(Solved): Problem \#1-Transfer Functions for Filter Circuits: For the two circuits given above, determine: a) ...

Problem \#1-Transfer Functions for Filter Circuits: For the two circuits given above, determine: a) The transfer function in terms of the given circuit parameters ( $R,L,C$, etc.). b) Identify any breakpoint frequencies* $\left({?}_{cl},{?}_{c2}\dots \right)$ in terms of the transfer function circuit parameters. *note that breakpoint frequencies are a more general term for cut-point or resonance frequency for higher-order filter circuits. Dimensionless groups that define breakpoint frequencies include: ${?}_c=1/\mathrm{RC},{?}_c=\mathrm{L}/\mathrm{R},{\left({?}_c\right)}^2=1/\mathrm{LC}$ c) Write both transfer functions in terms of the breakpoint frequencies you defined in part $b$. Problem \#2-Determine Gain and Phase: For the two transfer functions from Problem \#1, determine: a) The Gain in terms of frequency ratios $\left(?/{?}_i\right)$. e) The Phase in terms of frequency ratios $\left(?/{?}_c\right)$. f) Evaluate Limit(Gain) as $??0$ and $???$ for both circuits. Based on your results, determine the filter types (low-pass, high-pass, band-pass, or band-stop) for circuits A and B. a) For filter circuit A, you want to design the lower breakpoint frequency to be $3.2\mathrm{Hz}$, and the upper breakpoint frequency to be $8\mathrm{Hz}$. You have a capacitor with capacitance of 5 milliFarads. Determine the values of: i) the resistor (Ohms) and ii) the inductor (milli-Henrys) b) For filter circuit $\mathrm{B}$, you want to design the breakpoint frequency to be $8\mathrm{Hz}$, and the maximum value of gain (at any frequency) to be 5. You have a capacitor with a capacitance of 5 milli-Farads. Determine the values of: i) the input resistor, $R_{/}$(Ohms), ii) the feedback resistor, $R_2(\mathrm{Ohms})$ Problem \#4-Graphing the Circuit Performance: For both filter circuits, use Excel, Matlab, Python, etc. to compute and plot the Gain and Phase from a frequency range from $0.1\mathrm{Hz}$ to 100 $\mathrm{Hz}$. a) Plot the Gain in Decibels of power using a ${\log }_{10}$ scale for frequency $(\mathrm{Hz})$ for filters A \& B. b) Plot the Phase in Degrees using a ${\log }_{10}$ scale for frequency $(\mathrm{Hz})$ for filters A \& B. Problem \#5-Applying Filter Circuit A: The input signal $V_{in}(t)$ has the following function: $V_{in}(t)=3\sin 42?t+{?}_3)+7\cos (5?t)?5\sin 413?t+{?}_8)$ We will consider the $5\mathrm{Hz}$ cosine term to be the 'noise', and the two sine terms the 'signal'. Using the circuit design from problem 3 a, determine: a) The equation for the output signal, $V_{\text{owr }}(\mathrm{t})$ b) The signal-to-noise ratio $(\mathrm{S}/\mathrm{N})$ of both $V_{\text{in }}$ and $V_{\text{out }}$ in decibels of power $(\mathrm{dB})$. We will define $\mathrm{S}/\mathrm{N}$ ratio as: $S/N=\frac{V_{RMS\&si(nal}}{V_{RMS\&\text{ noise }}}$ Problem \#6-Applying Filter circuit B: The input signal $V_{im}(t)$ has the following function: $V_{in}(t)=3\sin 42?t+{?}_8)+7\cos (5?t)?5\sin 413?t+{?}_{8_4})$ We will consider both the $2\mathrm{Hz}$ sine term and the $5\mathrm{Hz}$ cosine term to be the 'noise', and the 13 $\mathrm{Hz}$ sine term the 'signal'. Using the circuit design from problem $3\mathrm{b}$, determine: a) The equation for the output signal, $V_{\text{our }}(\mathrm{t})$ b) The signal-to-noise ratio $(\mathrm{S}/\mathrm{N})$ of both $V_{in}$ and $V_{ou}$ in decibels of power $(\mathrm{dB})$. We will define $\mathrm{S}/\mathrm{N}$ ratio as: $S/N=\frac{V_{RMS\&si(nal}}{V_{RMS\&noise}}$