BEE Signals and Control Systems 5 — Questions and Answers
Question 1: The transfer function of an ideal differentiator in the Laplace domain is:
- 1/s
- s (Correct answer)
- s²
- 1/s²
Correct answer: s
Differentiation in the time domain corresponds to multiplication by s in the Laplace domain, so the transfer function of a differentiator is H(s) = s.
Question 2: A minimum-phase system is one in which:
- All poles are in the left-half s-plane
- All zeros are in the left-half s-plane
- Both poles and zeros are in the left-half s-plane (Correct answer)
- The system has no zeros
Correct answer: Both poles and zeros are in the left-half s-plane
A minimum-phase system has all its poles AND zeros in the open left-half s-plane, ensuring the system and its inverse are both stable and causal.
Question 3: For a discrete-time LTI system to be BIBO stable, the impulse response h[n] must satisfy:
- h[n] → 0 as n → ∞
- Σ|h[n]| < ∞ (absolutely summable) (Correct answer)
- Σ|h[n]|² < ∞ (square summable)
- h[n] = 0 for all n < 0
Correct answer: Σ|h[n]| < ∞ (absolutely summable)
BIBO stability requires that the impulse response be absolutely summable: Σ|h[n]| < ∞, which guarantees bounded output for any bounded input.
Question 4: The phase margin of a control system is measured at:
- The phase crossover frequency
- The gain crossover frequency (Correct answer)
- The natural frequency
- The half-power frequency
Correct answer: The gain crossover frequency
Phase margin is measured at the gain crossover frequency (where |G(jω)H(jω)| = 1, or 0 dB) and equals 180° plus the actual phase angle at that frequency.
Question 5: Which of the following correctly represents the relationship between the DTFT X(e^jω) and the z-transform X(z)?
- X(e^jω) = X(z)|_{z=jω}
- X(e^jω) = X(z)|_{z=e^jω} (Correct answer)
- X(e^jω) = X(z)|_{z=1/ω}
- X(e^jω) = X(z)|_{z=σ+jω}
Correct answer: X(e^jω) = X(z)|_{z=e^jω}
The DTFT is the z-transform evaluated on the unit circle |z| = 1, i.e., X(e^jω) = X(z)|_{z=e^jω}.
Question 6: In Mason's Gain Formula, a 'non-touching loop' pair refers to:
- Two loops that share at least one node
- Two loops that share no common nodes (Correct answer)
- Two loops with opposite loop gains
- A forward path and a feedback loop
Correct answer: Two loops that share no common nodes
Non-touching loops in Mason's formula are loop pairs that share no nodes or branches, and their product appears in higher-order cofactor terms of the formula.
Question 7: If the open-loop gain of a feedback amplifier is 1000 and the feedback fraction is 0.099, the closed-loop gain is approximately:
- 10 (Correct answer)
- 100
- 1000
- 9.9
Correct answer: 10
Closed-loop gain = A/(1 + Aβ) = 1000/(1 + 1000×0.099) = 1000/100 = 10.
BEE Signals and Control Systems 5
The transfer function of an ideal differentiator in the Laplace domain is: