AC and impedance: resistance that depends on frequency
The resistance of capacitors and inductors depends on frequency. With phasors and impedance, AC circuits are calculated like DC ones.
AC signals
v(t) = Vp × sin(2πft + φ). Described by three values: amplitude, frequency and phase. The RMS value is Vp / √2; a 220 V outlet is 220 V RMS with a peak of about 311 V.
Reactance
| Element | Reactance | Low frequency | High frequency |
|---|---|---|---|
| Resistor | R | R | R |
| Capacitor | 1 / (2πfC) | Blocks | Passes |
| Inductor | 2πfL | Passes | Blocks |
A 100 nF capacitor is about 1.6 kΩ at 1 kHz and about 1.6 Ω at 1 MHz. This is how a decoupling capacitor sends high-frequency noise to ground.
RoadmapSee where this note sits in Circuit Theory→Impedance and phasors
Resistance and reactance combined as a complex number is the impedance Z. Treat a sine wave as an arrow with a magnitude and an angle (a phasor) and AC circuits can be solved with Ohm's law, V = IZ, instead of differential equations.
Phase
In an inductor the current lags the voltage by 90°, and in a capacitor it leads by 90°. A motor is strongly inductive, so the current follows late when voltage is applied. This is one reason FOC controls current directly and not voltage.