Midterm
This Midterm covers Weeks 1-4. Show all required calculations, MATLAB code and MATLAB plots for full credit.
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- Determine the partial fraction expansion for V(s) and compute the inverse Laplace transform. The transfer function V(s) is given by:
V(s) = 400 / (s2 + 8s +400)
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- A second-order system is Y(s)/R(s) = T(s) = (10/z)(s + z) / ((s + 1)(s + 8))
Consider the case where 1 < z < 8. Obtain the partial fraction expansion, and plot y(t) for a step input r(t) for z = 2, 4, and 6.
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- Determine whether the systems with the following characteristic equations are stable or unstable:
- s3 + 4s2 + 6s + 100 = 0
- s4 + 6s3 + 10s2 + 17s + 6 = 0
- s2 + 6s + 3 = 0
- A single-loop negative feedback system has the loop transfer equation:
- Determine whether the systems with the following characteristic equations are stable or unstable:
L(s) = Gc(s)G(s) = K(s + 2)2 / (s(s2 + 1)(s + 8))
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- Sketch the root locus for 0 ≤ K ≤ infinity to indicate the significant features of the locus.
- Determine the range of the gain K for which the system is stable.
- For what value of K in the range of K ≥ 0 do purely imaginary roots exist? What are the values of these roots?
- Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain (K ≥ 50)?