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% 0.1 BCBrown 03-Aug-1998 Initial entry.
% 0.2 BCBrown 07-Aug-1998 Strength ratio info added
% 0.21 BCBrown 04-Sep-1998 Minor checking
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\date{9/4/98
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\title{Some Additional Information for Dipole and Quadrupole Power Supply Control
}
\author{Bruce C. Brown \\ Beams Division, Main Injector Department\\
{\em Fermi National Accelerator Laboratory }
\thanks{Operated by the Universities
Research Association under contract with the U. S. Department of Energy}
\\ \em P.O. Box 500 \\ \em Batavia, Illinois 60510\\
}
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\begin{abstract}
The efforts to describe the dipole and quadrupole magnet performance
for Main Injector operation continue. In this note we will examine
the relations between tune and magnet strengths, use measured data to
give explicit, if crude numbers for magnet operation with various ramp
conditions and examine the some crude fits to hysteresis measurements.
\end{abstract}
\section{Introduction}
The requirements for focusing and chromaticity control in the Main
Injector were reviewed in Fermilab-Conf-97/147\cite{PAC97:MI_PS_Control}
with additional details developed in MI-0211\cite{BCBrown:MIPSCIMH}.
Work continues to provide explicit guidance for Main Injector power
supply control programming. This note will provide an update on
ongoing efforts.
\section{Quadrupole Strength and Tune Control Model}
To provide the correct gradient strengths for Main Injector operation,
we examine the relation between gradient strength on focusing and
defocusing buses {\em vs.} the tune achieved in Lattice MI19. In
general terms, we know that the the horizontal and vertical tunes are
related to the quadrupole strength of two families of quadrupoles. If
we express this focusing strength in geometric terms (momentum
independent) and use a linear expansion of this relation about the
operating point, we should describe it with a linear matrix equation.
Let us determine this function using the MI19 lattice
model\footnote{Some of the following material has been circulated
privately as BCB-98-001, 6/9/98.}.
\begin{table}[tbh]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$\nu_x$ & $\nu_y$ & $k_f$ & $k_d$ \\ \hline
26.425 & 25.415 & 0.0406811 & -0.039808 \\ \hline
26.435 & 25.415 & 0.0406913 & -0.03981 \\ \hline
26.425 & 25.425 & 0.0406830 & -0.039818 \\ \hline
26.435 & 25.425 & 0.0406932 & -0.03982 \\ \hline
26.415 & 25.415 & 0.0406709 & -0.039806 \\ \hline
26.425 & 25.405 & 0.0406793 & -0.039798 \\ \hline
\end{tabular}
\caption{Calculated quadrupole strengths $k_f$ (focusing bus), $k_d$
(defocusing bus) of IQB quadrupoles in the MI19 lattice for the
specified tune values shown in Columns 1 and 2. $k_f$ and $k_d$
are specified in unit of m$^{-2}$. These results
were obtained by Dave Johnson from MAD simulations. }
\label{Tab:MI19TuneCalc}
\end{table}
Dave Johnson has used the MAD lattice modeling code
and established the relations between tune and focusing ($k_f, k_d$)
for an array of tunes ($\nu_x$ , $\nu_y$) near the operating point.
The design lattice used is identified as MI19. These results were
described in MI-0185\cite{DEJohnson:MIQLRCTP}. He provided a
description of these results there, but we choose instead to
re-examine the calculated results and provide a description which we
believe to be more generally useful. The tune results which Dave
obtained\footnote{Private communication
from David E. Johnson, MI\#3 Logbook, p.118.} are reported in
Table~\ref{Tab:MI19TuneCalc}. We fit them to a bilinear equation,
\begin{equation}
\left (\!\!\! \begin{array}{c} \nu_x\\ \nu_y \end{array} \!\!\! \right ) =
\left (\!\!\! \begin{array}{cc} Q_{11}&Q_{12}\\Q_{21}&Q_{22} \end{array} \!\!\! \right )
\left (\!\!\! \begin{array}{c} k_{1f}\\k_{1d} \end{array} \!\!\! \right ) +
\left (\!\!\! \begin{array}{c} \nu0_{x}\\\nu0_{y} \end{array} \!\!\! \right ).
\label{Eq:tune1}
\end{equation}
Fits to these data\footnote{Stan Pruss used DataDesk
%% \textcircled{r}
to produce these fit results. The
fits show 100\% correlation which implies that the linear dependence fully accounts for
the data over the limit range of variation which was explored.} give the parameters
of this equation as follows:
\begin{equation}
\left (\!\!\! \begin{array}{c} \nu_x\\ \nu_y \end{array} \!\!\! \right ) =
\left (\!\!\! \begin{array}{cc} 1014.17&186.132\\ -197.2& -1028.26 \end{array} \!\!\! \right )
\left (\!\!\! \begin{array}{c} k_{1f}\\k_{1d} \end{array} \!\!\! \right ) +
\left (\!\!\! \begin{array}{c} -7.42307\\\-7.4951 \end{array} \!\!\! \right ).
\label{Eq:tune2}
\end{equation}
Since this is a linear expansion about the operating point, the inverse relation
is also easily available in the form:
\begin{equation}
\left (\!\!\! \begin{array}{c} k_{1f}\\k_{1d} \end{array} \!\!\! \right ) =
\left (\!\!\! \begin{array}{cc} K_{11}&K_{12}\\K_{21}&K_{22} \end{array} \!\!\! \right )
\left (\!\!\! \begin{array}{c} \nu_x\\ \nu_y \end{array} \!\!\! \right ) +
\left (\!\!\! \begin{array}{c} k0_{1f}\\ k0_{1d} \end{array} \!\!\! \right ).
\label{Eq:tune3}
\end{equation}
Fit results in this form take the values
\begin{equation}
\left (\!\!\! \begin{array}{c} k_{1f}\\k_{1d} \end{array} \!\!\! \right ) =
\left (\!\!\! \begin{array}{cc} 0.001022&0.000185\\-0.000196&-0.001008 \end{array} \!\!\! \right )
\left (\!\!\! \begin{array}{c} \nu_x\\ \nu_y \end{array} \!\!\! \right ) +
\left (\!\!\! \begin{array}{c} 0.008973\\ -0.00901 \end{array} \!\!\! \right ).
\label{Eq:tune4}
\end{equation}
The two matrices are inverses so their product should be the unit
matrix. This has been confirmed. The errors are of order $10^{-6}$.
We note that a matrix which is accurate for small tune deviations need
only be supplemented by an offset (constants) to fully describe the
required strength in the useful operating region. We then examine
ways to apply corrections based on machine measurements in order to
create a coherent plan to use for machine operation.
\section{Applying Corrections for Tune Control}
The actual machine will only approximately match the design described
by $\underline{k_1} = \underline{K} \, \underline{\nu} +
\underline{k0_1}$ or its inverse. Effects due to component placement
as well as the cumulative effect of small magnet measurement errors
assure that we will not achieve precisely the beta functions of the
design lattice. The response of the tune to the quadrupole focusing
will be different than the predictions of the design\footnote{The MI19
Lattice model which Dave Johnson used employed the design properties
of the magnet, without knowledge of the magnet to magnet variation.
The strength ratio between 84$^{\prime \prime}$, 100$^{\prime \prime}$
and 116$^{\prime \prime}$ magnets is different at low fields and at
high fields (above 120 GeV/c).}. The above equations will need to be
supplemented by machine measurements to achieve the desired precision
of tune control.
The desired working point, ($\nu_x,\nu_y$), will be determined by observed
machine operation (probably losses and emittance). It will probably be
close to but different from the design tune of (26.425, 25.415).
We would like to examine how measured tune values can be most effectively
analyzed to provide control parameters for setting the quadrupole
currents.
We can, at least in principle, make measurements at fixed momentum of
the tune function at a set of tunes near the design operating point.
We would operate the machine at various tune values and then
determine the actual tune achieved by various ($k_f, k_d$) values.
Suppose we use these measurements to establish the measured tune
equation, $\underline{k_{1m}} = \underline{K_m} \,\underline{\nu_m}+
\underline{k0_{1m}}$. How shall we employ these measured results
in combination with the design parameters to minimize the sensitivity
of this operation to various unknown factors?
Let us consider differences between measured and design parameters.
Let $\delta \underline{k_1} = \underline{k_{1m}}
- \underline{k_{1d}}$, $\delta \underline{\nu} = \underline{\nu_m} -
\underline{\nu_d}$, and $\delta \underline{k0_1} = \underline{k0_{1m}}-
\underline{k0_1}$. We begin with
\begin{equation}
\underline{k_{1m}} = \underline{K_m} \,\underline{\nu_m}+ \underline{k0_{1m}}
\end{equation}
\begin{equation}
\underline{k_{1d}} = \underline{K} \,\underline{\nu_d} + \underline{k0_{1d}}
\end{equation}
where the d subscript signifies the design point. Taking differences we have
\begin{equation}
\delta \underline{k_1} = \underline{K_m} \,\underline{\nu_m}-\underline{K} \,\underline{\nu_d}+\delta \underline{k0_1}
\end{equation}
\begin{equation}
\delta \underline{k_1} = \underline{K_m} \,(\underline{\nu_m}- \underline{\nu_d} )
+(\underline{K_m}-\underline{K}) \,\underline{\nu_d}+\delta \underline{k0_1}
\end{equation}
letting $\underline{K^{\prime}}=\underline{K_m}-\underline{K}$ we have
\begin{equation}
\delta \underline{k_1} = \underline{K_m} \,\delta \underline{\nu}
+\underline{K^{\prime}} \,\underline{\nu_d}+\delta \underline{k0_1}
\label{Eq:kerror_m}
\end{equation}
Alternatively, we can recombine terms to show
\begin{equation}
\delta \underline{k_1} = \underline{K} \,\delta \underline{\nu}
+\underline{K^{\prime}} \,\underline{\nu_m}+\delta \underline{k0_1}
\label{Eq:kerror_m1}
\end{equation}
\subsection{Main Ring Traditions}
The quadrupole control in the Main Ring was based on a sensitivity
matrix (same principle as $\underline{K_m}$ or $\underline{K}$) for
relating desired tune changes to the required current changes, and a
set of measured tunes and currents which were stored in a table keyed
on the dipole current which was called the {\em calibration} table.
Compared with our understanding of Main Injector requirements, this
has the disadvantage, in principle, that the relation between fields
(and the resulting $\underline{k_1}$) is dependent on magnet history
due to hysteresis. We are committed to attempting to make the
hysteretic effect repeatable among different ramps so perhaps we can
follow this example. Let us assume that the {\em calibration} table
gives a set of tunes for given dipole and quadrupole currents. We
can express this result in the notation used here as
\begin{equation}
\underline{k_{1m}} = \underline{K_m} \,\underline{\nu_m}+ \underline{k0_{1m}}
\end{equation}
and we assume that if the specified operational tune sought is
$\underline{\nu_s}$, we can achieve this tune by changing the focusing by
\begin{equation}
\delta \underline{k_1} = \underline{K_m} \, (\underline{\nu_s}- \underline{\nu_m})
\end{equation}
In succeeding sections we will examine some hysteresis data. The
limitations of the calibration table approach due to hysteretic
differences in the relation between field and current will be made
explicit there. A principle limitation of the {\em calibration} table
for tune control is that it is defined as a single current-dependent
table which applies to all ramps. To add further control, one will
need to explicitly subtract the results which this feature generates
to permit a time-dependent control function to be implemented.
\subsection{Considerations for Tune Control Software Design}
In defining the software for Main Injector ramps, the following features
have been identified for consideration:
\begin{enumerate}
\item It is very important that the tune achieved for a specified ramp
be very nearly the tune specified and displayed.
\item One might wish to have easy access also to the
\begin{enumerate}
\item the design nominal tune
\item the measured nominal tune
\item the specified tune change
\end{enumerate}
so that the the relation between the model, measurements and specified tune
are well understood.
\item The matrix relating tune to focusing can only be measured at at
most a few momenta. It is likely that the differences between the
design ($\underline{K}$) and measured ($\underline{K_m}$) matrices are
small such that the design matrix is sufficient for initial
implementation.
\item It is desirable to preserve in the control information some
clarity as to the degree to which the underlying lattice model
is matched by the observed machine properties. Similarly, it is
desirable to see what important features change as a function of
momentum. For these reasons, we should employ momentum-independent
focusing functions where possible (rather than the measured magnetic
fields) and should display $\delta \underline{k_1}$, $\delta \underline{k0_1}$
and $\underline{K^{\prime}}$ as a function of momentum.
\item Although measurements at several tunes settings at a fixed
momentum are likely to be available at only a few settings, using
equations~\ref{Eq:kerror_m}~or~\ref{Eq:kerror_m1}, we can analyze
measurements at a single tune setting, extracting only a value for
$\delta \underline{k0_1}$ which can be tabulated as a function of
momentum and used to achieve a good description of the machine.
\end{enumerate}
Efforts to complete the tune control software are now underway.
\section{Observations on Saturation and Hysteresis}
Although measurements exist on all MI dipole and quadrupole magnets
and hysteresis studies have been performed on one or more magnets
of each type, the analysis of this information is still incomplete.
In order to expose some of the issues which will affect power supply
regulation and control design, we will examine here some of the
current ratios and differences which are important. For simplicity,
we choose to interpolate measured data (simple, linear interpolation)
rather than use the fitted data shown later in this document. Since
we are using only one magnet of each type, we should not expect to
match precisely the final results which will represent the whole ring.
The magnets are all like to about 0.5\% worst case and typically much
better. As will be noted, there is some data in which we will use
which is not quite right but the general properties which are of interest
will still be apparent.
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{Quad_o_dipole_1.eps}} }}}
\caption{Ratio of Quadrupole to Dipole current for F and D quad buses.}
\label{Fig:DoQ}
\end{figure}
Hysteresis studies of IDA114-0 and IQB310-1 will be used for these
illustrations. Ramp tables were constructed in a spreadsheet program
using 0.5 GeV/c steps in momentum from injection to 20 GeV/c and
5.0 GeV/c steps from 20 - 150 Gev/c. For each momentum and for the
design tune, (26.425, 25.415), the strengths required for IDA and
IQB magnets were calculated. The dipole strength was calculated using
Equation 3 of MI-0211 while the quadrupole strength used Equation~\ref{Eq:tune4}
along with Equation~4 of Fermilab-Conf-97/147\cite{PAC97:MI_PS_Control}.
$L_{eff} = 2.1176 m$ was used for the Quadrupole effective length per
MI-0185\cite{DEJohnson:MIQLRCTP}. The required current for each desired
strength was obtained by interpolating between the values measured.
Three ramp conditions were considered: Upramp reset at 0 A (dipole
and quadrupole), Upramp reset at 400 A for dipoles and 150 A for
quadrupoles, and downramp reset at the measurement peak current of
9500 A for dipoles and 4000 A for quadrupoles. In addition to these
primary values (p, 3 strengths, and 9 currents), a number of ratios
and differences were tabulated in the spreadsheet.
\def\topfraction{.99}
\def\bottomfraction{.99}
\def\textfraction{0.01}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{MI_quad_transductor.eps}} }}}
\caption{Signal in Amperes expected on 2:5 ratio transductor for Quadrupole
Control. Transductor uses outputs of dipole and quadrupole buses with
winding ratios of 2 turns for Dipole to 5 turns for F and D quad buses.
Left graph has signal in amperes divided by dipole current.}
\label{Fig:QtransD}
\end{figure}
To understand the control requirements we consider the relations
between the dipole bus current and the two quadrupole bus currents as
a function of momentum. In Figure~\ref{Fig:DoQ} we plot the ratio of
quadrupole to dipole current for 3 ramp conditions. These are simply
example conditions, not selected as particularly desirable. We see
that when viewed as a ratio, the up ramp and down ramp ratios are
quite similar ( differing by $4 \times 10^{-4}$ or less over most of
the momentum range). However, for the data with higher resets, the ratio
changes (differences between the ratios with 0 reset and the higher
resets are as large as $30 \times 10^{-4}$ near injection, approaching
no difference at about 20 GeV/c).
This information can be displayed to illustrate another feature of the
planned hardware. For control of the quadrupole current, a
transductor is provided which compares 2 times the dipole current with
5 times the quadrupole current. In the Main Ring the comparable
transductor compared directly the dipole and quadrupole currents,
which were comparable at all operating conditions. This provided two
benefits: the quadrupole current was regulated to match excursions in
the dipole regulation, permitting improved tune control, while the
close matching of the two currents permitted one to regulate the
quadrupole current from a source whose observed range was small
compared to the quadrupole dynamic range. Presumably, the
cross-regulation feature will be preserved. However, saturation
differences will limit the ability to maintain a very small range for
the regulation signal. In Figure~\ref{Fig:QtransD} we show the
transductor signal for the focusing quad bus on the right. On the
left we show the transductor signal for various ramps for both the
focusing and defocusing buses but we divide by the dipole current to
reduce the required range of the plot. We see that the match works
well at low field. A fifth of the transductor signal should be
compared to the quadrupole current. At low field it is one about
1.4\% of the quad signal for the focusing bus and 0.7\% for the
defocusing bus. But as saturation sets in we see that the quads
saturate faster than the dipoles. If we only had Main Ring Quads
(IQB's) we would find that the high field transductor signal
corresponds to 150 or 230 A at 150 GeV/c which is 4\% to 6\% of the
respective quadrupole current. There is more saturation in the IQC
and IQD quadrupoles due to the lower permeability of the Main Injector
project steel. This will reduce this saturation difference between
dipoles and quadrupole since the long magnets do provide a significant
fraction of the focusing in the ring.
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{Hys_curr_diff.eps}} }}}
\caption{Difference (up - dn) in current required for the same momentum and tune
on up ramp and down ramp. Note that the dipole downramp data is not adequate.}
\label{Fig:UminusDCur}
\end{figure}
Another way to explore the effects which must be accounted for in
controlling the dipole and quadrupole currents is to examine the
differences in currents which produce the same fields. We do this at
a variety of momenta, requiring again that the design tune be
achieved. We plot this results in Figure~\ref{Fig:UminusDCur}. Note
that the data for dipole downramp which was selected in this study is
of poor quality so the results plotted above about 60 A should be
discounted. However below that point for dipoles and at all currents
for quadrupoles, the difference grows slowly and monotonically with
momentum (current). In planning the control strategies, we should
be sure that this feature is not masked by assumptions which could
add difficulty to the task of deceleration.
\section{Hysteresis Fitting}
To relate magnetic fields to the current required to produce them, the
power supply control application will use an analytic model of the
relation between magnetic field integral and current history. A
preliminary fit to the hysteresis studies on BQB310-1 and IDA114-0 has
been carried out and will be reported here. Improved fits are required.
\subsection{General Properties}
If one subtracts from the measured integrated strength, the field
generated by the current in the existing geometry by ideal iron (this
term is linear in $I_{meas}$), the non-linear term remaining is
related to the $H$ of the steel by geometric constants. The
measurements we have made show two simplifications from the most
general hysteretic properties which are reported on magnet steel.
Since we only power the magnets to positive currents, we have major
hysteresis loops which only go from near $H = 0$ to a maximum $H$.
Our minor loops seem to asymptotically approach these curves even when
we reverse current at arbitrary points between the extreme values of $H$.
\begin{itemize}
\item The first simplicity we observe is that the differences in
asymptotic loops can be ignored and we can fit for curves (we
designate them as Hysteresis Curves) with only one up ramp curve
and one down ramp curve.
\item The other simplifying factor is in the transitional curves
which join up ramp and down ramp Hysteresis functions. We designate
these as Interjacent Curves. We find that these curves have
the same general shape for up to down transitions as for down to up
transitions, they are all of a shape which is roughly exponential
and to a useful degree, all are characterized by the same parameters.
Only when we have a satisfactory detailed fit will we know if the
parameters are completely independent of the transition direction
or of the current where the change in ramp direction is made.
\end{itemize}
Hysteresis studies have been done systematically with ramps which
explore various reset currents for each ramp direction.
\subsection{Measured Hysteresis Response and Fits}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IDA114-0_HYS_dn.eps}} }}}
\caption{Measured and Fit non-linear integrated strength
of IDA114-0 with various downramp reset currents.}
\label{Fig:Hyst_IDA114_d}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IDA114-0_HYS_up.eps}} }}}
\caption{Measured and Fit non-linear integrated strength
of IDA114-0 with various upramp reset currents.}
\label{Fig:Hyst_IDA114_u}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IQB310-1_HYS_dn.eps}} }}}
\caption{Measured and Fit non-linear integrated strength
of IQB310-1 with various downramp reset currents.}
\label{Fig:Hyst_IQB310_d}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IQB310-1_HYS_up.eps}} }}}
\caption{Measured and Fit non-linear integrated strength
of IQB310-1 with various upramp reset currents.}
\label{Fig:Hyst_IQB310_u}
\end{figure}
A set of measurements of magnet strength were carried out for IDA114-0
and IQB310-1 which employed a set of currents with increasing (or
decreasing) levels of reset current before upramp (downramp)
measurements. The data were spaced following a fixed pattern of
offset current from the reset current value. Measurements were
carried out at MTF using the CHISOX\cite{PAC95:MTF_l} measurement
system. Data from the harmonics.harmonics\_red\_runs table were
extracted and organized into sets of ramps using a perl script. These
magnets were also subjected to the standard set of measurements for
Main Injector magnets of this series. The linear coefficient of a fit
to $Bdl$ {\em vs.} $I$ was obtained from a fit to the lower current
points of the standard downramp measurement. This term was then
subtracted from the measured field integral using the measured current
for $I$ except where zero current was requested in which case the
power supply is assumed to have produced zero current. The scale of
the data is made suitable for presentation when the linear term is
subtracted. Figures~\ref{Fig:Hyst_IDA114_d},~\ref{Fig:Hyst_IDA114_u},~\ref{Fig:Hyst_IQB310_d},~and~\ref{Fig:Hyst_IQB310_u}
show the measured non-linear fields in the lower left plot (upper left
for quadrupoles) and on a less expanded scale on the lower right.
Parameterizations of the three curves are as follows:
\begin{itemize}
\item Linear (Electromagnet) Curve
\begin{equation}
Bdl_{lin}(I) = (Slope) * I
\end{equation}
\item Hysteresis Curve
\begin{equation}
Bdl_{Hyst}(I, (dir)) = [C_0 + C_2 I_s] + [H_1 I_s - \sqrt{(H_1-H_2)^2 I^2 + C}]
\end{equation}
where $I_s = I / I_{scale}$, and the coefficients $C_0, C_2, H_1, H_2$, and $C$ have distinct
values for upramp and down ramp hysteresis curves.
\item Interjacent Curve
\begin{equation}
Bdl_{Inj}(I) = A_{Inj} e^{-(I - I_{reset})/ichar(dir)}
\end{equation}
where $I_{reset}$ is the most recent current at which the ramp direction was changed
and for the up ramp, $A_{Inj}= Bdl_{Hyst}(I, up) - Bdl_{Hyst}(I, dn)$
is the difference in field between the downramp and upramp hysteresis curves
evaluated at the reset current. For downramps, $A_{Inj}$ has the opposite sign.
\end{itemize}
For IQB310, a subset of the data was fit using the NFIT program (a GUI
interface to MINUIT). The full data set for IDA114 was fit with this
parameterization using a FORTRAN program which called the MINUIT
subroutine package. Using a perl script to manipulate the data and
xmgr for creating plots, the fit parameters were used to plot the
fitted results the residuals (measured minus fit) for both magnets at
each of the measured currents.
The parameters obtained for these fits and used for the plotting program
are shown in Tables~\ref{Tab:IDA114-0_param}~and~\ref{Tab:IQB310-1_param}.
\begin{table}[tbh]
\centering
\begin{verbatim}
Magnet IDA114-0
Hysteresis HypPar1
1 Slope 0.12041E-02 fixed NaN
2 Iscale 10000. fixed NaN
3 Idn 7431.3 0.88319 0.12412E-01
4 Hdns1 -1.6843 0.10970E-02 -0.14469E-04
5 Hdns2 -0.35229E-01 0.63556E-03 -0.13378E-05
6 Cdn 0.28904E-01 0.12671E-03 0.79892E-06
7 Coeff0dn 0.22405E-01 0.24135E-03 0.27126E-06
8 Coeff2dn -0.44337E-01 0.51163E-03 -0.20101E-05
9 ichardn 453.22 0.67519 -0.35841E-02
10 Iup 7885.5 0.56305 -0.64630E-02
11 Hups1 -2.2452 0.61632E-03 0.69469E-05
12 Hups2 0.47087E-01 0.81470E-03 -0.14389E-05
13 Cup 0.10944 0.28929E-03 -0.18210E-05
14 Coeff0up 0.66404E-01 0.33678E-03 -0.13116E-05
15 Coeff2up -0.72650E-03 0.64762E-03 -0.20016E-06
16 icharup 350.08 0.74211 0.13127E-02
\end{verbatim}
\caption{Parameters for crude fit to IDA114-0 hysteresis studies. }
\label{Tab:IDA114-0_param}
\end{table}
\begin{table}[tbh]
\centering
\begin{verbatim}
Magnet IQB310-1
Hysteresis HypPar1
1 Slope 0.12254E-01 fixed NaN
2 Iscale 10000. fixed NaN
3 Idn 3874.6 0.88319 0.12412E-01
4 Hdns1 -14.229 0.10970E-02 -0.14469E-04
5 Hdns2 -0.77145 0.63556E-03 -0.13378E-05
6 Cdn 0.28661 0.12671E-03 0.79892E-06
7 Coeff0dn -0.12485E-01 0.24135E-03 0.27126E-06
8 Coeff2dn -1.4203 0.51163E-03 -0.20101E-05
9 ichardn 180 0.67519 -0.35841E-02
10 Iup 3746.6 0.56305 -0.64630E-02
11 Hups1 -10.300 0.61632E-03 0.69469E-05
12 Hups2 -1.3797 0.81470E-03 -0.14389E-05
13 Cup 0.12005 0.28929E-03 -0.18210E-05
14 Coeff0up -0.17799 0.33678E-03 -0.13116E-05
15 Coeff2up -2.4447 0.64762E-03 -0.20016E-06
16 icharup 180 0.74211 0.13127E-02
\end{verbatim}
\caption{Parameters for crude fit to IQB310-1 hysteresis studies. }
\label{Tab:IQB310-1_param}
\end{table}
\subsection{Discussion of Fit Results}
In MI-0211\cite{BCBrown:MIPSCIMH}, a specification was given for the
fidelity with which the fitted results needed to match the magnet
performance. This crude fit provides a very useful description of
the data and permits one to see important effects, but falls short
of the desired fidelity at both low and high fields by a factor of
several. Nevertheless, we can draw a number of useful conclusions.
\begin{itemize}
\item This parameterization has the correct general properties
at all field levels.
\item Using a single exponential for the Interjacent Curve seems
to leave a residual which has a characteristic shape in common
for various reset currents for both up and down ramps. It appears
that the fit would be improved with a second exponential term
with larger characteristic current.
\item It appears that at this level of examination, the use of
the same parameters for all Interjacent Curves is satisfactory.
Further examination will be required to determine if the differences
between upramp and downramp are significant.
\item The high field fits appear to be limited by the use of only
hyperbola and parabola terms. Addition of one or more terms
will likely permit a substantially better fit at high fields.
\end{itemize}
It is expected that efforts to modify the program so that more
complex fitting can be attempted will begin soon.
\section{Conclusions}
Description of the magnetic field properties of Main Injector magnets
in a way which permits the power supply control program to properly
take into account the saturation and hysteresis of the magnets has
achieve a substantial degree of success. The present characterization
of the measured data fall short of the precision which would permit
one to use it without any tuning parameters. However, the residual
errors in this description have a pattern which suggests that the
analytic approach will be adequate for machine control.
%\bibliography{compute,accelerators,magnets,mathscieng,mtf,minote}
\bibliography{magnets,minote}
\appendix
\end{document}