This assignment is just a review of Laplace Transform concepts and techniques from E12.

The circuit below is used in the problems that follow.

For the circuit shown:

- Find the differential equation relating v
_{in}and v_{out}. - For particular values of R, L and C, show that the system has the
transfer
function
with ω
_{0}=4 and ζ=0.25.

For the system from problem 1b:

- By hand (calculators are OK), find the pole and zero locations and draw a pole zero diagram
- Create a transfer function object (Matlab's "tf"). Repeat a with Matlab's "zpkdata" and "pzmap" functions.
Check ζ and ω
_{0}with "damp." The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10]. We'll use these limits later - keeping the limits consistent makes it easier to compare different cases. - Repeat b if ω
_{0}=4 and ζ=0.025. - Repeat b if ω
_{0}=8 and ζ=0.25. - Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
- How do ω
_{0}and ζ affect the pole and zero locations (one or two quick sentences).

For the system from problem 1b:

- Zero-State Response
- Find an expression for the unit step response (Laplace Transform table)
- Use Matlab to plot the result from part a,i for 15 seconds.
- Use Matlab's "step" command to plot the step response for 15 seconds.

- Zero-Input Response: Find an expression for the zero input response if initial conditions are

x(0-)=0, ẋ(0-)=-1. - Complete Response: Find an expression for the output of the system from problem 2 if the input
is a unit step and initial conditions are

x(0-)=0, ẋ(0-)=-1

- Repeat problem 3,a,iii if ω
_{0}=4 and ζ=0.025. Keep max time the same as before, to make comparisons easier. - Repeat problem 3,a,iii if ω
_{0}=8 and ζ=0.25. Keep max time the same as before, to make comparisons easier. - How do ω
_{0}and ζ effect the step response? (1 or 2 brief sentences).

For the system from problem 1:

- Use Matlab's "bode" command to make a Bode plot and "pzmap" for a pole zero diagram.
- If the input is v
_{in}(t)=2sin(4·t+45°) use the Bode plot to find an approximate expression for the sinusoidal steady state output. This requires no complicated mathematics. - Repeat b if v
_{in}(t)=2sin(t+45°) - Repeat a if ω
_{0}=4 and ζ=0.025 (i.e., the system from problem 4a). The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10]. - Repeat a if ω
_{0}=8 and ζ=0.25 (i.e., the system from problem 4b). The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10]. - Explain a, d, and e in terms of pole locations (i.e., why are some outputs larger than others?). (1 or 2 brief sentences).
- Classify the system as lowpass, bandpass or highpass.
- Explain g in terms of pole and zero locations.

If the resistor and capacitor in the original circuit are swapped the transfer function becomes:

- Draw a pole-zero diagram with ω
_{0}=4 and ζ=0.25. The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10]. - Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
- Draw a Bode diagram.
- Classify the system as lowpass, bandpass or highpass
- Explain d in terms of pole and zero locations. (1 or 2 brief sentences).
- If the input is v
_{in}(t)=2sin(4·t+45°) use the Bode plot to find an approximate expression for the sinusoidal steady state output. - Repeat part f if v
_{in}(t)=2sin(t+45°)

If the resistor and inductor in the original circuit are swapped the transfer function becomes :

- Draw a pole-zero diagram with ω
_{0}=4 and ζ=0.25. The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10]. - Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
- Draw a Bode diagram.
- Classify the system as lowpass, bandpass or highpass.
- Explain d in terms of pole and zero locations. (1 or 2 brief sentences).
- If the input is v
_{in}(t)=2sin(4·t+45°) use the Bode plot to find an approximate expression for the sinusoidal steady state output. - Repeat part f if v
_{in}(t)=2sin(t+45°)