Power, Choke Coils & Hot-Wire Meters

In an AC circuit, current times voltage isn’t the whole story: if the two are out of step, some energy just sloshes back and forth. The power factor measures how much is really used.

Average power#

With ℰ = ℰ₀ sin ωt and i = i₀ sin(ωt + φ), the work done in time dt is ℰi dt = ℰ₀i₀(sin²ωt cos φ + sin ωt cos ωt sin φ) dt. Over a full cycle the second term averages to zero:

• Purely resistive: φ = 0, cos φ = 1, P = ℰ_rmsi_rms.
• Purely reactive (only L and/or C): φ = ±π/2, cos φ = 0, no power is absorbed, even though current flows.

The choke coil#

A choke is a coil of large inductance and small resistance. Old fluorescent tube fittings put one in series with the tube to cut down the voltage across it. Treating the tube as a resistance R and the choke as an ideal inductor, the circuit is an LR circuit with Z = √(R² + ω²L²), and the tube’s rms voltage is

The advantage over a series resistor: an ideal inductor consumes no power (its voltage and current are 90° apart), so no energy is wasted as heat.

Hot-wire instruments#

A moving-coil meter can’t read AC: its torque reverses every half cycle and averages to zero. AC meters use an effect that depends on i², which is never negative. In a hot-wire ammeter, current heats a platinum–iridium wire AB stretched between fixed points; it expands and sags, and a spring pulls a thread wound round a small drum, turning a pointer. The rise in temperature, and so the deflection, depends on i_rms²; the scale is marked to read i_rms directly. A small shunt keeps the meter’s resistance low. A hot-wire voltmeter is the same but with a large series resistance instead of a shunt.