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For a type-2 control system, evaluate the magnitude and phase of the factor (jω)^2 appearing in the denominator as ω → 0.

Difficulty: Medium

Correct Answer: infinity and - 180°

Explanation:


Introduction / Context:
System type is defined by the number of pure integrators 1/s in the open-loop transfer function. A type-2 system has two integrators overall, leading to distinct low-frequency behavior in magnitude and phase of the closed-loop frequency response.


Given Data / Assumptions:

  • Type-2 implies a denominator factor proportional to (jω)^2 in the frequency response.
  • We consider the effective contribution of 1/(jω)^2 to the overall magnitude and phase as ω → 0.


Concept / Approach:
At low frequency, |jω| → 0 and thus |(jω)^2| → 0. However, because this term is in the denominator, the net magnitude contribution of 1/(jω)^2 tends to infinity and its phase is −2 × 90° = −180°. This reflects the −40 dB/dec slope and −180° phase contribution at low frequencies for two ideal integrators.


Step-by-Step Solution:

Start from (jω) = ω∠+90°; square → ω^2∠+180°.In the denominator, the factor acts as 1 / (jω)^2 → magnitude ∞ as ω → 0.Phase contribution of reciprocal is −180°.


Verification / Alternative check:

Bode low-frequency asymptotes for a type-2 loop show −40 dB/dec and −180° phase shift.


Why Other Options Are Wrong:

Options with 0 magnitude ignore the reciprocal role in the denominator.+180° is incorrect for the reciprocal; the sign flips.


Common Pitfalls:

Treating (jω)^2 itself rather than its effect through 1/(jω)^2 on the response.


Final Answer:

infinity and - 180°
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