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}

• A. \( \frac{1}{s+1} \)
• B. \( \frac{2}{s+1} \)
• C. \( \frac{1}{s+20} \)
• D. \( \frac{2}{s+20} \)

Hint:

For first-order approximations, neglect insignificant poles while maintaining the DC gain.

Answer 1

The Correct Option is B

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).