E84 Home Work 6

1. The resistance $R$ of a circuit is a real value which can be measured by a multimeter. However, the impedance $Z$ of a component in the circuit is complex which cannot be measured directly. Instead, one can use an oscilloscope to find sinusoidal voltage $v(t)$ across and current $i(t)$ through the component, and then obtain the impedance as the ratio between the complex representations of the voltage and current. Suppose we find:
   
   $$v(t)=12 cos(1000t-30^\circ),\;\;\;\; i(t)=6\;cos(1000t+15^\circ)$$
   
   
   Find the impedance (both resistance and reactance) and the admittance (both conductance and susceptance) of the circuit.

2. A voltage $v(t)=120\sqrt{2} cos(1000t+90^\circ) V$ (volt) is applied to a resistor $R=15\Omega$, a capacitor $C=83.3\mu F$ and an inductor $L=30\; mH$ connected in parallel. Find the over all steady state current $i=i_R+i_C+i_L$ by phasor method.

3. A voltage $v(t)=12\sqrt{2} \cos 5000 t$ (volt V) is applied to a circuit composed of two branches in parallel. One branch has a capacitor $C=10\mu F$, while the other has a resistor $R=20\Omega$ and an inductor $L=3 mH$ in series. Using phasor method, find the impedances $Z_C$ and $Z_{RL}$ of the two branches, and then the overall combined impedance $Z_{all}$ of the circuit. Then find the steady state current $i(t)$ through the circuit.

4. Solve the problem above again but this time use the admittances $Y_C=1/Z_C$, $Y_{RL}=1/Z_{RL}$, $Y_{all}=1/Z_{all}$ (instead of the impedances $Z_C$, $Z_{RL}$, $Z_{all}$). Recall that Ohm's law becomes $\dot{I}=\dot{V}/Z=\dot{V}Y$. (Make sure all impedances you found in previous problem are correct before you find the admittances as their reciprocals.)

5. Find the output voltage $v_{out}(t)$ across the right most branch containing $R_2$ and $C$, when $\omega=0$ and $\omega\rightarrow \infty$ and the input $v_{in}(t)=V=10\;cos(\omega t)$, assuming $R_1=100\Omega$, $R_2=100\Omega$, $C=10\mu F$ and $L=10\;mH$.