Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
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electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2023/03/09 13:36] mexleadmin |
electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2024/02/08 15:40] (aktuell) mexleadmin |
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| - | <panel type=" | + | <panel type=" |
| <WRAP right> {{: | <WRAP right> {{: | ||
| Zeile 5: | Zeile 5: | ||
| The circuit shown right is given with the following data: | The circuit shown right is given with the following data: | ||
| - | * $U = 10 ~V$ | + | * $U = 10 ~{\rm V}$ |
| - | * $I = 4 ~mA$ | + | * $I = 4 ~{\rm mA}$ |
| * $R_1 = 100 ~\Omega, R_2 = 80 ~\Omega, R_3 = 50 ~\Omega, R_4 = 10 ~\Omega$ | * $R_1 = 100 ~\Omega, R_2 = 80 ~\Omega, R_3 = 50 ~\Omega, R_4 = 10 ~\Omega$ | ||
| - | * $C = 40 ~nF$ | + | * $C = 40 ~{\rm nF}$ |
| - | 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=" | <button size=" | ||
| Zeile 30: | Zeile 30: | ||
| <button size=" | <button size=" | ||
| - | 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 $R_1$, $R_2$, and $C$ 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 ~{\rm nF} | ||
| + | \end{align*} | ||
| </ | </ | ||
| - | <button size=" | + | <button size=" |
| + | \begin{align*} \tau = 7.2 ~{\rm µs} | ||
| + | \end{align*} \\ | ||
| + | </ | ||
| - | 2. What is the value of the voltage $u_C(t)$ drop over the capacitor $C$ at $t=10 ~\mu s$? | + | 2. What is the value of the voltage $u_C(t)$ drop over the capacitor $C$ at $t=10 ~{\rm µs}$? |
| <button size=" | <button size=" | ||
| \begin{align*} | \begin{align*} | ||
| - | U_C(t) = U \cdot (1 - e^{-t/ | + | U_C(t) = U |
| - | U_C(t) = 10 ~V \cdot (1 - e^{-10 ~\mu s/7.2 ~\mu s}) | + | U_C(t) = 10 ~{\rm V} \cdot (1 - e^{-10 ~{\rm µs}/7.2 ~{\rm µs}}) |
| \end{align*} | \end{align*} | ||
| Zeile 49: | Zeile 57: | ||
| <button size=" | <button size=" | ||
| - | \begin{align*} U_C(t) = 7.506 ~V \rightarrow 7.5 ~V \end{align*} \\ </ | + | \begin{align*} U_C(t) = 7.506 ~{\rm V} \rightarrow 7.5 ~{\rm V} \end{align*} \\ </ |
| - | 3. What is the value of the energy, when the capacitor is fully charged? | + | 3. What is the value of the stored |
| <button size=" | <button size=" | ||
| Zeile 57: | Zeile 65: | ||
| \begin{align*} | \begin{align*} | ||
| W_C &= \frac{1}{2} C U^2 \\ | W_C &= \frac{1}{2} C U^2 \\ | ||
| - | &= \frac{1}{2} \cdot 40~nF \cdot (10~V)^2 | + | &= \frac{1}{2} \cdot 40~{\rm nF} \cdot (10~{\rm V})^2 |
| \end{align*} | \end{align*} | ||
| Zeile 63: | Zeile 71: | ||
| <button size=" | <button size=" | ||
| - | \begin{align*} W_C = 2 ~\mu J \end{align*} \\ | + | \begin{align*} W_C = 2 ~{\rm µJ} \end{align*} \\ |
| </ | </ | ||
| Zeile 73: | Zeile 81: | ||
| \begin{align*} | \begin{align*} | ||
| \tau &= (R_2 + R_3) \cdot C \\ | \tau &= (R_2 + R_3) \cdot C \\ | ||
| - | & | + | & |
| \end{align*} | \end{align*} | ||
| </ | </ | ||
| - | <button size=" | + | <button size=" |
| + | \begin{align*} \tau = 5.2 ~{\rm µs} | ||
| + | \end{align*} \\ </ | ||
| - | 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 $t = 1 ~\mu s$? | + | 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 $t = 1 ~ {\rm µs}$? |
| <button size=" | <button size=" | ||
| - | * Through the current source there is a continuous flow of elctric | + | * Through the current source there is a continuous flow of electric |
| * The resistors passed by the current on the way to the capacitor are irrelevant. They only increase the voltage of an ideal current source to guarantee the current. | * The resistors passed by the current on the way to the capacitor are irrelevant. They only increase the voltage of an ideal current source to guarantee the current. | ||
| Zeile 91: | Zeile 101: | ||
| <button size=" | <button size=" | ||
| - | The voltage $U_C$ is in general: $U_C = \frac{Q}{C}$. In this case the constant current I results in $Q = \int I dt = I \cdot t$ | + | The voltage $U_C$ is in general: $U_C = \frac{Q}{C}$. In this case, the constant current I results in $Q = \int I {\rm d}t = I \cdot t$ |
| \begin{align*} | \begin{align*} | ||
| - | U_C(t) | + | U_C(t) |
| - | 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} \\ | + | U_C(t) |
| + | U_C(1μs) &= \frac{4~{\rm mA} \cdot 1~{\rm µs}}{40~{\rm nF}} | ||
| + | | ||
| \end{align*} | \end{align*} | ||
| Zeile 101: | Zeile 113: | ||
| <button size=" | <button size=" | ||
| \begin{align*} | \begin{align*} | ||
| - | U_C(1~\mu s) &= 1~V \\ | + | U_C(1~{\rm µs}) &= 1~{\rm V} \\ |
| \end{align*} \\ | \end{align*} \\ | ||
| </ | </ | ||
| </ | </ | ||
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