From 61ec132aeed71d7bfe0fdde3513189097b84f086 Mon Sep 17 00:00:00 2001
From: Joshua Drake <joshua.ellis.drake@gmail.com>
Date: Tue, 5 Nov 2024 03:19:56 -0600
Subject: Finished Electric Motor Section.

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 background.tex | 23 ++++++++++++++++++++++-
 1 file changed, 22 insertions(+), 1 deletion(-)

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diff --git a/background.tex b/background.tex
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@@ -42,6 +42,13 @@ Turboshaft style engines are most often used in helicopters, and are characteriz
 \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
+At a fundamental level, an electric motor can be thought of mechanically as a relationship between the stator and rotor, and electrically as a relationship between field magnets and the armature. The rotor section of the motor undergoes rotation, while the stator houses the stationary elements of the motor. Less semantically obvious are the functions of the electric motor's field magnets and armature. The field magnets, be they contained within the stator or rotor, create an electric field that passes through the aramture \cite{scarpino2015motors}. The armature is comprised of multiple windings or coils, which, when exposed to an electric current and the magnetic field of the field magnets, results in the production of rotational force. A highly simplified illustration of the union of these parts can be seen in figure \ref{basicem}.
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=.65\textwidth]{img/basicem.png}
+	\caption{\label{basicem}Basic Cross Section of an Electric Motor}
+\end{figure}
+\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}
@@ -54,7 +61,21 @@ where $E$ is the back-EMF from two conducting motor phases, and $\kappa_E$ is th
 \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}.
+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}. The back-EMF output of the generator is rectified by a full bridge rectifier with stages equal to the number of phases possessed by the generator, after which it enters the DC sytem.
+\par
+Electric motors are utilized to take the electric energy produced by and stored witin the hybrid aircraft and convert it into rotational energy, effectively serving an opposite purpose to the generator. This process first requires the conversion of the direct current that is distributed throughout the elctrical system of the aircraft to be converted back to alternating current to achieve this end. This is a non-trivial task when compared to the simplicity of diode rectification, and thus requires explanation. The device responsible for this task is called the inverter, which is most commonly configured in accordance with the diagram shown in figure \ref{inverter}, where each phase of the motor is driven by half bridge circuit.
+\begin{figure}[h!]
+	\centering
+	\includegraphics[width=.5\textwidth]{img/inverter.png}
+	\caption{\label{inverter}Three Phase Inverter Schematic Showing Conducting Loops \cite{designpmm}}
+\end{figure}
+Of note is the component symmetry between the three phase inverter and a full bridge rectifier of equal phase number. The body diodes contained within the inverter, as denoted by $D1$-$D6$ in figure \ref{inverter}, allow it to function as a rectifier when uncontrolled. There are many control schemes by which an inverter can be operated to deliver desired motor performance. One such scheme is depicted in figure \ref{inverter}, and functions as follows: $Q1$ is the control, or chopping transistor, as well as the ``incoming'' transistor. This means that $Q1$ is responsible for modulating its duty cycle in an effort to reconstruct an AC signal, while also being the switch through which current first enters the inverter respectively. $Q6$ is the outgoing transistor, and remains enabled for entire base cycle, or first $60^\circ$ of the control cylce, after which it is disabled. As might be expected, $Q3$ assumes the role of $Q1$ after the first $120^\circ$ of the control cycle, followed by $Q5$ at $240^\circ$. The same is true of $Q2$ and $Q4$ in their relationship to $Q6$. The current sensor operating $Q1$ can either be in line ``A'' or the DC supply line, allowing the inverter's controller to prevent the current through phase ``1'' from exceeding the set-point current.\cite{designpmm}
+\par
+The current supplied by the inverter to the windings of the motor generate a rotational force on its mechanically coupled propulsor in accordance with equation \ref{torqueemf}. The torque constant of an electric motor varies based on its design and topology. Hybrid electric aircraft commonly use variations of three phase, brushless AC motors, the torque constant for which can be computed as follows.
+\begin{equation}\label{3ptc}
+	 \kappa_T = \frac{3}{2}k_{w1}N_{ph}\Phi_1
+\end{equation}
+Where $k_{w1}$ is the fundamental harmonic winding factor, and represents the distribution, pitch, and skew factors of the windings; $N_{ph}$ is the number of turns per phase; and $\Phi_1$ is the fundamental flux of the stator. \cite{designpmm} The mechanical and electrical systems of hybrid aircraft influence the torque constants required of its electric motors, after which equations such as \ref{3ptc} can be used to inform the rest of their designs.
 \section{Battery Theory}
 \begin{equation}\label{battC}
 	I_{Battery}=\frac{V_{Battery}-V_{Supply}}{R_{Battery}}
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