Consider a DC motor that changes the angular position of some load (t) as a function of an applied voltage Va(t). Friction in the motor causes interference with the electromotive force applied to the load through the armature system. The inner workings of this system are well beyond the scope of IBEHS 4A03 and are left to the electrical engineers in the group. For our purposes, the applications of Kirchhoff's and Faraday's Laws result in the following open loop block diagram for this system: V 4170 Field Amate Disturbance Ta(s) Armature Кт T(S) T₁(s) 1 Va(s) R₁ + Las Js + b Kb Back electromotive force Speed w(s) Roto: windings, Brusa St Laertia J Friction-b Ineti Load Angle 6 91 Position 0(s) Stator winding Brush Com Bearings Where Km is a motor constant, Ra is the armature resistance, La is the armature inductance, I is the inertia of the load, b is the friction of the load on the motor, and K is the gain of the interfering back electromotive force caused by friction in the motor. V (s) is the INPUT for this process, and e(s) is the OUTPUT. You do not need to know anything about electrical systems to answer these questions. 1. Using the block diagram as it presented above, show clearly that the open loop transfer 8(s) function G(s) = is described via the following third-order transfer function. You are Va(s) to assume that the disturbance signal Ta(s) = 0. e(s) G(s) = = Km 2. Va(s) s[(Ra + Las)(b + Js) + K₂Km] After substituting typical process parameters representing a disk drive reading system, the above transfer function takes the following form: 5000 e(s) G(S) = Va(s) s(s+20) (s + 1000) Based on your understanding of process dynamics and time constants, argue that this transfer function can be safely approximated as a second-order transfer function: G(S) ≈ G (s) 5 s(s +20) 3. 4. Suppose that this process is placed under closed-loop proportional feedback control with some controller gain K that prescribes the input Va(s) to achieve a desired (s). If we approximate the system dynamics using the expression Ĝ(s) above, draw a block diagram of this process tracking a setpoint R(s) and determine the closed-loop transfer function GCL(S) = 8(s) R(s) Regardless of what you found above, you are to assume that the closed-loop transfer function is: GCL(S) = 8(s) R(s) = 5Kc s² + 20s + 5K Based on this transfer function B(s) R(s) a. Show using any method that the closed-loop response is stable for any positive Kc. b. Determine the value of K for which the closed-loop response will be critically damped. c. Sketch the response (t) responding to a step in R(t) of magnitude 0.5 for some value of K, LESS than you found in part (b).