Exercise E8 Impedances at Frequencies
(written test, approx. 14 % of a 60-minute written test, SS2023)
At which frequencies show the following components the given values?
1. An inductor with $X_{L1} = 60 ~\rm m\Omega$ and $L_1 = 15.9 ~\rm \mu H$.
\begin{align*} X_{L1} &= \omega_1 \cdot L = 2\pi f_1 \cdot L_1 \\ \rightarrow f_1 &= {{X_{L1}} \over {2\pi \cdot L_1}} \\ &= {{60 ~\rm m\Omega} \over {2\pi \cdot 15.9 ~\rm \mu H}} \\ &= {{60 \cdot 10^{-3}} \over {2\pi \cdot 15.9 \cdot 10^{-6}}} \rm {{ {{V}\over{A}} } \over { {{Vs}\over{A}}}} \\ \end{align*}
\begin{align*} f_1 = 600.58...~{\rm Hz} \rightarrow f_1 = 600~{\rm Hz} \end{align*}
2. A capacitor with $C_2 = 5.2 ~\rm nF$, where an AC voltage of $U_2 = 6.8 ~\rm V$ generates a current $I_2 = 1 ~\rm mA$.
\begin{align*} X_{C2} &= {{1} \over{\omega_2 \cdot C_2} } = {{1} \over{2\pi f_2 \cdot C_2}}={{U_2} \over {I_2}} \\ \rightarrow f_2 &= {{I_2} \over{2\pi \cdot U_2 \cdot C_2}} \\ &= {{1 \cdot 10^{-3}} \over{2\pi \cdot6.8 \cdot 5.2 \cdot 10^{-9}}} \rm {{A} \over{V \cdot {{As} \over{V}} }} \end{align*}
\begin{align*} f_2 = 4'500.9...~{\rm Hz} \rightarrow f_2 = 45.0 ~{\rm kHz}\end{align*}
3. An inductor with $L_3 = 50 ~\rm \mu H$, which shows the same absolute value of the impedance as a capacitor with $C_3 = 5.6 ~\rm nF$.
\begin{align*} X_{L3} &=X_{C3} \\ \omega_3 \cdot L_3 &= {{1} \over{\omega_3 \cdot C_3} } \\ 2\pi f_3 \cdot L_3 &= {{1} \over{2\pi f_3 \cdot C_3} } \\ \rightarrow f_3 &= {{1} \over{2\pi}} \sqrt{{{1} \over {C_3 \cdot L_3}}} \\ f_3 &= {{1} \over{2\pi}} \sqrt{{{1} \over {5.6\cdot 10^{-9} \cdot 50\cdot 10^{-6}}}. {{1}\over\rm {V/As \cdot A/Vs}} } \\ \end{align*}
\begin{align*} f &= 300'774.5 ... ~\rm Hz \rightarrow f &= 300 ~\rm kHz \end{align*}