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GATE 2016 EE – Question 40

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Multiple choice

Consider the following asymptotic Bode magnitude plot ($\omega$ is in rad/s).

[Figure: A Bode magnitude plot. The slope is +20 dB/dec, crossing 0 dB at $\omega = 0.5$, then the plot is flat at 12 dB, and finally the slope is -40 dB/dec, crossing 0 dB at $\omega = 8$.]

Which one of the following transfer functions is best represented by the above Bode magnitude plot?

Diagram for GATE 2016 EE question 40
  1. $\frac{2s}{(1 + 0.5s)(1 + 0.25s)^2}$
  2. $\frac{4(1 + 0.5s)}{s(1 + 0.25s)}$
  3. $\frac{2s}{(1 + 2s)(1 + 4s)}$
  4. $\frac{4s}{(1 + 2s)(1 + 4s)^2}$

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Show answer and explanation

Correct answer: (A) $\frac{2s}{(1 + 0.5s)(1 + 0.25s)^2}$

Explanation

Option A has a zero at the origin, so it rises at +20 dB/dec and equals 1 (0 dB) at $\omega = 0.5$ because $2 \times 0.5 = 1$. The pole at $\omega = 2$ makes the plot flat at $2 \times 2 = 4$, which is 12 dB. The double pole at $\omega = 4$ then gives -40 dB/dec, which drops 12 dB over the factor 2 between 4 and 8, reaching 0 dB at $\omega = 8$. All three features match the plot.