RLC Circuit Theory

RC circuits introduced first-order dynamics and exponential relaxation due to energy dissipation. By adding an inductor, we obtain RLC circuits, which are the simplest electrical systems capable of oscillatory behavior. In these circuits, energy is exchanged between electric and magnetic fields while resistive elements dissipate energy as heat.

Inductors

An inductor is a circuit element that stores energy in a magnetic field. Physically, inductors resist changes in current, since a changing current produces a changing magnetic field, which induces a voltage opposing that change.

The voltage across an inductor is given by

V_L = L \frac{dI}{dt},

where \(L\) is the inductance, measured in henries (H). The energy stored in an inductor is

E_L = \tfrac{1}{2} L I^2.

LC Circuits

Before introducing resistance, it is useful to consider an ideal LC circuit. In this case, energy oscillates between the electric field of the capacitor and the magnetic field of the inductor without loss.

Let \(Q(t)\) denote the charge on the capacitor. Applying Kirchhoff’s Voltage Law gives

L\frac{d^2 Q}{dt^2} + \frac{1}{C} Q = 0.

This equation describes simple harmonic motion with natural frequency

\omega_0 = \frac{1}{\sqrt{LC}}.

In an ideal LC circuit, the total energy remains constant and oscillates between electric and magnetic forms.

RLC Circuits

Real circuits contain resistance, which dissipates energy and damps oscillations. Including a resistor in series with the inductor and capacitor leads to the RLC circuit.

Applying Kirchhoff’s Voltage Law yields the governing equation

L\frac{d^2 Q}{dt^2} + R\frac{dQ}{dt} + \frac{1}{C} Q = 0.

This is a second-order linear differential equation analogous to the equation of motion for a damped mass–spring system. The resistor plays the role of damping, while the inductor and capacitor correspond to inertia and restoring force, respectively.

Damping Regimes

The qualitative behavior of an RLC circuit depends on the relative strength of resistance. Define

\alpha = \frac{R}{2L}, \qquad \omega_0 = \frac{1}{\sqrt{LC}}.

The system exhibits three distinct regimes:

Underdamped (\(\alpha < \omega_0\)): oscillatory behavior with exponentially decaying amplitude.
Critically damped (\(\alpha = \omega_0\)): fastest return to equilibrium without oscillation.
Overdamped (\(\alpha > \omega_0\)): slow, non-oscillatory return to equilibrium.

These damping regimes appear throughout physics and engineering and play a central role in circuit design, control systems, and signal processing.

RLC circuits therefore provide a canonical example of second-order linear dynamical systems, illustrating how energy storage, dissipation, and oscillation interact in physical systems.

Inductors in Series and Parallel

When multiple inductors are connected together, their arrangement determines the effective inductance seen by the circuit. This affects how strongly the circuit resists changes in current and therefore influences the overall dynamics.

Inductors in Series

Inductors connected end-to-end carry the same current. The total voltage across the series combination is the sum of the individual voltages, leading to an equivalent inductance

L_{\text{eq}} = L_1 + L_2 + \dots + L_n.

Series inductors increase the total inductance, making changes in current more difficult.

Inductors in Parallel

Inductors connected in parallel experience the same voltage, while the current divides among the branches. The equivalent inductance is given by

\frac{1}{L_{\text{eq}}} = \frac{1}{L_1} + \frac{1}{L_2} + \dots + \frac{1}{L_n}.

Parallel inductors reduce the effective inductance, allowing current to change more easily.