Linear Controller for Maglev Motor with Axial Flux

Linear Controller for Maglev Motor with Axial Flux F. A. BARATA

J. C. QUADRADO DEEA Instituto Superior de Engenharia de Lisboa R. Conselheiro Emídio Navarro, 1, 1950-072 Lisboa PORTUGAL

In this paper, in order to stimulated the develop active bearings (AMB) systems we propose a linear controller design for maglev motors, based on forces and voltages required for stable control of the rotor in the radial direction. The control currents intentionally break force symmetry, resulting in unbalanced radial forces. The system employs two stators to effectively remove the rotational frequency modulation effect in the radial control forces. Key-Words: Maglev Motor, Linear Controller

1 Introduction In recent years, the need for precise control and clean operation of rotating machinery has been extensively increased in the areas of semiconductor and material processing [1]. It has stimulated the development of active magnetic bearing (AMB) systems that levitated the rotating parts by electromagnetic forces. The main advantages of AMB system are: there is no mechanical contact; it is possible to control the radial position of rotating part. These advantages inspired great progress in the area of conventional magnetic bearings, but in spite of the various advantages of AMB systems, they have some limitations in applications to industrial fields because they require relatively long shaft, large and complex structure. Maglev motors are electric motors that levitate by electromagnetic forces that come from a single stator coil group. The first idea of magnetically levitated motor was given in the paper “Development of a bearingless electric motor” by Bosch in 1988 [2]. “Bearingless motor”, meaning that it has no mechanical bearing, is another widely used expression. Chiba [3, 4, 5], Okada [6, 7, 8] and Schöb [9, 10, 11] independently developed magnetically levitated motors in early stages, utilizing the mature knowledge on AMB systems. In this work, the basic structure of magnetically levitated motor was to use different number of poles for stators and rotors. The mismatch in the numbers of the poles for stators and rotors produces radial forces. In this paper, a maglev motor having 8 stator coil groups and 8 permanent magnet rotor poles is

develop, having axial flux path formed along the axial direction. There is no negative stiffness of conventional AMB systems because the system adopted the electric motor with axial flux path. This property improves the system stability. The maglev motor has two phase windings to remove the rotating frequency modulation characteristics of actuator force witch comes from the rotation of rotor flux. Stator windings have two phases placed on either sides of the rotor disk like a sandwich but with 22.5º spatial angle difference. The resultant force of the two forces generated from two stator windings makes up control force of desired frequency.

2 Basic Principle of Maglev Motor with Axial Flux Figures 1, 2 and 3 conceptually depict how radial forces of the maglev motor with axial flux are generated [7]. Figure 2 shows the basic principle of generating the radial force in a coil and Figure 3 shows the procedure for generating the total radial force and torque in a stator and rotor. The dot in the circle means the magnetic flux coming out of the page, and the cross in the circle means the magnetic flux going into the page. Each flux comes from the permanent magnet in the rotor. The arrow in the coil indicates the direction of the current flow. From the Lorenz’s law, the forces produced in the left and right sides of the coil are (1)

where Nc is the number of coil turns, lc is the effective coil length, Im is the motoring current, ic is the control current, and Bm is the flux density from the magnet. As shown in Figure 1, since the effective force in a coil is the sum of the forces at the left and right sides of the coil, the total force in a single coil is (2) where FL, FR are the forces produced at the left and right sides of the coil. The supplied current into each coil has the motoring and control components, while rotates and levitates the rotor respectively.

Fig. 1. Force Generation in a Coil For conventional electric motors with only the motoring current Im , there always exists force symmetry or balance. On the other hand, the magnetically levitated motor adds unbalanced control currents ic and jc to the motoring current Im to intentionally break the force symmetry. A pair of facing coils is engaged with the control current ic or jc such that the control current is added to the motoring current in one coil but subtracted in the opposite coil, as shown in Figure 2.

Substituting Equation (1) into Equation (2), we can get the forces in the coils A and C as

(3) where αc is the angle between each side of coil. The forces in the coils B and D are determined in a similar manner. The resultant force generated from the coils A and C, B and D determines the vertical force Fz and the horizontal force Fy. The forces in coils can be decomposed into two parts: the rotating force FIm and the levitating force, Fic or Fij, as shown in Figure 3, i.e.

(4) Every rotating force contributes to torque to the rotor, and two pairs of unbalance forces contribute to levitation of the rotor. The coil forces result in two couple’s moments and two radial forces in the y and z directions as shown in Figure 3, where the pair of coils A and C, B and D produce Fy and TAC, Fz and TBD. The total motoring torque generated by the two pairs of facing coils is 2TAC plus 2TBD, which can be expressed as (5) where r0 is the distance from the center of the stator to the center of the coil. And the total radial forces becomes

(6) At least in theory as shown in Figure 3, we can obtain the radial levitating force and the motoring torque independently. However, the problem of frequency modulation in the radial forces is inevitable with a single pair of rotor and stator because the magnetic field rotates with the rotor. This problem will is addressed in [13]. Fig. 2. Generated Radial Forces in Each Coil

supplied control currents for producing the given control forces. The system equation of motion without actuator dynamics is given in (7).

(7) where

In (7), M, x , f are the mass matrix, the radial displacement vector of the rotor, the control force vector acting on the rotor including the controlled magnetic force, respectively. Considering the block diagram shown in Figure 4 except the shaded part, the control force vector can be written as (8). (b) Fig. 3. Generated Global Force and Torque (8) where

3 Controller Design of Maglev Motor Figure 4 is the block diagram of the maglev motor control system, where f* and v* mean the required forces and voltages for stable control of the rotor in the radial direction.

Here Kc, Ka, Ks, Kp, Kd are the controller matrix, the gain matrix of the power amplifier, the gain matrix of the sensor amplifier, the proportional gain matrix, and derivative gain matrices, respectively. Substituting Equation (8) into Equation (7), the system equation can be rewritten as (9).

(9)

Fig. 4. Block Diagram of Maglev Motor System

or, in the state-space form, as Dynamics of the rotating part is straightforward, because the maglev motor system has the shape of simple pendulum. However, since the force-current relation matrix has trigonometric functions of the rotating speed as discussed in [13], we can not easily determine the controller gain with the system including the force-current relation. Let’s consider a simple control loop that excludes the shaded blocks in Figure 4, then the actuator dynamics is removed in the control loop, and the control gain in this simple control loop can be readily determined. Then, the determined control gain will directly give the required force for stable motion in the radial direction. Multiplying the demodulation matrix to given control forces, we can easily determine the

(10) where

Here, A is the real valued 4x4 system matrix and p is the 4x1 state vector. The controller gain matrices are scalar quantities, because the system has the symmetric property in the radial direction. The gain matrices can be rewritten as (11)

The eigenvalue problem associated with Equation (9) and (11) gives the eigenvalues as

(12) Table 1 shows the parameters of the designed maglev motor system. Table 2 gives the equivalent mass m, sensor and power amplifier gains Ks, Ka based on the values of Table 1. Using the root loci with the parameters Kp and Kd varied, the stable control gains were determined such that the system displays a damping ratio ζ 0.8. Table 2 also summarizes the determined eigenvalues and control gains. Table 1. Specifications of the Maglev Motor

Fig. 5. Cross Sectional View of Maglev Motor with Axial Flux [9] The rotor of the maglev motor was designed to operate in its upright configuration, in order to be free of the gravity effect in the radial motion. The upper bearing supports the rotor in the axial direction, and a self-aligning bearing was used so that the lower part of the rotor can move freely in the radial direction. The sensor plane was designed as a rim type and placed around the rotor to measure the exact radial gap. The full system implementation follows the schematic diagram present in figure 6. The characteristics of the digital control are presented in table 3. Table 3. Specifications of the Digital Control System

Table 2. System Parameters of the Maglev Motor

4 Implementation setup Figure 5 shows the structure of the maglev motor.

Fig. 6. Schematic Diagram of Maglev Motor System [9]

5 Conclusion This work describes the design of a synchronous type maglev motor with axial flux. The required current and radial force relationships of the maglev motor were derived based on the frequency demodulation scheme [13]. The proposed controller design, allows a straight forward application of these machines. Future work includes experimental validation of this approach.

[10]

[11]

[12] References: [1] C.S. Kim, Dynamic Analysis and Isotropic Optimal Control of Active Magnetic Bearing System, Ph.D. Thesis, KAIST, 1995, pp1-2 [2] R. Bosch, "Development of a Bearingless Electric Motor", Proc. ICEM '88, Vol. 3, pp.331-335 [3] A. Chiba, D. T. Power and M. A. Rahman, "No Load Characteristics of a Bearingless Induction Motor", IEEE Industry Applications Society Annual Meeting Part 1 (of 2), 1991, pp.126-132 [4] A. Chiba, D. T. Power and M. A. Rahman, "Analysis of No-Load Characteristics of a Bearingless Induction Motor", IEEE Transactions on Industry Applications, Vol.31, No.1, January/February 1995, pp.7783 [5] A. Chiba, T. Deido, T. Fukao and M. A. Rahman, ''An Analysis of Bearingless AC Motors'', IEEE Transactions on Energy Conversion, Vol. 9, No. 1, March 1994, pp.61-68 [6] T. Ohishi, Y. Okada, K. Dejima, “ Analysis and Design of a Concentrated Wound Stator for Synchronous-type Levitated Motor” , Fourth International Symposium on Magnetic Bearings, 1994, pp.201-206 [7] Y. Okada, T. Ohishi, Y. Taenaka, T. Yamane, C. Chen, “Design of Centrfugal type Artificial Blood Pump Using Combined Motor-Bearing” [8] Y. Okada, C. W. Lee, H. Konishi, W. S. Han, “Disc type Self-Bearing Motor with 5 Control Degrees of Freedom”, The First Korea-Japan Symposium of Frontiers in Vibration Science and Technology, September 1998, KAIST, Taejon, Korea, pp.130-135 [9] R. Schöb and N. Barletta, "Principle and Application of a Bearingless Slice Motor," Fifth International Symposium on Magnetic

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