E84 Home Work 8

1. In the circuit below, $R_1=20\Omega$, $R_2=10\Omega$, $C=318\mu F$, and the sinusoidal voltage source is $v_0(t)=14.1 \cos(314 t-45^{\circ})$. Find the complete voltage response $v(t)=v_C(t)$ across $C$ and $R_2$ after the switch closes at $t=0$.

   

2. An RCL series circuit composed of $R=10\Omega$, $L=10 mH$ and $C=1 \mu F$ is connected to an input AC voltage $v_{in}(t)=cos\omega t$.
   • Find the quality factor $Q$ and resonant frequency $\omega_0$.
   • Assume the voltage $v_R(t)$ across $R$ is taken as the output voltage. Find the bandwidth of the circuit. Also, use any software (e.g., Matlab) to plot the ratio of the magnitudes between the output and input voltages $v_R/v_{in}$ as a function of frequency $\omega$.
   • Assume the voltage $v_L(t)$ across $L$ is taken as the output voltage. plot the ratio $v_L/v_{in}$ as a function of frequency $\omega$.
   • Assume the voltage $v_C(t)$ across $C$ is taken as the output voltage. plot the ratio $v_C/v_{in}$ as a function of frequency $\omega$.

3. A series circuit composed of a capacitor and an inductor is to be resonant at 800 kHz with voltage input. Specify the value of $C$ for the capacitor required for the given inductor with $L=40\mu H$ and an internal resistance $R_L=4.02\Omega$, and predict the bandwidth. Assume the capacitor is ideal, i.e., it introduces no resistance.

4. Design a parallel circuit to be resonant at 800 kHz with a bandwidth of 32 kHz. The inductor has $L=40\mu H$ and $R_L=4.02\Omega$. Find the capacitance $C$ needed for the desired resonant frequency. In order to satisfy the desired bandwidth, you may also need to include a resistor in the circuit.

5. The function of a loudspeaker crossover network is to channel frequencies higher than a given crossover frequency $f_c$ into the high-frequency speaker (``tweeter'') and frequencies below $f_c$ into the low-frequency speaker (``woofer''). One such circuit is shown below. Assume the resistances of the tweeter is $R_1=8\Omega$ and that of the woofer is $R_2=8\Omega$, the voltage amplifier can be modeled as an ideal voltage source, and the crossover frequency is $f_c=2000\; Hz$. Design the network in terms of $L$ and $C$ so that $f_c$ is the corner freqnency or half-power point of each of the two speaker circuits. Give the expression of the power $P_1(f)$ and $P_2(f)$ of the speakers as a function of frequency $f$ and crossover frequency $f_c$, and sketch them. Assume the RMS of the input voltage is 1V.