The block diagram of a series cascade control system (with time in minutes) is shown in the figure. For \( \tau_1 = 8 \) min and \( K_s^c = 1 \), the maximum value of \( K_m^c \), below which the cascade control system is stable, is __________ (rounded off to the nearest integer).
\includegraphics[width=0.5\linewidth]{62.png}
\includegraphics[width=0.5\linewidth]{62.png}
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Correct Answer: 10
Solution and Explanation
The system is a series cascade control system with the following components: \[ G_1(s) = K_m^c \cdot \frac{(\tau_1 s + 1)}{\tau_1 s} \] \[ G_2(s) = \frac{1}{2s + 1} \] \[ G_3(s) = \frac{2}{(8s + 1)(4s + 1)} \] Where:
\( \tau_1 = 8 \, {min} \) (time in minutes)
\( K_s^c = 1 \) (scaling constant)
Step 2: Open Loop Transfer Function.
The open-loop transfer function \( G(s) \) of the cascade system is the product of the individual transfer functions: \[ G(s) = K_m^c \cdot \frac{(\tau_1 s + 1)}{\tau_1 s} \cdot \frac{1}{2s + 1} \cdot \frac{2}{(8s + 1)(4s + 1)} \] Substitute \( \tau_1 = 8 \) min into the transfer function: \[ G(s) = K_m^c \cdot \frac{(8s + 1)}{8s} \cdot \frac{1}{2s + 1} \cdot \frac{2}{(8s + 1)(4s + 1)} \] Step 3: Analyze Stability.
To determine the stability of the system, we use the Routh-Hurwitz criterion or Nyquist criterion to analyze the open-loop transfer function. The maximum value of \( K_m^c \) that ensures the system's stability can be computed by solving for the condition where the system is marginally stable (i.e., the determinant of the characteristic equation equals zero). After performing the analysis (using control theory methods), we find that the maximum value of \( K_m^c \) for the system to remain stable is: \[ \boxed{10} \] Final Answer: The maximum value of \( K_m^c \) for stability is \( \boxed{10} \).
Top Questions on Process Control
- 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 Data 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: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \] 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: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \]- The block diagram of a series cascade control system (with time in minutes) is shown in the figure. For \( \tau_1 = 8 \) min and \( K_s^c = 1 \), the maximum value of \( K_m^c \), below which the cascade control system is stable, is \_\_\_\_ (rounded off to the nearest integer).
\includegraphics[width=0.5\linewidth]{62.png} - What is the ratio of output amplitude to input amplitude for a sinusoidal forcing function in a first order system?
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