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 initial interpretation of the system, with forward path gain \(G(s) = \frac{10/s}{1 + (10/s)} = \frac{10}{s+10}\) and feedback \(H(s) = 10\), doesn't represent a standard unity feedback system. The block diagram is re-evaluated.The forward path is redefined as \(G(s) = \frac{10}{s}\), and the feedback path as \(H(s) = 10\).The resulting 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 indicates a first-order system, which contradicts the expectation of a second-order system. The initial interpretation of the diagram may be incorrect. The diagram is re-examined.The system features a feedforward path \(10/s\) summed with the output \(Y(s)\), and this sum is then multiplied by \(10/s\), a non-standard configuration. The possibility of an inner feedback loop is considered.The inner loop transfer function is calculated as: \(G_{inner}(s) = \frac{10/s}{1 + 10/s} = \frac{10}{s+10}\).However, this interpretation remains problematic.Assuming a standard feedback configuration with forward path \(G(s)\) and feedback \(H(s)\), let's define the first block as \(G_1=10/s\) and the second as \(G_2=10\). It's questioned whether \(G = G_1 G_2\), but this is deemed incorrect.The possibility of the feedback path being \(s\) instead of 10 is explored but discarded, as the problem states it's 10.The configuration \(Y(s) = \frac{10}{s} [ \frac{10}{s} R(s) - Y(s) ] \) is also considered but deemed non-standard.The common interpretation error of a minor feedback loop after the first block is investigated.With forward path \(G(s) = 10/s\) and feedback path \(H(s) = 1\), followed by a series connection with \(10/s\). This is considered.The assumption \(G(s) = \frac{10/s \cdot 10}{s} = \frac{100}{s^2}\) and \(H(s)=1/10\) is made, but also rejected.Adhering to 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 consistently results in a first-order system, suggesting an error in the problem statement.Assuming a typo where the feedback path is \(s\):\(G(s) = 10/s\), \(H(s)=s\). This represents rate feedback.\[ T(s) = \frac{10/s}{1 + (10/s)(s)} = \frac{10/s}{1+10} = \frac{10}{11s} \] Still a first-order system.Considering the second block \(10/s\) in the forward path with unity feedback: \(G(s) = \frac{10}{s} \cdot \frac{10}{s} = \frac{100}{s^2}\). \(H(s) = 1\).\[ T(s) = \frac{100/s^2}{1 + 100/s^2} = \frac{100}{s^2+100} \]Comparing with the standard second-order form \( \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \).\(\omega_n^2 = 100 \implies \omega_n = 10\).\(2\zeta\omega_n = 0 \implies \zeta = 0\). This indicates an undamped oscillator.The diagram likely contains a typo. A standard second-order system form is \(G(s) = \frac{\omega_n^2}{s(s+2\zeta\omega_n)}\).Assuming the second block is \(10/(s+a)\) and the feedback is unity is not productive.Considering feedback as \(1+as\). \(G(s)=100/s^2\).Assuming 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 aligns with options A and C and is the most probable intended problem setup.
