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<fs medium>(written test, approx. 10 % of a 120-minute written test, SS2024)</fs> #@TaskText_HTML@#

A real capacitor behaves like an $RLC$ resonant circuit, with an equivalent series resistance $R$ and an equivalent series inductance $L$. 

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A given capacitor shall have the following values: 
  * $C=10 ~\rm nF$
  * $R=20 ~\rm m\Omega$ 
  * $L=1.6 ~\rm nH$

1. What is the impedance $Z_{RLC}$ of this real capacitor for $f_0=44 ~\rm MHz$? (Phase and magnitude)

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The impedance is based on the resistance $R$ and the reactance $X_{LC}= {\rm j}\cdot (X_L - X_C)$:
\begin{align*}
\underline{Z}_{RLC} &= R + {\rm j}\cdot (X_L - X_C) \\
                    &= R + {\rm j}\cdot (\omega L - {{1}\over{\omega C}}) \\
                    &= R + {\rm j}\cdot (2\pi f \cdot L - {{1}\over{2\pi f \cdot C}}) \\
\end{align*}

The reactive part is
\begin{align*}
X_{LC} &= 2\pi f \cdot L - {{1}\over{2\pi f \cdot C}} \\
       &= 2\pi 44 \cdot 10^{6} {~\rm MHz} \cdot  1.6 \cdot 10^{-9} {~\rm H} - {{1}\over{2\pi \cdot 10^{6} {~\rm MHz} \cdot 10 \cdot 10^{-9} {~\rm F}}} \\
       &= +0.08062... ~\Omega \\
\end{align*}

To get the magnitude of the impedance $|\underline{Z}_{RLC}|$ one can use the Pythagorean Theorem:
\begin{align*}
|\underline{Z}_{RLC}| &= \sqrt{R^2 + X_{LC}^2} \\
                      &= \sqrt{(0.020~\Omega)^2 + ( 0.08062... ~\Omega )^2} \\
                      &= 0.0830 ... ~\Omega \\
\end{align*}

For the phase $\varphi$ the $\arctan$ can be applied:
\begin{align*}
\varphi &= \arctan \left( {{X_{LC}}\over{R}} \right) \\
        &= \arctan \left( {{0.08062... ~\Omega}\over{0.020 ~\Omega}} \right) \\
        &= 1.3276 ... \hat{=} +76° \\
\end{align*}

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  * $|\underline{Z}_{RLC}| = 83.0  ~\rm m \Omega$
  * $\varphi = +76°$
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2. What is the resonance frequency $f_r$ for the given capacitor? What is the impedance in this case?

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The formula for the resonance frequency $f_r$ is:
\begin{align*}
f_r &= {{1}\over{2\pi \sqrt{LC}}} \\
    &= {{1}\over{2\pi \sqrt{1.6 \cdot 10^{-9} {~\rm H} \cdot 10 \cdot 10^{-9} {~\rm F}}}} \\
    &= 39.788... ~\rm MHz \\
\end{align*}

The impedance at resonance is purely the resistance.

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  * $f_r = 39.79 ~\rm MHz$
  * $|\underline{Z}_{RLC}(f_r)| = 20.0  ~\rm m \Omega$
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3. For an application, the component shall be used in resonance on a supply of $5 ~\rm V$.
What is the voltage on the ideal capacity $C$ in the shown circuit?

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The voltage on the ideal capacity is the input voltage by the $Q$-factor increased:
\begin{align*}
U_C &= U_{\rm s} \cdot Q \\
    &= U_{\rm s} \cdot \sqrt{ {{L}\over{C}} } \cdot {{1}\over{R}}\\
    &= 5 {~\rm V} \cdot \sqrt{ {{ 1.6 \cdot 10^{-9} {~\rm H} }\over{ 10 \cdot 10^{-9} {~\rm F} }} } \cdot {{1}\over{0.020~\Omega}}\\
    &= 100 ~\rm V
\end{align*}

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$U_C = 100 ~\rm V$
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