Servo otors Classification Based on the Accelerating Factor Hermes GIBERTI, Simone CINQUEANI echanical Engineering Department, Politecnico di ilano, Campus Bovisa Sud, via La asa 34, 0156, ilano, Italy ABSTRACT This work is focused on the analysis of the so-called accelerating factor (α) [9,10] defined, for each motor, as the ratio between the square of the motor nominal torque and its momentum of inertia. The coefficient α is exclusively defined by parameters related to the motor and therefore it doesn t depend on the machine task: it can be calculated for each motor using the information collected in the manufacturer catalogues. Actually there is not any theoretical study that investigates the dependence of the accelerating factor on the electro-mechanical characteristics of the motor. One way to investigate these relationships is to collect information from catalogues of a significant number of motors produced by different manufacturers. This allows to have a statistical population on which perform appropriate analysis. For this reason a database containing more than 300 brushless motors has been created containing, for each record, information on the most important electro-mechanical characteristics. Using the collected information, some graphs are produced showing how motors having the same size have different accelerating factors. Keywords: Electric servo-motor; accelerating factor; continuous duty power rate 1. INTRODUCTION The need to increase production capability, while maintaining the quality standards, requires the implementation of automatic machines performance ever higher. In this context, it is of strategic importance in the machine design phase the correct selection of the motor-reducer unit. Unfortunately, the choice of the electric motor required to handle a dynamic load, is closely related to the transmission choice. This operation, in fact, is bound by the limitations imposed by the motor working range and it is subjected to a great number of constraints that depend indirectly on the motor (through its inertia) and on the reducer (through its transmission ratio, its mechanical efficiency and its inertia), whose selection is the object of the design. In literature there are many procedures for the selection of a motor-reducer unit [1-8] that, while all start from the same theoretical basis, they differ from their approach to the problem. This work is focused on the analysis of the continuous duty power rate (also called accelerating factor ) [8] defined, for each motor, as the ratio between the square of the motor nominal torque and its momentum of inertia. Each manufacturer of brushless synchronous motors adopts its own technological solutions and its constructive layout, which is generally different from those of another producer. However, the designer of an automatic machine who has to choose a motor, can consider all the motors as black boxes characterized by their accelerating factor. Actually there is not any theoretical study that investigates the dependence of the accelerating factor on the electro-mechanical characteristics of the motor, therefore the comparison of motor performances in terms of their accelerating factor is possible only in relative terms and not in absolute. The aim of this paper is to put the groundwork for a deeper analysis of the accelerating factor, in order to give to the designer of an automatic machine a tool to critically evaluate the performance of motors, not only as compared to the other available, but also in absolute terms. Table 1 Nomenclature T motor torque motor momentum of inertia T,rms motor root mean square torque T,N motor nominal torque TH, max motor theoretical maximum torque T,max servo-motor maximum torque ω motor angular speed ω,n motor nominal angular speed ω& motor angular acceleration P N motor nominal power m motor mass V N motor nominal voltage p motor poles T L load torque L load momentum of inertia T L generalized load torque T L,rms generalized load root mean square torque T L,max load maximum torque ω L load angular speed ω& L load angular acceleration ω& L,rms load root mean square acceleration transmission ratio η transmission mechanical efficiency α accelerating factor β load factor ω,max maximum speed achievable by the motor ω L,max maximum speed achieved by the load t a cycle time C th motor thermal capacity R th motor thermal resistance th motor thermal constant K T torque constant i current flowing in motor windings
. THE OTOR Brushless motors (Fig.1) are the most widespread electrical actuators in automation field, which working range (Fig.) could be approximately subdivided into a continuous working zone (called S1, bound by motor rated torque) and in a dynamic one (called S6, bound by the maximum motor torque T,max ). Usually the motor rated torque decreases with the motor speed ω. To simplify the rated torque trend and to have a cautionary approach, the continuous working range is approximated to a rectangle, identifying two values T,N and ω,max (Fig.3). Note how the approximation used to make the S1 field rectangular, actually has consequences on the value of T,N and ω,max. Information on catalogues are often poor and, in the best case, when speed/curve torque is available, they should be managed to obtain the interesting parameters. Note how the maximum torque achieved by the servo-motor T,max strongly depends on the drive associated with it and it is generally different from the motor theoretical maximum torque TH, max. T T,max T,N T Fig.1 - Commercial brushless motor T,max T,N ω,max Figure 3 Approximated speed/torque curve ω,max Fig. - Speed/torque curve of a common brushless motor ω At low speed, the constraint introduced by the drive systems is related to the maximum current supplied to the motor. Since torque depends on the current, this limit translates into a horizontal line on the motor working field corresponding to a maximum torque different from the theoretical one. At higher speed, this constraint is overcome by the condition on the maximum voltage endurable, which causes a reduction of the motor maximum torque with its speed. 3. THE THERAL PROBLE OF ELECTRICAL OTORS The thermal problem is of great importance in electric motors, and is generally the most binding condition in the choice of an electric motor for industrial applications. During their operation, in fact, motors waste power W d as heat: this is primarily because the windings are affected by the current flow (copper losses), but also for the eddy currents (iron losses) and mechanical effects.
The power lost as heat determines an increase in temperature of the motor. Heat is partially removed from the environment at least until a stationary condition is determined. Naming θ(t) the difference of temperature at time t between motor and environment, C th the motor thermal capacity and R th its thermal resistance, the differential equation for the equilibrium of power is: that can be rewritten as: where: d C θ th + θ = Wd (1) dt Rth θ d th + θ = Wd Rth () dt th = RthCth (3) is the motor thermal time constant (usually defined by the manufacturer and available on catalogues). Observance of the constraints relating to thermal problem requires, when selecting a motor, that the maximum temperature reached during the operation does not exceed the maximum permissible. This requires solving Eq. (1). However, if the task operation is cyclic, with period t a << th the problem can be simplified. In this case, the motor is not able to follow the fast thermal fluctuations of the power dissipation, due to high heat resistance. The temperature of the motor, then, evolves as if it were subject to constant power dissipation W d equal to the average power dissipated in the cycle. Assuming that the dissipation is related mainly to the oule effect due to the resistance R, it is: t a R W d = i dt t a 0 (4) = KT i (5) where K T is the torque constant. By substituting eq.(5) in eq.(4), it is possible to reach the value of the so called motor root mean square torque: The motor torque T can be written as: where: = T L + TL = TL + Lω& L & ωl is the generalized resistant torque at the load shaft. When selecting the motor-reducer unit, the transmission ratio and the motor inertia are still unknown. In this phase, transmission is considered ideal (η=1). Equation (8) highlights the dependence of the motor torque on this variables, while from eq.(9) it s possible to observe that all the terms related to the load are known. The root mean square torque is obtained from: ta t a 1 & ωl, rms = dt = TL + dt ta ta 0 0 (8) (9) (10) Developing the term in brackets and using the properties of the sum of integrals, it s possible to reach the root mean square torque as: ( T L & ω L ) mean rms TL rms &, =, + ω L, rms + (11) and inequality (7) can be written as: ( T L & ω L ) mean T N TL rms &,, + ω L, rms + (1) 4. THE OTOR ACCELERATING FACTOR Since T,N is positive by definition, one can gets: ( T L & ω L ) mean N TL rms &,, ωl, rms + + (13) Let s introduce the accelerating factor of the motor:, rms = t a 1 dt ta 0 (6), N α = (14) namely the torque, acting steadily over the cycle, which is attributable to the total energy dissipation really occurred in the cycle. The condition on the thermal problem becomes: T,rms < T,N (7) where motor torque T,N is obtainable from catalogues given by motor manufacturers and it is defined as the torque that can be supplied by the motor for an infinite time, without overheat. describing the performances of each motor, and the load factor: β [& ω T ( T & ω ) ] = L, rms L, rms + L L mean (15) defining the performances required by the task. The unit of measurement of both factors is W/s. The coefficient α is exclusively defined by parameters related to the motor and therefore it doesn t depend on the machine task: it can be calculated for each motor using the information collected in the manufacturer catalogues. oreover it could be reported on them, to provide a classification of the commercial motors on
the basis of this standard. Otherwise, the coefficient β depends only on the working conditions (applied load and law of motion) and it s a measure that defines the power required by the system. Substituting α and β in inequality (13) we reach:,, α β + T L rms & ωl rms (16) Data analysis Figure 4 represents the trend of the accelerating factor (y axis) for the entire population of considered motors (x axis). otors are identified by a unique growing index. Notice how α can assume values really different, and how some motors have an accelerating factor extremely high compared to the considered population. Since the term in brackets is always positive, or null, the load factor β represents the minimum value of the right hand side of eq.(16). It means that the motor accelerating factor α must be sufficiently greater than the load factor β, so that inequality (7) is verified. A motor must be rejected if α<β, while if α β the motor can have enough rated torque if is chosen properly. The preliminary motor choice is conducted comparing only the values α and β; these values are easily calculated knowing the mechanical properties of the motor and the load features. 5. SERVO OTOR COPARISON The aim of this work is to put the basis of a detailed analysis of the accelerating factor (or continuous duty power rate). The starting point is the answer to the question: Let s assume that motors with different sizes are hard to be compared, may similar motors have accelerating factors α extremely different? A negative answer to this question would make unnecessary any subsequent consideration, indicating that manufacturing parameters marginally influence the accelerating factor. That means the commercial brushless motors currently on the market have similar electromechanical features, presumably best suited to obtaining high values of α. On the opposite, a positive answer would open a research field to find which are the electromechanical features of a motor that most influence the accelerating factor and which is (if it exists) the theoretical or technological value of α whose overtaking is impossible or technically not convenient. A way to answer the question is to collect enough information from different manufacturers catalogues for a significant number of motors. The resulting database will be a useful instrument to compare different commercial devices and a suitable tool to highlight how the accelerating factor can t be the only parameter to describe the performance of a motor and how all motors features influence the design of a machine. The database The database collects the main information available on catalogues of about 300 motors whose power is between 15[W] and 15[kW]. Information collected relate to: brand, model, type of motor (AC or DC), torque coefficient, winding electrical resistance, number of poles, geometrical dimensions and, naturally, motor nominal torque and the rotor momentum of inertia. The momentum of inertia includes the inertia of the rotor and the one of the positioning sensor, a needed component for the machine functioning and thus a part of it. The inertia of any brake systems, or related to any additional sensors is neglected. Figure 4 Accelerating factor (α) for the motors in the database The graph can not highlights if this high values of the accelerating factor are due to a high nominal torque, or to small rotor inertia, or to the combination of the two factors. It is also unclear whether the accelerating factor is related to the motor size or not. For this reason, values of α, motor nominal torque T,N and motor momentum of inertia are reported on the same chart for all the motors in the database (Fig.5). For ease of consultation, motors are ordered with increasing momentum of inertia. The three series of data are normalized on their respective maximum value to allow a comparison between series. Figure 5 Normalized accelerating factors (α), motor nominal torques (T,N) for the considered motors ordered with increasing momentum of inertia Figure 6 depicts the trends of motor weight () and nominal torques (T,N ) for the considered motors ordered with increasing momentum of inertia.
Actually, this conclusion is the starting point to investigate what are the electromechanical characteristics that allow a motor to be more performing. Figure 6 Normalized masses (Μ), motor nominal torques (T,N) for the considered motors ordered with increasing momentum of inertia Looking the chart in Fig.5, 6 some interesting consideration can be done: 1. High values of accelerating factor can be obtained, even with high values of, because of increased motor nominal torque;. otor nominal torques and rotor inertia seem to be proportional each others; 3. otors with same momentum of inertia can have accelerating factors extremely different; 4. otors momentum of inertia and mass seem to be proportional These considerations give a first answer to the question done: commercial brushless motors are built with different designs and generally have different performances. It means that some motors are better than others. Table shows, as example, the main features of motors classified as no.44 and no.30. Despite their different characteristics and dimensions, the two motors are identical at least as regards their accelerating factors. Table Comparison between two selected motors Let s now consider the transmission that could be coupled with each motor, such that condition on thermal problem is verified. The range of suitable transmission ratios can be calculated by solving eq.(16). It results: = T L,rms α β (17) where T L,rms and β do not depend on the motor. otors in table have similar accelerating factors but rotor momentum of inertia really different. Suppose their accelerating factors were higher, for a given task, than the load factor. Then motor no.30, with a greater moment of inertia, would have a wider range of useful transmission ratios than motor no.44. 6. CONCLUSIONS The accelerating factor, or continuous duty power rate, it is a parameter characterizing the performance of a motor and it is defined as the ratio between the motor rated torque and the square of its rotor momentum of inertia. The higher is the accelerating factor α, the wider is the range of transmission ratios that can be used for coupling the motor to the load to be moved. The designer who is choosing the motor reducer unit, however, has difficulties in understanding whether the choice done is the best or not, because there are no absolute references on the accelerating factor on which perform the selection. In other words, it is impossible, at now, to evaluate if the chosen motor is the best solution for an application, or if a smaller one with the same accelerating factor, and therefore better for weight and dimensions, is available in commerce. The analysis reveals how motors for automation field (taking into account only synchronous brushless motors) are extremely heterogeneous in terms of performance and highlights the needing to define benchmarks for the accelerating factor to help the designer in selecting the best motor-transmission coupling. 7. REFERENCES [1] Pasch K.A.,Seering W.P. On the drive systems for Highperformance achines, Transactions of ASE Vol.106, 1984, pp.10-108 [] Van de Straete H., Degezelle P., de Shutter., Belmans R., Servo otor Selection Criterion for echatronic Application, IEEE/ASE Transaction on mechatronics Vol.3, 1998, pp.43-50 [3] Van de Straete H., de Shutter., Belmans R., An Efficient Procedure for Checking Performance Limits in Servo Drive Selection and Optimization, IEEE/ASE Transaction on mechatronics Vol.4, 1998, pp.378-386 [4] Cusimano G., A procedure for a suitable selection of laws of motion and electric drive systems under inertial loads, echanism and achine Theory Vol.38, 003, pp.519-533
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