Linear Systems and DSP

Circuit analysis via the Laplace transform method can be algebraically challenging. It is quite useful to be able to verify your theoretical results. In this project we explore two different ways to verify your results – circuit simulation using LTSpice and numerical integration using Octave/MATLAB. Part I Use Laplace transforms to solve problem 4.4-7 in the text book. (Shown below, this problem was assigned as homework.) Be sure to use the provided initial conditions at t = 0– in finding the solution. Include the theoretical expression for the output, y ( t ), and the theoretical expression for the transfer function, H ( s ) in your report. Part II Use LTSpice to simulate the circuit over the time interval 0 ≤ t ≤ 3. Include the initial conditions on the inductor current and the capacitor voltage. Plot the simulated output voltage, y(t). Plot your theoretical expression for y(t) from Part I on the same waveform graph as the simulated voltage. Refer to the handout provided in lecture for helpful tips on LTSpice simulation. Include a copy of the circuit schematic and the waveform graph (showing both the simulated and theoretical results) in your report. (Set the Color Preferences under the Tools menu to set the graph background to white instead of the default black. This will use less ink and look better when printed. You can also set the “Data trace width” to 2 on the Waveforms tab of the Tools → Control Panel menu to plot using thicker lines.) Part III Use your transfer function from Part I to derive the differential equation relating the output voltage y(t) and the input current x(t). Create an Octave/MATLAB m-file that uses the ode45 function to solve the differential equation via numerical integration over the time interval 0 ≤ t ≤ 3. Plot the result. In order to numerically solve the differential equation you will also need the initial conditions on y(t) and y'(t) at t = 0+ . You can find these conditions using your theoretical result for y(t) from Part I. (For an extra challenge, try to find the initial conditions on y(t) at t = 0+ using circuit analysis of the t = 0+ equivalent circuit.) Plot your theoretical expression for y(t) from Part I on the same graph as your numerical solution. Include a listing of your m-file and your graph (showing both the numerical integration and theoretical results) in your report.

I attached question and the answer for the question solving

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