windings of the stator operate and the ferromagnetic core is often in the condition of magnetic saturation. The simultaneous and comprehensive 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 consists 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 considered 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’s 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, 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. The issue of the mechanical motion of a rotor is not a complex phenomenon since a typical induction motor has only a single degree of mechanical freedom - angle of rotation of the rotor. 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 don’t take into consideration 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’s most often followed in the issues associated 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 studies devoted to the tolerance of the engineering structure of the machine to its characteristics 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 The presented three categories of models can be described as follows: a mathematical 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 considered 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. 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 calculations in 2D space without affecting the precision of the results. The calculations apply professional software suites using Finite Elements Method (FEM) or Edge Elements Method (EEM). Such software contains procedures making it possible 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 calculation. As one can conclude from this description, field models are applicable 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 parameters 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 determination 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. The mathematical models applied in the issues associated with drive control tend to be the most simplified models. As a principle, they disregard 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 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 resulting from parameter estimation using signals that are easily accessible by way of measurement. From the point of view of the current guide the principal interest focuses on the mathematical models designed for determination of the characteristics of the drives and the ones applied for the purposes of control.

Number of Degrees of Freedom in an Induction Motor

The question of the number of the degrees of freedom is encountered in systems with lumped parameters whose motion (dynamics) is described by a sys tem of ordinary differential equations. For the case of an induction machine this means one degree of freedom of the mechanical motion sm = 1, for variable theta_r denoting the angle of rotor position and the adequate number of the degrees of freedom se for electric circuits formed by the phase windings. For the case when both the stator and rotor have three phases and the windings are independent the number of electric degrees of freedom is sel = 6. The assumption that electric circuits take the form of phase windings with electric charges Qi as state variables does not exclude the applicability of a field model for the calculation of magnetic fluxes ?i linked with the particular windings of the motor. This possibility results from the decoupling of the magnetic and electric fields in the machine and the consideration of electric currents

+-+-+-4 Diagram with cross-section of induction motor:

a) 3-phase stator and rotor windings b) rotor squirrel-cage windings _ = in the machine as sources of magnetic vector potential, (functions that are responsible for field generation).

In this case we have to do with field-circuit models, in which the model with lumped parameters describing the dynamics of an electromechanical system (in this study the induction motor) is accompanied by an interactively produced model of the electromagnetic field in which the present flux linkages ?i are defined. Hence, the model of an induction motor whose diagram has ...

... degrees of freedom. In this place one can start to think about the state encountered in the windings of a squirrel cage motor, which does not contain a standard three-phase winding, but has a cage with m = Nr number of bars. The squirrel cage winding responds to the MMFs produced by stator winding current.

The induced EMFs in squirrel cage windings display the same symmetry proper ties on condition that the squirrel cage of the rotor is symmetric in the range of angular span corresponding to a single pole of the stator winding or its total multiple.

Hence, the resulting number of degrees of freedom ssq for a symmetrical squirrel cage winding is expressed by the quotient ... where: m - the number of bars in the symmetrical cage of a rotor u,v - relative prime integers

The number of the degrees of freedom of the electric circuits of a rotor's squirrel cage winding ssq = u corresponds to the smallest natural number of the bars in a cage contained in a span of a single pole of the stator's winding or its multiple.

This is done under the silent assumption that the stator's windings are symmetrical. If the symmetry is not actually the case, the maximum number of the degrees of freedom of a cage is equal to ... ... which corresponds to the number of independent electric circuits (meshes), in accordance with, in the cage of a rotor.

For the motor, we have p = 2, Nr = m = 22, hence the quotient:

... and, as a result, the number of electric degrees of freedom for a squirrel cage winding amounts to ssq = u = 11. This means that in this case the two pole pitches of the stator contain 11 complete slot scales or slot pitches of the rotor, after which the situation recurs. The large number of the degrees of freedom of the cage makes it possible to account in the mathematical model for the parasitic phenomena, For example parasitic synchronic torques.

However, if we disregard deformations of the magnetic field in the air gap and assume that it’s a plane-parallel and monoharmonic one with the single and basic harmonic equal to ? = p, then in order to describe such a field we either need only two coordinates or two substitutive phase windings, in most simple cases orthogonal ones. For such an assumption of mono-harmonic field the number of the degrees of freedom decreases to ssq = 2 regardless of the number of bars in the rotor's cage. In the studies of induction motor drives and its control the principle is to assume the planar and mono-harmonic field in the air gap. Nevertheless, at the stage when we are starting to develop the mathematical models of induction machines, it’s assumed for the slip ring and squirrel cage machines that the rotor's winding is three-phased for the purposes of preserving a uniform course of reasoning. Hence, as indicated earlier, under the assumptions of a planar and mono-harmonic field in the air gap, slip-ring and squirrel-cage motors are equivalent and can be described with a single mathematical model with the only difference that the winding of a squirrel-cage motor is not accessible from outside, in other words, the voltages supplying the phases of the rotor are always equal to zero. Lagrange's function for a motor with three phase windings in the stator and rotor can take the form: ... and the virtual work (2.198) expressing the exchange of energy is equal to: where: q = (Q1, Q2,…,Q6, theta_r) - vector of generalized coordinates J - moment of inertia related to the motor's shaft, Tl - load torque on the motor's shaft, D - coefficient of viscous damping of the revolute motion, Rk - resistance of k-th phase winding, electric current and supply voltage of k-th phase winding, - magnetic flux linked with k-th winding.

The model in this form already contains two simplifications, i.e. it disregards iron losses associated with magnetization of the core and changes of the windings' resistances following a change in their temperature.

From the above the equations of motion for electric variables follow in the form:

... with the capacitors missing from the system ... Whereas using the designation for currents ...obtain ...From the comparison it results that for the simplicity of notation we should treat the equation in, which is currently considered, as the final one.

In this case the equation for the circuits of an induction motor takes the form ...After the differentiation of the left-hand side we obtain ...The left-hand side expressions denote electromotive forces induced in k-th phase winding as a result of the variations in time of flux linkage. The first terms are derived from the variations of the currents and are called electromotive forces of transformation, while the final term is related to the angular speed of the rotor and is called the electromotive force of rotation. The equation for the torque expressed with variable theta_r takes the following form ...and, consequently, ...

Which, in case of constant moment of inertia J, can be denoted alternatively as ....

... where:... is the electromagnetic torque produced by the induction motor. In spite of the fact that the resulting equations of motion are stated for a system with lumped parameters and in this case for 7 variables corresponding to 7 degrees of freedom of the motor, they can find a very broad application. This results from the general form of the flux linkage associated with the particular phase windings ?k = (i1,…,in, theta_r). It could be gained by various methods accounting for the saturation and various engineering parameters of the magnetic circuit. For a squirrel-cage motor, for the case if one needed to account for the existing parasitic torques, it would be necessary to abandon the starting assumption of the monoharmonic image of the field in the air gap and, hence increase, the number of equations for the phases of the rotor from 3 to ssq, as it results from.

Mathematical Models of an Induction Motor with Linear Characteristics of Core Magnetization

Coefficients of Windings Inductance

In a majority of issues associated with the motion of an induction machine, in particular in the issues associated with the control of drives with induction motors, we can assume a simplification involving an approximation of the characteristics of motor magnetization using linear relation. Hence, the definition of the coefficients of self-inductance and mutual inductance of the machine's windings follows in the form ...For k = l this coefficient is named the self-inductance coefficient and includes two terms:... where Lsk is the leakage inductance, which results from the magnetic flux linked solely to k-th phase winding. In contrast, for this coefficient is denoted as mutual inductance coefficient ...

The determination of the coefficient of the inductance of an induction machine could be derived from measurements on an existing machine or could be based on calculations, which is already possible at the stage of motor design. The experimental studies, which can serve in order to determine the coefficients of winding inductance, involve the measurements of the characteristics of the idle running of a motor, short circuits - for the purpose of stating the inductance of the leakage and other tests - for instance of the response to a voltage step function. On this basis it’s possible to establish the approximate parameters of a machine, including the inductance of the windings as well as to apply the methods for the estimation of the parameters from selected measurement characteristics. The calculation methods involve the calculation of the field in the machine using field programs, which provides information regarding integrated parameters, including inductance. For an induction motor it’s sufficient to assume calculations of plano-parallel field (2D) with supplementary data and corrections regarding the boundary section of the field in the machine. In particular this concerns leakage inductance of the end winding section of the windings. Moreover, a number of analytical methods has been developed for the calculation of the field and inductance coefficients in an induction motor, thus providing valuable information for induction motor models. However, these tend to be less precise than the ones that result from field calculations since they account only for the major term of the energy of the magnetic field, i.e. the energy of the field in the machine's air gap. In the fundamental notion, under the assumption of monoharmonic distribution of the field in the gap, the coefficients of mutual inductance take the form:.... where: M - value of inductance coefficient for phase coincidence … ak, al - angles which determine the positions of the axes of windings k,l In accordance these angles are:... for the stator's windings and rotor's windings, respectively.

The number of the pole pairs p reflects p-time recurrence of the system of windings and spatial image of the field along the circumference of the air gap. On the basis of relations in, the matrix of the inductance coefficients of stator's windings takes the form ...and similarly, for the windings in the rotor ...

At the same time, mutual inductance matrices between stator and rotor windings are relative to the angle of rotation…

In the above equations: Lss, Lsr - are leakage inductance coefficients of stator and rotor windings Ms, Mr - main field inductance coefficients of stator's and rotor's windings Msr - mutual inductance coefficient of stator's and rotor's windings for full linkage between the windings (aligned position of windings' axes) 13 - unitary matrix with dimension 3.

Model with Linear Characteristics of Magnetization in Natural (phase) Coordinates

At the beginning it’s necessary summarize the simplifying assumptions for this model of the induction motor, starting with the most important ones:

- complete geometrical and material symmetry of the electromagnetic structure of the motor

- linear characteristics of magnetization of the electromagnetic circuit

- planar and monoharmonic distribution of the field in the air gap, resulting in a single harmonic with the number

- disregarding of the losses in the iron

- disregarding of the external influence ( For example temperature) on the parameters of the motor.

Since, in accordance with the second assumption, we consider a linear case of magnetization, on the basis of the following relation is satisfied

The equation concerning the revolute motion remains unchanged, it’s necessary only to apply a more detailed expression to determine electro magnetic torque of the motor.

Transformation of Co-ordinate Systems

Most of the dynamic curves and solutions with regard to control are not conducted in phase coordinates, such as the case of the mathematical model but they are stated in the transformed coordinates. There are several reasons for that: an adequately selected transformation is capable of transforming the system that contains periodically variable coefficients (trigonometric functions)

into a system with constant parameters. Thus, there is no need to apply large computing power and the cost thereof is reduced, which is particularly relevant in the issues of drive control in real time. In addition, as one can conclude from the form of mutual inductance matrix of the windings (), that their order is not 3 but only 2. They have one dimension too many, which can be concluded by adding up all the rows in each of the matrices. The physical reason is self-evident: in order to describe a monoharmonic planar field it’s sufficient to use two variables.

Hence, the field can be produced by currents in two phase windings that are not situated along a single axis (perpendicular axes are most applicable). The early applications of the transformation of the coordinate systems originate from Park and served in order to analyze the operation of synchronous generators. The general theory of transformation of coordinate systems in multi-phase electric machines is based on the Floquet's theorem. From it results that for linear systems of ordinary differential equations with time periodic coefficients it’s possible to identify such a transformation for which in the new defined coordinates the machine's equations are independent of the angle of rotation of the rotor. In the particular cases (monoharmonic field distribution) it’s possible to gain this result by the application of transformation T( theta_r), i.e. only relative to the position of the rotor. The examples of solutions and applications in this area are multiple and can be found in the bibliography. From the technical point of view it’s only sensible to apply orthogonal transformations in electric drive. This means the ones whose matrices fulfill the condition that ... This is the case when the vectors which form the matrix of transformation T are orthogonal and have an elementary length, that is: where vi is the column (row) of the matrix T. This property is indispensable since orthogonal transformations preserve the scalar product and square form for the transformed vectors. The scalar product for the case of an electric machine corresponds to the instantaneous power delivered to the clamps of the machine ...

As a result of transformation of the above expression, using orthogonal transformation T, we obtain:

... are transformations of the vectors of the voltage and current of a three-phase electric machine. Concurrently, electromagnetic torque in the multi-phase machine is in the algebraic sense expressed by the quadratic form ...... is the transformed matrix of the derivatives of mutual inductance between the windings. As one can conclude from, the variables have been transformed into the form * i , while due to the orthogonality of the matrix of transformation T the electromagnetic torque remains unchanged. Under the assumptions adopted at the beginning of this section the machine has a mono-harmonic field in the air gap, which results in the fact that the matrix of mutual inductance between the stator and rotor is relative only to argument of the periodic functions. This enables one to easily identify the orthogonal transformation of T such that orders the mathematical model in the sense of leading to the constant coefficients of the differential equations. This study applies the following orthogonal matrices of the transformation:

- for the quantities relative to 3-phase windings in the stator ...

- for quantities relative to 3-phase windings in the rotor

where: _c - is an arbitrary pulsation that constitutes the degree of freedom of the planned transformation:

a = 2p/3 - argument of the symmetric phase shift.

The transformation of the particular phase variables occurs in the following way:

Next, we will proceed to see how this works for a system of symmetric 3-phase sinusoidal voltages of frequency f supplying stator's windings:

... where: _s = 2pfs.

As a result of the transformation the voltage us0 = 0, and voltages usu and usv form an orthogonal system. In general, transformations of Ts,Tr lead to the restatement of the phase variables in the stator's or rotor's windings for two perpendicular axes 'u,v' which are in revolute motion with the arbitrary angular speed of ?c . The third of the transformed axes - axis '0' is perpendicular to axes 'u,v' and acts in the axial direction; hence, it does not contribute to the planar field of the machine.

+-+-+-5 Illustration of the orthogonal transformation of 3-phase windings of AC machine into rotating 'u,v' axes

Transformed Models of Induction Motor with Linear Characteristics of Core Magnetization

Model in Current Coordinates

Firstly, we transform equations for electric variables - currents is, i theta_r. For the equations of the stator windings

The models of the induction motor presented in 'xy' system are particularly applicable in the control, which comes as a consequence of the simplicity of the equations for the motor in this system. This is especially discernible for Field Oriented Control (FOC) techniques.

Mathematical Models of Induction Motor with Untransformed Variables of the Stator/Rotor Windings

There is a number of practical reasons why it’s beneficial to preserve untransformed variables on one of the sides of an induction motor. This means that the variables which define the electric state in the stator's or rotor's windings remain in the form of natural variables, while the state of the connections between the windings is maintained by the introduction of adequate equations with constraints resulting from Kirchhoff's laws. The preservation of the untransformed variables and the resulting equations of motion for a single side enables one to de rive the so-called internal asymmetry within windings, which are defined using natural variables. Hence, it’s possible either to incorporate arbitrary lumped elements in the particular phase windings or apply asymmetrical supply voltages with arbitrary waveforms. In particular, as a result of this, it’s possible to perform the calculations for the braking with direct current for any connections between phase windings, operation under single-phase supply, operating with an auxiliary phase for capacitor starting a single-phase motor, analysis of a series of emergency is sues and select safety measures.

However, the most important application of the mathematical models of this type is in the modeling of electronic power converters in combination with the supplied machine, in which power transistors or silicon controlled rectifiers (SCRs) are designed for the control of voltages and currents in the particular windings. This type of modeling, which has been the object in numerous research, can be most effectively conducted in the circumstances of pre serving a fixed structure of an electric system by the introduction of resistances with variable values corresponding to the state of the examined power electronic switches in the particular branches of electric circuits. In the blocking state they assume high values limiting the flow of current across a certain branch, while in the conducting state the values are small, i.e. ones which correspond to the parameters of conduction calculated on the basis of the data for such components taken from manufacturers' catalogues. This type of modeling is associated with a number of impediments in numerical calculations of the curves of examined variables due to time constants in the particular electric circuits, whose values differ by several orders of magnitude, as a result of application of high value blocking resistance. For this reason these models apply stiff methods of integration for Ordinary Differential Equations (ODFs).

Another solution to be applied involves the use of simple single-step procedures for the solution of stiff systems with a small step size. In any case, however, power semiconductor switches encountered in the branches of electric motors are more easily modeled for the case when variables for a given circuit are the phase variables. The reason is associated with the fact that the state of a given power semiconductor switch is relative to the control signals and forward current in this element. In the issues of control of squirrel-cage induction motors semiconductor systems are members of the circuits of the stator in a machine, while the variables concerning squirrel-cage windings of the rotor are transformed to the orthogonal axes u,v . However, if control occurs in the rotor of a slip-ring motor and the control elements are situated there (including the converter), it’s beneficial to have untransformed variables (phase currents) in the rotor. But then the electric variables of the stator's windings could as well be transformed into orthogonal axes u,v for the purposes of succinct notation.

Model with Untransformed Variables in the Electric Circuit of the Stator

We shall assume that we deal with a 3-phase motor with windings connected in a star. As a result, when the currents of the phase windings 1 and 2 are considered as state variables, the system of stator windings is characterized with constraints.

+-+-+- Star-connected 3-phase stator windings

Model with Untransformed Variables of Electric Circuit of the Rotor

This model finds application in the issues regarding internal asymmetries and control of drives with a slip-ring induction machine. For example, it’s applied in the calculations of the start-up of a slip-ring motor with asymmetric resistances during the start-up, atypical systems of connections between phase windings of the rotor, analysis of cascade systems with a slip-ring motor, for instance the Bi-directional cascaded system. In the discussed example the electric variables of the stator's circuits undergo axial transformation along u,v axes, while the variables of the rotor's circuits remain untransformed. Under the assumption of the star connection of rotor windings, we have to introduce equations of constraints. This time to en sure symmetry of the resulting equations the reference phase is the one denoted with the number 2 and, as a consequence, the current constraints offer the elimination of the current i theta_r^2, while voltage constraints refer inter-phase voltages to the terminals of phase winding 2:

... which results in the symmetry of matrices in equations. The assumption of the similar course of action leads to the transformation of the system of equations in the following way...

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