An LC AC circuit is the foundation of RF tuning, audio crossovers, and switching power supply filters. By pairing an inductor (L) and a capacitor (C) without a deliberate series resistor, you create a system that stores and transfers energy between magnetic and electric fields. At the resonant frequency ($f_r$), the inductive reactance ($X_L$) and capacitive reactance ($X_C$) are equal in magnitude but opposite in phase, effectively canceling each other out.
Textbook diagrams treat these components as ideal, but on the bench, parasitic resistance and component tolerances dictate your actual performance. This guide strips away the abstractions and walks through real-world topology selection, exact component sizing, and failure-mode analysis for LC networks.
Series vs. Parallel LC AC Circuit Topology
Before selecting components, you must choose your topology. For both configurations, we will define three standard nodes: Node A (AC Source/Input), Node B (The L-C Junction/Output), and Node C (Ground/Return).
In a series LC AC circuit, the inductor and capacitor are daisy-chained between Node A and Node C, with Node B acting as the junction between them. In a parallel LC AC circuit (often called a tank circuit), both L and C are connected between Node A and Node C, and Node B is typically the shared input rail.
Why choose one over the other? Series topologies are used when you need a bandpass filter or impedance matching, as they present minimum impedance at resonance. Parallel topologies are used for notch (bandstop) filters, oscillator tanks, and RF amplifiers, presenting maximum impedance at resonance.
| Characteristic | Series LC AC Circuit | Parallel LC AC Circuit |
|---|---|---|
| Impedance at Resonance ($Z_r$) | Minimum (equals parasitic ESR/DCR) | Maximum (equals $L / (C \times R_{parasitic})$) |
| Current at Resonance | Maximum (limited only by source and ESR) | Minimum from source; high circulating current inside the L-C loop |
| Phase Angle at Resonance | 0° (Purely resistive) | 0° (Purely resistive) |
| Voltage at Node B (Resonance) | Drops to near zero (shorted to ground) | Peaks at maximum (open circuit to ground) |
| Primary Application | Bandpass filters, series impedance matching | Tank circuits, notch filters, RF oscillators |
Design Walkthrough: Sizing a 10 kHz Resonant Tank
Let’s design a parallel LC AC circuit tuned to exactly 10 kHz. The resonant frequency formula is $f_r = \frac{1}{2\pi\sqrt{LC}}$. To find our component values, we pick a standard inductor first, as inductor choices are physically larger and less flexible than capacitor arrays.
Step 1: Select the Inductor
We choose a Bourns 78F103J-RC radial inductor. It is 10 mH, has a 5% tolerance, a DC resistance (DCR) of 2.1 Ω, and a maximum current rating of 180 mA. As of 2026, these cost roughly $0.80 each in small quantities.
Step 2: Calculate the Capacitor
Rearranging the formula to solve for C: $C = \frac{1}{(2\pi \cdot f_r)^2 \cdot L}$
$C = \frac{1}{(2\pi \cdot 10,000)^2 \cdot 0.01} = 253.3 \text{ nF}$
Step 3: Select Real-World Capacitors
A single 253.3 nF capacitor doesn't exist in standard E12/E24 series. We will parallel a WIMA MKS2 220nF and a WIMA MKS2 33nF polyester film capacitor to achieve exactly 253 nF. Film capacitors are mandatory here; ceramic MLCCs suffer from severe capacitance drop-off under DC bias and exhibit piezoelectric microphonics that will inject noise into your tank.
| Component | Part Number | Value | Parasitic ESR/DCR | Max Voltage / Current |
|---|---|---|---|---|
| Inductor | Bourns 78F103J-RC | 10 mH (±5%) | 2.1 Ω (DCR) | 180 mA RMS |
| Capacitor 1 | WIMA MKS2C042201K | 220 nF (±10%) | < 0.5 Ω (ESR) | 63 VDC |
| Capacitor 2 | WIMA MKS2C033301A | 33 nF (±10%) | < 0.2 Ω (ESR) | 63 VDC |
| Combined Network | — | 10 mH / 253 nF | ~2.4 Ω Total Loss | 10 kHz @ 180mA max |
According to Georgia State University HyperPhysics, the theoretical Q-factor of this circuit would be infinite. In reality, the 2.1 Ω DCR of the inductor limits our Q-factor to roughly $Q = \frac{X_L}{R} = \frac{2\pi(10000)(0.01)}{2.1} \approx 299$. This is an excellent Q for a passive bench circuit.
Behavior Matrix: Parameter Shifts and Circuit Response
When you tune an LC AC circuit on the bench, you rarely hit the exact mathematical target on the first try. You will need to swap components. Here is exactly how the circuit behaves when you alter specific parameters.
| Parameter Changed | Effect on Resonant Frequency ($f_r$) | Effect on Q-Factor | Effect on Bandwidth (BW) |
|---|---|---|---|
| Increase L | Decreases ($f_r \propto 1/\sqrt{L}$) | Decreases (higher DCR in larger coils) | Widens |
| Decrease C | Increases ($f_r \propto 1/\sqrt{C}$) | Remains mostly stable | Narrows slightly |
| Increase Parasitic ESR | No change | Drops heavily | Widens significantly |
| Increase Source Impedance | No change | Decreases (loads the tank) | Widens |
Failure Modes: What Breaks at the Extremes
Understanding how an LC AC circuit fails is critical for troubleshooting dead boards. The failure modes contrast sharply between series and parallel topologies.
Series LC AC Circuit Extremes
- Capacitor Shorts: The circuit becomes a pure inductor. At low frequencies, it passes signals easily; at high frequencies, it chokes them. The resonant notch disappears entirely.
- Inductor Shorts: The circuit becomes a pure capacitor. It blocks DC and low frequencies but passes high frequencies. Again, resonance is destroyed.
- Capacitor Opens: Infinite impedance. The circuit is dead. No AC current can flow to Node C under any condition.
Parallel LC AC Circuit Extremes
- Capacitor Shorts: Creates a direct dead short from Node A to Node C. If your AC source lacks current limiting or a fuse, you will burn out the source's output stage or melt the breadboard traces.
- Inductor Shorts: Identical to a shorted capacitor—a dead short across the AC source.
- Capacitor Opens: The tank circuit is broken. The inductor now acts as a simple low-impedance path to ground for low frequencies, and the high-impedance resonance peak vanishes.
Breadboard Testing Step-by-Step
Testing an LC AC circuit requires more care than a simple DC resistor network. High-frequency resonance is easily corrupted by probe capacitance and ground loops. Follow this sequence to verify your 10 kHz design.
- Prepare the Breadboard: Insert the Bourns 10mH inductor and the two WIMA film capacitors. Wire the capacitors in parallel, then wire that combined capacitance in parallel with the inductor. Node A is the top rail, Node C is the bottom ground rail.
- Isolate the Source: Connect your function generator to Node A. Critical step: Solder a 10 kΩ carbon film resistor in series with the function generator's center conductor. This prevents the generator's internal 50 Ω impedance from killing your tank's Q-factor.
- Configure the Oscilloscope: Connect Channel 1 to Node A (to monitor input) and Channel 2 to Node A as well (to measure the parallel tank's voltage response). Use 10x probes and ensure they are properly compensated using the scope's cal square-wave output.
- Set Initial Parameters: Set the function generator to a 1 Vpp sine wave at 5 kHz. Set the oscilloscope timebase to 50 µs/div to clearly see the 10 kHz waveform.
- Sweep for Resonance: Slowly increase the generator frequency from 5 kHz toward 15 kHz. Watch Channel 2. As you approach 10 kHz, the voltage amplitude on the scope will spike dramatically (potentially ringing up to 5 Vpp or more due to the Q-factor).
- Verify the Peak: Once you find the maximum voltage peak, use the scope's cursor function to measure the exact frequency. If it reads 9.8 kHz instead of 10 kHz, your inductor's actual value is likely 10.4 mH due to the 5% manufacturing tolerance.
- Measure Bandwidth: To find the Q-factor empirically, note the peak voltage. Adjust the frequency up and down until the voltage drops to 70.7% (-3dB) of the peak. The difference between these two frequencies is your bandwidth. Divide the center frequency by the bandwidth to get your real-world Q.
For a deeper mathematical breakdown of how phase angles shift across the resonance curve, the All About Circuits textbook chapter on resonance provides excellent phasor diagrams that complement this physical testing method.
By treating the LC AC circuit not just as a math equation, but as a physical system bound by parasitic resistance and component tolerances, you transition from textbook theory to reliable bench design. Always respect the current limits of your inductor, buffer your AC sources, and trust your oscilloscope over your calculations.
