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\author{Chris Payette, Neil Edelman, Margaux Brisco and Emily Hache}
\title{Hysteresis}
\date{3 February 2003}

\begin{document}

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\maketitle

\abstract{This experiment investigated the hysteresis curve of a toroi\-dal iron ring and found the
energy required to bring the material around a hysteresis loop for a given maximum magnetic field.
By varying the resistance in the circuit, the relation between magnetic flux density, $B$, and magnetic
field in the ring, $H$, was observed.  The graphs of the hysteresis curves $B$ {\it vs} $H$ show the lagging
effect due to a changing magnetizing force.  The area enclosed by these curves is equal to the work
per unit volume.  The plot of total work versus voltage indicated a linear relationship with slope
$9[1]\times 10^{-6} J V^{-1}$ and $y$ intercept $0.00011[1] J$. The Steinmetz coefficient was calculated for
each trial.  These were found to be constant within error at $56800[500]$.

--

This experiment investigated the hysteresis curve of a toroidal iron ring and found
the energy required to bring the material around a hysteresis loop for a given maximum
magnetic field. By varying the resistance in the circuit, the relation between current
and deflection was observed. The results show the lagging effect due to a changing
magnetizing force.

--

This experiment investigated the hysteresis curve of a toroidal iron ring and found
the energy required to bring the material around a hysteresis loop for a given maximum
magnetic field. By varying the resistance in the circuit, the relation between magnetic
flux density ($B$) and magnetic field in the ring ($H$) was observed. The graphs of the
hysteresis curves ($B$ {\it vs} $H$) show the lagging effect due to a changing magnetizing
force. The area enclosed by these curves is equal to the work per unit volume.  The
plot of total work versus voltage indicated a linear relationship with slope $X$ and $Y$
intercept $X$.  The Steinmetz coefficient was calculated for each trial.  These were
found to be constant within error at $57000$.}

\section{Introduction}

A demagnetized metal ring can easily be magnetized by the presence of an external magnetic field.
If we were to plot the magnetization of the ring, $B$, {\it vs} the applied magnetic field, $H$,
we would quickly see that the metal becomes saturated as in Figure \ref{mag-iron}. Let us suppose that
having magnetized the ring, we now wish to demagnetize it.

\fig{mag-iron}{Magnetization of Soft Iron \cite{lab-app-3}}

If the metal ring happens to be made of a ferromagnetic material, like Iron, the magnetization
will in fact be permanent, meaning that the sample will remain magnetized even as the external
magnetic field is brought back to zero. This is not what we would have expected; it is this phenomenon that
is known as hysteresis, which  can
be described by noting that $B$ lags behind the value of $H$.

There are several reasons that the hysteresis loop is important.
Firstly, the loop is unique to the metal that the ring is composed of.
In fact, hysteresis was once a method used to determine the identity of unknown metals and alloys.
However, a more practical application of the hysteresis data's uniqueness is to determine what kind of
situation a magnet made from a given substance is most suited for.  A very narrow hysteresis curve
indicates that a given substance is a temporary or soft magnet, while fatter curves mean a more
permanent, hard magnet.  The difference between these two types of magnets is that a soft magnet
will respond to a small magnetic field, while a hard magnet is permanent and will remain magnetized
after a strong magnetic field has been applied and then removed.  Soft magnets are often used in
circuits found in places like motors, generators and transformers where energy loss due to
magnetization should be minimized.  Magnetic materials which lie between hard and soft are
used a lot in recording devices where the stability of a hard magnet is required so that data
is not lost, but a strong magnetic field must be able to erase the previously recorded data if desired.

--

The hysteresis loop is a signature of a magnetic material, and is a plot of
magnetization against applied magnetic field. Slender loops indicate temporary, or
soft magnets, while fatter loops indicate more permanent, hard magnets. In this
experiment, the magnetic ring was initially unmagnetized. With a current send through
the ring, it produces

The coercivity of a magnet is the magnetic field required to reduce its magnetization
to zero. In 1743, Bernou\-lli developed the steel bar and horseshoe-shaped magnets who\-se
shape increased their coercivity, but modern magnetic materials have a coercivity
greater than their saturation magnetization and therefore resist self-demagnetization
in any shape. The strength of a permanent magnet is given by its maximum energy
product, twice the energy stored in the magnetic field around a magnet of optimum
shape. This value doubled every twelve years during the 20th century, but current
technology is at the point where another double would be difficult.

--

Hysteresis is used to describe the behavior of something that lags behind the
cause that effects it. For example, when a paper is cumpled it slowly returns
towards its original form once released but never completely regains it. When
magnetized, the iron ring shows this type of behavior as its magnetization
lags behind the magnetizing field applied. When the field is removed, the
material retains some of its magnetization. The area of the hysteresis loop is
a measure of the work required to overcome the this molecular magnetic
friction. A larger area shows that a material behaves as more of a permanent
magnet because it resists changing its magnetic alignment. The linear
relationship between voltage and work suggests that the energy required to
overcome the resistance of the iron core to changing polarity is proportional
to the strength of the magnetic field applied. The strength of the field
induced in the iron that opposes the magnetizing field increases
proportionally to its strength.

\section{Theory}

When a current is sent through the magnetizing coil, a magnetizing field, $H$ is created
in the iron and the ring is magnetized.  This gives rise to a magnetic flux $\phi$ (in Webers, $Wb$)
in the ring, which has density $B$ (in Teslas, $T$).  Changes in the current cause a change in $B$.
A second coil (search coil) is wound around the magnetizing coil and connected to a galvanometer.
When $B$ changes, a momentary current is created in the search coil, which passes through the
galvanometer and is directly proportional to the deflection in the galvanometer.  $H$
(in Ampere-turns per meter) is calculated using

\begin{equation}
H = \frac{N I}{l}
\end{equation}

where $N$ is the number of turns in the magnetizing coil, $I$ is the current (in Amperes, $A$) and
$l$ is the circumference of the iron ring ($m$).  $B$ is calculated using the equation,

\begin{equation}
B = \frac{N_s \phi_s d}{n A d_s}
\end{equation}

here $N_s \phi_s$ is the calibrated value for the Hibbert standard (in this case the value
is $1.72mWb$), $d$ is the galvanometer reading ($m$), $n$ is the number of turns on the
search coil, $A$ is the area of the search coil ($m$) and $d_s$ is the initial
half-amplitude reading of the galvanometer ($m$).

To obtain a hysteresis loop, the magnetic field, $H$, is increased to saturation, then
decreased to zero, then increased to saturation in the reverse polarity, then brought back
to zero and finally fully re-magnetized in the original direction.  $H$ is increased and
decreased gradually (by increasing and decreasing the resistance in the circuit) and the
deflection of the galvanometer needle (used to calculate $B$) is recorded with every change
in $H$.  When $B$ is plotted against $H$, it is found that after the first magnetization,
$B$ lags behind $H$.  This is hysteresis and is the result of a certain permanent magnetism
in the iron.  The area enclosed by the $B$--$H$ curve represents the work per unit volume.

The work done throughout the whole process can be found by,

\begin{equation}
W = A V
\label{wav}
\end{equation}

where $A$ is the area of the hysteresis loop ($AWbm^{-3}$) and $V$ is the volume of the
ring ($m^3$).  The loss of energy (the work) is due to molecular magnetic friction in the ring.

A relationship, discovered by Charles Steinmetz, exists between the area of the loop and
the maximum applied magnetic flux density.  The relationship is,

\begin{equation}
A = \eta \cdot (B_{max})^{1.6}
\label{anb}
\end{equation}

where $A$ is the area of the $B$--$H$ curve in $AWbm^{-3}$, $B_{max}$ is the maximum value
of the flux density ($T$), and $\eta$ is the Steinmetz coefficient.  The Steinmetz
coefficient is unitless, because it represents a strictly empirical relationship.

--

With a current send through the ring, a magnetizing field, H, which magnetizes the iron.
This results in a high flux density, B, in the ring. Changes in this current cause a
momentary current in a second coil around the ring which is connected to a galvanometre
used to measure the flux density.

\section{Apparatus}

\begin{center}
\begin{table}[t]
\small
\begin{tabular}{l}
Ballistic Galvanometer-Wall-mounted, Leeds \& Northrup (\#11)\\
Damping Key (\#7)\\
Hibbert standard flux -- S2H  (\#6)\\
Plug Resistance Box (\#2)\\
Resistance board of 14 resistors in series (\#4)\\
Rowland Ring, cast iron $F$ -- $C$ (\#4)\\
Simpson DC Ammeter, Model 19, $0$ -- $1 A$\\
Power Supply, $0$ -- $24 V$ (\#9) trials 1,2,3  (\#1) trials 4,5,6,7,8,9\\
Reversing Switch $S$ -- $R$ (S8H)
\end{tabular}
\caption{Equipment}
\label{equipment}
\end{table}
\end{center}

\section{Procedure}

\subsection{Demagnetizing the Coil}

Hook up the coil to the demagnetizer between every trial, according to
Figure \ref{demag-circuit}.  Start at the left of the rheostat, the coil is demagnetized
when it has reached the right.  This should take about ten minutes.

\fig{demag-circuit}{Demagnetizing Circuit}

\subsection{Setup and Trials}

Wire the circuit according to Figure \ref{demag-circuit}.  Once the galvanometer
has settled, adjust it to zero.  Drop the Hibbert standard and record the displacement.
Always include information on the sense of the deflection as the sign of the data.  On
our equipment, this was around $12 cm$.  This measurement is the half-amplitude reading
of the galvanometer. With the reversing switch set to OPEN and all the switches on the
resistance board open, the galvanometer should be at zero.

Turn on the power supply to $12 V$.  Set the reversing switch to DIRECT and immediately
record the current and deflection.  One by one, close the switches from left to right,
recording the current and deflection for each resistor. This is point {\bf a} (Figure \ref{hys}).
One by one, open the switches from right to left, recording each measurement.
With all the switches open, set the reversing switch to OPEN and record to deflection
(the current is zero).  This is point {\bf b} (Figure \ref{hys}).

Taking current and deflection reading between each step, set the reversing switch to
REVERSE and successively close all the resistance switches from left to right.  When they
are all closed, this is point {\bf c} (Figure \ref{hys}).  Now open the switches
from right to left.  Set the reversing switch to OPEN.  This is point {\bf d} (Figure \ref{hys}).

Set the reversing switch to DIRECT then close all the switches from left to right.
This point should be close to point {\bf a} (Figure \ref{hys}).

Repeat steps 1, 2 and 3 for the following voltages: $6$, $7$, $9$, $10.5$, $12$, and $13.5$ volts.

\subsection{Plotting Graphs}

For each trial, use the half-amplitude displacement to calculate $k_b$ and use it, along with
the data collected to calculate and plot $B$ versus $H$.  The displacement used in
calculating $B$ is cumulative so each displacement measurement is added to the sum
of all the preceding measurements, taking the sign of each reading into account. The
area enclosed in the Hysteresis loop represents the energy lost per unit volume.

\fig{hys}{Sketch of Hysteresis Loop}

\subsection{Data and Analysis}

\begin{center}
\begin{table}[t]
\begin{tabular}{llrl}
Calibrated Hibbert Standard&  $N_s\phi_s$& $1.72$&          $mWb$\\
Number of Turns Search Coil&  $n$&         $24$&            \\
Diameter of the Ring&         $d$&         $0.16$&          $m$\\
Area of Search Coil&          $A$&         $0.020$&         $m^2$\\
Ballistic Constant&           $kb$&        $27.85\pm 0.05$& $mT m^{-1}$\\
No. of Turns on Rolland Ring& $N$&	       $2385$&          \\
Circumference of Ring&        $l$&	       $0.50265$&       $m$\\
Volume of Ring&               $V$&         $0.0000503$&     $m^3$
\end{tabular}
\caption{Equipment Data}
\label{equipment-data}
\end{table}
\end{center}

The data in Table \ref{equipment-data} summarizes the values of various properties of the equipment. These
values were all used in order calculate $B$ and $H$ from the measured current
and deflection. (See Appendix 2 for sample calculations.)

The extensive data and all the plots of $B$ vs. $H$ (hysteresis loops) can
be found in Appendix 1 (Tables A1--A7 and graphs A1--A7).  See Appendix 3 for
the raw data. As a prototypical sample, the plot for the $10.10 V$ trial can be seen below.

\fig{trial-9}{Trial 9 ($10.10 V$)}

\fig{trial-9-e}{Analysis of Error on B (Trial 9)}

The values of $B$ are based on the sum of the previous deflection measurements, thus
the error on each successive point on the plots grows.  (See \ref{trial-9-e} and also
see sample calculations in Appendix 2 for a detailed explanation of how this error is
calculated.)  This effect is shown in Figure \ref{mag-iron} by the increasing size of
the error bars around the hysteresis loop.  The integrals of the hysteresis loops
(area of the loops) were needed to find the work.  These values were calculated by
interpolating between points with a cubic spline and connecting the beginning and end
of the loop with a line segment to form a closed curve.  Details can be found in
Appendix 2.  These areas are multiplied by the volume of the ring to find work (Equation \ref{wav}),
and the results are summarized in Table \ref{area-work}.

\begin{center}
\begin{table}[t]
\begin{tabular}{rrrrrr}
Trial&	Voltage ($V$)&	Area ($A Wb m^{-1}$)&	$\sigma_{Area}$&	Work ($J$)&	$\sigma_{Work}$\\
\hline
1&	11.89&	4.3&	0.2&	0.000214&	0.000008\\
2&	9.00&	3.81&	0.14&	0.000191&	0.000007\\
3&	6.02&	3.2&	0.1&	0.000162&	0.000005\\
5&	10.48&	4.17&	0.15&	0.000210&	0.000008\\
6&	13.50&	4.5&	0.2&	0.000227&	0.000009\\
8&	7.00&	3.4&	0.1&	0.000172&	0.000005\\
9&	10.10&	3.9&	0.1&	0.000197&	0.000006
\end{tabular}
\caption{Area and Work Data}
\label{area-work}
\end{table}
\end{center}

The calculations required for determining the error on the area are very detailed and
can be found in Appendix 2.  Uncertainty in calculating area is due partly to the
discrepancies of the endpoints.  Since six of the seven used trials yielded graphs
where the endpoints did not meet, the additional area had to accounted for.  Some
graphs crossed over each other, leaving a closed loop and an additional open-ended
area, while the other graphs left a single open loop.  We drew a straight line joining
point a (Figure \ref{hys}) to the final point in every trial, then took the total
enclosed area.

The error on the work is depends on the error on the area (see Appendix 2).  The
plot of the work {\it vs} voltage is presented in Figure \ref{wth}.  This graph shows
a linear relationship between the voltage and the work with all the points within
experimental error of the linear least-squares fit. Once the area of each curve
was found, the Steinmetz constant could be calculated using Equation \ref{anb}.
The results are summarized in Table \ref{steinmetz}.

\begin{center}
\begin{table}[t]
\begin{tabular}{rrrrr}
Voltage&	Area ($A Wb m^{-3}$)&	$B_{max}$ ($T$)&	Steinmetz's constant&	$\sigma_{Steinmetz's constant}$\\
\hline
11.89&	4.2574&	0.00265&	57000&	2000\\
9.00&	3.8062&	0.00252&	55000&	2000\\
6.02&	3.2146&	0.00220&	58000&	2000\\
10.48&	4.1725&	0.00258&	58000&	2000\\
13.50&	4.5207&	0.00275&	57000&	2000\\
7.00&	3.4168&	0.00234&	55500&	1600\\
10.10&	3.9205&	0.00245&	59000&	2000
\end{tabular}
\caption{Steinmetz's Constant}
\label{steinmetz}
\end{table}
\end{center}

The average value for the Steinmetz constant was found to be $56800\pm 500$, and
it is represented by the straight line on Graph \ref{foo}. The error value is based
on a weighted average of the data; see Appendix \ref{foo} for details.

\section{Discussion}

Hysteresis is used to describe the behavior of something that lags behind the cause
that affects it. For example, when a paper is crumpled it slowly returns towards its
original form but doesn't completely regain it once released. When magnetized, the
iron ring shows this type of behavior as its magnetization lags behind the magnetizing
field applied. When the field is removed, the material retains some of its magnetization.
The area of the hysteresis loop is a measure of the work required to overcome the
molecular magnetic friction. A larger area shows that a material behaves as more of
a permanent magnet because it resists changing its magnetic alignment. The linear
relationship between voltage and work suggests that the energy required to overcome
the resistance of the iron core to changing polarity is proportional to the strength
of the magnetic field applied. The strength of the field induced in the iron that
opposes the magnetizing field increases proportionally to its strength.

There were two main difficulties encountered when taking measurements.  The first was
in reading the galvanometer.  Since the measurement is viewed, it is dependent on the
person who reads it.  The first difficulty here is that it is a moving scale which
therefore needs to be read at the right time.  The other is that there is a discrepancy
between different people in reading very small numbers (what some people would read as
four others may read as five, as well as a probable psychological tendency for one to read
certain numbers more often than others).

The second difficulty encountered was reading the ammeter.  The current reading should
be taken immediately after the switch has been opened or closed because the current values
jump, then continue to climb a few more seconds.  The reason this is difficult is because
sometimes the reading changes very quickly while the digital meter displays the current at
discrete intervals. It was therefore difficult to be certain that the readings were taken
at the right time. In order to get accurate readings, we would need a method of reading
the first current displayed, before it changes, such as a graphical meter that plotted the
current.  In order to minimize this random error we always had the same person reading the
current although this contributed towards more systematic error.  Another source of
systematic error was the equipment we used. We had to assume the values given were absolute
since no error was given.  It is also possible that the galvanometer or the Hibbert flux
standard were not calibrated.  Since they are quite old, the galvanometer may not have been
producing accurate deflections.  In order to ensure that the equipment was accurate, we
tested the resistance box to ensure it was $130\Omega$ as indicated but we could also
have tested the voltmeter and ammeter.  One large source of error would be the mislabeling
of the flux standard.  While it was labeled as S2H which corresponded with a value of
$1.72 mWb$ as our value for $N_s \varphi_s$.  A different value here would have a significant
effect on all of our calculated values.

Due to a fluctuating voltage source, we had to change our power supply after the first
three trials.  We do not know how constant it was during these trials and therefore this
may account for some error in these trials.

\section{Conclusion}

This experiment measured the magnetization curve of an iron coil -- known as a
hysteresis loop -- and analyzed its shape and properties.  When a magnetic field is
applied to the coil, it causes it to become magnetized to a saturation point; the
polarity of the current is then reversed twice, magnetizing the iron to saturation
in opposite polarities each time and thereby completing the loop. The area of this
loop is equal to the work done per unit volume of the coil, indicating the loss of
energy due to molecular magnetic friction.  Analysis showed that change in the work
was proportional to change in the voltage with a coefficient of
$9[1]\times 10^{-6} J V^{-1}$. The fit line of work {\it vs} voltage should pass
through zero since no voltage implies no work; instead our fit had an intercept of
$0.00011[1] J$.  This was caused by systematic error because the fit line
passed cleanly through the points within the calculated random error.  Possible
explanations are additional energy loss not caused by hysteresis, small equipment
miscalibrations, and ambient magnetic fields.  Also, the endpoints of our loops
were not cleanly defined -- the most probable significant source of systematic error.
We ran the experiment to a certain resistance and assumed that this accomplished
the magnetic saturation, when in fact our high readings and crossing loops suggest
that perhaps this may have passed the saturation point. In theory this would not
affect the area, but in practice the lines after the saturation point do not coincide,
leading to a consistently high area.

There are several points where the results from this experiment could be improved.
The galvanometer should have been read to one tenth of a millimeter every reading.
Equipment could have been calibrated against a known standard to ensure accuracy.
Original values read from the equipment were taken without error; some of these values
might have been remeasured.  Because the error on the measured deflection was cumulative,
accuracy decreased with more measurements.  On the other hand, more measurements gave
a greater resolution to the curve.  The experiment could be repeated once with a fewer,
optimal number of points to give the most accurate result, then again with more, smaller
resistance changes to give a more precise shape of the curve.  Combining these two
results would then achieve greater accuracy and precision.

--

This experiment measured the magnetization curve of an iron coil -- known as a
hysterisis loop -- and analyzed its shape and properties. When a magnetic field is
applied to the coil, it causes it to become magnetized to a saturation point; the
polarity of the current is then reversed and the coil magnetized to saturation
in the opposite polarity. After being reversed again, the coil returns to saturation
in the original polarity, completing the loop. With voltages of the order of $10V$
and a resistance of $130\Omega$, the loops showed a magnetic saturation magnitude of
the order of $2.5mT$ at a maximum magnetizing field magnitude of the order of
$4000 A m^{-1}$. From this information, a Steinmetz coefficient of $56800\pm 500$
was found for the iron ring.

The area of this loop is equal to the work done per unit volume of the coil, indicating
the loss of energy due to molecular magnetic friction. Analysis showed that change in
the work was proportional to change in the voltage with a coefficient of
$8.89\times 10^{-6}J V^{-1}$. We expect that the fit line of work vs. voltage should pass
through zero since with zero voltage there is no work done; instead our fit had an
intercept of $0.0001 J$. It is expected that this is caused by a systematic error
because the fit line passed cleanly through the points within the calculated random
error but was offset by this amount. Due to the small values, this error could be
caused by small equipment miscalibrations or even ambient magnetic fields. Also,
the endpoints of our loops were not cleanly defined, the most probable significant
source of systematic error. We ran the experiment to a certain resistance and assumed
that this accomplished the magnetic saturation, when in fact our high readings and
crossing loops suggest that perhaps this may have passed the saturation point.
In theory this would not affect the area, but in practice the lines after the
saturation point do not coincide, leading to a consistently high area.

There are several points where the results from this experiment could be improved.
The galvanometre should have been read to one tenth of a millimetre every reading,
rather than only the first set. Equipment, specifically the galvanometre, could
have been calibrated against a known standard to ensure accuracy. Original values
read from the equipment were taken without error; since the equipment was old, some
of these values might have been remeasured. Because the error on the measured
deflection was cumulative, accuracy decreased with more measurements. On the other
hand, more measurements gave a greater resolution to the curve. The experiment
could be repeated once with a fewer, optimal number of points to give an accurate
result, then again with more, smaller resistance changes to give the more precise
shape of the curve. Combining these two results might achieve more accuracy and
precision to the loop.

\nocite{lab}
\nocite{lab-app-3}
\nocite{lucas}
\nocite{trinity}
\nocite{winch}
\nocite{steinmetz}
\nocite{harnwell}
\bibliography{hysteresis}

\section{Acknowledgements}

We would like to thank Mrs. E. Engelberg for her continuous aid and advice, Mr.
Mario Della Neve for his help with the apparatus as well as Prof. M. Hilke for his
advice on the analysis.  A special thanks to Mrs. Engelberg's number crunchers for
answering questions about Steinmetz coefficient units.

\onecolumn

\appendix

\section{Sample Calculations}

Trial 1 calculations; examples using the first data point --
$V = 11.89V$; $d_s = 12.80cm$; $d_1 = 0.2cm$; $I_1 = 0.017A$:

\begin{align*}
k_b &= \frac{N_s \varphi_s}{n A d_s} \\
k_b &= \frac{1.72mW}{(24)(\pi 0.0064 m^2)(0.128 m)} \\
k_b &= 27.84693 mW m^{-3}
\end{align*}

\begin{align*}
\sigma_{d_s} &= 0.00025 m \\
\sigma_{k_b} &= k_b \cdot \left(\frac{\sigma_{d_s}}{d_s}\right) \\
\sigma_{k_b} &= 27.84693 \cdot \left(\frac{0.00025}{0.1280}\right) \\
\sigma_{k_b} &= 0.05 \\
k_b &= 27.85\pm 0.05 mW m^{-3}
\end{align*}

\begin{align*}
d_n &= \sum_{m = 1}^n d_m \\
\sigma_{d_n} &= \sqrt{n(\sigma_{d_s})^2} = \sigma_{d_s}\sqrt{n} \\
\sigma_{d_1} &= 0.00025 m \cdot \sqrt{1} \\
\sigma_{d_1} &= 0.00025 m
\end{align*}

\begin{align*}
B_n &= k_b\cdot d_n \\
B_1 &= 27.84693 mW m^{-3}\times 0.002 m \\
B_1 &= 0.0557386 mT
\end{align*}

\begin{align*}
\sigma_{B_n} &= B_n \sqrt{
	\left(\frac{\sigma_{k_b}}{k_b}\right)^2 +
	\left(\frac{\sigma_{d_n}}{d_n}\right)^2} \\
&= k_b d_n \sqrt{
	\left(\frac{\sigma_{k_b}}{k_b}\right)^2 +
	i \left(\frac{\sigma_{d_n}}{d_n}\right)^2} \\
\sigma_{B_1} &= 27.84693 mW m^{-3} \times 0.002 m \sqrt{
	\left(\frac{0.05 mW m^{-3}}{27.84693 mW m^{-3}}\right)^2 +
	(1) \left(\frac{0.00025 m}{0.1280 m}\right)^2} \\
&= 0.0001 mT \\
B_1 &= 0.0557\pm 0.001 mT
\end{align*}

\begin{align*}
H &= \frac{N I}{l} \\
H_1 &= \frac{2385 \cdot 0.017 A}{\pi 0.16 m} \\
H_1 &= 80.6617 A m^{-1}
\end{align*}

\begin{align*}
\sigma_I &= 0.001 A \\
\sigma_H &= \frac{N}{l}\sigma_I \\
\sigma_H &= \frac{2385}{\pi 0.16 m} \cdot 0.001 A \\
\sigma_H &= 5 A m^{-1} \\
H_1 &= 81\pm 5 A m^{-1}
\end{align*}

(H is in Ampere-turns per metre.)

Computing the accumulated error on $B_n$:

\begin{align*}
\sigma_{B_n} &= \sqrt{i \sigma_B^{\phantom{B}2}} \\
\sigma_{B_1} &= \sqrt{1 \cdot (0.00025 m)^2} \\
&= 0.00025 m
\end{align*}

\begin{wrapfigure}{r}{3in}
\includegraphics{linear-cubic.jpg}
\end{wrapfigure}
The top and bottom curves of the hysterisis loop were fit to parametric Catmull-Rom splines.
This is similar to connecting a line between each point, but instead a continuous cubic curve
is smoothly interpolated between them. For some parameter $s$ from zero to one used to
interpolate $p$ between two points, $p_n$ and $p_{n+1}$, where
$p_n = \begin{pmatrix} H_n \\ B_n \end{pmatrix}$.

Linear interpolation:

\begin{equation*}
p = \begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix}
\begin{pmatrix} s \\ 1 \end{pmatrix}
\cdot \begin{pmatrix} p_n \\ p_{n+1} \end{pmatrix}
\end{equation*}

Hermite cubic interpolation that we used:

\begin{equation*}
p = \begin{pmatrix} 2 & -3 & 0 & 1 \\ -2 & 3 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 1 & -1 & 0 & 0 \end{pmatrix}
\begin{pmatrix} s^3 \\ s^2 \\ s \\ 1 \end{pmatrix}
\cdot \begin{pmatrix} p_n \\ p_{n+1} \\ t_n \\ t_{n+1} \end{pmatrix}
\end{equation*}

where $t_1$ and $t_2$ are the tangents at each point. These are calculated as:

\begin{equation*}
t_n = \frac{1}{2}(p_{n+1} - p_{n-1})
\end{equation*}

except at the endpoints where $p_n$ is replaced for the overflowing value and the result is doubled.

For example, the curve between points 16 and 17 for trial one was parameterized as:

\begin{equation*}
p_{16} = \begin{pmatrix} 2 & -3 & 0 & 1 \\ -2 & 3 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 1 & -1 & 0 & 0 \end{pmatrix}
\begin{pmatrix} s^3 \\ s^2 \\ s \\ 1 \end{pmatrix}
\cdot \begin{pmatrix}
	\begin{pmatrix} 3795.845 A m^{-1} \\ 2.589765 mT \end{pmatrix}\\
	\begin{pmatrix} 3103.104 A m^{-1} \\ 2.53407 mT \end{pmatrix}\\
	\frac{1}{2}\begin{pmatrix} 2434.086 A m^{-1} \\ 2.2.45053 mT \end{pmatrix} -
		\frac{1}{2}\begin{pmatrix} 4332.009 A m^{-1} \\ 2.645459 mT \end{pmatrix} =
		\begin{pmatrix} -614.452 A m^{-1} \\ -0.05569 mT \end{pmatrix}\\
	\frac{1}{2}\begin{pmatrix} 3795.845 A m^{-1} \\ 2.589765 mT \end{pmatrix} -
		\frac{1}{2}\begin{pmatrix} 2434.086 A m^{-1} \\ 2.45053 mT \end{pmatrix} =
		\begin{pmatrix} -680.88 A m^{-1} \\ -0.06962 mT \end{pmatrix}
\end{pmatrix}
\end{equation*}

To begin calculating the area of between the curves, each segment of the spline is integrated:

\begin{align*}
A_n &= \int_0^1 H'Bds \\
&= \int_0^1
	\left[\frac{d}{ds}\left(
		(2s^3 - 3s^2 + 1) H_n + (-2s^3 + 3s^2) H_{n+1} + (s^3 - 2s^2 + s) t H_n + (s^3 - s^2) t H_{n+1}
	\right)\right] \cdot \\
&\phantom{=\int_0^1}\quad
	\left(
		(2s^3 - 3s^2 + 1) B_n + (-2s^3 + 3s^2) B_{n+1} + (s^3 - 2s^2 + s) t B_n + (s^3 - s^2) t B_{n+1}
	\right)
ds \\
&= \frac{1}{10}(-H_ntB_n + H_ntB_{n+1} + tH_nB_n - tH_nB_{n+1} + H_{n+1}tB_n - H_{n+1}tB_{n+1} - tH_{n+1}B_n + tH_{n+1}B_{n+1}) + \\
&\phantom{=}\quad \frac{1}{2}(-H_nB_n - H_nB_{n+1} + H_{n+1}B_n + H_{n+1}B_{n+1}) + \\
&\phantom{=}\quad \frac{1}{60}(tH_ntB_{n+1} - tH_{n+1}tB_n)
\end{align*}

For example, the segment 16 to 17 in trial one had an area of:

\begin{flalign*}
A_{16} &= \frac{1}{10}(
	- (3595.845Am^{-1})(-0.5569mT) + (3595.845Am^{-1})(-0.06962mT) \\
&\phantom{=\frac{1}{10}(}
	+ (-614.452Am^{-1})(2.589765mT) - (-614.452Am^{-1})(2.534071mT) \\
&\phantom{=\frac{1}{10}(}
	+ (3103.104Am^{-1})(-0.5569mT) - (3103.104Am^{-1})(-0.06962mT) \\
&\phantom{=\frac{1}{10}(}
	- (-680.88Am^{-1})(2.589765mT) + (-680.88Am^{-1})(2.534071mT)) + \\
&\phantom{=}
\frac{1}{2}(
	- (3595.845Am^{-1})(2.589765mT) - (3595.845Am^{-1})(2.534071mT) \\
&\phantom{=\frac{1}{2}(}
	+ (3103.104Am^{-1})(2.589765mT) + (3103.104Am^{-1})(2.534071mT)) + \\
&\phantom{=}
\frac{1}{60}(
	(-614.452Am^{-1})(-0.06962mT) - (-680.88Am^{-1})(-0.5569mT)
) \\
&= -1775.02 mT A m^{-1}
\end{flalign*}

\begin{wrapfigure}{r}{3in}
\includegraphics{joined.jpg}
\end{wrapfigure}
Because the endpoints of the curve don't line up, a line segment needs to be added to close the loop.
This adds another integral to be computed. Note that the signs used in this calculation make this segment
add with the same sign as the start of the top curve:

\begin{align*}
A_t &= \int_{H_{bottom}}^{H_{top}} \left[
	B_{bottom} + \frac{B_{top} - B_{bottom}}{H_{top} - H_{bottom}} (h - H_{bottom})
\right] dh \\
&= \frac{1}{2} (H_{top} - H_{bottom}) (B_{top} + B_{bottom})
\end{align*}

For trial one:

\begin{align*}
A_t &= \frac{1}{2} \left(
	(4332.009Am^{-1}) - (4469.608Am^{-1})
\right) \left(
	(2.645459mT) + (2.81254mT)
\right) \\
&= -375.5087 mT A m^{-1}
\end{align*}

\begin{wrapfigure}{r}{3in}
\includegraphics{self-intersecting.jpg}
\end{wrapfigure}
This gives a closed curve, but some of our loops were not simple ({it ie} they
were self-intersecting.) Because we only consider positive area, the `top'
and `bottom' of the curves must be swapped over intervals where they cross
over. This leaves calculating the area for the interval on which the two curves
intersect. Finding the intersection of two cubics then integrating between
variable parametres turns out to be a very complicated calculation. Since part
of the area on either side of the intersection will cancel out, the area of
these intervals will be much smaller than that of other intervals. Doing linear
interpolation for this one segment of the spline has no significant effect on
the net result (doing linear interpolation on the entire curve has only a small
effect.)

First, the indices of the points ${\bf a}$, ${\bf a'}$, ${\bf z} = {\bf a} + 1$,
and ${\bf z'} = {\bf a'} + 1$ on either side of the intersection are determined
by inspection of the curve; next, $H$ of the intersection point, $p$, is
calculated:

\begin{align*}
H_p &= \frac{
	B_{a'} - B_a + \left(\frac{
		B_z - B_a
	}{
		H_z - H_a
	}\right) H_a - \left(\frac{
		B_{z'} - B_{a'}
	}{
		H_{z'} - H_{a'}
	}\right) H_{a'}
}{
	\left(\frac{
		B_z - B_a
	}{
		H_z - H_a
	}\right) - \left(\frac{
		B_{z'} - B_{a'}
	}{
		H_{z'} - H_{a'}
	}\right)
}
\end{align*}

For trial one, ${\bf a}$ was 18 and ${\bf a'}$ was 72:

\begin{align*}
H_p &= \frac{
	B_{72} - B_{18} + \left(\frac{
		B_{19} - B_{18}
	}{
		H_{19} - H_{18}
	}\right) H_{18} - \left(\frac{
		B_{73} - B_{72}
	}{
		H_{73} - H_{72}
	}\right) H_{72}
}{
	\left(\frac{
		B_{19} - B_{18}
	}{
		H_{19} - H_{18}
	}\right) - \left(\frac{
		B_{73} - B_{72}
	}{
		H_{73} - H_{72}
	}\right)
} \\
&= \frac{
	(2.478377mT) - (2.45053mT) + \left(\frac{
		(2.394836mT) - (2.45053mT)
	}{
		(2040.267Am^{-1}) - (2434.086Am^{-1})
	}\right) (2434.086Am^{-1}) - \left(\frac{
		(2.617612mT) - (2.478377mT)
	}{
		(3112.593Am^{-1}) - (2448.32Am^{-1})
	}\right) (2448.32Am^{-1})
}{
	\left(\frac{
		(2.394836mT) - (2.45053mT)
	}{
		(2040.267Am^{-1}) - (2434.086Am^{-1})
	}\right) - \left(\frac{
		(2.617612mT) - (2.478377mT)
	}{
		(3112.593Am^{-1}) - (2448.32Am^{-1})
	}\right)
} \\
&= 2256.624Am^{-1}
\end{align*}

Calculate the integral, swapping signs on either side of point ${\bf p}$:

\begin{align*}
A_I &= \int_{H_a}^{H_p} \left[ B_a + \frac{B_z - B_a}{H_z - H_a}(h - H_a) \right] dh -
\int_{H_p}^{H_z} \left[ B_a + \frac{B_z - B_a}{H_z - H_a}(h - H_a) \right] dh -
\int_{H_{a'}}^{H_p} \left[ B_a + \frac{B_z - B_a}{H_z - H_a}(h - H_a) \right] dh +
\int_{H_p}^{H_{z'}} \left[ B_a + \frac{B_z - B_a}{H_z - H_a}(h - H_a) \right] dh \\
&= \frac{1}{2} (B_{a'}H_{z'} + B_{z'}H_{z'} + B_{a}H_{z} + B_{z}H_{z} - B_{a}H_{a} - B_{z}H_{a} - B_{a'}H_{a'} - B_{z'}H_{a'})
\end{align*}

For trial one:

\begin{align*}
A_I &= \frac{1}{2} (
	(2.478377mT)(3112.593Am^{-1}) + (2.617612mT)(3112.593Am^{-1}) +
	(2.45053mT)(2040.267Am^{-1}) + (2.394836mT)(2040.267Am^{-1}) -
	(2.45053mT)(2434.086Am^{-1}) - (2.394836mT)(2434.086Am^{-1}) -
	(2.478377mT)(2448.32Am^{-1}) - (2.617612mT)(2448.32Am^{-1})
) \\
&= 22.99033 mT A m^{-1}
\end{align*}

Note that integrals along the top curve give negative results because we
are integrating from a higher $H$ to a lower $H$. With this in mind, the
final area function is given by summing the integrals under each segment
and the closing segment. If the curve intersects, the two integrals $A_a$
and $A_a'$ are replaced by the area for the intersecting segment, $A_I$:

\begin{align*}
\oiint_{loop} DH \cdot DB &= \begin{cases}
	-\sum_{i=1}^{\mbox{end}} A_i - A_t, & \mbox{simple} \\
	\sum_{i=1}^{a-1} A_i \sum_{i=a+1}^{a'-1} A_i \sum_{i=d+1}^{\mbox{end}} A_i + A_I + A_t, & \mbox{intersecting}
\end{cases}
\end{align*}

Since trial one was intersecting, its area was:

\begin{align*}
\oiint_{loop} DH \cdot DB &=
	(-1403.27 mTAm^{-1} - 1775.02 mTAm^{-1} - 1667.97 mTAm^{-1}) -
	(-864.89 mTAm^{-1} + \ldots + 839.90 mTAm^{-1}) +
	(1693.71 mTAm^{-1} + 2030.30 mTAm^{-1} + 1656.57 mTAm^{-1}) +
	(22.99033 mTAm^{-1}) +
	(-375.5087 mTAm^{-1}) \\
&= 4257.42 mTAm^{-1}
\end{align*}

\begin{wrapfigure}{r}{3in}
\includegraphics{error.jpg}
\end{wrapfigure}
The error on the area is calculated by first finding the errors on each
term of the area sum. Using the method of partial derivatives on these
formulae directly yields error equations several scores of printed pages
in length for each term. Because this is an error calculation, approximations
are definitely appropriate. For finding the error on the integral between
each point, we use the following approximation:

\begin{align*}
A_n \int_{H_n}^{H_{n+1}} h B_n dh &= B_n (H_{n+1} - H_n)
\end{align*}

This gives the following error calculation (the error on $H$ is constant
for each trial):

\begin{align*}
\sigma_{An} &= \sqrt{
	\left(\sigma_{Bn}\frac{\del A}{\del B_n}\right)^2 +
	\left(\sigma_{H}\frac{\del A}{\del H_n}\right)^2 +
	\left(\sigma_{H}\frac{\del A}{\del H_{n+1}}\right)^2
} \\
&= \sqrt{
	2 \sigma_H^{\phantom{H}2} \sigma_Bn^{\phantom{Bn}2} +
	\sigma_{Bn}^{\phantom{Bn}2}(H_{n+1} - H_n)^2
}
\end{align*}

In trial one, for example, one such error was:

\begin{align*}
\sigma_{A16} &= \sqrt{
	2 (4.744807 Am^{-1})^2 (0.028303 mT)^2 +
	(0.028303 mT)^2 \left(
		(3103.104 Am^{-1}) -
		(3795.845 Am^{-1})
	\right)^2
} \\
&= 26.19918 mTAm^{-1}
\end{align*}

The total error on the area is computed by taking the error on the sum
of the integrals of each segment. Error on the closing and intersecting
segments is ignored because they make up a very small portion of the
total area (which is compensated for by including error on the segments
they would have replaced.) The final approximation is well within 5\% of
the error that would have been obtained without taking any approximations:

\begin{align*}
\sigma_A &= \sqrt{\sum \sigma_{An} ^2}
\end{align*}

For trial one this was:

\begin{align*}
\sigma_A &= \sqrt{(23.06029 Am^{-1})^2 + \ldots + (40.34052 Am^{-1})^2} \\
165.2311 mTAm^{-1}
\end{align*}

Work Calculations - volume:

\begin{align*}
V &= 2 \pi^2 R^2 r
&= (\mbox{cross-sectional area} \cdot \mbox{circumference}) \\
&= (1\times 10^{-4} m^2) \cdot (0.16 m \pi) \\
&= 5.02655\times 10^{-5} m^3
\end{align*}

Work (where $V$ is volume):

\begin{align*}
W &= V \cdot \mbox{area of hysterisis loop} \\
&= (5.02655\times 10^{-5} m^3)(4.25742 AWbm^{-1}) \\
&= 2.14\times 10^{-4} J
\end{align*}

Error on work:

\begin{align*}
\sigma_W &= W \cdot \sqrt{
	\left(
		\frac{\sigma_{\mbox{area}}}{\mbox{area}}
	\right)^2 + \left(
		\frac{\sigma_V}{V}
	\right)^2
} \\
&= 2.14\times 10^{-4} J \cdot \sqrt{
	\left(
		\frac{0.165231 AWbm^{-1}}{4.25742 AWbm^{-1}}
	\right)^2 + \left(
		\frac{0 m^3}{5.02655\times 10^{-5} m^3}
	\right)^2
} \\
&= 8\times 10^{-6} J
\end{align*}

Steinmetz constant for each trial:

\begin{align*}
\eta_n &= \frac{\mbox{area}}{B_{\mbox{max}}^{\phantom{{\mbox{max}}}} 1.6} \\
\eta_1 &= \frac{4.25742 AWbm^{-1}}{(0.002645 T)^{1.6}} \\
&= 56642.824
\end{align*}

Individual error on Steinmetz constant:

\begin{align*}
\sigma_{\eta n} &= \eta_n \cdot \sqrt{
	\left(
		\frac{\sigma_{\mbox{area}}}{\mbox{area}}
	\right)^2 + \left(
		\frac{\sigma_{B\mbox{max}}}{B_{\mbox{max}}}
	\right)^2
} \\
\sigma_{\eta 1} &= 56642.824 \cdot \sqrt{
	\left(
		\frac{0.165231 AWbm^{-1}}{4.25742 AWbm^{-1}}
	\right)^2 + \left(
		\frac{2.74533\times 10^{-5} T}{0.0026455 T}
	\right)^2
} \\
&= 2275.54783
\end{align*}

Error of Steinmetz constant:

\begin{align*}
\sigma_{\eta} &= \sqrt{\sum \frac{1}{\sigma_{\eta n}^2}} \\
&= \sqrt{\frac{1}{(2275.547)^2} + \ldots + \frac{1}{(1835.748)^2}} \\
&= 535.665
\end{align*}

Weighted average of Steinmetz constant:

\begin{align*}
\eta &= \sigma_{\eta}^{\phatom{\eta} 2} \sum \frac{\eta^n}{\sigma_{\eta n}^2} \\
&= (535.66)^2 \cdot \left(
	\frac{(56642.824)}{(2275.547)^2} +
	\ldots +
	\frac{(58982.244)}{(1835.748)^2}
\right) \\
&= 56783.740
\end{align*}

\end{document}

The top and bottom curves were fit to separate qubic splines as functions of $H$,
$B_{upper}(H)$ and $B_{lower}(H)$.

The errors on the top and bottom of the curve were fit to separate qubic splines
as functions of $H$, $\sigma_{B_{upper}}(H)$ and $\sigma_{B_{lower}}(H)$.

The weighted values for $B$ on the upper curve were computed with:

\begin{equation}
\hat{B}_i = B_{min} +
\left(\frac{B_i}{B_{upper}(H_{max})}\right)
\left(\frac{\sigma_{lower}(B_i)}{\sigma_{lower}(B_i) + \sigma_{upper}(B_i)}
	B_{lower}(H_{max}) +
\frac{\sigma_{upper}(B_i)}{\sigma_{lower}(B_i) + \sigma_{upper}(B_i)}
	B_{upper}(H_{max})\right)
\end{equation}
