LC Resonant Frequency

Connect a charged capacitor across an inductor and the energy does not simply drain away — it sloshes. The capacitor discharges into the coil, building a magnetic field; when the capacitor is empty the field is at its peak, and a collapsing field keeps the current going, so it charges the capacitor back up the other way round. The cycle repeats at one natural frequency. It is the electrical twin of a mass bouncing on a spring, with inductance playing the part of mass — the reluctance to change velocity, or current — and $1/C$ playing stiffness. The frequency comes from setting the two reactances equal: $2\pi f L = 1/(2\pi f C)$, solve for $f$, and you have $f = 1/(2\pi\sqrt{LC})$.

A 100 µH coil with a 250 pF capacitor rings at $1/(2\pi\sqrt{10^{-4} \times 2.5 \times 10^{-10}}) = 1.01\ \text{MHz}$, the middle of the AM broadcast band. The square root makes the circuit reassuringly insensitive: to double the frequency you must quarter the product $LC$, so a 10% error in a capacitor moves the resonance by only 5%. Turning the tuning knob on an old radio physically rotated the vanes of a variable capacitor, sliding this frequency across the dial, and the whole receiver amounted to that one circuit picking one station out of the air.

Every radio transmitter, every RFID tag, every induction hob and every crystal oscillator descends from this arrangement. Heinrich Hertz used a spark-excited LC circuit in 1887 to generate and detect the first deliberate radio waves, confirming Maxwell's prediction; Marconi turned the same circuit into a business within a decade. The relation also explains why antennas have a length: a resonant antenna is an LC circuit whose inductance and capacitance are distributed along the conductor rather than lumped into parts.

Four things go wrong. The commonest by far is the factor of $2\pi$: $f$ is in hertz, $\omega = 2\pi f = 1/\sqrt{LC}$ is in radians per second, and the two differ by 6.28. Half the confusion in filter design traces to a formula quoted in one and used as the other. Second, prefixes — microhenries and picofarads must both be converted before multiplying, and an error here moves the answer by orders of magnitude, not percent. Third, this is the undamped natural frequency. Real circuits have resistance, which shifts the actual resonance slightly lower and, more importantly, sets how sharp it is: the quality factor $Q = (1/R)\sqrt{L/C}$ governs the bandwidth, and a lossy coil gives a broad, useless peak at the right frequency. Fourth, series and parallel LC circuits share this formula and behave in opposite ways at it. A series LC becomes a near short circuit at resonance, with the voltage across each component rising to $Q$ times the supply — a genuine hazard, since a modest input can put hundreds of volts across a capacitor. A parallel LC becomes a near open circuit. Choosing the wrong one gives a circuit that does the exact opposite of what was intended at precisely the frequency you designed for.