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6ISSN 0136-4545 !Ж!урнал теоретической и прикладной механики. №1 (86) / 2024.
©2024. B.I. Konosevich1, Yu.B. Konosevich2
LOCAL AND GLOBAL STABILITY PROPERTIES OF A MODEL OF THE ASYNCHRONOUS ELECTRIC MOTOR
The subject matter of the article is dynamics of the asynchronous electric motor. The investigation is based on the mathematical model of this motor in the form of a third order system of ordinary differential equations, which includes equation for the angular velocity of the rotor and equations for two currents in the windings of the rotor. It is assumed that the load moment is a continuous odd function of the angular velocity of the rotor relatively the stator, and this function permits a linear estimate. Such system of equations of motion of the asynchronous electric motor has a unique stationary solution, describing the steady rotation mode of the rotor. In the article, sufficient conditions of local asymptotic stability is obtained for this solution. Also, it is established that, under these conditions, any solution of the equations of motion tends with time to their stationary solution.
Keywords: asynchronous electric motor, local and global stability, steady rotation.
Introduction. Two base types of electric motors are used in practice, they are synchronous and asynchronous motors. The operation of an electric motor can be described by various systems of differential equations, their structure is determined by the design of the electric motor and by the accepted degree of detailing of the processes occurring in it [1].
In the case of the synchronous electric motor, its operation is described by a system of differential equations with respect to the angle, determining the position of the rotor relatively to the rotating magnetic field in the stator, the time derivative of this angle and electric currents in the windings of the rotor. These system belong to the type of phase systems, that is, it is periodic in the angular variable. Steady
1 Konosevich Boris Ivanovich - doctor of physical and mathematical sciences, Chief Researcher, Institute of Applied Mathematics and Mechanics, Donetsk, Department of Applied Mechanics, email: [email protected].
Коносевич Борис Иванович - доктор физ.-мат. наук, главный научный сотрудник отдела прикладной механики ФГБНУ "Институт прикладной математики и механики", Донецк.
2 Konosevich Yuliya Borisovna - Candidate of Physical and Mathematical Sciences, Researcher, Institute of Applied Mathematics and Mechanics, Donetsk, Department of Applied Mechanics, email: [email protected].
Коносевич Юлия Борисовна - канд. физ.-мат. наук, научный сотрудник отдела прикладной механики ФГБНУ "Институт прикладной математики и механики", Донецк.
МЕХАНИКА ТВЕРДОГО ТЕЛА
UDK 531.36
doi:10.24412/0136-4545-2024-1-5-14 EDN:CUIKIP
state operation mode of the synchronous electric motor is the rotation of its rotor with the angular velocity of rotation of the magnetic field in its stator. If a phase system of differential equations, describing the operation of the synchronous electric motor, has a stationary solution, then it has the countable set of stationary solutions. They can be obtained from the starting solution by displacements on the periods.
An important problem is to establish conditions, when any solution of a phase system tends over time to its stationary solution.
In the case of synchronous motor this property means that each solution of its equations of motion tends over time to one of their asymptotically stable solutions, corresponding to the steady-state rotation mode. To obtain conditions of global stability in this case, Leonov's method of nonlocal reduction can be used [2,3] This method makes it possible to derive the global stability property of a multidimensional ODE phase system from the global stability property of one second order differential equation of a special kind. Within the framework of this method, an effective sufficient global stability condition is obtained in [4] for the multiple-current model of the synchronous electric motor, which was proposed in [5].
The operation of the asynchronous electric motor is described by a system of differential equations with respect to the difference between the angular velocities of rotation of the magnetic fields in rotor and in stator and, also, with respect to electric currents in the windings of the rotor. Such system has a unique stationary solution, where a fixed angular velocity of the rotor is less than the angular velocity of magnetic field in the stator. In the case of asynchronous electric motor, its global stability means that this unique stationary solution of its equations of motion is locally asymptotically stable, and each solution of these equations tends over time to this solution.
In the present article, a sufficient global stability condition is established for the asynchronous electric motor on the basis of its two-current model. Like [4], the load moment is assumed to be a continuous odd function of the angular velocity of the rotor, having a linear estimate.
1. Two-current model of the asynchronous electric motor. This article uses the simplest adequate model of the asynchronous electric motor with two electric currents, which is in the following way obtained in [6] from a two-current model of the synchronous electric motor.
This starting model of the synchronous electric motor corresponds to the case when there are two identical windings in the rotor, made in the form of rectangular frames perpendicular to each other. This is a closed damper winding and an excitation winding, to which a constant voltage is applied through carbon brushes. A uniformly rotating magnetic field is created in the stator. This model is described by the following fourth-order system of differential equations [6]
CY = —SB(i\ sin y + i2 cos y) + Md,
Li{ = -Ri\ + SBY sin y + u, (1)
Li2 = -Ri2 + SBY cos y,
having the phase vector (y, Y, ii, i2). Here y = y — ut is the angle between the plane of the frame with current i1 and the plane perpendicular to the intensity vector of the rotating magnetic field of the stator, y is the angle of rotation of the rotor relative to the stator, u > 0 is the constant angular velocity of rotation of the magnetic field in the stator, i1 and i2 are the currents in the excitation winding and in the damper winding, u > 0 is the constant voltage in the excitation winding, Md is the load moment, C is the axial moment of inertia of the rotor, L and R are the inductance and the resistance of both frames, B is the magnetic field strength in the stator, S is the area of each of the frames. The load moment Md is assumed to be a continuous monotonously decreasing odd function Md = Md(y) of the angular velocity of the rotor y = u + Y.
Two-current model of the asynchronous electric motor can be described by the fourth-order system of differential equations, which is formally derived from the system (1) at u = 0 [6]:
CY = —SB(i1 sin y + i2 cos y) + Md(y),
Li{ = —Ri1 + SBY sin y, (2)
Li2 = —Ri2 + SBY cos Y-
Following [6], we transform system (2) to a third order system. Instead of currents i1 ,i2, we introduce variables x,y according to the formulas
x = (ncos7 - ¿2 sin7), y = -gjjih sin7 + i2 cos 7). (3)
From the second formula (3) we find ¿isin7 + i2 cos 7 = ^-y. Substituting this expression into the first equation (2), we bring it to the form
CY = -f3{-^y + Md(v). (4)
Further from formulas (3) we obtain expressions of currents i1,i2 through x,y, y:
SB SB
%i = —— (a;cos7 + y sin 7), i2 = —— (—a; sin 7 + y cos 7). (5)
LL
Substituting (5) into the second and the third equations (2), we find the expressions of the derivatives i{, i2 through x, y, y, Y:
RSB
Li\ =---— (a; cos 7 + y sin 7) + 6.07 sin 7,
L (6) RSB
Li2 = —-— (x sin 7 — y cos 7) + 6B7 cos 7. L
Differentiating now (3) by t and using (3), (6), in addition to (4) we obtain expressions for X,y. As a result of the change of variables (3), equations (2) of the two-current model of the asynchronous electric motor are converted to the following equations
(SB)2 ^ ... . . R ... ^ R C7 =--—y + Md(<p), x = -jy-—x, y = rf{x + l)-—y.
Introducing the notation a = (SB)2/L, b = R/L, we write down equations of the two-current model of the asynchronous electric motor in the form
C'Y = -ay + Md(w + Y), x = -bx - Yy, y = -by + y(x + 1), (7)
where a,b > 0 are constant parameters. These equations are equivalent to the third-order normal system with the phase vector (Y, x, y). Thus, as the result of the change of variables (3), the closed third-order system (7) of ordinary differential equations is obtained for the two-current model of the asynchronous electric motor, and this system does not contain the angular variable y.
2. Stationary mode of uniform rotation of the asynchronous electric motor. The operating mode of uniform rotation of the asynchronous electric motor corresponds to the stationary solution of equations (7). The following theorem of the existence and uniqueness of such solution is valid.
Theorem 1. Let
1) the function Ma(Y), called the static characteristic of the asynchronous electric machine [6], is defined by the formula
Ma(7) = (8)
2) in the equations (7), the dissipative moment Md(fi) is a continuous odd monoto-nically decreasing function of the angular velocity ^, and in the case when w - b > 0 this moment, being negative on the interval (0,w - b], satisfies the condition of smallness on it
Md(<£) >Ma(^ - w), (9)
where Ma(fi -w) = Ma(j) is the static characteristic (8) of the asynchronous electric motor, w > 0 is the the angular velocity of rotation of the magnetic field in its stator.
Then
a) the differential equations (7) of the two-current model of the asynchronous electric motor have a stationary solution
Y = Y°,x = x°,y = y°, (10)
when the number triplet Y°,x°,y0 is the solution of the following system of finite equations
-ay0 + Md(w + Y0) = 0, bx° + j°y° = 0, -by0 + j°(x° + 1) = 0; (11)
b) when conditions 1), 2) are satisfied, equations (11) have the unique solution (10); in this solution, the constant Y0 is expressed by the formula
Y0 = w0 - w, (12)
where w0 (0 < w0 < w) is the unique solution to the equation
Md(<p) = Ma& - w), (13)
relative to y, and constants x ,y are equal to
y° = ^Md(uj0), x° = ^(uj-uj0)Md(uj0). (14)
Proof. Substituting expressions (10) into equations (7), we obtain the conditions for the existence of a stationary solution of these equations in the form of relations (11). To derive an equation defining Y0, let's express y° through 70, using the first of these relations:
y° = ±Md(u+ 70), (15)
and substitute it into the second one:
bx° + lfMd(co + 70) = 0.
From here we find x0 through 70:
x° = -±fMd(u + f). (16)
Then expressions (15), (16) are substituted into the third relation (11). As a result, we obtain the equation for 70
Md(w + Y ) = Ma(Y), (17)
where the odd function Ma(Y) is defined by formula (8).
When Y = ^b, this function takes its minimum and maximum values ^a/2. It increases monotonously on the segment [-b, b] between the points of its minimum and maximum, and outside this segment the function Ma(Y) monotonously tends to zero from below and above at Y ^<x>. The marked properties of the functions Md(y) and Ma(Y) make it possible to draw their graphs versus y.
Let's write down equation (17) in the form of equation (13) for the variable y. Graph of the function Ma(y — w) of the variable y comes from the graph of the function Ma(Y) by shifting to the right by the amount w > 0. Hence, its properties are as follows: 1) the function Ma(y — w) increases monotonously on the segment [w — b,w + b] between the points of its minimum and maximum, vanishing in the middle of this segment, the point y = w, 2) outside the segment [w — b,w + b] the function Ma(y — w) monotonously decreases in modulus with increasing \y\, tending to zero at \y\ ^ to.
In the case when w—b < 0, the minimum point y = w—b of the function Ma(y—w) does not lie in the right half-plane, and then, as it is easy to see, the graph of this function has a single intersection point with the graph of the dissipative moment Md(y) at some value y = w0 £ (0,w] from the left side [w — b,w] of the increment interval of this function.
In the case when w — b > 0, we will assume that the dissipative moment Md(y) is so small on the interval (0, w — b] that its graph lies above the graph of the function
Ma(y — u) on this interval, that is, condition (9) is fulfilled . Then the graph of the dissipative moment intersects the graph of the function Ma(y — u) at a certain value y = u0 G [u — b,u) from the left side [u — b, u) of the interval of increase of this function.
In both cases, equation (13) has a unique solution y = u0 (0 < u0 < u). After this solution of equation (13) is found, the stationary value of Y0 is determined by formula (12), and the stationary values x0, y0 are determined by formulas (15), (16) in the form (14). □
3. Lyapunov function for equations of perturbed motion of the electric motor. Taking the stationary solution (10) of equations (7) as undisturbed, we introduce the perturbations 71, xi, yi, assuming
Y = y0 + Yi, x = x0 + xi,y = y0 + yi. (18)
To derive the differential equations for perturbations, substitute expressions (18) into equations (7), taking into account relations (11). We obtain the system of equations of perturbed motion
CYi = —ayi + AMd(Yi),
xi = — bxi — Y0 yi — y0Yi — Yiyi, (19)
yi = —byi + Y0xi + (x0 + i)Yi + Yixi. Here, the disturbance of the dissipative moment
AMd(Yi) = Md(u° + Yi) — Md(u°) (20)
is a continuous monotonically decreasing function of the variable Yi of the sign, opposite to the sign Yi. Thus,
Yi AMd(Yi) < 0 (Yi =0), AMd(0) = 0. (21)
Equations of perturbed motion (19) have the zero solution
Yi = 0, xi = 0, yi = 0, (22)
that corresponds to the stationary solution (10) of equations (7).
Lemma 1. If conditions of theorem 1 are fulfilled, then the zero solution (22) is the unique stationary solution to the equations of perturbed motion (19).
Proof follows from the fact that the conditions for the existence of a stationary solution Yi = Y0,xi = xi,yi = y° of the system of equations of perturbed motion (19) are the relations
—ay0 + AMd (Y0) = 0,
bx0 + Y 0y0 + y0Y0 + Y0y0 = 0, (23)
—by0 + Y0 xi + (x0 + 1)y0 + Y0x0 = 0,
which, by the change Yi = Y — xi = x — x0, yi = y — y0, inverse to the change (18), can be reduced to the conditions of existence (11) of the stationary solution (10) of the undisturbed system (7). □
Consider the following positive definite function of perturbations
Vi(7i,zi,2/i) = ^(Cjf + axj + ayi). (24)
Its time derivative, taken by virtue of the system of equations (19), is equal to
Vi(Yi ,xi ,yi) = Yi AMd(Yi) + aYi (—y0xi + x0yi) — ab(x? + y?). (25)
For Yi = 0, xi = 0, yi = 0 we have Vi(0,0,0) = 0. The product YiAMd(Yi), included in the right side of formula (25), is, according to (21), the negative definite function of the variable Yi. In order to obtain sufficient conditions for the positive definiteness of the derivative Vi(Yi,xi,yi) with regard to variables Yi,xi,yi, some additional conditions are imposed on the function AMd(Yi) in the following lemma.
Lemma 2. Let
1) the conditions of theorem 1 be fulfilled;
2) the positive definite function Vi(Yi,xi,yi) of the phase variables of the system of equations (19) of the perturbed motion is determined by formula (24), and Vi(Yi,xi,yi) is its time derivative by virtue of this system of equations, expressed by formula (25);
3) there exist a constant k > 0 such that the following inequalities are satisfied for the perturbation (20) of the dissipative moment
AMd(Yi) < —kYi (Yi > 0), AMd(Yi) > —kYi (Yi < 0); (26)
4) the following inequality is fulfilled
abk -\[Md{^)f [1 + ~hf)2] > 0, (27)
where a,b > 0 are parameters of the system of equations (19), y = w > 0 is the angular velocity of rotation of the magnetic field in the stator, Md(w0) < 0 is the moment of dissipative forces at the stationary angular velocity of rotation of the rotor y = w0, which under conditions of theorem 1 is uniquely determined by equation (13).
Then the derivative Vi(Yi,xi,yi) is a negative definite function of the variables
Yi, xi, yi.
Proof. Suppose there exist a constant k > 0 such that inequalities (26) are satisfied. Geometrically, they mean that at Yi = 0 the straight line — kYi separates the graph of the function AMd(Yi) from the axis OiYi, where Oi is the point of intersection of the graphs of the dissipative moment Md(y) and static characteristic Ma(y — w). Two inequalities (26) are equivalent to one inequality
Yi[AMd(Yi) + kYi] < 0 (Yi = 0).
From here it follows that
YiAMd(ji) < —kYi, Y1 = 0, and then from (25) we obtain the inequality for VV1
Vi(ji,xi,yi) < -ki2 - ay Yixi + ax Yiyi - ab(x^ + yj).
Its right-hand side is the quadratic form F(Yi,xi,yi) of the variables Yi, xi, yi. This quadratic form is definitely negative when the quadratic form
-F(Yi,xi,yi) = kYj + ay0Yixi - ax0Yiyi + ab(x\ + yj)
is definitely positive.
According to Sylvester's criterion, for the definite positivity of the quadratic form -F, the necessary and sufficient condition is that the determinant A3 of this quadratic form and its principal diagonal minors A2, Ai
A3
k
2 C^y 2 ^^
\ay°
_1. ft
2 U/tv
ab
ab
Aj
k
\ay°
ab
Ai = k
are positive.
The condition A1 > 0 is met, since k > 0. Having calculated the determinants A2, A3 and using formulas (14), we obtain the condition A3 > 0 in the form (27). The condition A2 > 0 is equivalent to the inequality abk — |[M<i(w0)]2 > 0, which is fulfilled when inequality (27) is fulfilled. □
4. Global stability of the asynchronous electric motor. The positive definite function V1('Y1,x1,y1), defined by formula (24), has the time derivative V1(Y1 ,x1 ,y{) by virtue of the system of equations (19), which is expressed by formula (25), and it is negative definite under conditions of Lemma 2. According to the well known Lyapunov theorem [7, Theorm 4.2], the asymptotic stability of the stationary mode of the steady rotation of such electric motor immediately follows from here.
Under conditions of Lemma 2, for the stationary solution of equations (19), more strong property of its global stability is fulfilled. This property is formulated below in the form of Theorem 3. It follows from definitions 1, 2 and Theorem 2.
In Russian mathematical literature, the term "global stability" is in use only for phase systems of ordinary differential equations, having a countable set of stationary solutions. In the case of ODE systems, having a unique stationary solution, the term "stability in whole" is used in Russian literature. In English literature, the term "global stability" is taken in both cases. So, we use the term "global stability" below.
Definition 1 [7, definition 12.1]. Zero solution of the system
x = f (x), f (0) = 0, x = (xi,x2,...,xn ),
0
is called globally stable (or asymptotically stable under any initial perturbations), if it is stable in the Lyapunov sense and if every other solution x(t) of this system has the property ||x(t)|| — 0 at t -ж.
Definition 2 [7, с. 45]. The Lyapunov function V is called infinitely large, if for any number A > 0 there exists a number R > 0 such that the inequality V > A is
n
fulfilled outside the sphere ^ x2 = R2.
i=l
Theorem 2 [7, теорема 12.1]. If there exists a positive definite infinitely large function V that has a negative definite derivative in the whole phase space, then the zero solution of the system X = f (x) is globally stable.
This theorem is a special case of more general Theorem 12.2 of [7], which is one of variants of known Barbashin-Krasovsky theorem. Theorem 2 follows also from La Salle invariance principle in the formulation of Theorem VIII in [8].
Function (24) is infinitely large, and this leads to the following theorem. Theorem 3. Let conditions of lemma 2 be fulfilled. Then the zero solution (22) of equations (19) of perturbed motion of the two-current model of the asynchronous electric motor, which corresponds to the stationary solution (10) of equations (7) of the two-current model of an asynchronous electric motor, is globally stable.
Conclusion. Sufficient conditions are obtained for local and global stability of the steady rotation mode of the asynchronous electric motor on the basis of its two-current mathematical model. The load moment is assumed to be a continuous odd function of the angular velocity of the rotor relatively the stator, and this function permits a linear estimate.
The research was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation within basic part of the state order in the field of science, topic No. 1023020900001-4-1.1.2;1.1.1.
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Б.И. Коносевич, Ю.Б. Коносевич
Свойства локальной и глобальной устойчивости модели асинхронного электромотора.
Темой работы являются вопросы устойчивости режима стационарного вращения асинхронного электромотора. Исследование основано на математической модели такого электромотора в виде системы обыкновенных дифференциальных уравнений третьего порядка, содержащей уравнение для угловой скорости ротора и уравнения для двух токов в обмотках ротора. Предполагается, что момент нагрузки является непрерывной нечетной функцией угловой скорости ротора относительно статора, и эта функция допускает линейную оценку. Такая система уравнений движения асинхронного электромотора имеет единственное стационарное решение, описывающее режим равномерного вращения ротора. В статье получены достаточные условия локальной асимптотической устойчивости этого решения. Установлено также, что при этих условиях любое решение уравнений движения с течением времени стремится к их стационарному решению.
Ключевые слова: асинхронный электромотор, локальная и глобальная устойчивость, равномерное вращение.
Received by the editorial board 25.07.2024; revised 26.08.2024;
recommended for publication 06.09.2024.
