An elementary magnetic suspension scheme is depicted in Fig. 5.61. For small motions

Chapter 5, Problem 5.29

(choose chapter or problem)

An elementary magnetic suspension scheme is depicted in Fig. 5.61. For small motions near the reference position, the voltage e on the photo detector is related to the ball displacement x (in meters) by e = 100x. The upward force (in newtons) on the ball caused by the current i (in amperes) may be approximated by f = 0.5i + 20x. The mass of the ball is 20 g and the gravitational force is 9.8 N/kg. The power amplifier is a voltage-tocurrent device with an output (in amperes) of i = u + V0 . (a) Write the equations of motion for this set-up. (b) Give the value of the bias V0 that results in the ball being in equilibrium at x = 0. (c) What is the transfer function from u to e? (d) Suppose that the control input u is given by u = Ke. Sketch the root locus of the closed-loop system as a function of K. Figure 5.62 Block diagram for rocket-positioning control system (e) Assume that a lead compensation is available in the form . Give values of K, Z, and p that yield improved performance over the one proposed in part (d).