AC Through Resistor, Inductor and Capacitor: Phase Guide

When alternating voltage is applied separately to a resistor, an inductor and a capacitor, each element responds in a distinctly different way. The resistor keeps voltage and current rising and falling together, the inductor makes the current lag behind, and the capacitor makes the current lead ahead. This behaviour, especially the phase relationship in each case, is the foundation of all AC circuit analysis.

The topic matters because every real device is a mixture of these three behaviours. A motor coil behaves mainly like an inductor, a cable has capacitance, and heating elements are resistors. Knowing how each pure element treats AC lets us predict phase difference, power consumption and the opposition offered to the current in any practical circuit.

Key Concepts

• Resistive circuit: voltage and current are in phase; the opposition is the resistance R in ohms.
• Inductive reactance (XL): the opposition offered by an inductor, XL = ωL, measured in ohms; it increases with frequency.
• Capacitive reactance (XC): the opposition offered by a capacitor, XC = 1/(ωC), measured in ohms; it decreases with frequency.
• Phase in resistor: current and voltage reach their peaks together, phase difference zero.
• Phase in inductor: current lags the voltage by 90 degrees, that is, by π/2 radians.
• Phase in capacitor: current leads the voltage by 90 degrees, that is, by π/2 radians.

Formulas

For a resistor, i = (V₀/R) sin(ωt) when v = V₀ sin(ωt), so I₀ = V₀ / R and the average power is P = Vrms Irms. For an inductor, XL = ωL = 2πfL and I₀ = V₀ / XL, with current lagging by π/2. For a capacitor, XC = 1/(ωC) = 1/(2πfC) and I₀ = V₀ / XC, with current leading by π/2.

The average power over a full cycle is Pav = Vrms Irms cos φ, where φ is the phase difference. For a pure inductor or pure capacitor, φ = 90° and cos φ = 0, so the average power consumed is zero: energy is borrowed and returned each cycle, not used up.

Worked Example

A 230 V RMS, 50 Hz supply is connected first to a pure inductor of 0.5 H and then to a pure capacitor of 20 μF. Find the RMS current in each case.

1. Inductor: XL = 2πfL = 2π × 50 × 0.5 = 157 Ω. Current Irms = 230 / 157 = 1.46 A, lagging the voltage by 90°.
2. Capacitor: XC = 1 / (2πfC) = 1 / (2π × 50 × 20 × 10⁻⁶) = 1 / (6.28 × 10⁻³) = 159 Ω. Current Irms = 230 / 159 = 1.45 A, leading the voltage by 90°.

The two currents are almost equal in size but opposite in phase behaviour: one lags, the other leads. This opposition is exactly what makes resonance possible when an inductor and capacitor are combined.

Common Misconceptions

• An inductor blocks DC completely. Wrong. A pure inductor offers zero reactance to steady DC because XL depends on frequency. It opposes only changing current, and a real coil still has some wire resistance.
• A capacitor blocks AC. Wrong. It is DC that a capacitor blocks after charging. AC passes because the capacitor charges and discharges continuously, giving an alternating current whose size grows with frequency.
• Current through an inductor is wasted as heat like in a resistor. Wrong. In a pure inductor the average power is zero. Energy is stored in the magnetic field for part of the cycle and returned to the supply in the next part.

Real-World Applications

Inductive behaviour dominates in motor and transformer windings, which is why industrial supplies need power factor correction. Capacitors are used in fan regulators, filters and for improving power factor in factories. The frequency dependence of XL and XC is used in radio tuning, audio crossovers that send bass to woofers and treble to tweeters, and in filters that remove the 50 Hz hum from audio equipment. Choke coils exploit high XL at high frequency to block interference while passing the wanted supply.

A resistor keeps AC voltage and current in phase and consumes real power, while a pure inductor lags the current by 90 degrees and a pure capacitor leads it by 90 degrees, both consuming zero average power. These three simple phase rules combine to explain every AC circuit that follows.