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EE 475 Fall 2011 HW #3 Due: 9/14/2011 1. In Lecture 7 on slides 3 and 4, We assumed that Kt*R2/R1=1, R4=R2, and R3=R1. With these assumptions, verify that the op amp circuit and the v1 to v2 block on slide 3 reduce to the summing junction and the gain block on slide 4. Then find the input output transfer function for the system on slide 4. 2. Modify the op amp circuit on slide 3, which is a proportional controller, by inserting a capacitor in series with R2. Show that the resulting op amp circuit is a PI (proportional plus integral) controller. Show that there is an appropriate capacitor voltage to provide any desired biasing voltage v10. Find the right capacitor voltage corresponding to v10=1.5V. 3. With the modified op amp circuit as in the previous problem and the biasing voltage for v1 as a symbol v10, replace the v1 to v2 block by a gain. (The derivative evaluated at v1 = v10 symbolically.) Find the transfer function from vin to . 4. Now introduce another change. Add an integrator (1/s) block after so that the final output is . The feedback block is now connected from instead of Replace Kt by Kp. Use the PI op am circuit from 3, find the transfer function from vin to . 5. Slides 5 to 8 provided the detailed derivation of a state space model of an armature controlled DC motor. Now suppose the motor is a field controlled motor. Re-do the derivation to get a state space model for the field controlled DC motor. 6. On page 72, take M=10, m=1, l=1, g=10 in eq(3-22) and (3-23). Enter the state space model into Matlab. Obtain the transfer function from u to y1 and y2. Find the poles of the system. 7. Consider the transfer function given in eq(3-7). Expand the denominator into a polynomial of s in descending order. Give a state space model for this transfer function. Use the simplest form as given on slide 20. 8. Re-do prob 1 part b and part c from HW1. 9. Do B-2-12 10. Do B-2-13. Linearize at the center: x0=3, y0=11. What is z0?