\section{Induction Accelerators for the Phase Rotator System}
\label{DandPR:ind-linac}
%\centerline{Lou Reginato, Simon Yu, Dave Vanecek}

%\centerline{\it Lawrence Berkeley National Laboratory}
%\subsubsection{Introduction}
The principle of magnetic induction has often been applied to the
acceleration of high-current beams in betatrons and in a variety of
induction accelerators~\cite{IL:ref1}.  The induction linac (IL)
consists of a simple nonresonant structure where the drive voltage is
applied to an axially symmetric gap that encloses a toroidal
ferromagnetic material. The change in flux in the magnetic core
induces an axial electric field that provides particle acceleration.
This simple nonresonant (low-Q) structure acts as a
single-turn transformer that can accelerate beams of hundreds of amperes
to tens of kiloamperes, limited only by the drive
impedance. The IL is typically a low-gradient structure that can
provide acceleration fields of varying shapes and time durations from
tens of nanoseconds to several microseconds. The efficiency of the IL
depends on the beam current, and can exceed 50\% if the beam current
exceeds the magnetization current required by the ferromagnetic
material. The acceleration voltage available is simply given by the
expression $V=A dB/dt.$ Hence, for a given cross sectional area {\it A} of material,
the beam pulse duration influences the energy gain. Furthermore, there is a
premium put on minimizing the core diameter, as this impacts the total
weight, or cost, of the magnetic material. Indeed, the diameter doubly impacts
the cost of the IL, since the power to drive the cores is
proportional to the volume as well.  

To meet the waveform requirements during
the beam pulse, we make provisions in the pulsing
system to maintain the desired $\frac{dB}{dt}$ during the useful part of the
acceleration cycle. This can be done in either of two ways: by using the
final stage of the pulse forming network (PFN) or by using the pulse-compensation network in close proximity to the acceleration
cell.  

The choice of magnetic materials is made by testing
various materials, both ferromagnetic and ferrimagnetic;  not only to determine the properties that
are essential in this application, possible
materials will include the nickel-iron, silicon steel, amorphous, and
various types of ferrites, but on the energy losses in the
magnetization process, which directly impact the cost.

\subsection{Accelerator Waveforms}
Parameters and pulse shapes from this study, compared with Study-I~\cite{IL:study1} 
have evolved
toward considerably improved physics performance and less demanding
accelerator waveforms, avoiding the need for multipulsing.

%The Feasibility Study 2 of the Neutrino Factory and Muon Collider have
%resulted in the architecture on Fig.~\ref{CPR:fig01} and the acceleration waveforms
%shown on Fig.~\ref{CPR:fig02}. The waveforms for Induction 1(Ind1), Induction
The present baseline design  
results in the acceleration waveforms
shown in Fig.~\ref{CPR:fig02}. The waveforms for IL1, IL2, and IL3 
are all unipolar.

%\begin{figure}
%\begin{center}
%\centerline{\includegraphics*[width=5in]{Fig01.eps}}
%\caption{Neutrino Factory and Muon Collider Specifications}
%\label{CPR:fig01}
%\end{center}
%\end{figure}
\begin{figure}
\begin{center}
%\vspace{0.1in}
\input{ind2-new.fig} % to use the ssme as in Capture and Phase Rotation
%\centerline{\includegraphics*[width=4.5in]{CPR-Fig02.eps}}
\caption[Acceleration waveforms]{Acceleration waveforms for induction 1, 2 and 3.}
\label{CPR:fig02}
\end{center}
\end{figure}


Although IL1 and IL3 could possibly be combined,
it became preferable for
technical reasons, and to reduce the risk factors, to separate the two
functions (with a small penalty in economics). (A combined IL2 and IL3
would require the application of \textsl{branched magnetics} to achieve
two waveforms that are independently controllable in shape and
timing. The branched magnetics approach could lead to a 5-10\% cost
savings, but at more risk, since this approach has only been applied to
small benchtop prototypes but not to presently operating induction
accelerators. This approach will be examined in Appendix~\ref{Chap:App-c}.)

\subsection{Magnetic Material}
A number of induction linac have been constructed in the past
that cover the pulse duration of the three units
required by the Neutrino Factory. None of these
accelerators, however, has gradients and energy gains that are as
high. To satisfy the requirements in an economically reasonable design,
it is imperative to choose a magnetic material, and a pulsing system,
that minimize the cost but still achieve the reliability and
performance required.  

In the past two decades, great strides have
been made in the development of a magnetic material that is replacing
all previous ones in the 60-Hz power industry because of its low loss,
ease of manufacturing, and low cost. Several alloys are made in ribbon
form by rapidly quenching a stream of molten material on a cold
rotating drum.  The ribbon thickness is typically 25~$\mu$m and can be of
any width from 5 to 20~cm. Because the ribbon is so thin, and has higher
resistivity than other ferromagnetic materials, it is directly
applicable to short pulse applications. In short pulse applications
where the rate of magnetization $(dB/dt)$ is very high, tens of volts
are generated between the layers of ribbon when it is wound into a
toroid. Thin insulation such as 2-4~$\mu$m Mylar must be used between
layers to insure that the ribbon layers are sufficiently insulated to
hold off the voltage generated.  

The soft magnetic properties can be
improved by annealing. Unfortunately, this procedure, although well 
below the
crystallization temperature, embrittles the material, making it nearly
impossible to wind into a toroid. Annealing can be done after winding
if the insulating material between layers has a sufficiently high melting 
point. Annealing is not an option when Mylar is used. Coatings have been
developed that allow annealing after winding, but at the present time
they are not fully developed and do not hold off sufficient voltage
per turn. Because the losses at high magnetization rates are almost
entirely due to the eddy current losses, very little is lost in our 
application using
the material ``as cast'' or unannealed.  

To choose the appropriate alloy of this
amorphous material it is important to measure the properties such as
flux swing ($\Delta B$) and magnetization ($\Delta H$) at the appropriate
pulse duration or magnetization rate ($dB/dt$). Figure~\ref{CPR:fig03} 
shows the losses
in J/m$^3$, at different rates of magnetization. It can be seen that
above $T/\mu$s the losses increase linearly with
magnetization rate. From Fig.~\ref{CPR:fig03} it appears that the lowest loss
material is the alloy 2705M with the lowest $\Delta B$ of about 1.4 T
while the highest loss material is 2605CO with a $\Delta B$ of about
3.3 T. The optimum material is selected by considering the $\Delta B$, 
the losses (J/m$^3$) and the cost per kilogram. Two alloys which
are not plotted on this chart (Fig.~\ref{CPR:fig03}) are the 2605SA1, used
exclusively in the 60-Hz power industry, and the 2605S3A, which is used
in pulse transformer applications. The SA1 material offers the
potential for greatest savings since it is mass produced for the power
industry, but it has not been investigated as thoroughly as the SC or S3A
materials at the very short pulse regime of interest here. For this reason 
the SC
alloy is chosen here since it has been used recently in an induction
accelerator for radiography at the Los Alamos National Laboratory and
extensive technical and cost data exist. The S3A also offers a good
choice since it has been used extensively in the AVLIS program at
the Lawrence Livermore National Laboratory. The SA1 alloy will be
investigated in the near future as part of our R \& D program since, 
as mentioned
previously, it offers the greatest possibility for cost savings.


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=4in]{../template/report/ps-and-eps/CPR-Fig03.eps}}
\caption[Losses of amorphous alloys]{Losses (J/m$^3$) of several amorphous alloys 
as a function of magnetization rate $(dB/dt).$}
\label{CPR:fig03}
\end{center}
\end{figure}
%Fig.3-Losses (J/m3) of several amorphous alloys as a function of magnetization rate (dB/dt).

\subsection{Induction Linac 1 Cell}

%From the Study II specifications shown on Fig.~\ref{CPR:fig01}, Induction
From the specifications~\ref{CHAP:intro} in this Study,  IL1
is 100~m in length and has the acceleration waveform shown in
Fig.~\ref{CPR:fig02}. The waveform has a full-width-half-maximum (FWHM) of 
about 180~ns,
with an approximately exponential rise time of 100~ns and a fast fall
time. The significant portion of the acceleration cycle is during the
rise time; the fall time is unimportant. In fact, the fall time
will be longer than indicated by the waveform since the energy stored
in the cell inductance will decay with a time constant $t=L/R$ of the
drive circuit. The $L/R$ of the fall time will be similar to the
rise time, hence, the actual FWHM will be about 250~ns. Since a 1-m
section must allow axial space for the cryogenic feed lines
and for vacuum pumping, the maximum allowable space for the core is
712~mm. To obtain the lowest cost for the amorphous
material, the alloy should be cast in widths of 101.6~mm
(4") or greater. Manufacturing limits, therefore, dictate a
maximum number of cells that is a multiple of 101.6~mm. 
For our case, we select 7 cells.  

From the required
gradient we now have a basis for calculating the cross-sectional area
of the magnetic material for IL1. From $V=A(dB/dt)$(PF) we can
calculate the $\Delta R$ knowing the $\Delta Z$ and the packing factor
(PF=0.75).  The hysteresis loop for the two alloys,
2605SC and 2605S3A, are shown in Fig.~\ref{CPR:fig04}.


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig04.eps}}
\caption[Hysteresis for 2605SC and 2605S3A]{Hysteresis for 2605SC and 2605S3A at two different magnetization rates.}
\label{CPR:fig04}
\end{center}
\end{figure}
%Fig. 4-Hysteresis for 2605SC and 2605S3A at two different magnetization rates.

Although the total flux swing to saturation is over 2.5~T, the
actual working flux swing ($\Delta B$) is chosen as 2.0~T so that the
pulse generator drives into a more linear load. The required voltage
for each of the seven cells that constitute one meter of acceleration
is 214.3~kV. The actual cross-section is 
\begin{equation}
A=(r_2-r_1)*w=\frac{V\Delta t}{\Delta B(PF)}
\end{equation}
so
\begin{equation}
(r_2-r_1)=frac{(214.3 \times 10^3)(250 \times 10^{-9})}{(2.0)(0.75)
  (r_2-r_1)}=0.325~\textrm{m}.
\end{equation}
The inside radius of the core is set by the outside radius of
the superconducting solenoid at a minimum of 0.4~m. Preliminary
calculations of the leakage flux at the solenoid gaps indicate that
this flux, which is orthogonal to the magnetization flux, can be of the
order of a few thousand gauss at the 0.4 m radius. From previous
tests for the Advanced Test Accelerator (ATA) and the
Dual-Axis Radiographic Hydro Test (DARHT)~\cite{IL:indref1a} accelerator, 
this is acceptable. Nonetheless, this issue should be investigated further with
laboratory tests to insure that the flux swing of the induction cell
is not reduced by this stray flux. To be conservative, we set the inside
radius of the actual amorphous material at 500~mm. The
magnetizing current and the losses can now be calculated. The 
magnetizing force $\Delta H =
\Delta I/\pi d$ where $d$ is the average diameter or $d=r_2+r_1=1.35$~m. From
Fig.~\ref{CPR:fig04} for a 0.25~$\mu$s saturation time we find that 
the magnetizing force is 
$\Delta H$~=~1200~A/m or $\Delta I$~=~5,087~A, and the loss is 
$U=V\   \Delta I \  \Delta t$  or
$U=(214.3 \times 10^3)(5.087 \times 10^3)(250\times 10^{-9}) = 272.5$ 
J/cell. The magnetic
material volume, including the mylar insulation is 
$V=\pi(r_2^2-r_1^2)(\Delta z),$ or
$V$=0.151~m$^3.$ With a packing factor of 0.75, the actual volume of
amorphous material is 0.113~m$^3$, and at a density of 7290~kg/m$^3$ 
it weighs 825~kg. The
core losses could also have been calculated from Fig.~\ref{CPR:fig03}, which
shows that the losses per cubic meter at a magnetization rate
$dB/dt=2.0$~T/0.25~$\mu$s, are 2~kJ/m$^3$ for a total of 262~J, slightly 
lower than the estimate
above. The loss calculations above determine the drive power
required by the pulse generator for one cell. For seven cells $P=7VI,$ and 
with $V=214.3$~kV, $I=5.09$~kA, $P=7.63$~GW and the impedance $Z=V/7I=6.0\Omega$.


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=4.25in]{../template/report/ps-and-eps/CPR-Fig05.eps}}
\caption[Induction cell cross-sections]{(a) Cross section of a 
single cell with compensation network boxes;
(b) cross section of a 2 m section.}
\label{CPR:fig05}
\end{center}
\end{figure}

%Fig.5-          (a) Cross-section of a single cell with compensation network
%        (b) Cross-section of a two meter section


\subsubsection{High Voltage Design of Cell}

Figure 5.8 shows a cross section of the induction cell. The cell is driven
by two high voltage cables at 180$^\circ$. The high voltage cables plug into
two connections, of the type used on the DARHT accelerator, that are part of
the compensation network box. The acceleration gap is 1~cm and is oil
filled. From Fig.~\ref{CPR:fig06}, which shows the voltage breakdown in oil for
different pulse durations and surface areas, it appears that the
safety factors are more than adequate, that is, the actual breakdown
is about twice the operating voltage. The highest voltage stress
occurs at the outside radius of the core where one half of the driving
voltage appears from each side of the core to ground. Insulation
is done with ten layers of 50~$\mu$m mylar with oil
impregnation.



\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig06.eps}}
\caption[Pulse voltage breakdown]{Short-pulse voltage breakdown in oil.}
\label{CPR:fig06}
\end{center}
\end{figure}
%Fig. 6- Short Pulse Voltage Breakdown in Oil

The oil-to-vacuum interface insulator is designed so that on the
vacuum side the field lines form a 30$^\circ$ or greater angle with the
insulator to achieve the highest possible voltage holding. The
empirical curve in Fig.~\ref{CPR:fig07} shows the voltage flashover for
different angles. For our design, the maximum surface gradient on the
insulator is nearly one order of magnitude lower.


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig07.eps}}
\caption[Flashover voltage]{Flashover voltage with a 30-ns  pulse for 
different cone angles.}
\label{CPR:fig07}
\end{center}
\end{figure}
%Fig. 7  Flashover Voltage with a 30~ns  pulse for different cone angles


The highest voltage gradient occurs between the solenoid
housings. Here the spacing is 100~mm and the radius is 30~mm. Using a
cylindrical geometry, the maximum gradient is about 150--200~kV/cm. Figure~\ref{CPR:fig08}
shows field emission after 200~ns for different types of surfaces. 
A standard electropolished stainless steel surface
is marginally acceptable for our purposes.  To be prudent, the surfaces 
should be greened.  In a  
subsequent optimization, the gradient will be reduced somewhat by redesigning the nose pieces.



\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=4.5in]{../template/report/ps-and-eps/CPR-Fig08.eps}}
\caption[Current density]{Current density after 200~ns  for 
different surface preparations.}
\label{CPR:fig08}
\end{center}
\end{figure}
%Fig 8  Current Density after 200~ns  for different surfaces



\subsection{Induction Linac 2 Cell}

From the specifications shown in Fig.~\ref{CPR:fig02} for IL2, the deceleration
pulse has an unspecified rise time (from zero to a negative value) and
a fall time of about 50 ns (from a negative value back to zero) that is a
significant portion of the waveform.

Induction accelerators with pulse durations of less than 100~ns have
traditionally used nickel-zinc ferrites as the magnetic material of
choice. This choice was the appropriate one a
decade or two ago when the last short-pulse induction accelerator was
built, since the amorphous materials at that time were not of a very high quality
and were more expensive than they are today. The choice of ferrites also was
logical if one compares their losses to those of amorphous materials
at saturation times of 50~ns. We can see from Fig.~\ref{CPR:fig09} that, if full
saturation is achieved in 50~ns, the losses for ferrites (CMD 5005)
are about 800~J/m$^3$ while the losses for amorphous materials (2605SC)
are about one order of magnitude higher. That is, even though the flux swing
for amorphous materials is five times greater than those of the
ferrites, the losses are more than ten times greater (at
full saturation). On the other hand, the cost of ferrites has quadrupled
in the past two decades while the cost of amorphous materials has
decreased considerably. This makes it imperative to take another look at using
amorphous materials of the same cross section (volume) as the ferrites.
Then the flux swing would be much lower than
that at full saturation, as would be the magnetization rates, and hence,
the losses. To make the best comparison, designs were made using
both the ferrites and the amorphous materials. Using the standard
101.6~mm width $(w)$ and a $\Delta B$ (from Fig.~\ref{CPR:fig10}) for the
ferrite CMD-5005 the area $A=w(r_2-r_1)$ = $V \Delta t/\Delta B$. From 
Fig.~\ref{CPR:fig02} the
significant part of the acceleration waveform is the fall time while
the rise time is unspecified and is determined by the pulse
generators. Because of the large gap capacitance and the impedance of
the pulse generator, the rise time will be nearly the same as the fall
time so that the FWHM will be about 100~ns.


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=3.5in]{../template/report/ps-and-eps/CPR-Fig09.eps}}
\caption[Magnetic materials losses]{Magnetic material losses for different saturation times.}
\label{CPR:fig09}
\end{center}
\end{figure}


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=3.0in]{../template/report/ps-and-eps/CPR-Fig10.eps}}
\caption[Hysteresis curves]{Hysteresis curves for various ferrites.}
\label{CPR:fig10}
\end{center}
\end{figure}

Using 100~ns as $\Delta t$ and the voltage per cell $ V$ = 188~kV, the
outside radius $r_2$ = 870~mm. From Fig.~\ref{CPR:fig10}, the hysteresis curve 
for CMD-5005           
indicates that $\Delta H$ = 1000~A/m and the losses will be 500~J/m$^3$. The 
ferrite
volume, $ V=\pi(r_2^2-r_1^2)$ = 0.162~m$^3$ will result in 81~J of losses per
cell requiring a drive current $I$ = 4.3~kA. Taking the same cross-sectional
area, but using the proporties of amorphous materials, we can compare the
losses. Because the packing factor of the amorphous material will be
0.75 instead of 1, the flux swing will be 0.667~T and the
magnetization rate $dB/dt$ = 6.67~T/$\mu$s (for, 100~ns saturation). From
Fig.~\ref{CPR:fig03}, the losses for 2605SC are about 1400~J/m$^3$. The
total losses, $U$, for the amorphous material will be
$U=(1400$~J/m$^3$)(0.162~m$^3$)(0.75) or $U=170$~J/cell. We conclude 
that losses using the
amorphous material are about twice as high as those of the ferrites
and, therefore, the cost of the pulse generator will be that much
greater. Surprisingly, the economics still favor the amorphous material
because its cost is about one-fourth of that of the ferrites. Even
though the pulse generator doubles in cost, the net result is a saving
of about 10\%.  

It is interesting to note that the design for IL2
is nearly identical to IL1, that is, $r_2$ = 0.85~m for IL1 and
$r_2$ = 0.87~m for IL2, so in actuality, for manufacturing and design cost
saving, the two cells can be identical. Since IL2 is a decelerating
gradient, the induction cells for this accelerator are simply
installed rotated 180$^\circ$ from those of Induction 1.


\subsection{Induction Linac 3 Cell}

Applying the same arguments used in the design of IL1, the
FWHM for IL3 is 380~ns and the acceleration voltage
$V$ = 143~kV. The outside radius of this cell can now be calculated from
$V \  \Delta t = A \Delta B(\textrm{PF})$. The magnetization rate for this cell 
is lower
since the saturation time is longer. For the same $\Delta B$ as in
IL1, or $\Delta B$ = 2.0 T,  $dB/dt$ = 5.26~T/$\mu$s, and the magnetization
will be the average between the two cases shown in Fig.~\ref{CPR:fig04} or
$\Delta H$ = 900~A/m. Applying these parameters to the design of IL3,
from $w(r_2-r_1)$ = $\frac{V\Delta t}{\Delta B (PF)}$ we get $r_2$ = 857~mm. 
The magnetizing
current is $I=\pi H (r_2 + r_1) =3.84$~kA and the cell losses 
$U=V I  \Delta t$ =
($143 \times 10^3$)($3.84\times 10^3$)($380\times 10^{-9}) =209$~J. 
The volume of material is $V=\pi (r_2^2
-r_1^2)w=0.155$~m$^3,$ and the weight with a $\textrm{PF}$ = 0.75,  is 846~kg. 
Table~\ref{CPR:tb1} 
summarizes important parameters for the three induction accelerators.



\begin{table}
\begin{center}
\caption[Induction accelerator parameters]{Induction accelerator parameters.}
\label{CPR:tb1}
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Unit&MV/m&Length&$\Delta t$&IR&OR&$V_{cell}$&W$_{cell}$&$\Delta B$&$\Delta H$&Voltage&Current&Energy&Weight\\
& &(m)&(ns)&(m)&(m)&(m$^3$)&(kg)&(T)&(A/m)&(kV)&(kA)&(J)&(tons)\\
\hline
IL1
&1.5
&100
&250
&0.5
&0.85
&0.151
&826
&2.0
&1200
&214
&5.09
&273
&578\\
IL2
&1.5
&80
&100
&0.5
&0.87
&0.162
&886
&0.67
&2100
&188
&9.05
&170
&496\\
IL3&
1.0&
80&
380&
0.5&
0.86&
0.155&
846&
2.0&
900&
143&
3.84&
209&
474\\
\hline
\end{tabular}
\end{center}
\end{table}


It is evident from Table~\ref{CPR:tb1} that the three induction accelerators can
be mechanically identical as far as induction cells are concerned. Each pulsing
system will, of course, be different.  

The amorphous alloy taken for 
this Study was not optimized but was chosen because the most
reliable information exists for it in the short-pulse applications and the
most accurate cost data was available from a recent induction
linac constructed at LANL for radiography. It is very likely
that the alloy being mass produced for the power industry, 2605SA1,
can be substituted for the 2605SC. The SA1 material has great
potential for cost reduction, since it is mass produced in very
large quantities. We will pursue testing samples of SA1 in the near
future as part of the R\&D program, and will begin negotiations 
with the scientific staff at
Honeywell (Allied Signal) to explore making the material
in thinner ribbon and less expensive than the SC material.

\subsection{Pulsing System}
IL1, IL2, and IL3 are driven by pulse generators
with output voltages from 100--200~kV and currents in the tens of
kiloamperes at pulse durations from 50~ns to 300~ns.  The peak power
levels exceed 1~GW and, except for spark gaps, no switches
exist that are capable of operating reliably at the required
repetition rates and power levels. 
The prefered option, then, is the nonlinear magnetic pulse
compression modulator.  

The use of saturable reactors for generating
very high peak power levels was described by Melville~\cite{IL:ref2} in 1951. The
basic principle behind magnetic pulse switching~\cite{IL:indref2} is to use the large
changes in permeability exhibited by saturating ferri-(ferro) magnetic
materials to produce large changes in impedance. The standard
technique for capitalizing on this behavior is shown on Fig.~\ref{CPR:fig11}. By
using multiple stages, as shown, it is possible to compress a pulse of
relatively low power and long duration into a pulse of very high peak
power and very short duration, maintaining the same energy (except for
a small core loss) per pulse. This is exactly the technique that
allows us to use available thyratrons or solid-state devices to
initiate the pulse and then pulse compress it to the desired peak
power levels.
\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig11.eps}}
\caption[Magnetic pulse compression]{Principle of magnetic pulse compression.}
\label{CPR:fig11}
\end{center}
\end{figure}
%Fig. 11  Principle of Magnetic Pulse Compression

The principle of operation of the magnetic pulse compressor has been
covered extensively in the literature but is briefly described here
for completeness. Referring to Figure 5.14, capacitor $C_1$ charges 
through inductance $L_0$ until
inductance $L_1$ saturates, becoming much less than $L_0$. Once this
happens, $C_2$ will begin to charge from $C_1$  through $L_{1sat}$ but since
$L_{1sat}$ is much less than $L_0$, $C_2$ charges more rapidly than $C_1$ did. This
process continues through the successive stages until C$_n$ discharges
into the load through $L_{nsat}$.  

To make this process efficient, we
design each of these successive stages so that saturation occurs at
the peak of the voltage waveform. Segment 1 to 2 in the hysteresis
loop of Fig.~\ref{CPR:fig11} is the active, or high-permeability, region during which
the inductor impedes current flow; the leveling off of the curve at
point 2, reached at the peak of the voltage waveform, indicates core
saturation when the inductor achieves a low impedance. During segment
2 to 4, the core is reset to its original state, ready for the next
cycle.
\subsubsection{IL1 pulse compressor}
The requirement for IL1 is to generate an acceleration pulse
shape and gradient shown in Fig.~\ref{CPR:indfg1}. Each accelerator cell
previously described produces a voltage of 214~kV; after the
beam traverses 700 of these cells it has gained 150~MV of
energy. From Table~\ref{CPR:tb1}, the necessary drive current for one cell is
5.09~kA for a duration (FWHM) of 250~ns. As
previously mentioned, no switches exist that can produce this type of
pulse directly. By investigating the optimum operating voltage and
current of the switches, the required stages of compression are
decided. Since thyristors have limits in $dI/dt$ of several
kA/$\mu$s and voltage limits of a few kV it can
be seen that a large number of them in series and parallel combination
will be required. Thyratrons also have limits on $dI/dt$ and voltage but
these limits are at least one order of magnitude greater than
thyristors. Thyristors have practically unlimited life while thyratrons 
have an operating life of the order of 20,000~hours. Even
taking into consideration replacement costs, the thyratrons offer a
simpler and more economical pulse compression system (fewer stages).

For technical and economic reasons, the pulse compression system is
designed to drive one 
meter or seven induction cells. The total energy required is
$U = 273 \times 7 = 1.9$~kJ plus 
that needed to make up the additional losses incurred in the 
pulse
compression scheme. The 500 J pulse compression system
(Fig.~\ref{CPR:fig13}), designed to replace the Advanced Test Accelerator 
spark gaps,
achieved efficiencies greater than 90\%. Allowing for 5\% losses in the 
thyratron switches, 5\% 
losses in the resonant charging and 5\% in the power supply, the total 
input energy per pulse  needed is 
2.5~kJ and, at 15~Hz average repetition rate, the power for
seven cells $P$ = 38.2~kW; the total power for IL1 is $P_t$ = 3.82~MW.

\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig12.eps}}
\caption[IL1 7-cell pulse generator]{Simplified diagram of IL1 7-cell pulse generator.}
\label{CPR:fig12}
\end{center}
\end{figure}
%Fig.~\ref{CPR:fig12}- Simplified Diagram of Ind. 1 7-Cell Pulse Generator


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=2.5in]{../template/report/ps-and-eps/CPR-Fig13a.eps} \includegraphics*[width=2in]{../template/report/ps-and-eps/CPR-Fig13b.eps}}
\caption[Pulse compression modulator]{500 J Mag 1-D magnetic pulse 
compression modulator driving the ETA II accelerator.}
\label{CPR:fig13}
\end{center}
\end{figure}

%Fig.13- 500 Joule Mag 1-D Magnetic Pulse Compression Modulator Driving the ETA II Accelerator



\subsubsection{IL1 7-Cell pulse generator}
Figure~\ref{CPR:fig12} shows a simplified diagram of the pulse generator 
that will
drive 7~cells of IL1 with a voltage pulse of 214~kV, 35.6~kA and a pulse
duration FWHM = 250~ns. The resonant charger initiates the sequence by
charging capacitor $C_0$ to $2\times V_{DCPS}$ or 30--40~kV. The charging current
through $C_0$ will have the effect of partially resetting the first stage
compression and the step-up transformer. The reset of the other stages
and the induction cells will be done by a separate pulse generator
just prior to initiating the pulse sequence. The optimum saturable
reactor is obtained by designing a time compression of about 3:1 and
with three stages the total compression will be about 27:1. The
thyratron switch will discharge $C_0$ in about 6.8~$\mu$s. As the magnetic
switch, $S1,$ saturates, it will discharge $C_1$ into the transformer
primary with a time period of 2.25~$\mu$s. This primary voltage of 30~kV
will be stepped up to 428~kV and charge $C_2$ with a $l - \cos(\omega t)$ 
waveform. The magnetic switch $S_2$ is designed to saturate at the peak of this
waveform, which will charge the pulse forming network in
750~ns. Finally, the saturable reactor $S_3$ switch the PFN energy at
the peak of that waveform, delivering the energy to the seven
cells. The desired waveform will be achieved by tailoring the temporal
impedance of the PFN to that of the nonlinear load of the
cells. Further waveform tailoring is done with series inductors and
an $RC$ compensation network in the boxes on each side of the cells.

The total energy that must be switched by the thyratrons includes the
system losses, and amounts to 2.5 kJ. This energy is stored in
capacitor $C_0=5.66~\mu$F and is switched into $C_1$ through inductor 
$L_1$ = 0.828~$\mu$H
with a series impedance $Z_0=0.382$~$\Omega$,  resulting in a peak 
half-sine-wave current of 28~kA, for a peak power of 2.3~GW. Several thyratron 
options are available. The highest continuous-power thyratrons
are the ceramic-envelope units, while the glass-envelope units are
capable of nearly as high a peak power with low average power
capability. Since the average power is moderate (38~kW) the
appropriate choice for technical and economic reasons is the glass-
envelope unit. To carry the 78~kA peak current, twelve
parallel devices are used. To insure current sharing, each
thyratron will switch its own capacitor which is $C_0/12=0.47~\mu$F.  Except
for thyratron replacement every 20,000 hours or more of operation, the
pulse compression systems should be maintenance free since all
components are passive devices.


\subsubsection{IL2 pulse compressor}

The pulse for the IL2 accelerator has a duration
(FWHM) of 100~ns. Assuming an additional 5\% loss (since the pulse compression
system has to go one step further), the total input energy for seven
cells would be $U$ = 1.6~kJ and, at 15~Hz, would  result in a power requirement of
24~kW. The total power requirement for IL1 is 1.9~MW, and its
pulse duration is 100~ns FWHM. The shorter pulse duration would dictate an 
additional stage of pulse compression on the system
described for IL1. However, since the energy for IL2 is 68\% of IL1, 
it is possible to 
achieve the shorter pulse duration with the same number of stages
simply by initiating the compression process with a shorter
pulse. The design of each stage, of course, will be different
and the transformer will have a step-up of 12:1. For IL2,
$C_0$ = 3.78~$\mu$F and the compression for three stages is 36 for an
initial discharge time of 3.6~$\mu$s with $L_0$ = 0.347~$\mu$H and $Z_0 =0.303
 \Omega.$ The peak current required of the twelve thyratrons is 99~kA,
or 8.25~kA each.

\subsubsection{IL3 pulse compressor}

The pulse duration for IL3 is 380~ns FWHM. A pulse compression
similar to IL1 with three stages is used. The 
transformer will have a step-up of about 10:1, and the three saturable
reactors will be similar to those in IL1. The energy at the input is $U=2$~kJ 
and, at 15~Hz average repetition rate, the input power per seven
cells is 29.3~kW and the total imput power is $P_t=2.34$~MW. With the input
energy of 2 kJ, the capacitor $C_0$ = 4.34~$\mu$F and with an initial
discharge time $t_0=10.3\mu$s, the inductor $L_0$ = 2.46~$\mu$H, and the series
impedance $Z_0=0.753\Omega,$ which results in a peak current of 40~kA.  The
total power required for IL1, IL2 and IL3 at 15~Hz average
repetition rate is
8.2~MW.  Including  the efficiency of 90\% for the DC charging power 
supplies, the total 60-Hz power requirement (see Table.~\ref{CPR:tb2}) will be very nearly 9~MW.
\begin{table}
\caption[Energy and power requirements]{Energy and power requirements. ${}^*$ Assuming 90\% efficiency.}
\label{CPR:tb2}
\begin{center}
\begin{tabular}{|lcccc|}
\hline
Unit  & Length  & Pulser energy & Total energy & $P_{total}$ (15 Hz)${}^*$ \\
&(m) & (J/m)& (kJ) & (kW) \\
\hline
IL1&
100&
2548&
254.8&
4247\\
IL2&
80&
1590&
126.9&
2115\\
IL3&
80&
1951&
156.1&
2602\\
\hline
\multicolumn{4}{|l}{Total power required from grid} &
8964 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Mechanical Systems}
In order to achieve the desired gradient for the three induction linacs, the induction cells 
are driven by the pulsing system in units of seven.  Hence, these cells 
are mechanically 
assembled into one module by bolting together seven cells.

The individual cores would be assembled at the plant. The 
mandrel on which the amorphous material is wound supports the complete core. 
An 
additional support cradle is included on the OD of the core to insure that there is no sagging 
(Fig.~\ref{CPR:fig14}). As specified under electrical requirements, the cores are wound with
101.6~mm wide ribbon with 3~$\mu$m mylar 
between layers and protruding 3~mm beyond the ribbon. 
As assumed earlier in this section, the 
complete core has a packing factor (PF) of about 75\%.

 
\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig14.eps}}
\caption[Metglas assembly]{Metglas single-cell assembly.}
\label{CPR:fig14}
\end{center}
\end{figure}
%Fig. 14  Metglas Single Cell Assembly


The high-voltage insulator, which is the oil-to-vacuum interface, is assembled in seven 
sections for each module and voltage grading of each section is provided by making contact with 
the appropriate cell. Each section has a gradient ring, which insures that 
the field lines 
enter the insulator at an angle of about 30$^\circ$ to provide maximum 
voltage holding (Fig.~\ref{CPR:fig07}). The 
seven section insulator is made of ``Mykroy/Mycalex'' and will be glued together as in the 
DARHT accelerator (Fig.~\ref{CPR:fig15}).
The induction module housing is fabricated and assembled using seven 
large rings 
fastened together by outside fixtures similar to those used in the Relativistic Two-beam 
Accelerator (RTA) at LBNL. The whole module is supported on the OD from these rings by a 
six-strut support system (Fig.~\ref{CPR:fig16}). The support system allows 
for excellent alignment of each 
module with respect adjacent modules and the absolute beam line.

The vacuum system will consist of turbo pumps and cryopumps located every 5-10 modules. 
These 
pumps will be connected to a roughing line alongside the accelerator. Beam position and total 
current diagnostics will also be located at the pump-out station.
Each module with its downstream SC solenoid magnet is assembled and aligned prior to installation 
in the beamline. After installation in the beamline, the module is aligned and the vacuum seal is 
fastened.

 
\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig15.eps}}
\caption[Seven cell housing]{Seven-cell housing that forms one 
accelerator module.}
\label{CPR:fig15}
\end{center}
\end{figure}
%Fig.15-Seven Cells housing which forms one Accelerator module


\begin{figure}
\begin{center}
\centerline{\includegraphics*[width=5in]{../template/report/ps-and-eps/CPR-Fig16.eps}}
\caption[Six strut module]{Six-strut module support system.}
\label{CPR:fig16}
\end{center}
\end{figure}
%Fig. 16-Six strut module support system








