LRC Circuits

Definition

An LRC circuit is an electrical system consisting of an inductor $L$, resistor $R$, and capacitor $C$. The charge $q(t)$ on the capacitor satisfies: $Lq'' + Rq' + \frac{1}{C}q = E(t)$ where $E(t)$ is the electromotive force (voltage).

Why It Matters

LRC circuits are the foundation of all oscillating electronics; a failure to understand their resonance and damping leads to ‘broken’ radio systems, unstable power grids, and the inability to process analog signals.

Core Concepts

• The LRC Circuit Equation $L q'' + R q' + \frac{1}{C} q = E(t)$

  • How to read: “The value L q double prime plus R q prime plus the fraction one over C times q equals E of t.”
  • Meaning / mechanical analogy: $E(t)$ is the applied voltage. The equation is mathematically identical to the forced damped harmonic oscillator $m x'' + b x' + k x = F(t)$.
    • L (inductance) plays the role of mass m (inertia against change in current).
    • R (resistance) plays the role of damping b (energy dissipation).
    • 1/C (inverse capacitance) plays the role of spring constant k (restoring “force” from charge buildup).
    • q(t) is charge on the capacitor (analogous to position x).
    • i(t) = q’(t) is current (analogous to velocity). The equation comes from Kirchhoff’s voltage law applied around the loop.

• Current form: Differentiating gives the equation in current: L i” + R i’ + (1/C) i = E’(t).

• Transient vs Steady-State:

  • Transient (homogeneous solution): Decays exponentially when R > 0. This is the “ringing” or dying out after the circuit is disturbed.
  • Steady-state (particular solution): The long-term behavior that follows the driving voltage E(t) (e.g., sinusoidal if E is AC).

• Resonance: When the driving frequency matches the natural frequency (determined by L and C), amplitude peaks — exactly like mechanical resonance.

Connected Concepts

• Calculus
• Systems Thinking
• RL Circuits