Instantaneous & Average Power

In DC circuits, power is simply P = VI — a constant. In AC circuits, both voltage and current vary with time, so the power delivered at any instant is the product of the instantaneous voltage and instantaneous current:

The instantaneous power p(t) tells you exactly how much energy per second is flowing into the load at moment t. For a resistor, this power is always non-negative (the resistor never returns energy). For reactive elements like capacitors and inductors, p(t) swings positive and negative — energy flows in during part of the cycle and back out during another part.

When voltage and current are sinusoids — v(t) = V_m cos(ωt + θ_v) and i(t) = I_m cos(ωt + θ_i) — we can expand the product using the trigonometric identity cos(A)cos(B) = ½[cos(A−B) + cos(A+B)]. This splits p(t) into two distinct terms with very different physical meanings.

The constant term, (V_m I_m/2) cos(θ_v − θ_i), represents the net energy per cycle that is converted to heat, light, mechanical work, or otherwise dissipated. This is the real or average power.

The oscillating term, (V_m I_m/2) cos(2ωt + θ_v + θ_i), swings positive and negative at twice the source frequency. Over one complete period T = 2π/ω, the cosine of 2ωt goes through two full cycles, so its integral — and hence its average — is exactly zero. It represents energy sloshing back and forth between source and load without net delivery.

Because the double-frequency term integrates to zero over one period, the average power is simply the constant term from the expansion:

The factor cos(θ_v − θ_i) is called the power factor. It ranges from 0 to 1 and tells you what fraction of the maximum possible power is actually delivered. A purely resistive load has θ_v = θ_i (zero phase difference), giving a power factor of 1. A purely reactive load (ideal capacitor or inductor) has a 90° phase difference, giving a power factor of 0 — no net power is delivered, even though current flows.