Reference no: EM131252027

The project tasks are outlined on the following pages. The student must produce a typed report, including equations, figures and tables, that includes discussion of the results and any conclusions reached. The report may contain a comparison of techniques, a discussion of the behavior of the circuit as a filter, and any positive outcomes of the efforts of the project. The guidelines for good technical report writing need to be followed. We are particularly interested in reports demonstrating abilities related to the following ABET Criteria 3 outcomes: (a) an ability to apply knowledge of mathematics, science, and engineering; (b) an ability to design and conduct experiments as well as analyze and interpret data; (g) an ability to communicate effectively; (k) an ability to use techniques, skills, and modern engineering tools necessary for engineering practice.

1) A preliminary report, consisting of Part I, tasks 1), 2) and 3), is due Wednesday, October 26, at the beginning of class lecture. For this preliminary due date, turn in only the results, the MATLAB commands, and a signed copy of the pledge below; no typed report is needed. Your results should present a derivation of the differential equation and the state space model. This preliminary report is worth 10 points out of the total 100 points on the project grade. You may turn in Part I on Friday, October 28 for a 20% reduction in that part of the grade (i.e., for a maximum of 8 points). The final report must also include all of Part I.

2) Any evidence of collaboration with other students on either the final project report or the preliminary results will result in a grade of zero for the project for all collaborators. Additionally, the violation will be reported to the School of EECS and the University.

3) Write the report in your own words; do not use sentences or paragraphs from any other source. Grading will be based, in part, on content and grammar, so be sure to proof read your report. If significant portions of the report are found to be copied from another source, without proper attribution, a grade of zero will be assigned for the project and the plagiarism reported to the School and University.

4) The pledge below must be signed and turned in with both the preliminary results and the final project report. Final project reports without the signed pledge will receive a grade of zero.

A circuit with input voltage v_i(t), and output voltage v₀(t) , is shown below.

Figure 1. Circuit with input voltage, and output voltage

Part 1.

1. Use s-domain techniques and determine the transfer function

H(s) = V_o(s) + V_i(s)

2. Using Kirchhoff's voltage and currents laws, derive a 3^rd order differential equation for vC₂ (t) in the circuit above. Assume that there is no energy stored in the capacitors or inductor at time t = 0- . From the differential equation, find a state variable representation, and specify the state variable matrices A, B, C, D, to generate the "outputs," voltages v₀ (t) , v_L (t) , and vC₂ (t) .

3. Let the circuit elements have parameter values L₁ = 0.15 H, 1 = 0.2 , 1 = 2000 , 2 = 0.15 , and 2 = 1,500 + × 2,000 , where = 0. , with xyz the final three digits of your student ID number. The value of 0.xyz satisfies 0 ≤ 0.xyz ≤ 1, so the 2 resistance value lies in the range [1.5, 3.5] kΩ. Use Matlab to determine the response of the circuit ( v0 (t) ) over a suitable time interval (roughly [0, 0.010] sec) to input vi (t) = u(t) V, where u(t) is the unit- step signal. Accurately plot the response, v0 (t) , and determine (from the numerical response) the 100% rise time, percent overshoot, and 2% settling time. (Note that the default rise time value that some Matlab functions provide is the 10% to 90% rise time, so make sure your numerical result matches your plot and corresponds to the 100% rise time.) Also, accurately plot vL (t) and vC2 (t) . Verify that v0 (t) , vL (t) , and vC2 (t) approach the correct steady-state values as t → ∞ . Find, in addition, the 100% rise time, percent overshoot, and 2% settling time for vC₂ (t). Comment on the relative shapes of the v₀(t) and vC₂ (t) waveforms.

Part II.

4. Using the circuit and transfer function from task 1), find all finite poles and zeros, and plot their location in the s-plane (e.g., using the Matlab pzmap function). Also, accurately plot the transfer function frequency response (use Bode plots). Verify that the steady-state output level of the unit step-response matches the "dc gain" of the frequency response. Use Matlab to find the response to input ( ) = 60 sin(5,500 ) ( ). Evaluate the response for a suitable length of time so that the response reaches steady state. Then, determine the magnitude and phase of the output v0 (t) , and compare to the magnitude and phase of your Bode plots at frequency 5,500 rad/sec.

5. Set the value of resistor 2 = 0 for the remainder of the project and let the inductance value be = 0.01 + 0.04 × 0. H, where again .xyz is the last three digits of your student ID number. (The inductance then lies in the range [0.01 to 0.05] H). Design the filter to be a 3rd-order Butterworth filter with - 3 dB cutoff frequency 12 kHz ( 24,000π radians/sec). This will involve, for your specified value of L, properly selecting the values of C1, C2 , and 1. Note that the desired Butterworth filter transfer function must have the form H (s)= ω³/s3 + 2ω_cs² + 2ω_c²s + ω_c³, where ω_c is the cutoff frequency. After designing the filter, complete the following. (Note: Use the expression for ( ) above, rather than the numerical valued based on your design, in the tasks below.)

a. Find the pole locations (e.g., using MATLAB), and plot the pole locations in the s-plane. Identify the angle of the complex poles in the s-plane and verify that they correspond to a 3rd-order Butterworth filter.

b. Use state variables and MATLAB to determine the unit step response. Accurately plot the step response and determine the 100% rise time, maximum percent overshoot, and 2% settling time.

c. Accurately plot the filter frequency response (use Bode plots). Verify that the filter response matches the desired Butterworth filter frequency response.

d. Without computer assistance (that is, by hand calculation using partial fraction expansion), determine the equation for the impulse response, h(t), the output voltage in response to an impulse input, vi (t) = δ (t) . Use the initial and final value theorems to check your result for = 0+ and → ∞.

e. Without computer assistance (using partial fraction expansion), determine the equation for the unit-step response. Verify that your equation for unit-step response matches the MATLAB state variable solution in task 5b).

f. The steady-state sinusoidal response of the filter is determined by the frequency response, evaluated at the sinusoid frequency. Let the circuit input be vi (t) = cos(2π12,000t) u(t) . Use Matlab to compute the filter response for a suitably long duration so that the response reaches (approximately) the steady-state signal. Plot in a single figure both v_i(t) and v₀(t), and identify in your figure that the expected filter magnitude and phase are realized. One way to show this would be to plot the theoretical sinusoidal steady state solution (also in the same figure) and show that v0 (t) approaches this as time gets sufficiently large. Verify that the results match your Bode plots.

Approximately how long does it take for the (Matlab) response to reach steady-state? Compare this (observed) time to the settling time found in part 5b).

Part III.

6. Return to the circuit shown in Figure 1, again with 2 = 0. Now, replace each capacitor with an inductor, and each inductor with a capacitor, keeping the resistor as a resistor. Determine the transfer function of the resulting filter (this will be a high-pass filter).

a. Let the resistor value be 1 = 100 + 100 × (0. ) ohms, where .xyz is the last three digits of your student ID number. Design the filter to be a Butterworth high pass filter with -3 dB cutoff frequency 100 Hz. The resulting filter transfer function is then of the form

H_(highpass)(s) =  s, where ω is the high-pass filter cutoff (-3 dB) frequency.

b. Use Matlab to generate the Bode frequency response plots for your filter. Verify that your filter has the correct cutoff frequency.

c. Use Matlab to generate the response to the input vi (t) = sin(2π50t) u(t).

Compare this response to the sinusoidal steady-state response, and relate the observed magnitude and phase (at steady state) to the filter frequency response.

Report Preparation.

In your report, present the results of the above tasks along with the discussion mentioned on the first page, and supporting derivations, analysis, design equations, plots, and MATLAB code. As a rule-of-thumb, the report should be sufficiently detailed so that if you were to refer to it in one year, you could easily follow the derivations, discussion, and results. Since the report must be typeset, select a suitable word processing system for your work. The LaTeX system is available for free download and is excellent, but requires some learning.

Note: It is common practice for students to use this project report as part of their written work submitted to satisfy the University's Writing Portfolio requirement. So, keep in mind that the report might, later, be evaluated for that purpose.

Bonus Credit

Implement the Butterworth low-pass filter using a Digilent board (or equivalent). Use magnitude or frequency scaling, as necessary, to adjust the task 5) inductance to a value readily available in a Digilent parts kit, and then select resistor and capacitor values to design the implemented filter frequency response to be as close as possible to the design requirement. Include design component values, frequency response plots based on measured values, and a photo of your circuit implementation in the final report. Arrange a time with the instructor to demonstrate the proper functioning of your implemented circuit.