Posted: October 29th, 2022

Principles of Electrical Engineering II

Principles of Electrical Engineering II
332:222 Spring 2022
Project #1
Please submit to Canvas by Friday March 11, 2022 at 11:59PM.
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 your name at the top of the page as
well as your 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 text on the title page shall be typed. 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 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 in 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. Class projects should be your individual work only!
Project Description
This project is intended to introduce you to the concept of series resonance. This project
will help to prepare you for Lab Experiment #3 in the associated laboratory course. All the
mathematical tools that you need you have already learned in Principles of Electrical Engineering I last semester. Please examine the schematic below.
This is a real circuit that I put together and performed measurements on. I used a function
generator (AC voltage source where the amplitude and frequency can be varied) to introduce
a fixed 2 volt peak sine wave to the circuit. This source was used as a reference voltage so it
will always have a phase angle of zero degrees associated with it. Then, using an oscilloscope (a test instrument which plots voltage versus time) I measured the resistor, capacitor,
and inductor voltages with their associated phase angles. See the four photos at the end of
this document. These values are recorded in the table below. To help improve the theoretical
results I measured all of the components using a good quality LCR meter. Note that I also
measured the internal resistance of the inductor so this should be taken into account in your
calculations.
TEST RESULTS
Freq (kHz) T (uS) |VR| (Volts) Angle VR
(Deg)
|VC| (Volts) Angle VC
(Deg)
|VL| (Volts) Angle VL
(Deg)
50 20 0.476 +72 3.38 -15 1.48 +160
70 14.3 1.22 +35 6.6 -57 5.6 +121
76.4 13.1 1.52 0 7.35 -89 7.1 +88
90 11.1 0.97 -46 3.88 -136 5.4 +39
100 10 0.7 -60 2.52 -151 4.2 +26
The theory behind this circuit is that the (complex) impedance of inductors ZL = j 2π f L
and capacitors ZC =−
j
2π f C
are both frequency dependent. Furthermore, the impedance
of these two components are in opposite directions on the complex plane. So, at one special
frequency, which is called the “resonant frequency” ω 0
or f
0
the reactive component of the
circuit impedance from the inductor will cancel out the reactive component of the circuit impedance from the capacitor and the total impedance of the circuit will be purely resistive. At
the resonance frequency, the current in this series circuit will be at a maximum and the phase
angle between the source voltage and the circuit current will be zero. The voltage across the
resistor will also be at a maximum because v R = i×R and R has an angle of zero degrees
associated with it.
Task #1 10 pts
Starting with |XL
|=|XC
| derive the formula for the resonant frequency f
0
(you have
seen this many times before as ω 0 = 2π f
0 =
1
√LC
).
Task #2 30 pts
Find the formulas for ztotal(f), vR(f), vC(f), vL(f), and i(f) (note that these are complex quantities that are a function of frequency – remember phasors from PEE1) in the series RLC circuit
above. Enter the real component values into these formulas (don’t forget the inductor resistance) so the only variable in your formulas will be the frequency f and everything else will be
a constant.
Task #3 50 pts
Calculate the theoretical values of VR, VC, and VL (both magnitude and phase angle) at the
test frequencies used in the table above. That is 50kHz, 70kHz, 76.4kHz, 90kHz, and
100kHz. Create a table for these calculated values, in the same order as in the table above,
so they can be easily compared to the experimental results. Include this table in your report.
Hint: You can use an equation solver or programming language, that can handle complex
numbers, for your calculations. It is a lot faster then doing your calculations using a scientific
calculator. I used the MATLAB programming language to do my calculations for this project.
Question: Can the voltage magnitude at resonance, across any of the circuit components, be
greater than the magnitude of the source voltage? 10 pts

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