Final answer:
An RLC circuit with R=2Ω and L=1H displays an overdamped response for C=1/2 F, a critically damped response for C=1 F, and an underdamped response for C=2 F.
Step-by-step explanation:
To determine the type of response exhibited by the network in a series RLC circuit, we can compare the damping factor (which depends on resistance R, inductance L, and capacitance C) to the characteristics of underdamped, critically damped, and overdamped responses. The general criterion for these responses is based on the relationship between the damping factor (ζ) and the resonant frequency (ω0) of the circuit.
For an RLC circuit, the damping factor (ζ) is given by ζ = R / (2 *× √(L/C)). The resonant frequency is ω0 = 1 / √(LC). A circuit is underdamped if ζ < 1, critically damped if ζ = 1, and overdamped if ζ > 1.
Given R = 2Ω and L = 1H, and varying C:
- For C = 1/2 F, ζ = 2 / (2 × √(1/(1/2))) = 2 / √2 ≈ 1.414, which is greater than 1. This means the response will be overdamped.
- For C = 1 F, ζ = 2 / (2 × √(1/1)) = 1, meaning the response is critically damped.
- For C = 2 F, ζ = 2 / (2 × √(1/2)) = 1 / √2 ≈ 0.707, which is less than 1. This indicates an underdamped response.
Therefore, the different capacitance values result in different damping behaviours for the RLC circuit.