Principles of Electrical Engineering II
332:222 Spring 2022
Project #2
Please submit to Canvas by April 15, 2022 at 11:59 PM
Project format
For all the projects assigned in this course the following format is to be used
1. Each project is to have a title page, which will include the student name at the top of the
page as well as their student ID number. The project number will be centered on the title
page along with the submission date. At the bottom of the title page please write “Principles
of Electrical Engineering II 332:222” and “Spring 2022.” The page format should be based on
8.5″ x 11″ (American A sized) plain white paper for all the pages in your report.
2. The title page will be followed by a brief introduction section, which will be one or two short
paragraphs long. After the introduction section the various project tasks will be answered.
Text must be typed. Schematic diagrams and graphs will be drafted and plotted using a
computer. Mathematical formulas may be neatly printed using either blue or black ink and
then scanned or typed using a word processor.
3. Class projects will be submitted to Canvas in PDF format. Please verify that your project
has been uploaded properly. Canvas has been set two allow 2, and only 2, submissions for
Project #2. The extra submission is just in case of internet connection issues. If you cannot
submit your project to Canvas and you have used up both of your submission attempts
immediately attach your Project #2, in pdf format, to an email and send it to Professor
McGarvey at [email protected] and Head TA Yichao Yuan at [email protected].
Project Description
An unmarked inductor, typically used in a switched mode power supply, is shown in the photo
below. For reference this component is about the size of a US quarter.
We want to find the inductance value of this component using basic electronic test
equipment which includes a function generator (an AC voltage source that can produce a
variety of waveforms over a wide range of voltages and frequencies), an oscilloscope (an
instrument that plots an input voltage versus time), a known capacitor, and various cables and
connectors. We also want to know the inductor’s internal resistance.
The circuit below was used to test the unknown inductor. A 200 Hz square wave with a
voltage that varies between 0 volts and 4.3 volts was supplied to the circuit using the function
generator. This acts as a repeating step input. Both the square wave input and the step
response were displayed using the oscilloscope. See the series of photos below. The input
square wave is the yellow trace and the step response is the blue trace. As you can see from
the oscilloscope photos the response is underdamped with minimal damping. The period of
the input square wave was selected to ensure that the response from the parallel LC circuit
had adequate time to settle to a steady-state value of zero volts before a new step input was
applied.
a) Use the approximation ω0≈ωn
, for a lightly damped system, to find the value of the
inductor L. As can be seen from the oscilloscope measurements, the time period for five
cycles of oscillation is 195.5 μs.
b) Use the envelope data, from the last oscilloscope photo, to find the resistance in the RLC
parallel circuit. Note that this resistance will be frequency dependent due to the skin effect.
This resistance can be calculated based on the generic formula for the voltage response
v (t) = e
−α t
(C1 cos(ωn
t)+C2sin(ωn
t)) where e
−αt
represents the envelope of the voltage
response.
Note that finding the inductor resistance R requires going through a very difficult derivation
which I would consider beyond the scope of this course. For this reason, I have included the
formula that I derived for α below.
I have included the needed information below along with an extra “hint” as a present.
1. First, even though the test circuit looks simple, α is more complicated then the simple
series and parallel RLC cases that we studied in the lectures. The formula for α for this
test circuit is
α =
1
2 RsC
+
R
2 L
where
R = the unknown inductor internal wire resistance
Rs = 1050Ω source resistance ( 1k Ω + 50Ω function generator resistance)
L = parallel inductance
C = parallel capacitance
The formula for α makes sense in that this test circuit has both series and parallel parts.
Inductors always have some internal resistance associated with them because the wire they
are wound with has some resistance depending on its size. Also, wire resistance is frequency
dependent due to something called the “skin effect” where the current flowing through a
conductor crowds to the outside at higher frequencies. The inductor resistance you will get is
at the RLC resonance frequency.
2. When you are calculating α you just want to look at the envelope of the waveform
which is represented by the e
−α t
term in the formula for the output voltage. The time
and voltage values for the two time points in the output voltage’s envelope are
At t
1
vout = 188mV and t
1 = 10μ S
At t
2
vout = 28mV and t
2 = 400 μ S
I read these values off of the oscilloscope plot.
Hint: Divide the equation for the output voltage at t2 by the equation for the output
voltage at t1 to find α. Some algebra will be needed to get an equation to solve for alpha.
