LC Resonant Frequency
Calculate the resonant frequency of an LC (tank) circuit from its inductance and capacitance using the Thomson formula. Enter L and C with their units to get frequency, period, and angular frequency.
Find the Resonant Frequency of an LC Circuit
LC Resonant Frequency Formula
Where:
- \(f_0\) — resonant frequency (Hz)
- \(L\) — inductance of the coil (henries, H)
- \(C\) — capacitance of the capacitor (farads, F)
- \(T\) — period of one oscillation (seconds)
- \(\omega_0\) — angular frequency (rad/s)
This is the Thomson (or tank-circuit) formula. It gives the natural frequency at which an ideal LC circuit oscillates, where the energy stored in the capacitor and the energy stored in the coil exchange back and forth with zero net reactance.
LC Resonance – Calculation Example
Example: a tank circuit uses a 100 µH coil and a 100 pF capacitor, the default values of the calculator.
\[ L = 100\,\mu\text{H} = 1\times10^{-4}\ \text{H}, \qquad C = 100\,\text{pF} = 1\times10^{-10}\ \text{F} \]
\[ f_0 = \frac{1}{2\pi\sqrt{(1\times10^{-4})(1\times10^{-10})}} = \frac{1}{2\pi\times10^{-7}} \approx 1.592\times10^{6}\ \text{Hz} \]
\[ T = \frac{1}{f_0} \approx 6.283\times10^{-7}\ \text{s}, \qquad \omega_0 = 2\pi f_0 = 10^{7}\ \text{rad/s} \]
Result: the circuit resonates at approximately 1.592 MHz, with a period of 628.3 ns and an angular frequency of 10,000,000 rad/s.
How to Use the LC Resonance Calculator
- Enter the coil inductance in the Inductance (L) field and select the matching unit (µH, mH, or H) from the dropdown.
- Enter the capacitor value in the Capacitance (C) field and choose its unit (pF, nF, µF, or F).
- Read the Resonant Frequency result, shown with an automatically selected SI prefix (Hz, kHz, MHz, or GHz).
- Check the Period and Angular Frequency results if you need the oscillation time or the value of \(\omega_0\) in rad/s.
What Is LC Resonance?
An LC circuit is made of an inductor (a coil) and a capacitor connected together, either in series or in parallel. When the capacitor is charged and the circuit is left to itself, current flows back and forth between the coil and the capacitor: the capacitor discharges through the coil, the coil's magnetic field builds up, then collapses and recharges the capacitor with the opposite polarity. This exchange repeats at one fixed natural frequency — the resonant frequency — determined only by the values of L and C.
At resonance, the inductive reactance \(X_L = 2\pi f L\) and the capacitive reactance \(X_C = \frac{1}{2\pi f C}\) are equal in magnitude and cancel each other. The impedance of an ideal parallel tank circuit is then purely resistive and at its maximum, while in a series circuit it is at its minimum. This is why a tuned circuit can select one frequency out of many: it responds strongly at \(f_0\) and barely at all elsewhere.
This behavior is the working principle behind radio tuners, filters, oscillators, and impedance-matching networks. The classic AM radio tuner, for instance, uses a variable capacitor precisely so you can sweep \(f_0\) across the broadcast band and pick out one station. In an oscillator, the tank circuit sets the frequency of the generated signal; in a filter, it marks the center or corner frequency of the passband.
Interpreting the result is straightforward: \(f_0\) tells you where the circuit tunes, \(T\) tells you how long one full oscillation cycle takes, and \(\omega_0\) is the form most often used in AC circuit analysis and differential equations. Larger values of L or C both push the frequency down, and halving either one raises \(f_0\) by a factor of \(\sqrt{2}\).
Assumptions, Limitations, and Practical Tips
- Ideal components: the Thomson formula ignores resistance and losses. Real coils have winding resistance and real capacitors have equivalent series resistance, which slightly shift the actual resonance and damp the oscillation.
- Stray reactance: parasitic capacitance of the coil, lead inductance, and PCB trace effects add to the nominal L and C values. This matters most at high frequencies — for example, a 100 pF capacitor with 2 pF of stray capacitance resonates about 1% lower than calculated. To estimate the inductance contributed by a straight conductor, see the Straight Wire Inductance calculator.
- Self-resonance: every real inductor also has a self-resonant frequency set by its own parasitic capacitance. Above it, the component stops behaving as an inductor, so keep \(f_0\) well below the coil's rated self-resonance.
- Units matter: the most common source of error is a unit mismatch. Always double-check whether the value is in µH or mH, pF or nF, before entering it.
- Practical tuning: real tank circuits are rarely used in isolation. To match a resonant circuit to a source or load, a T matching network or a PI network impedance matching design is usually built around the LC values you compute here.
- Combining components: if several coils are used, remember that their effective inductance depends on the connection — series inductance adds directly, while a parallel inductance combination follows the reciprocal rule. Enter the total effective L into this calculator.
Quick Reference: Common LC Combinations
| Inductance | Capacitance | Resonant Frequency |
|---|---|---|
| 1 µH | 1 pF | 159.2 MHz |
| 10 µH | 100 pF | 5.03 MHz |
| 100 µH | 100 pF | 1.59 MHz |
| 1 mH | 1 µF | 5.03 kHz |
| 10 mH | 100 µF | 159.2 Hz |
Use this table for a sanity check: radio-frequency work (kHz–GHz) uses small L and C, while audio and power applications use much larger values.
Frequently Asked Questions
How do I calculate the LC resonant frequency by hand?
First convert both values to base units: henries and farads. Then multiply them, take the square root of the product, multiply by \(2\pi\), and divide 1 by the result. For example, with \(L = 220\ \mu\text{H} = 2.2\times10^{-4}\ \text{H}\) and \(C = 10\ \text{nF} = 10^{-8}\ \text{F}\), you get \(\sqrt{LC} = 4.69\times10^{-6}\), so \(f_0 \approx 33.9\ \text{kHz}\). The calculator automates exactly these steps.
Can I find the required L or C for a target frequency?
Yes. Rearranging the Thomson formula gives \(LC = \frac{1}{(2\pi f_0)^2}\). Pick a convenient value for one component and solve for the other. For instance, to resonate at 7 MHz with a 100 pF capacitor: \(L = \frac{1}{(2\pi\cdot 7\times10^6)^2 \cdot 10^{-10}} \approx 5.2\ \mu\text{H}\). In practice, choose L and C values that give a realistic coil size and a component you can actually buy.
Does it matter whether the coil and capacitor are in series or in parallel?
For the ideal case, no — both connections resonate at the same \(f_0 = \frac{1}{2\pi\sqrt{LC}}\) with the same L and C. The difference is behavior at resonance: a series LC has minimum impedance (a short at \(f_0\)), while a parallel tank has maximum impedance (an open circuit). Real components with parasitic resistance shift series and parallel resonance slightly apart, but the difference is tiny when losses are low.
Why does my measured resonance differ from the calculated value?
The Thomson formula assumes ideal, lossless components. Real coils have winding resistance and self-capacitance, capacitors have ESR, and wiring adds stray inductance and capacitance — all of which pull the true resonance slightly lower or higher. Damping from resistance also lowers the frequency by roughly a factor \(\sqrt{1 - 1/(4Q^2)}\). Measure with a dip meter, VNA, or grid-dip oscillator if accuracy matters, and add trimmer capacitance for fine tuning.
What resonant frequencies do common circuits use?
Typical landmarks: AM broadcast tuning covers 530–1700 kHz; the classic AM intermediate frequency is 455 kHz; FM IF is 10.7 MHz; the 27 MHz CB band and 13.56 MHz RFID use values in that range; WiFi and similar radios work in the 2.4 GHz range. If your result lands far outside the MHz range for a radio project, double-check units — a µH/mH or pF/nF mix-up shifts the answer by 1000×.
Does the supply voltage or charge affect the resonant frequency?
No — in an ideal LC circuit the frequency depends only on L and C, not on the amount of energy stored. A higher initial capacitor voltage produces larger oscillation amplitude, but the same \(f_0\). This independence is exactly why LC tanks make stable oscillators and tuners. The main real-world exception is components whose value changes with voltage or current, such as varactor diodes (used deliberately for electronic tuning) or coils with saturable cores.
What's the most common mistake when using this calculator?
Unit mix-ups, by far. The classic error is entering microhenries while the dropdown reads mH, or picofarads while it reads nF — either one makes the result off by a factor of 1000. Also make sure you enter the total effective L and C: series inductors add, parallel capacitors add, and stray board capacitance (often 2–10 pF) should be included when working at high frequencies with small capacitor values.