Summary: Alternating Current

\[ \begin{gathered} u(t) = \hat{u} \cdot \sin(\omega t + \varphi) \\[6pt] \omega = 2\pi f = \frac{2\pi}{T},\; f = \frac{1}{T} \\[6pt] \text{Calculating the phase angle: } \varphi = \pm \frac{\delta t}{T} \cdot 2\pi \\[6pt] u(\omega) = |u| \cdot e^{j\varphi} = \hat{u} \cdot e^{j\varphi} \\[6pt] |u| = \hat{u} = \sqrt{\mathrm{Re}\{u\}^2 + \mathrm{Im}\{u\}^2} \\[6pt] \varphi = \arctan\left(\frac{\mathrm{Im}\{u\}}{\mathrm{Re}\{u\}}\right) \end{gathered} \]

\[ \begin{gathered} \text{Example: } u(\omega) = 1\,\mathrm{V} + j1\,\mathrm{V} = \sqrt{2}\,\mathrm{V} \cdot e^{j\frac{\pi}{4}},\; |u| = \sqrt{2}\,\mathrm{V},\; \varphi = \pi/4 \\[6pt] \text{Impedances: } u(\omega) = Z \cdot i(\omega) \text{ with } Z_R = R,\; Z_L = j\omega L,\; Z_C = \frac{1}{j\omega C} = -j\frac{1}{\omega C} \\[6pt] \text{For sinusoidal curves: } \overline{u} = 0\,\mathrm{V},\; u_{\mathrm{eff}} = \frac{\hat{u}}{\sqrt{2}} \\[6pt] \text{Power: } s = u \cdot i^*;\; p = \mathrm{Re}\{s\},\; q = \mathrm{Im}\{s\};\; s = p + jq \\[6pt] \varphi = \varphi_u - \varphi_i,\; \cos(\varphi) = \frac{p}{|s|} \\[6pt] s_{\mathrm{eff}} = \frac{u}{\sqrt{2}} \cdot \frac{i^*}{\sqrt{2}} = \frac{s}{2} \\[6pt] \text{Resistors: only active power; energy stores: only reactive power} \end{gathered} \]