Resistance for alternating current
With direct current, a resistor is all that limits the current. With alternating current, inductors and capacitors also resist the flow, in a way that depends on frequency. The combined opposition is the impedance, Z, measured in ohms. It decides how much current a circuit draws, how filters behave and how audio crossovers split sound between speakers.
The reactances
- Inductive reactance: XL = 2π × f × L. It grows with frequency, so inductors block high frequencies.
- Capacitive reactance: XC = 1 ÷ (2π × f × C). It shrinks with frequency, so capacitors block low frequencies and direct current.
Series and parallel impedance
In series, the reactances subtract and then combine with the resistance like the sides of a right triangle:
Z = √(R² + (XL − XC)²), with phase angle φ = arctan((XL − XC) ÷ R)
In parallel, the admittances (1 ÷ Z) add instead:
1 ÷ Z = √((1 ÷ R)² + (1 ÷ XC − 1 ÷ XL)²)
Example
A series circuit with R = 100 Ω, L = 10 mH and C = 1 µF at 1 kHz:
- XL = 2π × 1,000 × 0.01 = 62.8 Ω
- XC = 1 ÷ (2π × 1,000 × 0.000001) = 159.2 Ω
- Z = √(100² + (62.8 − 159.2)²) = 138.8 Ω
- φ = arctan(−96.3 ÷ 100) = −43.9°: the circuit is capacitive and the current leads the voltage.
- With 10 V applied, the current is 10 ÷ 138.8 = 72 mA, and the power factor is cos(−43.9°) = 0.72.
The resonant frequency, f₀ = 1 ÷ (2π√(LC)), is about 1,592 Hz. Above it, the same circuit would become inductive.
Where this matters
- Filters: an RC low-pass filter passes frequencies below 1 ÷ (2πRC) and attenuates those above.
- Power systems: motors are inductive loads with a lagging power factor; utilities and factories add capacitors to bring it back toward 1.
- Audio: speaker crossovers use inductors and capacitors to send bass to the woofer and treble to the tweeter.
- Tuned circuits: radios use an LC circuit at resonance to pick out one station.
Limits
The calculator treats components as ideal. Real inductors have winding resistance, real capacitors have some leakage and series resistance, and at very high frequencies stray capacitance and inductance matter. Use the RMS voltage for AC calculations, not the peak voltage.