(Solved): Calculations 1. For the circuits in Figs. 3(a) and 3(b), derive the transfer functions in the follo ...

Calculations 1. For the circuits in Figs. 3(a) and 3(b), derive the transfer functions in the following forms: \[ \begin{array}{l} H_{L}(s)=\frac{V_{L}}{V_{i}}(s)=K_{l} \frac{1}{1+\frac{s}{N_{l}}} \\ H_{\text {sep }}(s)=\frac{V_{\text {me }}}{V_{i}}(s)=K_{N_{N}} \frac{s}{s+\omega N} \end{array} \] and express \( w_{L}, \omega_{H}, K_{t} \) and \( K_{H} \) in terms of resistors and capacitors. Figure 3: First order (a) lowpass filter (b) highpass filter The frequencies \( w_{l} \) and \( \omega_{n} \) are known as: - Pole frequency, defined as the root of the denominator of \( H(s) \) - Corner frequency, defined as the frequency at which the gain is \( 0.707 \) times the passband value - Half power frequency (compared to passband), since \( 0.707^{2}=0.5 \) - 3dB frequency (compared to passband), since \( 20 \log _{10}(0.707)=-3 \mathrm{~dB} \) Figure 4: Magnitude plot of a first order (a) lowpass filter (b) highpass filter 2. Find \( R_{1}, R_{2}, R_{1}, C_{2}, C_{2} \) and \( C_{3} \), such that \( f_{6}=f_{w}=5 \mathrm{kHz} \) and \( K_{\mathrm{t}}=K_{w}=0.5 \). 3. Sketch the magnitude and phase Bode plots for \( H_{L P}(s) \) and \( H_{M P}(s) \). 4. Calculate the output voltages \( V_{1 P}(t) \) and \( V_{1 e}(t) \) for \( V_{i}(t)=0.4 \sin (2 \pi 4000 t) \). 5. Repeat step 4 for \( V_{f}(t)=0.3 \sin (2 \pi 6000 t) \).