159.1549
Hz
99.9969
Ω
1
Ω
99.5007
Ω
1
15.9155
Hz
159.1549
Hz
99.9969
Ω
1
Ω
99.5007
Ω
1
15.9155
Hz
The RLC circuit — containing a resistor (R), inductor (L), and capacitor (C) — is one of the most important and widely studied circuit topologies in electrical engineering. Its behavior exhibits resonance, frequency-dependent impedance, and energy oscillation between magnetic and electric fields, making it the foundation of filters, oscillators, tuned amplifiers, impedance matching networks, and wireless communication systems.
At the resonant frequency f₀ = 1/(2π√(LC)), the inductive reactance XL equals the capacitive reactance XC. In a series RLC circuit, these opposing reactances cancel, and the total impedance drops to its minimum value of R — the circuit becomes purely resistive. Maximum current flows, and the voltage across the inductor and capacitor can individually be much larger than the source voltage (by a factor of Q, the quality factor). In a parallel RLC circuit, the resonant impedance reaches its maximum, and the circuit presents maximum opposition to current flow at f₀.
The quality factor Q = (1/R)√(L/C) = f₀/BW quantifies how 'sharp' or selective the resonance is. A high-Q circuit has a narrow bandwidth and strong frequency selectivity — useful in radio tuners, crystal filters, and precision oscillators. A low-Q circuit has broad bandwidth and is useful for broadband filters, impedance matching, and snubber designs. Bandwidth BW = R/(2πL) = f₀/Q defines the frequency range around resonance where the response is within 3 dB of the peak.
RLC circuits appear in virtually every electronic system: LC tank circuits in radio frequency oscillators maintain oscillation at f₀ through resonant energy exchange; bandpass and band-reject (notch) filters select or reject specific frequency bands; impedance matching networks in RF power amplifiers transform source impedance for maximum power transfer; and resonant converters in switching power supplies achieve high efficiency through soft switching (ZVS/ZCS) at the resonant frequency.
In power systems, RLC resonance is a concern for harmonic distortion. Variable frequency drives and other non-linear loads inject harmonic currents that can excite resonance in power factor correction capacitor banks combined with transformer leakage inductance, causing destructive overvoltage. Power quality engineers must assess resonant frequencies of distribution systems when installing capacitor banks.
The damping factor α = R/(2L) relates to how quickly oscillations decay in a transient response. When α > ω₀ (overdamped): exponential decay without oscillation. When α = ω₀ (critically damped): fastest exponential decay to steady state. When α < ω₀ (underdamped): oscillatory ringing before settling. This behavior determines the transient response of filters, amplifiers, and power converter output stages.
Resonant frequency f₀ = 1/(2π√(LC)). Inductive reactance XL = 2πfL. Capacitive reactance XC = 1/(2πfC). Impedance Z = √(R² + (XL - XC)²). Quality factor Q = (1/R)√(L/C). Bandwidth BW = R/(2πL) = f₀/Q. All at the specified operating frequency f, with resonant frequency f₀ computed independently.
When XL = XC (at f = f₀), Z = R (minimum for series RLC). Q > 10: high selectivity (sharp resonance, narrow filter bandwidth). Q = 0.5–2: broadband response (overdamped). At f < f₀: circuit is capacitive (XC > XL). At f > f₀: circuit is inductive (XL > XC). Bandwidth BW tells you the -3 dB frequency range around resonance.
Inputs
Results
An AM radio tank circuit with 250 µH (250 mH in µH = use 0.25 mH) — use actual values: 250 µH = 0.25 mH. Q ≈ 393 gives bandwidth of ~1.3 kHz, suitable for selecting AM stations 10 kHz apart.
Inputs
Results
A series RLC notch filter tuned to 1 kHz (60 Hz hum harmonic) with Q ≈ 6.3 and BW ≈ 159 Hz. At resonance, impedance drops to 100 Ω minimum, providing maximum signal rejection at exactly 1 kHz.
Series RLC: at resonance, impedance is minimum (= R), current is maximum. Used as a bandpass filter when output is taken across R, or as a notch filter with other configurations. Parallel RLC: at resonance, impedance is maximum (= L/RC = Q²R for high-Q circuits), current from source is minimum. Used in parallel-resonant tank circuits and oscillators.
Below f₀: XC > XL, the circuit is net capacitive — current leads voltage. Above f₀: XL > XC, the circuit is net inductive — current lags voltage. At f₀: XL = XC, net reactance = 0, the circuit is purely resistive with zero phase shift. Impedance is minimum (series) or maximum (parallel) exactly at f₀.
At series resonance, V_L = I × XL = (V/R) × XL = V × Q. Similarly V_C = V × Q. For Q = 10, voltages across L and C are 10× the source voltage! They are equal and opposite, so they cancel in KVL, but individual components must be rated for Q × V_source to avoid insulation breakdown.
When a step voltage is applied to a series RLC circuit, current oscillates at the damped resonant frequency ωd = √(ω₀² - α²), where α = R/2L. The oscillation decays with time constant τ = 2L/R. Underdamped circuits ring; overdamped circuits exponentially settle without oscillation. This behavior is critical in filter design, power converter control, and transmission line pulse response.
A tank circuit is a parallel LC circuit (often with small R representing losses) that stores oscillating energy, alternating between magnetic energy in L and electric energy in C. Oscillators use positive feedback to replenish losses and sustain oscillation at f₀. The word 'tank' evokes the reservoir of oscillating energy — high-Q tanks 'hold' energy for many oscillation cycles.
Wireless power transfer (WPT) systems use resonant coupling between transmitter and receiver RLC circuits both tuned to the same resonant frequency. At resonance, power transfer efficiency peaks. The Qi wireless charging standard (5–15 W) and high-power EV charging systems (3.3–11 kW) both use resonant RLC coupling for efficient inductive power transfer over small air gaps.
Quality factor Q = ω₀/(2α) = f₀/BW — higher Q means less damping and sharper resonance. Damping ratio ζ = 1/(2Q) = α/ω₀. ζ > 1: overdamped, ζ = 1: critically damped, ζ < 1: underdamped. Control engineers prefer ζ (damping ratio) while RF engineers prefer Q — they convey the same information from different perspectives.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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