ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR STĘPIEŃ, GRZEGORZ SZYMAŃSKI Chair of Computer Engineering, Poznań University of Technology Piotrowo 3a, 60-965 Poznań, Polan e-mail: Jakub.Kolota@put.poznan.pl (Receive: 0.07.00, revise: 9..00) Abstract: This research presents a metho of moeling an numerical simulation of a reluctance stepper motor using reuce finite-element time-stepping technique. In presente moel, the circuit equations are reuce to non-stationary ifferential equations, i.e. the inuctance mapping technique is use to fin relationship between coil inuctance an rotor position. A strongly couple fiel-circuit moel of the stepper motor is presente. In analyze moel the magnetostatic fiel partial ifferential equations are couple with rotor motion equation an solve simultaneously in each iterative step. The nonlinearity problem is solve using Newton-Raphson metho with spline approximation of the B-H curve. Key wors: stepper motor, finite element metho, control, inuctance lookup table. Introuction A stepper motor, as its name suggests, moves one step at a time, unlike those conventional motors, which spin continuously. If we comman a stepper motor to move some specific number of steps, it rotates incrementally that many number of steps an tops. Because of this basic nature of a stepper motor, it is wiely use in low cost, open loop position control systems. Open loop control means no feeback information about the position is neee. This eliminates the nee for expensive sensing an feeback evices, such as optical encoers. Motor position is known simply by keeping track of the number of input step pulses. They are use in consumer proucts, computer technology, complex rives an machines etc. Classically, the analysis of fiel-circuit systems is compose of two parts: the analysis of electric circuits escribe by ifferential equations, where magnetic flux is a function of fiel potentials an magnetic fiel escribe by bounary value problem [, 6]. Many researchers propose numerical proceure to investigate the couple system of equations [5, 8]. Unfortu-
6 J. Bernat, J. Kołota, S. Stępień, G. Szymański Arch. Elect. Eng. nately, strongly couple fiel-circuit moels have isavantages relate with goo efinite of global matrices of equations system. Sometimes they lea to stability loss of numerical solution. In the work a stanar moel with electric circuit equations consiere as a function of fiel potentials is compare with a surrogate moel, where the circuit equations are functions of non-stationary inuctances. In this paper the moeling metho an its numerical approach is examine using the fou-phase variable reluctance stepper motor. The time stepping finite element approximation incluing inuctance maps is applie to calculate the couple fiel-circuit problem also consiering mechanical motion. The valiity of this metho is also verifie by the experiments.. Moelling technique Stanar analysis of the electromagnetic systems is compose of three parts: analysis of n-phase electric circuit [3]: M t t A(t) l + R i (t) = u (t) l A(t) l + R i (t) = u (t) n An above equations are expresse by impeance matrix, where n =... 4 (four-phase motor moel). The analysis of electromagnetic fiel constructe as the following bounary value problem [6]: n n n () A Ω = j Ω. μ Ω Ω () The magnetic vector potential A is the magnetic fiel variable, μ is a permeability an j is a current ensity of the wining. In this moel the ey current problem is ignore, because the stator an rotor are laminate. The secon orer ifferential motion equation (3) consists of magnetic moment M which is a mechanical excitation, rotor inertia J, friction b an rotor angle Θ [, 3]. Analysis of the mechanical motion is following [, 3]: Θϕ Θϕ J + b = M. ϕ (3) t t t 0 Θϕ = ωϕ 0 Θ 0 ϕ b + M ωϕ J J ϕ. (4)
Vol. 60(0) An inuctance lookup table application for analysis 7 In case of the rotational system, the isplacement of the rotor can be assume as a one egree of freeom motion problem which is performe along axis φ. Assuming that the magnetic circuit is linear an the coil inuctances are inepenent of the stator current excitation, a relationship between the rotor position an winings inuctance can be etermine an written in lookup tables which are exactly calle maps of inuctance. In this fact creating a surrogate non-stationary circuit moel, incluing inuctance maps can be very helpful an useful in simulation. In many electromechanical systems, the torque generation can be explaine using the principle of electromechanical energy conversion in a solenoi. The solenoi has N turns, an when it is excite with a current i, the coils sets up a flux Φ. The incremental mechanical energy in terms of the electromagnetic torque an change in rotor position is written as [, 5]: W = T θ, (5) m where W m is energy converte into mechanical work, T e is the electromagnetic torque, an is the incremental rotor angle. If the inuctance is linear vs. the rotor position (i.e. inuctance oes not epen on the current), then the torque can be erive as [, ]: an numerically: T e e L ( i, θ ) i = (6) θ L ( i, θ ) θ L ( θ θ, i) L (, i) θ θ i = const. (7) The ifferentiation of the inuctance can be consiere in evaluate moel with quasi-constant torque expresse in Nm/A. It means, the surrogate moel of reluctance stepper motors is equippe with non-stationary equivalent circuit [,, 4, 7]. In this way the circuit equation () is simplifie to non-stationary initial value problem: t ( L (Θ ( t)) i ( t)) = R i ( t) u ( t). (8) n n + In these types of motors the mutual inuctances are small enough (Fig. 4) to be exclue from calculation []. n 3. Numerical experiment A variable reluctance stepper motor is consiere in numerical experiment. The evice consists of four-phase winings with eight slots. There is examine D problem using 3D first orer approach, in which the ey current problem is neglecte. The ata of the motor are presente in the Table.
8 J. Bernat, J. Kołota, S. Stępień, G. Szymański Arch. Elect. Eng. The simulation is performe for the moel with 45 440 noes. The mesh of the couple moel is shown in Fig. an Fig.. Table. Motor parameters Quantity Value Unit Outer iameter of stator 39 mm Inner iameter of stator 34 mm Outer iameter of rotor 6.96 mm Inner iameter of rotor 5 mm Air gap with 0.0 mm Motor length 30 mm Resistance/phase 5 Ω Rotor inertia 8.5 gcm Friction 0-5 Nms Fig.. 3D mesh of the reluctance stepper motor Fig.. Quarter part of the moel mesh of soli magnetic parts
Vol. 60(0) An inuctance lookup table application for analysis 9 The stepper motor consists of four winings with 08 turns each excite from pulse current source. The mechanical parameters of the rotor are: moment of inertia J = 8.5 gcm an friction b = 0-5 Nms. In the moel the rotor as well as the stator are consiere as magnetic material characterize by B-H curve. The significant changes of inuctance profile are etermine in terms of the stator an rotor pole arcs an number of rotor poles. Phase inuctances versus rotor position are shown in Fig. 3 for a constant phase current. There also exists relatively small mutual inuctance between phase winings (Fig. 3). So, in the moelle problem they can be excepte in simulations. Fig. 3. Inuctance vs. rotor position for research moel Fig. 4. An input current in motor wining f (t)
0 J. Bernat, J. Kołota, S. Stępień, G. Szymański Arch. Elect. Eng. Fig. 5. Inuctance vs. rotor position for research moel In analyze stepper motor moel, the inuctance epens on rotor position such as is presente in Figure 5. When the functions are place in lookup tables as maps, then motor circuits are strongly simplifie an makes the numerical solution easier an faster. The isplacement responses presente in Fig. 6 proves that the accepte technique is correct. Step response signals are ientically an the time consumption of the calculation is significantly reuce. For computation one step isplacement which is presente in Fig. 6 the gain is about 70 percent. Fig. 6. Comparison of linear istribute moel an simplifie non-stationary moel using inuctance lookup table metho 4. Conclusion This paper shows how to create useful an simplifie stepper motor moel base on reuce fiel-circuit escription. The simulation experiment shows that all properties of stepper
Vol. 60(0) An inuctance lookup table application for analysis motor are obtaine. Step response has oscillations an steay-state position is equale to basic step. This approach is helpful, when the equivalent circuits are built with the conition of the magnetic circuit linearity (without saturation). An avantage of this approach is reuce time consumption uring simulation, especially in case of close-loop analysis with controller or optimization nees. The propose metho can be use to esign electromagnetic rives an may contribute to the improvement of the rives ynamics. The phase inuctance L of a SRM is etermine by the phase current I an the rotor position θ an is a highly nonlinear function of both rotor position an phase current. It is well known that the ANFIS (Aaptive Neuro- Fuzzy Inference System) is a very powerful approach for builing a complex an nonlinear relationship between a set of input an output ata. For this reason, in the next work, the phase inuctance of a SRM will be calculate by using a metho base on the ANFIS. References [] Krishnan R., Switche reluctance motor rives Moeling, Simulation, Analysis, Design, an Applications. CRC Press (00). [] Lee B.S., Bae H.K., Vijayraghavan P., Krishnan R., Design of a linear switche reluctance machine. IEEE Inustry Appl. Conf. (IAS 99) 4, Oct. 3-7, Phoenix, AZ: 67-74 (999). [3] Bernat J., Kołota J., Stępień S., Dynamic of Reluctance Stepper Motor Depening on Different Air Gap. Monograph: Information Systems Architecture an Technology Moel-base Decision. pp. 79-9 (008). [4] Bae H.K., Krishnan R., A novel approach to control of switche reluctance motors consiering mutual inuctance. IEEE In. Electronics Conf., Nagoya, Japan: 369-374 (000). [5] e Oliveira A., Antunes R., Kuo-Peng P. et al., Electrical machine analysis consiering fiel circuit movement an skewing effects. Compel: The International Journal for Computation an Mathematics in Electrical an Electronic Engineering 3(4): 080-09 (004). [6] Stępień S., Patecki A., Moeling an Position Control of Voltage Force Electromechanical Actuator. Compel: The International Journal for Computation an Mathematics in Electrical an Electronic Engineering 5(): 4-46 (006). [7] Omekana A.M., A new technique for multiimensional performance optimization of switche reluctance motors for vehicle propulsion. IEEE Trans. In. Appl. 39(3): 67-676 (003). [8] Dunlop Wu W., Stephen J.B., Collocott J., Kalan B., Design optimization of a switche reluctance motor by electromagnetic an thermal finite-element analysis. IEEE Trans. Magn. 39(5): 3334-3336 (003).