Method of estimating transients in induction machines

The present paper is concerned with induction motors with wound and squirrel-cage rotors. It is assumed that the magnetic field produced by the stator windings is constant in magnitude and rotates with constant angular velocity. Differential equations that describe relations between the electromagnetic torque and main electric and mechanical quantities of the induction motors under consideration are derived in detail, the geometry of the rotors of motors being fully taken into account. A special nonsingular transformation is applied to the initial systems of equations (with angular coordinates) to split them up and reduce to a lower-order (in fact, third-order) system. A local stability analysis of the resulting equations is carried out. The stable equilibrium states that correspond to the operating modes of induction motors are determined. Methods of speed control of induction motors with wound and squirrel-cage rotors are considered. The limit load problem on motors and the control problem of the speed of motors are discussed; these problems lead to the necessity to estimate the transients in induction motors. The method of estimating transients that occur due to changes of the operating parameters of motors is developed based on a modification of the nonlocal reduction method. Using this method in combination with Barbashin and Tabueva’s methods to apply to the obtained systems made it possible to find analytical estimates of the ultimate permissible loads on induction motors and the control ranges of the parameters of the system that correspond to additional external active and inductive resistances. Moreover, estimates for the region of the attraction of stable equilibrium states of systems that describe the dynamics of induction motors are obtained.


INTRODUCTION
At present, induction motors are the most common AC electric motor and the major electrical energy consumer. Because of the simplicity of the design and high operational reliability, induction motors are used in most modern electric drives, e.g., in rolling mills, metal cutting machines, conveyers, excavators, drill rigs, and mills. The practical application of these motors involves a number of applied problems related to variations in the operation modes of a motor and the operational parameters. These problems include, first of all, the control problem of the rotation speed of induction motor and the limit load prob lem. In practice two methods of controlling the rotation speed are possible, i.e., stepless (required in elec tric trains and hoisting equipment) and steplike (applied in metalworking machines, rolling mills, mills, and in many industrial electric drives). In this paper, we will be concerned with the stepwise control of the rotating speed of a rotor of a motor and stepwise variation of the external load torque, which leads us to the problem of estimating transients in induction electric machines.
The configuration of the windings of rotors of induction electric machines is quite difficult to describe mathematically. In the present paper, we describe the windings and perform the analysis of rotors dynam ics under the natural (in our opinion) simplification of the model [1][2][3][4]. We assume that the magnetic field produced by the stator windings is constant in value and rotates with a constant angular speed. This assumption, which dates back to classical ideas by Tesla [5] and Ferraris [6], allows to describe in more details the electromechanical models of induction motors. An investigation of transients in induction machines suggests the consideration of differential equations, to which are applied some modification of the nonlocal reduction method [3,[7][8][9]. LEONOV et al.

ELECTROMECHANICAL AND MATHEMATICAL MODELS OF INDUCTION MOTORS
The basic constructive elements of induction machines are the fixed stator and the rotating rotor. Windings are placed on the stator, with the rotating magnetic field being produced by the winding when supplying the AC current.
Depending on the construction of the rotor windings, the induction electric machines subdivide into two types, i.e., with a wound rotor (Fig. 1a) and squirrel cage rotor (Fig. 1b). The winding of a squirrel cage rotor has the form of a "squirrel cage," which consists of n bars and end rings connected to the bars. A three phase winding made of three coils is placed in slots of a wound rotor. Each coil is made of some turns of an insulated wire. Some ends of coils are connected in star, and another ends are connected to the slip rings. Using these rings and the brushes laid on them, the circuit of the rotor winding is either short circuited cage or is connected to external devices like rheostat, inductors, or windings of other electrical machines.
Assume that the magnetic field produced by the stator winding is constant in value and rotates clock wise with constant angular speed n 1 . We shall consider the motion of rotors in the system of coordinates rigidly connected with the magnetic induction vector B that rotate with it. We assume that the positive direction of the rotor axis coincides with the direction of the rotation of the magnetic induction vector.
The motion of a rotor of an induction motor in the chosen coordinate system is described by the equa tion of the rotation of a rigid body around the fixed axis as follows: where θ is the mechanical angle of rotation of the rotor, J is the torque of inertia of the rotor with respect to the shaft, M em is the electromagnetic torque and M l is the external load torque. In what follows, the external load moment will be assumed to be constant in magnitude.
We first study an induction motor with wound rotor with a rheostat in the winding circuit of the rotor (Fig. 2). In this case, the electromagnetic torque is produced by the electromagnetic forces (Fig. 3a), which take place as a result of the interaction of coils that carry a current and rotating magnetic field in accordance with the law of electromagnetic forces. The value of the electromagnetic force in a coil is determined by Ampere's law as follows: where l is the length of the coil side, B is the amplitude value of the magnetic induction vector, and i is the current in the coil.
Let us determine the torque produced by the electromagnetic forces F k , k = 1, …, 6. The projections of forces F 1pr and F 2pr (Fig. 3, b), which act on the coil carrying a current i 1 , are calculated by Hence, taking into account the number of turns in the coil and the positive direction of the axis of the rotor, we find that the resulting electromagnetic torque, which acts on the coil with current i 1 , is as follows: Here, n is the number of turns in each of the coils, l 0 is the length of the radius vector, α is the angle between the radius vector pointing towards the coil with current i 1 and the coil plane, S = 2l 0 lcosα is the area of one coil turn, and i 1 is the current in the first coil. The electromagnetic torques that act on the coils with currents i 2 , i 3 are defined similarly. Thus, we have Here, i k is the current in the kth coil. Hence, Under the action of the electromagnetic torque, the rotor begins to rotate with frequency n 2 , where the direction of rotation coincides with that of the magnetic field.
Let us find the currents in each of the coils. For this purpose, we consider the circuit of the winding of a wound rotor as shown in Fig. 4; this circuit is equivalent to the one shown in Fig. 2.
Using Kirchhoff's first law for junction 1, we obtain the following relation for the currents:  Using Kirchhoff's second law and describing the contour in the positive (clockwise) direction, we obtain the following differential equations for closed contours of the winding circuit of a wound rotor: (3) where R and L are the active and inductive resistances of each coil, respectively; r is the variable external resistance supplied to the slip rings of a wound rotor; and ε k is the EMF induced in the kth coil by the rotating magnetic field.
According to the law of electromagnetic induction, the EMF that arises in the first coil that moves in the magnetic field is calculated as follows: ; here v 1 , v 2 are the velocities of the coil relative to the magnetic field, and their directions are depicted in Fig. 5a; ζ 1 , ζ 2 are the angles between the velocity vector and the magnetic induction vector. The angle ζ 1 is defined in Fig. 5b. Thus, taking into account the number of turns, the EMF in the first coil equals The expressions for the EMF in the remaining coils are obtained similarly, i.e., Fig. 4. Equivalent electrical circuit of winding of a wound rotor. The system of differential equations (4) describes the currents i 1 , i 2 , i 3 in the winding of a wound rotor. Thus, the system of differential equations (5) describes the dynamics of an induction motor with wound rotor with a rheostat in the winding. Applying the nonsingular transformation of coordinates (6) system (5) may be transformed to (7) where The variables x, y, z define electric quantities in the winding of a wound rotor, and s is responsible for the rotor slip. The slip is used to compare the rotor speed of a loaded induction machine with the rotation speed of the magnetic field as follows: Hence, the rotor of an induction motor stops at s = n 1 . In the case s > n 1 , the rotor starts to rotate back wards. This means that the load torque exceeds the electromagnetic torque produced by the motor. Thus, the slip under operating modes will not exceed the rotation speed of the magnetic field, i.e., Note that, if the winding circuit of a wound rotor involves inductors, then the behavior of an induction motor is described by the same system of differential equations (7) with the parameters (9) where l is the variable external inductive resistance supplied to the slip rings of a wound rotor.
Note that the first and last equations of system (7) can be integrated independently of the rest of the system and has no effect on its stability. Hence, below, we will be concerned with the system of third order differential equations: (10) Consider the structural scheme of a squirrel cage rotor consisting of n bars and the rings connected to the bars at the ends (Fig. 6).
Let us find the motion equation of the given rotor relative to the rotating magnetic field. Taking into account the positive direction of the rotation of the rotor axis, the value of the electromag netic force in the kth bar (its direction is indicated in Fig. 6) and the torque acting on the kth bar are cal culated as follows: (11) , where i k is the current strength in the kth bar. Thus, the motion equation of a squirrel cage rotor with respect to the rotating magnetic field has the following form: The EMF induced in the cages bars (the direction of the EMF is shown in Fig. 7) is as follows: (14) where B is the amplitude value of the magnetic induction vector, v is the velocity, α is the angle between the magnetic induction vector and the velocity vector, and l 0 is the length of a bar. If the sign of sinα changes, then direction of the EMF in the bar will also change. For the kth bar, where θ is the angle between the magnetic induction vector and the radius vector directed towards the nth bar. Taking into account the positive direction of the rotor axis, the velocity is calculated as follows: Here, l is the distance from the center to the kth bar (the length of the radius vector). Hence, the formula for the EMF of the kth bar (14) becomes (15) We now proceed from Fig. 1b to Fig. 8, which shows the circuit of the squirrel cage rotor. It has 2n junctions.
Using Kirchhoff's first law for junctions 1, 2, …, n, we obtain the following relations for the currents: Here, i k denotes the current that conventionally flows in (out) the junction k (the junction k'); the currents conventionally flowing from the kth to the (k + 1)th junction are denoted by i k, k + 1 . It is worth noting that all directions of the currents depicted in Fig. 8 are conventional. If the direction of the current was chosen incorrectly, then the current will be negative in solving the system.
We now express i 1 of (16) in terms of i k as follows: Using Kirchhoff's second law and traversing the contour in the positive direction, we arrive at differ ential equations for closed contours of the squirrel cage circuit as follows: (18) where l and l 0 are the radius and length of the squirrel cage, respectively, and θ is the angle between the radius vector pointing towards the bar with current i n and the magnetic induction vector B.
Substituting the expression for the current i 1 , as given by (17), into the first equation (18), we obtain (19) Then, from Eq. (19), we will in turn subtract the last equation of (18), subtract the next to last equation twice, and so on. At the last (n -2)th step, we get (20) Thus, we have Applying (21) and (18), we obtain the system of differential equations (22) that describes the currents in bars of a squirrel cage rotor as follows: The system of differential equations (22), (13) is a system of equations of an induction motor with a squirrel cage rotor. Let us transform system (22), (13) to more convenient form. For this purpose, we introduce the addi tional assumption that n = 4m. This assumption is legitimate, since the number of bars used in modern squirrel cage rotors is usually divisible by four. Consider the following transformation of coordinates: We proof that this transformation is nonsingular. We denote β = -2L/(nl 0 lB), The transformation of coordinates (23) is nonsingular, if the Jacobian is nonzero: We show that the determinant of the matrix D is independent of θ. For this, we expand the determinant of the matrix D along the first row as follows: where is the complementary minor to the element (1, i) of the matrix D. Next we expand each com plementary minor along the first row as follows: Clearly, for the complementary minors , the following property holds: (25) We set: Hence, expression (24) can be written as follows: It easily follows that Hence, As a result, the determinant of the matrix D is independent of θ. Thus, the determinant of the matrix M is constant, which depends only on the parameter n. Numerical analysis using MATLAB has shown that detD = 4β 2 for n = 4, detD = 241.137β 2 for n = 8, detD = 3.4 × 10 5 β 2 for n = 12, detD = 2.8 × 10 9 β 2 for n = 16, detD = 8.24 × 10 13 β 2 for n = 20, detD = 6.02 × 10 18 β 2 for n = 24, detD = 9.42 × 10 23 β 2 for n = 28, detD = 2.8 × 10 29 β 2 for n = 32. The numerical calculation of detD can be continued further; however, in the main, the number of bars in a squirrel cage rotor is at most 32.
No. 4 2013 Therefore, this proves the nonsingularity of the transformation of variables (23) in the case n < 32. Then, using the equations for the currents (22), we see that Then system (22), (13) can be transformed into the form: In this new system (27), the variables x, y, z k define the electrical values in the rotor bars, the variable s defines the rotor slip. Note that, in the last system of differential equations, the last n -2 equations can be integrated independent of the rest of the system and have no effect on its stability. The remaining equa tions, except for the first one, are independent of θ and, hence, as in the case of a wound rotor, we will study the system of third order differential equations (10). Thus, the system of differential equations (10) describes the dynamics of induction motors with a squirrel cage and a wound rotor. Under the condition 0 ≤ γ < a/2, this system has two equilibrium states; in the case γ > a/2, it has no equilibrium states (its stationary set is empty).
The equilibrium state is asymptotically stable. It corresponds to the statically stable steady state operational mode of an induc tion motor with squirrel cage rotor or with wound rotor. This means that, after small, short term varia tions in the external forces (under a quite small short term perturbation of the steady state mode) the motor goes back to the steady state mode. We refer to this mode as operating modes.
The following equilibrium state is unstable: It corresponds to a statically unstable operation mode of the motor; in other words, under an arbitrary small perturbation, the motor does not go back to the previous steady state mode. This mode is referred to as physically unrealizable. For γ = 0, system (10) has a unique asymptotically stable equilibrium state (x = 0, y = 0, s = 0), which corresponds to the operating mode of an induction motor at no load.
Note that the value s is determined from The function (31) is called the static mechanical characteristic of an induction machine with squirrel cage and wound rotors. It is depicted in Fig. 9 for a wide slip range, which includes all possible modes of an induction machine. The static mechanical characteristic ϕ(s) makes it possible to describe the behavior of system (10) upon variations in the load parameter γ or slip s, to determine the stability region, and to find out the critical values of these parameters [10,11]. The operating modes of an induction motor correspond to the period during which the mechanical characteristic increases (s ∈ (0, c)). The physically unrealizable mode cor responds to the interval in which the static mechanical characteristic (s > c) decreases. At a constant speed of rotation of the magnetic field n 1 , the rotation speed of an induction motor n 2 = n 1 -s (or the slip s) are determined using a given external load torque M l from the form of the static mechanical characteristic of a motor. Indeed, since the steady state operating mode of a motor is characterized by the equality ϕ(s) = γ, the motor operates with a given slip depending on the form of the static mechanical characteristic.
Expression (31), which is the slip function s, shows that, by varying the parameters a and c, it is possible to change the static mechanical characteristic of induction motors with wound and squirrel cage rotors and, hence, to control their rotation speed. Consider the principal control cases of these parameters.
It follows from (8), (9), and (28) that parameter a can be controlled by varying the amplitude value of the magnetic induction vector B by changing the voltage supplied to the stator of a motor. This way of con trolling the rotation speed is applicable both for induction motors with squirrel cage rotors and for induc tion motors with wound rotors.
An induction motor with a wound rotor has more possibilities for controlling the rotation speed due to including external devices in the rotor circuit. Introducing a rheostat into the circuit of a wound rotor changes the resistance to change the parameter c of system (10). Note that expression (8) shows that the maximum of the static mechanical characteristic of an induction machine is independent of the value of the resistance of the circuit of the rotor winding. At the same time, the slip s m = c, under which the max imum is attained, is proportional to the resistance. Hence, for increased values of the resistance, the max imum of the static mechanical characteristic of an induction machine shifts in the direction of greater slips, its value being the same (Fig. 10). Thus, an increase in the resistance of the circuit of a wound rotor results in an increase of the slip s m , which extends the control range of the rotation speed. However, this type of control of the rotation speed involves depositing a significant electric power in the rotor circuit of the motor, with most of this power being lost in the rheostat. Instead of a rheostat, one may introduce an inductor in the circuit of a wound rotor to change its induc tive resistance. In this case, it follows from (9) that parameters a and c of system (10) will simultaneously change. Sometimes, instead of a rheostat, the windings of other induction machines are connected to the connecting rings of a wound motor [10,11]. In this case, the effect of an auxiliary machine will also be simultaneously felt on parameters a and c.
Another way of controlling the rotation speed of induction motors depends on the variations in the rotation speed of the magnetic field n 1 . This approach involves using power sources with controlled fre quency and, hence, it is appropriate to use this type of control only for groups of motors, e.g., to control the motors of powerful rolling mills [10,12]. In what follows, we will consider only the first way of con trolling the rotation speed, i.e., controlling the slip s of a motor at a constant rotation frequency of a mag netic field.

METHOD OF ESTIMATING TRANSIENTS
In the application of induction motors, changes in the parameters of the system may result in tran sients, e.g., due to variations in the external load to the rotor shaft or change in the grid voltage [13]. The study of transients in induction motors is related to the determination of the regions of attraction of stable equilibrium states of systems that describe the dynamics of induction motors.
Transients are characteristic of transitions from one mode to another. Assume that an induction motor, which is described by the system of differential equations (10) with parameters a = a * , c = c * , γ = γ * , works in the operating mode. The asymptotically stable equilibrium state of the system s = s * , x = x * , y = y * , which is defined by (29), corresponds to this mode. Assume that, at some time instant, the operation parameters of an induction motor change. Hence, the parameters of system (10) change from a = a * , c = c * , γ = γ * to a = a 0 , c = c 0 , γ = γ 0 . In what follows, for brevity, we omit the subscript 0 on parameters of the system. It is necessary to find the conditions under which the motor pulls in a new operating mode after a transient. The mathematical statement of the problem is as follows: to find the conditions under which the solution of system (10) with initial data s = s * , x = x * , y = y * would be in the region of attraction of the stable equilibrium state s 0 , x 0 , y 0 ; that is, the following relations must be satisfied: The values of s 0 , x 0 , y 0 are determined similarly to those of s * , x * , y * using formulas (29), where instead of * 's, we use 0's. The next theorem makes it possible to estimate the transients that arise in induction motors and is a modification of the nonlocal reduction method [1,2,7,8]. We introduce the notation Theorem 1. Assume that γ < 2c 2 , s * < s 1 and for solution of the equation with initial data F(s 1 ) = 0, the condition is satisfied. Then, the solution of the system of differential equations (10) with initial data s = s * , x = x * , y = y * satisfies relations (32).
Proof. Changing the variables reduces system (10) to the form (35) and transforms the initial data s = s * , x = x * , y = y * of system (10) to the initial data s = s * , η = ay * + γ, Under the condition 0 < γ < a/2, the stationary set of system (35) consists of two points, i.e., one asymptotically stable (s = s 0 , η = 0, z = 0) and one unstable (s = s 1 , η = 0, z = 0). Then, for the new system (35), relations (32) take the following form: We claim that any bounded solution of system (35) tends to the equilibrium state. Consider the func tion for which the inequality (37) holds for the solutions of system (35), provided that γ < 2c 2 .
Let x(t) = (s(t), η(t), z(t)) be the solution of system (35), which is bounded for t ≥ 0. Then, the function V(s(t), η(t), z(t)) is also bounded for t ≥ 0. It follows from (37) that the function V(s(t), η(t), z(t)) does not increase in t for t ≥ 0. Hence, we have the finite limit Since the trajectory x(t) is bounded, it follows that the set Ω of its ω limit points is nonempty. Let ξ be some ω limit point. The set Ω is invariant, and therefore the trajectory going from the point ξ is contained in Ω for all t. Hence, for all t ∈ ‫,ޒ‬ Using (37), we obtain the identities η(t, ξ) ≡ 0, and z(t, ξ) ≡ 0. Relations (35) and (37) imply that (t, ξ) ≡ 0. Hence, s(t, ξ) ≡ const and the set Ω is a subset of the stationary set of system (35).
Thus, any bounded solution of system (10) tends to an equilibrium state. Then, in view of (34), the two following cases are possible for the solution F(ω) of equation (33): (1) either there exists a number s 2 such that, for F(ω), (2) or the following inequality is satisfied: Consider the function Using (34), we obtain Hence, the points (s * , ay * + γ, -x * -γ/(ac)) and (s 0 , 0, 0) lie in Ω and Ω 1 . It follows from condition (39) that Ω 2 also contains these points. Thus, since the sets Ω and Ω * are bounded and positively invariant and, furthermore, since it follows that relations (36), which are equivalent to relations (32), hold.
4. THE LIMIT LOAD PROBLEM Consider the limit load problem [2,[14][15][16][17][18][19] for the system of differential equations (10), which describes induction motors with squirrel cage and wound rotors. This problem arises in the use of induc tion motors under abrupt variation of the load torque on the shaft, e.g., when a motor is used in the drives of a metal cutting machine or of a drill rig. A load on may be long or short in time. In both cases, there is a problem of determining the admissible load under which a motor pulls in a new stable operation mode after a transient.
As was shown above, the stationary solution s = x = y = 0 of system (10) corresponds to the synchro nous no load operation mode (without the load, γ = 0) of a motor. Then, at some instant of time t = τ, there is an instantaneous load on γ > 0. It is required to determine the limit permissible load under which an induction motor pulls in a new operation mode. In other words, one must find the conditions under which relations (32) are satisfied for the solution of system (10) with initial data s = x = y = 0.
The following corollary to Theorem 1 allows one to estimate the limit instantaneous load on an induc tion motor working under no load. Corollary 1. [3,20]. Assume that γ < 2c 2 and for solution of the equation with initial data F(s 1 ) = 0, the condition (41) is satisfied. Then, relations (32) are satisfied for solution of the system of differential equations (10) with initial data s = x = y = 0.
Using Corollary 1, we obtain an analytical estimate of the limit permissible loads on induction motors with squirrel cage or wound rotors. Corollary 2. [3] If 5c 2 ≥ 2a and if then the load surge γ is permissible.

CONTROLLING ROTATION SPEED OF INDUCTION MOTOR
Consider the control problem of the rotation speed of an induction motor. Assume that the stable equi librium state s = s * , y = y * , x = x * of system (10) with parameters a = a * , c = c * corresponds to the oper ating mode of an induction motor under the load (γ > 0), with this state being defined by formula (29). Then, at some time instant τ, it is necessary to change the rotation speed of an induction motor. For this purpose one changes the voltage supply to the stator or, in the case of an induction motor with wound rotor, the additional active resistance is changed for r = r * or the additional inductive resistance is changed for l = l * in the rotor circuit. Hence, the parameters of system (10) will change from a = a * , c = c * to a = a 0 , c = c 0 (for brevity, we drop the subscript 0). As a result, a transient arises that is related to changes in currents in circuits and in the slip. The operating mode of the motor changes to s = s 0 , x = x 0 , y = y 0 .
In this case, the problem of controlling the rotation speed of an induction motor with a wound rotor is posed as follows: to find for which values of the supply voltage, additional active resistance or additional inductive resistance an induction motor pulls in a new operation mode after a transient. The mathematical statement of the control problem of the rotation speed of an induction motor with a wound rotor under a fixed load is as follows: determine the conditions under which the solution of system (10) with initial data s = s * , y = y * , x = x * would be in the region of attraction of the stationary solution of the system; that is, relations (32) must be satisfied.
First, we study the control of the rotation speed by means of an external inductive resistance: i.e., when a * ≠ a, c * ≠ c. We set Corollary 3. Assume that γ < 2c 2 , s * < s 1 , s 0 < n 1 and for solution of the equation with initial data F(s 1 ) = 0, the condition is satisfied. Then, the solution of system (10) with initial data s = s * , y = y * , x = x * satisfies relations (32).
The proof of this corollary is similar to that of Corollary 1. Corollary 3 to Theorem 1 allows one to estimate the control range of the parameters ρ a and ρ c . In the case when only parameter ρ c varies (when controlling by the additional active resistance, a * = a, c * ≠ c), formula (42) assumes the following form: In the case when only parameter ρ a varies (controlling by varying the voltage through the stator of the motor, a * ≠ a, c * = c), formula (42) reads as follows: (44) Using Corollary 3, we obtain the control ranges of the parameters ρ a and ρ c . Proof. The following estimate for the solution F(σ) was obtained in [21]: Hence, the relations (47) imply that condition (43) holds and, thus, ρ c is a permissible control. We claim that conditions (45) and (46) imply condition (47). To do this, we use the equality c * = to obtain It easily follows that, if the following relations are satisfied: then so is (47). The last two conditions are equivalent to conditions (45) and (46), respectively. Therefore, this proves the corollary. Corollary 5. Assume that c < s 1 , s 0 < n 1 and that Then, ρ a is a permissible control.