Figure $3$: Second Order Active Sallen$-$Key Filter

$3.$ Consider the second order Sallen$-$Key circuit configuration illustrated in $3.$ Let $C_1=C_2=1F.$ Use whatever computer resource you prefer for the following tasks. Note that the cutoff frequency for this circuit is given by ${}_c=\frac{1}{\sqrt[\phantom{2}]{R_1{R}_2{C}_1{C}_2}}.$

a$.)$ Longrightarrow Calculate the voltage transfer function for this circuit in terms of $R_1,R_2,C_1,$ and $C_2.$

b$.)$ Longrightarrow Calculate values of $R_1$ and $R_2$ that give you a cutoff frequency, $f_c,$ of $500$ Hz $.$

c$.)$ Longrightarrow Compute the DC gain of this circuit.

d$.)$ Longrightarrow Over a frequency range from $10$ Hz to $100$ kHz plot the magnitude bode plot $($using dB$)$ and, of course, a $lo{g}_{10}$ axis for frequency. Label the halfpower frequencies on your plot.

e$.)$ Longrightarrow What is the gain at $f_c?$ How can you determine the circuit order from your plots? Comment on your results.