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\chapter[BACKGROUND]{BACKGROUND}
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%\section{Advanced Air Mobility}
\section{Turbine Engines}
A cursory understanding of turbine engines is necessarry to contextualize this work, as their improved power to weight ratio and performance at altitude when compared to piston engines make them an ideal choice for use in hybrid electric aircraft. The following is a description of how a general jet engine with a single inlet and exhuast functions. This description corresponds to the station numbering found in \ref{EoPturbojet} and is applicable to the subcategories of turbine engines discuessed later.
\begin{figure}[h]
	\centering
	\includegraphics[width=\textwidth]{img/EoPturbojet.png}
	\caption{\label{EoPturbojet}Ideal Turbojet with station numbering}
\end{figure}
\par
The Inlet is the first section of the gas turbine engine, denoted by station numbers 0-2, and its operation and design are described in terms of the efficiency of the compression process, the external drag of the inlet, and the mass flow into the inlet. \cite{EoPGTR2} Inlet design is most heavily influenced by whether the air entering it is subsonic or supersonic. Subsonic inlet design is simple, and typically involves selecting an operating velocity at which air compression is most efficient at the expense of performance at other velocities. Supersonic inlets must take the shockwaves endemic to supersonic flow into account for optimal performace. This is accomplished by adjusting inlet geometry to reduce flow velocity while adding as little weight to the system as possible. Variable inlet geometry will allow for increased efficiency accross many velocities.
\par
Compressors, denoted by station numbers 2-3, increase the pressure of the flow obtained by the inlet such that the combustion and exhaust processes can be conducted more efficiently. Increasing the pressure of an initial volume of air results in the reduction of its volume, allowing for the combustion of the air/fuel mixture to occur within a smaller volume than it would otherwise. Turbine engines most commonly employ centrifugal or axial compressors. Figure \ref{EoPturbojet} appropriately depicts an axial compressor in the makeup of the common turbine engine by virtue of their superiority. However, centrifugal compressors find use in smaller, less expensive engines due to their simple design. Centrifugal compressors are comprised of an impeller, which serves to increase flow velocity through rotation; a diffuser, which decreases the velocity of the flow thereby increasing its pressure; and a manifold which directs the compressed air into the combustor. Axial compressors are made of a series of stator vanes and rotor blades that are concentric to the axis of rotation. Each set of these stators and rotors is referred to as a stage. ''The flow path in an axial compressor decreases in cross-sectional area in the direction of flow." \cite{EoPturbojet} Each stage of the compressor results in an increase in air density. Thus, multiple stages are used in the design of high compression ratio turbine engines. Many turbines, including that which is depicted in figure \ref{EoPturbojet}, are equiped with dual axial compressors. Dual axial compressors allow for a more uniform loading of compressor stages, as well as for improved flexibility in the balancing between the initial and later stages.
\par
The combustor, as illustrated in figure \ref{EoPturbojet} between station numbers 3 and 4, is responsible for burning a mixture of compressed air and fuel and delivering the resulting exhaust gases to the turbine stage at a consistent temperature. The air that enters the combustion chamber is characterized as either primary air, meaning that it mixes with fuel and burns, and secondary air, which cools the extremity of the combustion chamber as well as exhaust gases to ensure optimal temperature within the turbine. The air to fuel ratio varies from 30 to 60 parts of air to one part of fuel by weight, depending on the design and type of engine. \cite{EoPGTR2} The types of combustion chambers found within tubine engines are can, which consist of multiple circular chambers arranged in a similarly circular fashion; annular, a large single chamber design around a center casing; and can-annular, a combination of the previous architectures in which can chambers are organized within an annular cavity.
\par
The turbine section of the engine, denoted by station numbers 4 through 5, is responsible for taking the energy generated in the combustion chamber and turning it into shaft horsepower to drive the compressor stages and external loads. Almost 75 percent of the energy generated from the combustion process is required to drive the compressor alone.\cite{EoPGTR2}The axial-flow turbine is similar to the axial compressor, and is likewise comprised of a series of stages of rotors and stators. However, the turbine has the opposite effect of the compressor: it turns the energy contained within flow into shaft rotation. The stage quantity of the turbine section of a given turbine engine is typically lower than that of its compressor, as the flow is expanding rather than compressing. Axial turbines are either impulse design, which maintain flow velocity across their rotor and decrease pressure across their stator, whereas reaction stages increase pressure across their rotor blades and direct flow within their stator. Most turbines use a combination of these two stage designs, and must be dual or split commensurately with the design of the compressor.
\par
The final stage of the turbine engine, the exhaust nozzle, denoted by station numbers 5 through 9, is responsible for increasing the velocity of the exhaust gas before discharge such that ample thrust can be generated by the engine. Ideally, the exit pressure of the flow leaving the nozzle should equal ambient pressure, otherwise the engine will operate less efficiently than it is capable. Nozzles are typically either convergent, or convergent-divergent, meaning a convergent duct followed by a divergent duct. Simple convergent ducts are used in the case where the ratio of turbine exit pressure to nozzle exit pressure is less than 2. The convergent-divergent duct is employed in instances where this nozzle pressure ratio is in excess of 2. Such ducts incorporate more sophisticated aerodynamic features and variable geometry in certain applications.\cite{EoPGTR2}
\par
\begin{figure}[h]
	\centering
	\includegraphics[width=\textwidth]{img/faaturbofan.png}
	\caption{\label{faaturbofan}Turbofan Engine Cross Section}
\end{figure}
Gas turbine engines fall into four categories: turbofan, turboprop, and turboshaft, and turbojet. Turbojets make use of a propelling nozzle to create thrust by allowing the heated exhaust created by a gas turbine to expand, without extracting rotational power from the engine. \cite{nasa_turbojet}
Turbofans make use of a front mounted fan to extract as much as 80 percent of thrust from the engine, significantly more than their turbojet counterparts. The inlets of turbofans differ from other topologies by virtue of their inlet design, as can be visualized in figure \ref{faaturbofan}.
The air driven by the fan will generally bypass the core, the amount of which contributes to the engine's bypass ratio. This ratio is simply the amount of flow through the engine bypass ducts over the flow through its core. The turboprop engine, that which is employed in this paper, drives a propeller through a reduction gearbox.
Turboshaft style engines are most often used in helicopters, and are characterized by their transfer of power to a shaft which later connects to another implement such as a propeller transmission or auxilary power unit. \cite{faa_engines}
\begin{figure}[h]
	\centering
	\includegraphics[width=\textwidth]{img/tp100cutaway.png}
	\caption{\label{tp100cutaway}PBS TP100 Cutaway}
\end{figure}
\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}
\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}. 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}}
\end{equation}
\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]
	\centering
	\includegraphics[width=\textwidth]{img/turbosystems.png}
	\caption{\label{turboarch}Turboelectric Architectures}
\end{figure}
The last manner in which turboelectric systems can be categorized is with respect to their inclusion of additional power sources. For example, just as is illustrated in figure \ref{turboseriesparallel}, systems with battery supplementation are called "parallel," whereas those which source power exclusively from their turbine engine are "seires." \cite{nasa_prop_overview}
Both systems constructed for this research are partial by virtue of their turboprop engines. However, configuration 1 is parallel due to its inclusion of a battery, whereas configuration 2 is devoid of additional power sources and is thus series.
\begin{figure}[h]
	\centering
	\includegraphics[width=.6\textwidth]{img/turboseriesparallel.png}
	\caption{\label{turboseriesparallel}Parallel Turboelectric Design}
\end{figure}
\section{Previous Work}



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