FE Electrical and Computer Linear Systems Practice Problems

The current FE Electrical and Computer specification assigns 5-8 of the 110 exam questions to Linear Systems. Form A contains 6 original problems in this area. The sample below is published in full, with the same worked-solution format used throughout the book.

A unity-feedback position servo has closed-loop transfer function $G (s) = \frac{400}{s ^{2} + 12 s + 400}$ , with $s$ in rad/s. For a step command, the percent overshoot of the output is most nearly:

(A)9.5%
(B)13.9%
(C)37.2%
(D)95.4%

Show worked solution

Answer: (C)

A damping ratio of 0.3 governs the peak entirely, giving roughly 37 percent overshoot regardless of the 20 rad/s natural frequency.

Matching the denominator against the standard second-order form fixes both parameters at once.

s^{2} + 2 ζ ω_{n} s + ω_{n}^{2} = s^{2} + 12 s + 400

ω_{n} = 20 rad/s, ζ = \frac{12}{2 ( 20 )} = 0.3

Overshoot of an underdamped step response is set by the damping ratio alone, since $ω_{n}$ only rescales time.

M_{p} = 100 e^{- π ζ / 1 - ζ^{2}}

With $ζ = 0.3$ the exponent falls just below unity, leaving a peak more than one-third above the final value.

M_{p} = 100 e^{- 0.9425/0.9539} = 100 (0.3723) = 37.2%

FE Reference Handbook — Control Systems: Second-Order Control-System Models, Percent Overshoot

Why the other choices appear

(A)Setting $ζ ω_{n} = 12$ instead of $2 ζ ω_{n} = 12$ gives $ζ = 0.6$ and $100 e^{- 1.885/0.8} = 9.5%$ .
(B)Using the successive-peak exponent $2 π ζ / 1 - ζ^{2}$ squares the correct factor: $(0.3723)^{2} = 13.9%$ .
(D)Taking $ω_{n} = 400$ rad/s directly from the constant term gives $ζ = 12/800 = 0.015$ and $95.4%$ .