A process is simulated by the second-order passive circuit, shown in Figure P5.52, where the feedback amplifier, controller, and final control element are represented by op-amp circuits.

a. Denoting the input and output as R(s) = V_{i} (s) and C(s) = V_{o}(s), with R(s) - C(s) = E(s), and noting that the feedback amplifier has a unity gain, draw a block diagram for this feedback control system, where G_{C}(s), G_{F}(s), and G_{P}(s) are the transfer functions of the controller, final control element, and the process, respectively.

b. Find the value of R_{P} that makes the circuit representing the process critically damped.

c. Noting that the proportional controller is simply an amplifier, G_{C}(s) = K_{P}, find the value of its gain K_{P} that results in dominant closed-loop poles with a damping ratio, ζ = 0:5, and a settling time, T_{s} = 4 ms. Verify that the other pole is non dominant. What would be the appropriate value of the controller potentiometer, RF, given that its tolerance is ±10%?