# 交流电功率

## 公式

### 实數

• 有功功率、平均功率或实功率 P：瓦特（W）
• 无功功率或虛功率 Q：无功伏安（var）
• 电压-电流相位差或功率角（power angle） ${\displaystyle \theta =\theta _{v}-\theta _{i}}$ 角度（°）或弧度（rad）
• 视在功率 S：伏安（VA）
• 功率因數（power factor）：無單位

${\displaystyle p(t)=v(t)\,i(t)}$ ${\displaystyle ={\frac {V_{m}I_{m}}{2}}\cos \,\theta \,(1+\cos \,2\omega t)-{\frac {V_{m}I_{m}}{2}}\sin \,\theta \,\sin \,2\omega t}$ ${\displaystyle ={\frac {V_{m}I_{m}}{2}}\cos \,\theta +{\frac {V_{m}I_{m}}{2}}\cos \,(\theta +2\omega t)}$  ${\displaystyle =P+|S|\,\cos \,(\theta +2\omega t)}$

${\displaystyle S=V\,I={\sqrt {P^{2}+Q^{2}}}}$

${\displaystyle pf={\frac {P}{S}}=\cos \,\theta }$

${\displaystyle Z={\sqrt {R^{2}+X^{2}}}}$
${\displaystyle V=IZ}$

### 複數

${\displaystyle j^{2}=-1}$

${\displaystyle |S|={\sqrt {P^{2}+Q^{2}}}}$
${\displaystyle S=VI^{*}=P+jQ}$
${\displaystyle S=I^{2}Z={\frac {V^{2}}{Z^{*}}}}$

${\displaystyle pf={\frac {P}{|S|}}=\cos \,\theta }$

${\displaystyle Z=R+jX}$
${\displaystyle V=IZ}$

${\displaystyle V=V_{pk}\angle \theta _{v}}$
${\displaystyle I=I_{pk}\angle \theta _{i}}$
${\displaystyle S=V_{pk}I_{pk}\angle (\theta _{v}-\theta _{i})=VI^{*}}$

## 參考資料

1. ^ Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. The Analysis and Design of Linear Circuits 8. Wiley. 2016: 812–813. ISBN 978-1-119-23538-5.
2. ^ IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions. IEEE. 2010. ISBN 978-0-7381-6058-0. doi:10.1109/IEEESTD.2010.5439063.
3. ^ 《Electric Circuits》10th edition by James W. Nilsson & Susan A. Riedel (2014)