I-12. OPTIMIZATION OF A LINEAR BRUSHLESS DC MOTOR DRIVE Ph. Dessante 1 , J.C. Vannier 1 and Ch. Ripoll 2 1 Service EEI Ecole Sup´erieure d’´electricit´e (Supelec), Plateau de Moulon, 91192 Gif sur Yvette, France, philippe.dessante@supelec.fr jean-claude, vannier@supelec.fr 2 Renault Research Center—Guyancourt, christophe.ripoll@renault.com Abstract. The paper describes the design of a drive consisting of a voltage supplied brushless motor and a lead-screw transformation system. In order to reduce the cost and the weight of this drive an optimization of the main dimensions of each component considered as an interacting part of the whole system is conducted. An analysis is developed to define the interactions between the elements in orderto justifythe methodology.A specificapplication inthen presentedand comparisonsare made between different solutions depending on different cost functions(max power, weight, cost, ).With this procedure, the optimization is no longer limited to the fitting between separated elements but is extended to the system whose parameters are issued from the primitive design parameters of the components. Introduction The system studied in this paper is a linear electrical drive system realized with a voltage supplied brushless motor whose shaft is mechanically connected to a lead-screw drive device. The aim of this system is to drive a load along a linear displacement. The specifications concerning the load consist mainly in two parts. Firstly, it has to apply a rather high static force at standstill as for instance to overcome some static friction force. Secondly, it has to be driven from one point to another point in a given time. This second part implies a dynamic force and a maximum speed depending on the kind of displacement function is chosen. A discussion to chose the displacement function is important because as the motor will have a limited torque capacity, it may be necessary not to accelerate neither too early nor too late when it is entering the constant power region. Consequently this definition can have consequences on the system size. At the very beginning it may be considered sinusoidal or corresponding to a bang-bang acceleration. System modeling A general presentation of the system is given in Fig. 1 where the power source is supposed to be a battery bank. S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 127–136. C 2006 Springer. 128 Dessante et al. Brus hl ess Motor Speed Reducer Lead-Screw Device Load Battery Bank Power Supply Figure 1. System main components. Concerning the kinematic model, the lead-screw is represented by its transformation ratio deduced from the screw pitch while the speed reduction system introduces a speed transformation ratio.These two componentscan berepresented by theglobal transformation ratio, ρ, between the motor shaft angle and the linear load displacement: θ = ρx (1) This association between a speed reducer of a given ratio, N, and screw of a given pitch, τ , gives the resulting value for the transformation ratio: ρ = 2π N τ (2) This ratio is used to convert the load specifications in motor specifications. The load displacement is directly changed in angle variation and the forces are converted in torques taking into account the efficiency of each component. During acceleration the motor inertia leads to a difference between the output torque and the electromagnetic torque. In this application two sorts of torques are to be generated by the drive system. A static torque (at zero speed) can be necessary to reach the breakaway force on the load just before it starts to move. It can either represent the torque needed to maintain the load in a position when an external force is applied. With a given force, F sta , the static torque is given by: C sta = F sta ρ (3) When the load speed is increased, generally the motor has to generate a torque with two components. This second sort of torque is called the dynamic torque. It contains a part corresponding to the force required to accelerate the load and a second part to accelerate the rotor and the transmission system. This part is represented by the inertia, J mot , of the motoring part. With a given dynamic resistive force, F dyn , an equivalent mass of the load m , a friction coefficient f , the dynamic torque for an acceleration γ at a speed v on the load can be expressed as follows (4). C dyn = J mot ρ + m ρ γ + f v + F dyn ρ (4) The two types of torque are dependent on the transformation ratio level. For the static torque it is obviously interesting to use a high value of the transformation ratio because the corresponding torque value will decrease and this will reduce the motor constraint (Fig. 2). I-12. Optimization of Linear Brushless DC Motor Drive 129 Motion transformation ratio Static Dynamic Figure 2. Torques vs. transformation ratio (5). Forthedynamictorque,theincreaseofthe transformationratiowillreducethecomponent of the torque needed to drive the load but it will increase the torque required to accelerate the motoring parts mainly constituted of the rotor of the electrical motor. Consequently a first limitation appears when choosing the value of the transformation ratio. It is not possible to retain a high value without having to generate a high dynamic torque. If we first considerthe situation illustratedin Fig.2for a fixed rotor inertia, itcorresponds to the case of the total force required by the load in dynamic mode, F dyn tot , whose value is referenced to the static force as: F dyn tot < F sta /2 (5) In this case, a good value for ρ could the one observed at the intersection between the two curves [1–3]: ρ i = F sta − F dyn tot J mot γ (6) With that value the torque to be generated by the motor is minimal. Secondly we consider the case of a greater relative value for the total dynamic force needed by the load as: F sta /2 < F dyn tot < F sta (7) As it can be observed in Fig. 3, the dynamic torque will be minimal after the intersection between the two curves. For this reason, a good value for the transformation ratio could be in that case the one corresponding to the minimization of the dynamic torque: ρ 0 = F dyn tot J mot γ (8) 130 Dessante et al. Motion transformation ratio Static torque Dynamic torque Torque (Nm) Figure 3. Torques vs. transformation ratio (7). In that case, the torque to be generated by the motor is minimal with this choice. As thedynamic torque alsodepends on the value oftherotor inertia whichwill be defined during the motor design the situation is more complex and will be discussed. Other constraints [4] are also to be considered. The load duty cycle is generally defined and leads via the rms and the average values of the load dynamic to the definition of the corresponding rms torque: C 2 rms = F rms ρ 2 + f v ave ρ 2 + 2F dyn f v ave ρ 2 +γ 2 rms ρ J mot + m ρ 2 (9) Among the limits concerning the motor, there could be a maximal rotor speed and the actual speed has to be considered: = ρv (10) This expression clearly indicates that the augmentation of the mechanical transformation ratio will need higher rotor speed for the motor. The motor supply and the battery tank characteristics introduce a limitation of the power consumption. This finally depends on the efficiency reached by the motor and on the power consumed by the load. The efficiency of a motor can be estimated from its main character- istics and the peak consumed power can then be defined: P dyn = C dyn ˆ = (J mot ρ 2 + m )γv+( f v + F dyn )v (11) For the motor design, different levels of complexity in modeling are available. To simplify, it is possible to define the main dimensions by using the peak torque, the rms torque, and I-12. Optimization of Linear Brushless DC Motor Drive 131 the rotor inertia as follows [5]: J mot = 1 2 πμ v R 4 L (12) C p = 4pH 0 BRLE (13) C n = 2πABγ p R 2 L (14) Consequently, these three relationships introduce three main dimensions parameters for the design: the rotor radius R, the rotor length L, and the permanent magnet thickness E. The remaining parameters are more or less constant or weakly dependent on the motor size. They are defined as: p = pole’s number. H 0 = magnet’s peak magnetic field. B = airgap flux density. A = stator excitation level. γ p = pole’s overlapping factor. μ v = mass density. Concerning the converter, the volume of silicon can be linked to the maximum power value needed by the motor to drive the load. Optimization Theestablishedrelationshipsare usedtodefinethe constraintsintheoptimization procedure. The motor peak torque has to be greater than the static and the dynamic torques. The nominal torque is also greater than the required rms torque. C p > C sta (15) C p > C dyn (16) C n > C rms (17) The maximum power consumption is to be kept below the maximal value supplied by the battery bank. The mechanical transformation device introduces inertia in the system equations. Furthermore it needs a volume that will be a part of the total volume allowed to the system. P max > P dyn (18) Some technical constraints have to be added in order to be able to define a feasible motor. It concerns the maximal rotor speed and the ratio between the rotor length and the diameter. max > ˆ (19) aR > L > bR (20) A minimum relative value is needed for L to be kept in the domain of validity of the previous expressions (12–14). A maximum value is settled to avoid the definition of a too thin rotor with a high length to diameter ratio as it could be required to reduce the rotor 132 Dessante et al. inertia (12). A quasi fixed ratio can also be imposed by the choice of the coefficients a and b. Depending on the application, different cost functions can be minimized. For instance, if the weight is the principal criteria, the motor size will be reduced. If the volume is to be kept as low as possible, the mechanical transformation system size will be an issue. Results We present here the results concerning the definition of the motor and the motion transfor- mation ratio whose dimensions are optimized for a given load. In this example, the load characteristics are the followings: F sta = 900 N ˆγ = 1m/s/s F dyn = 450 N ˆv = 35 mm/s F rms = 90 N γ rms = 0.1m/s/s f = 0N/s v ave = 28 mm/s m = 1kg The optimizationprocedure uses theconstraints (15)–(20) andsearches asetof values for R, L, E, and ρ which minimizes the motor peak torque. It appears that the mass is minimized as a consequence. As boundaries are used to limit the variation of these parameters to feasible values it appears that the result is always for the upper boundary value for the transformation ratio. In Fig. 4, the evolution of the main rotor dimensions with the maximum authorized transformation ratio value are presented. 0 0.5 1 1.5 2 2.5 3 x 10 4 0 5 10 15 20 25 R (O): L(*) & 10×E (+) in mm Figure 4. Rotor dimension R, L, E vs. ρ max per meter. I-12. Optimization of Linear Brushless DC Motor Drive 133 0 0.5 1 1.5 2 2.5 3 x 10 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Csta (o); Cdyn (*) & 2xCrms (.) in Nm Figure 5. Motor torques vs. ρ max per meter. The rotor mass as its inertia are decreasing as long as the maximum value for ρ is increased. In Fig. 5 the evolution of the torques is presented too. As it was observed before, the static torque diminishes when the ratio increases. But in that procedure, it is observed for the dynamic torque as well and the good value for ρ is the maximum permitted value. This main difference is due to the fact that the rotor inertia changes its value when the ratio does so. This could be a very important constraint for the motor design. In the presenteddesign procedure,someconstraints(20) havebeenintroduced toavoidsuch design difficulties. For every value of the maximum ratio, the rotor inertia can be evaluated and the previous good values for ρ (6) and (8) can be calculated too. They give the corresponding torques presented in Fig. 6. In that particular case the values are almost the same because the total dynamic force is near half the static force. We can notice that the “good” ratio value is much more important than the permitted ratio value. Consequently, the torques values are lower than the values obtained at the boundary of the domain. Finally, among the different values proposed by the design procedure, it is necessary to retain one of them to design the motor. A criterion can be the maximum rotor speed. In Fig. 7, the evolution of the maximum rotor speed with the maximum transformation ratio is presented. These speed values are rather common values for electrical motors. For small motors the choice of a maximum speed of 6,000 or 9,000 rpm is reasonable. When the optimization procedure succeeds in defining a feasible motor, a more complex model is used to calculate all the dimensions. In Fig. 8 is presented a view of one of these motors. 134 Dessante et al. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 10 5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Co (o) & Ci (.) in Nm Figure 6. Former minimum torques vs. ρ opt per meter. The airgap diameter is 8 mm and the outer diameter is close to 19 mm. The rotor length is 12 mm and the inertia is 0.022 kgmm 2 . NdFeB magnets are used to magnetize the airgap with gives a flux density equals to 0.8 T. The resulting active mass is 20 g. With the housing the resulting mass will be slightly higher. The original commercial motor used to drive this application had a mass equal to 100 g. 0 0.5 1 1.5 2 2.5 3 x 1 0 4 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1 0000 Omega max (rpm) Figure 7. Maximum rotor speed vs. ρ max per meter. I-12. Optimization of Linear Brushless DC Motor Drive 135 -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 -8 -6 -4 -2 0 2 4 6 8 x 10 -3 Figure 8. Resulting motor dimensions. In Fig. 9, a simulation of the flux lines distribution is obtained with FEM analysis. This permits to verify the values expected from the design procedure. The nominal torque is 10 mNm and the peak torque is at least 50 mNm. The maximum speed should be 6,600 rpm to drive the load at its maximum speed. At maximum power, the motor efficiency is about 50% if it is assumed that the joule losses are predominant. -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 9. Flux lines at load. 136 Dessante et al. A value of 20,000/m for the motion transformation ratio can be obtained with a lead- screw pitch of 3 mm and a speed reduction gearbox with a 9.5 ratio. It can be observed that when the motor size decreases, the rotor speed increases which leads to the definition of a larger mechanical transformation system. This is another con- straint which can be considered. Conclusion In this paper, an electromechanical conversion system is analyzed resulting in a modeling of the components.The modelhas to beinversed tolink the dimensionsto theperformancesfor each component involved in the power conversion system. Consequently the whole system dimensions are available for the aggregate optimization of the system. This procedure permits a correct association between the components and can lead to a smaller volume or a smaller weight than it could be defined with a separated element optimization. The results presented have shown theinterest to optimize simultaneously the rotormain dimensions and the transformation. Actually, this procedure avoids therisk of havingto design a nonfeasible motor with a too low inertia for a given torque. As it needs the complete specific design of a dedicated motor, it is reserved for rather expensive application (aircraft, space, . ) with severe criteria or for very large scale appli- cation (automotive, . ). As forthistype of application,the total mass ofthe system is tobe considered, acomplete modeling of the transformation system is needed as for the electronic converter. This could be presented in a further work. References [1] E. Macua, C. Ripoll, J C. Vannier, “Optimization of a Brushless DC Motor Load Association”, EPE2003, Toulouse, France, September 2–4, 2003. [2] E. Macua, C. Ripoll, J C. Vannier, “Design, Simulation and Testing of a PM Linear Actuator for a Variable Load”, PCIM2002, N¨urnberg, Germany, May 14–16, 2002, pp. 55–60. [3] E. Macua, C. Ripoll, J C. Vannier, “Design and Simulation of a Linear Actuator for Direct Drive”, PCIM2001, N¨urnberg, Germany, June 19–21, 2001, pp. 317–322. [4] M. Nurdin, M. Poloujadoff, A. Faure, Synthesis of squirrel cage motor: A key to optimization, IEEE Trans. Energy Convers., Vol C6, pp. 327–335, 1991. [5] C. Rioux, Th´eorie g´en´erale comparative des machines ´electriques ´etablie `a partir des ´equations du champ ´electromagn´etique, Revue g´en´erale de l’Electricit´e (RGE), Vol. t79, No. 5, pp. 415– 421, mai 1970. . vs. ρ max per meter. I-12. Optimization of Linear Brushless DC Motor Drive 135 -0 .01 -0 .008 -0 .006 -0 .004 -0 .002 0 0.002 0.004 0.006 0.008 0.01 -8 -6 -4 -2 0 2 4 6 8 x 10 -3 Figure 8. Resulting. that the joule losses are predominant. -1 -0 .5 0 0.5 1 -1 -0 .8 -0 .6 -0 .4 -0 .2 0 0.2 0.4 0.6 0.8 1 Figure 9. Flux lines at load. 136 Dessante et al. A value of 20,000/m for the motion transformation. a lead-screw transformation system. In order to reduce the cost and the weight of this drive an optimization of the main dimensions of each component considered as an interacting part of the whole