Reactance, impedance and phasor diagrams

Key ideas

• In AC, capacitors and inductors oppose current without dissipating energy. This frequency-dependent opposition is reactance (unit: ohms):
  • Capacitive reactance — falls as frequency rises (more frequent recharging = more current):

$X_C = \frac{1}{\omega C}$

• Inductive reactance — rises with frequency (faster current changes = more back-EMF):

$X_L = \omega L$

• Phase: in a resistor, voltage and current rise and fall together. In L and C they are a quarter-cycle (90°) apart:
  • inductor — voltage leads the current by 90°
  • capacitor — voltage lags the current by 90°
  • (memory aid: CIVIL — in a C, I leads V; V leads I in an L)
• A phasor diagram draws each voltage as an arrow rotating at $\omega$; lengths are the peak (or rms) values, angles are the phases. In a series circuit every component shares the same current, so the current phasor is the reference:
  • $V_R$ along the current
  • $V_L$ at 90° ahead; $V_C$ at 90° behind
  • the supply voltage is the vector sum
• Because $V_R$ and $V_{L}$ (or $V_C$) are perpendicular, the sum uses Pythagoras — voltages in AC series circuits don't simply add.
• Impedance $Z$ is the circuit's total opposition, defined through $V = IZ$. For a series LR circuit: $Z = \sqrt{R^2 + X_L^2}$ (and likewise with $X_C$ for CR).

Tips

• Frequency dependence is the favourite explain question: high frequency → inductor dominates ($X_L\uparrow$), capacitor fades ($X_C\downarrow$); low frequency → the reverse. A capacitor blocks DC entirely ($X_C \to \infty$ at $f = 0$).
• In a series circuit, always take the current as the phasor reference — every component shares it. Trying to reference the supply voltage first makes the diagram unbuildable.