AC Power

Magnitude & Root-Mean-Square Voltage & Current

magnitude voltage $\large V_m$ magnitude current $\large I_m$ are defined as peak values of voltage and current respectively.

root-mean-square voltage $\large V_{rms}$ root-mean-square current $\large I_{rms}$ are then defined as:

\begin{matrix} V_{r m s} = \sqrt{\frac{1}{T} \int_{0}^{T} (v (t))^{2} d t} \\ I_{r m s} = \sqrt{\frac{1}{T} \int_{0}^{T} (i (t))^{2} d t} \end{matrix}

$\large v(t)$ $\large i(t)$ are voltage and current as functions of time respectively.

For a sinusoidal voltage and current:

\begin{matrix} V_{r m s} = \frac{1}{\sqrt{2}} V_{r m s} \\ I_{r m s} = \frac{1}{\sqrt{2}} I_{r m s} \end{matrix}

$\large \vec{V}$ $\large \vec{I}$ flowing through it, such that:

\begin{matrix} \vec{V} = V_{m} ∠ θ_{v} = \sqrt{2} V_{r m s} ∠ θ_{v} \\ \vec{I} = I_{m} ∠ θ_{i} = \sqrt{2} I_{r m s} ∠ θ_{i} \end{matrix}

$\large V_m$ $\large I_m$ $\large V_{rms}$ $\large I_{rms}$ $\large \theta_v$ $\large \theta_i$ are voltage and current phase angles respectively.

apparent power $\large \vec{S}$ of the system is given by:

\vec{S} = P + j Q = \vec{V} {\vec{I}}^{*}

$\large P$ $\large Q$ are the active power and reactive power respectively. They are given by:

\begin{matrix} P = \frac{V_{m} I_{m}}{2} c o s (ϕ) = V_{r m s} I_{r m s} c o s (ϕ) \\ Q = \frac{V_{m} I_{m}}{2} s i n (ϕ) = V_{r m s} I_{r m s} s i n (ϕ) \\ \frac{Q}{P} = t a n (ϕ) \end{matrix}

$\large \phi$ is the phase difference between voltage and current:

ϕ = θ_{v} - θ_{i}

power factor $\large pf$ is given by:

\begin{matrix} p f = c o s (ϕ) = \frac{P}{| S |} \\ θ_{v} - θ_{i} < 0 ⟹ leading power factor \\ θ_{v} - θ_{i} > 0 ⟹ lagging power factor \end{matrix}