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ee2:task_d9io924n0e3du21g_with_calculation [2024/07/16 00:17] (aktuell) mexleadmin angelegt |
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+ | {{tag> | ||
+ | |||
+ | # | ||
+ | <fs medium> | ||
+ | A symmetric and balanced three-phase motor is driven with a $230 ~\rm V$ / $400 ~\rm V$ / $50 ~\rm Hz$ three-phase power net. | ||
+ | Each single string has a resistor $R=5 ~\Omega$ and an inductance of $L=10 ~\rm mH$. | ||
+ | |||
+ | {{drawio> | ||
+ | |||
+ | 1. Calculate the $\cos \varphi$, and the magnitude of the impedance $|Z|$ for a single string. | ||
+ | |||
+ | # | ||
+ | |||
+ | The phase $\varphi$ is given by: | ||
+ | \begin{align*} | ||
+ | \varphi &= \arctan \left( {{X_{L}}\over{R}} \right) \\ | ||
+ | &= \arctan \left( {{2\pi \cdot f \cdot L}\over{R}} \right) \\ | ||
+ | &= \arctan \left( {{2\pi \cdot 50 {~\rm Hz} \cdot 10 \cdot 10^{-3} {~\rm H}}\over{5 ~\Omega}} \right) \\ | ||
+ | &= 0.5609 ... \hat{=} +32° \\ | ||
+ | \end{align*} | ||
+ | |||
+ | With this, the $\cos \varphi$ becomes | ||
+ | \begin{align*} | ||
+ | \cos \varphi | ||
+ | &= 0.84673...\\ | ||
+ | \end{align*} | ||
+ | |||
+ | The impedance is given by: | ||
+ | \begin{align*} | ||
+ | |\underline{Z}_{RL}| | ||
+ | &= \sqrt{( 2\pi \cdot f \cdot L )^2 + R^2} \\ | ||
+ | &= \sqrt{( 2\pi \cdot 50 {~\rm Hz} \cdot 10 \cdot 10^{-3} {~\rm H})^2 +(5 ~\Omega)^2} | ||
+ | &= 5.905... ~\Omega\\ | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | * $|\underline{Z}_{RL}| = 5.90 ~\Omega$ | ||
+ | * $\cos\varphi = 0.84673$ | ||
+ | # | ||
+ | |||
+ | 2. Calculate the true power, the apparent power, and the reactive power of the motor. | ||
+ | |||
+ | # | ||
+ | The apparent power $S$ is given by | ||
+ | \begin{align*} | ||
+ | S &= 3 \cdot U_s \cdot I_s \\ | ||
+ | &= 3 \cdot {{U_s^2}\over{Z_{RL}}} \\ | ||
+ | &= 3 \cdot {{(230~\rm V)^2}\over{ 5.90 ~\Omega }} \\ | ||
+ | &= 26.898... ~\rm kVA \\ | ||
+ | \end{align*} | ||
+ | |||
+ | The active power is | ||
+ | \begin{align*} | ||
+ | P &= S \cdot \cos \varphi \\ | ||
+ | &= 26.898... \cdot 0.84673 ~\rm kW \\ | ||
+ | &= 22.775... ~\rm kW \\ | ||
+ | \end{align*} | ||
+ | |||
+ | The reactive power is | ||
+ | \begin{align*} | ||
+ | Q &= \sqrt{S^2 - P^2} \\ | ||
+ | &= \sqrt{ (26.898... ~\rm kVA)^2 - (22.775... ~\rm kW)^2} \\ | ||
+ | &= 14.310...~\rm kVAr \\ | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | * active power: | ||
+ | * reactive power:\\ $Q=14.310~\rm kVAr$ \\ \\ | ||
+ | * apparent power:\\ $S=26.898~\rm kVA$ \\ \\ | ||
+ | # | ||
+ | |||
+ | # |