PX157 - E5 - AC power

power in circuits with inductors and capacitors
typically, $I$ and $V$ are $\pm 90 °$ out of phase:

\begin{aligned} V (t) & = V_{0} \cos (ω t) \\ I (t) & = I_{0} \sin (ω t) \end{aligned}

instantaneous power:

P (t) = V (t) I (t) = V_{0} I_{0} \cos (ω t) \sin (ω t)

average power over a cycle:

\begin{aligned} ⟨ P ⟩ & = \frac{1}{T} \int_{0}^{T} P (t) d t \\ = \frac{1}{T} \int_{0}^{T} V_{0} I_{0} \cos (ω t) \sin (ω t) d t \\ = \frac{V_{0} I_{0}}{2 T} \int_{0}^{T} \sin (2 ω t) d t \\ = 0 \end{aligned}

no electrical energy dissipated over a cycle in pure inductors and capacitors

circuits with a resistive load

instantaneous power:

P (t) = R I^{2} (t) = R I_{0}^{2} \cos^{2} (ω t)

average power:

\begin{aligned} ⟨ P ⟩ & = \frac{1}{T} \int_{0}^{T} R I_{0}^{2} (\frac{1}{2} + \frac{1}{2} \cos (2 ω t)) d t \\ = \frac{R I_{0}^{2}}{2} \end{aligned}

root-mean-square current and potential:

\begin{aligned} I_{r m s} & = \frac{I_{0}}{\sqrt{2}} \\ V_{r m s} & = \frac{V_{0}}{\sqrt{2}} \\ ⟨ P ⟩ & = R I_{r m s}^{2} \\ = V_{r m s} I_{r m s} \\ = \frac{V_{r m s}^{2}}{R} \end{aligned}

general power dissipation

it is expected that the power delivered by the source is equal to the power dissipated in the resistor

power from source

\begin{aligned} V (t) & = V_{0} \cos (ω t) \\ I (t) & = I_{0} \cos (ω t + ϕ) \\ P (t) & = V_{0} I_{0} \cos (ω t) \cos (ω t + ϕ) \\ = V_{0} I_{0} \cos (ω t) [\cos (ω t) \cos (ϕ) - \sin (ω t) \sin (ϕ)] \\ = V_{0} I_{0} \cos^{2} (ω t) \cos (ϕ) - V_{0} I_{0} \sin (ω t) \cos (ω t) \sin (ϕ) \\ ⟨ P ⟩ & = \frac{1}{2} V_{0} I_{0} \cos ϕ \\ = V_{r m s} I_{r m s} \cos ϕ \end{aligned}

$\cos ϕ$ is the power factor

power dissipated in the source

\begin{aligned} P (t) & = V_{R} (t) I (t) \\ I (t) & = I_{0} \cos (ω t + ϕ) \\ V_{R} (t) & = V_{R_{0}} \cos (ω t + ϕ) \\ ⟨ P ⟩ & = ⟨ V_{R} I ⟩ \\ = V_{R_{0}} I_{0} ⟨ \cos^{2} (ω t + ϕ) ⟩ \\ = \frac{1}{2} V_{R_{0}} I_{0} \\ = \frac{1}{2} R I_{0}^{2} \\ = R I_{r m s}^{2} \end{aligned}

the power factor has disappeared!

\begin{aligned} ⟨ P ⟩ & = \frac{1}{2} V_{0} I_{0} \cos ϕ \\ = \frac{1}{2} I_{0}^{2} Z \cos ϕ \\ = \frac{1}{2} R I_{0}^{2} \end{aligned}