Series LCR circuit
In an AC circuit, current and voltage usually aren't in step with each other — until you hit exactly the right frequency. Sweep the frequency and watch the phasors turn. Every reading — reactance, impedance, current, phase angle — updates live from the real equations, and you can see resonance happen the moment XL = XC.
The rules you're seeing
Reactance vs frequency
XL = ωL grows with frequency — inductors resist faster changes more. XC = 1/(ωC) shrinks with frequency — capacitors let high frequencies through more easily. Opposite behaviours that must eventually cross.
Resonance
At f₀ = 1/(2π√(LC)), XL exactly equals XC, so they cancel in the impedance formula. That leaves Z = R, its minimum possible value, so Irms is at its maximum — the peak of the I-vs-f curve.
Phase angle
Current and voltage are not generally in phase when L or C is present. φ tells you exactly how far apart they are, and whether voltage leads (inductive, XL>XC) or current leads (capacitive, XC>XL).
Why resonance matters
A resonant LCR circuit is how a radio tuner selects one station out of the whole spectrum — only signals near f₀ produce a large current in the circuit, so every other frequency is effectively rejected.
Part of the Alternating Current chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
Why frequency changes everything in an AC circuit
Three components, three behaviours
In a DC circuit only resistance matters. In AC, the current is constantly reversing, and the inductor and capacitor respond to that reversal in opposite ways.
A resistor behaves identically at any frequency: current and voltage stay in step, and R is constant. An inductor opposes changes in current, so the faster the alternation the harder it resists — its reactance XL = 2πfL rises with frequency. A capacitor does the reverse: at high frequency it barely finishes charging before reversing, so it passes current easily, and XC = 1/2πfC falls as frequency rises. At DC, where f = 0, an inductor is a plain wire and a capacitor is an open circuit — a useful sanity check on both formulas.
Why reactances subtract
The voltage across an inductor leads the current by a quarter cycle; across a capacitor it lags by a quarter cycle. Being half a cycle apart from each other, they act in opposition, which is why they subtract rather than add. The total opposition, the impedance, is
Z = √(R² + (XL − XC)²)
The Pythagorean form is not decorative — it comes from adding the voltages as perpendicular vectors in a phasor diagram, because they peak at different moments. Adding them arithmetically would overstate the total, and it is the reason individual component voltages in a series LCR circuit can each exceed the supply voltage without violating anything.
Resonance
Sweep the frequency and the current peaks sharply at one value. That is resonance, and it occurs when XL = XC, so the two cancel completely and Z falls to its minimum value of just R. The condition rearranges to f₀ = 1/(2π√(LC)).
At resonance the circuit behaves as if the inductor and capacitor were not there: current is maximum, the phase angle is zero, and the power factor is 1. This is precisely how tuning works. Turning a radio dial adjusts the capacitance until f₀ matches one station's frequency, so that station drives a large current while all others do not.
Mistakes that cost marks
Adding component voltages arithmetically. VR + VL + VC does not equal the supply voltage. Combine them as vectors: V = √(VR² + (VL − VC)²).
Assuming reactance dissipates power. Only the resistor consumes energy. Inductors and capacitors store it and return it each cycle, which is why average power is VI cos φ, and why a purely reactive circuit consumes nothing on average.
Mixing peak and RMS values. Meters read RMS, and Vrms = V₀/√2. Substituting a peak value into a formula expecting RMS inflates the answer by about 41%.