LR Circuits

An LR circuit contains a resistor (R) and an inductor (L) connected together, typically with a voltage source and a switch. Unlike a purely resistive circuit where current responds instantaneously to voltage changes, an LR circuit has a 'memory' — the inductor resists any sudden change in current. This produces a fascinating time-dependent behaviour known as a transient response: current either grows gradually toward a final value or decays gradually toward zero, following a smooth exponential curve rather than jumping abruptly. The physical reason is rooted in Faraday's law of electromagnetic induction. When current through an inductor changes, it creates a changing magnetic flux inside its coils, which in turn induces a back-EMF that opposes the change (Lenz's law). Think of it like pushing a heavy flywheel: you can apply force instantly, but the wheel speeds up slowly because its inertia resists acceleration. The inductor's inductance L plays the role of inertia for electric current. LR transient circuits are everywhere in real engineering: they appear in the switching circuits of power supplies, in the ignition coils of car engines, in radio transmitters, and in any system where electromagnetic relays or solenoids are switched on and off. Understanding how current rises and falls exponentially — and how quickly this happens — is essential for designing safe, reliable electronic and electrical systems.

Key takeaways
  • Setting Up the Circuit Equation
  • Solving for Current Growth (Switch Closed)
  • The Time Constant τ
  • Current Decay (Switch Opened / Source Removed)
  • Energy Considerations

Setting Up the Circuit Equation

When a DC voltage source V is connected in series with resistance R and inductance L, Kirchhoff's voltage law gives: V = iR + L(di/dt). The term L(di/dt) is the back-EMF of the inductor. This is a first-order linear ordinary differential equation (ODE) in the unknown current i(t). The presence of di/dt is what makes the response time-dependent.

Solving for Current Growth (Switch Closed)

Starting from i(0) = 0 (no initial current), the solution to the ODE is i(t) = (V/R)(1 − e^(−Rt/L)). As t → ∞, i approaches the steady-state value V/R — exactly what Ohm's law gives when the inductor acts like a plain wire (since di/dt → 0, its voltage drop vanishes). The exponential term shows that the approach is gradual, never instantaneous.

The Time Constant τ

The combination τ = L/R has units of seconds (H/Ω = s) and is called the time constant of the circuit. After one time constant (t = τ), the current has reached (1 − e^(−1)) ≈ 63.2% of its final value. After 5τ, the current is within 0.7% of V/R and is considered essentially steady. A large L or small R means a slow response; a small L or large R means a fast response.

Current Decay (Switch Opened / Source Removed)

If the circuit is then opened while carrying a steady current I₀, the inductor drives current through any available path (such as the resistor). With no source, the ODE becomes 0 = iR + L(di/dt), giving i(t) = I₀ e^(−Rt/L). The current decays exponentially with the same time constant τ = L/R. Critically, the voltage across the inductor during rapid switch-off can be enormous — L × (di/dt) — which is why inductive 'kickback' can destroy transistors and why flyback diodes are used in practice.

Energy Considerations

An inductor stores energy in its magnetic field: E = ½LI². During current growth, the source does work partly against R (heat) and partly to build the magnetic field. During decay, the stored energy is fully released into the resistor as heat. This energy accounting confirms that no energy is created or destroyed — the inductor is a temporary energy store, not a source.

The Key Equation

i(t)=VR(1−e−t/τ),τ=LRi(t) = \frac{V}{R}\left(1 - e^{-t/\tau}\right), \quad \tau = \frac{L}{R}
  • i(t) — Instantaneous current (A (amperes)): The current flowing in the circuit at time t after the switch is closed.
  • V — Source voltage (V (volts)): The EMF of the DC voltage source driving the circuit.
  • R — Resistance (Ω (ohms)): The total series resistance in the circuit; determines the final steady-state current.
  • \tau — Time constant (s (seconds)): The characteristic time scale of the transient; equal to L/R. After one τ, current is 63.2% of its final value.
  • L — Inductance (H (henries)): The self-inductance of the coil, measuring how strongly it opposes changes in current.
  • e — Euler's number (dimensionless): The base of the natural logarithm, approximately 2.718; appears because the ODE solution is an exponential function.

Worked Example

Problem. A 12 V battery is connected in series with a 40 Ω resistor and a 200 mH inductor. The switch is closed at t = 0. (a) Find the time constant τ. (b) Find the current at t = 5.0 ms. (c) Find the initial rate of change of current, di/dt, at t = 0.

τ=5.0 ms,i(5.0 ms)≈0.190 A,didt∣t=0=60 A/s\tau = 5.0\text{ ms},\quad i(5.0\text{ ms}) \approx 0.190\text{ A},\quad \left.\frac{di}{dt}\right|_{t=0} = 60\text{ A/s}

Answer. Time constant τ = 5.0 ms; current at t = 5.0 ms is approximately 0.190 A; initial rate of change is 60 A/s.

Figure
Diagram
LR Circuit: Current vs. Timeti(t)V/Rτ63%Growth: i=(V/R)(1−e^(−t/τ))Decay: i=I₀e^(−t/τ)CircuitVRL
Origins · 1 minJoseph Henry and how this was discovered
JH
Joseph Henry
1797–1878 · American

Joseph Henry independently discovered electromagnetic induction at nearly the same time as Michael Faraday, and made crucial advances in understanding self-inductance. He constructed powerful electromagnets and discovered that a coil carrying current can induce a voltage in itself when the current changes — the very phenomenon that gives inductors their transient behaviour. The SI unit of inductance, the henry (H), is named in his honour. Henry discovered self-induction before Faraday announced electromagnetic induction publicly, but he was so busy teaching students at the Albany Academy that he delayed publishing — a lesson in the importance of promptly sharing scientific results!

  • RC Circuits — RC circuits show analogous exponential charging and discharging of a capacitor; comparing RC and LR transients deepens understanding of both time constants and dual electrical quantities.
  • Faraday's Law of Induction — The back-EMF that drives LR transient behaviour arises directly from Faraday's law; a firm grasp of induced EMF is essential for understanding why inductors resist current changes.
  • Self-Inductance — Inductance L is the key parameter in LR circuits; understanding how coil geometry, turns, and core material determine L explains why different inductors produce different time constants.
  • RLC Circuits and Resonance — Adding a capacitor to an LR circuit produces an RLC circuit capable of oscillations and resonance; LR transient analysis is a prerequisite for that richer behaviour.