RL Circuit Current Calculator (Transient Simulator)

TL;DR: This calculator simulates the transient current i(t) of a series RL circuit driven by a constant (step) voltage. It reports the key summary values (τ = L/R, i_final = V/R) and numerically integrates the differential equation so you can plot the full curve and export samples to CSV.

What this simulator calculates

A series resistor–inductor (RL) circuit appears in motor windings, solenoids/actuators, relays, and power electronics. When a DC source is applied, the inductor resists abrupt changes in current, so the current rises (or falls) smoothly toward a steady value rather than stepping instantly.

This page models an ideal series RL connected to a constant voltage source at t = 0 with an optional initial current I₀. You choose the simulation time step Δt and total duration T to control the plotted timeline and numerical resolution.

Variables (inputs)

• V (V): applied (step) voltage, assumed constant during the simulation.
• R (Ω): series resistance (winding resistance + any explicit series resistor).
• L (H): inductance.
• I₀ (A): initial current at t = 0.
• Δt (s): numerical integration time step (smaller usually means higher accuracy, more points).
• T (s): total simulated time window.

Governing equation (RL step response)

Applying Kirchhoff’s Voltage Law (KVL) around the loop for a series RL driven by a constant voltage V gives:

Rearrange to the standard first-order ODE:

di/dt = (V − R·i) / L

Analytical solution (useful checkpoints)

For constant V, the closed-form solution is:

i(t) = i_final + (I₀ − i_final)·e^−t/τ

where:

• i_final = V/R (steady-state current as t → ∞, for R > 0)
• τ = L/R (time constant, for R > 0)

Interpretation of τ: after one time constant, the current has moved about 63.2% of the way from I₀ to i_final. After about 5τ, it is very close to steady state (~99.3%).

How to use the simulator

1. Enter V, R, L, and an optional I₀.
2. Choose Δt small enough to capture the curve shape (a common rule is Δt ≲ τ/100 for smooth plots).
3. Set T to the horizon you care about (often 3τ to 5τ to see settling).
4. Press Play to animate; Pause to inspect; Reset to restart.
5. Use CSV export to download time/current samples for spreadsheets or lab comparison.

Interpreting results

• Final current (V/R): the plateau the curve approaches for a DC step (assuming R is not zero).
• Rise speed (set by τ = L/R): larger L slows the change; larger R speeds the settling but lowers the final current.
• Energy in the inductor: the magnetic energy stored is E = ½·L·i². If the UI shows an energy indicator, it should track this relationship (quadratic with current).
• Effect of I₀: if I₀ is not zero, the waveform starts from that value and exponentially moves toward V/R.

Worked example (using the defaults)

Inputs: V = 5 V, R = 2 Ω, L = 0.5 H, I₀ = 0 A.

• Final current: i_final = V/R = 5/2 = 2.5 A
• Time constant: τ = L/R = 0.5/2 = 0.25 s

Analytical checkpoints:

• At t = τ = 0.25 s: i(t) = 2.5·(1 − e⁻¹) ≈ 1.58 A
• At t = 3τ = 0.75 s: i(t) = 2.5·(1 − e⁻³) ≈ 2.38 A
• At t = 5τ = 1.25 s: i(t) ≈ 2.5·(1 − e⁻⁵) ≈ 2.48 A

If you set T to at least ~1.25 s, you’ll see the curve nearly reach steady state. If you set T = 0.5 s (the default), you’ll still see a substantial rise (about 86.5% of final by 2τ).

Quick comparisons

Assumptions & limitations (important)

• Ideal inductor (constant L): real inductors can saturate; L can drop at high current.
• No parasitics: ignores winding capacitance and leakage, so it will not show ringing/oscillation.
• Constant step voltage: source impedance, PWM drives, and supply droop are not modeled unless you fold them into an effective R or adjust V.
• Resistance is constant: temperature rise can increase R and reduce i_final over time.
• R > 0 required for τ and i_final: as R → 0, τ → ∞ and i_final becomes unbounded in the ideal model; real circuits are always limited by supply and parasitics.
• Numerical accuracy depends on Δt: too-large steps can distort the curve and energy estimate. Reduce Δt if the plot looks jagged or overshoots.

FAQ

What does the time constant mean in plain terms?

τ = L/R sets how quickly current changes. Roughly: 1τ gets you 63% of the way to the final value; 5τ is “basically settled.”

Why does Δt matter?

The simulator integrates di/dt in discrete steps. Smaller Δt tracks the continuous solution more closely, especially when τ is small or you want accurate energy values.

Can I use this for AC?

Not directly. This tool is for a DC step (transient) with constant V. AC steady-state RL analysis uses impedance and phase, which is a different model.