Series RLC — impedance is minimum at resonance
Figure 1: At resonance Xₗ = Xᴄ, so the reactive components cancel. Series impedance drops to R; parallel impedance rises to maximum.
Enter the target f₀ and one known component — get the other.
Table of Contents
- Series vs Parallel Resonance
- Q Factor and Selectivity
- Damping and Transient Response
- Practical Considerations
What Is Resonant Frequency?
Resonant frequency is the natural oscillation frequency of an LC circuit — the frequency at which energy swings back and forth between the inductor’s magnetic field and the capacitor’s electric field with minimum loss. At resonance, inductive reactance equals capacitive reactance, and the two cancel each other out.
Every circuit containing both inductance and capacitance has a resonant frequency. In a series circuit, impedance drops to a minimum at resonance. In a parallel circuit, impedance rises to a maximum. This behaviour is the foundation of radio tuning, filters, oscillators, and impedance matching. The Impedance Calculator shows the full impedance picture; this calculator focuses on the resonance point itself.
The Resonance Formula
Angular form: ω₀ = 1 / √(LC)
At resonance: XL = XC = ω₀L = 1/(ω₀C)
Q factor (series): Q = XL/R = (1/R)√(L/C)
Bandwidth: BW = f₀/Q
The resonant frequency depends only on L and C — resistance affects the sharpness (Q) and bandwidth but not the frequency itself.
Worked Example — AM Radio Tuning Circuit
Step 1 — f₀ = 1/(2π√(250×10−6 × 100×10−12)) = 1.007 MHz
Step 2 — XL at f₀ = 2π × 1.007×106 × 250×10−6 = 1581 Ω
Step 3 — Q = 1581/10 = 158.1
Step 4 — BW = 1.007×106 / 158.1 = 6.37 kHz
A Q of 158 gives a very narrow bandwidth of 6.4 kHz — perfect for selecting a single AM station (9 kHz channel spacing). Turning the variable capacitor changes C and sweeps f₀ across the AM band (530 kHz to 1700 kHz).
Worked Example — Audio Bandpass Filter
Step 1 — f₀ = 1/(2π√(0.0253 × 1×10−6)) = 1000 Hz
Step 2 — XL = 2π × 1000 × 0.0253 = 159 Ω
Step 3 — Q = 159/100 = 1.59
Step 4 — BW = 1000/1.59 = 629 Hz
With Q = 1.59, this is a broad filter passing frequencies from about 686 Hz to 1315 Hz. Audio crossover networks often use low-Q resonant circuits deliberately to get a smooth frequency response. For dedicated crossover design see the Crossover Calculator.
Worked Example — Power Line LC Trap
Step 1 — f₀ = 1/(2π√(0.1 × 101.3×10−6)) = 50.0 Hz
Step 2 — |Z| at 50 Hz = R = 5 Ω (minimum)
Step 3 — Q = 31.4/5 = 6.28, BW = 7.96 Hz
A series LC trap tuned to 50 Hz presents minimum impedance at the mains frequency, shunting harmonic currents to ground. This technique is used in power factor correction and harmonic filtering on industrial power lines.
Series vs Parallel Resonance
In a series resonant circuit, the inductor and capacitor are in the current path. At resonance their reactances cancel, impedance drops to R, and current peaks. This is used in bandpass filters and traps where you want maximum current flow at a specific frequency.
In a parallel (tank) resonant circuit, L and C form a loop. At resonance the circulating current inside the tank is very large, but the impedance seen from outside is very high. This is used in oscillators, amplifier loads, and frequency-selective networks where you want to reject a specific frequency. The RLC Circuit Calculator provides full transient analysis for both configurations.
Q Factor and Selectivity
Q factor (quality factor) measures how “sharp” the resonance is. A high-Q circuit has a narrow bandwidth and high selectivity — it strongly favours the resonant frequency over nearby frequencies. A low-Q circuit has a wide bandwidth and responds to a broader range.
For a series circuit: Q = XL/R = (1/R)√(L/C). For a parallel circuit: Q = R/XL = R√(C/L). In both cases, reducing resistance increases Q. The voltage or current at resonance is magnified by a factor of Q — a Q of 100 means the voltage across the capacitor is 100 times the source voltage, which can be dangerous in high-power circuits.
Damping and Transient Response
The damping ratio ζ = 1/(2Q) determines how quickly oscillations decay after a transient. When ζ < 1 (Q > 0.5) the circuit is underdamped and rings. When ζ = 1 (Q = 0.5) it is critically damped. When ζ > 1 (Q < 0.5) it is overdamped and does not oscillate at all.
For filter design, critically damped (Butterworth) or slightly underdamped responses are often preferred for a flat passband. For oscillator design, very light damping (high Q) is needed so the circuit sustains oscillation with minimal energy input.
Practical Considerations
Real inductors have winding resistance, core loss, and parasitic capacitance. Real capacitors have ESR and ESL. These parasitics limit the achievable Q and can shift the actual resonant frequency from the calculated ideal. At high frequencies, the inductor’s self-resonance (where its own parasitic capacitance resonates with its inductance) sets an upper frequency limit.
The Inductive Reactance Calculator and Capacitive Reactance Calculator can help you understand how each component contributes to the resonance.
Frequently Asked Questions
Does resistance affect resonant frequency?
Can I have resonance without an inductor?
What is a tank circuit?
Why does my circuit not resonate at the calculated frequency?
How do I design for a specific bandwidth?
What is the difference between resonant frequency and natural frequency?
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