Question

In the open-loop process shown in the figure, the input \(U(s)\), the transfer function \(G_p(s)\) and the output \(Y(s)\) are given in the Laplace domain in terms of the Laplace variable \(s\). For this process, which of the following is true?

(where \(M, \tau_p, K_P\), are the magnitude of the input, the characteristic time and the gain for the process, respectively)

• A. \(y(t) = K_p\left(1 - e^{-t/\tau_p}\right)\)
• B. \(y(t) = MK_p\left(1 - e^{-t/\tau_p}\right)\)
• C. \(y(t) = K_p\left(M - e^{-t/\tau_p}\right)\)
• D. \(y(t) = K_p\left(1 - Me^{-t/\tau_p}\right)\)

Hint:

Use \(Y(s)=G_p(s)U(s)\) and take the inverse Laplace transform of a first order lag driven by a step of size \(M\).

Answer 1

The Correct Option is B

Step 1: Write the transfer function relation.
The block diagram gives \(Y(s) = G_p(s)\,U(s)\), where \(U(s) = \dfrac{M}{s}\) is the Laplace transform of a step input of size \(M\), and \(G_p(s) = \dfrac{K_p}{\tau_p s + 1}\) is a first order process.

Step 2: Substitute and set up partial fractions.
\[ Y(s) = \frac{K_p}{\tau_p s + 1} \cdot \frac{M}{s} = \frac{K_p M}{s(\tau_p s + 1)} \]
Split this into partial fractions in \(s\):
\[ \frac{K_p M}{s(\tau_p s+1)} = \frac{A}{s} + \frac{B}{\tau_p s + 1} \]
Multiplying both sides by \(s(\tau_p s+1)\) and matching coefficients: setting \(s=0\) gives \(A = K_p M\); setting \(\tau_p s+1=0\), i.e. \(s=-1/\tau_p\), gives \(B = -K_p M \tau_p\).

Step 3: Take the inverse Laplace transform.
\[ Y(s) = \frac{K_p M}{s} - \frac{K_p M \tau_p}{\tau_p s + 1} = \frac{K_p M}{s} - \frac{K_p M}{s + 1/\tau_p} \]
Using \(\mathcal{L}^{-1}\left[\frac{1}{s}\right] = 1\) and \(\mathcal{L}^{-1}\left[\frac{1}{s+a}\right] = e^{-at}\):
\[ y(t) = K_p M - K_p M e^{-t/\tau_p} = K_p M \left(1 - e^{-t/\tau_p}\right) \]

Step 4: Check why the other options fail.
Option (A) drops the input magnitude \(M\) entirely, so it only holds for a unit step, not a general step of size \(M\).
Option (C) wrongly places \(M\) inside the bracket subtracting the exponential term, which does not come from the correct partial fraction expansion above.
Option (D) multiplies only the exponential term by \(M\) instead of multiplying the whole bracket, which again does not follow from the algebra.

Final Answer:
The response of a first order process to a step of magnitude \(M\) is \(y(t) = K_p M (1 - e^{-t/\tau_p})\), which is option (B).
\[ \boxed{y(t) = K_p M \left(1 - e^{-t/\tau_p}\right)} \]