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%**** Fermilab Main Injector Department ****
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% Document Cross References: FERMILAB-Conf-97-147, MTF-94-0078
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% 0.1 BCBrown 4-Jun-1997 Initial entry. Based on ver from 6-Feb
% 0.11 BCBrown 5-Jun-1997 Serious Rewrite due to PAC97 Paper
% 0.12 BCBrown 9-Jun-1997 Add sections on ramps, various fixes.
% 0.18 BCBrown 18-Jun-1997 Try to recover to Ver 0.2 of June 12
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% 0.19 BCBrown 19-Jun-1997 Small Typo's fixed, only.
% 0.3 BCBrown 26-Jun-1997 More Small Typo's fixed.
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\date{ 6/26/97
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\title{MI Power Supply Control Issues for Magnets with Hysteresis
}
\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\\
}
\begin{document}
\bibliographystyle{unsrt}
\mtfdocnv{\mtfdocnum}{\mtfdocver}
\maketitle
\newpage
\tableofcontents
\newpage
\begin{abstract}
The power supply control system for the Main Injector will interact
with the machine hardware and the operators through sets of programs
which share information on the state of the machine. The required
currents are determined by the desired beam parameters, lattice
properties and the properties of the magnets. A PAC97 paper\cite{PAC97:MI_PS_Control}
provides a physics description of the accelerator and magnet issues.
This paper will discuss the assumptions which will permit the application
of those equations and a model of the program structure for this interaction.
\end{abstract}
\section{Introduction}
Control of the momentum ($p$), tune ($\nu_x, \nu_y$) and
chromaticity ($\xi_x, \xi_y$) of the accelerated beam is
maintained through the interaction of several power supply
systems and the rf system. Within the controls systems one must
describe these variables as well as the currents and perhaps the
voltages in the power supply loops. To simplify the
interactions among these systems, we will attempt to rely on the beam
variables rather than such secondary properties as the magnet
currents or rf phases. This should allow us to deal with
subtleties, such as the history dependent hysteresis of the
magnets, in only one place.
The combination of lattice design, accelerator survey and
measured magnet properties provide an initial model of the
accelerator. Beam-based accelerator measurements may provide a
refinement of that model.
This document will provide
\begin{enumerate}
\item An overall model of the program system
\item The beam physics equations which describe particle motion
\item A structure for specifying the operational variables
\item A specification of some of the database parameters
which will be required to implement this system.
\end{enumerate}
As a preliminary specification document we assume that, at most,
only the fundamental equations are specified in final form.
The discussion will be divided into a magnet overview; hardware
assumptions; a program overview; a ramp table specification;
specifics for momentum, tune, and chromaticity; a discussion of
issues; a sensitivity analysis; and a concluding
section with some opinions and action items. A section which
motivates the design, based on significant differences between
Main Ring and Main Injector magnet systems is provided as an
appendix.
\section{Magnet Overview}
The magnet systems\cite{MI:TDH1997} which must be considered in this
context include the main dipole (IDA, IDB, IDC and IDD) system, the
main quadrupole (BQB, IQC, IQD) system, which is divided into a
focusing and a defocusing systems, and the chromaticity sextupole
(ISA) system which also has two families of sextupoles. We will
assume that the average value (0th harmonic) of the correction dipole
(IDH and IDV) system is maintained small enough that it can be ignored
in the momentum calculation.
Measurements on a variety of ramps have been performed on the magnets
listed above. Most measurements have been performed with
`quasi-static' ramps\cite{BCBrown:DcurVD}\cite{Sim:DrampVD} in which
a ramp segment is executed to create a current change, then the
current is maintained at a constant value while the field strength and
shape is measured. The actual fields are modified substantially by
eddy current effects. We will directly account for the sextupole
created by dipole eddy currents. Eddy currents will modify the
dominant fields in these magnets but these can be treated as primarily
a time delay and, while they can be treated within the context of the
systems we describe, they will be ignored in this description.
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\epsfbox{hr_bnlin.eps}}
%\resizebox{3.25in}{!}{
%\includegraphics{hr_bnlin.eps}}
\caption{Nonlinear portion of integrated dipole strength for four 6-m Main
Injector dipoles as measured by the Harmonics measuring system.}
\label{Fig:hr_bnl}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IQC020-0_bnl.eps}}}}}
\caption{Nonlinear portion of quadrupole strength.}
\label{Fig:quad_hyst}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{2P008f2.eps}}}}}
%\resizebox{3.25in}{!}{\rotatebox{270}{
%\includegraphics{2P008f2.eps}}}
\caption{Nonlinear portion of sextupole strength.}
\label{Fig:sext_hyst}
\end{figure}
For an electromagnet, a field which is proportional to the drive
current provides most of the design features of the magnet\footnote{In
note MTF-94-0078\cite{BCBrown:AMSSBHCG}, the strength contribution to
the magnet field due to iron saturation and hysteresis has been
extensively explored.}. To achieve the precision required for
accurate control of beam variables, we must also account for
saturation and remanence of the iron yokes. Iron saturation reduces
the available field by a few percent at the design maximum fields of
these magnets. Fields due to steel remanence are roughly constant,
while the direction of field (current) change is constant, and provide
a contribution of a few per mil (.001) of the full field. However,
they are hysteretic, depending upon the magnet excitation history.
Since the dipoles operate over a dynamic range of 17, the hysteretic
contribution is as large as a few percent at low fields.
Figures~\ref{Fig:hr_bnl},~\ref{Fig:quad_hyst}~and~\ref{Fig:sext_hyst} show the
contributions from these fields in the dipole, quadrupole, and sextupole magnets.
Note how the hysteresis depends upon the ramp history.
The field strength profile achieved at a given current profile
will depend upon (at least)
the peak field in the previous cycle, the minimum current (reset
current) in the
present cycle and the ramp rate. However, the range of peak
currents and ramp rates which will be used in the Main Injector
are limited. We believe that we may achieve the desired field
profile for any given ramp cycle with a current profile which
is not dependent upon the history of previous ramp cycles, provided we
``prepare'' the magnet with a suitable reset profile at the
end of each ramp. This cannot be exact, but it is possible
that such a reset strategy can allow ramps with sufficient
precision, independent of previous ramp history. We set that
as a goal for this design.
\section{Hardware Assumptions}
We will assume the following criteria can be met with the hardware
we will construct and the measurements we will have on that hardware:
\begin{enumerate}
\item We assume that a magnet current ramp I(t) can be specified
and the power system will produce that ramp to an accuracy
sufficient for the present purposes. The small differences
between that ramp and the ramp which is actually achieved is only
important in a small number of systems and will be controlled in
real time by those systems as required. We assume that the
calculated ramp is achieved.
\item \label{It:MagBusses} The accelerator parameters which are to be
controlled by the major magnet systems are the momentum (p), the tunes
($\nu_x, \nu_y$), and the chromaticities ($\chi_x, \chi_y$).
Corresponding to this are 5 magnet current busses: dipole, quad
focusing, quad defocusing, h-sextupole and v-sextupole. We expect to
represent the basic equations which relate these with a fundamental
set of equations which assume that the design lattice has been
created, and perhaps, a second related set of equations which account
for lattice imperfections with offsets and slightly modified
coefficients.
\item We assume that the magnetic fields are functions of I,
dI/dt and history. Sufficient measurements will be taken with
adequate precision that for a given ramp I(t), one will know the
fields in each magnet. For the measurements to be sufficient may
require certain ramp shape strategies to avoid unacceptable
sensitivity of results on details of previous history achieved.
\item Two effects create time dependent fields, {\em i.e.} fields
which are not in phase with the magnet current. Eddy currents (mostly
in the beam pipes) produce small amplitude reductions and significant
time delays. We could account for these in the control algorithms but
that is not described in this document. In addition, there are time
dependent effects in the response of the iron (such as magnetic
viscosity) which can modify accelerator magnetic fields. Some small
effects in Fermilab accelerators have been attributed to these
phenomena. We believe these are small effects and we have no
measurements of them at present. They will not be discussed further
here.
\item We assume that some strategy, such as the reset strategy
described above, will permit one to describe the beam parameter
and magnet current ramps with sufficient precision, independent
of the previous ramp history design.
\item We divide the magnet systems into major ramped systems
(dipole, quadrupole, and sextupole) with which control of the
accelerator parameters shown in item~\ref{It:MagBusses} are
maintained, and correction systems (skew quad, steering dipole,
harmonic quadrupole...) which we assume are not significant in
the control of those parameters. The exception is the dipole
correction system, whose interaction with machine momentum may
require monitoring.
\item The correction quadrupoles, skew quadrupoles, additional
correction sextupoles as well as the higher order correction elements
do not interact directly with the five beam parameters which we are
controlling. In particular, we will consider a machine in which the
horizontal and vertical motions are uncoupled. The control of skew
quad may need to reference the tables described here in order to
maintain decoupling through the ramp.
\item The closed orbit of the working accelerator will not pass
through the centers of quadrupoles and sextupoles. We will assume
that the dipole (steering) and quadrupole (tune) effects of these
do not require explicit consideration in momentum and tune calculations.
\end{enumerate}
\section{Program Overview}
Based on the assumptions below, we can specify the following interactions
to create the control of the magnet current:
\begin{center}
\fbox{\parbox{12ex}{{\Large Operator Interface}}} \hfill\mbox{\Huge$\Leftrightarrow$}\hfill \fbox{\parbox{16ex}{{\Large Application Program}}} \hfill\mbox{\Huge$\Rightarrow$}\hfill \fbox{\parbox{27ex}{{\Large Ramp \\Hardware/Software}}} \hfill
\end{center}
%\begin{picture}(\textsize,6ex)
%\end{picture}
%\begin{verbatim}
%
%Operator Interface ==> Application Program ==> Ramp Hardware/Software
%
%\end{verbatim}
where all subtle and/or complex calculations (which depend on
details of magnet and lattice properties) are hidden in the
Application Program. Both the Operator interface and the Ramp
system (MECAR...) will be presented with a machine description
which is simple and intuitive. The working description of the
ramps is captured in two sets of tables which reference a common
set of time markers. One set of tables describes accelerator
parameters, another set describes the currents in various busses.
The Application Program(s) creates these tables based on input
from the Operator Interface including selected `calibration'
information about the lattice, magnets, and power systems which
it will obtain from the accelerator database. The same time
slots (or perhaps all being at most subsets of a single most
detailed time list) are used for all tables. By assumption, these
are `played' with sufficient fidelity to permit most systems to
assume that they are exact. Real time feedback based on actual
current achieved will be required within the current control
loop, but otherwise, if at all, only by the RF systems.
Various signals will be broadcast including, {\em e.g.}, p and
dp/dt (at nominal radius) from the power supply system and $f_{rev}$
or $f_{rf}$ from the llrf system. Curve generator systems which
require such input will have it available.
Significant properties of the accelerator and magnets will be
required to perform the needed calculations. These will not be
coded into the application, but will be identified to the
application by names and values will be retrieved from the database.
\section{Ramp Specification}
By accepting a few rules, the description of the system and its
implementation can be facilitated. In Appendix~\ref{RampDesc} we will
discuss possibilities for the actual specification options
which may be used for passing information (points for an f.i.r.
filter, points and derivatives for a ``Taylor Series'' ramp
specification...). But we will assume that for any breakpoint
at which a machine operator wishes to specify any beam property,
there will exist (or he can trivially create) a momentum
specification. It is proposed that these be locked to some
suitably high frequency clock, assumed for the purposes of
discussion to be a 1440 Hz.
Ramped parameters to be specified are described as basic ramp which
can be (perhaps) described by the machine designer from design studies
and offset ramp which is the (small) change in the design characteristics
selected by the machine operators to tune for optimal performance.
Some may be grouped in sets with the assumption
that the operator interface program will maintain consistent sets
based on a mixture of specification inputs. We assume the
following list of beam parameters require ramp specification:
\begin{enumerate}
\item $p$ (momentum), $\dot{p}$ (pdot), $\ddot{p}$ (pdoubledot) and momentum offset
\item $\nu_x,\nu_y$ for nominal tune and $\delta\nu_x,\delta\nu_y$ tune offset
\item Design chromaticity $\chi_x, \chi_y$ and $\delta\chi_x, \delta\chi_y$ offset
\item $r_{pos}$ (radial position offset).
\end{enumerate}
With these the following control device parameters are specified:
\begin{enumerate}
\item DCUR (for dipole current)
\item FQCUR, DQCUR (for horizontally focusing and defocusing quad current)
\item FSCUR, DSCUR (for horizontal and vertical sextupole current)
\item RPOS (radial position offset)
\end{enumerate}
We observe that these can be represented either as many separate time-property
tables or as a single large table (with perhaps some empty entries).
We designate this table as the MI Acceleration Profile Table.\\
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c||}
\hline
time & p & $\nu_x$&$\nu_y$&$\chi_x$&$ \chi_y$\\ \hline
1.0000 &8.9 &26.42 &25.40 &-4.8 &-4.8 \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
%\end{tabular}
%\begin{tabular}{|c|c|c|c|c|c|}
\hline\hline
time &DCUR&FQCUR&DQCUR&HSCUR&VSCUR \\ \hline
1.0000 &1000 &201 &200 &1.5 &1.9 \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
& & & & & \\ \hline
\end{tabular}
\end{center}
\caption{MI Acceleration Profile Table}
\end{table}
Note that, with the assumption that all ramps lock to a specified
common time base, the above table can be added to with any/all
properties and devices, thereby representing, conceptually, all
of the needed ramped properties. Interpolations or other
operations which are needed for specifying either beam or device
ramps only interact with points which can be shown on this table.
We note that the number of columns which can be included in this
table is quite large. It is likely that the operator interface
will need to allow flexibility in selecting which columns are
displayed or manipulated.
\section{Basic Equations}
\label{BasicEq}
In Reference~\cite{PAC97:MI_PS_Control}, we have presented the basic equations for
particles on the design orbit of a synchrotron. We will be satisfied
here to supplement them with equations for the momentum effect of a
radial offset created by the RF system. The llrf system controls the
momentum achieved in the accelerator using feedback on beam
parameters. If a horizontal (radial) position detector is located at
a ring location, $s$, with dispersion $D(s)$, the measured position
change from that of the nominal closed orbit is related to the
difference in momentum from the nominal closed orbit momentum by
\begin{equation}
\delta r = \delta x = D(s) \; \frac{\Delta p}{p}
\end{equation}
from which one concludes that if $p_{o}$ is the momentum of beam on
the offset orbit and $p_{co}$ is the momentum on the unperturbed closed orbit
\begin{equation}
p_{o} = p_{co}(1+\frac{\delta r}{D})
\end{equation}
While this relation correctly describes the effect of requesting
the llrf system to produce a change of $\delta r$, great care will
be required to integrate this into the control of accelerator parameters,
principally because a local orbit bump with steering dipoles can also
modify the x position of the beam at the position detector used by
the llrf system. We will assume that effort which is not currently
available will be required before one will consider using RPOS offsets
as part of operationally tuning the machine. We assume that the
llrf system is directed to tune beam to the center of the aperture
except for special studies\footnote{Thanks to Dave Capista for interesting
discussions of this point.}.
Please refer to Reference\cite{PAC97:MI_PS_Control} for the relations between
accelerator parameters, magnet properties and currents.
\section{Momentum and the Dipole System}
The dipole field controls the momentum. The input to calculating the
required dipole ramp is the specified p vs t curve. We may have
dipole measurements on the exact ramp which we require\footnote{The
Magnet Test Facility is unable to match the design downramp ramp rate.
It is thought that that is not relevant.} If we haven't measured the
exact ramp, we understand the hysteresis and saturation effects well
enough that we can calculate the $\int B\; dl$ {\em vs.} $ I$ anyway.
We will use appropriate averages of the installed dipoles if that is
required, or perhaps of all dipoles, if that is sufficiently accurate.
The mapping from $\int B \;dl$ to momentum (p) is linear. Since the
`input' is a curve of p vs t, several steps of interpolation may be
required. One will construct some correct curve of $\int B \; dl$ vs
$I$. One may have to interpolate on both axes to provide the $\int
B\; dl$ and I at the specified times. The resulting curve should be
`absolute' since the momentum (on the design orbit) is determined
`simply' from the integral field around the ring.
The energy and momentum of the accelerator are governed by two
basic equations.
\begin{equation}
p = \frac{e}{2 \pi} \int_C B_y \; ds = e (B \rho)
\label{Eq:momentum1}
\end{equation}
\begin{equation}
f_{rf} = h f_{rev} = \frac{h \beta \gamma}{2 \pi R}
\end{equation}
where $p$ is the proton momentum in GeV/c, $e$ is the proton electric
charge, $\int B_y \; ds$ is the integrated dipole field in Tesla-m,
$f_{rf}$ is the acceleration frequency and $f_{rev}$ is the revolution
frequency in Hertz and $2 \pi R$ is the path length of the protons in
meters. These equations apply to protons (and antiprotons) on the
design orbit for the machine. We wish to associate the field integral
with the measured properties of the dipoles. For this, we need to
assume
\begin{enumerate}
\item The zeroth harmonic of the horizontal correction dipoles is
sufficiently small.
\item The net steering due to the quadrupoles is small.
\item The time delay between the field and the current can be neglected.
\item The RF acceleration system has accelerated the beam to keep the
protons on the design orbit\footnote{In Section~\ref{BasicEq}, we
showed how the momentum is affected by a finite radial offset. I
assume that this is not to be taken into account in the calculation of
dipole current since the operators of the machine will avoid such
manipulations during routine operation. Is that correct? Will this
aid or interfere with diagnostics such as measurement of chromaticity?
}.
\end{enumerate}
The value of $\int B \, dl$ which is used in
Equation~\ref{Eq:momentum1} is constructed from the magnet properties
suitably averaged. Based on the lattice (or the installed magnets),
one linearly sums the required curves of $\int B \, dl$ {\em vs.} I
and history based on the number of IDA's, IDB's, IDC's and IDD's in
the lattice\footnote{This should be constructed outside of this
framework and referenced from the database. We need to specify
how such curves are stored and referenced.}.
\subsection{Momentum Specification}
Successful operation of the accelerator demands that care be taken
in the demands which are placed on subsystems at critical times. Both
the RF and dipole power supply system have ranges of parameters which
demand great care in specifying the momentum. Initial acceleration
at the end of the injection period has traditionally required great
care. A typical solution has utilized a parabola for the initial
acceleration. The operator interface will have to provide techniques
to specify a sufficiently smooth and precise curve, including, one
assumes, no acceleration, linear acceleration and parabolic acceleration.
But, whatever the desired ramp, the application program will create
a momentum profile and any required derivatives of that profile which
are useful.
\subsection{Dipole Current Calculation}
The calculation of the desired current consists of matching the
required field profile to the specified current profile in 6 regions
of the ramp: injection, acceleration, peak field, de-acceleration,
ejection, and ramp reset. The required algorithm will be described
elsewhere. The output of this algorithm is a dipole current profile
in the time slots specified in the MI Accelerator Profile Table.
\section{Tune }
Quadrupole fields are used to set the accelerator tunes ($\nu_x,
\nu_y$). Operationally, the tunes reflect ratios of quadrupole fields
to the momentum (set by the dipole field). Since both the dipole and
quadrupole have significant saturation, it simplifies implementation
to use the momentum as the link between them. The actual tune
equations depend on the lattice which is actually achieved. The ratio
of tune to field is not linear so we will describe the coefficients
for achieving the design tune separately from coefficients to achieve
small corrections. Additionally, a `tune modifier' which operates
with respect to a set of measured tune offsets may prove useful.
Whether this or some different system is finally worked out, we assume
that the operator interface can provide the desired tune ramps to the
Application Program. Let me here describe a scheme. We use the
lattice design program to relate integrated quadrupole fields to the
momentum for a specified `design tune' which is taken as energy
independent. The Application Program is provided with this design
tune and could calculate quad fields which correspond. The Operator
Interface also provides a pair of tune vs time (or momentum) tables
which are used as tune offset parameters. To determine quadrupole
field requirements based on this tune offset information, the
Application Program may use both lattice calculations and/or measured
machine parameters which have already been corrected by the actual
lattice which was achieved. Whatever the combination of inputs, the
application program provides to the power supply hardware (and back to
the user interface) a pair of current vs time curves for the two
quadrupole buses.
We assume that the design tune is determine by
\begin{equation}
\underline{\nu}=\underline{Q}\,\underline{k_1}
\end{equation}
where $\underline{\nu}$ is a column vector of $ \nu_x$ and $\nu_y$,
$\underline{k_1}$ is a column vector of momentum normalized quadrupole
strengths and $\underline{Q}$ is the required sensitivity matrix (See
Reference~\cite{PAC97:MI_PS_Control}). Precise control of the tune
will be achieved by applying linear corrections based on the tune
offsets demanded by the operators and by the tune correction learned from
accelerator measurements to correct the model for the achieved properties.
Both can be represented as
\begin{equation}
\delta\underline{\nu}=\delta\underline{Q}\, \delta\underline{k_1}
\end{equation}
\section{Chromaticity}
For the sextupole, one wishes to provide, from the user interface, a
simple nominal chromaticity (this may require a ramp-based
specification, since a single value is not normally acceptable), and
an offset curve for $\chi_x$ and $\chi_y$ vs time. One will wish to
create the desired sextupole field but here it depends on both the
sextupole created by the sextupole circuit, and the sextupoles induced
by the dipole (and perhaps quadrupole ???) magnets. For the dipole
terms, one has history-dependent hysteretic fields, ramp-rate
dependent eddy current fields and very important saturation effects.
In addition the sextupole strength of the sextupole magnets is
hysteretic. Given a dipole and quadrupole ramp, the properties of the
dipole eddy current term, the sextupole magnet measurements and a
suitable algorithm, one can calculate the desired current ramp for the
sextupole magnets. This is not trivial, but if we accept this model
in which all calculations are for specified momentum ramps, a
self-consistent set of results can be obtained in a straightforward
manner independent of what form the relations between fields and
currents are presented in. If we wished to calculate this
'on-the-fly', it would be less obvious unless the the relations are
both analytic and invertible.
The Main Injector design uses two chromaticity sextupole circuits:
MISEXF and MISEXD (or should we call them MISEXX and MISEXY)
(with magnets at the horizontally and vertically focusing quadrupoles).
As discussed in Bogacz and Peggs\cite{Bogacz:CCMISS}, the chromaticity which
results is described by the equation:
\begin{equation}
\underline{\chi}=\underline{\chi_0}+
\underline{S_D}{K_{2D}}+\underline{S}\underline{k_2}
\end{equation}
One can think of this as expressing the contributions due the natural
chromaticity, sextupole fields at dipole locations, and sextupole
fields at quadrupole locations. As above, we will expect the sensitivity
parameters $\underline{S}$ and $\underline{S_D}$ to be determined by first calculating
their values from a lattice model,
then measuring them in the machine
and adding a difference (tuning) term
to the model term.
\section{Concerns and Issues}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IDB100-0_nlfrac.eps}}}}}
%\resizebox{3.25in}{!}{\rotatebox{270}{
%\includegraphics{2P008f2.eps}}}
\caption{Nonlinear portion of IDB100-0 strength as a fraction
of the linear field.}
\label{Fig:dip-fracnl}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{IQC020-0_fracnl.eps}}}}}
%\resizebox{3.25in}{!}{\rotatebox{270}{
%\includegraphics{2P008f2.eps}}}
\caption{Nonlinear portion of IQC020-0 strength as a fraction
of the linear field.}
\label{Fig:quad-fracnl}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{ISA010-0_fracnl.eps}}}}}
%\resizebox{3.25in}{!}{\rotatebox{270}{
%\includegraphics{2P008f2.eps}}}
\caption{Nonlinear portion of ISA010-0 strength as a fraction
of the linear field showing full range of current. }
\label{Fig:sext-fracnl}
\end{figure}
\begin{figure}[tbh]
\centerline{\epsfxsize=4.5in
\rotate{\rotate{\rotate{\epsfbox{ISA010-0_fracnl_hires.eps}}}}}
%\resizebox{3.25in}{!}{\rotatebox{270}{
%\includegraphics{2P008f2.eps}}}
\caption{Nonlinear portion of ISA010-0 strength as a fraction
of the linear field with limit range of current to illustrate
low field behavior. }
\label{Fig:sext-fracnl_hr}
\end{figure}
\begin{enumerate}
\item We assume (but must confirm) that the currents required for each
operational ramp (150 GeV to Tevatron, 120 GeV Pbar production...) is
sufficiently independent of previous ramp(s) as to allow one to
fully specify the ramp and AT MOST have a separate small correction
to apply which depends upon the previous history.
\item The language to describe reset current issues needs to be
resolved and perhaps some issues in hardware are involved. We believe
that the quadrupole and dipole bussed will require 'reset' currents
below the injection current. It would be 'natural' to include the
reset portion of the ramp with the up-ramp portion, since that pairing
may permit more flexibility within a simple system. But clock
definitions may preclude that. It should be examined. Note that a
special concern will be the \$2D cycles which remain at the 8 GeV
injection level. For now we may wish to assume that the reset portion
of the ramp occurs just prior to the end.
\item The plan described for including measured tune and chromaticity
effects has been presented quite casually. Considerable thought is
required to permit a realistic system using measured tune and
chromaticity effects, which will be time dependent, to be inserted
into the framework above. I speculate that it can be done without
compromising the fundamental plan. This requires thought.
\end{enumerate}
\section{Importance of Nonlinear Fields}
The proposed mechanisms for dealing with hysteresis and saturation
which have been described will require substantial new analysis and
software effort. Is it required? In Figures~\ref{Fig:dip-fracnl},
\ref{Fig:quad-fracnl},
\ref{Fig:sext-fracnl}~and~\ref{Fig:sext-fracnl_hr}, we plot the
non-linear field as a fraction of the linear field.
We note that for the MI Dipole, high field saturation has the largest
non-linear effect. But the hysteretic low field non-linear terms are
much to large to ignore in setting the momentum precisely. For the
sextupole, the saturation creates only about a 1\% change in the
field. However, the hysteresis at low fields is very important.
Since the current at which the fields are reset varies with the ramp,
there are effects which are typically 10\% of the linear field. To
see what effect this will have on machine operation, one needs to
examine the sensitivity results presented in
Reference~\cite{PAC97:MI_Sextupoles}.
A quantitative measure of the importance of hysteresis can be obtained
by observing that the hysteresis is characteristically weakly current
dependent and approximately the same magnitude as the integrated
remanent field (See Figures~\ref{Fig:hr_bnl}, \ref{Fig:quad_hyst}, and
\ref{Fig:sext_hyst}). We can calculate the current required to create
such a change. By comparing this characteristic current ($I_{Rem}$)
to the currents of interest (such as the current required at injection
or transition), we can readily ascertain the relative significance of
hysteresis and gauge with what care we must set the hysteretic fields.
These results are shown in Table~\ref{Tab:HysSensitivity}\footnote{
The remanent fields shown in Table~\ref{Tab:HysSensitivity} for
dipoles are from Reference~\cite{MT14:Dipole}. Quadrupole and
sextupole remanent field averages were calculated from measurements
extracted from the {\bf results} database. Quadrupole sensitivity is
calculated using results in Reference~\cite{DEJohnson:MIQLRCTP}.
Sextupole sensitivity is calculated with results from
Reference~\cite{PAC97:MI_Sextupoles}.}
\begin{table}[tbh]
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& & & & &Injection \\ \hline
N &Magnet & $B_N^{Rem}$ & $B_N^{Lin}/I$ & $\delta I_{Rem}$& Effect \\ \hline
1 & IDA & .01335 T-m & .001203 T-m/A & 11.09 A & $\delta p/p\approx 2\%$ \\ \hline
2 & IQC & .06654 T-m/m & .01459 T-m/m/A & 4.56 A & $\delta \nu \approx 0.89$ \\ \hline
3 & ISA & .3311 T/m & .2022 T/m/A & 1.637 A & $\delta \chi \approx 6.5 (3.3)$ \\ \hline
\end{tabular}
\caption{Characteristic effects of hysteresis. The difference between
up ramp and down ramp fields is typically with 30\% of the magnitude
of the remanent field. $\delta I_{Rem}$ characterizes the corresponding
change in current on the up ramp to achieve the field obtained at the
same nominal current on the down ramp. The effect on machine parameters is
characteristically proportional to $1/p$. The magnitude of the effect
at injection is shown in the final column.}
\label{Tab:HysSensitivity}
\end{table}
\section{Summary}
The following items would be labeled conclusions, were they that, but instead
can be appropriately labeled opinions:
\begin{enumerate}
\item The portions of strengths of magnetic fields for the Main Injector
magnets which are not linear with magnet current are sufficiently large as to
demand careful attention in specifying the magnet current ramps. The deceleration option
has made this significantly more important.
\item A hysteresis model of the sextupole which make no error larger
than 5\% of the remanent field is sufficient. At low momentum, it will be useful
if the dipole and quadrupole models make errors which are typically smaller than 0.5\%.
\item The assumption that the current control system achieves the specified ramp
is critical to the design which is outlined above. Since the beam dynamics requirements for
current control are quite tight, unless we choose very aberrant ramps, the
hysteretic effects should not create a more demanding requirement for
current regulation control.
\end{enumerate}
Issues which have arisen in this context include:
\begin{enumerate}
\item It is known that the saturation differences between Main Ring
84$^{\prime\prime}$ quadrupoles and the IQC and IQD quadrupoles which
were fabricated from the new steel for the Main Injector result in
strength ratios which change as a function of current. One would
expect that this will result in changes in the lattice at high
fields. What will be required to permit sufficiently well controlled
operation? Since the effects will predominately affect the lattice
above 120 GeV only, can we ignore them for acceleration and make only
a final correction just prior to 150 GeV extraction? OR will we need
to provide a series of lattices which describe operations at different field
levels?
\item In like manner, one may wish to add to the calculated machine
description with parameters determined from measurements. The
description presented here assumes that the accelerator description is
momentum independent. what mechanism will be used to permit one to
use a set of several (many) lattice descriptions in place of one.
Will matching at points be sufficient or will derivatives also need to
match, for example?
\item The calculation of the required ramps to achieve the magnetic
fields which match a specified set of machine parameters must involve
maintaining information about the hysteresis state of each set of
magnets. A single application which does the conversion from required
field to ramp is desirable.
\item Description of the ramps must be passed from the applications
programs to the power supply control software. it will soon be
appropriate to select the description which is to be used. (see
Appendix~\ref{RampDesc}).
\end{enumerate}
Finally, we know of one effect which violates the assumptions of this
discussion. The coherent Laslett tune shift~\cite{Gerig:MCCLTSFMR}
depends upon the beam current and thus cannot be pre-specified from
the ramp design. The correction will have to be applied in real time
by the current control system (MECAR). this is likely to be
sufficiently small as to not change the hysteretic state, thereby not
modifying the basic assumptions from which we are working. Are there
other such effects?
\appendix
\section{Options for Ramp Description}
\label{RampDesc}
The precise description of a time sequence of values is a problem
which does not have a unique solution within the Fermilab Accelerator
control system. Generally, to provide sufficient information one can
pass lessor information on a more detailed set of values [(time,value)
pairs, for example], or pass fewer points (less data) but with more
detailed information or a more complex data processing scheme for that
data. We will describe schemes which imply an increasingly
complex data processing for interpretation of the data passed.
\begin{description}
\item[Value Tables] One passes complete sets of times and values as (t,v)
ordered pairs with sufficient time resolution to permit the complete specification
of the ramp with the required accuracy.
\item[Linear Interpolation] One passes a set of times and values as (t,v)
ordered pairs. The receiving program is instructed to provide linear
interpolation between points. Simple variations could include a spline
interpolation.
\item[Taylor Series Interpolation] This system, used for some existing
controls problems passes the ramp definition as sets of ($t, v, \dot{v}, \ddot{v}
\ldots$) with the specification that the receiving program interpret them
as coefficients for a Taylor series.
\item[Finite Impulse Response] One passes a set of times and values as (t,v)
which are to be processed by an FIR filter program whose output describes
the desired ramp.
\end{description}
The correct choice of ramp specification has elicited some vigorous discussion.
It has been said that the clear engineering choice is an FIR. Some limitations
in fidelity of the ramp have been observed with the Taylor Series specification
as implemented for the Main Ring. This document will not resolve this discussion.
\section{Differences from Main Ring System Operation}
\label{MRdifference}
This construct of the overall system is different from the present
Main Ring system primarily by accounting for the non-linear behaviors of
magnets in a coherent and simple way. This is demanded due to several
differences:
\begin{enumerate}
\item For 150 GeV operation, the MI Dipoles experience a much larger
($\approx 10\%$) saturation field error (when compared to a linear B
vs. I. This is mostly because there are 301 1/3 {\em vs.} 774 dipoles
of $\approx$ 6 m length so the Main Ring dipoles were operating at
only 0.7 T whereas the Main Injector requires more than 1.7 T. The
dipole control of the Main Ring uses a saturation table which is
important for operation at energies above 400 GeV, but other systems
relied upon scaling the dipole current to determine momentum.
\item Deceleration is expected by design in the MI and differences
of up to 1\% are expected (current dependent) in the down ramp vs up
ramp behavior.
\item Main Ring operation determines the eddy current correction for
chromaticity on-the-fly. The higher ramp rate of the main injector
creates sufficient eddy current sextupole that explicitly accounting
for it using well-confirmed algorithms and more direct determination
of the field sources is likely to be more important than for the Main
Ring.
\item Hysteretic differences associated with different ramp
reset values can be accommodated in a straightforward manner in such a
system.
\end{enumerate}
Since these dominant issues are to be accounted for explicitly, the
parameters which are provided for operator tuning will be free to solve
smaller or more subtle effects without being forced to provide for
larger, well understood effects.
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