Series RLC Resonance: Frequency, Q, and Bandwidth Worked Example
Calculate resonant frequency, quality factor, impedance at resonance, and bandwidth for a practical series RLC circuit.
Why this calculation matters
At series resonance, an ideal inductor and capacitor have equal and opposite reactance, leaving the resistor to dominate the net impedance. The resonance frequency is easy to calculate, but real components add winding resistance, ESR, tolerance, and parasitics.
This example uses component values that produce a moderate Q so the relationship between resonance and bandwidth is easy to see.
What you will calculate
- Calculate ideal series resonant frequency.
- Calculate Q and the approximate 3 dB bandwidth.
- Understand why current is maximized at series resonance.
- Recognize how component losses and tolerances shift the measured response.
Given values
- Series resistance R = 10 Ω
- Inductance L = 50 mH
- Capacitance C = 10 µF
- Ideal L and C for the first-pass calculation
Governing equations
Resonant angular frequency
ω0 = 1/√(LC)At this frequency XL = XC in the ideal series circuit.
Resonant frequency
f0 = 1/(2π√(LC))Frequency in hertz.
Series-circuit Q
Q = ω0L/REquivalent to 1/(ω0CR) for the ideal series case.
Approximate bandwidth
BW = f0/Q = R/(2πL)Difference between the two half-power frequencies for the ideal series model.
Worked solution
1. Calculate resonance
The LC product is 5×10⁻⁷. Taking the reciprocal square root gives about 1414 rad/s, corresponding to 225.1 Hz.
ω0 ≈ 1414.2 rad/s; f0 ≈ 225.1 Hz2. Calculate Q
At resonance the inductor reactance magnitude is about 70.7 Ω. Dividing by the 10 Ω series resistance gives Q ≈ 7.07.
Q = 1414.2×0.05/10 ≈ 7.073. Calculate bandwidth
The ideal 3 dB bandwidth is approximately 31.8 Hz. A higher series resistance would reduce Q and broaden the response.
BW ≈ 225.1/7.07 ≈ 31.8 Hz4. Interpret the actual circuit
At ideal series resonance the reactive terms cancel and the impedance magnitude is approximately R, so current is maximized for a fixed source voltage. In practice, inductor winding resistance and capacitor ESR add to R, reducing Q and changing the measured bandwidth.
Engineering interpretation
For R = 10 Ω, L = 50 mH, and C = 10 µF, the ideal series resonance is about 225.1 Hz, Q is about 7.07, and bandwidth is about 31.8 Hz.
A bench measurement should use the actual component values and include inductor DCR, capacitor ESR, source resistance, and measurement loading.
Sanity checks
- Increasing L or C should lower resonant frequency.
- Increasing series R should reduce Q and increase bandwidth.
- At the ideal resonance frequency, |XL| and |XC| should match.
- A measured resonance shifted from the nominal value can be explained by component tolerance and parasitic elements.
Common mistakes
- Using millihenries or microfarads without converting prefixes.
- Confusing series-resonant minimum impedance with parallel-resonant behavior.
- Ignoring source impedance in a low-resistance circuit.
- Assuming ideal Q when inductor DCR dominates the intended resistor.
References and model boundaries
- Standard series-RLC relationships from introductory circuit-analysis references.
- Measured response depends on component tolerance, parasitic resistance/inductance/capacitance, source impedance, and instrument loading.
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