by Jeff Anderson
Last updated: Thursday August 01, 2024
Dedication: This work is dedicated to my loving parents, Darcy and Richard Anderson. I could not have created this project without their support and the many sacrifices they’ve made to help me earn my education. I am eternally grateful for everything they’ve done. This work is as much a testament to their strength as parents as it is to the giants on whose shoulders I stand.
Introduction
Have you ever wondered how practicing engineers use linear algebra in their jobs? Do you want to know how to build nonsingular matrices that are used in the real world? Would you like to get hands-on experience using linear algebra to better understand the world around you?
In this laboratory project, you study the Linear Algebraic Nodal Analysis (LANA) algorithm which provides a novel approach to the classic nodal analysis method for electric circuits. Specifically, in this project, you learn how to build nonsingular matrices that are directly applicable to real-world problems in electrical engineering.
This math is centrally important to practicing engineers who often use Computer Aided Design (CAD) software to test their design. Such software runs mathematical algorithms to calculate and model the electrical behavior of proposed circuit designs. The LANA algorithm is a fundamental component of circuit analysis software and provides a great introduction to CAD design as a tool in electrical engineering processes.
This project is great for students majoring in electrical engineering, physics, or applied mathematics or for any student who wants an authentic modeling experience applying linear algebra in an applied context.
Welcome
Welcome to the companion website for the paper Linear Algebraic Nodal Analysis: An Applied Project for a First Course in Linear Algebra. Interested readers can download a pdf copy of this article by clicking the link below:
Linear Algebraic Nodal Analysis: An Applied Project for a First Course in Linear Algebra (.pdf) [uploaded 10/10/2024]
This is the Accepted Manuscript of an article published online on July 31, 2024 by Taylor & Francis in the journal PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. For more information, please note that this article is also referenced online here: https://doi.org/10.1080/10511970.2024.2369984. We owe a big thank you Taylor & Francis for their friendly policies that allow us to share our work in this form.
This webpage is designed to help college mathematics and engineering students and professors enhance linear algebra and circuit analysis curriculum with authentic modeling exercises. The goal of this work is to empower students to create and solve their own nonsingular linear-systems problems within the context of a useful mathematical modeling paradigm.
To achieve this goal, I show you how to prototype or simulate real electric circuits using resistors, dc voltage sources, and dc current sources. I also provide a modeling scheme that supports learners in analyzing the electrical behavior of these systems using a nonsingular linear-systems problem. The entire laboratory set-up and measuring system is easy-to-build, inexpensive, simple-to-assemble, safe, and of appropriate size for in-class demonstrations and student laboratory explorations.
This support website includes links to online videos that introduce relevant curriculum, laboratory project prompts that can be assigned to students, sample experiment videos, MATLAB (or Octave) code that automates the analysis for the modeling process, and many other resources to support you in applying nonsingular linear-systems theory to a collection of useful real-world problems.
For electrical engineers, I should say that this project makes an explicit connection between classical nodal analysis taught in most introductory circuit analysis classes and the more recent modified nodal analysis algorithm created in the mid-1970s. More importantly, this project positions these nodal analysis algorithms in the intersection of electrical engineering and linear algebra. With this in mind, I refer to the algorithm presented in this work as the Linear-Algebraic Nodal Analysis (LANA) algorithm.
For folks interested in learning more about my thought process, please watch my YouTube video of my first academic talk to the public about this project.
Build real-world circuits
I begin by providing a collection of resources that introduce the large volume of prerequisite content necessary to understand how the LANA algorithm works. Specifically, I provide draft copies of two workbooks in which I develop relevant theory using a practical approach.
I believe that the best way for people to learn theory is through their lived experiences and observations. Accordingly, in the introductory workbooks found below, I show you how to build a collection of circuits and make measurements on those circuits as the starting point for all learning. I then present the major results of circuit theory by guiding the learner to make relevant observations about the her own collected data.
This approach is designed to meet the needs of the many students in introductory circuit analysis classes who have zero experience building real circuits before their course begins. To help these students, I believe in starting with the practice of experimentation before diving into mathematical algorithms that simulate physical prototyping.
To help instructors who similarly value a practical approach to circuit analysis, I also provide YouTube videos available free-of-charge that can be assigned for homework prior to in-class meeting times. These resources may be very helpful for instructors who want to use a flipped learning model and dedicate large portions of in-class meetings to hands-on experimentation. Links to those video playlist are provided in the table below.
Create a mathematical modeling framework
Now that we have seen how to build and measure real circuits, I continue this introduction with a general discussion of the linear algebraic nodal analysis algorithm. First, I present all steps for this algorithm in a quick reference guide:
Quick reference guide to the linear-algebraic nodal analysis algorithm (.pdf)
While this short synopsis is a great resource for fast access to major steps in the LANA algorithm, motivated readers might desire a more detailed overview of each step which can be found in the step-by-step guide below:
Step-by-step guide to the linear-algebraic nodal analysis algorithm (.pdf)
The LANA algorithm sits at the intersection of electrical engineering and numerical linear algebra. As I created and refined this algorithm, I made many choices about notation based on my knowledge of both fields. Because I believe that notation matters, I share a document to highlight the decisions I made about the various matrices and equations that arise in the LANA algorithm. You can find this variable guide in the link below:
Linear-algebraic nodal analysis algorithm variable guide (.pdf)
In that document, I provide possible names for every major equation that results from the LANA algorithm. I also explain why I chose each variable name and present other important considerations. Please note that I am not suggesting that my notation hold for the rest of time. I hope that other people will refine my work and improve the notation we use to describe these ideas. However, to do so, I want to ensure my intuition and reasoning is quite clear since many of the choices I made are subtle and took me many hours of thought and research.
Apply the LANA Algorithm to Real Circuits
One of the best ways to learn a new skill is to watch other people perform that skill. To help you decipher this work, my students and I have created a library of example circuits that can be analyzed using the LANA algorithm. In the table below, I provide not only links to .pdf write ups for these examples but also a selection of other resources. These include MATLAB .m script files that can be used to solve the resulting nonsingular linear-systems problems, LaTeX files that show how to typeset the mathematical equations and draw the corresponding circuits, and YouTube video playlists that illustrate the entire modeling process for each of these examples. Hopefully this work makes your learning process easier.
I should note that for the full solution write ups (.pdf) files below, I run through the LANA algorithm in great detail to show each step. This work is based on both the Quick reference guide and the Step-by-step guide to help readers follow along. I also translate the very compact equations into corresponding entry-by-entry form. I thank my students Tommer, Chris, Hayden, Derek, and Misha for the hours of work they put into coaching me about how much detail I should show in these solution sets.
For instructors, please feel free to use this work in your classrooms. I ask you to kindly email me to let me know you are using this work. Even though I grant implicit permission by posting this material freely on this website, I have a strong interest building community with other educators who want to figure out the best way(s) to bring this to students. Also, if I know you are using this material, I am happy to consider specific suggestions for new content, revisions, edits, or any other comments you want to share.
| ELECTRIFY THE LINEAR-SYSTEMS PROBLEM: ADDITIONAL EXAMPLE CIRCUITS | |
| Example 1 circuit with 7 resistors, 3 dc v-sources, and 3 dc i-sources | LANA EXAMPLE 1 PLAYLIST |
|---|---|
| Example 2 circuit with 8 resistors, 2 dc v-sources, and 2 dc i-sources | LANA EXAMPLE 2 PLAYLIST |
| Example 3: circuit with 11 resistors, 3 dc v-sources, and 2 dc i-sources | LANA EXAMPLE 3 PLAYLIST |