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1 PROJECT 6 (100 points total) ENGIN 211L Engineering Math Submission: • Submit this project via Blackboard as a single pdf or Word document is preferred. • If some portion of your project is handwritten, submit it as a black and white scan (not a color scan). Scans must be high contrast with no shadows. Use a dark pen or pencil on unlined paper for best scanning results. • Clearly label all Parts (a) – (e) in your submission file(s). • For Parts (b) and (c), you must show all work with intermediary steps to receive full credit. • For Part (e), submit the plot as an image (jpg/pdf/png) incorporated into your Project pdf/Word document or as separate image files . MATLAB fig files will not be accepted, and the project will be returned with a zero score. • For partial extra credit in case of point deductions, include any MATLAB code you used to do this project either as a separate m-file or pasted into your Project pdf/Word document. PARALLEL RLC CIRCUIT (total 100 points) Consider the parallel RLC circuit below with a 50-Hz AC current source: 𝑖 ac (?) = 100 cos(100𝜋?) A Assume that no initial energy is stored in the circuit at the instant when the switch, which is shorting the dc current source, is opened. The problem is to find the time- domain expression for 𝑣(?) when ? ≥ 0 and determine if the voltage rating threshold of the capacitor has been exceeded when the dc current source is first switched on. Use the Laplace transform method to determine the complete response of the voltage 𝑣(?) . To begin, the integrodifferential equation that governs the circuit response is the following: 𝑣(?) 𝑅 + 1 𝐿 ∫ 𝑣(𝑥) 𝑑𝑥 𝑡 0 + 𝐶 𝑑𝑣(?) 𝑑? = 𝑖 ac (?) Equation 1

2 Notes: • Initial condition: The initial voltage across the circuit is 𝑣(0 − ) = 0 because the initial energy stored in the circuit is zero. • The AC current source is 𝑖 ac (?) = 100 cos(100𝜋?) A. • Use the specific values for 𝑅, 𝐿, 𝐶, and the capacitor’s voltage rating threshol d assigned to you in the RLC Circuit Parameter Assignments on the last page . You can introduce those values right away in Part (a) or stay with variables up until plotting in Part (e), your choice. (a) Using Laplace transforms, transform the integrodifferential equation (Equation 1) into an algebraic equation in the ? -domain in which 𝑉(?) is the unknown variable. Don’t move terms around yet, just take Laplace transforms term-by- term keeping the order from Equation 1. (10 points) (b) Solve the algebraic equation for the unknown 𝑉(?) . The final result will be a 𝑉(?) = a rational function. Set all ? 2 terms to have a coefficient of one. You must show all work with intermediary steps to receive full credit. (10 points) (c) Find the partial fraction expansion of 𝑉(?) . You must show all work with intermediary steps to receive full credit. i. Find the poles (roots) of the polynomial in the denominator. (5 points) ii. Solve for the unknown coefficients for each partial fraction factor. (22 points) (d) Take the inverse Laplace transform of the partial fraction expansion of 𝑉(?) to find the time-domain expression for the voltage response: 𝑣(?) = ℒ −1 {𝑉(?)} . (10 points) (e) Plot the voltage response 𝑣(?) in the time domain using MATLAB. Submit the plot as jpg/pdf/png incorporated into a (Word) document that contains your response to questions in Part (f) below. Code (m-files) alone or MATLAB fig files will not be accepted. i. Plot 𝑣(?) over the time interval 0 ≤ t ≤ 0.1 second using a blue solid line of width 2. (10 points) ii. Plot the capacitor’s voltage rating threshold as horizontal black dashed lines of width 2. (10 points) iii. Label both axes with the appropriate physical quantity as well as units. The minimum font size is 16. (6 points) iv. The minimum font size for all text and numbers in the figure is size 16. (5 points)