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 para – graphs 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 Engi- neering 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 oscillo- scope (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 2 π f L

and capacitors ZC = − j

2 π 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 im – pedance 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 vR = 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 = 2 π 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 quanti- ties 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 resis – tance) 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