Step Response of First-Order RL Circuits
The step response of a first-order RL circuit refers to how the current and voltage change over time when a sudden voltage (step input) is applied. In such a circuit, the inductor initially resists changes in current, causing the current to rise gradually rather than instantly. The current increases exponentially, approaching its maximum value determined by Ohm’s Law, with a time constant defined by the ratio of inductance to resistance (L/R).
Step Response of First-Order RL Circuits
The step response of a first-order RL circuit refers to how the current and voltage change over time when a sudden voltage (step input) is applied. In such a circuit, the inductor initially resists changes in current, causing the current to rise gradually rather than instantly. The current increases exponentially, approaching its maximum value determined by Ohm’s Law, with a time constant defined by the ratio of inductance to resistance (L/R).
💡 Key Takeaways
- Understand how a step voltage applied to a series RL circuit causes the current to rise from 0 toward its steady-state value V/R.
- Learn that the speed of the rise is set by the time constant tau = L/R.
- Connect the time-domain current to the step response: i(t) = (V/R) [1 − exp(−tR/L)] for a ideal step input.
- See how the inductor voltage decays with time as v_L(t) = V exp(−tR/L) while the current approaches steady state.
❓ Frequently Asked Questions
What is the time constant of a first-order RL circuit and how does it affect the step response?
tau = L / R. It sets how fast the current rises: after one tau, i ≈ 0.63 of its final value; after ~5 tau, the response is near final.
What is the expression for the current i(t) in a series RL circuit after a DC voltage V0 is applied at t = 0 (initial current is zero)?
i(t) = (V0 / R) [1 − e^(−tR/L)].
What is the voltage across the inductor v_L(t) during the step response?
v_L(t) = V0 e^(−tR/L). It starts at V0 at t = 0 and decays to 0 as t increases.
What is the voltage across the resistor v_R(t) during the step response?
v_R(t) = i(t) R = V0 [1 − e^(−tR/L)].
What are the final steady-state values for current and voltages in a series RL circuit with a DC step V0?
I∞ = V0 / R; v_R∞ = V0; v_L∞ = 0.