Name that Filter

The questions below are due on Monday November 18, 2024; 10:00:00 PM.
 
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In lab , we saw examples of low-pass filters, i.e., filters that let low-frequency signals through to their output but heavily attenuate high frequencies. The opposite is a high-pass filter, which allows high frequencies through to its output but heavily attenuates low frequencies.

For each of the circuits below, find analytic expressions for the magnitude and phase of the ratio v_{\rm O}\over {V_{\rm I}} as functions of \omega when V_{\rm I}(t) = Ae^{j\omega t}, and specify whether each acts as a low-pass or high-pass filter.

In all cases, specify your answer as a Python expression in terms of sin, cos, tan, atan, pi, e, ln, j (the imaginary unit), sqrt, omega, and/or any other values specified in the circuit.

1) Circuit A

\displaystyle{\left|{v_{\rm O}\over {V_{\rm I}}}\right|} =~

\displaystyle{\angle\left({v_{\rm O}\over {V_{\rm I}}}\right)} =~

What kind of filter is this? 

2) Circuit B

\displaystyle{\left|{v_{\rm O}\over {V_{\rm I}}}\right|} =~

\displaystyle{\angle\left({v_{\rm O}\over {V_{\rm I}}}\right)} =~

What kind of filter is this? 

3) Circuit C

\displaystyle{\left|{v_{\rm O}\over {V_{\rm I}}}\right|} =~

\displaystyle{\angle\left({v_{\rm O}\over {V_{\rm I}}}\right)} =~

What kind of filter is this? 

4) Circuit D

\displaystyle{\left|{v_{\rm O}\over {V_{\rm I}}}\right|} =~

\displaystyle{\angle\left({v_{\rm O}\over {V_{\rm I}}}\right)} =~

What kind of filter is this?