ECE 213 Spring 2019 Continuous-Time Signals & Systems
Computing Exercises
Exercise SS-C2: Using Laplace to find the impulse response
[20pts] Consider the RLC circuit shown to the right. The “input” voltage is x(t), and the “output” voltage is y(t).
We will use the Laplace transform to find the impulse response for different values of R, L, and C, which will allow us to find the convolution integral in the next exercise.
Design Specifications
1. Use the Laplace transform to find the transfer function associated with this circuit, i.e.,
H(s) = Y(s)/X(s). Explain why H(s) is also the Laplace transform of the impulse response h(t). There are three dampings possible. For each, rewrite H(s) so that you can transform it back to the time domain, and then write h(t). Indicate how you will determine each unspecified constant in h(t). [5]
2. Use MATLAB to take user input for R, L, and C. Determine the damping, then construct and plot h(t) vs. t from t = 0 to t = 120ms. Each function h(t) has two terms, therefore, plot each of the individual terms with a dotted line, then draw the sum with a solid line. The three sets of values to test and include are (R, L, C) = (50Ω, 250mH, 625μF), (50Ω, 500mH, 800μF), and (60Ω, 625mH, 250μF). [10]
3. Include a meaningful title and legend. For instance, include the damping and the values of R, L, and C in the title, and include the functional forms of the individual terms in the legend, with known values wherever possible. [5]
4. Your script should be well organized and easy to understand. Include your name, a context (e.g., ECE 213, Exercise C2), the date you started, and a description. Add in-line comments to help the reader, and add sectioning to help organize your script. Have the user input R in Ω, LinmH,andCinμF. [+2]
5. The figure should be easy to read. For instance, use LaTeX in the legend. Use ms for the time axis, and make the font sizes and line widths appropriate for the size of the figure. [+2]
6. Your script should be robust and efficient. For instance, you only need one set of IF statements to determine the damping and calculate everything. You can also write some of the legend text within the same set of IF statements. In other words, you do not need one set of IF statements to determine the damping, another set to create the impulse response, and a third set to set up the legend and plot. You only need one PLOT command and one LEGEND command. You only need one time array in seconds and one time array in ms. That is, these do not need to be defined inside the IF statements. Use one time array to define the other. Use parameters wherever possible, e.g., tmax. If there is a common factor, define it once and use it multiple times. [+3]
[continued]
ECE 213 Spring 2019 Computing Exercise SS-C2 Continuous-Time Signals & Systems Using Laplace to find the impulse response
NOTES AND SUGGESTIONS:
• Use the Examples of Laplace Transform Pairs in your textbook (Table 3-2 on page 97). You will need to adapt its notation to this notation, but this is typical.
• It is strongly recommended to convert the denominator of H(s) to the following standard form:
D(s)=s2+2αs+ω02
This form is recommended because the roots of D(s) can be written easily, and it’s also
easier to identify the damping:
s12=−α± α2−ω02
• For all three cases, once you have written the partial fraction expansion, multiply both sides by D(s) and match coefficients. That is, the coefficients of s on both sides of your equation must be equal to each other, and the constant terms on both sides must be equal to each other. (This is the only way that the left side can equal the right side.)
• For the overdamped case (α > ω0), use a matrix equation in MATLAB to find the coefficients. (For the other two cases, you can solve for the coefficients yourself.)
• For the critically damped case (α = ω0), you should find that there is a repeated pole in your expression for H(s). The partial fraction expansion is unusual (and you might not
have done this in quite a while):
N(s) = a1 + a2 +!+ an
(s+α)n s+α (s+α)2
where N(s) is a polynomial in s having an order smaller than n.
(s+α)n
• For the underdamped case (α < ω0), you will need to compute: ω= ω02−α2 in rad/s, then rewrite the denominator as:D(s)=(s+α)2+ω2so that you can match the forms for the damped sinusoidal functions in your table ofLaplace transform pairs.• Successful completion of Design Specs #1–3 is worth 20 points, plus 5 Mastery points, if appropriate. (The Mastery points fade away after the due date.)• Design Specs #4–6 are worth Bonus points added onto your score. If your solution is poor, then you might earn Bonus points that are later taken away once you have fixed your solution. In other words, you might successfully meet Design Specs #4–6 without meeting all of Design Specs #1–3, and then lose those Bonus points later, once you have successfully met Design Specs #1–3.ece213 C2 19-0311.doc
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