Principles of Electrical Engineering II332:222 Spring 2022

Project #2Please 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 thepage as well as their student ID number. The project number will be centered on the titlepage along with the submission date. At the bottom of the title page please write "Principlesof Electrical Engineering II 332:222" and "Spring 2022." The page format should be based on8.5" 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 shortparagraphs 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 acomputer. Mathematical formulas may be neatly printed using either blue or black ink andthen scanned or typed using a word processor.

3. Class projects will be submitted to Canvas in PDF format. Please verify that your projecthas been uploaded properly. Canvas has been set two allow 2, and only 2, submissions forProject #2. The extra submission is just in case of internet connection issues. If you cannotsubmit your project to Canvas and you have used up both of your submission attemptsimmediately attach your Project #2, in pdf format, to an email and send it to ProfessorMcGarvey 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 testequipment which includes a function generator (an AC voltage source that can produce avariety of waveforms over a wide range of voltages and frequencies), an oscilloscope (aninstrument that plots an input voltage versus time), a known capacitor, and various cables andconnectors. 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 avoltage that varies between 0 volts and 4.3 volts was supplied to the circuit using the functiongenerator. This acts as a repeating step input. Both the square wave input and the stepresponse were displayed using the oscilloscope. See the series of photos below. The inputsquare wave is the yellow trace and the step response is the blue trace. As you can see fromthe oscilloscope photos the response is underdamped with minimal damping. The period ofthe input square wave was selected to ensure that the response from the parallel LC circuithad adequate time to settle to a steady-state value of zero volts before a new step input wasapplied.

a) Use the approximation ω0≈ωn , for a lightly damped system, to find the value of theinductor L. As can be seen from the oscilloscope measurements, the time period for fivecycles of oscillation is 195.5 μs.

b) Use the envelope data, from the last oscilloscope photo, to find the resistance in the RLCparallel 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 responsev(t) = e−α t(C1 cos (ωn t)+C2 sin(ωnt)) where e

−αt represents the envelope of the voltageresponse.

Note that finding the inductor resistance R requires going through a very difficult derivationwhich 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 thistest circuit is

α = 12 RsC

+R

2 L

where

R = the unknown inductor internal wire resistanceRs = 1050 Ω source resistance ( 1k Ω + 50Ω function generator resistance)L = parallel inductanceC = 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 theyare wound with has some resistance depending on its size. Also, wire resistance is frequencydependent due to something called the “skin effect” where the current flowing through aconductor crowds to the outside at higher frequencies. The inductor resistance you will get isat the RLC resonance frequency.

2. When you are calculating α you just want to look at the envelope of the waveformwhich is represented by the e−α t term in the formula for the output voltage. The timeand voltage values for the two time points in the output voltage’s envelope are

At t1 vout = 188mV and t1 = 10 μ SAt t2 vout = 28mV and t2 = 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.