{{fa>pencil?32}} {{:elektrotechnik_1:schaltung_klws2020_3_2_1.jpg?400|schaltung_klws2020_3_2_1.jpg}} The circuit shown right is given with the following data: * $U = 10 ~{\rm V}$ * $I = 4 ~{\rm mA}$ * $R_1 = 100 ~\Omega, R_2 = 80 ~\Omega, R_3 = 50 ~\Omega, R_4 = 10 ~\Omega$ * $C = 40 ~{\rm nF}$ At first, the voltage drop on the capacitor $u_C = 0$, and all switches are open. The switch S1 will be closed at $t = 0$. {{icon>eye}} Simulation {{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjOB0AMt-CwFMC0B2E1IGYAsBOaPAJnwFY1cA2MADlzCpDIjJF22dTDACgA3EMWK4Q2MsSEjwaJtEzt5YeSsyQyvAM5TRYWUNq0Zc8CABmAQwA2mpLwBOBo3qbZ6xhcugOx7l79FiMhNabwBjAKFgyPFJeVxUIxVeAHcYiR0xDO9tN119XFCPJXNrWx9CpQKi2IUybzS8rMlK5sxU9hqMilds3gBLZn1iNEketpUYaFwfcZGx-TBiJPAGoaYlowIN5fbtDklN9nwdlYhLGzs0g-BdpqO1+7u-fW8Ae3ZTJWJsaDip6AQLBwIFCT6cACuAH0AMK8IA noborder}} 1. Determine the time constant $\tau$ for this charging process. {{icon>eye}} Tips * What equivalent circuit can be found for the mentioned states of the switches? * What parameter do you need to determine $\tau$? * The charging current is flown through which component? {{icon>eye}} Solution 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*} {{icon>eye}} Final value \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 $t=10 ~{\rm µs}$? {{icon>eye}} Solution \begin{align*} U_C(t) = U \cdot (1 - e^{-t/\tau}) \\ U_C(t) = 10 ~{\rm V} \cdot (1 - e^{-10 ~{\rm µs}/7.2 ~{\rm µs}}) \end{align*} {{icon>eye}} Final value \begin{align*} U_C(t) = 7.506 ~{\rm V} \rightarrow 7.5 ~{\rm V} \end{align*} \\ 3. What is the value of the stored energy in the capacitor, when it is fully charged? {{icon>eye}} Solution \begin{align*} W_C &= \frac{1}{2} C U^2 \\ &= \frac{1}{2} \cdot 40~{\rm nF} \cdot (10~{\rm V})^2 \end{align*} {{icon>eye}} Final value \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. {{icon>eye}} Solution 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*} {{icon>eye}} Final value \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 $t = 1 ~ {\rm µs}$? {{icon>eye}} Tips * Through the current source there is a continuous flow of electric charge into the capacitor. * 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. {{icon>eye}} Solution 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) &= \frac{Q}{C} \\ U_C(t) &= \frac{I \cdot t}{C} \\ U_C(1μs) &= \frac{4~{\rm mA} \cdot 1~{\rm µs}}{40~{\rm nF}} = \frac{4 \cdot 10^{-3}~{\rm A} \cdot 1\cdot 10^{-6}~{\rm s}}{40\cdot 10^{-9}~{\rm F}} \\ \end{align*} {{icon>eye}} Final value \begin{align*} U_C(1~{\rm µs}) &= 1~{\rm V} \\ \end{align*} \\