Derive an expression for the average power consumed in an AC circuit. Define power factor and wattless current, and state the power factor for a pure resistor and a pure inductor.

1. 1 · Write v and i. v = vₘ sin ωt; i = iₘ sin(ωt + φ).
2. 2 · Write the instantaneous power. p = vi = (vₘiₘ/2)[cos φ − cos(2ωt + φ)].
3. 3 · Average over a cycle. The cos(2ωt + φ) term averages to zero, so P = (vₘiₘ/2) cos φ = VI cos φ.
4. 4 · Define power factor. cos φ; it is 1 for a pure resistor and 0 for a pure inductor or capacitor.
5. 5 · Define wattless current. I sin φ, the component at π/2 to V; it uses no power.
6. 6 · Real life. Factories add capacitor banks to raise their power factor.
7. 7 · Close in one line. So, only the in-phase component I cos φ does useful work.

Scoring tip: Write the trigonometric identity you use; it earns the derivation step.