A series RLC circuit places a resistor (R), inductor (L), and capacitor (C) in a single continuous conductive loop, forcing the exact same alternating current through all three components. At the resonant frequency, the inductive reactance and capacitive reactance perfectly cancel each other out, leaving only the resistor to limit current flow. This topology is the foundational building block for bandpass filters, intermediate frequency (IF) stages in radios, and impedance matching networks.
The Series RLC Topology: Nodes, Current, and Voltage Drops
To analyze this circuit on the bench, we define four distinct nodes in the loop. Assume an AC voltage source ($V_{in}$) drives the circuit:
- Node A: AC Source positive terminal, connected to the Resistor.
- Node B: Junction between the Resistor and the Inductor.
- Node C: Junction between the Inductor and the Capacitor.
- Node D: Junction between the Capacitor and the AC Source negative (ground) terminal.
Because it is a series topology, Kirchhoff’s Current Law dictates that the current is identical through every component ($I_R = I_L = I_C$). However, Kirchhoff’s Voltage Law requires that the vector sum of the voltage drops across each component equals the source voltage. The total impedance ($Z$) is calculated as:
Z = √(R² + (X_L - X_C)²)
Where $X_L = 2\pi fL$ and $X_C = 1 / (2\pi fC)$. At resonance, $X_L = X_C$, the reactance terms zero out, and $Z = R$.
Series vs. Parallel RLC: Why Choose Series?
When designing a reactive network, you must choose between series and parallel topologies. The decision hinges entirely on how you want the circuit to behave at resonance and how it should fail.
| Criterion | Series RLC | Parallel RLC |
|---|---|---|
| Impedance at Resonance | Minimum (Z = R) | Maximum (Z = L / (RC)) |
| Current at Resonance | Maximum (Limited only by R) | Minimum (Line current drops) |
| Primary Filter Type | Bandpass (Accepts resonant freq) | Bandstop / Tank (Rejects resonant freq) |
| Open Component Failure | Entire circuit dies (I = 0) | Only that branch dies; circuit continues |
| Short Component Failure | Component bypassed; filter shifts type | Dead short across source; blows fuse |
Design Walkthrough: Tuning a 15.9 kHz Bandpass Filter
Let’s design a practical series RLC bandpass filter targeting a resonant frequency ($f_r$) of roughly 15.9 kHz, a common intermediate frequency for ultrasonic sensors and older sonar equipment.
1. Pick the Capacitor (C):
Capacitors have tighter manufacturing tolerances and lower parasitic effects than inductors. We will select a standard 100 nF (0.1 µF) X7R ceramic capacitor. Avoid Y5V dielectrics here; their capacitance drops drastically with applied AC voltage.
2. Calculate the Inductor (L):
Using the resonance formula $f_r = 1 / (2\pi\sqrt{LC})$, we solve for L:
L = 1 / ((2π × 15915)² × 100×10⁻⁹) = 1 mH
We will use a standard 1 mH axial leaded inductor (e.g., Bourns 78F series, approx. $1.20). Ensure its DC resistance (DCR) is low (under 2 ohms) so it doesn't secretly act as your primary resistor.
3. Set the Q-Factor with the Resistor (R):
The Quality factor (Q) determines the bandwidth. A higher Q means a narrower, sharper peak.
Q = (1/R) × √(L/C)
If we want a moderate Q of 4.5 to allow some frequency drift without losing the signal:
R = (1/4.5) × √(0.001 / 100×10⁻⁹) = 22.2 Ω
We will use a standard 22 Ω, 1/4W carbon film resistor.
With these real-world values, our bandwidth (BW = $f_r / Q$) is approximately 3.5 kHz, meaning the filter will pass signals strongly between 14.1 kHz and 17.6 kHz.
Element Sensitivity and Failure Extremes
Understanding how a series RLC circuit behaves when pushed to its extremes or when a component fails is critical for troubleshooting and protection design.
| Component Change / Failure | Effect on Circuit Behavior | Physical Consequence on the Bench |
|---|---|---|
| R increases (or Open) | Q drops to zero; resonance peak flattens entirely. | Signal amplitude drops. If open, current ceases completely. |
| R shorts (R = 0 Ω) | Q approaches infinity; impedance at resonance drops to just the wire/parasitic resistance. | Massive current spike at resonance. Will likely burn out the inductor windings or trip the function generator's short-circuit protection. |
| L shorts | Circuit becomes a simple RC high-pass filter. | Resonance is destroyed. Low frequencies are blocked, high frequencies pass to the capacitor. |
| C shorts | Circuit becomes a simple RL low-pass filter. | Resonance is destroyed. The inductor may overheat at low frequencies due to low reactance and lack of current limiting. |
| L or C opens | Current path is broken. | Total circuit failure. Zero current flows regardless of frequency. |
How to Breadboard and Test a Series RLC Circuit
Testing reactive circuits on a breadboard introduces parasitic capacitance (usually 2-5 pF between rows) and contact resistance. Follow this exact sequence to validate your 15.9 kHz design.
- Wire the Series Loop: Connect your function generator's output to Node A. Wire the 22 Ω resistor from Node A to Node B. Wire the 1 mH inductor from Node B to Node C. Wire the 100 nF capacitor from Node C to Node D (Ground). Keep lead lengths under 2 cm to minimize stray inductance.
- Account for Source Impedance: Most bench function generators have a 50 Ω output impedance. If your generator is set to 'High-Z' mode but physically outputs 50 Ω, your total circuit resistance is actually 72 Ω (50 + 22). This will drastically lower your Q factor. Fix: Set the generator to '50 Ω Load' mode, or factor the 50 Ω into your math.
- Probe the Current: Connect Oscilloscope Channel 1 across the 22 Ω resistor (Nodes A and B). Because $V = I \times R$, the voltage waveform across the resistor is a perfect, scaled representation of the circuit's current.
- Sweep the Frequency: Set the function generator to output a 2 Vpp sine wave. Sweep the frequency logarithmically from 1 kHz to 50 kHz.
- Identify the Peak: Watch Channel 1. The voltage amplitude across the resistor will peak sharply. Note the frequency at this peak—this is your actual measured $f_r$. It should read close to 15.9 kHz. If it reads 14.2 kHz, your inductor likely has a +10% tolerance drift, which is standard for off-the-shelf ferrite chokes.
- Measure Reactive Voltage Magnification: Move the oscilloscope probe to measure across just the inductor (Nodes B and C). At resonance, you will observe a voltage across the inductor that is higher than your 2 Vpp source voltage. This is normal and dictated by the Q factor.
Frequently Asked Questions
What happens to a series RLC circuit at DC?
At DC (0 Hz), the capacitive reactance ($X_C$) becomes infinite, acting as an open circuit. After a brief initial transient where the capacitor charges through the inductor and resistor (an RLC step response), steady-state current drops to absolute zero. The inductor acts as a short circuit (limited only by its DCR), but the capacitor blocks all continuous current flow. Therefore, a series RLC circuit cannot be used to pass DC signals.
Can I use a series RLC circuit for power factor correction?
No. Power factor correction (PFC) on AC mains requires a parallel capacitor (or parallel LC bank) placed across the load. A series RLC circuit would insert impedance directly into the line, causing a massive voltage drop to your load and wasting real power as heat in the resistor and inductor windings. Furthermore, if a series PFC circuit accidentally hit resonance with the 50/60 Hz mains frequency, it would create a dead short across the utility supply, tripping breakers or causing a fire.
Why does the voltage across the inductor or capacitor exceed the source voltage?
This phenomenon is called resonance voltage magnification. At resonance, the source voltage is entirely dropped across the resistor. However, the current flowing through the circuit is at its maximum ($I = V_{source} / R$). This same massive current flows through the inductor and capacitor. The voltage across the inductor is $V_L = I \times X_L$. If the reactance of the inductor is larger than the resistance (which is true for any Q > 1), the resulting $V_L$ will mathematically and physically exceed the source voltage. The inductor and capacitor are continuously exchanging stored magnetic and electric energy back and forth, 'sloshing' energy between them, which manifests as elevated voltage potentials across their individual terminals.
