In one line
In a pure resistor current is in phase with voltage, in a pure inductor current lags voltage by 90°, and in a pure capacitor current leads voltage by 90°; L and C consume no average power.
Explained simply
A resistor reacts instantly, like a light switch. An inductor resists change in current, like a heavy flywheel that is slow to speed up, so current comes late (lags). A capacitor must fill with charge before voltage builds up, like a bucket that fills before its level rises, so current comes first (leads). A handy memory word is CIVIL: in C, I leads V; V leads I in L.
Technically, opposition to AC by L and C is called reactance: \( X_L = \omega L \) rises with frequency, \( X_C = 1/(\omega C) \) falls with frequency.
How it works
| Circuit | Opposition | Phase of current | Average power | Power factor |
|---|---|---|---|---|
| Pure R | \( R \) | in phase | \( VI \) | 1 |
| Pure L | \( X_L = 2\pi fL \) | lags by 90° | 0 | 0 (lagging) |
| Pure C | \( X_C = 1/(2\pi fC) \) | leads by 90° | 0 | 0 (leading) |
- For \( v = V_m\sin\omega t \) across L: \( i = \frac{1}{L}\int v\,dt = \frac{V_m}{\omega L}\sin(\omega t - 90^\circ) \).
- Across C: \( i = C\,dv/dt = \omega CV_m\sin(\omega t + 90^\circ) \).
- In L and C, energy is stored in one quarter-cycle and returned in the next, so the average power is zero.
Key formula
\( X_L = 2\pi fL, \qquad X_C = \dfrac{1}{2\pi fC}, \qquad I = \dfrac{V}{X} \)
\( X_L, X_C \) = inductive and capacitive reactance (\( \Omega \)), \( f \) = frequency (Hz), \( L \) = inductance (H), \( C \) = capacitance (F), \( V, I \) = RMS voltage (V) and current (A).
Quick example
A 0.1 H inductor on 230 V, 50 Hz: \( X_L = 2\pi\times50\times0.1 = 31.4\,\Omega \), \( I = 230/31.4 = 7.32 \) A, lagging by 90°, and the average power taken is zero. On DC (\( f = 0 \)) the same inductor has zero reactance, while a capacitor has infinite reactance and blocks DC.
Common mistakes
- Saying an inductor blocks DC; it is the capacitor that blocks DC.
- Treating reactance like resistance that dissipates heat; reactance stores and returns energy.
- Writing \( X_C = 2\pi fC \) instead of its reciprocal.
How MU asks it
Asked as "Show that current lags voltage by 90° in a purely inductive circuit and that the average power is zero" (5 marks), or the same for a capacitor. For full marks: start from \( v = V_m\sin\omega t \), derive \( i \), draw waveforms of \( v \), \( i \) and \( p \), draw the phasor diagram, and show the power integral equals zero.