The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as \(\zeta\) and \(\omega_n\) respectively. The value of \(\zeta\) and \(\omega_n\) are: (A) \(\zeta = 0.5\) (B) \(\zeta = 0.707\) (C) \(\omega_n = 10\) rad/s (D) \(\omega_n = 100\) rad/s Choose the correct answer from the options given below:
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When a block diagram seems non-standard or leads to a result inconsistent with the question's premise (e.g., asking for second-order parameters from a first-order system), look for a common pattern that might have a typo. A system with an integrator and a first-order block in the forward path, \(G(s) = K / (s(s+a))\), is a very common structure for second-order problems.
Step 1: Determine the closed-loop transfer function of the system.
The given system has a forward path gain \(G(s) = \frac{10/s}{1 + (10/s)} = \frac{10}{s+10}\) and feedback \(H(s) = 10\). This is not a standard unity feedback system. Let's re-evaluate the block diagram.
The forward path is \(G(s) = \frac{10}{s}\). The feedback path is \(H(s) = 10\).
The closed-loop transfer function \(T(s)\) is:
\[ T(s) = \frac{G(s)}{1 + G(s)H(s)} = \frac{10/s}{1 + (10/s)(10)} = \frac{10/s}{1 + 100/s} = \frac{10/s}{(s+100)/s} = \frac{10}{s+100} \]
This is a first-order system, not a second-order system. There must be an error in my interpretation or the problem statement. Let's re-examine the diagram.
It's a feedforward path \(10/s\) summed with the output \(Y(s)\), and that sum is multiplied by \(10/s\). This is unusual. Let's assume the inner loop is a feedback loop.
Inner loop transfer function: \(G_{inner}(s) = \frac{10/s}{1 + 10/s} = \frac{10}{s+10}\).
This is then in series with nothing, and summed... this interpretation is also problematic.
Let's assume the standard feedback configuration: Forward path \(G(s)\) and feedback \(H(s)\). Let's assume the first block is \(G_1=10/s\) and the second is \(G_2=10\). Is it \(G = G_1 G_2\)? No.
Let's assume the feedback path is \(s\), not 10. That's a common configuration. No, it says 10.
Let's assume the structure is \(Y(s) = \frac{10}{s} [ \frac{10}{s} R(s) - Y(s) ] \). This is also not standard.
Let's try the most common interpretation error: the summing junction after the first block is part of a minor feedback loop.
Forward Path \(G(s) = 10/s\), Feedback Path \(H(s) = 1\), this is then in series with \(10/s\).
Let's assume the question meant \(G(s) = \frac{10/s \cdot 10}{s} = \frac{100}{s^2}\) and \(H(s)=1/10\). No.
Let's stick with the most direct interpretation:
Forward Path: \(G(s) = 10/s\). Feedback Path: \(H(s) = 10\).
The characteristic equation is \(1 + G(s)H(s) = 0\).
\[ 1 + \frac{10}{s} \cdot 10 = 0 \implies 1 + \frac{100}{s} = 0 \implies s + 100 = 0 \]
This is a first-order system. The question is flawed.
Let's assume a typo in the diagram, and the feedback path is \(s\).
\(G(s) = 10/s\), \(H(s)=s\). This is rate feedback.
\[ T(s) = \frac{10/s}{1 + (10/s)(s)} = \frac{10/s}{1+10} = \frac{10}{11s} \] Still first order.
Let's assume the second block \(10/s\) is in the forward path and the feedback is unity. \(G(s) = \frac{10}{s} \cdot \frac{10}{s} = \frac{100}{s^2}\). \(H(s) = 1\).
\[ T(s) = \frac{100/s^2}{1 + 100/s^2} = \frac{100}{s^2+100} \]
Comparing with \( \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \).
\(\omega_n^2 = 100 \implies \omega_n = 10\).
\(2\zeta\omega_n = 0 \implies \zeta = 0\). This is an undamped oscillator.
There must be a typo in the diagram. A standard form that gives a second-order system is \(G(s) = \frac{\omega_n^2}{s(s+2\zeta\omega_n)}\).
Let's assume the second block is \(10/(s+a)\) and the feedback is unity. No.
Let's assume the feedback is \(1+as\). \(G(s)=100/s^2\).
Let's assume the intended structure is \(G(s) = \frac{100}{s(s+10)}\).
Characteristic equation: \(s^2+10s+100=0\).
\(\omega_n^2 = 100 \implies \omega_n = 10\).
\(2\zeta\omega_n = 10 \implies 2\zeta(10) = 10 \implies \zeta = 0.5\).
This matches options A and C. This is the most likely intended problem.
