# Quaternion Mathematics in Electromagnetic Modeling and Simulation

Matthew David Marko
<EMAIL_ADDRESS>

Joe Schaff
<EMAIL_ADDRESS>
(22 September 2022)

## Abstract

The purpose of this effort is to investigate if the use of quaternion
mathematics can be used to better model and simulate the electromagnetic
fields that occur from moving electromagnetic charges. One observed deficiency
with the commonly used Maxwell’s equations is the issue of polar versus axial
vectors; the electromagnetic field E is a polar vector, whereas the magnetic
field B is an axial vector, where the direction of rotation remains the same
even after the axial vector is inverted. This effort first derived the
rotation matrix for quaternion geometry. This rotation matrix was then applied
to modeling the magnetic fields at a distance from a source, and comparing it
to traditional Maxwell’s equations. This effort was taken to model a series of
moving charges, an observation aircraft observing a submarine, as well as an
eddy current dynamic brake. It was clearly observed that when a moving
observer is studying a moving target, differences in the magnetic direction
and magnitude can be observed, demonstrating the effectiveness of quaternion
mathematics in such applications.

NAVAIR Public Release 2022-450 Distribution Statement A - ”Approved for public
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##  1 Introduction

The purpose of this effort is to investigate if the use of quaternion
mathematics [1, 2, 3, 4, 5, 6, 7, 8] can be used to better model and simulate
the electromagnetic fields that occur from moving electromagnetic charges. One
observed deficiency with the commonly used Maxwell’s equations is the issue of
polar versus axial vectors; the electromagnetic field E is a polar vector,
whereas the magnetic field B is an axial vector, where the direction of
rotation remains the same even after the axial vector is inverted. It can be
seen that the sense of rotation remains the same for the mirror image (axial
vector), in contrast, the direction of motion of a particle, shown by the
straight arrow, is inverted (polar vector). This illustrates the different
natures of the electromagnetic field E (polar) and magnetic field B (axial)
vectors.

The quaternion-based geometric algebra model of electromagnetism developed and
studied in this research provides better preservation of sign due to its non-
commutative nature, and can also be represented by a single equation. The
limitation of both this and the classical equations is that multiple charges
or magnetic poles are treated as an aggregate for computation. This does not
describe some of the subtle changes between the charges, or the differences
between localized eddy currents, and their observed clustering. One way to add
to the mathematical model / equations to provide better details of those
electromagnetic phenomena is to incorporate a fractal aspect to the equations.
Fractal structures have been observed in electromagnetic field distributions
in both natural and synthetic environments. Fractal patterns in synthetic
electronic environments were first observed by Mandelbrot [9] over 50 years
ago in his research on transmission line noise, and in large volumes of packet
data traffic in wide area network nodes by Paxson and Floyd [10] over 25 years
ago.

Magnetic anomalies have been observed in varying geophysical environments. The
general diffusion equations through a geological medium is not sufficient to
describe some coherent structures observed at multiple size scales. Moreover,
the electromagnetic eddy currents and respective fields that traverse a
geophysical medium have a fractal quality that has been observed in some
media.

These fractal aspects of EM fields in geophysical media [11] are due to
several factors that include: fractal diffusion layers which are caused in
part by erosion of striated mineral deposits; existing fluids embodying
fractional Brownian motion as described by Mandelbrot [9]. Random noise has
been eliminated due to the observance of intrinsic structures. While this may
describe most of the observed fractal structures frequently labeled as
anomalous, there is still the fundamental observation by Voyager 2 of the
magnetic fields at a significant distance of 25 AU from earth [12]. The
topology of these fields had a multifractal aspect to them, without the
attributes needed to describe geophysical erosion, or similar changes to the
medium. This was originally the motivation to try to rewrite the multipole
(electric or magnetic) models using fractal dimensionality, in order to
determine if such a model would have greater fidelity when compared to real-
world observations.

The discovery of fractal-based equations that better describe eddy currents
and multipole charges will be the focus of future research. The groundwork for
this has been started with this current research, specifically with the
incorporation of a fractal noise model as a baseline for the current new
equations. It is hoped that further investigations into the fractal nature of
these real-world anomalies would provide a more comprehensive insight into
anomalous EM field models.

Commonly, electromagnetic fields are represented with Maxwell’s equation. The
most fundamental term is the electromagnetic field, commonly denoted as E. The
force from a pair of electromagnetic charges is simply

$\displaystyle{F}_{12}$ $\displaystyle=$
$\displaystyle\frac{{Q_{1}}{\cdot}{Q_{2}}{\cdot}{{\bf{a}}_{12}}}{/}{{k_{0}}{\cdot}{R^{2}}},$
(1)

where $Q_{1}$ and $Q_{2}$ are the charges (Coulombs), the normalized vector
${\bf{a}}_{12}$, the distance in between the two charges _R_ (m) squared, and
$k_{0}$, where

$\displaystyle{k_{0}}$ $\displaystyle=$
$\displaystyle\frac{1}{4{\cdot}{\pi}{{\epsilon}_{r}}{\cdot}{{\epsilon}_{0}}}$
(2)

where ${{\epsilon}_{0}}$ is the universal electric permittivity
(${{10^{-9}/4{\cdot}{\pi}}}$), and ${{\epsilon}_{r}}$ is the relative
permittivity of the material; where ${{\epsilon}_{r}}$ = 1 in a vacuum. The
electric field is simply the force for a given charge

$\displaystyle{\bf{E}}=\frac{\bf{F}}{Q_{2}}={Q_{1}}{\cdot}{\frac{\bf{a}_{12}}{k_{0}}}{\cdot}{R^{2}}.$
(3)

This is the electric field for a single point charge; the total electric field
is the accumulation of all of the sources / fields.

The magnetic field is a function of the electric field, and due to Faraday’s
law, the rate of change in the magnetic field is the curl of the electric
field

$\displaystyle{\frac{{\partial}{\bf{B}}}{{\partial}{t}}}$ $\displaystyle=$
$\displaystyle-{\nabla}{\times}{\bf{E}}.$ (4)

Some parameters of the electric E and magnetic E field include that the
gradient of the magnetic field is consistently 0 (Gauss’ law of magnetism),

$\displaystyle{\nabla}{\bullet}{\bf{B}}$ $\displaystyle=$ $\displaystyle 0,$
(5)

the gradient of the electric field is proportional to the electric charge
density $\rho$ (Gauss’s law)

$\displaystyle{\nabla}{\bullet}{\bf{E}}$ $\displaystyle=$
$\displaystyle{\frac{\rho}{\epsilon_{0}}},$ (6)

and the curl of the magnetic field is proportional to the current density J,

$\displaystyle{\nabla}{\times}{\bf{B}}-{{\frac{1}{c_{0}^{2}}}{\cdot}{\frac{{\partial}{\bf{E}}}{{\partial}t}}}$
$\displaystyle=$ $\displaystyle{\mu_{0}}{\cdot}{\bf{J}},$ (7)

where ${\mu}_{0}$ is the magnetic permeability of a vacuum
(${\mu}_{0}=4{\cdot}{\pi}{\cdot}{10^{-9}}$), and $c_{0}$ is the speed of light
(${3{\cdot}10^{8}}$ m/s). One relationship with the speed of light, the
electric permittivity, and the magnetic permeability,

$\displaystyle c_{0}$ $\displaystyle=$
$\displaystyle{\frac{1}{\sqrt{{{\mu}_{0}}{\cdot}{{\epsilon}_{0}}}}}.$ (8)

Geometric algebra is another approach, that maintains the distinction between
axial and polar vectors. Each of the three directions are treated as distinct
algebraic imaginary variables ${e_{1}}$, ${e_{2}}$, and ${e_{3}}$. When
squared, they all equal a scalar

$\displaystyle{e_{1}^{2}}={e_{2}^{2}}={e_{3}^{2}}=1.$ (9)

When the three of them are all multiplied together, they are equal to the
imaginary value

$\displaystyle{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}={\sqrt{-1}}=j.$ (10)

These variables anti-commute, where

$\displaystyle{e_{1}}{\cdot}{e_{2}}$ $\displaystyle=$
$\displaystyle-{e_{1}}{\cdot}{e_{2}},$ (11)
$\displaystyle{e_{3}}{\cdot}{e_{1}}$ $\displaystyle=$
$\displaystyle-{e_{1}}{\cdot}{e_{3}},$ $\displaystyle{e_{3}}{\cdot}{e_{2}}$
$\displaystyle=$ $\displaystyle-{e_{2}}{\cdot}{e_{3}}.$

Given the trivector quantity is the imaginary number, and that the vectors
anti-commute, a dual relationship between the vectors and bivectors exist

$\displaystyle{e_{1}}{\cdot}{e_{2}}$ $\displaystyle=$
$\displaystyle{j}{\cdot}{e_{3}},$ (12) $\displaystyle{e_{3}}{\cdot}{e_{1}}$
$\displaystyle=$ $\displaystyle{j}{\cdot}{e_{2}},$
$\displaystyle{e_{2}}{\cdot}{e_{3}}$ $\displaystyle=$
$\displaystyle{j}{\cdot}{e_{1}}.$

When using Geometric Algebra the imaginary bivector represents a rotation, as
typically observed when comparing a magnetic field versus the vector electric
field.

Finally, when multiplying two vectors when using geometric algebra, it comes
out to the gradient of the two vectors, plus the bivector quantity of the curl
of the two vectors

$\displaystyle{\bf{vw}}$ $\displaystyle=$
$\displaystyle{{({{v_{1}}{\cdot}{e_{1}}}+{{v_{2}}{\cdot}{e_{2}}}+{{v_{3}}{\cdot}{e_{3}}})}{\cdot}{({{w_{1}}{\cdot}{e_{1}}}+{{w_{2}}{\cdot}{e_{2}}}+{{w_{3}}{\cdot}{e_{3}}})}}=$
$\displaystyle=$
$\displaystyle{{v_{1}}{\cdot}{w_{1}}}+{{v_{2}}{\cdot}{w_{2}}}+{{v_{3}}{\cdot}{w_{3}}}+{{({{v_{1}}{\cdot}{w_{2}}}-{{w_{1}}{\cdot}{v_{2}}})}{\cdot}{{e_{1}}{\cdot}{e_{2}}}}+{{({{v_{1}}{\cdot}{w_{3}}}-{{w_{1}}{\cdot}{v_{3}}})}{\cdot}{{e_{1}}{\cdot}{e_{3}}}}+{{({{v_{2}}{\cdot}{w_{3}}}-{{w_{2}}{\cdot}{v_{3}}})}{\cdot}{{e_{2}}{\cdot}{e_{3}}}}=$
$\displaystyle=$
$\displaystyle{v{\bullet}w}+{{({{v_{1}}{\cdot}{w_{2}}}-{{w_{1}}{\cdot}{v_{2}}})}{\cdot}{{j}{\cdot}{e_{3}}}}+{{({{v_{1}}{\cdot}{w_{3}}}-{{w_{1}}{\cdot}{v_{3}}})}{\cdot}{{j}{\cdot}{e_{2}}}}+{{({{v_{2}}{\cdot}{w_{3}}}-{{w_{2}}{\cdot}{v_{3}}})}{\cdot}{{j}{\cdot}{e_{1}}}}=$
$\displaystyle{\bf{vw}}$ $\displaystyle=$
$\displaystyle{v{\bullet}w}+{j{\cdot}{({\bf{v}}{\times}{\bf{w}})}}$

When modeling electromagnetic fields using geometric algebra, all four
Maxwell’s equations can be combined into a single equation. A common variable
_F_ is used for geometric algebra of electromagnetic modeling

$\displaystyle F$ $\displaystyle=$
$\displaystyle{\bf{E}}+{j{\cdot}{c_{0}}{\cdot}{\bf{B}}}.$ (14)

A singular equation for the electric field E, the magnetic field B, the
current density J, and the electric charge density $\rho$, can be written as

$\displaystyle{{[{{\frac{1}{c_{0}}}{\cdot}{(\frac{\partial}{{\partial}t})}}+{\nabla}]}F}$
$\displaystyle=$
$\displaystyle{{\frac{\rho}{\epsilon_{0}}}-{{c_{0}}{\cdot}{\mu_{0}}{\cdot}{\bf{J}}}}.$
(15)

This can be derived as

$\displaystyle{{[{{\frac{1}{c_{0}}}{\cdot}{(\frac{\partial}{{\partial}t})}}+{\nabla}]}F}$
$\displaystyle=$
$\displaystyle{{[{{\frac{1}{c_{0}}}{\cdot}{(\frac{\partial}{{\partial}t})}}+{\nabla}]}{[{\bf{E}}+{j{\cdot}{c_{0}}{\cdot}{\bf{B}}}}]}={{\frac{1}{c_{0}}}{\cdot}{\frac{{\partial}{\bf{E}}}{{\partial}t}}}+{j{\cdot}{\frac{{\partial}{\bf{B}}}{{\partial}t}}}+{{\nabla}{\bf{E}}}+{j{\cdot}{c_{0}}{\cdot}{{\nabla}{\bf{B}}}}$
$\displaystyle=$
$\displaystyle{{\frac{1}{c_{0}}}{\cdot}{\frac{{\partial}{\bf{E}}}{{\partial}t}}}+{j{\cdot}{\frac{{\partial}{\bf{B}}}{{\partial}t}}}+{{\nabla}{\bullet}{\bf{E}}}+{j{\cdot}{\nabla}{\times}{\bf{E}}}+{j{\cdot}{c_{0}}{\cdot}{{\nabla}{\bullet}{\bf{B}}}-{{c_{0}}{\cdot}{{\nabla}{\times}{\bf{B}}}}}.$

Note the following fundamental electromagnetic equations

$\displaystyle{{\frac{1}{c_{0}}}{\cdot}{\frac{{\partial}E}{{\partial}t}}}$
$\displaystyle=$
$\displaystyle{{{c_{0}}{\cdot}{{\nabla}{\times}{\bf{B}}}-{{c_{0}}{\cdot}{\mu_{0}}{\cdot}{\bf{J}}}}}$
$\displaystyle{\frac{{\partial}{\bf{B}}}{{\partial}t}}$ $\displaystyle=$
$\displaystyle{-{\nabla}{\times}{\bf{E}}}$
$\displaystyle{{\nabla}{\bullet}{\bf{B}}}$ $\displaystyle=$ $\displaystyle 0$

and therefore

$\displaystyle{{[{{\frac{1}{c_{0}}}{\cdot}{(\frac{\partial}{{\partial}t})}}+{\nabla}]}F}$
$\displaystyle=$
$\displaystyle{{c_{0}}{\cdot}{{\nabla}{\times}{\bf{B}}}+{{c_{0}}{\cdot}{\mu_{0}}{\cdot}{\bf{J}}}}-{j{\cdot}{\nabla}{\times}{\bf{E}}}+{j{\cdot}{\nabla}{\times}{\bf{E}}}-{{c_{0}}{\cdot}{\mu_{0}}{\cdot}{\bf{J}}}$
$\displaystyle=$
$\displaystyle{{\frac{\rho}{\epsilon_{0}}}-{{c_{0}}{\cdot}{\mu_{0}}{\cdot}{\bf{J}}}}.$

##  2 Rotating the Reference Plane with Quaternion Geometry

One of the advantages of quaternion mathematics for electromagetics is that
during a rotation, the sense of rotation is preserved for axial vectors (such
as the magnetic field), yet polar vectors (such as the electric field) get
inverted. For example, if inverting the reference frame entirely, where
${e_{1}}$ = -${e_{1}}$, ${e_{2}}$ = -${e_{2}}$, and ${e_{3}}$ = -${e_{3}}$,
when dealing solely with Maxwell’s equations, the electric field will invert
as it should. The magnetic field, however, ought to remain the same regardless
of rotating the reference, yet the traditional Maxwell’s equations will also
erroneously invert the direction of the magnetic field. With quaternions, the
axial magnetic field vector maintains its direction. Before inversion

$\displaystyle F$ $\displaystyle=$ $\displaystyle{\bf{E}}+{j{\cdot}{\bf{B}}},$
$\displaystyle=$
$\displaystyle{{E_{1}}{\cdot}{e_{1}}}+{{E_{2}}{\cdot}{e_{2}}}+{{E_{3}}{\cdot}{e_{3}}}+{j{\cdot}{({{B_{1}}{\cdot}{e_{1}}}+{{B_{2}}{\cdot}{e_{2}}}+{{B_{3}}{\cdot}{e_{3}}})}},$
$\displaystyle=$
$\displaystyle{{E_{1}}{\cdot}{e_{1}}}+{{E_{2}}{\cdot}{e_{2}}}+{{E_{3}}{\cdot}{e_{3}}}+{{{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}}{\cdot}{({{B_{1}}{\cdot}{e_{1}}}+{{B_{2}}{\cdot}{e_{2}}}+{{B_{3}}{\cdot}{e_{3}}})}},$

If inverting _F_ to get _F’_ ,

$\displaystyle F$ $\displaystyle=$
$\displaystyle{{E_{1}}{\cdot}{-e_{1}}}+{{E_{2}}{\cdot}{-e_{2}}}+{{E_{3}}{\cdot}{-e_{3}}}+{{{-e_{1}}{\cdot}{-e_{2}}{\cdot}{-e_{3}}}{\cdot}{({{B_{1}}{\cdot}{-e_{1}}}+{{B_{2}}{\cdot}{-e_{2}}}+{{B_{3}}{\cdot}{-e_{3}}})}},$
$\displaystyle=$
$\displaystyle-({{E_{1}}{\cdot}{e_{1}}}+{{E_{2}}{\cdot}{e_{2}}}+{{E_{3}}{\cdot}{e_{3}}})+{{{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}}{\cdot}{({{B_{1}}{\cdot}{e_{1}}}+{{B_{2}}{\cdot}{e_{2}}}+{{B_{3}}{\cdot}{e_{3}}})}},$
$\displaystyle=$ $\displaystyle{-\bf{E}}+{j{\cdot}{\bf{B}}},$

It is clear, by this inversion, that the electric field rotates with the
rotation, and the magnetic field stays intact. This rotation can be defined as
a rotation matrix $\bar{R}$

$\bar{R}=\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\ \end{bmatrix}\\\ $

It is observed that using quaternion geometric algebra, the magnetic field
direction is preserved during the rotation, simply by rotating the “imaginary”
magnetic field by the determinant of $\bar{R}$.

The method of multiplying the rotation matrix by the determinant for
quaternion geometry can be realized through the following derivation. To
understand how to best rotate a matrix with quaternion geometry, first the
rotation matrix is identified:

$\begin{bmatrix}{e_{1}^{\prime}}\\\ {e_{2}^{\prime}}\\\ {e_{3}^{\prime}}\\\
\end{bmatrix}$ = $\begin{bmatrix}{a_{11}}&{a_{21}}&{a_{31}}\\\
{a_{12}}&{a_{22}}&{a_{32}}\\\ {a_{13}}&{a_{23}}&{a_{33}}\\\ \end{bmatrix}$
${\cdot}$ $\begin{bmatrix}{e_{1}}\\\ {e_{2}}\\\ {e_{3}}\\\ \end{bmatrix}\\\ $

which has the affect of:

$\displaystyle{e_{1}^{\prime}}$ $\displaystyle=$
$\displaystyle{{a_{11}}{\cdot}{e_{1}}}+{{a_{21}}{\cdot}{e_{2}}}+{{a_{31}}{\cdot}{e_{3}}}$
$\displaystyle{e_{2}^{\prime}}$ $\displaystyle=$
$\displaystyle{{a_{12}}{\cdot}{e_{1}}}+{{a_{22}}{\cdot}{e_{2}}}+{{a_{32}}{\cdot}{e_{3}}}$
$\displaystyle{e_{3}^{\prime}}$ $\displaystyle=$
$\displaystyle{{a_{13}}{\cdot}{e_{1}}}+{{a_{23}}{\cdot}{e_{2}}}+{{a_{33}}{\cdot}{e_{3}}}$

If one wants to rotate a magnetic field component during quaternion geometry,
where $j={{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}}$, this is resolved by rotating
both and multiplying j with the vector. The rotation of the imaginary
component j is thus:

$\displaystyle{e_{1}^{\prime}}{\cdot}{e_{2}^{\prime}}{\cdot}{e_{3}^{\prime}}$
$\displaystyle=$
$\displaystyle({({{a_{11}}{\cdot}{e_{1}}}+{{a_{21}}{\cdot}{e_{2}}}+{{a_{31}}{\cdot}{e_{3}}})}{\cdot}{({{a_{12}}{\cdot}{e_{1}}}+{{a_{22}}{\cdot}{e_{2}}}+{{a_{32}}{\cdot}{e_{3}}})}{\cdot}{({{a_{13}}{\cdot}{e_{1}}}+{{a_{23}}{\cdot}{e_{2}}}+{{a_{33}}{\cdot}{e_{3}}})}$
$\displaystyle=$
$\displaystyle{({{a_{11}}{\cdot}{a_{12}}{\cdot}{e_{1}}{\cdot}{e_{1}}}+{{a_{11}}{\cdot}{a_{22}}{\cdot}{e_{1}}{\cdot}{e_{2}}}+{{a_{11}}{\cdot}{a_{32}}{\cdot}{e_{1}}{\cdot}{e_{3}}}+{{a_{21}}{\cdot}{a_{12}}{\cdot}{e_{2}}{\cdot}{e_{1}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{e_{2}}{\cdot}{e_{2}}}}+$
$\displaystyle{{a_{21}}{\cdot}{a_{32}}{\cdot}{e_{2}}{\cdot}{e_{3}}}+{{a_{31}}{\cdot}{a_{12}}{\cdot}{e_{3}}{\cdot}{e_{1}}}+{{a_{31}}{\cdot}{a_{22}}{\cdot}{e_{3}}{\cdot}{e_{2}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{e_{3}}{\cdot}{e_{3}}}){\cdot}{({{a_{13}}{\cdot}{e_{1}}}+{{a_{23}}{\cdot}{e_{2}}}+{{a_{33}}{\cdot}{e_{3}}})}$
$\displaystyle=$
$\displaystyle[({{a_{11}}{\cdot}{a_{12}}}+{{a_{21}}{\cdot}{a_{22}}}+{{a_{31}}{\cdot}{a_{32}}})+{({{a_{11}}{\cdot}{a_{22}}}-{{a_{21}}{\cdot}{a_{12}}}){\cdot}{e_{1}}{\cdot}{e_{2}}}+{({{a_{11}}{\cdot}{a_{32}}}-{{a_{31}}{\cdot}{a_{12}}}){\cdot}{e_{1}}{\cdot}{e_{3}}}+$
$\displaystyle{({{a_{21}}{\cdot}{a_{32}}}-{{a_{31}}{\cdot}{a_{22}}}){\cdot}{e_{2}}{\cdot}{e_{3}}}]{\cdot}{({{a_{13}}{\cdot}{e_{1}}}+{{a_{23}}{\cdot}{e_{2}}}+{{a_{33}}{\cdot}{e_{3}}})},$
$\displaystyle=$
$\displaystyle{({{a_{11}}{\cdot}{a_{12}}}+{{a_{21}}{\cdot}{a_{22}}}+{{a_{31}}{\cdot}{a_{32}}}){\cdot}{a_{13}}{\cdot}{e_{1}}}+{({{a_{11}}{\cdot}{a_{12}}}+{{a_{21}}{\cdot}{a_{22}}}+{{a_{31}}{\cdot}{a_{32}}}){\cdot}{a_{23}}{\cdot}{e_{2}}}+$
$\displaystyle{({{a_{11}}{\cdot}{a_{12}}}+{{a_{21}}{\cdot}{a_{22}}}+{{a_{31}}{\cdot}{a_{32}}}){\cdot}{a_{33}}{\cdot}{e_{3}}}+{{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{1}}{\cdot}({a_{11}{\cdot}{a_{22}}}-{a_{21}{\cdot}{a_{12}}}){\cdot}{a_{13}}}+$
$\displaystyle{{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}{\cdot}({a_{11}{\cdot}{a_{22}}}-{a_{21}{\cdot}{a_{12}}}){\cdot}{a_{23}}}+{{e_{1}}{\cdot}{e_{3}}{\cdot}{e_{1}}{\cdot}({a_{11}{\cdot}{a_{22}}}-{a_{21}{\cdot}{a_{12}}}){\cdot}{a_{33}}}+$
$\displaystyle{{e_{1}}{\cdot}{e_{3}}{\cdot}{e_{1}}{\cdot}({a_{11}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{12}}}){\cdot}{a_{13}}}+{{e_{1}}{\cdot}{e_{3}}{\cdot}{e_{2}}{\cdot}({a_{11}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{12}}}){\cdot}{a_{23}}}+$
$\displaystyle{{e_{1}}{\cdot}{e_{3}}{\cdot}{e_{3}}{\cdot}({a_{11}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{12}}}){\cdot}{a_{33}}}+{{e_{2}}{\cdot}{e_{3}}{\cdot}{e_{1}}{\cdot}({a_{21}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{22}}}){\cdot}{a_{13}}}+$
$\displaystyle{{e_{2}}{\cdot}{e_{3}}{\cdot}{e_{2}}{\cdot}({a_{21}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{22}}}){\cdot}{a_{23}}}+{{e_{2}}{\cdot}{e_{3}}{\cdot}{e_{3}}{\cdot}({a_{21}{\cdot}{a_{32}}}-{a_{31}{\cdot}{a_{22}}}){\cdot}{a_{33}}},$

and therefore,

$\displaystyle=$
$\displaystyle{e_{1}}{\cdot}({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{13}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{13}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{13}}}+{{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{23}}}-{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{23}}}-{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{33}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{33}}})+$
$\displaystyle{e_{2}}{\cdot}({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{23}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{13}}}-{{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{33}}}-{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{33}}})+$
$\displaystyle{e_{3}}{\cdot}({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{33}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{13}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{23}}}-{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{23}}})+$
$\displaystyle{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}{\cdot}({{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{33}}}-{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{23}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{13}}}).$

As a result, the rotation matrix of an imaginary value is rotated with the
following relationship

$\displaystyle{e_{1}^{\prime}}{\cdot}{e_{2}^{\prime}}{\cdot}{e_{3}^{\prime}}$
$\displaystyle=$
$\displaystyle{{\bar{E}_{1}}{\cdot}{e_{1}}}+{{\bar{E}_{2}}{\cdot}{e_{2}}}+{{\bar{E}_{3}}{\cdot}{e_{3}}}+{{\bar{E}_{x}}{\cdot}{e_{1}}{\cdot}{e_{2}}{\cdot}{e_{3}}}$
$\displaystyle=$
$\displaystyle{{\bar{E}_{1}}{\cdot}{e_{1}}}+{{\bar{E}_{2}}{\cdot}{e_{2}}}+{{\bar{E}_{3}}{\cdot}{e_{3}}}+{j{\cdot}{\bar{E}_{x}}},$

where

$\displaystyle{\bar{E}_{1}}$ $\displaystyle=$
$\displaystyle({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{13}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{13}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{13}}}+{{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{23}}}-{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{23}}}-{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{33}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{33}}})$
$\displaystyle{\bar{E}_{2}}$ $\displaystyle=$
$\displaystyle({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{23}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{13}}}-{{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{33}}}-{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{33}}})$
$\displaystyle{\bar{E}_{3}}$ $\displaystyle=$
$\displaystyle({{a_{11}}{\cdot}{a_{12}}{\cdot}{a_{33}}}+{{a_{21}}{\cdot}{a_{22}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{32}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{13}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{23}}}-{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{23}}})$
$\displaystyle{\bar{E}_{X}}$ $\displaystyle=$
$\displaystyle({{a_{11}}{\cdot}{a_{22}}{\cdot}{a_{33}}}-{{a_{21}}{\cdot}{a_{12}}{\cdot}{a_{33}}}+{{a_{31}}{\cdot}{a_{12}}{\cdot}{a_{23}}}-{{a_{11}}{\cdot}{a_{32}}{\cdot}{a_{23}}}+{{a_{21}}{\cdot}{a_{32}}{\cdot}{a_{13}}}-{{a_{31}}{\cdot}{a_{22}}{\cdot}{a_{13}}}).$

When the rotation of a magnetic field $j{\cdot}{c_{0}}{\cdot}B$ occurs with
quaternion geometry,

$\begin{bmatrix}{e_{1}^{\prime}}\\\ {e_{2}^{\prime}}\\\ {e_{3}^{\prime}}\\\
\end{bmatrix}$ = $\begin{bmatrix}{a_{11}}&{a_{21}}&{a_{31}}\\\
{a_{12}}&{a_{22}}&{a_{32}}\\\ {a_{13}}&{a_{23}}&{a_{33}}\\\ \end{bmatrix}$
${\cdot}$ $\begin{bmatrix}{j{\cdot}{e_{1}}}\\\ {j{\cdot}{e_{2}}}\\\
{j{\cdot}{e_{3}}}\\\ \end{bmatrix}$ =

$\begin{bmatrix}({{a_{11}{\cdot}{e_{1}}}+{{a_{21}{\cdot}{e_{2}}}+{{a_{31}{\cdot}{e_{3}}}){\cdot}({{\bar{E}_{1}}{\cdot}{e_{1}}}+{{\bar{E}_{2}}{\cdot}{e_{2}}}+{{\bar{E}_{3}}{\cdot}{e_{3}}}+{j{\cdot}{\bar{E}_{x}}}}}})\\\
({{a_{12}{\cdot}{e_{1}}}+{{a_{22}{\cdot}{e_{2}}}+{{a_{32}{\cdot}{e_{3}}}){\cdot}({{\bar{E}_{1}}{\cdot}{e_{1}}}+{{\bar{E}_{2}}{\cdot}{e_{2}}}+{{\bar{E}_{3}}{\cdot}{e_{3}}}+{j{\cdot}{\bar{E}_{x}}}}}})\\\
({{a_{13}{\cdot}{e_{1}}}+{{a_{23}{\cdot}{e_{2}}}+{{a_{33}{\cdot}{e_{3}}}){\cdot}({{\bar{E}_{1}}{\cdot}{e_{1}}}+{{\bar{E}_{2}}{\cdot}{e_{2}}}+{{\bar{E}_{3}}{\cdot}{e_{3}}}+{j{\cdot}{\bar{E}_{x}}}}}})\\\
\end{bmatrix}$ = $\begin{bmatrix}{Q_{1}}\\\ {Q_{2}}\\\ {Q_{3}}\\\
\end{bmatrix}$

$\displaystyle Q_{i}$ $\displaystyle=$
$\displaystyle{{E_{1}}{\cdot}{a_{1i}}{\cdot}{e_{1}}{\cdot}{e_{1}}}+{{E_{1}}{\cdot}{a_{2i}}{\cdot}{e_{1}}{\cdot}{e_{2}}}+{{E_{1}}{\cdot}{a_{3i}}{\cdot}{e_{1}}{\cdot}{e_{3}}}+{{E_{2}}{\cdot}{a_{1i}}{\cdot}{e_{2}}{\cdot}{e_{1}}}+{{E_{2}}{\cdot}{a_{2i}}{\cdot}{e_{2}}{\cdot}{e_{2}}}+{{E_{2}}{\cdot}{a_{3i}}{\cdot}{e_{2}}{\cdot}{e_{3}}}+$
$\displaystyle{{E_{3}}{\cdot}{a_{1i}}{\cdot}{e_{3}}{\cdot}{e_{1}}}+{{E_{3}}{\cdot}{a_{2i}}{\cdot}{e_{3}}{\cdot}{e_{2}}}+{{E_{3}}{\cdot}{a_{3i}}{\cdot}{e_{3}}{\cdot}{e_{3}}}+{{E_{X}}{\cdot}{a_{1i}}{\cdot}{j}{\cdot}{e_{1}}}+{{E_{X}}{\cdot}{a_{2i}}{\cdot}{j}{\cdot}{e_{2}}}+{{E_{X}}{\cdot}{a_{3i}}{\cdot}{j}{\cdot}{e_{3}}}.$
$\displaystyle=$
$\displaystyle{{E_{1}}{\cdot}{a_{1i}}}+{{E_{1}}{\cdot}{a_{2i}}{\cdot}{j}{\cdot}{e_{3}}}-{{E_{1}}{\cdot}{a_{3i}}{\cdot}{j}{\cdot}{e_{2}}}-{{E_{2}}{\cdot}{a_{1i}}{\cdot}{j}{\cdot}{e_{3}}}+{{E_{2}}{\cdot}{a_{2i}}}+{{E_{2}}{\cdot}{a_{3i}}{\cdot}{j}{\cdot}{e_{1}}}+$
$\displaystyle{{E_{3}}{\cdot}{a_{1i}}{\cdot}{j}{\cdot}{e_{2}}}-{{E_{3}}{\cdot}{a_{2i}}{\cdot}{j}{\cdot}{e_{1}}}+{{E_{3}}{\cdot}{a_{3i}}}+{{E_{X}}{\cdot}{a_{1i}}{\cdot}{j}{\cdot}{e_{1}}}+{{E_{X}}{\cdot}{a_{2i}}{\cdot}{j}{\cdot}{e_{2}}}+{{E_{X}}{\cdot}{a_{3i}}{\cdot}{j}{\cdot}{e_{3}}}$
$\displaystyle=$
$\displaystyle({{E_{1}}{\cdot}{a_{1i}}}+{{E_{2}}{\cdot}{a_{2i}}}+{{E_{3}}{\cdot}{a_{3i}}})+({{E_{2}}{\cdot}{a_{3i}}}-{{E_{3}}{\cdot}{a_{2i}}}+{{E_{X}}{\cdot}{a_{1i}}}){\cdot}{j}{\cdot}{e_{1}}+$
$\displaystyle(-{{E_{1}}{\cdot}{a_{3i}}}+{{E_{3}}{\cdot}{a_{1i}}}+{{E_{X}}{\cdot}{a_{2i}}}){\cdot}{j}{\cdot}{e_{2}}+({{E_{1}}{\cdot}{a_{2i}}}-{{E_{2}}{\cdot}{a_{1i}}}+{{E_{X}}{\cdot}{a_{3i}}}){\cdot}{j}{\cdot}{e_{3}}$

It was observed that a much simpler approach to rotating the magnetic field is
to multiply the rotation matrix by the determinant of the rotation matrix when
rotating the magnetic field; this is not an exact simplification but often a
good approximation. If rotating in units of 90∘, it matches exactly.

##  3 Model Development

The model (Figure 1) uses 360 point charges, spread in a circle in the X-Y
plane, 10 meters apart from the reference point. The charge varied with a
sinusoidal function

$\displaystyle Q$ $\displaystyle=$
$\displaystyle{10^{12}}{\cdot}{sin(2{\cdot}{\theta})},$ (16)

where $0<{\theta}<2{\cdot}{\pi}$. The velocity is only in the Z-direction,
going in both directions, at a velocity $V_{Z}$ (m/s) of

$\displaystyle V_{Z}$ $\displaystyle=$ $\displaystyle
5{\cdot}{cos({\theta})}{\cdot}[1+{0.5{\cdot}cos(2{\cdot}{\pi}{\cdot}f{\cdot}t)}],$
(17)

where _t_ (s) is the time, and _f_ (Hz) is the frequency of the velocity
fluctuation, defined as

$\displaystyle f={10^{12}}{\cdot}(1+0.5{\cdot}{sin(\theta)}).$ (18)

The simulation will run for 10 seconds, discretized into 1,000 time steps of
0.01 second.

Before focusing on quaternion’s, the simulation solved the Maxwell’s equations
for the electric field E (Volts / meter), the magnetic field B (Tesla), the
current density J (Amps/m2), and the charge volumetric density $\rho$ (C/m3).
The model worked by calculating the contribution of each charge to the
electric field both at the null reference point (0,0,0), but also in a
5$\times$5$\times$5 3-D grid of positions, separated by a small distance
increment of ${\delta}x$ = 1 cm. This distance increment is used to find the
change of electric field at the reference point, both at the reference point

$\displaystyle{\frac{{\partial}{E_{1}}}{{\partial}x}}{(0,0,0)}$
$\displaystyle=$
$\displaystyle{\frac{{E_{1}({\delta}x,0,0)}-{E_{1}{(-{\delta}x,0,0)}}}{2{\cdot}{\delta}x}},$

and at the neighboring points

$\displaystyle{\frac{{\partial}{E_{1}}}{{\partial}x}}{({{\delta}x},0,0)}$
$\displaystyle=$
$\displaystyle{\frac{{E_{1}(2{\cdot}{\delta}x,0,0)}-{E_{1}{(-{\delta}x,0,0)}}}{2{\cdot}{\delta}x}}.$

The differential of the electric field at the reference point is necessary to
find the magnetic field B at the reference point due to Faraday’s Law

$\displaystyle{\frac{{\partial}{\bf{B}}}{{\partial}t}}=-{\nabla}{\times}{\bf{E}}.$

The electric field differential at the neighboring grid points is used to find
the magnetic field at the neighboring grid points, so that the differential of
the magnetic field at the reference point can be determined, so that the curl
of the magnetic field can be used to find the current density J due to
Ampère’s Law

$\displaystyle{{\nabla}{\times}{\bf{B}}-{{\frac{1}{c_{0}^{2}}}{\cdot}{\frac{{\partial}{\bf{E}}}{{\partial}t}}}}$
$\displaystyle=$ $\displaystyle{{{\mu}_{0}}{\cdot}{\bf{J}}}.$ (19)

The results of this study are plotted in Figure 2.

Figure 1: Point-charge simulation parameters.
Figure 2: Point-charge results (Maxwell’s).

###  3.1 Parametric Study Initial Conditions

So far, the electric and magnetic fields were determined with Maxwell’s
equations, at a fixed reference point. The first study (Figure 3) was simply
inverted (${e_{1}}$ = -${e_{1}}$, ${e_{2}}$ = -${e_{2}}$, and ${e_{3}}$ =
-${e_{3}}$). The second study (Figure 4) will have the above inversion,
${e_{1}}$ = -${e_{1}}$, ${e_{2}}$ = -${e_{2}}$, and ${e_{3}}$ = -${e_{3}}$,
and then rotate the reference point by $\phi$ = 45∘.

R= $\begin{bmatrix}{-1}&{0}&{0}\\\ {0}&{-1}&{0}\\\ {0}&{0}&{-1}\\\
\end{bmatrix}{\times}\begin{bmatrix}{cos(\phi)}&{-sin(\phi)}&{0}\\\
{sin(\phi)}&{cos(\phi)}&{0}\\\ {0}&{0}&{1}\\\
\end{bmatrix}=\begin{bmatrix}{-{\sqrt{2}}/2}&{{\sqrt{2}}/2}&{0}\\\
{-{\sqrt{2}}/2}&{-{\sqrt{2}}/2}&{0}\\\ {0}&{0}&{-1}\\\ \end{bmatrix}$ .

Figure 3: Initial case of the point-charge study, with the inversion in all
three directions, where ${e_{1}}$ = -${e_{1}}$, ${e_{2}}$ = -${e_{2}}$, and
${e_{3}}$ = -${e_{3}}$.
Figure 4: The second case, with the inversion in all three directions, where
${e_{1}}$ = -${e_{1}}$, ${e_{2}}$ = -${e_{2}}$, and ${e_{3}}$ = -${e_{3}}$, as
well as the rotation along the X-Y plane of $\phi$ = 45∘. Note, with the
flipped electric field, ${{E_{X}}={E_{Y}}}$.

##  4 Moving Observer Study #1

A parametric study was done, to determine if one would observe a different
magnetic field between two moving observers, and if quaternion geometry could
detect these anomalies. In the moving observer study (Figure 5), the observer
has an X-Y-Z acceleration of (-10, 10, 1) m/s2, a velocity of (-2, 200, 0)
m/s, and an initial position of (-1000, +1000, 300) m. The target has no
acceleration, has a velocity of (1, 1, 0.1) m/s, and is initially located at
(0, 0, -20) meters. The goal was to roughly suggest an aircraft monitoring a
submarine. The target is treated as a solenoid with a steel core ($\mu_{R}$ =
2,000), 10 meters in radius, 100 meters in length, with an equivalent current
of 1 amp flowing through the solenoid (hull).

Fractal geometry is a component of this study, as a form of adding randomness
typically seen in nature, to confirm if the quaternion geometry can properly
capture it. A fractal geometry profile used typically for surface roughness
and friction [13] was used, where the variation $\alpha$ was determined

$\displaystyle{{\alpha}(x)}$ $\displaystyle=$
$\displaystyle{{(Ra^{D-1})}{\cdot}{{\Sigma_{j}}{\frac{cos(2{\cdot}{\pi}{\cdot}{\gamma^{j}}{\cdot}x)}{{{\gamma}^{(2-D){\cdot}j}}}}}},$

where _D_ = 1.6, $\gamma$ = 1.5, and _Ra_ = 108.221; these values are
arbitrary based on the roughness profile desired. The position in the _x_ ,
_y_ , and _z_ direction between the observer and the target was consistently
fluctuated by 50${\cdot}{\alpha}$ meters, with _x_ ranging from 1 to 100,000
before repeating itself, and _j_ ranging from 1 to 100.

Finally, one important step is to determine the rotation matrix for the
changing reference plane. This is determined by comparing the velocity
profiles of the observer (${V_{xo}}$, ${V_{yo}}$, ${V_{zo}}$) and the target
(${V_{xt}}$, ${V_{yt}}$, ${V_{zt}}$). The first step is to calculate the dot
product between the observer and the target; if it is negative, the target is
inverted for the time being. The following formula is used to find the
rotation matrix

$\displaystyle X_{12}$ $\displaystyle=$
$\displaystyle\sqrt{{\Sigma}{({V_{o}}{\times}{V_{t}})^{2}}},$

G= $\begin{bmatrix}{{V_{o}}{\bullet}{V_{t}}}&{X_{12}}&{0}\\\
{X_{12}}&{{V_{o}}{\bullet}{V_{t}}}&{0}\\\ {0}&{0}&{1}\\\ \end{bmatrix}$

$\displaystyle u$ $\displaystyle=$ $\displaystyle V_{o},$ $\displaystyle v$
$\displaystyle=$
$\displaystyle\frac{{V_{t}}-{({V_{o}}{\bullet}{V_{t}}){\cdot}{V_{o}}}}{\sqrt{{\Sigma}{V_{t}}-{(({V_{o}}{\cdot}{V_{t}}){\cdot}{V_{o}})^{2}}}},$
$\displaystyle w$ $\displaystyle=$ $\displaystyle{V_{o}}{\times}{V_{t}},$
$\displaystyle F$ $\displaystyle=$ $\displaystyle\frac{1}{u{\cdot}v{\cdot}w},$
$\displaystyle{\bf{RR}}$ $\displaystyle=$
$\displaystyle{F^{-1}}{\cdot}{({G}{\cdot}{F})},$

If the initial dot product was negative, and the target is inverted, then the
rotation matrix is made negative as well

${\bf{RR}}={F^{-1}}{\cdot}{({G}{\cdot}{F})}=\begin{bmatrix}{-1}&{0}&{0}\\\
{0}&{-1}&{0}\\\ {0}&{0}&{-1}\\\ \end{bmatrix}$ .

This simulation is run for 30 seconds with a time-step of a millisecond
(30,000 time-steps). The results of the magnetic field H (amp/meter) are
plotted in Figure 7, both for the Maxwell and Quaternion approach. It was
observed, with the varying direction, the magnetic fields in the X (Figure
7a), Y (Figure 7b), and Z (Figure 7c) direction are inverted; this phenomenon
is only observed when the velocities are inverted. The magnitude of both
approaches (Figure 7d) matches perfectly, as expected. This simulation shows
how quaternion geometry can accurately capture the polar vector properties of
magnetic fields.

Figure 5: Results of moving-observer study.

##  5 Moving Observer Study #2

###  5.1 Without fractal geometry

Another parametric study was developed, with an observer (ex an aircraft) and
a target (ex a submarine). The observer is flying back and forth in a pattern
as it is surveilling the magnetic fields from the target, which is moving in a
circular direction. The observer aircraft starts off at a position of _X_ =
100,000 meters, _Y_ = -100,000 meters, and a height of 1,000 meters. The
target submarine starts at _X_ = 100,000 meters, _Y_ = -50,000 meters, and a
depth of -20 meters. The target submarine treated as magnetic field in a
solenoid, with a current _I_ = 1, and a relative magnetic permeability of
$\mu_{R}$ = 2000. The observer travels in a straight path for 300 seconds
(with a time-step of 0.1 second),

$\displaystyle{V_{X,O}^{\prime}}$ $\displaystyle=$
$\displaystyle{D{\cdot}{100}},$

where _D_ is either 1 or -1, and then it turns over 300 seconds, at a velocity
of:

$\displaystyle{V_{X,O}^{\prime}}$ $\displaystyle=$
$\displaystyle{D{\cdot}{100}{\cdot}{cos({\frac{2{\cdot}{\pi}{\cdot}{t^{\prime}}}{300}})}},$
$\displaystyle{V_{Y,O}^{\prime}}$ $\displaystyle=$
$\displaystyle{{100}{\cdot}{sin({\frac{2{\cdot}{\pi}{\cdot}{t^{\prime}}}{300}})}}.$

where _t’_ (s) resets every 600 seconds. The Z-position of the observer is
also sinusoidal, following the velocity profile of

$\displaystyle{V_{Z,O}^{\prime}}$ $\displaystyle=$
$\displaystyle{{0.2}{\cdot}{cos({\frac{2{\cdot}{\pi}{\cdot}{t}}{1200}})}}.$

The target submarine, on the other hand, follows the following function for
${V_{X,T}^{\prime}}$ (m/s), ${V_{Y,T}^{\prime}}$ (m/s), and
${V_{Z,T}^{\prime}}$ (m/s)

$\displaystyle{V_{X,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{{40}{\cdot}{cos({\frac{2{\cdot}{\pi}{\cdot}{t}}{9000}})}},$
$\displaystyle{V_{Y,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{{40}{\cdot}{sin({\frac{2{\cdot}{\pi}{\cdot}{t}}{9000}})}},$
$\displaystyle{V_{Z,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{{0.1}{\cdot}{cos({\frac{2{\cdot}{\pi}{\cdot}{t}}{900}})}}.$

The velocities of both the target and the observer is rotated along the _Z_
-axis by 22.5∘. The _X_ and _Y_ position of the observer and target are
plotted in Figure 6-a, and the _Z_ position in Figure 6-b. The magnetic field
strength H (Amp/meter) is plotted in the _X_ (Figure 6-c), _Y_ (Figure 6-d),
and _Z_ (Figure 6-e) direction, as well as the magnetic field strength
magnitude (Figure 6-f). It is clear that using quaternion geometry will
provide a different direction for the magnetic field strength, but at the
correct _X-Y-Z_ position; using traditional Maxwell’s will give an erroneous
direction for the magnetic field strength. The magnitude of the magnetic field
H is consistent with both Maxwell’s and Quaternion Geometry.

Figure 6: (a) _X_ , _Y_ , and (b) _Z_ position of target and observer.
Magnetic field strength from target as detected by observer, in the (c) _X_ ,
(d) _Y_ , (e) _Z_ direction, as well as (f) the magnitude, without fractal
geometry.

###  5.2 With fractal geometry

This simulation was repeated, but with fractal geometry altering the velocity.

$\displaystyle{{\alpha}(x,y)}$ $\displaystyle=$ $\displaystyle
1+{{{0.01}{\cdot}{[{(Ra^{D-1})}{\cdot}{{\Sigma_{j=1}^{1}0}{\frac{{cos(2{\cdot}{\pi}{\cdot}{\gamma^{j}}{x})}{\cdot}{cos(2{\cdot}{\pi}{\cdot}{\gamma^{j}}{x})}}{\gamma^{(2-D){\cdot}{j}}}]}}}}},$

where _D_ = 1.6, $\gamma$ = 1.5, and _Ra_ = 80. When both the observer and the
target were moved in position, the value of $\alpha$ was altered to the
velocity, where $X={X_{0}}+{V{\cdot}{\alpha}{\cdot}{\delta}t}$. In addition,
the target used fractal geometry to generate an imperfect circular motion,
where ${\theta}={|\alpha|}^{1.1}/{\alpha}$.

$\displaystyle{V_{X,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{40}{\cdot}{cos(\frac{2{\cdot}{\pi}{\cdot}{\theta}{\cdot}t}{9000})},$
$\displaystyle{V_{Y,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{40}{\cdot}{sin(\frac{2{\cdot}{\pi}{\cdot}{\theta}{\cdot}t}{9000})},$
$\displaystyle{V_{Z,T}^{\prime}}$ $\displaystyle=$
$\displaystyle{0.1}{\cdot}{cos(\frac{2{\cdot}{\pi}{\cdot}{\theta}{\cdot}t}{900})}.$

The positions and magnetic field strength for this simulation with fractal
geometry are all plotted in Figure 7.

Figure 7: (a) _X_ , _Y_ , and (b) _Z_ position of target and observer.
Magnetic field strength from target as detected by observer, in the (c) _X_ ,
(d) _Y_ , (e) _Z_ direction, as well as (f) the magnitude, with fractal
geometry.

##  6 Eddy Current Brake Model

An eddy current brake [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]
involves a moving, conductive, non-ferrous material (ex. copper, aluminum,
gold) moving within a magnetic field. A finite difference model was built to
simulate this phenomenon. The first step is to calculate the magnetic field
strength _H_ (Amp / meter), using the Biot-Savart law (equation 20)

$\displaystyle{\bf{H}}$ $\displaystyle=$
$\displaystyle{\frac{I}{4{\cdot}{\pi}{R^{3}}}}{\cdot}{{\int}{{d{\bf{I}}}{\times}{\bf{R}}}}.$
(20)

If looking for the magnetic field strength of a single coil, the change in
length of the current ${d{\bf{I}}}$ is ${\rho}{\cdot}{d{\theta}}$, and the
equation for the magnetic field strength at a given position away is thus

$\displaystyle{\bf{H}}$ $\displaystyle=$
$\displaystyle{\frac{I{\cdot}{\rho^{2}}}{2{\cdot}{({R^{2}}+{\rho^{2}})^{\frac{3}{2}}}}}{\bf{a_{z}}},$
(21)

where $\rho$ (m) represents the radius of the coil, and _R_ (m) represents the
distance from the point of H to the center of the coil. In this model, the
user will specify the number of coils, and the number of coils per unit
distance, and the model will evenly space out incremental circular loops, and
combine the total magnetic field strength at every distance increment. After
this has been calculated, the model will determine the magnetic field B by
multiplying the magnetic permeability $\mu$. So long as the disk is a non-
ferrous conductive metal (ex copper, aluminum), it is reasonable to treat the
magnetic permeability as the magnetic permeability of a vacuum
${\mu_{0}}=4{\cdot}{\pi}{\cdot}{10^{-7}}$.

When a conductor moves through a magnetic field, electric eddy currents are
induced, and a Lorentz force is applied. The induced current density J
(Amps/m3) is

$\displaystyle{\bf{J}}$ $\displaystyle=$
$\displaystyle{{\sigma}{\cdot}{({\bf{b}}{\times}{\bf{B}})}},$ (22)

where v (m/s) is the velocity vector. The model is a rotating disk on the _X_
-_Y_ plane, therefore the velocity at a given point is thus

$\displaystyle{V_{X}}$ $\displaystyle=$
$\displaystyle{{\Omega}_{RPM}}{\cdot}{\frac{2{\cdot}{\pi}}{60}}{\cdot}{R_{XY}}{\cdot}{|cos(\theta)|}{\cdot}{\frac{xx}{|xx|}},$
(23) $\displaystyle{V_{Y}}$ $\displaystyle=$
$\displaystyle{{\Omega}_{RPM}}{\cdot}{\frac{2{\cdot}{\pi}}{60}}{\cdot}{R_{XY}}{\cdot}{|sin(\theta)|}{\cdot}{\frac{yy}{|yy|}},$
$\displaystyle{\theta}$ $\displaystyle=$
$\displaystyle{tan^{-1}(\frac{yy}{xx})}.$

The Lorentz force is simply applied as a power consumption _P_ (Watts) as

$\displaystyle P$ $\displaystyle=$
$\displaystyle{{\frac{J^{2}}{\sigma}}{\cdot}{V}},$ (24) $\displaystyle{J^{2}}$
$\displaystyle=$
$\displaystyle{J_{X}^{2}}+{J_{Y}^{2}}+{J_{Z}^{2}}={J_{X}^{2}}+{J_{Y}^{2}},$

where _V_ is the volume (m3) of a given grid node. Once the power for all of
the grid nodes is calculated and summed up, the torque is calculated simply by
dividing the angular torque $\omega$ (rad/s)

$\displaystyle T$ $\displaystyle=$ $\displaystyle{\frac{P}{\omega}},$ (25)
$\displaystyle{\omega}$ $\displaystyle=$
$\displaystyle{\Omega_{RPM}}{\cdot}{\frac{2{\cdot}{\pi}}{60}}.$

###  6.1 Limiting Speed

There is a limit on the effectiveness of the eddy current brake with
increasing rotor disk speed. As the disk moves through the magnetic field,
eddy currents are generated proportional to both the disk speed and the
conductivity of the disk material; these currents induce a magnetic field of
their own that counteracts the magnetic field of the solenoid. Previous
references have cited the limited linear speed $V_{lim}$ (m/s) before an eddy
current brake loses its effectiveness as

$\displaystyle{V_{lim}}$ $\displaystyle=$
$\displaystyle{\frac{2}{{d_{disk}}{\cdot}{{\sigma}_{disk}}{\cdot}{{\mu}_{disk}}}},$
(26)

where $d_{disk}$ (m) is the thickness of the disk, ${\sigma}_{disk}$ (1/ohms)
is the electrical conductivity of the disk material, and ${\mu}_{disk}$ (H/m)
is the magnetic permeability of the disk material. It is based on the limit
where the magnetic field induced by the eddy currents reach twice the magnet
field of the solenoid, or

$\displaystyle{\frac{{\bf{B}}_{eddy}}{{\bf{B}}_{solenoid}}}$ $\displaystyle=$
$\displaystyle 2.$ (27)

As demonstrated in the analytical derivation, the eddy currents $I_{eddy}$
(amps) is

$\displaystyle{I_{eddy}}$ $\displaystyle=$
$\displaystyle{{{\delta}t}_{disk}}{\cdot}{{\sigma}_{disk}}{\cdot}{|{\bf{v}}{\times}{\bf{B}}|},$
(28)

and the magnetic field induced by the eddy current is roughly
${\bf{B}}_{eddy}{\approx}{\mu_{0}}{\cdot}{I_{eddy}}$, and therefore a
relationship between ${\bf{B}}_{eddy}$ and ${\bf{B}}_{solenoid}$ can be found
as

$\displaystyle{B_{eddy}}$ $\displaystyle{\approx}$
$\displaystyle{{{\mu}_{0}}{\cdot}{{\delta}t_{disk}}{\cdot}{{\sigma}_{disk}}{\cdot}{V}{\cdot}{B_{solenoid}}}.$
(29)

If one sets the limit of ${B_{eddy}}/{B_{solenoid}}=2$ for the limiting speed,
then the above relationship can be rewritten as

$\displaystyle{B_{eddy}}/{B_{solenoid}}=2$ $\displaystyle=$
$\displaystyle{{\mu}_{0}}{\cdot}{{\delta}t_{disk}}{\cdot}{{\sigma}_{disk}}{\cdot}{V_{LIM}},$
(30) $\displaystyle{V_{LIM}}$ $\displaystyle=$
$\displaystyle{\frac{2}{{{\mu}_{0}}{\cdot}{{\delta}t_{disk}}{\cdot}{{\sigma}_{disk}}}}.$

This limit is purely theoretical, and many factors, including the
electromagnetic solenoid design and placement, will ultimately affect the
magnetic field. Single examples of a countering eddy current magnetic field
exceeding twice the solenoid magnetic field does not necessarily mean a total
reduction in torque. For this reason, a numerical model is the optimal way to
investigate this balance.

###  6.2 Basic Eddy Current Model

This model assumes an electromagnet, defined as a solenoid on both sides of a
disk.

The simulation depicted in Figure 8, Figure 9, Figure 10, and Figure 11 tests
an eddy current brake, utilizing a 5 inch diameter, 0.25 inch thick copper
disk, rotating at 1,000 RPM. One solenoid coil of 22 loops each, 0.260 inch in
diameter, over 3.15 inch of length with a 99.95% iron annealed in hydrogen
(${\mu}_{R}$ = 200,000, ${\epsilon}_{R}$ = 2) core, is situated 0.1525 inch
above and below the copper disk, at a radius of 4.5 inch on the X-axis
position. A current of 0.7 amps is flowing through the solenoid coil in this
simulation.

Figure 8: Velocity (m/s) profile of spinning 5” diameter copper disk at 1,000
RPM, in the (a) X and (b) Y direction.
Figure 9: Comparison of B, both (a) solely from the solenoid, versus (b)
magnetic field generated from the eddy currents.
Figure 10: Eddy Current generated (amps), in the (a) X and (b) Y direction.
Figure 11: (a) the cross product between the radial velocity vector v (m/s) of
the rotating disk (1,000 RPM) and the magnetic field B (T), and (b) the torque
as a function of disk speed (RPM).

As the velocity increases, the eddy currents increase in magnitude, and these
currents (Figure 10) result in a countering magnetic field (Figure 9-b). These
countering magnetic fields reduce the net magnetic field, and thus when as the
rotating speed of the disk increases, this countering effect gets greater and
greater, until eventually there is a limit to the torque generated with
greater speed.

The model demonstrated that for a constant speed, the torque is approximately
proportional to the solenoid current (and the ratio of the solenoid coils per
length) squared; for a constant current, the torque is approximately linearly
proportional to the disk speed (until the eddy-current induced magnetic field
becomes significant). An empirical equation (correlation coefficient _R_ =
1.00) for the numerically obtain torque _T_ (N$\cdot$m) with this designed
eddy current break is

$\displaystyle T$ $\displaystyle=$
$\displaystyle{3.3125}{\cdot}{10^{-6}}{\cdot}{{({I}{\cdot}{\frac{N_{coil}}{L_{coil}}})}^{2}}{\cdot}{{\Omega}_{RPM}}.$
(31)

This equation is applicable up till 1,000 RPM; at substantially higher speeds
(not realistic with the applications of interest) the torque increase will
decrease.

###  6.3 Eddy Current Probe Study

Another parametric study was conducted with the eddy current electric brake,
where the 40” diameter, 1/2” thick copper disk is spinning at 1,000 RPM. There
are two 4 inch radius magnetic solenoids, 180∘ apart, situated 1/16 inch above
and below the disk. The parametric study is analyzing the magnetic and
electric fields measured in the disk, as situated 1/16 inch _above_ the
spinning disk, parametrically moving in the _X_ and _Y_ direction
(53$\times$53 grid points). The parametric study is performed three times,
where the reference point is directly in the _Z_ direction (0, 0, 1), where
the reference point rotates with position (0, 0, 1), where the position
rotates with position (_X_ , _Y_ , 1), and when there is a random wobble
($\pm$0.05, $\pm$0.05, 1$\pm$0.05).

The parametric studies are run both without and with fractal geometry [13, 26,
27], to simulate random currents within the disk. This is conducted by apply a
weighting factor over the 53x53 grid, using the following function:

$\displaystyle{S(x,y)}$ $\displaystyle=$
$\displaystyle{[{{\Sigma}_{a=1}^{N}}{cos({\frac{2{\cdot}{\pi}{\cdot}{{\gamma}^{a}}{\cdot}{x}}{{\gamma}^{(2-D){\cdot}{a}}})}}]}{\cdot}{[{{\Sigma}_{b=1}^{N}}{cos({\frac{2{\cdot}{\pi}{\cdot}{{\gamma}^{b}}{\cdot}{y}}{{\gamma}^{(2-D){\cdot}{b}}})}}]}{\cdot}{Ra^{2{\cdot}{(D-1)}}},$
(32)

where _N_ = 1,000, $\gamma$ = 1.5, _D_ = 0.6, and _Ra_ = 1,000. This function
was then scaled, so as to have a mean of 1, and a standard deviation of 10.
The results are plotted in Figure 12.

Figure 12: Weighting function for Fractal Geometry.

The six parametric studies are as follows:

  * •

Study 1: No Fractal Geometry, No Rotating / Random Wobble (Figure 13)

  * •

Study 2: No Fractal Geometry, Rotating (Figure 14)

  * •

Study 3: No Fractal Geometry, Random Wobble (Figure 15)

  * •

Study 4: Fractal Geometry, No Rotating / Random Wobble (Figure 16)

  * •

Study 5: Fractal Geometry, Rotating (Figure 17)

  * •

Study 6: Fractal Geometry, Random Wobble (Figure 18)

In Figures 13-18, the measured electromagnetic and magnetic fields are plotted
in the (a) _X_ , (b), _Y_ , (c) _Z_ direction for both Maxwell’s and
Quaternion; the (d) magnitude of the magnetic field, for both Maxwell’s and
Quaternion; the ratio of Quaternion over Maxwell’s for the magnetic field in
the (e) _X_ , (f), _Y_ , (g) _Z_ direction, as well as (h) the absolute
magnitude.

Figure 13: Moving observer over eddy brake, Study 1 (No Fractal Geometry, No
Rotating / Random Wobble) results.

Figure 14: Moving observer over eddy brake, Study 2 (No Fractal Geometry,
Rotating) results.

Figure 15: Moving observer over eddy brake, Study 3 (No Fractal Geometry,
Random Wobble) results.

Figure 16: Moving observer over eddy brake, Study 4 (Fractal Geometry, No
Rotating / Random Wobble) results.

Figure 17: Moving observer over eddy brake, Study 5 (Fractal Geometry,
Rotating) results.

Figure 18: Moving observer over eddy brake, Study 6 (Fractal Geometry, Random
Wobble) results.

##  7 Conclusion

This effort demonstrated how quaternion geometry can better predict the
observed magnetic fields, as it overcomes the deficiency with the commonly
used Maxwell’s equations in separating polar and axial vectors. The
electromagnetic field E is a polar vector, whereas the magnetic field B is an
axial vector, where the direction of rotation remains the same even after the
axial vector is inverted; use of quaternion modeling helps to overcome this
error that leads to inverted magnetic results. The complex quaternion
mathematics were solved to a simplified approach, where the rotation matrix
was divided by its determinant, leading to identical results. Parametric
studies were analyzed, both of a series of charges, followed by a simulated
observer / target, and later an eddy current dynamic electric brake was
modeled and the magnetic properties at different positions were compared with
traditional Maxwell’s analysis. In all cases, discrepancies are observed,
suggesting that the quaternion approach is most accurate for magnetic modeling
and analysis of measured results from an observer.

Future work will address the mathematical augmentation of the geometric
algebra model with fractal components. The current work provided a fractal
”noise” model that was incorporated into the data. This was done for two
purposes: to produce the randomness actually found in nature, and determine if
the consequence would produce any instability or anomalous behaviors in the
quaternion equations derived for this research effort. There is sufficient
reason to expand the electromagnetic field equations to include a fractal
component which has been observed in both natural and synthetic environments.

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