1. Physical context

Series resistor–inductor (RL) circuits appear in power supplies, motor drives, and electromagnetic actuators. When a switch connects a voltage source to such a circuit, the current does not jump instantly; the inductor resists change by generating an opposing electromotive force. Instead, the current rises gradually toward a steady value. Traditional calculators report that final value and maybe the time constant, but they do not convey the shape of the transient. This upgraded calculator solves the differential equation in real time. As you vary the voltage, resistance, inductance, or initial current, the canvas redraws a smooth curve while a striped bar indicates magnetic energy. You can pause the evolution, step through time by adjusting the integration step, or export every time sample to a CSV file for comparison with oscilloscope traces.

2. Variables and assumptions

The system assumes a single ideal inductor of inductance $L$ in series with a resistor of resistance $R$, driven by a constant step voltage $V$. The starting current is $\mathrm{I\_0}$. The equations neglect parasitic capacitances, core saturation, temperature dependence, and supply resistance. The simulation time step $\mathrm{\Delta t}$ governs numerical accuracy, while the total time $T$ defines when the animation stops. All quantities are expressed in SI units: volts, ohms, henries, seconds, and amperes. Input validation guards against negative or non‑finite values; any such entry produces an error message instead of proceeding.

Because inductors generate magnetic fields, RL circuits also appear whenever conductors are wound into coils for sensing or actuation. Doorbells, relays, and even guitar pickups rely on the same transient current buildup. In such devices the resistance often represents both the coil wire and any series driving resistance, while the inductance reflects geometry and core material. Translating these physical attributes into the simplified R and L parameters helps connect hands-on hardware with the mathematical model used here.

3. Governing equations

Applying Kirchhoff’s voltage law around the loop gives $V=L\frac{\mathrm{di}}{\mathrm{dt}}+Ri$. Rearranging yields the first‑order differential equation $\frac{\mathrm{di}}{\mathrm{dt}}=\frac{V}{-}L$. The analytical solution for a constant $V$ is $\mathrm{i(t)}=\mathrm{i\_f}+(\mathrm{I\_0}-\mathrm{i\_f})e^{\frac{-t}{\tau }}$, where the final current is $\mathrm{i\_f}=\frac{V}{R}$ and the time constant $\tau =\frac{L}{R}$. For numerical integration we use the same differential equation directly, updating the state according to the chosen step size.

4. Numerical scheme

To animate the transient with high fidelity, the code employs a fourth‑order Runge–Kutta method. Defining $\mathrm{f(i)}=\frac{V}{-}L$, each step proceeds as $\mathrm{i\_\{n+1\}}=\mathrm{i\_n}+\frac{\mathrm{\Delta t}}{6}(\mathrm{k\_1}+2\mathrm{k\_2}+2\mathrm{k\_3}+\mathrm{k\_4})$, where $\mathrm{k\_1}=f(\mathrm{i\_n})$, $\mathrm{k\_2}=f(\mathrm{i\_n}+\frac{\mathrm{\Delta t}}{2}\mathrm{k\_1})$, $\mathrm{k\_3}=f(\mathrm{i\_n}+\frac{\mathrm{\Delta t}}{2}\mathrm{k\_2})$, and $\mathrm{k\_4}=f(\mathrm{i\_n}+\mathrm{\Delta t}\mathrm{k\_3})$. The method is stable for step sizes up to roughly $\mathrm{\Delta t}<\frac{\tau }{5}$; the interface therefore caps $\mathrm{\Delta t}$ at 0.1 s and warns if the ratio to $\tau$ grows too large. The algorithm simultaneously computes magnetic energy $U=\frac{1}{2}L{i}^2$ and tracks the maximum to scale the energy bar.

5. Worked example

Imagine energizing a 0.5 H coil through a 2 Ω resistor using a 5 V supply. The time constant is $\tau =\frac{0.5}{2}=0.25 s$, and the final current approaches $\frac{5}{2}=2.5 A$. Starting from zero current, the RK4 integration with $\mathrm{\Delta t}=0.001$ s and total time 0.5 s produces a smooth rise that reaches 2.16 A at 0.25 s and 2.42 A at 0.5 s. The energy bar climbs to 1.47 J at the end, matching $\frac{1}{2}L{i}^2$. Entering these numbers above reproduces the plot; pressing CSV exports the time and current columns for independent verification.

6. Comparison table

How do circuit parameters influence the transient? The following table compares the baseline example with two variations. Each row lists the time constant, current after 0.25 s, and magnetic energy after 0.5 s, values extracted directly from simulation output.

Doubling the inductance slows the rise but stores slightly more energy, while halving the resistance both increases the final current and doubles the time constant, illustrating how design choices trade speed for efficiency.

7. Reading the animation

The horizontal axis of the canvas spans from 0 to the chosen total time. The vertical axis scales automatically to cover the simulated current range. A blue curve traces the current, and a red circle marks the current at the present animation time. Press the space bar while the canvas is focused to toggle play and pause. Beneath the plot, the striped blue bar represents magnetic energy as a fraction of the maximum encountered; its changing length provides a color‑independent cue that the inductor is storing energy. The caption and the hidden text summary mirror the key numbers—time, current, and energy—for screen‑reader accessibility.

8. Limitations

This model presumes an ideal lumped inductor. Real coils have winding resistance, core losses, parasitic capacitance, and nonlinear inductance when driven near saturation. At very high frequencies the lumped‑element approximation breaks down entirely. Numerically, choosing a time step too large compared with the time constant will exaggerate error and may even produce oscillations, a sign that the integration scheme is being pushed beyond its stable region.

The simulator also ignores noise sources and quantization that appear in digital control. In microcontroller circuits, the finite resolution of analog-to-digital converters and the discrete update rate of pulse-width modulation can perturb the ideal exponential response. While these effects lie beyond the scope of the analytic RL model, being aware of them is crucial when applying results to precision instrumentation.

9. Possible extensions

A natural extension is to include a switch for supply removal and show the exponential decay when the inductor discharges through the resistor. Adding a capacitor would transform the model into an RLC oscillator, enabling resonance and phase plots. Another option is to allow arbitrary voltage waveforms by sampling a user‑defined function. Because the current code runs entirely in the browser, such experiments can proceed without servers or additional libraries. These additions would create a richer toolbox for electronics education.

10. References and related tools

For a thorough treatment of transient circuits, see P. Horowitz and W. Hill, The Art of Electronics, 3rd ed., Cambridge University Press, 2015. Additional derivations appear in many introductory physics texts such as D. Halliday et al., Fundamentals of Physics. To explore related phenomena, try our RC Circuit Charging Calculator, examine damping in the RLC Circuit Damping Calculator, or balance bridges with the Wheatstone Bridge Balance Calculator.