715. Transients in the electromagnetic actuator with the controlled supplier

Transcription

1 75. Tranient in the electromagnetic actuator with the controlled upplier Broniław Tomczuk, Andrzej Waindok, Dawid Wajnert Opole Univerity of Technology, Department of Indutrial Electrical Engineering Prozkowka tr. 76, Opole, Poland (Received September 20; accepted 4 February 202) Abtract. A calculation of tranient in the electromagnetic actuator with the controlled upplier ha been preented in the paper. The magnetic field model of the actuator ha been created with uing the finite element method (FEM) and verified experimentally. The mathematical model of the upplier and controller have been coupled with the field-circuit model of the actuator uing Matlab/Simulink package. The circuit parameter have been obtained from the FEM calculation. The algorithm of the proportionally-integral (PI) controller operation ha been implemented. The tranient of poition, current and force, for different controller parameter, have been obtained. The calculation reult have been verified by the meaurement tet. Keyword: control ytem, linear actuator, coupled field-circuit model, modelling of mechatronic ytem. Introduction The electromagnetic linear actuator and motor are ued a a linear drive in different application [, 2], e.g. upenion ytem, fatigue tand [3], generator of vibration [4] and ocillator [5, 6]. There are growing demand for uch pecial electric motor. In each application (e.g. CNC machine) the proper control and upply ytem are needed. In mot cae an ordinary PI or PID controller can be applied. The modelling of uch controller become more important due to deign cot. Intead making of the phyical control ytem, we can build and invetigate it mathematic model. Thu, we can tudy tranient in the ytem including changing the controller parameter. After thi, we can implement the parameter in the real prototype. The limit of the parameter could be obtained by calculation. Thu, we are able to introduce them in the tet without detroying the phyical model. In thi paper we conider the electromagnetic actuator (Fig. ) build in the controlled fatigue tand. The box of the control-upplying ytem i preented on the right ide of Fig.. We can ee that the controlled upplier i very mall. Fig.. The actuator in the fatigue tand with the controlled upplier VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 20. VOLUME 4, ISSUE. ISSN

2 Both, mathematical and phyical model have been done within the preented work. The parameter of the electromagnetic actuator, which i viible on he left ide of Fig., have been calculated with the finite element method (FEM) [7] through the magnetic field analyi. We have ued the FEMM oftware [8]. The whole device i.e. actuator and it control upplier ha been imulated with Matlab/Simulink oftware package. We have modelled the PI controller and the pule width modulation (PWM) power upplier and the mathematical model of the actuator (Fig. 2). The tree in the lever of the fatigue tand and the teted pecimen were calculated with Femap oftware [9]. Modelling of the actuator tranient For the modelling of the actuator tranient the voltage u excitation ha been forced. In our modelling, the runner poition z and the current i are the function to be calculated. The Lagrange method ha been adopted for the formulation of the ordinary differential equation [0, ], which decribe the unknown quantitie: 2 d z F( i, ( k = 2 dt + k ) z Dzɺ m L + mg () ( Ψ( i, ) u Ri v di = z (2) dt L ( i, d where the parameter are: k, k L the pring and fatigued pecimen contant repectively, D the friction coefficient, m ma of the runner, v= zɺ velocity of the runner, F(i, force characteritic, R coil electrical reitance, L d ( i, - dynamic inductance characteritic, Ψ ( i, - linkage flux characteritic. The characteritic of the function F(i,, L d ( i,, Ψ ( i, have been preented in Fig. 3 and 4. The value of the remaining parameter are given in Table. Product dpi(i,/dz 50 D Fcn f(u) Ld(i, 53000*u Specimen reilience /ma -C- -K- Velocity Poition Gravitation force Duty Voltage Current Product Current F(i, PWM R -K- Current Out In Dicrete PI Controller Add Sine Wave Force Poition Duty Fig. 2. The model of the actuator with the controlled upplier The integral parameter of the magnetic field are: thrut, magnetic flux and winding dynamic inductance. They have been obtained from the magnetic flux denity ditribution. We have ued FEM calculation. The geometry of the actuator can be preented in the axial 40

3 coordinate ytem r, φ, z. The thrut ha been obtained uing the expreion for the Lorentz force [2]: ( J ) ϕbzr JϕBr z F = 2π S rdrdz The flux value ha been calculated a the integral of the magnetic vector potential [3]. The dynamic inductance i very important in the imulation of the operation of the actuator under voltage upplying. It value ha been calculated a the current derivative of the flux linkage [4]: (3) Ψ = N k = lk A d (4) ϕ l k dψ( i, L d = (5) di where N i the number of turn in the actuator winding. The equation () and (2) have been olved in the Matlab/Simulink package [5]. The upplying through the pule width modulation (PWM) ytem and the control ytem were included in the mathematical model (Fig. 2). The force characteritic (Fig. 3a) i important for the control algorithm. The thrut value change linearly v. the current excitation value. The thrut characteritic i only lightly nonlinear for the high current intenity in the excitation coil. The flux inide the winding depend almot only on the mover poition (Fig. 3b). The dynamic inductance, which i very important value in the tranient model, i almot contant (Fig. 4a). The characteritic in Fig. 4b, which i ued in the calculation of the electromotive force (EMF), depend only on the mover poition. The mooth hape of the preented characteritic are very convenient for the quick and precie analyi of the actuator tranient. Fig. 3. The characteritic vero mover poition z and the excitation current I of: a) thrut F, b) flux Ψ The parameter in Table have been obtained both from meaurement and calculation. The reitance value of the coil and the mover ma have been meaured before the other parameter determination. The friction coefficient value ha been aumed taking into account the mover and tator material. The um of the pring and pecimen contant ha been calculated including the tre value (Femap oftware [9]). Table. Contant parameter of the field-circuit model Reitance Mover ma Friction coefficient Contant Parameter R [Ω] m [g] D [N/m 2 ] k + k L [N/mm] Value

4 Fig. 4. The characteritic of: a) dynamic inductance L d, b) poition derivative of the magnetic flux Calculation reult The electromagnetic actuator detination i to generate inuoidal wave of the runner poition. The amplitude of the runner troke z and the ocillation frequency f can be etablihed. The accomplihment of thi requirement demand a feedback loop from the runner poition. The difference between the reference poition value and the meaured poition of the runner i introduced to the PI controller (Fig. 2). The output of the controller i the PWM duty cycle. Value of the PI controller parameter were obtained by uing Contrain Signal block from Matlab/Simulink (Fig. 2). The optimization proce wa made for the following parameter of tep repone: rie time t r = 0.025, ettling time t = 0.05 and overhoot M p = 5% [6]. We obtained uitable factor for the control ytem: K p = and K i = 280 (Fig. 5a). Fig. 5. Calculation reult of the runner tep repone for: a) K p = 236.4, K i = 280, b) K p = 56.4, K i = 280 In Fig 5a the tep repone characteritic of the runner poition for optimal value of PI controller parameter i preented. The curve fulfill the aumed requirement. For comparion, in Fig. 5b the tep repone wave for non-optimal value of PI controller parameter i preented (K p = 56.4 and K i = 280). The lower value of K p parameter lead to increae the overhoot and ettling time. Many calculation of the actuator tranient have been carried out, uing the preented model (Fig. 2). In Fig. 6, the poition of the runner and current wave for aumed amplitude and frequency are preented. Meaured and calculated amplitude of the poition wave (Fig. 6a) are 42

5 lightly lower than the aumed one. Thi i due to the fact that the control ytem doen't track the given inuoidal wave. In Fig. 6b current wave for aumed amplitude z = 4 mm and frequency f = 0 Hz are preented. In both, meaured and calculated cae, the wave are imilar. Slight difference are viible between meaured and calculated value of generated force (Fig. 7). The meaured current and force wave are not mooth (Fig. 6b and 7). It i due to controller preence, which compenate the friction force and diturbance in the mechanical ytem. Fig. 6. Runner poition (a) and current wave (b) for f = 0 Hz and z = 4 mm Concluion Fig. 7. Thrut v. time for: a) f = 20 Hz and z = 2 mm, b) f = 0 Hz and z = 4 mm The reult obtained from the meaurement of the current value of runner poition and of force are in good agreement with the imulated numerical one. The difference do not exceed everal percent and are mainly in the force wave oberved. Mechanical and electrical characteritic of the controlled ytem have been determined. There are only mall nonlinearitie viible in the characteritic of the integral parameter v. runner poition and current value. The preented PI controller meet the requirement of tability and preciion in the runner poition controlling. The current and force wave are not mooth, which i mainly due to random diipation force, which diturb the runner movement. The field-circuit model can be ued in the tuning of the controller adjutment and in the prediction of the electrical and mechanical parameter of the fatigue tand with build in electromagnetic actuator and control ytem. One of the mot advantage of the model i hort calculation time. For the computer with 4 GB RAM, AMD Phenom II X4 955 proceor 43

6 (3.2 GH we need only few econd to imulate the complete tranient of the whole ytem (including PWM modeling). Acknowledgement Thi paper i partially upported by the Polih Minitry of Science and Higher Education under Grant No. N N and within the European Social Fund. Reference [] Tomczuk B., Waindok A. Linear motor in mechatronic achievement and open problem. Monograph: Tranfer of Innovation to the Interdiciplinary Teaching of Mechatronic for the Advanced Technology Need, Opole Univerity of Technology, OWPO, Opole, Poland, p [2] Giera J. F., Piech Z. J., Tomczuk B. Linear Synchronou Motor, Taylor & Franci, USA, 20 (in printing). [3] Karolczuk A., Kowalki M., Waindok A. Stand for fatigue tet of material uing electromagnetic actuator prototype. XXV Sympozjon Podtaw Kontrukcji Mazyn, Gdańk, 20 (in printing). [4] Bang Y. B., Lee K. M. Large thrut linear motor for low-duty-cycle operation. Mechatronic, No. 4, 2004, p [5] Enomoto H., Harada K., Ihihara Y., Todaka T. Optimal deign of linear ocillatory actuator uing genetic algorithm. Compumag, Rio de Janeiro, Brail, November 3-6, 997, p [6] Kciuk S., Pilch Z., Szczygieł M., Trawińki T. VCM motor for active vibroiolation theoretical background. Int. XV Symp. Micromachine and Servoytem, Soplicowo, Poland, September 7-2, 2006, p [7] Tomczuk B. Numerical Method for Field Analyi of Tranformer Sytem. Oficyna Wydawnicza Politechniki Opolkiej, Opole, 2007 (in Polih). [8] Meeker D. FEMM 4.0, Uer Manual, Univ. of Virginia, USA, [9] FEMAP Uer Guide, Verion 9.3., USA, [0] Tomczuk B., Waindok A. A coupled field-circuit model of a 5-phae permanent magnet tubular linear motor. Proc. of the XXI Sympoium Electromagnetic Phenomena in Nonlinear Circuit (EPNC 200), Dortmund und Een, Germany, June 29 July 2, 200, p [] Tomczuk B., Sobol M. A field-network model of a linear ocillating motor and it dynamic characteritic. IEEE Tran. on Magnetic, Vol. 4, No. 8, Augut 2005, p [2] Tomczuk B., Zimon J., Waindok A. Electromagnetic force calculation in the linear motion actuator with uing Finite Element Method (FEM). Pomiary Automatyka Kontrola (PAK), 0/2006, p [3] Tomczuk B., Waindok A. Integral parameter of the magnetic field in the permanent magnet linear motor. Monograph Intelligent Computer Technique in Applied Electromagnetic (in the erie Studie in Computational Intelligence), Springer Verlag, Heidelberg, Germany, Vol. 9, 2008, p [4] Waindok A. Computer Simulation and Meaurement Verification of the Permanent Magnet Tubular Linear Motor (PMTLM) Characteritic. PhD Thei, Opole Univ. of Technology, Opole, Poland, 2008 (in Polih). [5] Uing Simulink verion 4. The Mathwork Inc., [6] Franklin G. Feedback Control of Dynamic Sytem, Prentice Hall,