Question

Just for d and e

1. At the BRIC, the school of ECS has a laser etching machine for manufacuring printed circuit boards (PCBs). To see one in action, jump to the middle of this video LPKF - ProtoLaser S Supposed someone tells us that the 1-dimensional position (ie. ei- ther horizontal or vertical) of the laser beam p(t) is described by 4(s+40) R(s) s2 26s160 P(s)= where r(t) is the desired position and position is measured in mi crometers (um (a) [5] Draw the pole/zero diagram for this system (b) 5] Let r(t) be a unit step input (i.e. a single, constant location). Find the response p(t) (c) [5 What is the final value of p(t)? (d) 5] Use MATLAB to plot the solution p(t) for 0 t < 2 sec ес. (e) [5] Interpret this response in terms of the physical system. What is happening? Comment on the appropriateness of this response (i.e. Is this a good response for this system? Does it match what you see in the video?)

Answer 1

Given p(s) = 4 (s + 40)/ s^2 + 26s + 160 R(s) implies transfer function = p(s)/ R(s) = 4 (s + 40)/ s^2 + 26s + 160 find poles by equating Denominator to zero s^2 + 26s + 160 = 0 implies s^2 + 16s + 120s + 160 = 0 implies s^2 + 16s + 10s + 160 = 0 implies s (s + 16) + 10 (s + 16) = 0 s = -10,-16 are poles find zeros by equating numerator to zero s + 40 = 0 zero s = -40 pole zero plot: r(t) = unit step input = u(t) R(s) = 1/s p(s) = 4 (s + 40)/ s (s + 10)(s + 16) Apply partial fractions 4s + 160/ s (s + 10) (s + 16) = A/ s + B/ s + 10 + c/ s + 16 implies 4s + 160/ s (s + 10) (s + 16)