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2. Question 2 [25 marks] Consider a DC motor with armature control configuration described by the standard electro- mechanical equations of the DC motor with parameters: Ra = 822 (resistance), La = 0.86.10-3H (inductance), k, = 0.0202 NmA-? (torque constant), km = 0.0202 Vrad-ts (emf constant). Assume Im= 2.5. 10-4kg mạ for the inertia of the load. Note that km = km. Friction terms should be ignored. (a) Starting from the standard electro-mechanical equations, obtain the continuous-time transfer function from va (armature voltage) to (angular velocity). Express it in monic form (i.e. with the coefficient of the higher order term at the denominator equal to 1). [6 marks] (b) Knowing that the maximum voltage that can be applied to the motor is Vmax = 15 V, obtain the maximum load torque which can be sustained by the motor (stalling torque). [5 marks] (c) Under the approximation that the armature circuit inductance is null, obtain a simpli- fied version of the continuous-time transfer function from va to Omn (4 marks) (d) Show that the transfer function obtained in (a) and transfer function obtained in (c) have the same static gain [4 marks] (e) On the basis of the transfer function obtained in (c), select a suitable sampling time To such that the discrete-time transfer function, which describes the operation of the DC motor preceded by a Zero Order Hold, has a pole in 0.9. [6 marks]