Finished Electric Motor Section.HEADmaster

7 files changed, 36 insertions, 3 deletions

diff --git a/background.tex b/background.tex
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--- a/background.tex
+++ b/background.tex
@@ -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}}
diff --git a/dissertation_main.pdf b/dissertation_main.pdf
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diff --git a/img/basicem.png b/img/basicem.png
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diff --git a/img/cessna1.png b/img/cessna1.png
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diff --git a/img/inverter.png b/img/inverter.png
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diff --git a/introduction.tex b/introduction.tex
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--- a/introduction.tex
+++ b/introduction.tex
@@ -9,6 +9,7 @@ Motivation
 Objectives
 Questions Answered
 
+Hybrid electric aircraft present an attractive combination of energy density provided by the combustion of hydrocarbon fuels with the power density of batteries. The advantages conffered by energy density to the effectiveness of an aircraft are improved flight duration and efficiency, whereas power density allows for improved performance at takeoff and instantaneous power production. These benefits have motivated an increase in research into the field of hybrid electric aircraft.
 However, there is a distinct lack of practical knowledge
 associated with the physical construction of such systems. More
 public research in constructing distributed hybrid turbo-electric
@@ -17,8 +18,13 @@ detailing the real-world implementation of electrical systems,
 safety systems, experimental results, and mechanical–electrical
 powertrain interactions. These objectives are accomplished specifically through
 a relatively low voltage electrical system comprised of a pulley coupled generator, battery, distributed propulsors, and requsite mechanisms to enable safe operation. A second electrical configuration was implemented into the Cessna test rig \cite{melvincessna}to observe the transient performance of the mechainical elements of a turboelectic powertrain.
-\par
-The multifaceted nature of this work presents a unique opportunity to compare the disperate effects of two electrical configurations on the mechanical systems common to both, in addition to what has been gathered from their individual operation. Configuration one is more representative of real hybrid turboelectric aircraft by nature of its use of a battery and accompanying safety mechanisms, distributed propulsion, and full integration into an airframe. Thus, the results obtained from configuration one provides insight into the benefits afforded to hybrid turboelectric systems by the inclusion of batteries, the considerations necessarry to safely use these batteries, and the increased takeoff performance of electrically augmented aircraft. Configuration two presents a worst case scenario
+ \begin{figure}
+  \centering
+  \includegraphics[width=.7\textwidth]{img/cessna1.png}
+  \caption{\label{fig:cessna1}Cessna Test Rig with Wing Mounted Propulsors}
+ \end{figure}
+ \par
+The multifaceted nature of this work presents a unique opportunity to compare the disperate effects of two electrical configurations on the mechanical systems common to both, in addition to what has been gathered from their individual operation. Configuration one is more representative of real hybrid turboelectric aircraft by nature of its use of a battery and accompanying safety mechanisms, distributed propulsion, and full integration into an airframe. Thus, the results obtained from configuration one provides insight into the benefits afforded to hybrid turboelectric systems by the inclusion of batteries, the considerations necessarry to safely use these batteries, and the increased takeoff performance of electrically augmented aircraft. Configuration two presents a worst case scenario for the mechanical stresses induced through electric loading due to the system's lack of capacitance. This characteristic makes the effects of electrical loading on the turbine incredibly apparent, as all of the torque induced by the electrical system must be accounted for by the turbine engine.
 \par
 All sections pertaining to configuration one are recapitulated from research originally funded by the FAA and eventually published in ASME's Journal of Engineering for Gas Turbines and Power \cite{CessnaASME}. Similarly, sections over configuration two cover research funded by NASA. The author is pursuing publication of this work for presentation at ASME's 2025 Turboexpo Conference.
 % \begin{figure}
diff --git a/references.bib b/references.bib
index d331cb9..9c4369e 100644
--- a/references.bib
+++ b/references.bib
@@ -107,3 +107,9 @@ doi = {10.1007/s40313-021-00740-x}
 	author = {Miller, TJE and Hendershot, J.R},
 	publisher = {Magna Physics and Clarendon Press, Oxford},
 }
+@book{scarpino2015motors,
+  title={Motors for makers: a guide to steppers, servos, and other electrical machines},
+  author={Scarpino, Matthew},
+  year={2015},
+  publisher={Que Publishing}
+}