Resonance and Power in AC Circuits
At the frequency where the two reactances cancel, the impedance is smallest and the current peaks, which is how a radio picks one station. Average power depends on the power factor, so a purely reactive load consumes nothing.
What a learner can do afterwards
- Finds the resonant frequency of a series RLC circuit and describes the current peak
- Calculates average power using root mean square values and the power factor
- Explains why a capacitor or inductor alone takes no net energy from the supply over a cycle
1 · Read
Drive a series RLC circuit and sweep the frequency: current climbs to a sharp peak at omega0 equals 1 over sqrt(LC), then falls away. At that sweet spot the inductor and capacitor reactances cancel, impedance shrinks to bare R, and current marches in phase with voltage. Small resistance makes the peak tall and narrow, measured by the quality factor Q. A radio picks one station by parking its resonance on the wanted frequency.
Take a circuit with Irms equal to 2 A, Vrms equal to 10 V, and power factor 0.5. Average power is Pavg equals Irms times Vrms times cos phi, so 2 times 10 times 0.5 gives 10 W. Meters read rms, which is peak over sqrt(2), and cos phi equals R over Z. At resonance phi is zero and the factor is one, so the full Irms times Vrms turns into work.
A pure capacitor or inductor is the extreme case: current and voltage sit a quarter cycle apart, cos phi is zero, and every joule borrowed in one quarter cycle is refunded in the next. Over a full cycle the net energy taken is zero. Only current in step with voltage does real work. Utilities dislike low power factors because fat currents heat wires while delivering little usable power.
Judge any AC situation with this routine. Find omega0 and compare the drive frequency to it. Below resonance the capacitor rules and current leads; above it the inductor rules and current lags. Convert peaks to rms by dividing by sqrt(2), then multiply by the power factor for real power.
Current peaks at omega0 where reactances cancel, real power is Irms Vrms cos phi, and lone coils or capacitors borrow energy and hand it straight back.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.