Thomson Oscillation Formula

Use the Thomson formula in its frequency form f = 1/(2π·√(L·C)). First convert L and C into the SI units henry and farad; that is the most common source of error. Example: L = 10 mH = 0.01 H and C = 100 nF = 10⁻⁷ F give L·C = 10⁻⁹ s², the root is 3.16×10⁻⁵ s, so f = 1/(2π × 3.16×10⁻⁵) ≈ 5 kHz. The period is the reciprocal T = 1/f ≈ 0.2 ms. As a sanity check: a larger inductance or capacitance makes the circuit more sluggish, so the frequency must drop.

The energy oscillates periodically between two stores. Initially the capacitor is charged and all the energy sits in the electric field (½CU²). It discharges through the coil, and the growing current builds up a magnetic field there. When the capacitor is empty, the maximum current flows and all the energy sits in the magnetic field (½LI²). Due to its self-inductance the coil keeps the current going and recharges the capacitor with reversed polarity. Then the game reverses. A full period comprises two recharges. Without resistance this would run forever; in reality the ohmic resistance damps the amplitude exponentially.

First square the equation to remove the root. From f = 1/(2π√(LC)) you get f² = 1/(4π²LC), hence C = 1/(4π²f²L) and analogously L = 1/(4π²f²C). Using the period, L = T²/(4π²C). Example: a receiver circuit for 1 MHz with L = 100 µH needs C = 1/(39.48 × (10⁶)² × 10⁻⁴) ≈ 2.5×10⁻¹⁰ F, about 253 pF. A typical mistake is forgetting the root and rearranging linearly instead of quadratically; the frequency enters squared. Check: a higher target frequency demands a smaller capacitance.

Both obey the same differential equation of a harmonic oscillation, just with different quantities. The analogy is precise: charge Q corresponds to displacement s, current I to velocity v, inductance L to inertial mass m and the reciprocal capacitance 1/C to the spring constant D. From T = 2π√(m/D) you get T = 2π√(LC) directly. The energy balance transfers too: ½LI² corresponds to the kinetic energy ½mv², and ½Q²/C to the elastic energy ½Ds². This analogy is more than a mnemonic: whoever can solve one oscillation can automatically solve both, and exam questions frequently demand exactly this translation.

Every real circuit contains ohmic resistance in wires and coil winding. With each recharge part of the energy turns into heat (P = I²·R), and the amplitude decays exponentially, a damped oscillation. In addition the circuit radiates a small part as an electromagnetic wave, which is actually desirable for antennas. Damping lowers the natural frequency slightly. For a transmitter or clock to oscillate permanently, a feedback circuit must resupply energy in step with the oscillation, for instance via a transistor acting like the escapement of a clock (regeneration, Meissner oscillator). In exams a qualitative description of the damping causes usually suffices.