(Solved): Figure 1 shows a circuit diagram for a first-order low-pass filter, where V1 ...

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Figure 1 shows a circuit diagram for a first-order low-pass filter, where $V1$ indicates the input voltage of the circuit and $V2$ indicates the voltage across the capacitor as in this case also is the output voltage. 1. Laplace transform the circuit and determine $T_{LP}(s)=\frac{V_2(s)}{V_1(s)}$ 2. Show that the transfer function can be written as: $T_{LP}(s)=\frac{k}{s+a}$ Where $a=\frac{1}{R_1\cdot C_1}$ And what is K? 3. The low-pass filter is desired to be implemented to provide the following transfer function: $T_{LP}(s)=\frac{62832}{s+62832}$ $C_1=100nF$, what is $R_1$ ? 4. Draw a bode plot for $T_{LP}(s)$ that includes both the amplitude characteristic (in $\mathrm{dB}$ ) and the phase characteristic (in degrees), with a logarithmic frequency axis in the range $1\mathrm{KHz}$ to $1\mathrm{MHz}$ 5. What is the time constant of the circuit? 6. Draw a step response from time 0 to $5\tau$.