From 9e4faf7a5df2e6e733fd5d839d24f420028d942d Mon Sep 17 00:00:00 2001
From: Joshua Drake <joshua.ellis.drake@gmail.com>
Date: Mon, 4 Nov 2024 02:57:10 -0600
Subject: Worked on generator background.

---
 background.tex | 19 +++++++++++++++++--
 1 file changed, 17 insertions(+), 2 deletions(-)

(limited to 'background.tex')

diff --git a/background.tex b/background.tex
index 0bfe584..f8f6494 100644
--- a/background.tex
+++ b/background.tex
@@ -39,12 +39,27 @@ Turboshaft style engines are most often used in helicopters, and are characteriz
 	\includegraphics[width=\textwidth]{img/tp100cutaway.png}
 	\caption{\label{tp100cutaway}PBS TP100 Cutaway}
 \end{figure}
-\section{Generator Theory}
+\section{Electric Motor Theory}
+A basic overview of electric motors, often refered to as ``generators'' in the context of hybrid aircraft by viture of their function of generating electric power, is required just as with the preceeding section over turbine engines. As much information will be provided in this section as is necessarry to understand the function of electric motors and their operation within hybrid electric aircraft. As can be seen in figure \ref{img/turboarch}, electric motors are used generate electric power through a mechanical coupling to an engine. Additionally, electric motors are employed to convert distributed electrical energy into the torque necessarry to drive propulsors.
+\par
+The generator induces a load onto the turbine engine in order to produce electrical energy. The back-EMF, or the voltage of generator output in this context, can be determined through the multiplication of angular velocity and the rate of change of flux-linkage with rotor position.
+\begin{equation}\label{emfwave}
+	 e=\omega_m \frac{\partial\Psi }{\partial \O}
+\end{equation}
+This flux-linkage $\Psi$ is the product of the number of turns and the flux passing through the coil. \cite{designpmm} The most important expressions in the generator-turbine relationship are simple variations of equation \ref{emfwave}:
+\begin{equation}\label{emfconstant}
+	E=\kappa_E \omega_m
+\end{equation}
+where $E$ is the back-EMF from two conducting motor phases, and $\kappa_E$ is the EMF constant.
+\begin{equation}\label{torqueemf}
+	T=\kappa_EI
+\end{equation}
+Equations \ref{emfconstant} and \ref{torqueemf} showcase how a relationship between torque and voltage is obtained through the EMF constant. The implications of this relationship to the function of the turboelectric system as a whole will be elucidated in section \ref{turbotheory}.
 \section{Battery Theory}
 \begin{equation}\label{battC}
 	I_{Battery}=\frac{V_{Battery}-V_{Supply}}{R_{Battery}}
 \end{equation}
-\section{Turboelectric Theory}
+\section{Turboelectric Theory}\label{turbotheory}
 NASA defines turboelectric systems as being at the least a turboshaft coupled to an electric generator, which power electric motors which then drive propellers. This configuration can be further categorized in accordance with whether the turbine engine drives a load directly.
 These systems, refered to as "Partial Turbo Electric" \cite{nasa_prop_overview}, employ the use of either turbofans or turboprops in addition to being coupled to electric generators.
 \begin{figure}[h]
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