In an RL circuit, the time constant τ is defined by which relation?

In an RL circuit, the time constant τ is defined by which relation?

• A. τ = R/L
• B. τ = L/R
• C. τ = 1/(R L)
• D. τ = LR

Answer: (B)

In an RL circuit, the time constant is the characteristic time scale over which the current changes. The governing equation for a series RL circuit with a voltage source is V_in = L di/dt + R i. Looking at the natural response (no input, or after switching), L di/dt + R i = 0 leads to di/dt = -(R/L) i, so i(t) = i(0) e^{-(R/L) t}. The exponential decay can be written as i(t) = i(0) e^{-t/τ}, which identifies τ with L/R. This same τ also appears in the forced response for a step input: i(t) = (V_in/R)[1 - e^{-t/τ}]. So the time constant is the ratio of inductance to resistance. Larger L slows the response, while larger R speeds it up (in terms of the time constant). The correct relation is τ = L/R.

In an RL circuit, the time constant is the characteristic time scale over which the current changes. The governing equation for a series RL circuit with a voltage source is V_in = L di/dt + R i. Looking at the natural response (no input, or after switching), L di/dt + R i = 0 leads to di/dt = -(R/L) i, so i(t) = i(0) e^{-(R/L) t}. The exponential decay can be written as i(t) = i(0) e^{-t/τ}, which identifies τ with L/R. This same τ also appears in the forced response for a step input: i(t) = (V_in/R)[1 - e^{-t/τ}]. So the time constant is the ratio of inductance to resistance. Larger L slows the response, while larger R speeds it up (in terms of the time constant). The correct relation is τ = L/R.