Question:
Consider the cascaded system as shown in the figure. Neglecting the faster component of the transient response, which one of the following options is a first-order pole-only approximation such that the steady-state values of the unit step response of the original and the approximated systems are the same?
\includegraphics[width=0.5\linewidth]{22.2.png}
Consider the cascaded system as shown in the figure. Neglecting the faster component of the transient response, which one of the following options is a first-order pole-only approximation such that the steady-state values of the unit step response of the original and the approximated systems are the same?
\includegraphics[width=0.5\linewidth]{22.2.png}
\includegraphics[width=0.5\linewidth]{22.2.png}
Show Hint
For first-order approximations, neglect insignificant poles while maintaining the DC gain.
Updated On: Jan 23, 2025
- \( \frac{1}{s+1} \)
- \( \frac{2}{s+1} \)
- \( \frac{1}{s+20} \)
- \( \frac{2}{s+20} \)
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The Correct Option is B
Solution and Explanation
Step 1: Finding the original transfer function.
The given cascaded system has:
\[
T(s) = \frac{(s+40)}{(s+1)(s+20)}.
\]
Step 2: DC gain before approximation.
For DC gain (\( s = 0 \)):
\[
T(s) \big|_{s=0} = \frac{40}{1 \times 20} = 2.
\]
Step 3: First-order approximation.
Approximating \( s+20 \) as the dominant pole:
\[
T(s) \approx \frac{2}{s+1}.
\]
Hence, the correct option is (B).
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