The resonant frequency (f_r) of an LC circuit is the exact point where inductive reactance (X_L) and capacitive reactance (X_C) are equal in magnitude but 180 degrees out of phase, effectively canceling each other out. The governing formula is f_r = 1 / (2π√(LC)). For a circuit with a 1 mH inductor and a 100 nF capacitor, the resonant frequency is exactly 15.915 kHz. At this frequency, a series LC circuit acts as a dead short (passing the signal), while a parallel LC circuit acts as an open circuit (blocking the signal or sustaining oscillation).
Series vs. Parallel LC Topologies & Node Mapping
Before picking components, you must choose the topology based on what you want the circuit to do to the target frequency. Both configurations use the same f_r formula, but their impedance behaviors at resonance are exact opposites.
Series LC (Bandpass / Notch)
In a series configuration, the inductor and capacitor are daisy-chained. Current must flow through both to reach ground.
- Node A (Input): AC signal source connects to the first lead of Inductor L1.
- Node B (Midpoint): The second lead of L1 connects to the first lead of Capacitor C1. This is your measurement node for voltage division.
- Node C (Ground): The second lead of C1 connects to the system ground.
Why choose series? At f_r, the combined impedance drops to near zero (limited only by parasitic resistance). Use this when you need to pass a specific frequency to ground (creating a notch filter) or pass it to a load (creating a bandpass filter).
Parallel LC (Tank / Bandstop)
In a parallel configuration, the inductor and capacitor share the same input and ground nodes, forming a closed loop.
- Node A (Input): AC signal source splits, connecting to the first lead of both L1 and C1.
- Node B (Ground): The second leads of both L1 and C1 tie together to system ground.
Why choose parallel? At f_r, the circulating current between L and C maximizes, but the impedance looking into Node A spikes to infinity (limited by parallel leakage). Use this for RF oscillator tanks, impedance matching networks, or bandstop (reject) filters. For a deeper look at how these topologies interact with AC waveforms, refer to the All About Circuits guide on series resonance.
Resonant Frequency Behavior & Component Scaling
Understanding how component scaling shifts your target frequency is critical for tuning. Because the formula relies on the square root of the L and C product, quadrupling the inductance only halves the frequency. The table below maps real-world standard component values to their resulting f_r and primary impedance behavior at resonance.
| Inductor (L) | Capacitor (C) | Calculated f_r | Series Z at f_r | Parallel Z at f_r | Primary Application Band |
|---|---|---|---|---|---|
| 10 mH | 1.0 µF | 1.59 kHz | ~1 Ω (Low) | ~10 kΩ (High) | Audio crossovers, subsonic filtering |
| 1 mH | 100 nF | 15.9 kHz | ~2 Ω (Low) | ~50 kΩ (High) | Ultrasonic sensors, PWM filtering |
| 270 µH | 10 nF | 96.7 kHz | ~3 Ω (Low) | ~120 kΩ (High) | Inductive charging, metal detectors |
| 2.2 µH | 470 pF | 4.95 MHz | ~0.5 Ω (Low) | ~300 kΩ (High) | AM radio IF stages, RFID (13.56MHz harmonic) |
| 47 nH | 2.2 pF | 495 MHz | ~0.1 Ω (Low) | ~1 MΩ (High) | UHF RF matching, cellular bands |
f_r lower than the theoretical calculation. Always measure with a network analyzer or grid dip meter at UHF.
Design Walkthrough: Building a 100 kHz Tank Circuit
Let's design a parallel LC tank circuit tuned to exactly 100 kHz, a common frequency for ultrasonic transducers and basic metal detection loops. We will select real, purchasable components rather than theoretical ideals.
- Anchor the Capacitor First: Inductors are physically larger, more expensive, and have wider tolerances (often ±10% or ±20%). Capacitors are tighter. We select a 10 nF capacitor. Crucially, we specify a C0G (NP0) dielectric. Avoid X7R or Y5V ceramics; their capacitance drops severely under voltage bias and they exhibit high dielectric absorption, which destroys the Q-factor of your tank.
- Calculate Required Inductance: Rearranging the formula to solve for L gives
L = 1 / ((2π * f_r)^2 * C). Plugging in 100,000 Hz and 10e-9 F yields 253.3 µH. - Select the Standard Inductor: 253 µH is not a standard E12 value. The closest standard off-the-shelf part is 270 µH (e.g., Bourns 78F271K-RC, a shielded radial inductor rated for 355 mA DC).
- Recalculate Actual f_r: Using L = 270 µH and C = 10 nF, the actual resonant frequency shifts to 96.74 kHz. For most hobbyist applications, this 3.2% error is acceptable. If exact 100 kHz tuning is required, you would parallel the 10 nF C0G cap with a 300 pF trimmer capacitor to dial it in on the bench.
For further reading on how component parasitics affect high-frequency LC filters, review the Analog Devices LC filter design guide, which details how ESR and ESL alter the ideal transfer function.
Failure Modes: What Breaks at the Extremes?
Understanding how an LC circuit fails when a component shorts or opens is vital for troubleshooting and protecting downstream driver ICs. A shorted capacitor in a parallel tank behaves very differently than a shorted capacitor in a series filter.
| Topology | Component | Failure State | Circuit Behavior & Consequence |
|---|---|---|---|
| Series | Capacitor (C1) | Shorted | Circuit becomes purely inductive. Passes DC and low frequencies. If driven by a low-impedance DC source, the inductor will saturate and overheat. |
| Series | Inductor (L1) | Open | Total signal loss. The circuit acts as an infinite impedance block at all frequencies. Safe, but non-functional. |
| Parallel | Capacitor (C1) | Shorted | Catastrophic: Node A is shorted directly to Ground. This will blow the source fuse, trigger overcurrent protection, or destroy the driving op-amp/microcontroller pin. |
| Parallel | Inductor (L1) | Open | Circuit becomes purely capacitive. The tank loses its resonance and Q-factor. High frequencies pass unimpeded to ground; DC is blocked. |
| Parallel | Inductor (L1) | Shorted | Similar to a shorted capacitor, a shorted inductor ties Node A directly to Ground, causing an immediate overcurrent event at the source. |
Step-by-Step Breadboard Testing & Verification
Theory only gets you to the breadboard. Verifying the f_r requires sweeping the circuit with a function generator and observing the response on an oscilloscope. Follow this exact procedure to avoid the most common bench measurement errors.
Required Equipment
- Function Generator (e.g., Siglent SDG1032X)
- Oscilloscope with 10x passive probe (e.g., Rigol DS1054Z)
- LC Tank: 270 µH inductor, 10 nF C0G capacitor
- 1 kΩ series isolation resistor (to prevent loading the generator)
Testing Procedure
- Build the Circuit: Insert the 270 µH inductor and 10 nF capacitor in parallel on the breadboard. Connect the ground leads to the common bus.
- Insert Isolation: Connect a 1 kΩ resistor between the function generator's output (Node A) and the LC tank's input node. This prevents the tank's low-impedance off-resonance states from overloading the generator's output amplifier.
- Probe the Node: Connect your oscilloscope's 10x probe across the LC tank (input node to ground). Ensure the probe compensation is calibrated.
- Set the Sweep: Configure the function generator for a sine wave, 2 Vpp amplitude. Set the frequency to 50 kHz (well below our calculated 96.7 kHz
f_r). - Observe the Roll-Off: Slowly increase the frequency. In a parallel tank with a series isolation resistor, the voltage at the node will drop as you approach resonance because the tank's impedance spikes, forming a voltage divider with the 1 kΩ resistor. (If testing a series LC to ground, the voltage across the resistor will peak).
- Pinpoint f_r: Fine-tune the frequency in 100 Hz increments. The exact resonant frequency is where the voltage amplitude reaches its minimum (for parallel) or maximum (for series), and where the phase shift between the generator output and the tank node crosses exactly 0 degrees.
By anchoring your design in C0G capacitors, accounting for standard E12 inductor values, and respecting the 50-ohm output impedance of your bench gear, you can reliably design and verify LC resonant circuits without relying on trial and error. For comprehensive data on inductor core losses and saturation currents at resonance, consult Electronics Tutorials on AC Resonance.
