GATE EE 2019 Set 1 — Question 52

Given:
System voltage $V = 132$ kV
Series inductance $L = 50$ mH $= 50 \times 10^{-3}$ H
Shunt capacitance $C = 0.05\ \mu F = 0.05 \times 10^{-6}$ FThe critical value of resistance connected across the circuit breaker contacts to prevent transient oscillation (i.e., to achieve critical damping) is given by the critical damping resistance $R_c$.
For an RLC circuit, the damping factor $\zeta$ determines the nature of the transient response.
For critical damping, $\zeta = 1$.
The characteristic equation of an RLC circuit is $s^2 + \frac{R}{L}s + \frac{1}{LC} = 0$ (for series RLC) or $s^2 + \frac{1}{RC}s + \frac{1}{LC} = 0$ (for parallel RLC, if R is across LC).In the context of circuit breaker transient recovery voltage (TRV), the circuit formed by the inductance of the system and the capacitance at the breaker terminal is an LC circuit. When a resistance is connected across the breaker contacts, it forms a parallel RLC circuit with the system inductance and shunt capacitance.The natural frequency of oscillation is $\omega_n = \frac{1}{\sqrt{LC}}$.
The damping ratio for a parallel RLC circuit is $\zeta = \frac{1}{2R}\sqrt{\frac{L}{C}}$.
For critical damping, $\zeta = 1$.
$1 = \frac{1}{2R_c}\sqrt{\frac{L}{C}}$
$R_c = \frac{1}{2}\sqrt{\frac{L}{C}}$Substitute the given values:
$L = 50 \times 10^{-3}$ H
$C = 0.05 \times 10^{-6}$ F$R_c = \frac{1}{2}\sqrt{\frac{50 \times 10^{-3}}{0.05 \times 10^{-6}}} = \frac{1}{2}\sqrt{\frac{50}{0.05} \times 10^{3}} = \frac{1}{2}\sqrt{1000 \times 10^{3}} = \frac{1}{2}\sqrt{10^6}$
$R_c = \frac{1}{2} \times 10^3 = \frac{1000}{2} = 500\ \Omega$The critical value of resistance is $500\ \Omega$.Note on discrepancy: The calculated critical resistance is $500\ \Omega$. However, the specified answer range is $0$ to $0$. This indicates a significant discrepancy. The value of $0\ \Omega$ would imply a short circuit, which would not prevent transient oscillations in an LC circuit, but rather change the circuit behavior. Given the strict instruction to match the provided answer, we acknowledge this discrepancy. If the answer is $0$, it implies a different interpretation or a flaw in the problem statement/given answer.