Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
electrical_engineering_1:circuits_under_different_frequencies [2023/03/27 09:47] mexleadmin |
electrical_engineering_1:circuits_under_different_frequencies [2023/09/19 23:37] (aktuell) mexleadmin |
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- | ====== 7. Networks at variable frequency ====== | + | ====== 7 Networks at variable frequency ====== |
Further content can be found at this [[https:// | Further content can be found at this [[https:// | ||
Zeile 82: | Zeile 82: | ||
* the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and | * the amplitude response: $A = \frac {\omega L}{\sqrt{R^2 + (\omega L)^2}}$ and | ||
- | * the phase response: $\Delta\varphi_{u} = arctan \frac{R}{\omega L} = \frac{\pi}{2} - \arctan \frac{\omega L}{R}$ | + | * the phase response: $\Delta\varphi_{u} = \arctan \frac{R}{\omega L} = \frac{\pi}{2} - \arctan \frac{\omega L}{R}$ |
The main focus should first be on the amplitude response. Its frequency response can be derived from the equation in various ways. | The main focus should first be on the amplitude response. Its frequency response can be derived from the equation in various ways. | ||
Zeile 110: | Zeile 110: | ||
This can also be derived from understanding the components: | This can also be derived from understanding the components: | ||
* At small frequencies, | * At small frequencies, | ||
- | * At higher frequencies, | + | * At higher frequencies, |
- | * If the frequency becomes very high, only a negligible current flows through the coil - and hence through the resistor. The voltage drop at $R$ thus approaches zero and the output voltage $U_O$ tends towards $U_I$. | + | * If the frequency becomes very high, only a negligible current flows through the coil - and hence through the resistor. The voltage drop at $R$ thus approaches zero and the output voltage $U_\rm O$ tends towards $U_\rm I$. |
The transfer function can also be decomposed into amplitude response and frequency response. \\ | The transfer function can also be decomposed into amplitude response and frequency response. \\ | ||
Zeile 131: | Zeile 131: | ||
<WRAP centeralign> | <WRAP centeralign> | ||
- | $\large{\underline{A} = \frac {\underline{U}_{\rm O}^\phantom{O}}{\underline{U}_{\rm I}^\phantom{O}} | + | \begin{align*} |
- | = \frac {\omega L} {\sqrt{R^2 + (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}}$ | + | \large{\underline{A} |
- | $ \quad \quad \vphantom{\HUGE{I \\ I}} \large{\xrightarrow{\text{normalization}}} \vphantom{\HUGE{I \\ I}} \quad \quad \quad $ | + | = \frac {\omega L} {\sqrt{R^2 + (\omega L)^2}}\cdot {\rm e}^{{\rm j}\left(\frac{\pi}{2} - \arctan \frac{\omega L}{R} \right)}} |
- | $\large{\underline{A}_{norm} | + | |
- | = \frac {\omega L / R}{\sqrt{1 | + | \large{\underline{A}_{norm} |
- | $\large{ | + | = \frac {\omega L / R}{\sqrt{1 |
+ | \large{ | ||
+ | \end{align*} | ||
</ | </ | ||
Zeile 166: | Zeile 168: | ||
\begin{align*} | \begin{align*} | ||
\vphantom{\HUGE{I }} \\ | \vphantom{\HUGE{I }} \\ | ||
- | \underline{A}_{\rm norm} = \frac{x}{\sqrt{1 + x^2}} \cdot {\rm e}^{{\rm | + | \underline{A}_{\rm norm} = \frac{x}{\sqrt{1 + x^2}} \cdot {\rm e}^{{\rm |
= \frac{U_{\rm O}}{U_{\rm I}} \cdot {\rm e}^{{\rm j}\varphi} | = \frac{U_{\rm O}}{U_{\rm I}} \cdot {\rm e}^{{\rm j}\varphi} | ||
\end{align*} | \end{align*} | ||
Zeile 196: | Zeile 198: | ||
\begin{align*} | \begin{align*} | ||
- | R &= \omega L \\ | + | R |
- | \omega_{c} &= \frac{R}{L} \\ | + | \omega _{\rm c} &= \frac{R}{L} \\ |
- | 2 \pi f_{c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{R}{2 \pi \cdot L}} \end{align*} | + | 2 \pi f_{\rm c} &= \frac{R}{L} \quad \rightarrow \quad \boxed{f_{\rm c} = \frac{R}{2 \pi \cdot L}} \end{align*} |
==== 7.2.2 RL Low Pass ==== | ==== 7.2.2 RL Low Pass ==== | ||
Zeile 210: | Zeile 212: | ||
\begin{align*} | \begin{align*} | ||
- | \underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{-{\rm j} \; arctan \frac{\omega L}{R} } | + | \underline{A}_{\rm norm} = \frac {1}{\sqrt{1 + (\omega L / R)^2}}\cdot {\rm e}^{-{\rm j} \; \arctan \frac{\omega L}{R} } |
\end{align*} | \end{align*} | ||
Zeile 234: | Zeile 236: | ||
\begin{align*} | \begin{align*} | ||
- | \underline{A}_{\rm norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{\frac{\pi}{2}-{\rm j} \; arctan (\omega RC) } | + | \underline{A}_{\rm norm} = \frac {\omega RC}{\sqrt{1 + (\omega RC)^2}}\cdot {\rm e}^{\frac{\pi}{2}-{\rm j} \; \arctan (\omega RC) } |
\end{align*} | \end{align*} | ||