LC Resonance
Connect an inductor and a capacitor together in parallel, give the circuit a small kick of energy, and it will oscillate all by itself — no driving source needed — at a frequency fixed entirely by the two components you chose. This natural "ringing" is the starting point for every tuned circuit in radio and communications electronics.
Energy sloshing between a capacitor and an inductor
A capacitor stores energy in the electric field between its plates; an inductor stores energy in the magnetic field around its coil. Individually, each of these components just stores and releases energy passively. Connect them together in parallel, though, and something more interesting happens: the two components can pass energy back and forth between each other indefinitely, with the circuit's own current and voltage oscillating as they do it. This module covers parallel resonance only — a series LC arrangement behaves differently and isn't part of this specification, so keep the two arrangements distinct in your mind.
Picture starting the process with the capacitor fully charged and the inductor carrying no current. The capacitor begins to discharge through the inductor, and as it does, the growing current builds up a magnetic field in the inductor — energy is draining out of the capacitor's electric field and building up in the inductor's magnetic field. Once the capacitor is fully discharged, the current is at its maximum, and all the energy that was originally in the capacitor is now stored magnetically in the inductor. The inductor's magnetic field then can't sustain that current forever — as it starts to collapse, it drives current onward in the same direction, recharging the capacitor, but now with the opposite polarity. The whole process then repeats in reverse. This continuous back-and-forth exchange of energy, capacitor to inductor and back again, is what constitutes the oscillation of an LC circuit.
The resonant frequency
Because this exchange of energy repeats itself at a regular rate, the circuit has a natural oscillation frequency, called the resonant frequency, $f_0$. It depends only on the circuit's own inductance $L$ and capacitance $C$:
You are not required to derive this result — it follows from the way current and voltage are related in an inductor and a capacitor respectively, but for this course you simply need to be able to apply the formula confidently in both directions: calculating $f_0$ from given values of $L$ and $C$, and rearranging to find whichever of $L$ or $C$ is needed to produce a target resonant frequency.
The mass–spring analogy
The mathematics of an oscillating LC circuit is structurally identical to the mathematics of a mass oscillating on a spring, and leaning on that analogy is one of the most useful ways to build genuine intuition for what's happening electrically, since the mechanical picture is much easier to visualise.
An inductor's defining behaviour is that it opposes any change in the current flowing through it — exactly as a mass, by its inertia, opposes any change in its velocity. A large inductance behaves like a large, sluggish mass: hard to get moving (hard to establish current in), but equally hard to stop once it's moving (the current keeps flowing even as the driving push disappears). A capacitor, meanwhile, stores energy by building up charge and a potential difference across its plates — exactly as a spring stores energy by being stretched or compressed, building up a restoring force proportional to its extension. A large capacitance behaves like a "soft", low-stiffness spring: it can store a given amount of charge with only a small voltage built up, just as a soft spring stores a given amount of energy with only a small extension.
- Both systems have a single natural frequency of oscillation fixed by their own physical properties, not by how they were initially disturbed.
- Both continuously exchange energy between two forms — electric/magnetic field energy for the LC circuit, kinetic/elastic potential energy for the mass–spring system.
- In both cases, a "heavier" inertial element ($L$ or $m$) and a "softer" compliant element ($C$ or a low spring constant) each independently push the natural frequency down, which is exactly the shape of the $f_0 = \frac{1}{2\pi\sqrt{LC}}$ formula.
| Electrical quantity (LC circuit) | Mechanical quantity (mass–spring) |
|---|---|
| Inductance, $L$ | Mass, $m$ (inertia — resists change in motion) |
| Capacitance, $C$ | Compliance of the spring (inverse of stiffness — stores energy elastically) |
| Current through the inductor | Velocity of the mass |
| Charge / voltage on the capacitor | Extension (displacement) of the spring |
| Magnetic field energy in $L$ | Kinetic energy of the mass |
| Electric field energy in $C$ | Elastic potential energy in the spring |
An LC circuit left to oscillate on its own only tells half the story. What makes resonance genuinely useful in electronics is that a circuit can be driven at a range of frequencies, and it responds far more strongly when driven close to its own resonant frequency $f_0$ than when driven far from it. That selective response is exactly what allows a circuit to pick out one radio signal from a crowded band of frequencies — the sharpness of that selection is measured by the Q factor, covered in the next lesson.
Key takeaways
- This module covers parallel resonance only — a capacitor and inductor connected in parallel, not in series.
- Energy oscillates between the capacitor's electric field and the inductor's magnetic field, repeating at the circuit's own natural resonant frequency.
- The resonant frequency is $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$; you need to apply this formula, not derive it.
- The formula can be rearranged to find a required $L$ or $C$ for a target resonant frequency, as well as used directly to calculate $f_0$.
- The mass–spring analogy maps inductance to mass (inertia, resisting change in current/velocity) and capacitance to spring compliance (energy storage via charge/voltage or extension).
- Both the LC circuit and the mass–spring system have a single fixed natural frequency and continuously exchange energy between two storage forms.
- Real circuits exploit this resonant behaviour deliberately to select or reject particular frequencies, most notably in radio tuning — the subject of the next lesson.