Varactor — tuning by voltage

A reverse-biased junction is a capacitor whose plates move with voltage — so voltage tunes frequency. The law that seals is exact: since f ∝ 1/√C, halving the capacitance raises the frequency by exactly √2: https://narrowhighway.com/s/1ee0d3ed486453eee0967146a531984888709198b39fb16a7b2c2a5f58ef1b73 — and the tempting error 'halving C halves f' seals BROKEN with residual √2 − 1/2: https://narrowhighway.com/s/9ad4a43e7d3c0dd5cea7d1e8756669e79996fc98b70bf02bf35bc143d42108ac . Every phase-locked loop and electronically tuned radio rides this. The C(V) profile of a real varactor is an empirical junction property.