Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung Letzte Überarbeitung Beide Seiten der Revision | ||
|
electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2023/02/11 23:08] mexleadmin |
electrical_engineering_1:aufgabe_7.2.6_mit_rechnung [2023/11/30 00:07] mexleadmin |
||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| - | <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 $t=10 µs$? | + | 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*} U_C(t) = U \cdot (1 - e^{-t/ | + | \begin{align*} |
| + | U_C(t) = U | ||
| + | U_C(t) = 10 ~{\rm V} \cdot (1 - e^{-10 | ||
| + | \end{align*} | ||
| </ | </ | ||
| - | <button size=" | + | <button size=" |
| + | |||
| + | \begin{align*} U_C(t) = 7.506 ~{\rm V} \rightarrow | ||
| 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? | ||
| Zeile 50: | Zeile 63: | ||
| <button size=" | <button size=" | ||
| - | \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} | ||
| + | | ||
| + | \end{align*} | ||
| </ | </ | ||
| - | <button size=" | + | <button size=" |
| + | \begin{align*} W_C = 2 ~{\rm µJ} \end{align*} \\ | ||
| + | </ | ||
| 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. | ||
| Zeile 60: | Zeile 78: | ||
| <button size=" | <button size=" | ||
| - | 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 ~{\rm nF} | ||
| + | \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 $t = 1μs$? | + | 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 $t = 1 ~ {\rm µs}$? |
| <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 77: | 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 $Q = \int I dt = I \cdot t$ \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 $Q = \int I {\rm d}t = I \cdot t$ |
| + | \begin{align*} | ||
| + | U_C(t) | ||
| + | U_C(t) | ||
| + | U_C(1μs) &= \frac{4~{\rm mA} \cdot 1~{\rm µs}}{40~{\rm nF}} | ||
| + | | ||
| + | \end{align*} | ||
| </ | </ | ||
| - | <button size=" | + | <button size=" |
| + | \begin{align*} | ||
| + | U_C(1~{\rm µs}) & | ||
| + | \end{align*} \\ | ||
| + | </ | ||
| </ | </ | ||
| - | |||
| - | |||
| - | |||
| - | |||