The Core Formula for Resonant Frequency of RLC Circuits

The exact formula for the resonant frequency of an ideal RLC circuit is fr = 1 / (2π√(LC)). At this specific frequency, the inductive reactance (XL) and capacitive reactance (XC) are equal in magnitude but exactly 180 degrees out of phase, effectively canceling each other out. The circuit then behaves as a purely resistive load.

For this guide, we are analyzing the Series RLC Bandpass Topology. The signal path flows through three distinct nodes:

  • Vin (Input): The AC source or function generator output.
  • Node A (R-L Junction): The connection between the current-limiting resistor and the inductor.
  • Node B (L-C Junction / Vout): The connection between the inductor and capacitor. This is your output tap for a bandpass filter.
  • GND (Return): The ground side of the capacitor, completing the loop back to the source.
Expert Insight: The Parallel RLC Caveat
The formula fr = 1 / (2π√(LC)) is exact for series circuits and parallel circuits where the resistance is purely in series with the inductor (representing the coil's DC winding resistance). If you are designing a pure parallel RLC tank with a resistor in parallel across the L and C branches, the resonant frequency actually shifts slightly based on the resistance value. For 95% of hobbyist and bench filter designs, the series-derived formula is the standard starting point.

Component Behavior and Parameter Shifts

When tuning a filter, you rarely change all three components at once. Here is exactly how the circuit's behavior shifts when you isolate and alter a single variable. This table assumes a baseline series RLC bandpass configuration.

Parameter Changed Effect on Resonant Freq (fr) Effect on Q-Factor Effect on Bandwidth (-3dB) Practical Consequence
Increase R No change Decreases Widens Filter becomes less selective; peak amplitude drops.
Decrease R No change Increases Narrows Sharper peak, but risks high circulating currents and ringing.
Increase L Decreases Increases Narrows Shifts passband lower; increases physical footprint and DCR losses.
Increase C Decreases Decreases Widens Shifts passband lower; lowers the output impedance at resonance.

Why Choose Series Over Parallel Topology?

Use a Series RLC when you need a bandpass filter that passes a specific frequency to a low-impedance load, or when you want to block DC while passing an AC carrier. The impedance at resonance is at its minimum (equal to R).

Use a Parallel RLC (tank circuit) when you need a band-stop (notch) filter, or an oscillator tank where impedance at resonance is at its maximum. Parallel tanks are heavily used in RF front-ends and SMPS snubbers, but they are much more sensitive to load impedance variations than series configurations.

Design Walkthrough: Building a 10.7 kHz Bandpass Filter

Let’s move from abstract math to the workbench. We will design a series RLC bandpass filter targeting fr ≈ 10.7 kHz with a moderate Q-factor of ~4.5 to allow a clean audio-range passband without excessive ringing.

  1. Pick the Capacitor (C): Inductors are bulkier and have worse tolerances than capacitors. Start by selecting a standard capacitor value. Let's choose 22 nF. Use a polypropylene film capacitor (e.g., WIMA MKP10 series) rather than a Y5V ceramic, which suffers from severe capacitance drop under DC bias and voltage coefficients.
  2. Calculate the Inductor (L): Rearranging the resonant frequency formula to solve for L:
    L = 1 / ((2π × fr)² × C)
    L = 1 / ((2π × 10734)² × 22e-9) = 0.010 H (10 mH).
    Select a standard off-the-shelf part: the Bourns 78F103K-RC (10 mH axial inductor, ±10% tolerance).
  3. Calculate the Resistor (R) for Target Q: The Q-factor formula for a series circuit is Q = (1/R) × √(L/C).
    Rearranging for R: R = √(L/C) / Q.
    R = √(0.01 / 22e-9) / 4.5 = 674.2 / 4.5 = 149.8 Ω.
    Select a standard 150 Ω 1/4W metal film resistor (e.g., Vishay MRS25 series).
Hidden Resistance Gotcha:
Real inductors have DC Resistance (DCR). The Bourns 10 mH inductor has a DCR of roughly 14 Ω. Furthermore, standard bench function generators have a 50 Ω output impedance. Your total circuit resistance is actually R_physical + DCR + Z_source. If you want a true 150 Ω total resistance, your physical resistor should be 150 - 14 - 50 = 86 Ω (use an 82 Ω or 91 Ω standard value). If you ignore this, your Q-factor will be lower, and your bandwidth wider than calculated.

Failure Modes: What Breaks at the Extremes?

Understanding how the circuit fails is critical for debugging and protection design. Here is the failure-mode contrast for the series RLC topology:

  • Capacitor Shorts (C = 0 Ω): Node B is pulled directly to GND. At DC, Vout is 0V. At AC, the inductor and resistor now form a simple RL low-pass filter, but the massive current spike at turn-on can blow the function generator's output stage or overheat the physical resistor if it lacks adequate wattage headroom.
  • Capacitor Opens (C = ∞ Ω): The circuit is broken. No current flows. Vout reads 0V. The resonance is entirely destroyed.
  • Inductor Shorts (L = 0 Ω): The circuit becomes a simple RC high-pass filter. Resonance is lost, and the -3dB cutoff shifts dramatically upward based purely on R and C.
  • Inductor Opens (L = ∞ Ω): The circuit is broken. Vout reads 0V. (This is the most common real-world failure if the inductor is subjected to a current spike exceeding its saturation/current rating, melting the internal winding).
  • Resistor Opens (R = ∞ Ω): No current flows. Vout = 0V. The circuit is safely dead.

Step-by-Step Breadboard Testing Protocol

Do not trust the math until you verify it on the bench. Parasitic breadboard capacitance (usually 2-5 pF per node) won't affect a 10 kHz circuit, but poor connections will ruin your Q-factor.

  1. Prep the Board: Insert the 150 Ω metal film resistor, 10 mH Bourns inductor, and 22 nF WIMA film capacitor in series across the breadboard rails. Ensure the inductor leads are stripped cleanly and tinned if oxidized; poor contact resistance will artificially lower your Q.
  2. Connect the Source: Connect the function generator (e.g., Siglent SDG1032X) BNC-to-alligator clip to Vin and GND. Set the output to a 2 Vpp sine wave, 50 Ω output impedance enabled, starting at 1 kHz.
  3. Probe the Output: Connect Oscilloscope Channel 1 to Vin (for triggering and reference) and Channel 2 to Node B (Vout). Set both channels to 1X or 10X probe attenuation (ensure the scope menu matches the physical probe switch). Set the timebase to 50 µs/div.
  4. Sweep for Resonance: Slowly increase the function generator frequency from 1 kHz up to 20 kHz. Watch the amplitude of Channel 2. You will see the Vout sine wave grow in amplitude, peak, and then shrink.
  5. Find the Exact Peak: Use the scope's cursor measurement tools. Place Cursor A on the peak of Channel 1 and Cursor B on the peak of Channel 2. Adjust the frequency in 10 Hz increments until the phase difference between Ch1 and Ch2 is exactly 0 degrees (the waveforms perfectly overlap in time). This zero-phase crossing is your true fr.
  6. Measure Bandwidth: Note the peak Vout voltage (e.g., 1.5 Vpp). Multiply by 0.707 (the -3dB point). Sweep the frequency down until Vout hits 1.06 Vpp (mark as flow). Sweep up until it hits 1.06 Vpp again (mark as fhigh). The difference (fhigh - flow) is your measured bandwidth. Divide your measured fr by this bandwidth to find your real-world Q-factor.

For deeper theoretical background on the phasor math and impedance triangles governing these behaviors, the All About Circuits AC theory chapter on series resonance provides excellent interactive context. Additionally, Georgia State University's HyperPhysics database remains a gold standard for verifying the underlying differential equations that dictate RLC transient ringing and steady-state AC response.