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Exercise 1.1 : Complex Impedance Circuit
(written test, approx. 15% of a 60-minute written test, WS2022)
A circuit designed to filter the noise from a signal shall be analysed.
The input is given by a voltage source $u(t) = 3.0 V \cdot sin(2\pi \cdot 15 kHz \cdot t)$ with an internal resistance of $10 \Omega$.
This linear source is connected with an inductor of $330 \mu H$ and a capacitor of $0.22 \mu F$, all in series.
1. Draw the circuit diagram of the given circuit.
Label all components, voltages and currents.
2. Calculate the single impedance $|\underline{Z}_C|$, $|\underline{Z}_L|$ such as $|\underline{Z}|$ of the overall circuit.
\begin{align*} Z_L &= 2\pi \cdot f \cdot L\\ &= 2\pi \cdot 15 kHz \cdot 0.22 \mu F\\ \end{align*}
\begin{align*} Z_C &= {{1}\over{2\pi \cdot f \cdot C}}\\ &= {{1}\over{2\pi \cdot 15 kHz \cdot 330 \mu H}}\\ \end{align*}
\begin{align*} \underline{Z} &= R + \underline{Z}_L + \underline{Z}_C \\ &= R + j \cdot {Z}_L - j \cdot {Z}_C \\ &= R + j \cdot ({Z}_L - {Z}_C) \\ |\underline{Z}| &= \sqrt{R^2 + (\underline{Z}_L - \underline{Z}_C)^2 }\\ \end{align*}
3. Draw the three impedance phasors $|\underline{Z}_C|$, $|\underline{Z}_L|$ and $|\underline{Z}_R|$ in a diagram.
Choose and appropriate scaling factor and write it down.
4. Calculate the current $|\underline{I}|$.
With $I = {{1}\over{\sqrt{2}}}\cdot \hat{I}$: \begin{align*} I &= {{1}\over{\sqrt{2}}}\cdot {{\hat{U}}\over{Z}} \\ &= {{1}\over{\sqrt{2}}}\cdot {{3.0 V}\over{19.28 \Omega}} \\ \end{align*}