Test Problems (don't click until you start taking the test)

1. (33 points) A crystal oscillator is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with a precise frequency. The crystal can be modeled by the RLC circuit shown in the figure. We assume $R$ can be approximated to be zero.
   • Find the parallel resonant frequency $\omega_p$ at which the impedance of the model is maximized.
   • Find the series resonant frequency $\omega_s$ at which the impedance of the model is minimized.

   

2. (33 points) In the circuit below, the filter composed of $L$, $C_1$ and $C_2$ between the source $v(t)$ and the load $R_L=100\Omega$ is to pass the fundamental frequency $\omega_0=1000$ without attenuation but completely block the 2nd harmonic $2\omega_0=2000$. Given $L=25\,mH$, find $C_1$ and $C_2$.

   

3. (34 points) In the circuit below, $R_1=3\Omega$, $R_2=6\Omega$, $R_3=2\Omega$, $L=0.5\,H$, $V_1=6V$, $V_2=3V$. The circuit is in steady state when $t<0$. Find current $i_3(t)$ through $R_3$ when switch is closed at $t=0$.