Lab Project 1: RLC Circuit Analysis with MATLAB

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 $\omega / \omega_{0} “$.
    Add a legend to your graph.

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?

Linear Systems Tools:

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Fourier analysis代写



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