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Exercise E1 Resonant Circuit
(written test, approx. 4 % of a 120-minute written test, SS2021)

Multiphase systems (10 points) A symmetrical three-phase generator in a delta connection shall be considered in the following.
A voltage with the RMS value $U_{\rm RMS} = 110~\rm V$ is applied between the terminals of each winding.
Through each of the windings, there is a current with an RMS value $I_{\rm RMS} = 5 ~\rm A$ and a phase shift of $\varphi = +25°$ compared to the voltage.

a) Draw the circuit diagram.

Result

ee2:ezrkjzifcegttcpc_solution1.svg

b) Specify the RMS value of the phase voltage $U_\rm L$ and the string voltage $U_\rm S$.

Path

Since the given voltage of $U_{\rm RMS} = 110~\rm V$ is applied between the terminals of each winding, this is also the string voltage $U_\rm S$.
For delta configuration, the phase voltage $U_\rm L$ is equal to the string voltage $U_\rm S$.

Result

  • $U_{\rm S} = 110~\rm V$
  • $U_{\rm L} = 110~\rm V$

c) Specify the RMS value of the phase current $I_\rm L$ and the string current $I_\rm S$.

Path

Result

$f_0 = 205.5 \rm Hz$

d) Determine the active power.

Path

The resonant frequency $f_0$ is given as \begin{align*} f_0 = {{1}\over{ 2\pi \sqrt{LC} }} \end{align*}

With the values: \begin{align*} f_0 &= {{1}\over{ 2\pi \sqrt{20 \cdot 10^{-3} ~\rm H \cdot 30 \cdot 10^{-6} ~\rm F} }} \\ &= 205.4681... \rm Hz \end{align*}

Result

$f_0 = 205.5 \rm Hz$