A series RLC circuit places a resistor, inductor, and capacitor in a single continuous current loop. At its resonant frequency, the inductive and capacitive reactances cancel out, leaving only the resistance to limit current. For low-impedance voltage sources (like a 50-ohm function generator or an op-amp output), the series RLC is the default topology over a parallel tank circuit because it provides minimum impedance at resonance, making it ideal for bandpass, lowpass, highpass, and notch filtering without requiring a high-impedance current drive.
The Series RLC Topology and Node Map
To design or troubleshoot effectively, you must define your nodes. In a standard series bandpass configuration, the components are arranged sequentially from the source to ground:
- Node 0 (Vin): AC voltage source input.
- Node 1: Junction between the source and the Resistor (R).
- Node 2: Junction between the Resistor (R) and the Inductor (L).
- Node 3 (Vout): Junction between the Inductor (L) and the Capacitor (C). Note: For a bandpass filter, Vout is typically taken across the resistor, meaning Vout is between Node 1 and Node 2, or the component order is rearranged so R is at the bottom.
- Node 4 (GND): Ground return.
A parallel RLC circuit exhibits maximum impedance at resonance and requires a high-impedance current source (or a voltage source with a massive series resistor) to achieve a high Q-factor. If you drive a parallel tank with a low-impedance 50-ohm lab source, the source impedance swamps the tank, flattening the resonance peak entirely. The series RLC exhibits minimum impedance at resonance, perfectly matching low-impedance voltage sources. According to All About Circuits, series resonance is the standard for voltage-driven bandpass applications.
Component Behavior and Resonance Mechanics
The resonant frequency ($f_r$) is dictated strictly by L and C: $f_r = 1 / (2\pi\sqrt{LC})$. The Q-factor (quality factor) and bandwidth are controlled by R. Here is how altering one element shifts the circuit's behavior, assuming the other two remain constant:
| Parameter Changed | Resonant Freq ($f_r$) | Q-Factor | Bandwidth (-3dB) | Impedance at $f_r$ |
|---|---|---|---|---|
| Increase R | Unchanged | Decreases | Increases (wider) | Increases |
| Increase L | Decreases | Increases | Decreases (narrower) | Unchanged (equals R) |
| Increase C | Decreases | Decreases | Increases (wider) | Unchanged (equals R) |
Design Walkthrough: Building a 10 kHz Bandpass Filter
Let's design a series RLC bandpass filter targeting $f_r = 10 \text{ kHz}$ with a Q-factor of 10. We will select real, purchasable components rather than theoretical ideal values.
Step 1: Select the Inductor (L)
Inductors are harder to source in exact values and suffer from parasitic DC resistance (DCR). We start here. Choose a Bourns 78F-103K-RC (10 mH radial inductor). It costs about $1.20 and has a specified DCR of 1.5 $\Omega$.
Step 2: Calculate the Capacitor (C)
Using $C = 1 / (4\pi^2 f_r^2 L)$:
$C = 1 / (4 \times \pi^2 \times 10,000^2 \times 0.01) = 25.33 \text{ nF}$.
Standard E12 values don't hit 25.33 nF exactly. We will parallel a 22 nF and a 3.3 nF Wima MKP film capacitor to get 25.3 nF. Avoid cheap Y5V ceramics here; their capacitance drops drastically with applied AC voltage.
Step 3: Calculate the Resistor (R)
The formula for Q in a series circuit is $Q = (1/R) \times \sqrt{L/C}$.
Rearranging for R: $R = (1/Q) \times \sqrt{L/C}$.
$R = (1/10) \times \sqrt{0.01 / 25.3 \times 10^{-9}} = 0.1 \times 628.9 = 62.89 \Omega$.
However, our inductor already has 1.5 $\Omega$ of DCR. The intentional resistor must be $62.89 - 1.5 = 61.39 \Omega$. The closest standard 1% metal film value is 61.9 $\Omega$ (Vishay MRS25 series). Place this resistor at the bottom of the chain (Node 3 to GND) and tap Vout across it to create your bandpass response.
At resonance, the voltage across the inductor and capacitor is Q times the input voltage. If you drive this circuit with 10V RMS, the voltage across the 10 mH inductor will be $10 \times 10 = 100\text{V RMS}$. Ensure your capacitor's voltage rating exceeds this multiplied voltage, and keep your hands clear of the nodes during high-power testing.
Failure Modes: What Breaks at the Extremes?
When debugging a dead board, you need to know how component failures manifest. A single open or short drastically alters the topology.
| Component | Failure Mode | Circuit Result | Diagnostic Symptom |
|---|---|---|---|
| Resistor (R) | Opens | Broken loop | Zero output at all frequencies. |
| Resistor (R) | Shorts | Pure LC Tank | Massive current spike at $f_r$; Q approaches infinity; output drops to zero if Vout was across R. |
| Inductor (L) | Opens | Broken loop | Zero output at all frequencies. |
| Inductor (L) | Shorts | RC Lowpass | $f_r$ vanishes; circuit passes low frequencies and rolls off high frequencies. |
| Capacitor (C) | Opens | Broken loop | Zero output at all frequencies (blocks DC/AC). |
| Capacitor (C) | Shorts | RL Highpass | $f_r$ vanishes; circuit blocks DC but passes high frequencies directly to ground if R is at the top. |
Breadboard Testing and Verification Steps
Simulations (like LTspice) assume ideal ground planes. Breadboards introduce stray capacitance (~2 pF per contact) and inductance. Follow these steps to verify your physical build using a Siglent SDG1032X function generator and a Rigol DS1054Z oscilloscope.
- Build the Chain: Insert the Bourns inductor, Wima capacitors, and Vishay resistor in a single line on the breadboard. Keep leads as short as possible to minimize stray inductance.
- Connect the Source: Use a BNC-to-alligator cable from the function generator. Connect the center conductor to Node 0 and the ground clip to the ground rail. Do not use the long ground spring on your scope probe; use the short pigtail clip to avoid picking up radiated noise.
- Probe Vout: Attach Channel 1 to the function generator output (Vin) and Channel 2 across the 61.9 $\Omega$ resistor (Vout).
- Sweep for Resonance: Set the generator to a 1Vpp sine wave. Sweep manually from 1 kHz to 50 kHz. Note the frequency where the Vout amplitude peaks. It should read close to 10 kHz.
- Measure Bandwidth: Record the peak Vout voltage. Calculate the -3dB point (Peak $\times 0.707$). Sweep down in frequency to find the lower -3dB point, and sweep up to find the upper -3dB point. The difference between these two frequencies is your bandwidth. For Q=10 at 10 kHz, expect a bandwidth of roughly 1 kHz.
Decision Path: Series vs. Parallel RLC Selection
Use this decision matrix to lock in your topology before opening your CAD software or parts bin. For a deeper look at automated component sizing, the Analog Devices Filter Wizard is an excellent supplementary tool for active and passive networks.
| Design Condition | Choose Topology | Reasoning |
|---|---|---|
| Source impedance is low (< 50 $\Omega$) | Series RLC | Low source Z won't dampen the series resonance peak. |
| Source impedance is high (> 1 k$\Omega$) | Parallel RLC | High source Z acts as a current source, allowing parallel tank to ring. |
| Need a Bandpass filter with low insertion loss | Series RLC | At $f_r$, series Z is minimal (just R), passing maximum voltage. |
| Need a Bandstop (Notch) filter | Series RLC | Take output across the L+C series combination; at $f_r$, their combined Z is near zero, shorting the signal to ground. |
| Need to block DC but pass a specific AC band | Series RLC | The series capacitor inherently blocks DC bias from the source. |
The Default Recommendation: If you are working in a standard lab environment with 50-ohm equipment, or designing audio line-level filters driven by op-amps (which have < 100-ohm output impedance), default to the Series RLC topology. Specifically, use the configuration outlined in our walkthrough: source -> L -> C -> R(to ground), tapping Vout across the resistor. It provides predictable Q control, inherent DC blocking, and interfaces perfectly with modern low-impedance test gear and driver ICs.
