Describe the RL-transient response of current when a step voltage is applied.

Describe the RL-transient response of current when a step voltage is applied.

• A. i(t) = I_final (1 - e^{-t/τ}) with τ = L/R; current starts at 0 and rises to I_final.
• B. i(t) = I_final e^{-t/τ}.
• C. i(t) = I_final (1 + e^{-t/τ}).
• D. i(t) = I_final (t/τ).

Answer: (A)

This question tests how current through an inductor responds to a sudden applied voltage, showing an exponential rise toward a final dc value. When a step voltage V is applied to a series RL circuit, the inductor resists the sudden change in current, so the current starts at zero (assuming zero initial current) and climbs toward a steady value I_final = V/R. The time constant τ = L/R sets how quickly it approaches that value. The proper expression is i(t) = I_final [1 − e^(−t/τ)], which captures the rapid initial change (di/dt = V/L at t = 0) and the gradual approach to the final current as the inductor’s voltage decays to zero. The other forms don’t fit this behavior: a decaying exponential would describe current dying away, not rising from zero; a plus sign with the exponential would imply starting above the final value and decreasing toward it; a linear ramp would not reflect the exponential approach dictated by the inductor’s response.

This question tests how current through an inductor responds to a sudden applied voltage, showing an exponential rise toward a final dc value. When a step voltage V is applied to a series RL circuit, the inductor resists the sudden change in current, so the current starts at zero (assuming zero initial current) and climbs toward a steady value I_final = V/R. The time constant τ = L/R sets how quickly it approaches that value. The proper expression is i(t) = I_final [1 − e^(−t/τ)], which captures the rapid initial change (di/dt = V/L at t = 0) and the gradual approach to the final current as the inductor’s voltage decays to zero.

The other forms don’t fit this behavior: a decaying exponential would describe current dying away, not rising from zero; a plus sign with the exponential would imply starting above the final value and decreasing toward it; a linear ramp would not reflect the exponential approach dictated by the inductor’s response.