Instantaneous 76754

For a dipole, calculate the complex apparent power S and the instantaneous value of the current i(t), given: R=10 Ω, C=100uF, f=50 Hz, u(t)= square root of 2, sin( ωt - 30 °). Thanks for any help or advice.

Correct answer:

S =  0.0599 VA
i = 0.0424*sin(ωt−102.5594)

Step-by-step explanation:

$$\begin{gathered} R=10\Omega \\ C=100\cdot 10^{-6}=0.0001\text{F} \\ f=50\text{Hz} \\ U=\sqrt{2}=\sqrt{2}\text{V}\doteq 1.4142\text{V} \\ \alpha =-30{}^{\circ } \\ \omega =2\pi \cdot f=2\cdot 3.1416\cdot 50\doteq 314.1593 \\ {Z}_1=\frac{1}{\omega \cdot C}=\frac{1}{314.1593\cdot 0.0001}\doteq 31.831\Omega \\ Z=\sqrt{R^2+Z_1^2}=\sqrt{10^2+31.831^2}\doteq 33.3648\Omega \\ I=\frac{U}{Z}=\frac{1.4142}{33.3648}\doteq 0.0424\text{A} \\ S=I\cdot U=0.0424\cdot 1.4142=0.0599\text{VA} \end{gathered}$$

$$\begin{gathered} \tan \beta =Z_1:R=31.831:10\doteq 3.1831=165849:52103 \\ \beta =\frac{180°}{\pi }\cdot \arctan (Z_1/R)=\frac{180°}{\pi }\cdot \arctan (31.831/10)\doteq 72.5594{}^{\circ } \\ \phi =\alpha -\beta =(-30)-\beta \doteq -102.5594{}^{\circ } \\ i(t)=I\cdot \sin (\omega t+\phi ) \\ i=0.0424\cdot \sin (\omega t-102.5594) \end{gathered}$$

Try another example