RL Circuit Behavior

What “RL behavior” really means under sine and pulse excitation — and why the simulator is a safe analytic reference

The RL branch is the simplest inductive energy system: a resistor dissipates energy, and the inductor stores energy temporarily. If you understand RL clearly, then RLC behavior becomes a controlled extension (adding electric-field storage in C).

1) What an inductor “does” (physical meaning)

• Current cannot jump instantly unless voltage becomes infinite (idealized).
• The inductor voltage is the counter-EMF: \[ v_L=L\frac{di}{dt} \] It appears whenever current is changing.
• Magnetic stored energy is: \[ W_L=\frac{1}{2}Li^2 \]
• When excitation changes, the inductor exchanges energy with the rest of the circuit: it can return energy (negative instantaneous power) during parts of a cycle, while the resistor always dissipates.

2) Sine excitation (steady state)

With a sinusoidal source \(v(t)=V_{pk}\sin(\omega t)\), an RL branch reaches a steady periodic regime where the current is also sinusoidal, but lags the voltage by a phase angle \(\phi\).

Impedance, phase, and current amplitude

The RL impedance is: \[ Z_{RL}=R+j\omega L \] Magnitude and phase: \[ |Z|=\sqrt{R^2+(\omega L)^2},\qquad \phi=\tan^{-1}\!\left(\frac{\omega L}{R}\right) \] So the current is: \[ i(t)=I_{pk}\sin(\omega t-\phi),\qquad I_{pk}=\frac{V_{pk}}{|Z|} \] (These are the same definitions used in the simulator’s math documentation.)

What you should “see” mentally

• Low frequency (\(\omega L \ll R\)): \(\phi\approx 0^\circ\). The inductor is “weak”, current follows voltage.
• High frequency (\(\omega L \gg R\)): \(\phi\to 90^\circ\). Current strongly lags; most power is reactive (energy sloshing in/out of the magnetic field).
• Inductor voltage \(v_L=L\,di/dt\) is proportional to slope of current — so it becomes large when the waveform changes quickly.

Real and reactive power for sine RL

For pure sinusoidal steady state, using RMS values: \[ V_{rms}=\frac{V_{pk}}{\sqrt{2}},\qquad I_{rms}=\frac{I_{pk}}{\sqrt{2}} \] Real power and reactive power: \[ P=V_{rms}I_{rms}\cos\phi,\qquad Q=V_{rms}I_{rms}\sin\phi \] These match the simulator’s power definitions.

3) Pulse excitation (transient + steady-periodic)

A pulse train repeatedly turns a DC level on and off. The RL branch responds with exponential rise and decay governed by \(\tau\). For a constant voltage step \(v(t)=V\), the RL current is: \[ i(t)=\frac{V}{R}+\left(i_0-\frac{V}{R}\right)e^{-t/\tau} \] This is the “one-state” RL dynamic used for pulse operation.

Pulse train definitions

ON and OFF segments (piecewise solution)

ON: \(v(t)=V\)

OFF: \(v(t)=0\)

The inductor voltage is especially important in pulse work:

• ON: \(v_L(t)=V-Ri(t)\)
• OFF: \(v_L(t)=-Ri(t)\) (the inductor drives the current using its stored energy)

Steady-periodic pulse (why the simulator is stable)

After enough cycles, a pulse-driven RL reaches a repeating regime: \[ i(T^-)=i(0) \] This periodic closure condition is the key reason analytic steady-periodic solutions are safe and deterministic.

4) Worked example (with calculation steps)

We’ll do one sine example and one pulse example with realistic lab numbers. The goal is not “symbol pushing” — it’s to build intuition you can immediately compare to the simulator traces.

Example A — Sine RL (phase lag + P/Q)

Given: \(L=10\,\text{mH}\), \(R=2\,\Omega\), \(V_{pk}=10\,\text{V}\), \(f=100\,\text{Hz}\)

1. Angular frequency: \[ \omega=2\pi f=2\pi\cdot 100=628.319\,\text{rad/s} \]
2. Inductive reactance term: \[ \omega L=628.319\cdot 0.01=6.283\,\Omega \]
3. Impedance magnitude: \[ |Z|=\sqrt{R^2+(\omega L)^2}=\sqrt{2^2+6.283^2}=\sqrt{43.478}=6.595\,\Omega \]
4. Phase lag: \[ \phi=\tan^{-1}\!\left(\frac{\omega L}{R}\right)=\tan^{-1}\!\left(\frac{6.283}{2}\right)=72.34^\circ \]
5. Peak current: \[ I_{pk}=\frac{V_{pk}}{|Z|}=\frac{10}{6.595}=1.516\,\text{A} \]
6. RMS values: \[ V_{rms}=\frac{10}{\sqrt2}=7.071\,\text{V},\qquad I_{rms}=\frac{1.516}{\sqrt2}=1.072\,\text{A} \]
7. Real power: \[ P=V_{rms}I_{rms}\cos\phi =7.071\cdot 1.072\cdot \cos(72.34^\circ) \approx 2.30\,\text{W} \]
8. Reactive power: \[ Q=V_{rms}I_{rms}\sin\phi =7.071\cdot 1.072\cdot \sin(72.34^\circ) \approx 7.22\,\text{var} \]

Example B — Pulse RL (rise/decay + steady-periodic current)

Given: \(L=10\,\text{mH}\), \(R=2\,\Omega\), pulse amplitude \(V=10\,\text{V}\), \(f=1\,\text{kHz}\), \(D=20\%\)

1. Time constant: \[ \tau=\frac{L}{R}=\frac{0.01}{2}=0.005\,\text{s}=5\,\text{ms} \]
2. Period and on/off times: \[ T=\frac{1}{f}=1\,\text{ms},\qquad T_{on}=DT=0.2\,\text{ms},\qquad T_{off}=0.8\,\text{ms} \]
3. DC (infinite-time) current for ON: \[ I_\infty=\frac{V}{R}=\frac{10}{2}=5\,\text{A} \]
4. Exponential factors: \[ a=e^{-T_{on}/\tau}=e^{-0.2/5}=e^{-0.04}=0.9608,\qquad b=e^{-T_{off}/\tau}=e^{-0.8/5}=e^{-0.16}=0.8521 \]
5. Steady-periodic start current \(I_0=i(0)\) (solve \(i(T^-)=i(0)\)): \[ I_0=\frac{I_\infty(1-a)b}{1-ab} \] Numerically: \[ I_0=\frac{5(1-0.9608)\cdot 0.8521}{1-0.9608\cdot 0.8521} \approx 0.922\,\text{A} \]
6. End-of-ON current: \[ I_{end}=I_\infty+(I_0-I_\infty)a =5+(0.922-5)\cdot 0.9608 \approx 1.082\,\text{A} \]
7. Inductor voltage (ON): \[ v_L(0^+)=V-RI_0=10-2\cdot 0.922=8.156\,\text{V} \] \[ v_L(T_{on}^-)=V-RI_{end}=10-2\cdot 1.082=7.836\,\text{V} \]
8. Inductor voltage (OFF): \[ v_L(T_{on}^+)=-RI_{end}=-2\cdot 1.082=-2.164\,\text{V} \] and it decays exponentially as current decays.

5) Why this makes the simulator “safe” for RL (and trustworthy for RLC)

• RL is a one-state system (\(i\) only). Pulse steady state is defined by exact periodic closure \(i(T^-)=i(0)\).
• Sine RL is an explicit impedance/phase model — no hidden solver behavior, no tuning parameters.
• Steady-periodic pulse solutions are analytic, computed by boundary conditions, not by “integrating until it looks stable,” so you don’t get numeric drift artifacts.
• Energy relations are explicit (\(W_L=\tfrac12Li^2\), and \(\Delta W=\int vi\,dt\)) and are used as consistency checks in the documentation.

If you want the full formal derivations, state definitions, periodicity criteria, and limitations in one place, see the Mathematical Foundations page.