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Analytic Form for Fitting\\ Hysteretic Magnet Strength\\
\vspace{10ex}
Bruce C. Brown\\
IMMW XI at Brookhaven\\
21-September-1999
\end{center}
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
At IMMW X at Fermilab, I presented a strategy for studying hysteresis
effects in accelerator and beam line magnets. I would like to update
that with a report on my progress at finding an analytic form which
will fit this data to a precision of $3 \times 10^{-4}$ or so.
\vfill
A status report on this work was presented at PAC99 and can be found on my WWW site:\\
http://www-ap.fnal.gov/~bcbrown/Docs/p-Conf-99-096.ps
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{\textwidth}{!}{\rotatebox{0}{
%% \includegraphics{dipole_AL.idr}}}
\includegraphics{dipole_AL.ps}}}
}
To predict magnetic fields we employ Ampere's Law:
\begin{equation}
\int_g \frac{1}{\mu_0}\vec{B_g}\; \cdot d \vec{\ell} +
\int_{\cal L} {\vec{H}} \cdot d \vec{\ell} \; = N_g I,
\end{equation}
where $g$ represents the path in the air gap and ${\cal L}$ represents
the path through the steel.
We will use $N_g$ turns per gap as for loop on left.
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
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\begin{slide}
\small{
In a well designed multipole magnet (dipole, quadrupole....) the the
field in the gap is well represented by the dominant multipole
component. We integrate along a field line in the gap (to pole radius
$A$). To be ready to fit integrated field measurements, we integrate
along the beam path by multiplying our body field strength by an
effective length, $L_{eff}$. For the integral over the path in steel
we choose a typical path along a flux line.}
\normalsize
\begin{equation}
B_N L_{eff} = \frac{\mu_0 N N_g L_{eff} I}{2A^{N}}
- \frac{N{\cal L} L_{eff}} {2A^{N}} \mu_0 .
\label{Eq:Integ_field1}
\end{equation}
\small{
where $N$ is the harmonic number (1 for dipole), $N_g$ is the number
of turns per gap in the coil,
$A$ the pole tip radius ($g/2$ for a dipole), ${\cal L}$
is the length of a flux line in iron with average $H$ along the path of
$<\!\!\!H_{steel}\!\!\!>$. $I$ is the current through the coil. We note that
the first term is proportional to $I$ and it represents the field
created in idealized iron by the magnet current. The second term
describes the field lost in driving the iron. All saturation and
hysteretic terms due to iron remanence are described by $<\!\!\!H_{steel}\!\!\!>$.}
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{\textwidth}{!}{\rotatebox{270}{
\includegraphics{IQC020-0_b.eps}}}
}
Let us look at some typical data for magnet strength.
Note that there is both an up ramp and
a down ramp measurement on this plot.
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{5in}{!}{\rotatebox{270}{
\includegraphics{IQC020-0_bnl.eps}}}
}
\small
To see the effects of $<\!\!\!H_{steel}\!\!\!>$, we
subtract a term linear in current. It can be
from fitting the previous plot or by
calculation from pole geometry and
the number of turns. We see two sorts of
contributions in this plot. The upramp
strength is less than the downramp strength (hysteresis)
and there is a sharp change at high field (saturation).
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{5in}{!}{\rotatebox{270}{
%\resizebox{\textwidth}{!}{\rotatebox{270}{
\includegraphics{IDA114-0_uprmp_hyst.eps}}}
}
Before selecting an analytic fitting form, we need to examine
additional measurements which will guide our choices. If
we perform measurements with a series of different minimum (reset)
fields we find a family of similar curves for the non-linear
field.
\vfill
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\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{5in}{!}{\rotatebox{270}{
%\resizebox{\textwidth}{!}{\rotatebox{270}{
\includegraphics{IDA114-0_dnrmp_hyst.eps}}}
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The results for measurements with various peak excitations
again have an obvious pattern with shapes very suggestive
of the same form.
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
Our analytic form will need two types of terms. We have chosen to
call the form reached well after current changes as the 'hysteresis
curve'. There is an upramp hysteresis curve and a downramp hysteresis
curve. We initially note that it has some obvious similarity to a
hyperbola which has been suitably rotated and offset.
\begin{displaymath}
H(I, D) = -\sqrt{h_2}x - \sqrt{h_2x^2+h_0}
\end{displaymath}
The curves which transitions the strength between the upramp curve and
the downramp curve we call Interjacent curves. The exponential
character of these is apparent to the most casual observer.
\begin{displaymath}
J(I, I_{r},I_{p},D) = A( I_{r},I_{p},D)\;\;\;e^{-(\frac{I-I_{r}}{I_{C,D}})}
\end{displaymath}
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\small
Adding a parabola to provide a little freedom for fitting, we
applied this to the data and achieved a fit precision of about
0.3\% ($30 \times 10^{-4}$). We are using this prescription,
however inadequate, for Main Injector operation at this time.
\normalsize
To fit the data more precisely, we had to overcome a number
of problems:
\small{
\begin{itemize}
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\item The remanent field has a weak dependent on the peak
of the last ramp. This is likely to be unimportant for
ramps of operational interest, but in trying to get sufficient
range of data to constrain the fit parameters, we get enough
differences to make this significant.
\item The hyperbola is not sufficiently `rich' to represent the
hysteresis curves.
\item A single exponential falls too quickly to represent the data.
\item The current control was very good (a 10 kA system operating
at 500 A gave an RMS magnet strength deviation consistent with less
than 20 mA RMS current deviation) but the current readback was about
one order of magnitude worse. We `calibrate' the the control current
to get information for fitting.
\end{itemize}
}
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
We consider the magnet strength
M ($\int B_1 dl, \int B_2 dl$ or $\int B_3 dl$) to be comprised of
four terms, L (linear), R (remanent), H (hysteretic)
and J (interjacent) . We continue to explore suitable
expressions for these contributions but find useful fits
with the following functional relations:
\begin{eqnarray*}
\lefteqn{ M(I, I_{r},I_{p},D)=}\\
& & L(I)
+ R(I_{p},D)
+ H(I, D)
+ J(I, I_{r},I_{p},D)
\end{eqnarray*}
where $I$ is the magnet current during the measurement,
$I_{r}$ is the reset current (current at last sign change
in $dI/dt$), $I_{p}$ is the preset current (reset current
of last ramp), and
$D$ is the ramp direction with +1 for upramps and -1 for downramps.
We express the relations with normalized variables to provide
consistency of representation among magnets. Use $I_{scale}$ as a
maximum current of interest (rounded) and $I_S$ as a characteristic
current for saturation.
\begin{displaymath}
x = \frac{I - I_S}{I_{scale}} \,\,\,\,\,\, x_0 = \frac{ - I_S}{I_{scale}}
\end{displaymath}
\vfill
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\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
Expressions used for these terms are
\begin{displaymath}
L(I) = Slope * I
\end{displaymath}
\begin{displaymath}
R(I_{p},D) = RemStr_{D} + RemSlp_{D}*(I_{p}-I_{scale})
\end{displaymath}
\begin{eqnarray*}
\lefteqn{ H(I, D) =} \\
& & C_1 * \frac{I}{I_{scale}}\\
& & -\sqrt[4]{h_4}x
-\sqrt[4]{h_4x^4+h_3x^3+h_2x^2+h_1x+h_0}\\
& & + \sqrt[4]{h_4}x_0
+ \sqrt[4]{h_4x_0^4+h_3x_0^3+h_2x_0^2+h_1x_0+h_0}.
\end{eqnarray*}
Note that $H$ is defined to have the value 0 at $I$ = 0. Each parameter
is distinct for the upramp or downramp curve and could be
expressed as $h_{iD}$ or $C_{1D}$ .
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
Two
forms have been used for fitting $J$:
\begin{displaymath}
J(I, I_{r},I_{p},D) = A( I_{r},I_{p},D)(s e^{-\frac{I-I_{r}}{I_{C1,D}}}+(1-s)e^{-\frac{I-I_{r}}{I_{C2,D}}})
\end{displaymath}
\begin{displaymath}
J(I, I_{r},I_{p},D) = A( I_{r},I_{p},D)e^{-(\frac{I-I_{r}}{I_{C,D}})^N}
\end{displaymath}
where $N$ is a real number, typically less than 1. The amplitude function
$A$ is the difference in hysteresis curves at the reset current.
\begin{eqnarray*}
\lefteqn{ A( I_{r},I_{p},D) = }\\
& & H(I_r, -D) - H(I_r, D) + R(I_{p},-D) - R(I_{p},D).
\end{eqnarray*}
\vfill
\begin{flushright}
\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
\begin{slide}
\centerline{
\resizebox{\textwidth}{!}{\rotatebox{270}{
\includegraphics{IDA114-0_difplot_sel.eps}}
}}
\small
Selected data from the IDA114-0 hysteresis study
were fit with the interjacent curve described by 2 exponentials.
Top plot shows fits to the selected upramp data. Center and
lower plots show residuals (measured - fitted) on scales which
emphasize the low field and high field results.
\vfill
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\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
\end{flushright}
\end{slide}
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\begin{center}\begin{large}
Summary
\end{large} \end{center}
\small
\begin{itemize}
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\item Strength measurements of accelerator magnets, while dominated
by the linear strength term have important field components which
are not linear in excitation current. These non-linear terms
have surprisingly simple regularities which permit analytic descriptions.
\item To good accuracy, these non-linear terms exponentially
approach a common hysteresis curve following a sign change in $dI/dt$.
A small effect due to the reset (or preset) current may remain.
\item The Interjacent curves which characterize the fashion in which
the strength approaches the hysteresis curve is nearly exponential.
Fits using two exponentials or a modified exponential are sufficient
for present requirements.
%\end{itemize}
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%21-Sep-1999 IMMW \\
%Hysteresis Fits\\
%Bruce C. Brown
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\item Analytic fitting functions have been found which describe these
effects well enough to leave fitting residuals which are less than
$5 \times 10^{-4}$ relative to the magnet strength at each current.
\item Data have been measured on six or more magnet designs.
The same characteristics are apparent in all of them.
Efforts to get a complete software system which will
fit all of this measured data is continuing.
\end{itemize}
\vfill
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\tiny
21-Sep-1999 IMMW \\
Hysteresis Fits\\
Bruce C. Brown
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\end{slide}
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