E71 Assignment 0

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

 

The circuit below is used in the problems that follow.

Problem 1: Differential Equation and Transfer Function

For the circuit shown:

  1. Find the differential equation relating vin and vout.
  2. For particular values of R, L and C, show that the system has the transfer function with ω0=4 and ζ=0.25.

Problem 2: Pole Zero Plot

For the system from problem 1b:

  1. By hand (calculators are OK), find the pole and zero locations and draw a pole zero diagram
  2. Create a transfer function object (Matlab's "tf").  Repeat a with Matlab's "zpkdata" and "pzmap" functions.  Check ζ and ω0 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.
  3. Repeat b if ω0=4 and ζ=0.025.
  4. Repeat b if ω0=8 and ζ=0.25.
  5. Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
  6. How do ω0 and ζ affect the pole and zero locations (one or two quick sentences).

Problem 3: Zero State / Zero Input / Complete Response

For the system from problem 1b:

  1. Zero-State Response
    1. Find an expression for the unit step response  (Laplace Transform table)
    2. Use Matlab to plot the result from part a,i for 15 seconds.
    3. Use Matlab's "step" command to plot the step response for 15 seconds.
  2. Zero-Input Response: Find an expression for the zero input response if initial conditions are
    x(0-)=0, ẋ(0-)=-1.
  3. 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

Problem 4: The Step Response:

  1. Repeat problem 3,a,iii  if ω0=4 and ζ=0.025.  Keep max time the same as before, to make comparisons easier.
  2. Repeat problem 3,a,iii  if ω0=8 and ζ=0.25.  Keep max time the same as before, to make comparisons easier.
  3. How do ω0 and ζ effect the step response?  (1 or 2 brief sentences).

Problem 5: The Bode Diagram

For the system from problem 1:

  1. Use Matlab's "bode" command  to make a Bode plot and "pzmap" for a pole zero diagram.
  2. If the input is vin(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.
  3. Repeat b if vin(t)=2sin(t+45°)
  4. Repeat a if ω0=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].
  5. Repeat a if ω0=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].
  6. Explain a, d, and e in terms of pole locations (i.e., why are some outputs larger than others?).   (1 or 2 brief sentences).
  7. Classify the system as lowpass, bandpass or highpass.
  8. Explain g in terms of pole and zero locations.

Problem 6: Swap Capacitor and Resistor

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

  1. Draw a pole-zero diagram with ω0=4 and ζ=0.25.  The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10].
  2. Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
  3. Draw a Bode diagram.
  4. Classify the system as lowpass, bandpass or highpass
  5. Explain d in terms of pole and zero locations.   (1 or 2 brief sentences).
  6. If the input is vin(t)=2sin(4·t+45°) use the Bode plot to find an approximate expression for the sinusoidal steady state output.
  7. Repeat part f if vin(t)=2sin(t+45°)

Problem 7: Swap Inductor and Resistor

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

  1. Draw a pole-zero diagram with ω0=4 and ζ=0.25.  The pole zero map should have as axis limits [xmin xmax ymin ymax]=[-3 1-10 10].
  2. Identify the number of finite poles and zeros as well as the numbers of zeros as |s|→∞.
  3. Draw a Bode diagram.
  4. Classify the system as lowpass, bandpass or highpass.
  5. Explain d in terms of pole and zero locations.   (1 or 2 brief sentences).
  6. If the input is vin(t)=2sin(4·t+45°) use the Bode plot to find an approximate expression for the sinusoidal steady state output.
  7. Repeat part f if vin(t)=2sin(t+45°)