The fundamental RLC circuit formulas—resonant frequency, quality factor, and bandwidth—define the theoretical limits of your filter. But on the bench, parasitic resistance, dielectric absorption, and source impedance dictate reality. The core resonant frequency formula is fr = 1 / (2π√(LC)), but picking the right topology (series vs. parallel) and selecting components with appropriate parasitics determines if your filter actually works in hardware or just in SPICE.
This guide moves past abstract theory into practical circuit configuration, mapping real component values, node topologies, and failure modes for both series and parallel RLC designs.
Series vs. Parallel RLC Topologies and Node Mapping
Choosing between a series and parallel RLC topology depends entirely on your source impedance and the filter response you need.
Series RLC (Bandpass / Lowpass)
A series RLC circuit is best driven by a low-impedance voltage source. The components share the same current, and the voltage divides among them based on their frequency-dependent impedance.
- Node A (Vin): AC signal input.
- Node B: Junction between Capacitor (C1) and Inductor (L1).
- Node C: Junction between Inductor (L1) and Resistor (R1).
- Node D (GND): Ground reference.
Configuration: Vin → C1 → Node B → L1 → Node C → R1 → GND. Taking Vout across R1 (Node C to GND) yields a bandpass response. Taking Vout across C1 yields a lowpass response.
Parallel RLC (Tank / Notch)
A parallel RLC (tank) circuit requires a high-impedance current source (or a voltage source with a large series resistor). The components share the same voltage, and the current divides among them.
- Node A (Vin): AC signal input (typically fed through a high-value series resistor, Rs).
- Node B (Tank Node): Junction where L1, C1, and the load all connect in parallel.
- Node C (GND): Ground reference.
Configuration: Vin → Rs → Node B. L1 and C1 are connected between Node B and GND. Taking Vout at Node B yields a bandpass response. Inserting the parallel tank into the ground leg of a voltage divider yields a notch (bandstop) response.
Core RLC Circuit Formulas and Behavior Matrix
Before picking parts, you need to understand how changing one variable shifts the entire response curve. The governing equations for a series RLC circuit are:
- Resonant Frequency: fr = 1 / (2π√(LC))
- Quality Factor (Q): Q = (1/R) * √(L/C)
- Bandwidth (BW): BW = fr / Q
- Impedance at Resonance: Z = R (The reactive components cancel out, leaving only the resistive element).
For a comprehensive look at the mathematical derivations of these parameters, refer to the Electronics Tutorials guide on series resonance.
The table below maps exactly what happens to your circuit's behavior when you alter a single component value.
| Component Changed | Direction | Effect on fr | Effect on Q (Series) | Effect on Bandwidth | Real-World Consequence |
|---|---|---|---|---|---|
| Inductor (L) | Increase | Decreases | Increases | Narrows | Filter peaks lower; insertion loss may rise due to higher inductor DCR. |
| Capacitor (C) | Decrease | Increases | Increases | Narrows | Filter peaks higher; parasitic PCB capacitance becomes a larger error factor. |
| Resistor (R) | Increase | No Change | Decreases | Widens | Filter becomes 'flatter' and broader; peak output voltage drops significantly. |
| Source Impedance | Increase | No Change | Decreases | Widens | Invisible Q-killer; a 50Ω generator acts as extra series resistance. |
Design Walkthrough: Building a 15.9 kHz Series Bandpass Filter
Let's design a series bandpass filter targeting a resonant frequency of ~15.9 kHz with a Quality Factor (Q) of 5. We will use the formulas to select real, off-the-shelf components.
1. Selecting L and C for Resonance
We need fr ≈ 15,915 Hz. Let's pick a standard inductor value first, as inductors have fewer available values and higher parasitics than capacitors.
- Choose L: 10 mH. We will use the Bourns 78F103K-RC radial inductor. It has a DC Resistance (DCR) of 0.65Ω, which is low enough not to ruin our Q.
- Calculate C: C = 1 / ((2π * 15915)2 * 0.01) = 10 nF.
2. Selecting R for Target Q
We want Q = 5. Using the series Q formula: Q = (1/R) * √(L/C).
- 5 = (1/R) * √(0.01 / 10×10-9)
- 5 = (1/R) * √(1,000,000)
- 5 = 1000 / R → R = 200Ω
We will use a 200Ω 1% metal film resistor (e.g., Vishay MRS25 series). Note that the inductor's internal DCR (0.65Ω) adds to this, making the true R = 200.65Ω, which drops our actual Q to 4.98—a negligible error.
Component Spec Sheet
| Component | Value | Part Number / Spec | Role in Circuit |
|---|---|---|---|
| L1 | 10 mH | Bourns 78F103K-RC (DCR 0.65Ω) | Stores magnetic energy, sets fr with C1. |
| C1 | 10 nF | Kemet C0G/NP0 50V Ceramic | Stores electric energy, blocks DC offset. |
| R1 | 200 Ω | Vishay MRS25 1% Metal Film | Sets Q factor, provides Vout reference. |
Failure Modes: What Breaks at the Extremes
Understanding how series and parallel topologies fail when a component shorts or opens is crucial for troubleshooting and designing fail-safe circuits.
Series RLC Failure Contrast
- C1 Shorts: The DC blocking capability is lost. The circuit becomes an RL lowpass filter. High-frequency roll-off changes from -40dB/decade to -20dB/decade.
- L1 Shorts: The circuit becomes an RC highpass filter. Resonance is destroyed.
- R1 Opens: The circuit is dead. No current flows, Vout is 0V regardless of frequency.
- L1 Opens: Same as R1 opening. The series path is broken.
Parallel RLC (Tank) Failure Contrast
- C1 Shorts: Catastrophic failure. The tank node is shorted directly to ground. If driven by a low-impedance source without a series current-limiting resistor, the source will likely be damaged.
- L1 Opens: The tank loses its resonance. The circuit acts as a simple RC lowpass filter (if Rs and C1 remain).
- L1 Shorts: The tank node is shorted to ground through the inductor's DCR. Similar to C1 shorting, this will draw massive DC/low-frequency current from the source.
The fundamental difference is that an open component in a series circuit safely kills the signal, while a shorted component in a parallel tank circuit creates a dead short to ground, risking source damage.
Step-by-Step Breadboard Testing and Verification
Simulating an RLC circuit is easy; measuring it on a breadboard introduces parasitics that can completely mask your design. Follow this procedure to verify the 15.9 kHz bandpass filter.
- Layout for Minimal Parasitics: Place C1, L1, and R1 in a tight, continuous line on the breadboard. Do not use long jumper wires between components. The stray capacitance of breadboard rows (typically 2pF to 5pF per row) can detune a high-frequency tank, though at 15.9 kHz with 10nF, it will only shift fr by a few Hertz. Keep the ground return path for R1 as short as possible.
- Account for Source Impedance: Most function generators have a 50Ω output impedance. If you set the generator to output 1Vpp into a 'High-Z' load, it actually outputs 2Vpp open-circuit, dropping to 1Vpp when it sees a 50Ω load. Because our filter's input impedance at resonance is 200Ω, the generator's 50Ω impedance forms a voltage divider. Fix: Measure the actual voltage at Node A (Vin) with the oscilloscope, not at the generator's display.
- Probe Compensation and Grounding: Use a 10x oscilloscope probe. Before connecting to the circuit, attach the probe to the scope's calibration square wave and adjust the compensation trimmer until the square wave edges are perfectly flat. Use the short ground spring attachment rather than the long alligator ground lead; the long lead forms a loop antenna that will pick up switching noise and ring at high frequencies.
- Sweep and Measure the -3dB Points: Set the function generator to a 1Vpp sine wave. Sweep the frequency from 5 kHz to 30 kHz. Note the peak voltage at Vout (Node C). It should peak near 15.9 kHz. Calculate the -3dB voltage (Peak Voltage * 0.707). Find the lower and upper frequencies where Vout drops to this -3dB level. The difference between these two frequencies is your measured Bandwidth. It should be approximately 3.18 kHz (15,915 Hz / 5). For deeper theory on Q factor measurements, consult the MIT OpenCourseWare Circuits and Electronics materials.
By respecting the physical realities of component parasitics, source impedance, and dielectric behavior, you can bridge the gap between textbook RLC formulas and a working, predictable filter on your bench.
