In a series RLC circuit, which parameter governs the exponential decay rate of the transient response, and what is the formula?

• A. α = L/(2R); the decay is e^(−α t)
• B. α = R/(2C); the decay is e^(−α t)
• C. α = R/(2L); the decay is e^(−α t)
• D. α = 2√(L/C); the decay is e^(−α t)

Answer: (C)

In a series RLC circuit, the transient decay rate is set by the damping factor α that comes from the circuit’s second‑order equation. Writing the loop equation for charge q gives L d^2q/dt^2 + R dq/dt + q/C = 0, whose characteristic equation is L s^2 + R s + 1/C = 0. The roots are s = [-R ± sqrt(R^2 - 4L/C)]/(2L), and the real part of these roots is -R/(2L). That means the envelope of the transient decays like e^{-α t} with α = R/(2L). In the underdamped case, the full response is q(t) or i(t) = e^{-α t} [A cos(ω_d t) + B sin(ω_d t)], where ω_d = sqrt(1/(LC) - (R/(2L))^2. So the correct expression uses α = R/(2L) and the decay factor e^{-α t}.