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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 \( V_{f}(t)=0.3 \sin (2 \pi 6000 t) \).

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