## Low-pass RLC circuit:

(a) Find the transfer function, $H(s)=\frac{V_{\text {ouz }}(s)}{V_{\text {in }}(s)}$, in the above low-pass RLC circuit.
(b) The frequency response of the circuit can be written as $H(j \omega)=\frac{1}{1+2 \zeta \frac{j \omega}{\omega_{0}}+\left(\frac{j \omega}{w_{0}}\right)^{2}} .$ Write $\omega_{0}$ and $\zeta$ in terms of $R, L$, and $C$.
(c) Write a MATLAB script to calculate and plot the magnitude and phase of $H(j \omega)=$
$\frac{1}{1+2 \zeta \frac{3 \omega}{\omega_{0}}+\left(\frac{J \omega}{\omega_{0}}\right)^{2}}$ for $\omega_{0}=1$ and $\zeta=\left[\begin{array}{llllll}0.1 & 0.3 & 0.707 & 1.0 & 3.0 & 10.0\end{array}\right] .$
i. Create a vector named $w$ to store samples of $\omega$ between $0.01$ and $20(0.01<\omega<20)$.
Set the step size to be $0.01$.
ii. For every $\zeta$ value on the given list,

• Create a vector named $\mathrm{h}$ to store the samples of $H(j \omega)$ for the given $\omega$ values.
• Find $|H(j \omega)|$ for the given $\omega$ and store the values in a vector named $\mathrm{h} 2$.
• Using the plot function, plot $|H(j \omega)|$ vs $\omega$.
Use the hold on function to overlay the graphs.
iii. Label your graph: Add a proper title to your graph. Label the $\mathrm{x}$ -axis “frequency $\omega / \omega_{0} “$.

As you can see, it is hard to read the plot on linear axes. The standard form of plotting
the frequency response is with magnitude squared in $\mathrm{dB}$ vs $\log$ frequency and phase in
degrees vs log frequency. The plots of the magnitude in $\mathrm{dB}$ and phase in degrees are
called Bode plots. Next, you will use MATLAB to draw the Bode plots for $H(j \omega)=\frac{1}{1+2 \zeta \frac{1 \omega}{\omega_{0}}+\left(\frac{1 \omega}{\omega_{0}}\right)^{2}}$ for $0.1<\omega<20$ and $\omega_{0}=1$ and $\zeta=\left[\begin{array}{lllll}0.1 & 0.3 & 0.707 & 1.0 & 3.0 & 10.0\end{array}\right]$

## High-pass and Band-pass RLC circuits:

(a) Write the transfer functions $H_{H P}(s)$ and $H_{B P}(s)$ in terms of $\mathrm{R}, \mathrm{L}$, and $\mathrm{C}$. Put them in the
generalized form so the denominator is the same as the one for $H_{L P}(s)$ and the numerator
is written in terms of s, $\omega_{0}$ and $\zeta$. Of course $\omega_{0}$ and $\zeta$ are the same (in terms of $\mathrm{R}, \mathrm{L}$, and
C) as for the low-pass circuit because the loop current is unchanged by changing the order
of the elements. All that happens is you read the voltage across a different element. This
changes the numerator.
(b) Modify your script to make the Bode plots of $H(j \omega)$ for the band-pass and high-pass filters.

## Mystery Filter:

What happens if you take the output across both $\mathrm{C}$ and $\mathrm{L}$ together, rather
than just one of them? Write the transfer function $\mathrm{H}(\mathrm{s})$ and plot the Bode plots for $H(j \omega)$ as
you did for the band-pass and high-pass filters. What could you use this circuit for?