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Unformatted text preview: DESIGN OF HORIZONTAL SHAFT ACTIVE MAGNETIC BEARING SYSTEM Bostjan Polajzer , Drago Dolinar , Gorazd Stumberger , Joze Ritonja , Bojan Grcar , Kay Hameyer] University of Maribor, Faculty of Electrical Engineering and Computer Science, Smetanova 17, SI{2000 Maribor, Slovenia, KU Leuven, Dept. E.E., Div. ESAT/ELEN, Leuven, Belgium 1 INTRODUCTION Several principles of magnetic bearings operation are known 1], but the principle based on the use of electromagnets to provide the force necessary for the levitation of a rigid body is the most widely used. The magnetic eld has to be adjusted continuously to attain stable levitation and the required dynamics of the levitating body. This can be done only with controlled electromagnets. One application are Active Magnetic Bearings (AMBs) where two pairs of radial bearings controlling four DOFs are placed at each rotor end. The fth DOF is controlled by a pair of axial bearings. Rotation, i.e. the sixth DOF, is controlled by an independent driving motor. AMBs o er signi cant advantages due to their non-contact operation. Higher speeds, no friction, no lubrication, weight reduction, precise position control and active vibration damping make them far superior to the conventional bearings. AMBs are therefore a typical mechatronic product and are particularly appropriate for high-speed rotating machines. Commercial applications include pumps, compressors, ywheels, milling and grinding spindles, turbine engines, centrifuges, etc. The laboratory prototype of an AMB is presented in the paper. The mathematical model of an AMB is determined separately for the mechanical and for the electrical part. The modeling is restricted to the y{ axis. The dynamic model with lumped parameters is expressed in the time domain. The model is coupled and nonlinear. The di erential driving mode is introduced to avoid the redundancy of input variables. Also, the linearized equilibrium point deviation model is given. Its parameters are determined by the numerical analysis of the magnetic eld 2] and by measurements. The obtained model is used for control design 3]. Because of the decentralized control, the same controller parameters are used for the y{ axis and for the x{ axis. A comparison of experimental and simulation results for the control in y{ axis is shown along with experimental results of high speed rotation. Basic mechatronic components of the experimental system are brie y described as well. At the very end, some important ndings, di culties and suggestions with respect to the problem of active vibration damping are summarized. 2 MAGNETIC BEARING SYSTEM MODELING 2.1 Laboratory prototype In this subsection the mathematical model of the laboratory prototype is presented. The system consists of two axially allocated pairs of electromagnets, i.e. the vertical and the horizontal subsystem. In Fig. 1 a) we can see the schematic presentation of the horizontal{shaft magnetic bearing system with its geometry, and in Fig. 1 b) y{ axis of the AMB. The four input variables of the system are voltages on each electromagnet winding. If rotation and elasticity of the shaft are neglected, then the system has two DOFs. The two output variables of the system are shaft positions in the x{ and in the y{ axis. The determination of the mathematical dynamic model is separated into three steps. First, we write the equations of motion where several geometric relations occur due to the axial allocation of actuators, sensors and weight. The next step of modeling deals with the electromagnets. On the assumption that the iron core and coil windings are idealized we can write the voltage equation for each electromagnet. In the last step we write the equations for the y fy sy l lsy lay lsx lax fx sx i1 000 111 11 111 000 111 u1 000 00 111 000 000 111 f x x y mg A i2 000 00 111 11 111 11 u2 000 00 z a) y z mg b) Figure 1: a) Schematic presentation of the horizontal{shaft magnetic bearing system, b) y{ axis of the active magnetic bearings electromagnetic force generated by each particular electromagnet excited by the coil current. Their sum is the resultant electromagnetic force. The system has only two independent DOFs, so we will restrict our further discussion to the subsystem describing the y{ axis. It consists of two electromagnets with a serially connected pair of coils. The subsystem shown in Fig. 1 b) is described by the mechanical equation of motion (1), two voltage equations (2) where sign of the last term depends on the index (positive sign for h = 1, negative sign for h = 2) and the equation of the resultant electromagnetic force (3). R is the resistance of one electromagnet, L is the inductance of one electromagnet when the axis of the shaft is in the center. k Nm2 =A2 ] is the force coe cient and ku Vs/m] the coe cient of the back{EMF. The nominal air gap and equivalent shaft mass are denoted with and m respectively. 2 y m d 2 = f ; m g fl dt uh = ih R + L dih ku dy dt dt i2 f = k ( ;1y)2 ; ( (1) h=1 2 i2 2 (2) ! (3) + y )2 The AMB model given in the form of equations (1), (2) and (3) is multi-variable, coupled and nonlinear. Let us assume constant model parameters. Voltages u1 and u2 are system inputs, the position y is the output, and the common disturbance consists of the equivalent gravity force mg and the load force fl . If we bear in mind that the system is totally controllable the redundancy of input variables becomes evident. For its elimination we introduce the di erential driving mode. The bias current i0 is forced through the coils of both electromagnets. Considering the given assumptions the resultant electromagnetic force f0 is zero. But as we know there always exist the load force fl and the disturbances as well as the gravity force mg, we have to add the so-called control current (i i0) in the upper coil and subtract it in the lower coil (4). In this way a SISO system is obtained where the input variable is the so-called control voltage u or the control current i in case of a current fed system. i1 := i0 + i i2 := i0 ; i (4) 2.2 Linearized equilibrium point deviation model Among the model equations (1), (2) and (3), only equation (3) is nonlinear. It is linearized in the equilibrium point (i0 y0 ) where i0 is an arbitrary bias current and y0 the position of the shaft's axis in the center (y = 0). Taking into account the rst term of the Taylor expansion of equation (3) about the equilibrium point the electromagnetic force is given by the well-known linear equation (5). The current gain coe cient ki N/A] and the position sti ness coe cient ky N/m] are de ned as (6). After rearranging equations (1), (2) and (5) the linearized AMB model is obtained (7). The equivalent model in the input-output domain is de ned by the transfer function (8). f = ki i + ky y (5) 2 ky := @@f = 4k i0 3 y (i0 y0 ) 2 ki i + ky y ; m ddty2 = 0 Ri + L dit + ki dyt ; u = 0 d d Y G(s) = U (s) = mL s3 + mR s2 +kik2 ; k L) s ; k R (s) (i y y = 4k i0 ki := @@f 2 i (i0 y0 ) Y (s) I (s) (6) (7) (8) ms y I = m ski;ky Gel (s) = U ((ss)) = mL s3 +mR s2 +(;i2k;ky L) s;ky R (9) 2 k The parameters of the linearized equilibrium point deviation model of the laboratory prototype of active magnetic bearings are shown in Table 1. Current gain and position sti ness are calculated by the numerical analysis of the magnetic eld 2] using the nite element method (FEM). The calculated results have been compared with the measured values. Gmech(s) = 2 Table 1: Parameters of the linearized model in equilibrium point (i0 y0 ) data resistance inductance equivalent mass current gain position sti ness bias current nominal air gap parameter value determination R] 0.6 measured L H] 0.0048 measured m kg] 5 measured ki N/A] 32.7 FEM ky N/m] 108350 FEM i0 A] 2.5 free parameter ;3 construction data m] 0:6 10 3 CONTROL y r ef GconY i r ef GconI u Gel i Gmech y Figure 2: Control structure of the system This AMB represents unstable system, therefore we need a closed-loop control system to stabilize it. The cascade structure has been used (Fig. 2). In the inner loop the current controller GconI (s) is responsible for the best possible reference current tracking. Let us assume that the latter is perfect. Then it is enough to use only the mechanical transfer function Gmech(s) (9), de ned by two real poles (s 1 2 = 164:58), for the further position controller design. A PID controller (10) has been used for stabilization purposes. Its parameters are de ned by increasing the amplitude of the frequency characteristic at low frequencies, and by trying to attain an adequate phase margin for the chosen cross frequency in case of high frequencies. As the system consists of two independent DOFs decentralized control has been implemented. Therefore the same controller parameters has been used also for the x{ axis. The position controller was set up as follows: controller gain Kcon = 10000, integral time constant Ti = 0:03 s, derivative time constant Td = 0:003 s and parasitic derivative time constant Td0 = Td =10. + GconY (s) = Kcon s TiT 1 s Td0 + 1 (10) s i s Td + 1 4 SIMULATION AND EXPERIMENT In simulations (MATLAB { Simulink) all parts of the experimental system shown in Fig. 3 a) were considered in addition to the nonlinear actuator model (Fig. 3 b). Let us describe some parts of the system. First, the inductive position sensor with a sensitivity of 7.7 mV/ m and 20 kHz cut-o frequency was chosen very carefully. An additional lter for sensor cross-talk elimination was implemented afterwards at 5 kHz. Next, the analogue current controller and a 20 kHZ PWM switching ampli er were used with a 300 VA per channel. Finally, the digital PID position controller with anti-windup was implemented into the power PC environment with a sampling time of 100 s. a) b) Figure 3: a) Experimental system and b) laboratory prototype AMB Fig. 4 shows the comparison of calculations and measurements on the laboratory prototype. Only the comparison of control in the y{ axis is shown. It is obvious that the agreement of results is very good { excellent damping agreement and acceptable sti ness disagreement. However, the result for the stability test was quite di erent. The upper stability limit of the experimental system was much lower than the one we established theoretically. This means that we can not achieve a very high sti ness. The reason for this are the actuator limitations. As a result, insu cient forces are generated for a wide range of shaft positions. This conclusion is con rmed by FEM calculations and measurements of the f (i y) relation. In Fig. 5 the results of the rotation test, where the position error does not exceed 15 m, are also presented. In Fig. 5 b) the shaft elasticity problem turns up in addition to the rotor unbalance problem which is more evident in Fig. 5 c). 5 CONCLUSION The paper deals with the modeling and the analysis of AMB laboratory prototype. If we take into account that the system has two independent DOFs, then the analysis of the model is restricted to the y{ axis. The di erential driving mode is introduced and the linearized equilibrium point deviation model is written. Its parameters are measured and calculated by the numerical analysis of the magnetic eld. The PID controller design of the y{ axis is included. Because of the decentralized control, the same control design is used also for the x{ axis. 2.5 2 simulation 2 position [m] position [m] 3 experiment 1 1.5 simulation experiment 1 0.5 0 0 0.3 a) 0.35 time [s] 0.4 0.45 −0.5 b) 0.3 0.35 time [s] 0.4 0.45 Figure 4: The position response of y{ axis from the equilibrium point: a) to the reference step function (0.2 mm) and b) to the load step function (60 N) 2 y−position [m] y−position [m] −5 x 10 1 0 −1 −2 −2 a) −1 0 1 2 x−position [m] x 10−5 −5 x 10 2 y−position [m] −5 2 1 0 −1 −2 −2 b) −1 0 1 2 x−position [m] x 10−5 x 10 1 0 −1 −2 −2 c) −1 0 1 2 x−position [m] x 10−5 Figure 5: The experimental position response: a) at 3890 rpm, b) at 6600 rpm and c) at 8040 rmp The presented work represents one of the rst steps in the research of modeling, analysis and control design of AMB at our institution. Although we used one of the most simple control methods we came to the following important conclusions: the rigidity of the system is increased by a higher bias current i0 and controller gain Kcon the system dynamics is improved by an appropriate derivative time constant Td the in uence of disturbances is reduced by an appropriate integral time constant Ti the system sti ness insu ciency is caused by actuator limitations, therefore an appropriate actuator should be carefully chosen with respect to the expected load forces and available power supply or vice versa the non-modeled dynamics of the shaft elasticity and unbalanced rotor becomes evident at the so-called critical speeds and this is why the control design for active vibration damping should also employ the non-modeled rotor-dynamics. Acknowledgments The authors are grateful to the Belgian Federal o ce for scienti c, technical and cultural a airs for promoting the W&T cooperation with Central- and East Europe by giving the DWTC grant to D. Dolinar. Thanks are due to the Ministry of Science and Technology of Slovenia for the nancial support. References 1] G. Schweitzer, H. Bleuer, and A. Traxler, Active Magnetic Bearings { Basics, Properties and Applications of Active Magnetic Bearings. Zurich: VDF, 1994. 2] Olympos - Finite Element and Optimization Package, see, online help. 3] C. Knospe, \PID control for magnetic bearings," in Short Course on Magnetic Bearings, Lecture 7, (Alexandria, Virginia), 1997. ...
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