Determination of the rational shift value the center of the magnetic system electromagnetic bearing

Full Text

Electromagnetic bearings are designed for non-contact suspension of the rotor in the field of electromagnets. At the same time, like any bearing, this type of bearing is designed to maintain the rotor in a given position with the required rigidity. As a rule, two radial and one axial electromagnetic bearings are required for rotor suspension. These bearings perceive and transfer the load from the movable assembly to other parts of the structure, including the rotor gravity. The equation of rotor motion in the field of electromagnets of each bearing control channel is determined by the differential equation [1] , (1) where is the rotor displacement relative to the center of the magnetic system along the axis of the bearing; is the design coefficient of the electromagnetic bearing; and - currents in the windings of opposite electromagnets; - the gap between the stator and the rotor when the rotor is located in the center of the magnetic system; is the weight force per one axis of the bearing; - time. Equation (1) shows that the magnitude of the force acting on the rotor depends both on the ratio of currents in opposite electromagnets and on the displacement of the rotor from the center of the magnetic system of each axis of the bearing. This allows us to propose a way to compensate for the weight of the rotor by shifting the center of the magnetic system relative to the axis of rotation [2, 3]. At the same time, the actual task is to determine the rational value of the displacement, at which, with equal currents in opposite electromagnets, the weight force per one axis of the bearing will be completely compensated. This will reduce the current load on the electromagnet, which prevents the action of the weight force, and expand the control capabilities within the limits of current limitation.

Solution of the problem Equation (1) in static mode with and equal currents in opposite magnets will be written as follows: , (2) where is the reference supply voltage of the electromagnets; - active resistance of each of the windings; is the mass of the rotor per axle; – free fall acceleration. The solution of equation (2) will allow to find a rational value of the displacement of the center of the magnetic system, at which the weight force attributable to one axis of the bearing will be fully compensated. In this case, equal currents will be observed in opposite electromagnets controlled according to the differential law [4]. Equation (2) is transformed to the form:. (3) Performing simple algebraic transformations in (3), as a result, we obtain an equation relating the displacement of the center of the magnetic system with the parameters of the electromagnets and the part of the rotor mass per one axis: ; .Expression (4) is an incomplete fourth-order equation, the roots of which, in accordance with the Descartes-Euler solution, can be found as follows [5]: . (5) Here , and are the roots of the cubic equation, (6) where ; ; .The roots of equation (6) can be found using the Cardano solution [6]:; , (7) where ; ;. Substituting into and , the values and , expressed in terms of the coefficients , and , we get:; (8). (9) Taking into account the fact that formulas (5), (7) - (9) allow us to find a rational value of the displacement of the center of the magnetic system of the electromagnetic bearing relative to the axis of rotation: (10) Formula (10) is obtained from the analysis of the roots (5) and the physical meaning of the problem being solved. From (7) it follows that the roots are complex conjugate: , where ; .To calculate in (10) and without specialized software, you can use the formulas:; ,Where ; .A simpler approach to determining the rational value of the displacement of the center of the magnetic system is to use another mathematical description of the process of rotor movement in the field of electromagnets. Under the differential law of current control in opposite electromagnets, the following equation is valid [4, 7 - 9]: – coefficient of positive feedback on displacement. In statics at , from (11) a simple dependence of the rational value of the displacement of the center of the magnetic system on the weight force per one axis of the electromagnetic bearing follows: (12) Solution (12) is simpler, but it does not take into account the fact that the quantity changes its value depending on the displacement. In addition, it should be borne in mind that the exact values of the coefficients and can only be obtained by simulating the magnetic fields of an electromagnetic bearing in specialized programs. Calculation example For an example, we calculate the rational value of the displacement of the center of the magnetic system of radial electromagnetic bearings designed for suspension of the rotor of a prototype turbocharger 6TK-E diesel locomotive [10]. In the unit under consideration, the mass of the rotor per one electromagnetic bearing is kg, and the reference voltage of the pulse-width converter is V. In the central position of the rotor, the electromagnets are characterized by the following parameters: Ohm, N, N/m, the gap between the stator and the rotor is mm. The calculation according to formula (12) shows that the rational value of the displacement of the center of the magnetic system should be equal to: m, (13) that is, 124 μm. To calculate the rational value of the displacement using formulas (7) - (10), it is necessary to know the coefficient It can be determined from the results of modeling an electromagnetic bearing with varying ratios of currents in opposite electromagnets [10]:, (14) determined from the results of full-scale experiments on a real operating installation. For example, if it is known that when currents A, A are set in the considered radial electromagnetic bearing, the force acting on the rotor is equal to H [10], then the coefficient in accordance with formula (14) will be equal to: Nm2/A2. (15) Using the value obtained in (15), we use formulas (7) - (10) to calculate the rational value of the displacement of the center of the magnetic system of the considered radial electromagnetic bearing. In this case, the following values of the quantities necessary to determine the rational bias are obtained: , , , . With this in mind, the roots of the cubic equation (6) will be equal to: (16) Comparison of the results (13) and (16) shows that they are very close, since the discrepancy does not exceed 1.2%. Moreover, it should be noted that this completely coincides with the electromagnetic calculation of the radial bearings of the 6TK-E turbocharger [10]. However, the above example implies that the axis is located vertically. At the same time, in radial electromagnetic bearings, it is customary to rotate the coordinate system by 45 angular degrees in order to distribute the weight force to two electromagnets. This leads to a decrease in mass and, consequently, in the weight force per one axle by a factor of 1. Then, for the considered variant of the electromagnetic suspension of the turbocharger rotor, it is necessary to take kg in the calculations. With this in mind, the rational value of the displacement of the magnetic system along the axis in accordance with formulas (7) - (10) should be equal to 92 microns. The calculation according to formula (12) gives the result of µm, that is, the discrepancy has increased to 4.3%. This is due, according to the authors, to the neglect of the non-stationarity of the coefficient , the value of which, in turn, depends on the displacement of the rotor from the center of the magnetic system.

The displacement of the center of the magnetic system relative to the axis of rotation of the rotor allows, as shown in [2], to reduce the value of the reference supply voltage of the windings of electromagnets. This leads to a reduction in the consumption of electrical energy by electromagnetic bearings and the provision of a more favorable thermal regime for the operation of the windings. At the same time, the displacement of the center of the magnetic system does not affect the stability of the three-circuit control system of the electromagnetic bearing (Fig. 1) [11]. Fig. 1. 1 - Functional diagram of a three-loop control system of an electromagnetic bearing Indeed, when choosing a 12-bit pulse-width modulator with a transfer coefficient and a position sensor with discrete / m, the settings of the regulators of the control system of the electromagnetic bearing, determined for the central position of the rotor, will be as follows: (PD) of the controller s, the transfer coefficients of the proportional (P) and PD controllers , the time constants of the differentiating link and the integral (I) controller, respectively, are equal to s, s [12, 13]. When the center of the magnetic system is displaced relative to the axis of rotation, the values of the inductances of the windings of electromagnets take on the values Hn, Hn, Hn. In this case, the coefficients characterizing the induced EMF in the windings are equal to Vs/m, Vs/m. At the same time, the coefficients that determine the force acting on the rotor have the following values: N, N/m. The transfer function of a closed three-loop control system of an electromagnetic bearing has the form [4]:, where ; ; ; ;; ;; ;;;;;; ; ; ; .Consequently, the stability of the three-circuit control system of the electromagnetic bearing is determined by the characteristic equation: . (17) At A, equation (17) has the following values of the coefficients: с5, с4, с3, с2, с. Solution (17) shows that the roots of the characteristic equation are:,,,. Since all roots have negative real parts, the three-circuit control system of the electromagnetic bearing remains stable for the selected parameters of the controllers and the displacement of the center of the magnetic system relative to the axis of rotation by a rational value [14, 15].

Conclusions

The found analytical expressions allow us to determine the rational value of the displacement of the center of the magnetic system of the electromagnetic bearing relative to the axis of rotation, at which the force of the weight of the rotor is compensated for equal currents in opposite magnets.

The displacement of the center of the magnetic system of the electromagnetic bearing relative to the axis of rotation of the rotor by the calculated value does not affect the stability of the control system at the settings of the regulators selected for the center position of the rotor.

 

About the authors

Alexander V. Starikov

Samara State Technical University

Email: star58@mail.ru

(Dr. (Techn.)), Professor

Russian Federation, 244, Molodogvardeyskaya st., Samara, 443100

Vladislav D. Kostyukov

Samara State Technical University

Author for correspondence.
Email: kostyukovvlad@yandex.ru

Postgraduate Student

Russian Federation, 244, Molodogvardeyskaya st., Samara, 443100

References

Supplementary files