Circuit Visualization
Circuit Parameters
Component Values
Animation Controls
Display Options
Visualization Options
RLC Circuit Equations
Instructions
- Adjust R, L, C values to change circuit behavior
- Observe damping status: underdamped, critically damped, or overdamped
- Switch between waveform, phase portrait, frequency response, and energy views
- In phase portrait mode, watch the spiral trajectory decay toward origin
- At resonance (ω = ω₀), current reaches maximum amplitude
- Energy continuously transfers between capacitor (E-field) and inductor (B-field)
What is RLC Circuit Oscillation?
An RLC circuit consists of a resistor (R), inductor (L), and capacitor (C) connected in series. When the capacitor is initially charged and then connected to the circuit, energy oscillates between the electric field in the capacitor and the magnetic field in the inductor, while the resistor dissipates energy as heat. This creates a damped harmonic oscillation described by the differential equation Lq'' + Rq' + q/C = 0, where q is the charge on the capacitor and I = q' is the current.
Damping Modes
The behavior depends on the damping ratio ζ = γ/ω₀ = R·√(C/L)/2. For underdamped (ζ < 1), the circuit oscillates with exponentially decaying amplitude at frequency ωd = √(ω₀² - γ²). This is the most interesting case, showing clear oscillations. For critically damped (ζ = 1), the circuit returns to equilibrium as quickly as possible without oscillating, achieved when R = 2√(L/C). For overdamped (ζ > 1), the circuit returns to equilibrium slowly without oscillations, with two exponential decay time constants.
Resonance in RLC Circuits
When driven by an AC voltage source, an RLC circuit exhibits resonance at the natural frequency ω₀ = 1/√(LC). At resonance, the impedance is minimum (Z = R) and the current is maximum. The quality factor Q = ω₀L/R measures the sharpness of resonance; higher Q means narrower bandwidth and more selective frequency response. This principle is used in radio tuners, filters, and communication systems to select specific frequencies.
Energy Transfer
Energy in an RLC circuit continuously converts between electric potential energy in the capacitor (UE = q²/2C) stored in the electric field between its plates, and magnetic energy in the inductor (UB = LI²/2) stored in the magnetic field around its coils. The resistor dissipates this energy as heat at rate P = I²R. In the absence of resistance (LC circuit), total energy remains constant and oscillation continues forever. With resistance, the total energy decays exponentially as E(t) = E₀·e^(-2γt), eventually all energy is lost as heat.
Phase Portrait
The phase portrait plots charge (q) on the x-axis versus current (I) on the y-axis. For an undamped LC circuit, this creates a closed ellipse representing constant energy. With resistance, the trajectory spirals inward toward the origin as energy is dissipated, with each loop representing one oscillation cycle. The spiral's tightness depends on the damping ratio. This visualization reveals important system dynamics and stability properties that are not apparent from waveform plots alone.
Applications
RLC circuits have countless practical applications: tuning circuits in radio and TV receivers to select specific frequencies; filters in audio systems and signal processing; oscillators and clock generators in computers and communication devices; voltage regulation in power supplies; impedance matching networks; induction heating and wireless power transfer; damping systems to suppress unwanted vibrations; sensors and measurement devices; and as fundamental building blocks for understanding more complex electrical networks and control systems.