Figure 1: The damping ratio ζ determines how the circuit responds to a step input. ζ < 1 oscillates, ζ = 1 is the fastest settling without overshoot, ζ > 1 is slow and non-oscillating.
Table of Contents
What Is an RLC Circuit?
An RLC circuit contains three fundamental passive components: a resistor (R), an inductor (L) and a capacitor (C). The interaction between the inductor’s magnetic energy storage and the capacitor’s electric energy storage creates oscillatory behaviour, while the resistor dissipates energy and controls how quickly oscillations decay.
RLC circuits are the basis for filters, oscillators, tuning circuits, impedance matching networks and power supply smoothing. The Impedance Calculator computes the impedance at a specific frequency; this calculator focuses on the circuit’s natural behaviour: resonant frequency, Q factor, damping and transient response.
Series vs Parallel
In a series RLC circuit, current is common to all three components. At resonance, impedance drops to a minimum (just R) and current peaks. This makes series circuits ideal for bandpass filters where you want maximum signal throughput at a specific frequency.
In a parallel RLC circuit, voltage is common. At resonance, impedance rises to a maximum, effectively blocking current at the resonant frequency. Parallel (tank) circuits are used in oscillators and frequency-selective amplifier loads.
Worked Example — Underdamped Radio Tuner
f₀ = 1/(2π√(250×10−6 × 100×10−12)) = 1.007 MHz
Q = XL/R = 1581/10 = 158
ζ = 1/(2Q) = 0.00316 — heavily underdamped, rings for many cycles
BW = 1.007M/158 = 6.37 kHz
The high Q creates a very narrow bandwidth — perfect for selecting a single AM station. The Resonant Frequency Calculator provides a dedicated tool for tuning circuit design.
Worked Example — Critically Damped Filter
f₀ = 711.8 Hz
Q = 0.5, ζ = 1.0 — critically damped
A critically damped circuit settles to its final value as fast as possible without overshooting. This is the preferred response for measurement instruments and control systems where ringing would cause false readings.
Worked Example — Overdamped Power Supply
f₀ = 711.8 Hz (same L and C)
Q = 0.158, ζ = 3.16 — overdamped, no oscillation
The high resistance heavily damps any oscillation. The circuit responds sluggishly to changes but never rings. This is acceptable in power supply filtering where stability matters more than speed.
Damping Ratio and Transient Response
The damping ratio ζ = 1/(2Q) classifies the transient behaviour. When ζ < 1 (Q > 0.5) the circuit is underdamped — it oscillates with exponentially decaying amplitude. When ζ = 1 (Q = 0.5) it is critically damped — the fastest response without any overshoot. When ζ > 1 (Q < 0.5) it is overdamped — it settles slowly without oscillating.
The damped natural frequency is ωd = ω₀√(1 − ζ²), which is slightly lower than the undamped ω₀. The settling time (to within 2% of final value) is approximately τ = 4/(ζω₀) for underdamped systems.
Q Factor and Bandwidth
Q factor determines both the sharpness of the frequency response and the transient ringing. High Q means narrow bandwidth (BW = f₀/Q) and many cycles of ringing. Low Q means wide bandwidth and quick settling. For filter design, the choice of Q is a direct trade-off between selectivity and response speed.
The Phase Angle Calculator shows how the phase shifts rapidly near resonance in high-Q circuits, crossing through 0° at f₀.
Applications
RLC circuits appear in radio receivers (tuning), audio equipment (crossovers and equalisers), power supplies (filtering), telecommunications (impedance matching), and control systems (feedback loop compensation). The Power Factor Calculator uses RLC principles when sizing correction capacitors for inductive loads.
Frequently Asked Questions
What is the difference between this and the Impedance Calculator?
Can I use this for a circuit with only R and L or R and C?
How do I choose between underdamped and critically damped?
Does the resonant frequency depend on R?
What is a tank circuit?
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