Reference no: EM131285251

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

Figure 1. Circuit with input voltage v_i(t), and output voltage v₀(t).

Part 1.

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

H(s) = V₀(s)/V_i(s)

2. Using Kirchhoff's voltage and currents laws, derive a 3rd order differential equation for v_C2(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 v_C(t).

3. Let the circuit elements have parameter values L₁ = 0.15 H, C₁ = 0.2 μF, R₁ = 2000 Ω , C₂ = 0.15 μF, and R₂ = 1,500 + α × 2,000 Ω, where α = 0.xyz, with xyz the final three digits of your student ID number. The value of 0.xyz satisfies 0 ≤ 0.xyz ≤ 1, so the R2 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 v_i(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 v_L(t) and v_C(t). Verify that v₀(t), v_L(t), and v_C(t) approach the correct steady-state values as t → ∞ . Find, in addition, the 100% rise time, percent overshoot, and 2% settling time for v (t). Comment on the relative shapes of the v₀ (t) and v_C2(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 V_i(t) = 60 sin(5,500t) u(t). 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 R₂ = 0 for the remainder of the project and let the inductance value be L = 0.01 + 0.04 × 0.xyz 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 R₁. Note that the desired Butterworth filter transfer function must have the form

H (s) = ω_c³/(s³ + 2ω_cs² + 2ω_c²s + ω³_c , where ω_c is the cutoff frequency. After designing the filter, complete the following. (Note: Use the expression for H(s) 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, v_i(t) = δ(t). Use the initial and final value theorems to check your result for t = 0+ and t → ∞.

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 R₂ = 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 R₁ = 100 + 100 × (0.xyz) 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) = s³/(s³ + 2ω_cs² + 2ω_c²s + ω_c³), where ω_c 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 v_i(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.