Resonant Frequency
Frequency where an LC circuit resonates.
Formula
Variables
Example
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How it works
In an ideal LC circuit, energy repeatedly shuttles between the capacitor’s electric field and the inductor’s magnetic field. The resonant frequency is the natural rate of that exchange, determined by how quickly the capacitor can charge/discharge and how strongly the inductor resists changes in current. The formula comes from solving the differential equation for oscillation, where the circuit’s second-order dynamics produce an angular frequency \(\omega_0 = 1/\sqrt{LC}\), and converting to ordinary frequency gives \(f_0 = \omega_0/(2\pi)\).
When to use it
Use this when analyzing or designing resonant tanks, band-pass filters, oscillators, impedance-matching networks, and tuned RF circuits. It is especially useful for finding the natural frequency of an LC pair before adding real-world losses or source/load effects.
Worked example, step by step
Suppose an LC tank has: L = 10 \u03bcH = 10 \times 10^{-6} H C = 100 nF = 100 \times 10^{-9} F Step 1: Multiply L and C LC = (10 \times 10^{-6})(100 \times 10^{-9}) LC = 1000 \times 10^{-15} LC = 1.0 \times 10^{-12} Step 2: Take the square root \u221a(LC) = \u221a(1.0 \times 10^{-12}) = 1.0 \times 10^{-6} Step 3: Multiply by 2\u03c0 2\u03c0\u221a(LC) = 2\u03c0 \times 10^{-6} \approx 6.283 \times 10^{-6} Step 4: Invert to get frequency f_0 = 1 / (6.283 \times 10^{-6}) \approx 1.59 \times 10^5 Hz Final answer: f_0 \approx 159 kHz
Common mistakes to avoid
- Forgetting to convert microhenries and nanofarads into henries and farads before calculating
- Using this formula for a real circuit without remembering that resistance shifts the peak response slightly
- Confusing resonant frequency f0 with angular frequency \u03c90 and omitting the factor of 2\u03c0
Did You Know?
At resonance an LC circuit swaps energy between capacitor and inductor, like a perfectly timed swing.
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