Reference no: EM13875523

Part 1.

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

H(s) = V_o(s)/V_i(s)

in terms of the impedances Z₁, Z₂, Z₃, and Z₄.

2. Let Z₁(s) = sL₁, Z₂(s) = 1/sC₂, Z₃(s) = R₃, and Z₄(s) = 1/sC₄, corresponding to the circuit below. Using Kirchhoff's voltage and currents laws, derive the 3rd order differential equation for v₀(t). 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 voltages v₀ (t) and v_L(t) . From the differential equation, determine the transfer function, H(s) = V_o(s)/ V_??(s). Verify that it is equivalent to the result in 1) for the choice of impedances in 2).

3. Let the circuit elements have parameter values L₁ = 0.1 H, C₂ = 1.4 μF - α × 0.4 μF, R₃ = 1300 + α × 100 ohms, and C₄ = 1.0μ + α × 0.2μ, 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 C₂ capacitance value lies in the range [1, 1.2] μF, the C₄ capacitance value lies in the range [1, 1.2] μF, and the resistor value lies in the range [1300, 1400] ohms. 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 vL₁ (t) . Verify that v₀ (t) and vL₁ (t) approach the correct steady-state values as t → ∞

Part II.

4. Using the circuit and transfer function from task 2), find all finite poles and zeros and plot their location in the s-plane. 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) = 10 sin(2500t)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 2,500 rad/sec.

5. Set the inductance value to L₁ = 0.1 + 0.4 × 0.xyzH, where again .xyz is the last three digits of your student ID number. (The inductance then lies in the range [0.1 to 0.5] H.) Design the filter to be a 3rd-order Butterworth filter with -3 dB cutoff frequency 20 kHz(40,000Π radians/sec). This will involve, for your specified value of L₁, properly selecting the values of C₂, C₄,and R₂. Note that the desired Butterworth filter transfer function must have the form

H(s) = ω_c³/(s³ + 2ω_cs² + 2ω_c²s + ω_c³) where ω_c is the cutoff frequency. After designing the filter, complete the following.

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), determine the equation for the impulse response, h(t), the output voltage in response to an impulse input, v_i(t) = δ(t).

e. Without computer assistance, 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 v_i(t) = cos(2Π20000t) 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 v_o(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 v_o(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 generic circuit depicted in Figure 1, and now let Z₁ be a capacitor (Z₁(s) = 1/sC₁), let Z₂ and Z₄ be inductors (Z₂(s) = sL₂ and Z4(s) = sL₄), with Z3 still a resistor (Z₃(s) = R₃). Determine the transfer function of the resulting filter.

a. Let the resistor value be R₃ = 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 200 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(2Π200t) 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.

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.