typesetting and tidying up to the end of ch3

acknowledgements.tex

@@ -1,11 +1,14 @@
 %!TEX root = thesis.tex
 
-\section*{Acknowledgements}
+\section*{Acknowledgements \& Dedication}
 
+\begin{quote}
 \itshape
 Thanks to my supervisors Ben and Anthony, who gave me the freedom to be myself and who inspired me beyond my work.
 My appreciation couldn't be greater for the long leash they allowed me.
 With stricter guidance I may have finished sooner, but I believe neither I nor this thesis would have been the better for it.
 
-This thesis is dedicated to all those with whom I've shared this life over many years, especially to those who have come into the world and to those who have departed it during this period of my life.
-\upshape
+\medskip
+\noindent
+This thesis is dedicated to those many with whom I've shared this happy life, especially to those in this time who have come into this strange world and to those who have departed from it.
+\end{quote}

coil-design.tex

@@ -521,9 +521,7 @@ \subsection{Relationship between coil impedance and outer diameter}
 \end{dmath}
 where $\turnsRCoil=\mathord{\gp{\oradiusCoil-\iradiusCoil}}/\diamWire$ and $\turnsZCoil=\lengthCoil/\diamWire$ are the number of turns in the axial and radial directions respectively.
 While this relationship does not model any wire coating or the packing effect of how tightly-wound coils will sit,
-\note{
-  Taking into account of the packing factor will reduce the outer radius by around 10\% of the thickness of the coil \cite{yan2008-ietmx}.
-}
+\note{Taking into account of the packing factor will reduce the outer radius by around 10\% of the thickness of the coil \cite{yan2008-ietmx}.}
 this equation is simple and allows some conservatism in the quality of the construction of the electromagnet.
 Solving \eqref{coil-lengthwire} for $\oradiusCoil$, the outer radius of the coil for a coil of fixed inner radius and fixed total wire length is given by
 \begin{dmath}[label=coil-outerdiam]

front.tex

@@ -46,16 +46,16 @@
 \small
 \null
 \begin{center}
-\Large`\textit{Table that floats on magnets}'
+\large`\textit{Table that floats on magnets}'
 \end{center}
 \begin{quote}
 \nonfrenchspacing
 \parfillskip=0pt
-I certify that this work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint-award of this degree.
+I certify that this work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide.
 \PPP
 I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968.
 \PPP
-I also give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library catalogue and also through web search engines, unless permission has been granted by the University to restrict access for a period of time.
+I also give permission for the digital version of my thesis to be made available on the internet, via the University’s digital research repository, the Library catalogue, and also through web search engines.
 \end{quote}
 \vfill
 

magnet-theory.tex

@@ -9,7 +9,7 @@ \chapter{Magnetic and electromagnetic forces}
 
 \referpaper{The work presented in \secref{cyl-forces} is based on material that has been published as a journal paper~\cite{robertson2011-ietm}.}
 
-\section{Magnetic fields}
+\section{Whence magnetic fields}
 
 The following is a brief introduction to the physics behind magnets, largely to introduce the notation used later in the thesis.
 
@@ -56,7 +56,7 @@ \subsection{Magnetic parameters}
 \end{dmath},
 in which $\magB$ is referred to as the magnetic flux density, $\magH$ is the magnetic field strength, and $\magM$ is the magnetisation.
 \note{The names of these terms are not always consistent in the
-  literature. $\magM$ is also known as polarisation, and $\magB$ and $\magH$
+  literature. $\magM$ is also known as polarisation (often used to avoid statements such as `magnetisation of the magnet'), and $\magB$ and $\magH$
   are both sometimes known as the magnetic field in different contexts.}
 
 \Eqref{BHM} may now be used to describe the situation at all points in space (\figref*{BHM}).
@@ -85,12 +85,14 @@ \subsection{Relationship between magnetic parameters}
   \end{math},
 known as the relative permeability.
 
+\enlargethispage{-\baselineskip}
 The relative permeability of the vacuum is unity, and materials considered `non-magnetic' such as air, wood, water, and so on, have permeabilities very close to unity (within \num{1e-5}) as a result.
 (The consequences of permeability less than unity, diamagnetism, has been discussed on page~\pageref{sec:diamag}.)
-Materials which are more strongly affected by magnetic fields have greater permeabilities; \eg, within the soft iron core of an electromagnet, current in the coil generates an applied magnetic field on the core, which induces its own internal magnetic field as a result, and which increases the overall strength of the electromagnet. The larger the permeability, the larger this `amplification' effect.
+Materials which are more strongly affected by magnetic fields have greater permeabilities; \eg, within the soft iron core of an electromagnet, current in the coil generates an applied magnetic field on the core, which induces its own internal magnetic field as a result, and which increases the overall strength of the electromagnet (until the core saturates).
+The larger the permeability, the larger this `amplification' effect.
 Despite having strong remanence and coercivity values, rare earth magnets have a relative permeability of $\permrel=\num{1.05}$.
 
-Refering to \eqref{BHM} in free space where $\magM=0$ the relationship between $\magB$ and $\magH$ is quite simple, which is essentially the reason that there is a historical terminological confusion between the two.
+Refering to \eqref{BHM}, in free space where $\magM=0$ the relationship between $\magB$ and $\magH$ is quite simple, which is essentially the reason that there is a historical terminological confusion between the two.
 It can be seen that within a magnet, however, their relationship is more complex and important.
 Consider a magnetic material which has not yet been magnetised.
 As an external magnetic field is applied to it, the magnetic dipoles within the material begin to align along the direction of the applied field.
@@ -104,6 +106,7 @@ \subsection{Relationship between magnetic parameters}
 The \bhcurve/ of a magnetic material describes how the magnetic flux density generated by the magnet changes according to applied magnetic field.
 For the case of using magnetic fields for doing work (such as generating forces), only the second quadrant of the \bhcurve/ is considered.
 
+\enlargethispage{-\baselineskip}
 Two important features are shown in \figref{BHcurve}.
 First, the remanence of the magnet, $\remanence=\permvac\Msat$, is the value used to indicate the `strength' of the magnet alone corresponding to its saturation magnetisation.
 The remanence refers to the amount of magnetic flux density that is measured in the absence of an applied external magnetic field, and it is a common term in practice since the internal magnetisation of a magnet $\Msat$ cannot be measured directly.
@@ -121,7 +124,7 @@ \subsection{Relationship between magnetic parameters}
 
 \begin{figure}[htbp]
   \centerline{
-    \subbottom[Strong magnet with `linear' second-quadrant.]{\asyinclude{\jobname/bhcurve}}\quad
+    \subbottom[Strong magnet with `linear' second-quadrant.]{\asyinclude{\jobname/bhcurve}}\qquad
     \subbottom[Weak magnet with its knee in the second quadrant.]{\raisebox{5mm}{\asyinclude{\jobname/bhknee}}}
   }
   \lofcaption{Characteristic \bhcurve/s for a strong magnet and a weak magnet.}
@@ -134,7 +137,7 @@ \subsection{Relationship between magnetic parameters}
 
 As well as the remanence $\remanence$ and coercivity $\coerce$, two other parameters are often used to describe the `strength' of a permanent magnet.
 The first is known as the `maximum energy product' $\BHmax$, relating to the amount of potential energy that can be supplied by the magnetic field in the second quadrant of the \bhcurve/.
-In an ideal magnet with linear \bhcurve/, the maximum energy product is directly related to its saturation magnetisation via \cite{campbell1994}
+In an ideal magnet with a linear \bhcurve/, the maximum energy product is directly related to its saturation magnetisation via \cite{campbell1994}
 \begin{dmath}[label=bhmax]
   -\BHmax = \permvac\gp{\frac{\Msat}{2}}^2
 \end{dmath}.
@@ -144,7 +147,6 @@ \subsection{Relationship between magnetic parameters}
   \note{Since a permanent magnet is made up of a very large number of magnetic domains, the magnetic field strength used to initially magnetise a permanent magnet is said to be around $5\coerce{}_i$ as a rule of thumb.}
 For weak magnetic material, the intrinsic coercivity is low enough such that the `knee' of the \bhcurve/ enters the second quadrant as shown in \figref{BHcurve}(b), causing the magnet to be easily demagnetised.
 
-
 \subsection{Properties of magnetic flux}
 \seclabel{flux}
 
@@ -163,12 +165,13 @@ \subsection{Properties of magnetic flux}
 flowing through, whereas while magnetic flux is known to \emph{prefer}
 areas of greater permeability, it occasionally can deviate from these simple paths.
 
-\begin{figure}
+\begin{figure}[t]
 \includegraphics{PhD/Figures/Magnets/magflux}
 \caption{Lines of magnetic flux of a single magnet.}
 \figlabel{magflux}
 \end{figure}
 
+\enlargethispage{-\baselineskip}
 An analysis of how to derive the paths of magnetic flux is a beyond the scope of this document, but it is important to discuss the flux lines themselves.
 Typical flux lines for a rectangular cross-section magnet are shown in \figref{magflux}.
 It is more instructive for a basic understanding of how magnets behave to examine the ways their flux lines interact.
@@ -201,8 +204,11 @@ \subsection{Properties of magnetic flux}
 behaviour, \emph{not} for explaining the reasons behind it.
 \note{Just ask \textcite{sodano2006}, who received nitpicking comments about their terminology \cite{marneffe2007}.}
 
-\section{Permanent magnets and magnetic materials}
 
+\section{Magnet properties and selection}
+\seclabel{magprop}
+
+\enlargethispage{-\baselineskip}
 There are several materials from which permanent magnets can be
 made. Short attention will be placed on the cheaper, legacy magnetic
 materials such as ferrite magnets and alnico magnets due to their
@@ -363,7 +369,7 @@ \section{General technieques for calculating forces between magnets}
 This discretisation method is avoided in this thesis due to the limited advantage it has over using the semi-numerical approach that uses numerical integration directly.
 
 
-\section{Analytical expressions for calculating the magnetic flux density}
+\section{Equations for calculating the magnetic flux density}
 
 For the purposes of this work, the analytical calculation of the magnetic flux density $\magB$ is largely overlooked in favour of analytical force calculations, which will be addressed in the next section.
 However, since an analytical formulation for $\magB$ is a requirement for then calculating the force, a short literature review will be covered here for different magnet geometries.
@@ -397,7 +403,7 @@ \subsection{(Anti-)parallel alignment}
 geometries can be realised through superposition of the solutions
 \cite{bancel1999}.
 
-\begin{figure}
+\begin{figure}[b!]
   \asyinclude{\jobname/akoun}
   \caption
   [Geometry for calculating the force between parallel cuboid magnets.]
@@ -755,33 +761,23 @@ \subsection{Forces between magnets with relative rotation}
        \smash{\sum_{{i,j,k,l,p,q}\in\{0,1\}^6}}
         f_{z_2}\cdot\gp{-1}^{{i+j+k+l+p+q}}
 \end{dmath},
+where
 \begin{dmath}
 f_{y_2} = f_3\fn{u_0 , \cdy , \cdz , \mrot , \hwzX k , \hwzF q}
 \end{dmath},
 \begin{dmath}
 f_{z_2} =  \frac{f_3\fn{u_1,v_1,w_1,-\mrot,0,0}}{\sin\mrot}
          + \frac{f_3\fn{u_2,v_2,w_2,\mrot,0,0}}{\tan\mrot}
 \end{dmath},
-\begin{dgroup}
-\begin{dmath}
-u_0 = \cdx - \hwxX i + \hwxF l
-\end{dmath},
-\begin{dmath}
-u_1 = u_0-2\cdx
-\end{dmath},
-\begin{dmath}
-v_1 = -v_2\cos\mrot - w_2\sin\mrot
-\end{dmath},
-\begin{dmath}
-w_1 = v_2\sin\mrot - w_2\cos\mrot
-\end{dmath},
-\begin{dmath}
-v_2 = \cdy-\hwzF q\sin\mrot
-\end{dmath},
-\begin{dmath}
-w_2 = \cdz - \hwzX k + \hwzF q\cos\mrot
-\end{dmath}.
-\end{dgroup}
+and
+\begin{align}
+u_0 &= \cdx - \hwxX i + \hwxF l \,, &
+u_1 &= u_0-2\cdx \,, \\
+v_1 &= -v_2\cos\mrot - w_2\sin\mrot \,, &
+w_1 &= v_2\sin\mrot - w_2\cos\mrot \,, \\
+v_2 &= \cdy-\hwzF q\sin\mrot \,, &
+w_2 &= \cdz - \hwzX k + \hwzF q\cos\mrot \,.
+\end{align}
 The following auxiliary function is used in the above. All dashed variables are
 local to this function.
 \begin{align}
@@ -792,7 +788,10 @@ \subsection{Forces between magnets with relative rotation}
   &\quad +\half u' \pi \Sign{f_5}\Abs{f_6}
   +u' f_6 \ArcTan{\frac{u' f_4 - {u'}^2 -f_6^2}{f_5 f_6}}
   +\half f_4 f_5 \,,
-\end{split}\\
+\end{split}
+\end{align}
+with
+\begin{align}
 f_4 &= \sqrt{{u'}^2+f_5^2+f_6^2} \,, \\
 f_5 &= \gp{v'-\hwyX j}\cos\mrot'+\gp{w'-c'}\sin\mrot'+2\hwyF p \,, \\
 f_6 &= -\gp{v'-\hwyX j}\sin\mrot'+\gp{w'-c'}\cos\mrot'+C' \,.
@@ -860,7 +859,7 @@ \subsubsection{Experimental verification}
 An apparatus for the work of an honours project \cite{byfield2012-honoursthesis} was constructed to position two magnets relatively to each other with some fixed rotation and horizontal offset with unconstrained vertical motion (\figref*{magrig}).
 The vertical displacement was measured with a Wenglor CP35MHT80 laser sensor, and the base magnet was mounted to an ATI Mini85 SI-950-40 load cell to measure the reaction forces generated by the magnets.
 
-\begin{figure}
+\begin{figure}[b!]
 \includegraphics{PhD/Figures/Rig/magrig-label}
 \caption{Photo of the experimental apparatus to measure magnet forces.}
 \figlabel{magrig}
@@ -899,6 +898,7 @@ \subsubsection{Experimental verification}
   \figlabel{magrotmeas}
 \end{figure}
 
+\clearpage
 \subsection{Torques between cuboid magnets}
 
 \textcite{allag2009-ietm} have proposed a method, corrected later \cite{yonnet2011-ietm}, using the magnetic nodes approach to calculate the torques between cuboid magnets.
@@ -982,6 +982,16 @@ \section{Forces between cylindrical magnets}
 \seclabel{cyl-forces}
 \seclabel{magnetcoil-forces}
 
+
+\begin{figure}[b]
+\centering
+\asyinclude{\jobname/coil-mag-equiv.asy}
+\caption
+[The equivalence between a permanent magnet and a current-carrying coil.]
+{The equivalence between a permanent magnet of magnetisation $J=\remanence$ (left) in the positive vertical direction, and a current-carrying coil (right) with equivalent magnetisation $J_{\text{eq.}}=\permvac \turnsCoil \current/\lengthCoil$ for current $\current$ shown flowing anti-clockwise from the top through $\turnsCoil$ axial turns across length $\lengthCoil$.}
+\figlabel{coil-mag-equiv}
+\end{figure}
+
 The force equations between cylindrical magnets are more difficult to derive than for cuboid magnets since their integrals require a cylindrical coordinate system.
 In early work in this field, \textcite{cooper1973-ietm} presented an integral expression for calculating the force between two cylindical magnets.
 \textcite{nagaraj1988} investigated and compared the force between cuboid and cylindrical magnets with arbitrary displacements using numerical integration to calculate his results; \citeauthor{furlani1993-ietm}~\cite{furlani1993-ietm,furlani1993-ietm-coupl} calculated the force between radially-aligned ring magnets using a numerical discretisation of the magnet volume.
@@ -994,16 +1004,7 @@ \section{Forces between cylindrical magnets}
 This simplification, \eqref[vref]{simpl4}, results in a faster execution time and more convenient calculation with numerical software.
 
 
-\begin{figure}
-\centering
-\asyinclude{\jobname/coil-mag-equiv.asy}
-\caption
-[The equivalence between a permanent magnet and a current-carrying coil.]
-{The equivalence between a permanent magnet of magnetisation $J=\remanence$ (left) in the positive vertical direction, and a current-carrying coil (right) with equivalent magnetisation $J_{\text{eq.}}=\permvac \turnsCoil \current/\lengthCoil$ for current $\current$ shown flowing anti-clockwise from the top through $\turnsCoil$ axial turns across length $\lengthCoil$.}
-\figlabel{coil-mag-equiv}
-\end{figure}
-
-The equation for the force between cylindrical magnets can also be used to calculate the force between thin coils with many axial turns, as both magnet and coil can be modelled as a surface current density around a cylinder (see \figref{coil-mag-equiv}). In related work, \textcite{kim1996-ietm} presented a different integral equation for the radial force between (single-turn) circular coils with eccentric radial displacement, for which further application of their results is required to calculate the forces between coils with many turns.
+The equation for the force between cylindrical magnets can also be used to calculate the force between thin coils with many axial turns, as both magnet and coil can be modelled as a surface current density around a cylinder (see \figref[vref]{coil-mag-equiv}). In related work, \textcite{kim1996-ietm} presented a different integral equation for the radial force between (single-turn) circular coils with eccentric radial displacement, for which further application of their results is required to calculate the forces between coils with many turns.
 Little attention has been paid to the forces between rotated cylindrical magnets; \textcite{babic2011-ietm-incl-coil} presented equations for calculating the forces between rotated and eccentric (single-turn) circular coils.
 These expressions can be used with superposition to calculate the forces between rotated and inclined thin-wall solenoids or permanent magnets.
 
@@ -1040,10 +1041,10 @@ \subsubsection{Coaxial magnet force simplification}
 \end{dmath},
 with parameters
 \begin{align}
-\m1 &= z_i - z_j, \\
-\m2 &= \frac{\gp{r_1-r_2}^2}{\m1^2}+1,\\
-\m3^2 &= \gp{r_1+r_2}^2+\m1^2, \\
-\m4 &= \frac{4 r_1 r_2}{\m3^2}, \qquad 0<\m4\le 1.
+\m1 &= z_i - z_j \,, &
+\m2 &= \frac{\gp{r_1-r_2}^2}{\m1^2}+1 \,,\\
+\m3^2 &= \gp{r_1+r_2}^2+\m1^2 \,, &
+\m4 &= \frac{4 r_1 r_2}{\m3^2} \,, \qquad 0<\m4\le 1 \,.
 \end{align}
 This equation is particularly efficient to calculate as the complete elliptic integrals of the first, second, and third kind can all be calculated simultaneously with a single iteration of the arithmetic-geometric mean approach \cite[\S19.8(i)]{DLMF2010}.
 Additionally, with the use of the complete elliptic integrals this equation is straightforward to implement in numerical software (such as Matlab) that does not have in-built support for the incomplete elliptic integrals.