Series RLC: Filters

The questions below are due on Monday May 05, 2025; 10:00:00 PM.

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In this problem, we'll consider a series RLC circuit driven by a sinusoidal drive, V_i(t) = V_o\cos(\omega t):

For this problem, we'll ignore transient behaviors of the circuit and consider it as operating in sinusoidal steady state.

Capacitor Voltage

Let's start by considering the output of our circuit to be the drop across the capacitor, v_{\rm C}.

What is the frequency response \tilde{H}_c(\omega) = {\tilde{v}_{\rm C}\over \tilde{V}_i}? Enter your answer in terms of j, omega, C, R, and L.
\tilde{H}_c(\omega) =~

For frequencies much smaller than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_c(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_c(\omega) approach as \omega \to 0?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

For frequencies much bigger than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_c(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_c(\omega) approach as \omega \to \infty?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_c(\omega)|? Enter your answer in terms of C, R, and L:

\left|\tilde{H}_c\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_c(\omega)|? Enter your answer in terms of Q (the quality factor of this circuit):

\left|\tilde{H}_c\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the phase \angle\tilde{H}_c(\omega)?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

Inductor Voltage

Now let's consider the output of our circuit to be the drop across the inductor, v_{\rm L}.

What is the frequency response \tilde{H}_L(\omega) = {\tilde{v}_{\rm L}\over \tilde{V}_i}? Enter your answer in terms of j, omega, C, R, and L.
\tilde{H}_L(\omega) =~

For frequencies much smaller than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_L(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_L(\omega) approach as \omega \to 0?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

For frequencies much bigger than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_L(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_L(\omega) approach as \omega \to \infty?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_L(\omega)|? Enter your answer in terms of C, R, and L:

\left|\tilde{H}_L\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_L(\omega)|? Enter your answer in terms of Q (the quality factor of this circuit):

\left|\tilde{H}_L\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the phase \angle\tilde{H}_L(\omega)?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

Resistor Voltage

Now let's consider the output of our circuit to be the drop across the resistor, v_{\rm R}.

What is the frequency response \tilde{H}_R(\omega) = {\tilde{v}_{\rm R}\over \tilde{V}_i}? Enter your answer in terms of j, omega, C, R, and L.
\tilde{H}_R(\omega) =~

For frequencies much smaller than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_R(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_R(\omega) approach as \omega \to 0?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

For frequencies much bigger than 1/\sqrt{LC}, how does the magnitude |\tilde{H}_R(\omega)| change as \omega changes?

It remains roughly constant at 0 (-\infty{\rm dB})

It remains roughly constant at 1 (0{\rm dB})

It remains roughly constant at some other nonzero value

It trends upward as \omega increases, roughly proportional to \sqrt{\omega}

It trends upward as \omega increases, roughly proportional to \omega

It trends upward as \omega increases, roughly proportional to \omega^2

It trends downward as \omega increases, roughly proportional to 1/\sqrt{\omega}

It trends downward as \omega increases, roughly proportional to 1/\omega

It trends downward as \omega increases, roughly proportional to 1/\omega^2

What value does the phase \angle\tilde{H}_R(\omega) approach as \omega \to \infty?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_R(\omega)|? Enter your answer in terms of C, R, and L:

\left|\tilde{H}_R\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the magnitude |\tilde{H}_R(\omega)|? Enter your answer in terms of Q (the quality factor of this circuit):

\left|\tilde{H}_R\left(1/\sqrt{LC}\right)\right| =~

At \omega = 1/\sqrt{LC}, what is the phase \angle\tilde{H}_R(\omega)?

-5\pi/6

-3\pi/4

-2\pi/3

-\pi/2

-\pi/3

-\pi/4

-\pi/6

\pi/6

\pi/4

\pi/3

\pi/2

2\pi/3

3\pi/4

5\pi/6

\pi