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Problem 5.29PP
An elementary magnetic suspension scheme is depicted in Fig. For smail motions near the 
reference position, the voltage e on the photo detector is related to the ball displacement x (in 
meters) by e = 10Ox. The upward force (in newtons) on the ball caused by the current / (in 
amperes) may be approximated by f= 0.5/+ 20x. The mass of the ball is 20 g and the 
gravitational force is 9.8 N/kg. The power amplifier is a voltage-to-current device with an output 
(in amperes) of i = u + VO.
(a) Write the equations of motion for this set up.
(b) Give the value of the bias VO that results in the ball being in equilibrium at x = 0.
(c) What is the transfer function from u to e?
(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.
(e) Assume that a lead compensation is available in the form values
of K. z, and p that yield improved performance over the one proposed in part (d).
Figure Elementary magnetic suspension
Step-by-step solution
step 1 of 5
=0.5i+20rH3flg
Step 2 of 5
b. For Equilibriiun at X = 0, 0.5Vg^ng 
V„=2mg=0.392V
Step 3 of 5
c. sm X (s)=0.5V (s)+20X (s) 
0.5V (s)=X (s)[0.02s’ -20]
E ( s ) _ r
V (s)
50
0.02s^-20
Step 4 of 5
K50 2500K
0.02s"-20 s"-1000
Step 5 of 5
D (s ) = ^ f c ! ^ l let 1=31.6 
 ̂ ̂ s4p
p=50
for ^ . 7 ( H . K=0.665
D(s)=0.665^ ^ ^ '^ ^
'■ ' (s+50)