Difficulty: Intermediate | Prerequisites: Kirchhoff's laws, capacitor charging/discharging (RC circuits), basic inductance concepts.
Tags: RL circuit, inductor, time constant, exponential decay, LC circuit, energy oscillation, transient response, PHYS, University of Illinois
RL circuits are the inductor counterpart to the RC circuits you already know. Where RC circuits store energy in electric fields (capacitors), RL circuits store energy in magnetic fields (inductors). The mathematics is nearly identical, with exponential growth and decay governed by a time constant. LC circuits then combine both storage mechanisms, producing oscillations rather than simple decay. If you are comfortable with RC transient behaviour, RL circuits will feel familiar. If you are not, revisit RC circuits first.
An RL circuit's current rises or falls exponentially with time constant τ = L/R. At t = 0 an inductor blocks sudden current changes; at t → ∞ it behaves like a wire. LC circuits swap energy between inductor and capacitor with no resistive loss, producing sinusoidal oscillations.
Inductor
A passive component that stores energy in a magnetic field when current flows through it. Think of it as electrical inertia: it resists changes in current the same way a heavy object resists changes in velocity.
Time constant (τ) for an RL circuit
τ = L / R, where L is inductance and R is resistance. This is the time it takes for the current to reach about 63% of its final value (charging) or fall to about 37% of its initial value (discharging). In simple terms, bigger inductance or smaller resistance means the circuit takes longer to respond.
Transient response
The temporary behaviour of a circuit as it transitions from one steady state to another. After roughly 5τ, the transient is essentially over and the circuit has reached its new steady state.
LC circuit
A circuit containing only an inductor and a capacitor. Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor, analogous to a mass on a spring trading kinetic and potential energy.
Inductors combine the same way resistors do:
Series: L_total = L₁ + L₂ + L₃ + ...
Parallel: 1/L_total = 1/L₁ + 1/L₂ + 1/L₃ + ...
This is the opposite of capacitors, which combine like resistors in the inverse sense.
When a voltage source is connected to an RL circuit, current grows exponentially toward its maximum value:
I(t) = I(∞)(1 − e^(−t/τ))
Boundary behaviour:
At t = 0: I = 0 A. The inductor initially opposes any change, so no current flows yet.
At t → ∞: I = I(∞) = V/R. The inductor acts like a plain wire (zero voltage across it), and the circuit behaves as if only the resistor is present.
The voltage across the inductor starts at its maximum (equal to the source voltage) and decays to zero.
When the source is removed and the circuit is closed through a resistor, the stored magnetic energy drives a decaying current:
I(t) = I(0) e^(−t/τ)
Boundary behaviour:
At t = 0: the inductor acts like a current source, maintaining the current it had just before the switch.
At t → ∞: I = 0. The inductor acts as a wire with no energy left to sustain current.
Total energy in an LC circuit is conserved (no resistor to dissipate it):
U_total = U_inductor + U_capacitor
U_inductor = ½LI² and U_capacitor = ½CV² = Q²/(2C)
Energy sloshes back and forth between the two components:
When the capacitor is fully charged, all energy is in the electric field and current is zero.
When the current is at its peak, all energy is in the magnetic field and the capacitor voltage is zero.
The natural oscillation frequency is ω₀ = 1/√(LC), or equivalently f₀ = 1/(2π√(LC)).
Quantity | Expression |
|---|---|
RL charging current | I(t) = I(∞)(1 − e^(−t/τ)) |
RL discharging current | I(t) = I(0) e^(−t/τ) |
RL time constant | τ = L / R |
LC total energy | U_total = ½LI² + Q²/(2C) |
LC natural frequency | ω₀ = 1/√(LC) |
RL circuits appear in any system where current must ramp up or down smoothly, such as the ignition coil in a car (which uses the inductor's energy release to generate a spark) or surge-protection circuits that limit how fast current can change. LC oscillators are the basis of radio tuning circuits, where selecting a station means choosing the resonant frequency by adjusting L or C.
"At t = 0 the inductor acts like a wire." It does not. At t = 0 the inductor opposes any sudden current change, so it initially acts like an open circuit (for charging from zero) or a current source (for discharging). It is at t → ∞ that the inductor acts like a wire.
"The RL time constant is τ = RC." That is the RC circuit time constant. For RL circuits, τ = L/R. Mixing these up is one of the most common exam errors.
"LC circuits always lose energy over time." An ideal LC circuit has no resistance and oscillates forever. Energy loss only occurs when resistance is present (making it an RLC circuit).
"Inductors in parallel add directly." They do not. Inductors in parallel combine using the reciprocal rule, just like resistors in parallel.
⚠️ Be ready to sketch I(t) and V_L(t) for both charging and discharging, and to identify the boundary values at t = 0 and t → ∞.
⚠️ Know which component the inductor "acts like" at each extreme: open circuit / current source at t = 0, wire at t → ∞.
⚠️ LC energy conservation problems often ask you to equate ½LI² and Q²/(2C) at different instants.
True or false: In an RL charging circuit, the current is at its maximum value at t = 0.
Fill in the blank: The RL time constant is τ = ___ / ___.
True or false: At t → ∞ in a charging RL circuit, the voltage across the inductor is zero.
Fill in the blank: In an ideal LC circuit, the total energy equals U_inductor + ___.
True or false: Inductors in series add using the reciprocal formula.
Answers: 1. False (it is zero). 2. L / R. 3. True. 4. U_capacitor. 5. False (that is parallel; series inductors add directly).
Q: An RL circuit has L = 0.5 H and R = 10 Ω. What is the time constant, and how long until the current is approximately at its steady-state value?
A: τ = L/R = 0.5/10 = 0.05 s. The current is effectively at steady state after about 5τ = 0.25 s.
Q: In an RL discharging circuit, the initial current is 2 A. What is the current after one time constant?
A: I(τ) = I(0) e^(−1) = 2 × 0.368 ≈ 0.74 A.
Q: An LC circuit has a 4 mH inductor and a 1 µF capacitor. What is the oscillation frequency?
A: ω₀ = 1/√(LC) = 1/√(4×10⁻³ × 1×10⁻⁶) = 1/√(4×10⁻⁹) = 1/(2×10⁻⁴·⁵) ≈ 15,811 rad/s, giving f₀ = ω₀/(2π) ≈ 2,516 Hz.
Q: At the moment when the capacitor in an LC circuit is fully discharged, where is all of the circuit's energy?
A: All energy is stored in the magnetic field of the inductor (U = ½LI²), and the current is at its maximum value.
This material connects directly to RC circuits: the exponential charging/discharging equations have the same form, and swapping the roles of voltage and current gives a useful analogy. LC oscillations lead naturally into the next topic, AC (RLC) circuits, where a driving voltage source and resistance are added, producing forced oscillations and resonance. The energy methods used here also appear in electromagnetic wave theory, where E and B fields carry energy in a manner analogous to C and L storing it.
Related Terms / Search Tags: RL circuit, LR circuit, inductor transient, time constant tau, exponential decay current, LC oscillator, LC tank circuit, natural frequency, energy oscillation, inductor charging, inductor discharging, Kirchhoff inductance, PHYS 212, university physics electricity and magnetism