E84 Home Work 7

1. Find the current $i_1(t)$ through resistor $R_1$ after the switch is closed at $t=0$, assuming the input is a DC voltage and the circuit is already in steady state before $t=0$. (Hint: current through an inductor cannot change instantaneously.)

   

2. In the circuit below, $V_s=6V$, $R_1=6\Omega$, $R_2=3\Omega$, $L=0.5H$, $I_s=2A$. Assume before the switch is closed at $t=0$, the system is already stablized. Find current $i_L(t)$ through $L$ and voltage $v_{R_1}$ across $R_1$.

   

3. In the circuit below, $V_s=12V$, $R_1=5\Omega$, $R_2=20\Omega$, $R_3=6\Omega$, $C_1=10\mu F$, $C_2=30\mu F$. Assume before the switch is closed at $t=0$, the system is already stablized. Find voltages $v_1(t)$ and $v_2(t)$ across capacitors $C_1$ and $C_2$, respectively. (Hint, $C_1$ and $C_2$ are two capacitors in series with an equivalent capacitance is $C=C_1 C_2/(C_1+C_2)$. $C_1$ and $C_2$ have share the same time constant $\tau=RC$.)

   

4. In the circuit below, $R_1=100\Omega$, $R_2=150\Omega$, $R_3=100\Omega$, $H=0.1H$, $C=20\mu F$, $V=60V$. The circuit is in steady state initially when the switch is at position 2 (not connected). Find $v_C(t)$ and $i_L(t)$ for the following two independent cases:
   • after the switch is changed to position 1 at $t=0$;
   • after the switch is changed to position 3 at $t=0$;