Beide Seiten der vorigen Revision
Vorhergehende Überarbeitung
Nächste Überarbeitung
|
Vorhergehende Überarbeitung
Nächste Überarbeitung
Beide Seiten der Revision
|
electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2021/11/03 15:06] tfischer |
electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2023/03/09 13:36] mexleadmin |
<panel type="info" title="Excercise 5.2.6: Charging and Discharging of RC elements (exam task, ca. 11% of a 60 minute exam, WS2020)"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> | <panel type="info" title="Exercise 5.2.6: Charging and Discharging of RC elements (exam task, ca. 11% of a 60 minute exam, WS2020)"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> |
| |
<WRAP right> {{:elektrotechnik_1:schaltung_klws2020_3_2_1.jpg?400|schaltung_klws2020_3_2_1.jpg}}</WRAP> | <WRAP right> {{:elektrotechnik_1:schaltung_klws2020_3_2_1.jpg?400|schaltung_klws2020_3_2_1.jpg}}</WRAP> |
The circuit shown right is given with the following data: | The circuit shown right is given with the following data: |
| |
* $U = 10 V$ | * $U = 10 ~V$ |
* $I = 4 mA$ | * $I = 4 ~mA$ |
* $R_1 = 100 \Omega, R_2 = 80 \Omega, R_3 = 50 \Omega, R_4 = 10 \Omega$ | * $R_1 = 100 ~\Omega, R_2 = 80 ~\Omega, R_3 = 50 ~\Omega, R_4 = 10 ~\Omega$ |
* $C = 40 nF$ | * $C = 40 ~nF$ |
| |
At first the voltage drop on the capacitor $u_C=0$ and all switches are open. The switch S1 will be closed at $t=0$. | At first the voltage drop on the capacitor $u_C = 0$ and all switches are open. The switch S1 will be closed at $t = 0$. |
| |
| <button size="xs" type="link" collapse="Loesung_7_2_6_6_Simu">{{icon>eye}} Simulation</button><collapse id="Loesung_7_2_6_6_Simu" collapsed="true"> |
| |
| <WRAP>{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjOB0AMt-CwFMC0B2E1IGYAsBOaPAJnwFY1cA2MADlzCpDIjJF22dTDACgA3EMWK4Q2MsSEjwaJtEzt5YeSsyQyvAM5TRYWUNq0Zc8CABmAQwA2mpLwBOBo3qbZ6xhcugOx7l79FiMhNabwBjAKFgyPFJeVxUIxVeAHcYiR0xDO9tN119XFCPJXNrWx9CpQKi2IUybzS8rMlK5sxU9hqMilds3gBLZn1iNEketpUYaFwfcZGx-TBiJPAGoaYlowIN5fbtDklN9nwdlYhLGzs0g-BdpqO1+7u-fW8Ae3ZTJWJsaDip6AQLBwIFCT6cACuAH0AMK8IA noborder}} </WRAP> |
| |
| </collapse> |
| |
1. Determine the time constant $\tau$ for this charging process. | 1. Determine the time constant $\tau$ for this charging process. |
<button size="xs" type="link" collapse="Loesung_7_2_6_1_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_1_Lösungsweg" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_1_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_1_Lösungsweg" collapsed="true"> |
| |
The electrical components $R_1$, $R_2$ und $C$ are connected in series with a source $U$. The time constant $\tau$ is therefore: \begin{align*} \tau &= (R_1 + R_2) \cdot C \\ \tau &= 180 \Omega \cdot 40 nF \end{align*} | The electrical components $R_1$, $R_2$ und $C$ are connected in series with a source $U$. The time constant $\tau$ is therefore: \begin{align*} \tau &= (R_1 + R_2) \cdot C \\ \tau &= 180 ~\Omega \cdot 40 ~nF \end{align*} |
| |
</collapse> | </collapse> |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_1_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_1_Endergebnis" collapsed="true"> \begin{align*} \tau = 7,2 µs \end{align*} \\ </collapse> | <button size="xs" type="link" collapse="Loesung_7_2_6_1_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_1_Endergebnis" collapsed="true"> \begin{align*} \tau = 7.2 ~\mu s \end{align*} \\ </collapse> |
| |
2. What is the value of the voltage $u_C(t)$ drop over the capacitor $C$ at $t=10 µs$? | 2. What is the value of the voltage $u_C(t)$ drop over the capacitor $C$ at $t=10 ~\mu s$? |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_2_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_2_Lösungsweg" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_2_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_2_Lösungsweg" collapsed="true"> |
| |
\begin{align*} U_C(t) = U \cdot (1 - e^{-t/\tau}) \\ U_C(t) = 10 V \cdot (1 - e^{-10 µs/7,2 µs}) \end{align*} | \begin{align*} |
| U_C(t) = U \cdot (1 - e^{-t/\tau}) \\ |
| U_C(t) = 10 ~V \cdot (1 - e^{-10 ~\mu s/7.2 ~\mu s}) |
| \end{align*} |
| |
</collapse> | </collapse> |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_2_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_2_Endergebnis" collapsed="true"> \begin{align*} U_C(t) = 7,506 V -> 7,5 V \end{align*} \\ </collapse> | <button size="xs" type="link" collapse="Loesung_7_2_6_2_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_2_Endergebnis" collapsed="true"> |
| |
| \begin{align*} U_C(t) = 7.506 ~V \rightarrow 7.5 ~V \end{align*} \\ </collapse> |
| |
3. What is the value of the energy, when the capacitor is fully charged? | 3. What is the value of the energy, when the capacitor is fully charged? |
<button size="xs" type="link" collapse="Loesung_7_2_6_3_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_3_Lösungsweg" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_3_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_3_Lösungsweg" collapsed="true"> |
| |
\begin{align*} W_C &= \frac{1}{2}CU^2 \\ &= \frac{1}{2} \cdot 40nF \cdot (10V)^2 \end{align*} | \begin{align*} |
| W_C &= \frac{1}{2} C U^2 \\ |
| &= \frac{1}{2} \cdot 40~nF \cdot (10~V)^2 |
| \end{align*} |
| |
</collapse> | </collapse> |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_3_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_3_Endergebnis" collapsed="true"> \begin{align*} W_C = 2 µJ \end{align*} \\ </collapse> | <button size="xs" type="link" collapse="Loesung_7_2_6_3_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_3_Endergebnis" collapsed="true"> |
| \begin{align*} W_C = 2 ~\mu J \end{align*} \\ |
| </collapse> |
| |
4. Determine the new time constant when the switch $S_1$ will be opened and the switch $S_3$ will be closed simultaneously. | 4. Determine the new time constant when the switch $S_1$ will be opened and the switch $S_3$ will be closed simultaneously. |
<button size="xs" type="link" collapse="Loesung_7_2_6_4_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_4_Lösungsweg" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_4_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_4_Lösungsweg" collapsed="true"> |
| |
The capacitor $C$ discharges by the series connected resistors $R_2$ und $R_3$. \begin{align*} \tau &= (R_2 + R_3) \cdot C \\ \tau &= 130 \Omega \cdot 40 nF \end{align*} | The capacitor $C$ discharges by the series connected resistors $R_2$ und $R_3$. |
| \begin{align*} |
| \tau &= (R_2 + R_3) \cdot C \\ |
| &= 130 ~\Omega \cdot 40 ~nF |
| \end{align*} |
| |
</collapse> | </collapse> |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_4_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_4_Endergebnis" collapsed="true"> \begin{align*} \tau = 5,2 µs \end{align*} \\ </collapse> | <button size="xs" type="link" collapse="Loesung_7_2_6_4_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_4_Endergebnis" collapsed="true"> \begin{align*} \tau = 5.2 ~\mu s \end{align*} \\ </collapse> |
| |
5. When the capacitor is empty all switches will be opened. The switch $S_4$ will be closed at $t= 0$. \\ What is the voltage $u_C$ at the capacitor C after $t = 1μs$? | 5. When the capacitor is empty all switches will be opened. The switch $S_4$ will be closed at $t= 0$. \\ What is the voltage $u_C$ at the capacitor C after $t = 1 ~\mu s$? |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_5_Tipps">{{icon>eye}} Tips</button><collapse id="Loesung_7_2_6_5_Tipps" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_5_Tipps">{{icon>eye}} Tips</button><collapse id="Loesung_7_2_6_5_Tipps" collapsed="true"> |
<button size="xs" type="link" collapse="Loesung_7_2_6_5_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_5_Lösungsweg" collapsed="true"> | <button size="xs" type="link" collapse="Loesung_7_2_6_5_Lösungsweg">{{icon>eye}} Solution</button><collapse id="Loesung_7_2_6_5_Lösungsweg" collapsed="true"> |
| |
The voltage $U_C$ is in general: $U_C = \frac{Q}{C}$. In this case the constant current I results in $Q = \int I dt = I \cdot t$ \begin{align*} U_C(t) &= \frac{Q}{C} \\ U_C(t) &= \frac{I \cdot t}{C} \\ U_C(1μs) &= \frac{4mA \cdot 1μs}{40nF} = \frac{4 \cdot 10^{-3}A \cdot 1\cdot 10^{-6}s}{40\cdot 10^{-9}F} \\ \end{align*} | The voltage $U_C$ is in general: $U_C = \frac{Q}{C}$. In this case the constant current I results in $Q = \int I dt = I \cdot t$ |
| \begin{align*} |
| U_C(t) &= \frac{Q}{C} \\ U_C(t) &= \frac{I \cdot t}{C} \\ |
| U_C(1μs) &= \frac{4~mA \cdot 1~\mu s}{40~nF} = \frac{4 \cdot 10^{-3}~A \cdot 1\cdot 10^{-6}~s}{40\cdot 10^{-9}~F} \\ |
| \end{align*} |
| |
</collapse> | </collapse> |
| |
<button size="xs" type="link" collapse="Loesung_7_2_6_5_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_5_Endergebnis" collapsed="true"> \begin{align*} U_C(1μs) &= 1V \\ \end{align*} \\ </collapse> | <button size="xs" type="link" collapse="Loesung_7_2_6_5_Endergebnis">{{icon>eye}} Final value</button><collapse id="Loesung_7_2_6_5_Endergebnis" collapsed="true"> |
| \begin{align*} |
| U_C(1~\mu s) &= 1~V \\ |
| \end{align*} \\ |
| </collapse> |
| |
</WRAP></WRAP></panel> | </WRAP></WRAP></panel> |