Task 1. In steady state, di dt = 0, ω˙ = 0, show that the electrical power input that goes into torque production is Kbωi, and that the mechanical power output is Ktiω. Thereby, show that Kb = Kt . [3 marks] A. Determining motor parameters The motor parameters are determined through a dynamometer test [Nise, 2011], which is based on the observation of the relationship between the motor's torque and angular velocity. This relationship is revealed by substituting Eq. (3) into Eq. (2), giving us T = − KbKt R ω + Kt R v, (8) With v constant, the plot of Eq. (8) is called the torque-speed curve. If we move ω to the left-hand side, and T to the right-hand side of Eq. (8), then we get a speed-torque curve. For sample torque-speed curves, refer to the datasheet in Appendix A. Consider these two operating points: • Operating point where ω = 0: This is the "stall" operating point, where the torque-speed curve intersects the torque axis. This occurs when the rated voltage is applied and the rotor is stalled. To reach this stalling point, it is customary to couple an adjustable torque load such as a small particle brake to the motor shaft, and increase the torque load to the point where stall occurs [Micro Motion Solutions, 2014]. At stall, the torque from the brake is measured as the stall torque. From Eq. (6), we have Ktistall = Tfric + Tstall, (9) where Tfric is the friction torque for overcoming friction, and Tstall is the stall torque for matching the load. • Operating point where T = 0: This is the "no load" operating point, where the torque-speed curve intersects the speed axis. This occurs when no load is connected to the motor. In practice, the "no load" operating point does not correspond to T = 0, because the no-load torque needs to overcome friction, and is proportional to the nonzero no-load current, i.e., Tnoload = Tfric = Ktinoload. (10) Substituting Eq. (10) into Eq. (9), we get Kt = Tstall istall − inoload . (11) At no load, the motor spins at the no-load speed, ωnoload, thus v = Kbωnoload + Rinoload, (12) which we can use to calculate ωnoload from R, or R from ωnoload. Some datasheets provide the speed constant and/or the motor constant. The speed constant is the inverse of the back emf constant, whereas the motor constant, Km def = Kt/ √ R, measures the stall torque per Watt input. The motor constant can be understood as a measure of the motor's efficiency. Task 2. Referring to the datasheet in Appendix A, fill in Table I for the motor Maxon A-max 32 order number 236653. Hints: (i) pay attention to the required units; (ii) the last item in the table requires a simple calculation. 4 TABLE I: Motor parameters. Parameter Maxon A-max 32 order number 236653 Nominal voltage, v (V) Torque constant, Kt (N·m/A) Armature resistance, R (Ω) Armature inductance, L (H) Moment of inertia, J (kg·m 2 ) No-load speed, ωnoload (rad/s) No-load current, inoload (A) Max. speed, ωmax (rad/s) Max. continuous current, imax (A) Max. continuous torque, T max (N·m) Coefficient of viscous friction, D (N·m·s/rad) [6 marks] Task 3. Based on the information in Table I, verify that • the terminal voltage calculated using Kt , ωnoload, R, inoload in Table I and Eq. (12) is close to the nominal voltage in Table I; • the speed-torque gradient in rpm/mN·m calculated from R and Kt in Table I is close to the sleep-torque gradient on the datasheet; • the speed n the datasheet; • the speed constant in rpm/V calculated from ωnoload, R, inoload in Table I is close to the speed constant on the datasheet.