AC RESONANCE
At ω₀ the reactances cancel BITWISE and φ === 0 — driven Q equals free-ringing Q, one number.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
The same coil and capacitor the rlc module let ring freely — a tenth of a henry, ten microfarads, inherited and disclosed — but now a sinusoidal source drives the loop and we ask for the steady state. Free, this circuit rang at its own frequency and died away. Driven, it answers at whatever frequency you choose. The question is how loudly.
Impedance
The coil charges omega L of reactance and the capacitor one over omega C — ninety degrees apart in phase and OPPOSED, so on the phasor diagram they subtract head to tail. The impedance is the Pythagorean sum of R with their difference, and the phasor diagram at left is a new primitive built for this wing: three voltage arrows that must close on the source.
Resonance
At omega-nought equals one over root L C, the two reactances cancel to the last bit and the phase is exactly zero — the circuit is purely resistive at one frequency, and the current peaks at V-nought over R exactly. The sharpness is Q, omega-nought L over R — and this is the SAME Q the free-ringing module computes from its decay rate. Two definitions, one number, audited live against lcDerive itself.
Through resonance
The sweep drives the source from well below resonance to well above. Watch the phasors: the capacitor arrow dominates below — phase negative, current leading — and the coil above. At omega-nought they are equal and opposite, and the amber marker crests the resonance curve at exactly V-nought over R.
Audit
Audited: the reactances cancel bitwise at resonance — both exactly one hundred ohms — with phase identically zero and the peak confirmed by symmetric neighbors at five ratios; the quality factor here equals the free-ringing module’s to the bit at the defaults; the power ledger holds within one part in ten to the fifteen everywhere and lands residual zero on the dyadic cell; and the bandwidth is omega-nought over Q exactly as a closed form, two hundred radians per second.