MAE 2055: Mechetronics I
Mechanical and Aerospace Engineering
Fall 2013
Homework 6
1) A 1V step input signal, 1V∙ 𝑢(𝑡), is applied to the following RC circuit.
i(t)
R
vC(t)
2 KΩ
vs(t)
1V∙ u(t)
C
10 nF
a) What is the initial value of 𝑣𝑐 (𝑡), just after the input transition? What is the final value of 𝑣𝑐 (𝑡)?
b) What is the initial value of 𝑖(𝑡), just after the input transition? What is the final value of 𝑖(𝑡)?
c) What is the time constant of the circuit?
d) Determine an expression for the voltage across the capacitor as a function of time, 𝑣𝑐 (𝑡), for
𝑡 ≥ 0. (You may use the general-form solution we derived in class as your starting point.)
e) Determine an expression for the current through the circuit as a function of time, 𝑖(𝑡), for 𝑡 ≥ 0.
2) Using MATLAB, plot 𝑣𝑐 (𝑡) and 𝑖(𝑡) for t ≥ 0.
a) Plot both waveforms on the same figure, but on separate axes (use MATLAB’s subplot
functionality). Write a MATLAB program (m-file) to perform the plotting. Submit hard copies of
the plot and the m-file with your homework assignment.
b) Plot both curves with a line width of 2.
c) Ensure that the final values of the waveforms are not at the edge of the axes. For example, if
𝑣𝑐 (𝑡) eventually settles to 5V, the vertical axis should extend to maybe 6V or so, not just 5V.
d) Provide axis labels (with appropriate units) and titles on both subplots.
e) Plot over a long enough time period to show the signals reaching their final values, but not so
long that the transient behavior is not well displayed (perhaps ~7𝜏 - 10𝜏 or so).
f)
Place a marker on each plot at 𝑡 = 𝜏. Remember, the waveforms should have transitioned
through 63% of their full swing at 𝑡 = 𝜏. You may use the “Data Cursor” function in the figure
window (there is a button in the toolbar) to manually add markers to your plots. Hold the “Alt”
key while clicking your plot to add additional data markers.
3) You’ll now repeat 1) and 2) for the following RC circuit. Here, the input signal is a negative-going
voltage step, stepping from 3𝑉 to −1𝑉 at 𝑡 = 0. The input signal, 𝑣𝑠 (𝑡), can be expressed as:
𝑣𝑠 (𝑡) = 3𝑉 − 4𝑉 ∙ 𝑢(𝑡) = �
i(t)
R
+3𝑉, 𝑡 < 0
−1𝑉, 𝑡 ≥ 0
vC(t)
330 Ω
vs(t)
3V - 4V∙u(t)
C
1 μF
a) What is the initial value of 𝑣𝑐 (𝑡), just after the input transition? What is the final value of 𝑣𝑐 (𝑡)?
b) What is the initial value of 𝑖(𝑡), just after the input step? What is the final value of 𝑖(𝑡)?
c) What is the time constant of the circuit?
d) Determine an expression for the voltage across the capacitor as a function of time, 𝑣𝑐 (𝑡), for
𝑡 ≥ 0.
e) Determine an expression for the current through the circuit as a function of time, 𝑖(𝑡), for 𝑡 ≥ 0.
4) Using MATLAB, plot 𝑣𝑐 (𝑡) and 𝑖(𝑡) for 𝑡 ≥ 0.
a) Plot both waveforms on the same figure, but on separate axes (use MATLAB’s subplot
functionality). Write a MATLAB program (m-file) to perform the plotting. Submit hard copies of
the plot and the m-file with your homework assignment.
b) Plot both curves with a line width of 1. (NOTE: this is different than the first circuit.)
c) Ensure that the final values of the waveforms are not at the edge of the axes. For example, if
𝑣𝑐 (𝑡) eventually settles to 5V, the vertical axis should extend to maybe 6V or so, not just 5V.
d) Provide axis labels (with appropriate units) and titles on both subplots.
e) Plot over a long enough time period to show the signals reaching their final values, but not so
long that the transient behavior is not well displayed (perhaps ~7𝜏 - 10𝜏 or so).
f)
Place a marker on each plot at 𝒕 = 𝟓𝝉. (NOTE: this is different than the first circuit.) What
percentage of their final values have the signals reached by 𝒕 = 𝟓𝝉? (Again, use the “Data
Cursor” feature to manually add the markers.)