RLC Resonance Calculator

How RLC resonance is calculated

Resonance occurs in an RLC circuit at the frequency where inductive reactance and capacitive reactance are equal in magnitude, canceling each other's effect on impedance. This calculator finds that frequency, the circuit's characteristic impedance, and the quality factor — which depends on topology, since series and parallel RLC circuits respond to resistance in opposite ways (higher R lowers Q in a series circuit but raises Q in a parallel circuit).

From the quality factor, bandwidth and an approximate pair of half-power frequencies are derived, along with a damping classification describing whether the circuit would ring (oscillate) or settle smoothly if disturbed. If a source voltage is supplied, the calculator also estimates voltage or current magnification at resonance — a key safety consideration for resonant circuits.

Formula steps

Safety guidance: voltage and current magnification

Resonant circuits have a non-obvious safety implication: components can experience stress far beyond what the source voltage or current alone would suggest. In a series RLC circuit at resonance, the voltage across the inductor and the voltage across the capacitor are each roughly Q times the source voltage — a modest 12V source into a circuit with Q = 10 can put over 100V across the capacitor and inductor individually, even though they're 180° apart and cancel in the total loop voltage.

In a parallel RLC circuit at resonance, the equivalent effect appears as circulating current between the inductor and capacitor that can be roughly Q times the current drawn from the source. Component voltage and current ratings — including capacitor working voltage and inductor RMS/saturation current — must account for this magnification, not just the nominal source values, especially in higher-Q filter, tank, and resonant power circuits.

Code cautions

Unintended resonance between power factor correction capacitor banks and system inductance (transformers, cables, reactors) is a recognized power-system hazard, particularly when harmonic-producing loads such as variable frequency drives are present. Engineering guidance in this area — including harmonic limits addressed by IEEE 519 and code provisions for capacitor bank protection (such as NFPA 70/NEC Article 460) — generally calls for a harmonic/resonance study by a qualified engineer before adding or modifying capacitor banks on a system with significant non-linear loads. This calculator provides idealized single-frequency resonance math for design and educational use; it is not a substitute for a power system harmonic study or protective device coordination.

Design limitations of this calculator

• Ideal lumped components: the inductor's DC winding resistance and the capacitor's ESR are not separately modeled — the resistance you enter is treated as the only resistive element in the loop.
• Single-frequency, linear analysis: harmonic content, nonlinear loads, and time-varying component values are not modeled — this calculator finds one idealized resonant response, not a full frequency sweep.
• Half-power bandwidth approximation: the f0 ± BW/2 estimate for the −3dB frequencies is most accurate for higher-Q circuits (commonly Q ≥ 5) and becomes less precise for low-Q, heavily damped circuits.
• No parasitic coupling: stray capacitance, lead inductance, and electromagnetic coupling to nearby components/conductors are not modeled.
• Simple series or parallel topology only: more complex resonant networks with multiple L/C branches are not covered.

Frequently asked questions

• How do you calculate the resonant frequency of an RLC circuit?
  Resonant frequency is f0 = 1 ÷ (2π × √(L × C)), where L is inductance in henries and C is capacitance in farads. At this frequency, the inductive and capacitive reactances are equal in magnitude and cancel each other out, leaving a circuit that behaves as if it were purely resistive.
• What is the quality factor (Q) of an RLC circuit?
  The quality factor describes how "sharp" or narrow the circuit's resonance is relative to its energy losses. For a series RLC circuit, Q = (1 ÷ R) × √(L ÷ C). For a parallel RLC circuit, Q = R × √(C ÷ L). Higher Q means a narrower resonance peak, lower losses relative to stored energy, and — importantly — greater voltage or current magnification at resonance.
• What is bandwidth in a resonant RLC circuit?
  Bandwidth is the range of frequencies around resonance where the circuit response stays within about 70.7% (−3dB) of its peak value: BW = f0 ÷ Q. A high-Q circuit has a narrow bandwidth (sharp, selective resonance); a low-Q circuit has a wide bandwidth (broad, less selective resonance).
• What is the difference between series and parallel RLC resonance?
  In a series RLC circuit, impedance is minimum at resonance (equal to R), so current is maximized. In a parallel RLC circuit, impedance is maximum at resonance (ideally equal to R), so current drawn from the source is minimized while circulating current between L and C internally can be large. The two configurations also use different formulas for quality factor.
• What is damping in an RLC circuit?
  Damping describes how oscillations in an RLC circuit die out over time. A series RLC circuit is underdamped (oscillates) when R is less than 2×√(L ÷ C), critically damped when R equals that value (fastest settling without oscillation), and overdamped (settles slowly without oscillating) when R is greater than that value.
• Why can voltage across a capacitor or inductor exceed the source voltage at resonance?
  In a series RLC circuit at resonance, the voltage across the inductor and the voltage across the capacitor are each approximately Q times the source voltage, since the reactive voltages are equal and opposite but individually large — they cancel in the loop equation while each remaining large in magnitude across its own component. This voltage magnification can significantly exceed the supply voltage and is a common cause of component overstress if not accounted for in design.
• What causes resonance problems in power systems with capacitor banks?
  Power factor correction capacitor banks can form an unintended resonant circuit with the inductance of transformers, cables, and other system elements. If this resonance occurs near a harmonic frequency present in the system (often from non-linear loads like variable frequency drives), it can amplify harmonic voltages and currents well beyond normal levels, potentially damaging equipment. This is why utility and facility engineers evaluate harmonic resonance risk before adding capacitor banks.
• What are the half-power frequencies of a resonant circuit?
  The half-power (or −3dB) frequencies are the two frequencies, one below and one above resonance, where the circuit's power response drops to half its peak value. They are approximately f0 − BW/2 and f0 + BW/2 for circuits with a reasonably high Q (commonly Q ≥ 5); this approximation becomes less accurate for low-Q circuits.