A system has a transfer function
\[
G(s) = \frac{3e^{-4s}}{12s^2 + 1}
\]
When a step change of magnitude M is given to the system input, the final value of the system output is measured to be 120. The value of M is \(\underline{\hspace{1cm}}\) .
\[ G(s) = \frac{3e^{-4s}}{12s^2 + 1} \] When a step change of magnitude M is given to the system input, the final value of the system output is measured to be 120. The value of M is \(\underline{\hspace{1cm}}\) .
Show Hint
Correct Answer: 40
Solution and Explanation
\[ y(t) \to \lim_{s \to 0} s G(s) \cdot M \]
Substitute the given transfer function:
\[ \lim_{s \to 0} s \times \frac{3e^{-4s}}{12s^2 + 1} = \frac{3}{12} = 0.25 \]
Thus, the final output is:
\[ 0.25M = 120 $\Rightarrow$ M = 480 \]
Thus, the value of \( M \) is:
\[ \boxed{102.4} \]
Top Questions on Transfer functions and dynamic responses of various systems
- For the block diagram shown in the figure, the correct expression for the transfer function \( G_d = \frac{y_1(s)}{d(s)} \) is:
- GATE CH - 2024
- Instrumentation and Process Control
- Transfer functions and dynamic responses of various systems
- A cascade control strategy is shown in the figure. The transfer function between the output \((y)\) and the secondary disturbance \((d_2)\) is defined as \(G_{d2}(s)=\dfrac{y(s)}{d_2(s)}\). Which one of the following is the CORRECT expression for \(G_{d2}(s)\)?
- GATE CH - 2023
- Instrumentation and Process Control
- Transfer functions and dynamic responses of various systems
- A measuring instrument has transfer function \[ G_m(s)=\frac{1.05}{2s+1}\,e^{-s}. \] At \(t=0\), a step of \(+1\) unit is applied. Find the time taken for the output to increase by \(1\) unit.
- GATE CH - 2023
- Instrumentation and Process Control
- Transfer functions and dynamic responses of various systems
- Level (\(h\)) in a steam boiler is controlled by manipulating the flow rate (\(F\)) of the make-up water using a proportional (P) controller. The transfer function is \[ \frac{h(s)}{F(s)} = \frac{0.25(1-s)}{s(2s+1)}. \] The measurement and valve transfer functions are both 1. A process engineer wants the closed-loop response to give decaying oscillations under servo mode. Which one of the following is the CORRECT value of the controller gain?
- GATE CH - 2023
- Instrumentation and Process Control
- Transfer functions and dynamic responses of various systems
- A liquid surge tank has \(F_{in}\) and \(F_{out}\) as the inlet and outlet flow rates respectively, as shown in the figure. \(F_{out}\) is proportional to the square root of the liquid level \(h\). The cross-sectional area of the tank is \(20~\text{cm}^2\). Density of the liquid is constant everywhere in the system. At steady state, \(F_{in}=F_{out}=10~\text{cm}^3\text{s}^{-1}\) and \(h=16~\text{cm}\). The variation of \(h\) with \(F_{in}\) is approximated as a first-order transfer function. Which one of the following is the CORRECT value of the time constant (in seconds) of this system?
- GATE CH - 2023
- Instrumentation and Process Control
- Transfer functions and dynamic responses of various systems
Questions Asked in GATE CH exam
Consider a process with transfer function: \[ G_p = \frac{2e^{-s}}{(5s + 1)^2} \] A first-order plus dead time (FOPDT) model is to be fitted to the unit step process reaction curve (PRC) by applying the maximum slope method. Let \( \tau_m \) and \( \theta_m \) denote the time constant and dead time, respectively, of the fitted FOPDT model. The value of \( \frac{\tau_m}{\theta_m} \) is __________ (rounded off to 2 decimal places).
Given: For \( G = \frac{1}{(\tau s + 1)^2} \), the unit step output response is: \[ y(t) = 1 - \left(1 + \frac{t}{\tau}\right)e^{-t/\tau} \] The first and second derivatives of \( y(t) \) are: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \]- GATE CH - 2025
- Process Control
Choose the transfer function that best fits the output response to a unit step input change shown in the figure:
- GATE CH - 2025
- Control Systems
- Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?
An electrical wire of 2 mm diameter and 5 m length is insulated with a plastic layer of thickness 2 mm and thermal conductivity \( k = 0.1 \) W/(m·K). It is exposed to ambient air at 30°C. For a current of 5 A, the potential drop across the wire is 2 V. The air-side heat transfer coefficient is 20 W/(m²·K). Neglecting the thermal resistance of the wire, the steady-state temperature at the wire-insulation interface __________°C (rounded off to 1 decimal place).
GIVEN:
Kinematic viscosity: \( \nu = 1.0 \times 10^{-6} \, {m}^2/{s} \)
Prandtl number: \( {Pr} = 7.01 \)
Velocity boundary layer thickness: \[ \delta_H = \frac{4.91 x}{\sqrt{x \nu}} \]- GATE CH - 2025
- Fluid Mechanics
The first-order irreversible liquid phase reaction \(A \to B\) occurs inside a constant volume \(V\) isothermal CSTR with the initial steady-state conditions shown in the figure. The gain, in kmol/m³·h, of the transfer function relating the reactor effluent \(A\) concentration \(c_A\) to the inlet flow rate \(F\) is:
- GATE CH - 2025
- Fluid Mechanics