Unterschiede
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ee2:task_nyniewamxfshpuwt_with_calculation [2024/07/04 03:52] (aktuell) mexleadmin angelegt |
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+ | {{tag> | ||
+ | |||
+ | # | ||
+ | <fs medium> | ||
+ | Given a resonant circuit, that is fed by a linear voltage source (circuit diagram on the right). \\ | ||
+ | The inductance $L$ and capacitance $C$ are fixed. The resistance $R$ can be varied. | ||
+ | * $u_{\rm s} = 12{~\rm V} \cdot \sin (2 \pi \cdot f_0 \cdot t) $ | ||
+ | * $R_i = 200~\rm m\Omega$ | ||
+ | * $L = 20~\rm mH$ | ||
+ | * $C = 30~\rm \mu F$ | ||
+ | |||
+ | {{drawio> | ||
+ | |||
+ | a) What is the resonance frequency $f_0$? | ||
+ | | ||
+ | # | ||
+ | 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*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | $f_0 = 205.5 \rm Hz$ | ||
+ | # | ||
+ | |||
+ | b) How large must $R$ be, so that at resonance the voltage across the capacitor is $U_C = 4 \cdot U_{\rm s}$? (independent) | ||
+ | |||
+ | # | ||
+ | For the following calculation, | ||
+ | \begin{align*} | ||
+ | R_\Sigma = R_i + R \\ | ||
+ | \end{align*} | ||
+ | |||
+ | Here, either one knows that the gain factor $Q$ stands for $Q={{U_C}\over{U_{\rm s}}}$ and therefore can directly use the following formula: | ||
+ | \begin{align*} | ||
+ | Q = {{U_C}\over{U_{\rm s}}} &= {{1}\over{R_\Sigma}} | ||
+ | R_\Sigma | ||
+ | \end{align*} | ||
+ | |||
+ | < | ||
+ | When the gain factor is not known, one has to derive it: \\ | ||
+ | The voltage $I$ at resonance is only given by the total ohmic resistance $R_\Sigma$ and the source voltage $U_{\rm s}$: | ||
+ | \begin{align*} | ||
+ | I = {{U_{\rm s}}\over{R_\Sigma}} | ||
+ | \end{align*} | ||
+ | |||
+ | This current flow also through the impedance of the capacitor | ||
+ | \begin{align*} | ||
+ | U_C &= Z_C \cdot I \\ | ||
+ | &= {{1}\over{\omega C}} \cdot I \\ | ||
+ | &= {{U_{\rm s}}\over{\omega C R_\Sigma }} \\ | ||
+ | \end{align*} | ||
+ | |||
+ | At resonance, the angular frequency $\omega$ is given by $\omega= {{1}\over{\sqrt{LC}}}$. \\ | ||
+ | \begin{align*} | ||
+ | U_C &= {{U_{\rm s}}\over{{{1}\over{\sqrt{LC}}} C R_\Sigma }} \\ | ||
+ | &= {{U_{\rm s}}\over{\sqrt{{{C}\over{L}}} R_\Sigma }} \\ | ||
+ | &= {{U_{\rm s}}\over{R_\Sigma }} \sqrt{{{L}\over{C}}} | ||
+ | |||
+ | \end{align*} | ||
+ | </ | ||
+ | |||
+ | In both cases, we end up with the same formula, where we have to insert the physical values: | ||
+ | \begin{align*} | ||
+ | R_\Sigma | ||
+ | &= {{1}\over{4}} \sqrt{ {{20\cdot 10^{-3} ~\rm H}\over{30\cdot 10^{-6} ~\rm C}} } \\ | ||
+ | &= 6.4549...~\Omega \\ | ||
+ | \end{align*} | ||
+ | |||
+ | And so, the resistance $R$ is: | ||
+ | \begin{align*} | ||
+ | R &= R_\Sigma - R_i \\ | ||
+ | & | ||
+ | & | ||
+ | \end{align*} | ||
+ | |||
+ | # | ||
+ | |||
+ | # | ||
+ | $R = 6.255~\Omega$ | ||
+ | # | ||
+ | |||
+ | # |