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Control systems: board exam questions with solutions

On the ECE exam under Electronics Engineering and the REE exam under Electrical Engineering Professional Subjects. Try each question before you open its solution, then practise with fresh numbers below: 1 computed question type on this topic.

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Questions

1. For a negative-feedback system with forward path G(s) and feedback path H(s), what is the closed-loop transfer function?

  1. G/(1 − GH)
  2. GH/(1 + GH)
  3. 1/(1 + GH)
  4. G/(1 + GH)
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Answer: D. G/(1 + GH)

C/R = G/(1 + GH) for negative feedback.

2. A unity-feedback system has G(s) = 10/(s + 2). What is the steady-state error for a unit step input?

  1. 0
  2. 0.200
  3. 0.167
  4. 0.500
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Answer: C. 0.167

Kp = lim G(s) as s→0 = 10/2 = 5; ess = 1/(1 + Kp) = 1/6 = 0.167.

3. A second-order system has the characteristic equation s² + 4s + 25 = 0. What is its damping ratio?

  1. 0.8
  2. 0.4
  3. 0.16
  4. 2.0
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Answer: B. 0.4

Compare with s² + 2ζωn s + ωn²: ωn = √25 = 5 rad/s, 2ζωn = 4 → ζ = 0.4.

4. A unity-feedback system has closed-loop transfer function 36/(s² + 6s + 36). What is the percent overshoot of its step response?

  1. 16.3%
  2. 44.4%
  3. 50.0%
  4. 20.8%
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Answer: A. 16.3%

ωn = 6, ζ = 6/(2 × 6) = 0.5. %OS = 100·e^(−ζπ/√(1 − ζ²)) = 100·e^(−0.5π/0.866) = 16.3%.

5. For what range of K is the characteristic equation s³ + 2s² + Ks + 8 = 0 stable?

  1. K > 8
  2. K > 2
  3. 0 < K < 4
  4. K > 4
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Answer: D. K > 4

Routh array: s³: 1, K; s²: 2, 8; s¹: (2K − 8)/2; s⁰: 8. No sign change requires 2K − 8 > 0, so K > 4.

6. Where must all closed-loop poles of a continuous-time LTI system lie for it to be stable?

  1. In the right half of the s-plane
  2. On the imaginary axis
  3. In the left half of the s-plane
  4. Outside the unit circle
Show the answer and solution

Answer: C. In the left half of the s-plane

Poles with negative real parts produce decaying exponentials; any pole in the right half-plane gives a growing response.

7. Using the 2% criterion ts ≈ 4/(ζωn), what is the settling time of a system with characteristic equation s² + 4s + 25 = 0?

  1. 0.8 s
  2. 2.0 s
  3. 4.0 s
  4. 1.0 s
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Answer: B. 2.0 s

ζωn = 4/2 = 2, so ts ≈ 4/2 = 2.0 s.

8. At the phase-crossover frequency (where ∠GH = −180°), the open-loop magnitude |GH| is 0.25. What is the gain margin?

  1. 12.04 dB
  2. 6.02 dB
  3. −12.04 dB
  4. 4.00 dB
Show the answer and solution

Answer: A. 12.04 dB

GM = 20 log(1/|GH|) = 20 log 4 = 12.04 dB.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. A unity-feedback system has open-loop G(s) = 18/(s + 1). What is the steady-state error to a unit step?

  1. 5.26%
  2. 5.56%
  3. 94.7%
  4. 0%
Show the answer and solution

Answer: A. 5.26%

Type 0 system, unit step: ess = 1/(1 + Kp) = 1/(1 + 18) = 5.26%.

Other topics in the same subjects

DC circuits · AC circuits · Transients (RC, RL, RLC) · Semiconductors and diodes · BJT and FET circuits · Operational amplifiers · Oscillators and feedback · Power electronics · Digital electronics · Microprocessors and microcontrollers · Instrumentation and measurement · Three-phase systems · Transformers · DC machines · Induction motors · Synchronous machines · Power systems and transmission · Protection: fuses, breakers, relays · Illumination · Power plant engineering · PEC: branch circuits, ampacity and protection

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