RLC Series Circuit: Resonance, Q-Factor, Bandwidth

A series RLC circuit is the canonical second-order system in electronics — it shows up as the input stage of a radio receiver, the equivalent circuit of a crystal or ceramic resonator, the parasitic model of a via-decoupling-cap loop, and the textbook example for every filter and control concept that follows it. Understanding its resonance, Q-factor, and bandwidth is a prerequisite for reading any Bode plot, tuning any PLL loop filter, or predicting ringing on a PCB trace driven through a non-ideal source impedance.

Circuit and Impedance

Consider a resistor, inductor, and capacitor in series, driven by a sinusoidal source V(jω):

      R          L          C
o---[///]---[~~~~~~]---||---o
       series RLC, driven by V(jω)

The total impedance seen by the source is the sum of the three impedances (series elements add):

Z(jω) = R + jωL + 1/(jωC) = R + j(ωL − 1/(ωC))

The real part is fixed at R; the imaginary part (reactance) is X(ω) = ωL − 1/(ωC), which is negative (capacitive) at low frequency and positive (inductive) at high frequency. The magnitude and phase are:

|Z(ω)| = √(R² + X(ω)²)
θ(ω)   = atan(X(ω) / R)

Resonance

Resonance occurs at the frequency where the inductive and capacitive reactances cancel exactly: ωL = 1/(ωC). Solving:

ω₀ = 1/√(LC)          f₀ = 1/(2π√(LC))

At ω₀, X(ω₀) = 0, so:

• Z(ω₀) = R — impedance is purely resistive and at its minimum magnitude (a series RLC is a low-impedance network at resonance, unlike the parallel RLC "tank," which peaks in impedance there).
• The current I = V/Z is at its maximum, in phase with the source voltage (θ = 0°).
• The voltage across L and across C are individually large (often much larger than the source voltage) but equal in magnitude and 180° out of phase, so they cancel in the loop.

This last point is the practical hazard of series resonance: even a modest source voltage can produce large voltage swings across the inductor and capacitor individually, which matters for component voltage ratings in filter and matching-network design.

Q-Factor

The quality factor Q describes how underdamped the circuit is — equivalently, how much energy is stored relative to how much is dissipated per cycle:

Q = 2π × (energy stored) / (energy dissipated per cycle)

For a series RLC driven at ω₀, this reduces to the standard design formula:

Q = (1/R)·√(L/C) = ω₀L/R = 1/(ω₀RC)

All three forms are algebraically equivalent (substitute ω₀ = 1/√(LC) to verify). Q is dimensionless and, physically, equals the voltage magnification at resonance: |V_L(ω₀)| = |V_C(ω₀)| = Q·V_source. A high-Q series circuit used as a bandpass filter or antenna matching network can therefore see internal voltages many times the drive level — a real design constraint, not just a curiosity.

Q also sets the damping ratio ζ used in second-order transient analysis: ζ = 1/(2Q). Q > 0.5 (ζ < 1) corresponds to an underdamped, oscillatory/ringing step response; Q = 0.5 is critically damped; Q < 0.5 is overdamped.

Bandwidth

The bandwidth BW is the width of the frequency range over which the current (or power delivered to R) stays within −3 dB of its peak value at resonance — i.e., where |I| ≥ I_peak/√2, equivalently where power ≥ half the peak power. For a series RLC:

BW = ω₀/Q = R/L     (rad/s)
BW = f₀/Q           (Hz)

The two half-power (cutoff) frequencies ω₁ (below ω₀) and ω₂ (above ω₀) bracket this band and satisfy two useful identities:

ω₂ − ω₁ = BW = ω₀/Q
ω₁·ω₂  = ω₀²        (geometric mean of the two edges is ω₀)

This geometric-mean relationship is why resonance curves look symmetric on a log-frequency (Bode) axis even though ω₁ and ω₂ are not equidistant from ω₀ on a linear scale.

Worked Example

Design a series RLC with L = 10 mH and C = 100 nF, and find its resonant frequency, Q, and bandwidth for R = 10 Ω, then repeat for R = 100 Ω.

Step 1 — resonant frequency (independent of R):

ω₀ = 1/√(LC) = 1/√(10×10⁻³ × 100×10⁻⁹) = 1/√(1×10⁻⁹) = 1/(3.162×10⁻⁵) ≈ 31,623 rad/s
f₀ = ω₀/2π ≈ 5033 Hz

Step 2 — Q for R = 10 Ω:

Q = ω₀L/R = (31,623 × 0.01)/10 = 316.23/10 ≈ 31.6

Step 3 — bandwidth for R = 10 Ω:

BW = ω₀/Q = 31,623/31.6 ≈ 1000 rad/s   →   BW(Hz) = f₀/Q ≈ 5033/31.6 ≈ 159 Hz

Step 4 — check via R/L: BW(rad/s) = R/L = 10/0.01 = 1000 rad/s ✓ — matches Step 3 exactly, confirming the two bandwidth formulas agree.

Step 5 — half-power frequencies: using ω₂−ω₁ = 1000 and ω₁ω₂ = ω₀² = 31,623² ≈ 1.000×10⁹, solve the quadratic ω₁² + 1000ω₁ − 1.000×10⁹ = 0:

ω₁ = [−1000 + √(1000² + 4×1.000×10⁹)] / 2 = [−1000 + √(4.001×10⁹)] / 2
   ≈ [−1000 + 63,253.5] / 2 ≈ 31,127 rad/s
ω₂ = ω₁ + 1000 ≈ 32,127 rad/s

Check: ω₁·ω₂ ≈ 31,127 × 32,127 ≈ 1.000×10⁹ ≈ ω₀² ✓ — matches almost exactly, as expected.

Step 6 — repeat for R = 100 Ω: Q drops by 10× to ≈ 3.16, and bandwidth rises by 10× to ≈ 1590 Hz. This illustrates the core trade-off: lower R gives a sharper, higher-Q resonance with narrower bandwidth; higher R broadens the passband and damps the response.

Design Implications

• Component parasitics matter. Real inductors have winding resistance that adds directly to R, capping the achievable Q — a "high-Q" LC tank is often Q-limited by the inductor's ESR, not the intended series resistor.
• This same math governs unintended resonances. A decoupling capacitor's ESL/ESR in series with PCB trace/via inductance forms a series RLC; its resonant frequency defines the effective self-resonant frequency (SRF) of the "capacitor," above which it looks inductive.
• Q and bandwidth are reciprocal, so any spec on one implies the other once f₀ is fixed — there is no way to get both a sharp resonance and a wide passband from the same single series RLC section.
• Ringing on transmission lines and switching edges driven through source/load resistance and parasitic L-C often behaves exactly like this circuit's underdamped step response, with Q read directly off the overshoot.

Key Takeaways

• Series RLC impedance is Z = R + j(ωL − 1/(ωC)); resonance occurs where the reactance cancels, at ω₀ = 1/√(LC), where Z is purely resistive and minimum, and current is maximum.
• Quality factor Q = ω₀L/R = (1/R)√(L/C) = 1/(ω₀RC) measures energy storage vs. dissipation and equals the voltage magnification across L or C at resonance.
• Bandwidth BW = ω₀/Q = R/L (rad/s) defines the −3 dB (half-power) width of the resonance; the half-power frequencies satisfy ω₁ω₂ = ω₀².
• Q and BW trade off directly: lower R sharpens resonance and narrows bandwidth; higher R broadens bandwidth and damps ringing (ζ = 1/(2Q)).
• The same series-RLC math describes unintended resonances from real-world parasitics — decoupling capacitor SRF, trace/via inductance, and ringing on switched signal edges.