5th semester files

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-01/Lab01.aux

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-01/Lab01.tex

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+\documentclass[conference]{IEEEtran}
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+\hyphenation{op-tical net-works semi-conduc-tor}% correct bad hyphenation here
+
+\font\myfont=cmr12 at 15pt 
+\title{\myfont Getting Situated with Ansys Electronics Desktop for EEMAGS}
+\author{Aidan Sharpe 916373346}
+
+
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+\begin{document}
+
+\makeatletter
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+
+\section{Introduction \& Methods}
+The primary goals of this lab were to get situated with the \textit{Ansys Electronics Desktop} software and the Rowan Virtual Desktop environment as a whole. Within the \textit{Ansys Electronics Desktop} software, the primary goal was to learn the placements of specific tool bars and widgets by constructing objects in the primary view port and analyzing the electric field around some primitive objects.
+
+The lab consisted of two parts, with the first acting as a guide to using the basic features of the software, and the second allowing for further exploration into different object primitives and the electrical interactions between them.
+
+\section{Results \& Analysis}
+In part one, a perfectly conducting cylinder was charged and centered in a square-based rectangular prism with the same height. The electric field inside the rectangular prism was then analyzed. Since there were no reference voltages, the field lines were un-ordered and fairly chaotic. To resolve this, one of the vertical faces of the prism was defined as $0$V. Then, electric field intensity and direction were analyzed inside the rectangular prism. The resulting electric field, in accordance with predictions, extended from the cylinder to the grounded side of the rectangular prism. On the side of the rectangular prism opposing the grounded side, however, the electric field lines still exhibited some chaotic behavior, similar to the analysis before grounding. While quite simple in theory, this part of the lab still proved to be moderately challenging since some parts of the software were quite unintuitive. After completing part one, however, creating and analyzing objects should be much more straightforward in the future.
+
+While part one was a guided exercise, part two was much more open-ended. A similar process was to be followed with some other shape. For this we created a sheath-ground coaxial cable. By making a hollow copper cylinder defined as $0$V, and a much skinnier concentric solid copper core with charge $1000$C, the electric field in and around the coaxial cable could be measured.
+
+\section{Conclusions}
+While parts of this lab did not directly align with the course material, it definitely served its purpose as a crash course in the basic features of \textit{Ansys Electronics Desktop}. Going forward, using the software will require much less direction, and hopefully less fiddling around with. We found the software to be quite unintuitive to use at times, but we could not find any real suitable alternatives online. While it can be difficult to use at times, using visualization software at all is better than relying on calculations and analogy alone.
+
+\section{Further Comments}
+Unfortunately, I was unable to access Ansys to take screenshots for this lab. I have an IRT ticket and they are working it out. Thank you for understanding.
+\end{document}

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-02/Lab-02.aux

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-02/Lab-02.tex

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+\documentclass[conference]{IEEEtran}
+\usepackage[siunitx]{circuitikz}
+\usepackage{graphicx}
+%\usepackage{lipsum}
+\usepackage{color}
+\usepackage{enumitem}
+\usepackage{epsfig} 
+\usepackage{mathptmx} 
+\usepackage{times} 
+\usepackage{amsmath}
+\usepackage{amssymb}  
+\usepackage{float}
+\usepackage{hyperref}
+%\usepackage{setspace}
+%\usepackage{tikz}
+\usepackage{circuitikz}
+\usepackage{pgfplots}
+\usepackage{textcomp}
+%\usepgfplotslibrary{external}
+%\tikzexternalize
+%\usepackage[utf8]{inputenc}
+%\usepackage[english]{babel}
+%\usepackage{pgf}
+\usepackage{pgfpages}
+\usepackage[margin=0.4in]{geometry}
+\usepackage{lmodern}
+\usepackage{datetime}
+\usepackage{ragged2e}
+\input{border.tex}
+\pgfpagesuselayout{boxed}
+
+\hyphenation{op-tical net-works semi-conduc-tor}% correct bad hyphenation here
+
+\font\myfont=cmr12 at 15pt 
+\title{\myfont Simulating Simple Electrostatic Capacitors}
+\author{Aidan Sharpe 916373346}
+
+
+\providecommand{\keywords}[1]{\textbf{\textit{Keywords---}} #1}
+\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
+\setlength{\columnsep}{7mm}
+\pgfplotsset{compat=1.15}
+\begin{document}
+
+\makeatletter
+\@twocolumnfalse
+\maketitle
+\@twocolumntrue
+\makeatother
+\begin{abstract}
+The primary goal of this lab was to understand the electric fields between differently charged conductors. Since the definition of a capacitor is two separated conductors of differing charge, capacitance was also predicted and measured. Along with a simple parallel plate model, coaxial conductors were also analyzed and found to change the characteristics of the electric field.
+\end{abstract}
+
+\section{Introduction \& Methods}
+The idea of a charged object in a vacuum can be quite helpful when developing simple models and fundamental understandings. Unfortunately, this approach can only be taken so far before a system becomes \textit{oversimplified}. To better understand these complex interactions, more complex models must be made.
+
+Parallel plate capacitor with square plates of side length 0.5[mm], and spacing 0.1[mm]. The bottom plate was defined as 0[V], and the top plate was given a charge, $5 \times 10^{-6}$[C]. Since the measurements are being taken with air between the plates, and air has a relative permittivity of 1.0006, for air, $\varepsilon = 8.859 \times 10^{-12}$. The capacitance was calculated by plugging the known values for area and distance into equation \ref{eqn:capacitance-parallel-plate}. Doing so yields the theoretical capacitance of $2.215\times 10^{-14}$ [F].
+
+\begin{equation}
+    C = {\varepsilon A \over d}
+    \label{eqn:capacitance-parallel-plate}
+\end{equation}
+Where:\\
+$C$ is capacitance\\
+$A$ is the inside area of the plates\\
+$d$ is the distance between the plates\\
+$\varepsilon$ is the permittivity of the material between the plates.
+
+\section{Results \& Analysis}
+\subsection{Air as a Dielectric}
+After setting up the capacitor, analysis began. First, as seen in figure \ref{fig:field-air-edge} and figure \ref{fig:field-air-center}, the electric field between the plates was analyzed at the edge of the plates and through the center of the area between the plates. It is immediately apparent that changing the location of measurement inside the region between the plates does not affect the results of the measurement. For this reason, we can determine that the electric field strength between the plates is constant. Additionally, the strength of the electric field outside the region between the plates drops off quickly and is very small above and below the plates.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{BackAir.png}
+    \caption{Measuring electric field strength at the edge}
+    \label{fig:field-air-edge}
+\end{figure}
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{CenterAir.png}
+    \caption{Measuring electric field strength in the center of the capacitor}
+    \label{fig:field-air-center}
+\end{figure}
+
+Since one of the plates was charged and the other was set as a reference ground, there exists some voltage between the two plates. This voltage can be found using equation \ref{eqn:efield-line-voltage}. To calculate this voltage, the electric field strength was measured along a straight line from the center of one plate to the center of the other. The resulting voltage was 196[MV].
+
+\begin{equation}
+    V = \int\vec{E} \cdot d\vec{l}
+    \label{eqn:efield-line-voltage}
+\end{equation}
+
+To find the experimental capacitance, equation \ref{eqn:cap-charge-potential}. Using the predetermined charge of $5 \times 10^{-6}$[C] and the tool evaluated line integral for voltage, the experimental capacitance was determined to be $2.549 \times 10^{-14}$[F]. The assumption of a uniform electric field between the plates is likely the source of the error. Since the plates have thickness, modelling them as plane charges will cause some discrepancy. 
+
+\subsection{Changing the Dielectric}
+The region surrounding the plates was air in this context, however, changing the material between the plates will change the capacitance. However, changing this material into a conductor will short the plates and effectively ruin the capacitance. For this reason, the material was chosen to be diamond for its insulating and dielectric strengths. Diamond has a relative permittivity of 16.5. Therefore, it should be expected that the capacitance be multiplied by the same factor. Additionally, since capacitance is increasing and charge is remaining constant, the voltage between the plates should reduce by the same factor as well. Sure enough, the evaluated voltage between the plates is 11.86[MV], a factor of 16.53 smaller. 
+
+Similarly, since voltage is related to electric field strength by distance, and the distance is constant, the electric field strength should also decrease. This is verified by the test results shown in figure \ref{fig:diamond-dielectric}. With diamond as a dielectric, the maximum field strength was $1.192 \times 10^{11}$[V/m] compared to $2.004 \times 18^{11}$[V/m] with air as a dielectric.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{BackDiamond.png}
+    \caption{Using diamond as a dielectric}
+    \label{fig:diamond-dielectric}
+\end{figure}
+
+\subsection{Changing the Form of the Conductors}
+Finally, the dielectric was reset to air, and for this test, the parallel plates were replaced with concentric conductors in the shape of a coaxial cable. Since the surface area of the inner conductor is smaller than the surface area of the outer conductor, the electric field cannot be uniform between them. As seen in figure \ref{fig:coax-efield}, the electric field is stronger around the inner conductor, and drops off until the surface of the outer conductor. It is also apparent that there is no electric field beyond the outer conductor, so it has some shielding effects. Additionally, equation \ref{eqn:capacitance-parallel-plate} only applies to parallel plate capacitors, the capacitance of the coaxial conductors was evaluated experimentally.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Coax.png}
+    \caption{Electric field between coaxial conductors}
+    \label{fig:coax-efield}
+\end{figure}
+
+The charge of the inner conductor was set to the same $5 \times 10^{-6}$[C] as before, and the outer conductor was defined as a reference ground. The radius of the inner conductor was 0.05[mm], the inner radius of the outer conductor was 0.1[mm], and the outer radius of the outer conductor was 0.15[mm], leaving a 0.05[mm] gap between the conductors. The line integral evaluated voltage was 248[MV] between the conductors. Again using equation \ref{eqn:efield-line-voltage}, the capacitance was calculated to be $2.016 \times 10^{-14}$[F]. 
+
+\section{Conclusions}
+Using a parallel plate capacitor is quite limiting. Since capacitance is larger with high surface area, getting larger capacitance requires very large plates. Beyond the range of picofarads, other capacitor form factors are preferred for their compactness.
+
+Additionally, parallel plate capacitors, are superb for creating a uniform electric field. However, they are also create electromagnetic disturbances in the surrounding region. This can be reduced with shielding, but adding this shielding will affect the quality and uniformity of the internal electric field. This behavior is exemplified by the coaxial cable. There are no disturbances surrounding the central conductor, but this comes at the cost of lost uniformity.
+
+Overall, the primary takeaway is that the form factor makes a large contribution to a capacitors behavior. There are trade-offs to picking one design over another, so application must be considered.
+\end{document}

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-03/Lab-03.aux

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+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand\HyField@AuxAddToFields[1]{}
+\providecommand\HyField@AuxAddToCoFields[2]{}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction \& Methods}{2}{section.1}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Final coil design}}{2}{figure.1}\protected@file@percent }
+\newlabel{fig:coil}{{1}{2}{Final coil design}{figure.1}{}}
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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-03/Lab-03.tex

@@ -0,0 +1,140 @@
+\documentclass[conference]{IEEEtran}
+\usepackage[siunitx]{circuitikz}
+\usepackage{graphicx}
+%\usepackage{lipsum}
+\usepackage{color}
+\usepackage{enumitem}
+\usepackage{epsfig} 
+\usepackage{mathptmx} 
+\usepackage{times} 
+\usepackage{amsmath}
+\usepackage{amssymb}  
+\usepackage{float}
+\usepackage{hyperref}
+%\usepackage{setspace}
+%\usepackage{tikz}
+\usepackage{circuitikz}
+\usepackage{pgfplots}
+\usepackage{textcomp}
+%\usepgfplotslibrary{external}
+%\tikzexternalize
+%\usepackage[utf8]{inputenc}
+%\usepackage[english]{babel}
+%\usepackage{pgf}
+\usepackage{pgfpages}
+\usepackage[margin=0.4in]{geometry}
+\usepackage{lmodern}
+\usepackage{datetime}
+\usepackage{ragged2e}
+\input{border.tex}
+\pgfpagesuselayout{boxed}
+
+\hyphenation{op-tical net-works semi-conduc-tor}% correct bad hyphenation here
+
+\font\myfont=cmr12 at 15pt 
+\title{\myfont Simulating Simple Electrostatic Capacitors}
+\author{Aidan Sharpe 916373346}
+
+
+\providecommand{\keywords}[1]{\textbf{\textit{Keywords---}} #1}
+\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
+\setlength{\columnsep}{7mm}
+\pgfplotsset{compat=1.15}
+\begin{document}
+
+\makeatletter
+\@twocolumnfalse
+\maketitle
+\@twocolumntrue
+\makeatother
+\begin{abstract}
+This lab focuses on measuring the inductance of a rectangular coil of wire. The coil is modeled to be representative of a standard circuit board coil. The inductance was measured parametrically to understand the impact of trace width. 
+\end{abstract}
+
+\section{Introduction \& Methods}
+Lab setup was probably the most time-consuming portion of the process. Before taking any measurements, the model circuit board coil had to be created. This was done by drawing a path for the trace to follow, and then dimension was provided to create a trace with a rectangular cross section. 
+
+Due to a simpler geometry, measuring a coil following a rectangular path is less computationally intensive than measuring one with a circular path. To work with the computational restrictions imposed by the virtual machine's limited resources, the rectangular path was the obvious choice. It also happens that most PCB design packages default to eight trace directions, so rectangular path coils are somewhat common in the real world. 
+
+Every dimension of the coil plays some role in its inductance---its reaction to a changing current. The coil, as seen in figure \ref{fig:coil} has the dimensions shown in table \ref{tbl:coil-dimensions}. Width and height are the width and height of the rectangular section of the coil, terminal is the length of the terminals, via is the vertical via distance for bringing the second terminal back outside the coil. Trace is the width of the trace, thickness is the thickness of the trace, dist is the distance between parallel traces, and pitch is the distance between the centers of parallel traces.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{coil.png}
+    \caption{Final coil design}
+    \label{fig:coil}
+\end{figure}
+
+\begin{table}[h!]
+\caption{Initial coil dimensions}
+\begin{center}
+\begin{tabular}{c c c}
+    Dimension & Magnitude & Units \\
+    \hline
+    Width & 4 & mm \\
+    Height & 4 & mm \\
+    Terminal & 700 & $\mu$m \\
+    Via & 200 & $\mu$m \\
+    Trace & 200 & $\mu$m \\
+    Thickness & 50 & $\mu$m \\
+    Dist & 100 & $\mu$m \\
+    Pitch & trace + dist & -- \\
+\end{tabular}
+\end{center}
+\label{tbl:coil-dimensions}
+\end{table}
+
+After constructing the coil, a box of air, just larger than the coil, was created. Getting the box to fit properly around the coil took some messing with, but the final positioning and dimensions cab be seen in table \ref{tbl:box-dimensions}. This box of air defined the region over which the magnetic field was analyzed.
+
+
+\begin{table}[h!]
+\caption{Air box dimensions}
+\begin{center}
+\begin{tabular}{c c c}
+    Dimension & Magnitude & Units \\
+    \hline
+    X Position & -width/2 - trace & -- \\
+    Y Position & -1.3 & mm \\
+    Z Position & -1 & mm \\
+    X Size & 2 $\times$ trace + width & -- \\
+    Y Size & length + trace + terminal & -- \\
+    Z Size & 2 & mm \\
+\end{tabular}
+\end{center}
+\label{tbl:box-dimensions}
+\end{table}
+
+\section{Results \& Analysis}
+A 1[A] current was passed through the coil, and the magnetic field was analyzed inside the box of air. In doing so, the inductance of the coil could be found. Using the coil dimensions in table \ref{tbl:coil-dimensions}, the inductance was found to be 22.822[nH]. Given that it is a small coil with only two windings, a small, but non-zero inductance is expected.
+
+Analyzing equation \ref{eqn:inductance}, the inductance of the coil is directly proportional to magnetic flux. Therefore, inductance can be increased by either making the loop area larger or by increasing the number of turns of the coil.
+
+\begin{equation}
+    L = {\Phi_M(i) \over i}
+    \label{eqn:inductance}
+\end{equation}
+
+The goal of this lab was to determine the effect that trace width had on inductance. Originally, the trace width was increased from 100[$\mu$m] to 300[$\mu$m] at 100[$\mu$m] intervals to decrease analysis time, but the result of this simulation was too low resolution to make out any distinct patterns. To mitigate this, the range was increased to extend up to 400[$\mu$m] with a finer step value of 10[$\mu$m]. This drastically increased the resolution, and the results can be seen in figure \ref{fig:inductance}.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{inductance.png}
+    \caption{Inductance as it relates to trace width}
+    \label{fig:inductance}
+\end{figure}
+
+The magnetic field produced by the coil will add around parallel wires and cancel out between the wires. This means that as the coils get wider, the magnetic field has a smaller area to travel through. The net effects of the current in the coil can be seen in figure \ref{fig:magnetic-field}.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{magnetic-field.png}
+    \caption{Magnetic field around the coil}
+    \label{fig:magnetic-field}
+\end{figure}
+
+\section{Conclusions}
+Looking closer at figure \ref{fig:inductance}, increasing the trace width had a positive effect up until about 230[$\mu$m] before levelling off and trending downwards. Since the coil is made of copper and not a perfect conductor, it has some resistance. Therefore, increasing the trace width will increase cross-sectional area and thereby decrease resistance. This possibly explains why the inductance increases as resistance goes down. 
+
+However, as the traces get wider without increasing the size of the coil, the area of magnetic flux must decrease. This is because the area that contributes the most to the magnetic flux is inside the coil. This combination of behavior could explain the increase and decrease of inductance as a function of trace width.
+
+Making an inductor on a circuit board therefore poses a few challenges. Firstly, the current must be limited as the trace width impacts inductance. Second, the number of windings is limited by the area given to the coil. Typically, inductors are mounted components that use all three dimensions allotted. This is because they are defined by their geometry and can often be quite difficult to make compact.
+
+The PCB inductor has its benefits, however. If the required inductance is small, it can be quite easy to put together a PCB inductor to use as a small antenna. Unfortunately, this design takes up a lot of board space and makes the center more or less unusable. For this reason, a better, more compact design should probably be used for most applications.
+\end{document}

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-04/border.tex

@@ -0,0 +1,18 @@
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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-04/lab-04.aux

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+\relax 
+\providecommand\hyper@newdestlabel[2]{}
+\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
+\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
+\global\let\oldnewlabel\newlabel
+\gdef\newlabel#1#2{\newlabelxx{#1}#2}
+\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
+\AtEndDocument{\ifx\hyper@anchor\@undefined
+\let\newlabel\oldnewlabel
+\fi}
+\fi}
+\global\let\hyper@last\relax 
+\gdef\HyperFirstAtBeginDocument#1{#1}
+\providecommand\HyField@AuxAddToFields[1]{}
+\providecommand\HyField@AuxAddToCoFields[2]{}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Introduction \& Setup}{2}{section.1}\protected@file@percent }
+\newlabel{eqn:faraday-law}{{1}{2}{Introduction \& Setup}{equation.1.1}{}}
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+\gdef \@abspage@last{2}

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-04/lab-04.tex

@@ -0,0 +1,99 @@
+\documentclass[conference]{IEEEtran}
+\usepackage[siunitx]{circuitikz}
+\usepackage{graphicx}
+%\usepackage{lipsum}
+\usepackage{color}
+\usepackage{enumitem}
+\usepackage{epsfig} 
+\usepackage{mathptmx} 
+\usepackage{times} 
+\usepackage{amsmath}
+\usepackage{amssymb}  
+\usepackage{float}
+\usepackage{hyperref}
+%\usepackage{setspace}
+%\usepackage{tikz}
+\usepackage{circuitikz}
+\usepackage{pgfplots}
+\usepackage{textcomp}
+%\usepgfplotslibrary{external}
+%\tikzexternalize
+%\usepackage[utf8]{inputenc}
+%\usepackage[english]{babel}
+%\usepackage{pgf}
+\usepackage[final]{pdfpages}
+\usepackage{pgfpages}
+\usepackage[margin=0.4in]{geometry}
+\usepackage{lmodern}
+\usepackage{datetime}
+\usepackage{ragged2e}
+\input{border.tex}
+\pgfpagesuselayout{boxed}
+
+\hyphenation{op-tical net-works semi-conduc-tor}% correct bad hyphenation here
+
+\font\myfont=cmr12 at 15pt 
+\title{\myfont Simulating Simple Electrostatic Capacitors}
+\author{Aidan Sharpe 916373346}
+
+
+\providecommand{\keywords}[1]{\textbf{\textit{Keywords---}} #1}
+\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
+\setlength{\columnsep}{7mm}
+\pgfplotsset{compat=1.15}
+\begin{document}
+\makeatletter
+\@twocolumnfalse
+\maketitle
+\@twocolumntrue
+\makeatother
+\begin{abstract}
+    This lab applies Faraday's Law of Induction, which states that a changing magnetic field induces an EMF voltage in a conductor. In this specific case, the changing magnetic field was supplied by a permanent magnet with a constant, non-zero velocity moving through the center of a copper coil. The ends of the coil were connected through a resistor, which allowed the induced voltage to create a current by Ohm's law.
+\end{abstract}
+
+\section{Introduction \& Setup}
+This lab started as far from the simulation software as possible, locally in FreeCAD. While it is possible (and encouraged for most) to model a system directly in the simulation software, it is finicky at best. Furthermore, the documentation for it is almost as proprietary as the software itself. For this reason, it made sense to use an external tool for design purposes. The technical drawings are included after the appendix.
+
+All components were exported from FreeCAD into the simulation software, positioned properly and the simulation was set up. Currents were measured at the ends of the coil, and the magnet was moved along a path starting below the coil and ending above it. A 100$\Omega$ resistor was placed in series with the coil, so current could be measured through the coil.
+
+The magnet started 2[mm] below the coil and traveled upwards at 500[mm/s] for 10[ms]. The speed was initially 10[mm/s], but this needed to be increased as the magnet only travelled 0.1[mm]. This was so slow that the magnet did not even reach the bottom of the coil before the simulation ended. 500[mm/s] seemed like the most reasonable speed since it would travel a full 5[mm] over the course of the simulation. At this speed, the magnet travels from -2[mm] to +3[mm], which is above the top of the coil.
+
+The importance of it travelling through the coil is that the magnetic field is strongest close to the magnet, so it is critical to get the magnet as close as possible to the coil to see the best induction effects. The induced currents are in accordance with Faraday's Law of Induction seen in equation \ref{eqn:faraday-law}. For a coil of wire, the effects of Faraday's law are multiplied scaled up with each winding. The voltage across a coil with $N$ windings and magnetic flux, $\Phi_B$, through the center is given by equation \ref{eqn:solenoid-emf}. Therefore, by Ohm's Law, the current through the inductor is just the voltage given by equation \ref{eqn:solenoid-emf} divided by 100$\Omega$.
+
+\begin{equation}
+    \nabla \times \vec{E} = -{\partial \vec{B} \over \partial t}
+    \label{eqn:faraday-law}
+\end{equation}
+
+\begin{equation}
+    \mathcal{E} = -N {d\Phi_B \over dt}
+    \label{eqn:solenoid-emf}
+\end{equation}
+
+\section{Results \& Analysis}
+Passing the magnet through the coil creates a changing magnetic field from the coil's perspective. From the magnet's perspective field is static, and the coil is moving towards it. Therefore, when the animation is played, the magnetic field appears to move upwards at the same rate as the magnet. The shape of the magnetic field around the magnet, passing through the coils is seen in figure \ref{fig:magnetic-field}. The magnetic field is originating from the magnet, curling back on itself, and then terminating back at the magnet. This makes sense, since all magnets have a north and south pole, with the magnetic field lines travelling between them.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Magnetic-Field.png}
+    \caption{Magnetic field vectors in the region}
+    \label{fig:magnetic-field}
+\end{figure}
+
+As seen in figure \ref{fig:coil-current}, there are two spikes in current. As the magnet approaches the coil, it makes sense that the magnetic field increases rapidly. It also makes sense, that as the magnet leaves the coil, around 9[ms] or so, the current will decrease. However the dropout in the middle, may be unexpected. Faraday's Law of Induction relates voltage to the time rate of change of the magnetic field, and not directly to its strength. Following this line of thought, it must be that the most change in magnetic field strength happens when the magnet is either entering or leaving the coil.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{CurrentPlot.png}
+    \caption{Current through the coil over time}
+    \label{fig:coil-current}
+\end{figure}
+
+\section{Conclusion}
+The drop in current while the magnet is in the coil was quite unexpected at first, but after recalling Faraday's Law of Induction, the results made more sense. Since the change in magnetic field is highest when the magnet is approaching or receding from the coil, it makes sense that spikes in current occur here. However, when the magnet is centered in the coil, the magnetic field is symmetric with respect to the coil. For this reason, magnetic field changes very little at this point, and in a perfectly symmetric system, the current would probably go to zero. Thinking back to one dimensional kinematics, at the apex of a projectile being thrown straight into the air, the instantaneous velocity goes to zero. 
+
+Measuring the time rate of change of the magnetic flux is analogous to projectile motion in this way. Think about it as the magnetic field strength reaching a maximum in the coil before decreasing and going to zero at $t=\infty$. Since the strength reaches a maximum, the rate of change of strength is by definition, zero. 
+
+The voltage across the coil can be recovered since the resistor has a known value of 100$\Omega$. Simply by scaling the current results by a factor of 100 will yield the EMF voltage of the coil.
+Finally, by working backwards from Faraday's Law of Induction, since the number of coils is known to be $N=5$, by taking the negative of the integral of the EMF voltage with respect to time, the magnetic flux through the coil can be extrapolated.
+
+By knowing, the current, voltage, and magnetic flux of the coil with respect to time, much is learned about the system as a whole, and even more information, such as inductance characteristics, can be derived.
+\end{document}

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/Lab-06

@@ -0,0 +1 @@
+# Created by Octave 8.4.0, Mon Nov 27 11:05:07 2023 EST <sharpe@dhcp-150-250-217-15>

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/cemLaplace01.m

@@ -0,0 +1,129 @@
+% cemLaplace01.m
+% Solution of Laplace's equation
+
+clear all
+close all
+clc
+tic
+
+% INPUTS  ================================================================
+% Number of XY grid points (integer)
+     Nx = 10;  Ny = 10;
+% Range for X and Y values  [minX maxX minY maxY minZ maxZ]
+     XY = [0 2 0 1];
+% tolerance for ending iterations
+     tol = 0.1;
+% number of iteriations
+     maxN = 50;
+% limits for electric field plot
+   minXL = 0   ; maxXL = 2;
+   minYL = 0; maxYL = 1;
+
+% Boundary values for the potential
+    V0 = 10;
+    V = zeros(Ny,Nx);
+    V(:,1) = 20;
+    V(:,end) = 10;
+    V(1,:) = -5;
+    V(end,:) = -10;
+
+% SETUP ==================================================================
+
+    minX  = XY(1); maxX = XY(2);
+    minY  = XY(3); maxY = XY(4);
+    x = linspace(minX, maxX,Nx);
+    y = linspace(minY, maxY,Ny);
+    [xx, yy] = meshgrid(x,y);
+    hx = x(2) - x(1); hy = y(2) - y(1);
+    Kx = hx^2/(2*(hx^2+hy^2)); Ky = hy^2/(2*(hx^2+hy^2));
+
+
+
+    % CALCULATIONS ===========================================================
+
+% dSum difference in sum of squares  /  n  number of iterations
+dSum = 1; n = 0;  cc = 0;
+
+while n < maxN
+%while  dSum > tol
+    sum1 =  sum(sum(V.^2));
+
+    for ny = 2: Ny-1
+
+        for nx = 2: Nx-1
+            V(ny,nx) = Ky * (V(ny,nx+1) + V(ny,nx-1)) + Kx * (V(ny+1,nx) + V(ny-1,nx));
+       % end
+
+        %    V(ny,nx) = Ky * (V(ny,nx+1) + V(ny,nx-1)) + Kx * (V(ny+1,nx) + V(ny-1,nx));
+        end
+      %  V(ny,Nx) = Ky * (V(ny,Nx) + V(ny,Nx-2)) + Kx * (V(ny+1,Nx) + V(ny-1,Nx));
+    end
+
+   sum2 =  sum(sum(V.^2));
+   dSum = abs(sum2 - sum1);
+   n = n + 1; cc = cc+1;
+end
+  % electric field
+  [Exx, Eyy] = gradient(V,hx,hy);
+  Exx = -Exx;  Eyy = -Eyy;
+  E = sqrt(Exx.^2 + Eyy.^2);
+
+% GRAPHICS ===============================================================
+
+figure(1) % VECTOR FIELD V  -----------------------------------------
+    set(gcf,'units','normalized','position',[0.05 0.5 0.4 0.35]);
+    surf(xx(1:5:end,1:5:end),yy(1:5:end,1:5:end),V(1:5:end,1:5:end));
+    xlabel('x  [m]'); ylabel('y  [m]'); zlabel('V  [ V ]');
+    set(gca,'fontsize',14)
+    rotate3d
+    box on
+    axis tight
+    colorbar
+    view(43,54);
+
+figure(2)
+       set(gcf,'units','normalized','position',[0.5 0.5 0.4 0.35]);
+       c = 1; yStep = zeros(1,8);
+    for n = 1 : round(Ny/15): round(Ny/2)
+        xP = xx(n,:); yP = V(n,:);
+        plot(xP,yP,'b','linewidth',1.2);
+        hold on
+        c = c+1;
+        yStep(c) = yy(c,1);
+
+    end
+       tt = num2str(yStep,2);
+       text(0.2,95,'y  =');
+       text(0.4,95,tt,'fontsize',13)
+       xlabel('x  [m]'); ylabel('V  [ V ]');
+       set(gca,'fontsize',16)
+
+ figure(3)
+     set(gcf,'units','normalized','position',[0.05 0.05 0.4 0.35]);
+     index1 = 1 : Nx; index2 = 1 : Ny;
+     p1 = xx(index1, index2); p2 = yy(index1, index2);
+     p3 = Exx(index1, index2); p4 = Eyy(index1, index2);
+     h = quiver(p1,p2,p3,p4);
+     set(h,'color',[0 0 1],'linewidth',2)
+     xlabel('x  [m]'); ylabel('y  [m]');
+     set(gca,'xLim',[minXL, maxXL]);
+     set(gca,'yLim',[minYL, maxYL]);
+
+ figure(4)
+     set(gcf,'units','normalized','position',[0.5 0.05 0.4 0.35]);
+    surf(xx,yy,E);
+    shading interp
+    xlabel('x  [m]'); ylabel('y  [m]'); zlabel('|E|  [ V/m ]');
+    set(gca,'fontsize',14)
+    set(gca,'zTick',[0 1000 2000]);
+    rotate3d
+    box on
+    axis tight
+    colorbar
+    view(43,54);
+
+
+    cc
+    V
+
+toc

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/cemLaplace02.m

@@ -0,0 +1,204 @@
+% cemLaplace01.m
+
+% Solution of Laplace's equation
+clear
+clc
+tic
+
+% INPUTS  ================================================================
+% Number of XY grid points (integer)
+     Nx = 101;  Ny = 101;
+% Range for X and Y values  [minX maxX minY maxY minZ maxZ]
+     XY = [0 4 -1 1];
+% Boundary potential
+
+% tolerance for ending iterations
+tol = 1;
+% limits for electric field plot
+   minXL = 0; maxXL = XY(2);
+   minYL = 0; maxYL = XY(4);
+
+% boundary values for the potential
+    V0 = 100; V1 = -5; V2 = -10;
+    V = zeros(Ny,Nx);
+    V(:,1) = V0;
+   % V(:,end) = 0;
+   % V(1,:) = 0;
+   % V(end,:) = 0;
+
+    % linearly increasing potentials on boundaries
+       %enter INPUT details in setup
+
+
+% SETUP ==================================================================
+
+    minX  = XY(1); maxX = XY(2);
+    minY  = XY(3); maxY = XY(4);
+    x = linspace(minX, maxX,Nx);
+    y = linspace(minY, maxY,Ny);
+    [xx, yy] = meshgrid(x,y);
+    hx = x(2) - x(1); hy = y(2) - y(1);
+    Kx = hx^2/(2*(hx^2+hy^2)); Ky = hy^2/(2*(hx^2+hy^2));
+
+   % V(:,1) = V0 .* cos(2*pi*y/(4*maxY));
+
+% linearly increasing potentials on boundaries
+        m1 = 20/maxY; m2 = 20/maxX; m3 = 30/maxY; m4 = 10/maxX;
+        b1 = 0;       b2 = 20;      b3 = 10;   b4 = 0;
+
+        V(:,1)   = m1 .* y + b1;
+        V(:,end) = m3 .* y + b3;
+        V(end,:)   = m2 .* x + b2;
+        V(1,:) = m4 .* x + b4;
+
+% CALCULATIONS ===========================================================
+
+% dSum difference in sum of squares  /  n  number of iterations
+dSum = 10; n = 0;
+
+while  dSum > tol
+    sum1 =  sum(sum(V.^2));
+
+    for ny = 2: Ny-1
+        for nx = 2: Nx-1
+            V(ny,nx) = Ky * (V(ny,nx+1) + V(ny,nx-1)) + Kx * (V(ny+1,nx) + V(ny-1,nx));
+        end
+       % uncomment the next line of code if you need to calculate this column of values
+      % V(ny,Nx) = Ky * (V(ny,Nx) + V(ny,Nx-2)) + Kx * (V(ny+1,Nx) + V(ny-1,Nx));
+    end
+
+   sum2 =  sum(sum(V.^2));
+   dSum = abs(sum2 - sum1);
+   n = n+1;
+   dSum
+end
+  % electric field
+  [Exx, Eyy] = gradient(V,hx,hy);
+  Exx = -Exx;  Eyy = -Eyy;
+  E = sqrt(Exx.^2 + Eyy.^2);
+
+
+ % exact solution for boundary conditions
+     % left boundary V0, other boundaries V = 0
+    % Vexact = (2*V0/pi) .* atan(sin(pi*y(51)/maxX)./sinh(pi*x/maxX));
+   %  Vexact = (2*V0/pi) .* atan(1./sinh(pi*x/maxX));
+
+
+ % linearly increasing potentials on boundaries
+        m1 = 20/maxY; m2 = 20/maxX; m3 = 30/maxY; m4 = 10/maxX;
+        b1 = 0;       b2 = 20;      b3 = 10;   b4 = 0;
+
+        V(:,1)   = m1 .* y + b1;
+        V(:,end) = m3 .* y + b3;
+        V(end,:)   = m2 .* x + b2;
+        V(1,:) = m4 .* x + b4;
+
+% GRAPHICS ===============================================================
+
+figure(1) % VECTOR FIELD V  -----------------------------------------
+    set(gcf,'units','normalized','position',[0.02 0.52 0.3 0.32]);
+    surf(xx(1:5:end,1:5:end),yy(1:5:end,1:5:end),V(1:5:end,1:5:end));
+    xlabel('x  [m]'); ylabel('y  [m]'); zlabel('V  [ V ]');
+    title('potential','fontweight','normal');
+    set(gca,'fontsize',14);
+    rotate3d
+    box on
+    axis tight
+    colorbar
+    view(55,49);
+   % set(gca,'ZTick',[0 5 10]);
+
+%%
+   figure(2)
+       set(gcf,'units','normalized','position',[0.65 0.52 0.3 0.32]);
+       c = 0; yStep = zeros(1,6);
+
+    for n = 1:10:51
+        xP = xx(n,:); yP = V(n,:);
+        plot(xP,yP,'b','linewidth',1.2);
+        hold on
+        c = c+1;
+        yStep(c) = yy(n,1);
+
+    end
+       tt = num2str(yStep,2);
+       tm1 = 'y  =  ';
+       tm2 = tt;
+       tm = [tm1 tm2];
+       xlabel('x  [m]'); ylabel('V  [ V ]');
+       set(gca,'fontsize',16)
+       h_title = title(tm);
+       set(h_title,'fontsize',12,'FontWeight','normal');
+
+
+      % hold on
+      % plot(x,Vexact,'r');
+
+ %%
+ figure(3)
+     set(gcf,'units','normalized','position',[0.65 0.1 0.3 0.32]);
+
+     hold on
+     sx = x(5);
+         for sy = -1 : 0.1 : 1
+           h = streamline(xx,yy,Exx,Eyy,sx,sy);
+           set(h,'linewidth',2,'color',[1 0 1]);
+         end
+
+
+     index1 = 10 : 1: Nx; index2 = 10 : 1 : Ny;
+     p1 = xx(index1, index2); p2 = yy(index1, index2);
+     p3 = Exx(index1, index2); p4 = Eyy(index1, index2);
+     h = quiver(p1,p2,p3,p4);
+     set(h,'color',[0 0 1],'linewidth',2)
+     xlabel('x  [m]'); ylabel('y  [m]');
+
+     title('electric field','fontweight','normal');
+     hold on
+     axis equal
+     set(gca,'xLim',[minX,maxX]);
+     set(gca,'yLim',[minY,maxY]);
+     set(gca,'xTick',minX:1:maxX);
+     set(gca,'yTick',minY:1:maxY);
+     set(gca,'fontsize',14)
+     box on
+
+%%
+ figure(4)
+    set(gcf,'units','normalized','position',[0.02 0.1 0.3 0.32]);
+    surf(xx,yy,E);
+    shading interp
+    xlabel('x  [m]'); ylabel('y  [m]'); zlabel('|E|  [ V/m ]');
+    set(gca,'fontsize',14)
+    %set(gca,'zTick',[0 1000 2000]);
+    rotate3d
+    box on
+    axis tight
+    colorbar
+    view(43,54);
+    title('electric field','fontweight','normal');
+
+figure(5)
+    set(gcf,'units','normalized','position',[0.33 0.52 0.3 0.32]);
+    contourf(xx,yy,V,16);
+    shading interp
+    xlabel('x  [m]'); ylabel('y  [m]');
+    title('potential','fontweight','normal');
+    set(gca,'fontsize',14)
+    box on
+    colorbar
+    axis equal
+
+figure(6)
+    set(gcf,'units','normalized','position',[0.33 0.1 0.3 0.32]);
+    contourf(xx,yy,E,32);
+    shading interp
+    xlabel('x  [m]'); ylabel('y  [m]');
+    title('electric field','fontweight','normal');
+    set(gca,'fontsize',14)
+    box on
+    colorbar
+    axis equal
+
+
+toc

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/cemLaplace03.m

@@ -0,0 +1,68 @@
+% cemLaplace01.m
+
+% Solution of Laplace's equation
+clear
+clf
+tic
+
+% INPUTS  ================================================================
+
+% Number of XY grid points (integer)
+     Nx = 5;
+% Range for X and Y values  [minX maxX minY maxY minZ maxZ]
+     XY = [0 5];
+% Boundary potential
+     V0 = 100;
+% tolerance for ending iterations
+    tol = 0.1;
+
+% SETUP ==================================================================
+
+    V = zeros(Nx,1);
+    minX = XY(1);  maxX = XY(2);
+    x = linspace(minX, maxX,Nx);
+    hx = x(2) - x(1);
+    V(1) = V0;  V(Nx) = 0;
+
+% CALCULATIONS ===========================================================
+
+% dSum difference in sum of squares  /  n  number of iterations
+dSum = 10; n = 0;
+while dSum > tol
+    sum1 =  sum(sum(V.^2));
+
+    for nx = 2: Nx-1
+           V(nx) = 0.5*(V(nx+1) + V(nx-1));
+    end
+    sum2 =  sum(sum(V.^2));
+    dSum = abs(sum2 - sum1);
+    n = n+1;
+end
+
+% exact solution
+VT = -(V0/maxX)*x + V0;
+
+% electric field
+E = -gradient(V,hx);
+
+% GRAPHICS ===============================================================
+
+figure(1)
+    set(gcf,'units','normalized','position',[0.05 0.2 0.3 0.4]);
+    subplot(2,1,1)
+    plot(x,V,'b','lineWidth',2);
+    hold on
+    plot(x,VT,'r','lineWidth',2);
+    %set(gca,'xLim',[minX, maxX]);
+    %set(gca,'yLim',[minY, maxY]);
+    xlabel('x  [ m ]'); ylabel('V  [ V ]');
+    set(gca,'fontsize',14)
+
+    subplot(2,1,2)
+    plot(x,E,'b','lineWidth',2)
+    set(gca,'yLim',[0, 25]);
+    xlabel('x  [ m ]'); ylabel('E  [ V / m ]');
+    set(gca,'fontsize',14)
+
+ % =======================================================================
+    toc

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/lab-06.aux

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+\fi}
+\global\let\hyper@last\relax 
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+\providecommand\HyField@AuxAddToCoFields[2]{}
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+\gdef \@abspage@last{3}

5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/lab-06.out

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5th-Semester-Fall-2023/EEMAGS/Labs/Lab-06/lab-06.tex

@@ -0,0 +1,141 @@
+\documentclass[conference]{IEEEtran}
+\usepackage[siunitx]{circuitikz}
+\usepackage{graphicx}
+%\usepackage{lipsum}
+\usepackage{color}
+\usepackage{enumitem}
+\usepackage{epsfig} 
+\usepackage{mathptmx} 
+\usepackage{times} 
+\usepackage{amsmath}
+\usepackage{amssymb}  
+\usepackage{float}
+\usepackage{hyperref}
+%\usepackage{setspace}
+%\usepackage{tikz}
+\usepackage{circuitikz}
+\usepackage{pgfplots}
+\usepackage{textcomp}
+%\usepgfplotslibrary{external}
+%\tikzexternalize
+%\usepackage[utf8]{inputenc}
+%\usepackage[english]{babel}
+%\usepackage{pgf}
+\usepackage[final]{pdfpages}
+\usepackage{pgfpages}
+\usepackage[margin=0.4in]{geometry}
+\usepackage{lmodern}
+\usepackage{datetime}
+\usepackage{ragged2e}
+\input{border.tex}
+\pgfpagesuselayout{boxed}
+
+\hyphenation{op-tical net-works semi-conduc-tor}% correct bad hyphenation here
+
+\font\myfont=cmr12 at 15pt 
+\title{\myfont Simulating Boundary Conditions in Matlab}
+\author{Aidan Sharpe 916373346}
+
+
+\providecommand{\keywords}[1]{\textbf{\textit{Keywords---}} #1}
+\providecommand{\e}[1]{\ensuremath{\times 10^{#1}}}
+\setlength{\columnsep}{7mm}
+\pgfplotsset{compat=1.15}
+\begin{document}
+\makeatletter
+\@twocolumnfalse
+\maketitle
+\@twocolumntrue
+\makeatother
+\begin{abstract}
+\end{abstract}
+
+\section{Introduction and Background}
+This lab focused on approximating simple electromagnetic scenarios using finite element analysis techniques. Through an iterative process, the approximations could be refined to any arbitrary tolerance. Reaching ever tighter tolerances, however, comes at the cost of exponentially more computing power. After a certain point, computing more iterations yields diminishing returns. To decrease error by an order of magnitude requires an order of magnitude more computing power, or at least the relationship is of similar scale. Unfortunately, if the initial error is very large, the amount of computation goes up exponentially for each order of magnitude higher in error.
+
+\section{Part 1}
+Part one introduced approximating the electric potential and the electric field strength in one dimension. Since the endpoints were known, continuously iterating over the distance, taking the moving average at each point, a decent approximation could be honed in on to arbitrary precision. In one dimension, the amount of computing power required to evaluate the approximation is minimal. The tolerance was set to 0.1, and the process completed successfully in less than half a second. 
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part1.png}
+    \caption{Approximated and actual electrostatic potential}
+    \label{fig:part-1}
+\end{figure}
+
+The first graphic in figure \ref{fig:part-1} relates voltage to distance. The initial voltage at $x=0$ was 100[V] and the final voltage at $x=5$ was 0[V]. Since the tolerance was so low (0.1), only the red line, representing the approximated voltage is visible. Underneath the red line is a blue line that shows the actual voltage. In the second graphic in figure \ref{fig:part-1}, the actual electric field strength is shown in blue. Since the electric field strength is the negative gradient of electric potential, and voltage had a constant negative slope, the electric field strength is has a positive, constant value.
+
+\section{Part 2}
+Part two shifted from a one dimensional problem to a two dimensional one. Here, two metal plates with infinite length and a width of 4 meters were placed parallel to each other separated by two meters. They were connected along the left infinite edge by an insulating strip held at a potential of 100[V].
+
+The code for this part took a very long time to run. Using the inbuilt timing functions, it took exactly 2276.92 seconds. Since the initial error was around $4\times10^4$ and the desired tolerance was $1\times10^{-3}$. That is seven orders of magnitude difference. While running, each iteration took about a quarter of a second. Given the total time, we can compute the total number of iterations to be around 8000.
+
+After the algorithm completed, we were greeted with several figures depicting the electric potential and the electric field strength between the two plates.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part2-Heatmap-Potential.png}
+    \caption{Electric potential between the plates}
+    \label{fig:part-2-potential}
+\end{figure}
+
+Shown in figure \ref{fig:part-2-potential}, electric potential is clearly strongest close to the middle of the strip and goes to zero close to the plates or far from the strip.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part2-Heatmap-EField.png}
+    \caption{Electric field strength between the plates}
+    \label{fig:part-2-efield}
+\end{figure}
+
+While voltage is highest in the middle of the plates, the electric field is strongest close to the plates. This is because the voltage drops off most rapidly where the insulating strip meets the plates. Since electric field is higher when there is a sharper voltage drop, this makes intuitive sense.
+
+\section{Part 3}
+Part three analyzes the electric field and electric potential inside a rectangular region. In figure \ref{fig:part-3-efield}, the vectors are normal to the sides of the rectangle along the edges, and are strongest at the centers of each edge. This hints at a strong voltage drop near the centers of each boundary and a relatively constant voltage closer to the center. Based on the electric field alone, the electric potential would likely look like a dome made from an ellipse inscribed in the rectangle convolved with the surface of the rectangle itself.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part3-Vector-EField}
+    \caption{Electric field inside the rectangle}
+    \label{fig:part-3-efield}
+\end{figure}
+
+The approximated potential is shown in figure \ref{fig:part-3-potential}. The potential matches the dome shape, flat towards the middle, and dropping off more sharply towards the edges.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part3-Potential}
+    \caption{Potential inside the rectangle}
+    \label{fig:part-3-potential}
+\end{figure}
+
+\section{Part 4}
+Adjusting the voltages at the boundary of the rectangle to be 20[V] when $x=0$, -5[V] when $y=0$, 10[V] when $x$ is at its maximum value, and -10[V] when $y$ is at its maximum value, gives the electric field shown in figure \ref{fig:part-4-efield}. The electric field is again weak in the middle, much weaker than before, and it is stronger near the corners this time. It is also stronger along the left and right sides than it is along the top and bottom. This hints at the shape of the potential. It should be of a similar shape to the previous rectangle, but much weaker for longer in the middle, and it should not drop off as sharply near the edges as before.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part4-Vector-EField}
+    \caption{Electric field inside the new rectangle}
+    \label{fig:part-4-efield}
+\end{figure}
+
+Looking at figure \ref{fig:part-4-potential}, the shape matches prediction. It should also be noted that the slope is much larger on the left than on the right. Looking back at the electric field in figure \ref{fig:part-4-efield}, this corresponds to a stronger electric field on the left compared to the right. Also notice the sign of the slopes of the electric potential: a negative slope corresponds to a vector in the positive x direction, and a positive slope corresponds to a vector in the negative x direction. This makes sense since the electric field is the negative gradient of the potential.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part4-Potential}
+    \caption{Electric potential inside the new rectangle}
+    \label{fig:part-4-potential}
+\end{figure}
+
+\section{Part 5}
+Part five deals with linear varying potentials along a boundary. Now, rather than being constant, the potential will change along each boundary. Importantly, since this change is linear, the slope of the potential will be constant along the boundary. The potential is shown in figure \ref{fig:part-5-potential}.
+
+For most values, electric field will be constant, but at the corners, the slope changes, so the filed will take on some value. This can be seen in figure \ref{fig:part-5-efield}.
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part5-EField}
+    \caption{The electric field inside a rectangle with non-constant boundary potential}
+    \label{fig:part-5-efield}
+\end{figure}
+
+\begin{figure}[h]
+    \includegraphics[width=0.48\textwidth]{Part5-Potential}
+    \caption{The electric potential of a rectangle with non-constant potential}
+    \label{fig:part-5-potential}
+\end{figure}
+
+\end{document}

5th-Semester-Fall-2023/EEMAGS/Labs/My-Lab/Lab-Template.aux

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+
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+See the LaTeX manual or LaTeX Companion for explanation.
+Type  H <return>  for immediate help.
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+! LaTeX Error: Something's wrong--perhaps a missing \item.
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+Type  H <return>  for immediate help.
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+! LaTeX Error: Something's wrong--perhaps a missing \item.
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+See the LaTeX manual or LaTeX Companion for explanation.
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5th-Semester-Fall-2023/EEMAGS/Labs/My-Lab/Lab-Template.tex

@@ -0,0 +1,385 @@
+\documentclass{article}
+
+\begin{document}
+Engineering Electromagnetics
+
+Laboratory Exercise No. 4: Moving Magnetic Fields
+
+Objective
+
+In this lab, you will use Maxwell\textquotesingle s Transient Solver to
+model Faraday\textquotesingle s Law. You will model a
+
+magnetic field moving through and around a coil to induce current within
+the coil.
+
+\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
+
+A tutorial can be found at the following links:
+
+\begin{itemize}
+\item
+  \url{https://www.youtube.com/watch?v=LW9TcpkZaZY}~ (Part 1)
+\item
+  \url{https://www.youtube.com/watch?v=MruPdh_6hi0}~ (Part 2)
+\item
+  \url{https://www.youtube.com/watch?v=ZGn-gQPVn3Y} (Part 3)
+\end{itemize}
+
+1 Current Induced by Magnetic Field
+
+You will demonstrate how the electric and magnetic fields react to a
+magnet moving through an inductor.
+
+\textbf{Structure}
+
+\textbf{Coil}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  \begin{quote}
+  Create a 5-turn coil with an outer diameter 2 mm and wire thickness of
+  0.4 mm (diameter).
+  \end{quote}
+\item
+  \begin{quote}
+  Extend the faces of the coil ends
+  \end{quote}
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    \begin{quote}
+    Select the face
+    \end{quote}
+  \item
+    \begin{quote}
+    Right Click =\textgreater{} Select Surface =\textgreater Sweep Along
+    Normal (for 5 mm)
+    \end{quote}
+  \end{enumerate}
+\item
+  \begin{quote}
+  Select the coil and face extrusions and unite them
+  \end{quote}
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    \begin{quote}
+    Select all and Right-Click
+    \end{quote}
+  \item
+    \begin{quote}
+    Edit =\textgreater{} Boolean =\textgreater{} Unite
+    \end{quote}
+  \end{enumerate}
+\item
+  \begin{quote}
+  Change the material to copper
+  \end{quote}
+\end{enumerate}
+
+\includegraphics[width=4.62425in,height=3.6118in]{media/image1.png}
+
+\begin{quote}
+\textbf{Magnet}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Create a Regular Polyhedron
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    A shape of 8 segments
+  \item
+    A height of 1 mm and a diameter of 0.5 mm
+  \end{enumerate}
+\item
+  Change the material to NdFe36
+\end{enumerate}
+
+\includegraphics[width=1.4694in,height=1.7165in]{media/image2.png}
+
+\begin{quote}
+\textbf{Band}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Create a Regular Polyhedron
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    A shape of 12 segments
+  \item
+    A height of 25 mm and a diameter of 1 mm
+  \end{enumerate}
+\item
+  Change the material to vacuum
+\item
+  (Optional) Change the transparency to 75\%
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Select and Right-Click
+  \item
+    Edit =\textgreater{} Properties =\textgreater{} Transparent
+    =\textgreater{} 0.75
+  \end{enumerate}
+\end{enumerate}
+
+\includegraphics[width=3.67752in,height=4.24509in]{media/image3.png}
+
+\begin{quote}
+\textbf{Region}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Create a Region
+\item
+  The Region can have 0 padding in all directions.
+\end{enumerate}
+
+\includegraphics[width=3.54241in,height=3.94432in]{media/image4.png}
+
+\textbf{Simulation}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Transient for the Solution Setup
+\item
+  Add Motion Setup to the Band
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Select Band
+  \item
+    Go to Maxwell 3D =\textgreater{} Model =\textgreater{} Motion Setup
+    =\textgreater{} Assign Band
+  \item
+    Double-check that the motion will be along the z-axis
+  \end{enumerate}
+\item
+  Set the initial position
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Go to the Data tab
+  \item
+    Set the initial position between the bottoms of the band and coil
+  \item
+    Add the negative and positive limits above bottom of the band and
+    below the bottom of the coil
+  \item
+    Set the negative limit to the initial position
+  \item
+    Set the positive limit translational limit between the band and
+    above the coil
+  \end{enumerate}
+\item
+  Set the velocity
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Mechanical tab and set the velocity in mm/s.
+  \item
+    Set the velocity to 10 mm/s
+  \end{enumerate}
+\end{enumerate}
+
+\begin{quote}
+\textbf{Excitation}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Add a Coil Terminal Excitation
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Select the face of the end of the coil and Right-Click
+  \item
+    Assign Excitation =\textgreater{} Coil Terminal
+  \item
+    Number of conductors is one
+  \item
+    Do this for each face, they should both be going in the same
+    direction
+  \end{enumerate}
+\item
+  Add a winding
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Add the two coil terminals to it
+  \item
+    Go to the Properties
+
+    \begin{enumerate}
+    \def\labelenumiii{\roman{enumiii}.}
+    \item
+      Set the type to external solid
+    \item
+      Set the initial current to 0 A
+    \end{enumerate}
+  \end{enumerate}
+\end{enumerate}
+
+\begin{quote}
+\textbf{External Circuit}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Assign the RL Circuit to the coil
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Select one of the faces of the coil and Right-Click
+  \item
+    Assign Excitation =\textgreater{} External Circuit =\textgreater{}
+    Edit External Circuit =\textgreater Add Circuit
+  \end{enumerate}
+\item
+  Create the RL Circuit
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Go to Project and ``Insert Maxwell Circuit Design''
+  \item
+    Connect a 100 Ohm Resistor and an Inductor in series
+
+    \begin{enumerate}
+    \def\labelenumiii{\roman{enumiii}.}
+    \item
+      No power supply and ground one side
+    \end{enumerate}
+  \end{enumerate}
+\item
+  Export the netlist, Maxwell Circuit =\textgreater{} Export Netlist
+\end{enumerate}
+
+\includegraphics[width=2.92497in,height=1.83722in]{media/image5.png}
+
+\textbf{Eddy Currents}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Add an Eddy Circuit Excitations to the helix(Maxwell 3D → Excitations
+  → Set Eddy Effects).
+\end{enumerate}
+
+\begin{quote}
+\textbf{Mesh Operations}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Select the coil and Right-Click
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Assign Mesh Operations =\textgreater{} Inside Selection
+    =\textgreater{} Length Based
+  \item
+    Number of elements should be 10,000
+  \end{enumerate}
+\end{enumerate}
+
+\begin{quote}
+\textbf{Solution Setup}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Add a solution setup.
+\item
+  Assign the stop time to when your magnet will finish moving.
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Stop time at 10 ms with steps of 0.2 ms
+  \item
+    Do this in both the General and Save Fields tab
+  \end{enumerate}
+\item
+  Go the Save Field tab
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Change your step size to have at least 10 steps
+  \item
+    Click the replace list button
+  \end{enumerate}
+\end{enumerate}
+
+\begin{quote}
+\textbf{Magnetic Field}
+\end{quote}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Add the Magnetic Field Vector or Magnitude
+
+  \begin{enumerate}
+  \def\labelenumii{\alph{enumii}.}
+  \item
+    Select the Region and Right-Click
+  \item
+    Fields =\textgreater{} B =\textgreater{} Mag\_B or B\_Vector
+  \item
+    Analyze all and you should be good to go!
+  \end{enumerate}
+\end{enumerate}
+
+Calculator
+
+Use the Ansys calculator to get the current leaving the coil as a
+function of the magnet passing through the coil. You should be able to
+plot this as a function of distance or time since the magnet is moving
+at a constant speed.
+
+Variables
+
+Repeat the example twice with the magnet outside of the coil at
+different distances. Keep
+
+the magnet perpendicular to the coil.
+
+Laboratory Write-up
+
+Explain your results. Handing in excellent data without explanations is
+not enough. There should be an introduction to the lab, figures of
+results, Maxwell code (if necessary; preferably in an appendix), an
+explanation for each figure, and a summary of what you learned in the
+exercise.
+
+Use App Note format for this report. Templates can be found on Canvas.
+\end{document}