MAE 2055: Mechetronics I
Mechanical and Aerospace Engineering
Fall 2013
Homework 7
1) The following RL circuit is driven with a 2𝑉 step input voltage source, 𝑣𝑠 (𝑡) = 2𝑉 ∙ 𝑢(𝑡).
i(t)
R1
vo(t)
500 Ω
L
vs(t)
1 μH
2V∙ u(t)
R2
1.5 KΩ
a) What is the circuit time constant, 𝜏? (Hint: apply KVL around the loop to derive the differential
equation that describes the behavior of this circuit. You don’t necessarily need to solve this
equation, but compare it with the equation describing a similar circuit having only one resistor.
This should enable you to deduce the correct resistance to use to calculate 𝜏 = 𝐿/𝑅.)
b) Derive an expression for the output voltage as a function of time, 𝑣𝑜 (𝑡), for 𝑡 ≥ 0.
c) Derive an expression for the current through the circuit as a function of time, 𝑖(𝑡), for 𝑡 ≥ 0.
2) Using MATLAB, plot 𝑣𝑜 (𝑡) and 𝑖(𝑡) for 𝑡 ≥ 0.
As always, your plots should include titles and axis labels, and should be configured to adequately
display the responses. All of your plotting and plot configuration must be done in your m-file (as
opposed to being done manually, through the GUI).
Submit your plots and m-file with your homework.
3) For each of the following networks:
• Determine the input impedance as a function of frequency, 𝑍𝑖𝑛 (𝜔).
• For circuit (a), determine the value of the input impedance at 1 kHz. For circuit (b), determine
the input impedance at 10 MHz. Express the impedance in both Cartesian and polar forms.
R
C
500 Ω
R
L
0.1μF
Zina
5 KΩ
1 mH
Zinb
(a)
(b)
4) Using MATLAB, plot the magnitude and phase of the input impedance of each of the networks in 3).
a) Plot both the magnitude and phase, |𝑍𝑖𝑛 | and ∠𝑍𝑖𝑛 , 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 |𝑍𝑖𝑛 | and ∠Zin over a frequency range of 1Hz to 1GHz, with a log scale for the frequency
axis.
c) Use a log scale for the vertical axis of the |𝑍𝑖𝑛 | plots, and a linear scale for the vertical axis of the
∠𝑍𝑖𝑛 plots. Units of ∠𝑍𝑖𝑛 should be degrees.
Some additional MATLAB commands that will be useful for generating these plots:
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logspace()
abs()
angle()
semilogx()
loglog()
./
- this is the element-wise array division operator
5) What does |𝑍𝑖𝑛𝑎 | look like as f → 0? As f → ∞?
What does |𝑍𝑖𝑛𝑏 | look like as f → 0? As f → ∞?
That is, describe the impedance qualitatively. (e.g. like an open circuit, or like a 10Ω resistor, etc…)