RL Circuits

Definition

An RL circuit is an electrical circuit containing a resistor ($R$) and an inductor ($L$) connected in series to a voltage source ($V$). The current $i(t)$ in the circuit is governed by the first-order linear differential equation: $L \frac{di}{dt} + Ri = V(t)$

• How to read: “The L times the derivative of i with respect to t plus R times i equals V of t.”
• Meaning: Kirchhoff’s voltage law: applied voltage equals resistive drop ($Ri$) plus inductive back-EMF ($L\,di/dt$).

Why It Matters

RL circuits are the fundamental building blocks of modern power electronics; understanding their time-dependent behavior is critical for preventing surges that would otherwise destroy sensitive components in everything from smartphones to electric cars.

Core Concepts

• Inductance ($L$): Measured in henries, it represents the circuit’s opposition to changes in current.

• How to read: “The L.”

  • Meaning: Larger $L$ means current changes more slowly—the inductor “fights” sudden current changes.

• Resistance ($R$): Measured in ohms, it represents the opposition to the flow of current.

• How to read: “The R.”

  • Meaning: Dissipates energy as heat; sets the steady-state current limit.

• Time Constant ($\tau$): Defined as $L/R$, it determines how quickly the current reaches its steady state.

• How to read: “The tau equals L divided by R.”

  • Meaning / when to use: After one time constant $\tau$, current reaches ~63% of its final value. After $5\tau$, essentially at steady state.

• Steady-State Current: For a constant voltage, as $t \to \infty$, the current approaches $V/R$ (Ohm’s Law), as the inductor eventually acts as a simple wire.

• How to read: “As t approaches infinity, i approaches V divided by R.”

  • Meaning: At DC steady state, $di/dt = 0$, so the inductor is a short circuit and $i = V/R$.

Connected Concepts

• Classification of Differential Equations
• Superposition Principle (Differential Equations)
• Autonomous Differential Equations