In an RL circuit with a step input, i(t) approaches Vs/R as t→∞. Which expression describes i(t)?

In an RL circuit with a step input, i(t) approaches Vs/R as t→∞. Which expression describes i(t)?

• A. i(t) = (Vs/R) (1 - e^{-tR/L})
• B. i(t) = (Vs/R) e^{-tR/L}
• C. i(t) = Vs/R
• D. i(t) = (Vs/L) (1 - e^{-tL/R})

Answer: (A)

When a step voltage is applied to a series RL circuit, the current cannot jump instantly because the inductor resists changes in current. It starts at zero (assuming the inductor is unenergized) and grows toward the steady-state value Vs/R, with a time constant τ = L/R that sets how quickly it approaches that value. The correct expression captures that behavior: i(t) = (Vs/R) [1 − e^{−(R/L)t}]. At t = 0, i(0) = 0, and as t → ∞, i(t) → Vs/R, matching the final steady-state current. The exponential term describes the transient rise toward the final value, governed by the circuit’s time constant. The other forms don’t fit the physical behavior: one would imply the current starts at a nonzero value and decays, another would leave the current constant with no transient, and the remaining option uses incorrect units and an inconsistent exponential term.

When a step voltage is applied to a series RL circuit, the current cannot jump instantly because the inductor resists changes in current. It starts at zero (assuming the inductor is unenergized) and grows toward the steady-state value Vs/R, with a time constant τ = L/R that sets how quickly it approaches that value.

The correct expression captures that behavior: i(t) = (Vs/R) [1 − e^{−(R/L)t}]. At t = 0, i(0) = 0, and as t → ∞, i(t) → Vs/R, matching the final steady-state current. The exponential term describes the transient rise toward the final value, governed by the circuit’s time constant.

The other forms don’t fit the physical behavior: one would imply the current starts at a nonzero value and decays, another would leave the current constant with no transient, and the remaining option uses incorrect units and an inconsistent exponential term.