Simultaneous Tune and Coupling Feedback in RHIC, and Possible
Implications for LHC Commissioning

> P. Cameron, A. DellaPenna, L. Hoff, Y. Luo, A. Marusic, C.
> Schultheiss, and S. Tepikian

Brookhaven National Laboratory, Upton, NY, USA 11973

M. Gasior and R. Jones

CERN, European Organization for Nuclear Research, CH-1211 Geneva 23,
Switzerland

C.Y. Tan

Tevatron Department, Fermi National Accelerator Laboratory, Illinois,
USA

(Dated: July 16, 2006)

Abstract

The Relativistic Heavy Ion Collider **(**RHIC) is a versatile facility,
capable of operating with polarized protons and a variety of ion species
over a broad range of energies. Acceleration ramp requirements change
frequently to meet the evolving needs of the physics experiments. There
has been an ongoing effort at RHIC \[1\] to implement reliable betatron
tune feedback as a tool for ramp development, and possibly to control
tune during routine operations. Earlier efforts \[2,3\] were hampered by
large betatron coupling, as well as the requirement for large dynamic
range. Both problems have been addressed, the first by continuous
measurement of coupling \[4,5\], and the second by an improved analog
front end \[6,7\]. With these improvements, simultaneous tune and
coupling feedback were successfully implemented during RHIC Run 6. In
this paper we describe the measurement and control hardware and software
used to accomplish this, present some of the results, discuss areas that
require further investigation, and finally offer a few comments on
possible implications of these results for LHC commissioning.

PACS numbers: 2920Dh, 2927Eg, 2927Fh, 4185Qg

# Introduction

In hadron accelerators, to minimize emittance growth and the consequent
reduction in luminosity available to the physics experiments, the beam
excitation needed to reliably operate phase-locked loop (PLL) tune
measurement systems must be very small. The power in the resulting
betatron signal is typically of the order of femtowatts, while the power
delivered to the pickup unrelated to the betatron tune is of the order
of watts. In addition, transition crossing in RHIC introduces
complications resulting from bunch shortening and fast orbit, tune and
chromaticity changes. Much effort was devoted to solving this problem of
dynamic range, which was ultimately dealt with by the Direct Diode
Detection Analog Front End (3D AFE).

PLL tune tracking and feedback become impossible as coupling becomes
large. Coupling rotates the planes of betatron oscillation (the
eigenmodes) away from the planes in which the magnet portion of a tune
feedback loop applies corrections. When this rotation exceeds 45 degrees
the magnet loop applies corrections in the wrong plane and the feedback
is driven unstable. To deal with this, additional data acquisition
channels were added to the PLL to measure the amplitude and phase of
both eigenmode projections in both planes, making possible continuous
coupling measurement, and eventually feedback.

These improvements were incorporated into a new VME-based tune and
coupling measurement and feedback system that was assembled for RHIC Run
6. The system performed as expected 'out of the box', tracking tunes and
measuring coupling within minutes of the first circulating bunched beam.

The two RHIC rings are cooled to helium temperature sequentially, with
about one week elapsing between the time when the first 'blue' and
second 'yellow' rings are cold. During the set-up period of that one
week, preliminary tuning of the PLL system and integration with Controls
and the Magnet system was completed. At the end of the week, with both
rings cold, ramping commenced. The first ramp was done with feedbacks
off, to establish a baseline. For the next ramp feedbacks were turned
on, and decoupled beam was delivered to full energy. To the best of our
knowledge, this successful ramp was the world's first attempt at beam
acceleration in a synchrotron with simultaneous tune and coupling
feedback.

# Coupling formalism

A brief description of the coupling formalism \[8\] is presented here,
to assist in understanding material presented later in this paper.
Figure 1 shows the eigenmodes and their projections. In the most general
case the eigenmodes are ellipses, which are not orthogonal to each
other, as shown in the right portion of the figure. The left portion
shows the special case when the minor axes of the ellipses vanish. The
frequency of mode 1 is denoted by Q~1~, while A~1,x~ and A~1,y~
represent the amplitudes of this mode in the horizontal and vertical
plane respectively. Similarly φ~1,x~ and φ~1,y~ represent the phases of
this mode in the horizontal and vertical plane respectively. The same
notation applies for mode 2.

![](media/image1.jpeg){width="5.798611111111111in" height="2.3in"}

**Figure 1.** (color) Schematics showing the two eigenmodes rotated with
respect to the horizontal and vertical planes due to coupling

All of the quantities shown in the figure are measured by the PLL. From
this data the following parameters can be calculated:

$\begin{array}{r}
r_{1} = \ \frac{A_{1,y}}{A_{1,x}} = \ \sqrt{\frac{\beta_{y}}{\beta_{x}}} \cdot \frac{\left| C^{-} \right|}{2\nu + \Delta},\text{}r_{2} = \ \frac{A_{2,x}}{A_{2,y}} = \ \sqrt{\frac{\beta_{x}}{\beta_{y}}} \cdot \frac{\left| C^{-} \right|}{2\nu + \Delta},\text{} \\
{d\varphi}_{1} = \ \varphi_{1,y} - \ \varphi_{1,x},\text{}{d\varphi}_{2} = \ \varphi_{2,x} - \ \varphi_{2,y}
\end{array}$

where β~x~ and β~y~ are the horizontal and vertical beta functions at
the pickup, and

$\nu = \frac{1}{2}\sqrt{\Delta^{2} + \left| C^{-} \right|^{2}},\text{}\left| \ C^{-} \right| = \frac{2\ \sqrt{r_{1}\ r_{2}}\ \left| \ Q_{1} - Q_{2} \right|}{\left( \ 1 + r_{1}\ r_{2} \right)},\text{}\Delta = \frac{\left| \ Q_{1} - Q_{2} \right|\ \left( 1 - r_{1}\ r_{2} \right)}{\left( 1 + r_{1}\ r_{2} \right)}$

When the eigenmodes are decoupled they are in the horizontal and
vertical planes, the ratios r~1~ and r~2~ are equal to zero, the
ellipses collapse to straight lines, and the coupling phases dφ~1~ and
dφ~2~ are equal to zero. The coupling amplitude C^-^ is often referred
to as dQ~min~, or the minimum tune split. The parameter ∆ is the
difference between the fractional parts of the unperturbed, or 'set'
tunes. When the eigenmodes are decoupled the eigentunes (measured) and
the unperturbed (set) tunes are identical. For the purpose of presenting
the data in a form that is easily understood, the unperturbed tunes can
be calculated by:

$Q_{x,0} = \ Q_{1} + \ \frac{1}{2}\Delta\  - \ \frac{1}{2}\sqrt{\Delta^{2} + \left| C^{-} \right|^{2}},\text{}Q_{y,0} = \ Q_{2} - \ \frac{1}{2}\Delta\  + \ \frac{1}{2}\sqrt{\Delta^{2} + \left| C^{-} \right|^{2}}\text{}$

The utility of having available both set and measured tunes will become
clear later in this paper.

# System architecture and signal processing 

Figure 2 shows a block diagram of the tune and coupling measurement and
feedback system. Analog processing of signals from the pickup is
accomplished by the 3D AFE. After detection, filtering, and
amplification the tune signals from the two pickup planes are split and
digitized. Digital processing is accomplished in two signal paths that,
with the exception of coupling correction, are independent. The two
signal paths provide the data needed needed to maintain phase lock to
the two eigenmodes, and to correct coupling and tune, and eventually
chromaticity.

When the eigenmode PLLs are locked, the digitizer clocks are
phase-synchronous to the eigenmodes. As shown in the figure, the
digitizer clocks are generated by numerically-controlled oscillators
(NCOs), one for each eigenmode. The NCOs are beam-synchronously clocked
from the 28MHz RF. Their output is at four times the betatron frequency,
permitting simple I/Q demodulation.

![](media/image2.jpeg){width="5.802777777777778in"
height="3.3430555555555554in"}

**Figure 2.** (color) System Block Diagram

Data for each eigenmode is processed by a Front End Computer (FEC)
dedicated to that eigenmode. Processed data is passed to a Master FEC
that calculates magnet correction strengths. Corrections are written to
a reflected memory module that is mirrored via dedicated fiber to a
similar module in the power supply building, where a FEC applies the
corrections to the power supplies. Kicker and signal path gains, as well
as the 3D AFE gains and time constants, are controlled by additional VME
modules. The RHIC implementation accomplishes measurement and control
for both the 'blue' and 'yellow' rings in a single VME crate, and the
number of VME modules in that crate is approximately twice that shown in
the diagram.

A.  **Kicker and Pickup**

The kicker design originated at CERN, and was driven by two
considerations. The first was the need to excite the the rigid 7TeV LHC
beam without excessive kicker power, and second was to couple the kicker
effectively to the beam without presenting excessive impedance to the
beam when viewed in its reciprocal role as a pickup.

Figure 3 shows the relative contributions of the electric and magnetic
fields to the total field in the pickup as a function of the kicker
impedance, at constant power. It is clearly advantageous to favor the
magnetic field, and to operate at the lowest practical impedance. The
PLL operates at baseband, where transmission line effects can be
ignored, and the minimum practical impedance is then defined by the
resistance of the cables, connectors, feedthrus, and striplines, as well
as by the current carrying capability of the feedthrus. This approach
eases the task of exciting the beam without excessive kicker power.

  ----------------------------------------------------- -- -----------------------------------------------------
  ![](media/image3.jpeg){width="2.8493055555555555in"      ![](media/image4.jpeg){width="2.8493055555555555in"
  height="1.5944444444444446in"}                           height="1.5104166666666667in"}

  **Figure 3.** Kick as a function of impedance            **Figure 4.** Kicker block diagram
  ----------------------------------------------------- -- -----------------------------------------------------

Figure 4 shows the kicker layout adopted for RHIC. The kicker is a 1m
long dual plane device. The 50Ω striplines subtend 70 degrees, and their
inside diameter is 11cm. The design exploits the fact that kickers and
pickups are not truly reciprocal devices\[9,10\]. A kicker has response
at DC, but a pickup does not. The kicker is driven in series thru the
four 1μH inductors shown in the figure. Their impedance is \~0.1Ω at the
\~20KHz betatron frequency of RHIC, but more like 400Ω at the 75MHz
response peak of the 1m stripline when viewed as a pickup. The inductors
permit low impedance driving of the device as a kicker, and effectively
block its response as a pickup, protecting the driver amplifier. The 50Ω
resistors shown in the figure provide a matched termination for the
pickup response of the striplines.

The kicker is driven differential mode through shielded twisted pairs.
The system has been operated both with the 3D AFE in the tunnel, and
with beam signals cabled out on heliax. There are no problems with
bleedthru across the \~160dB difference between the kicker path and the
signal path, as long as single point grounding is maintained with the
beampipe serving as ground.

The pickup is identical in construction to the kicker. Signal is taken
from the upstream ports, and the downstream ports are terminated in 50
ohms. Beta functions at the pickup are β~h~ \~25m and β~y~ \~130m.
Motion control is available for centering the pickup on the beam if so
desired. Our experience so far has been that beam centering in the
pickup is not critical, and motion feedback has not been implemented.

B.  **Direct Diode Detection Analog Front End**

The crucial part of the 3D AFE is a peak detector. Two such detectors
connected to opposing electrodes of the pickup, as shown in Figure 5,
yield the amplitude modulation envelope of the beam signals. Such
signals, depicted in Figure 6, are superimposed on a DC voltage related
to the bunch amplitude. The signal difference, shown in Figure 7 for a
single bunch in the machine, contains almost the whole bunch modulation
amplitude, with a DC value related to the beam offset from the center of
the pick-up. Since the DC content can be suppressed using a series
capacitor, the corresponding revolution frequency (*f~r~*) background
can already easily be removed by some 50 dB before the first amplifying
stage.

  --------------------------------------------------- -- ---------------------------------------------------
  ![](media/image5.wmf){width="2.852777777777778in"      ![](media/image6.png){width="2.654861111111111in"
  height="1.0145833333333334in"}                         height="1.4097222222222223in"}

  **Figure 5.** Block diagram of a 3D AFE                **Figure 6.** Example of peak detector voltages

  ![](media/image7.png){width="2.532638888888889in"      ![](media/image8.wmf){width="2.657638888888889in"
  height="1.3555555555555556in"}                         height="1.35625in"}

  **Figure 7.** Difference of the signals in figure 6    **Figure 8.** RHIC AFE frequency characteristic
  --------------------------------------------------- -- ---------------------------------------------------

The 3D circuitry in Figure 5 can be thought of as two sample-and-hold
blocks, sampling bunch signals close to their maxima at the bunch
repetition rate and downmixing the wideband bunch spectrum into the
baseband. Note that a 3D AFE is "low frequency" only after the
detectors, due to the "time stretching" of the short beam pulses. Before
the detectors, the processed bandwidth can extend into the GHz range,
covering the full coherent spectrum in hadron machines. In the detection
process the spectral content from this wide bandwidth is converted to
the baseband, resulting in a system sensitivity in the nm range for high
intensity beams.

As sketched in Figure 5, the peak detector voltages with the DC content
removed by the series capacitors are subtracted by the differential
amplifier (DA). This further increases the suppression of *f~r~* and
improves the immunity of the system to interference. The filters further
attenuate *f~r~* by some 100 dB, as seen on the AFE characteristic in
Figure 8. The overall *f~r~* suppression from all system stages can be
conservatively estimated to be some 160 dB. Another advantage of the 3D
approach is that the output signal, which is in the audio range for
large machines, can be efficiently digitized with high resolution audio
ADCs.

Gain in the 3D AFE is adjustable over a range of 65dB in 5dB steps. In
addition, it is useful to have continuous gain control in both the
kicker and signal paths to stabilize the overall system loop gain under
varying beam conditions. Provision has been made to accomplish this by
monitoring signal strength and the signal-to-noise ratio. A feedback
loop can be closed in the signal path to maintain constant signal
strength, and a similar loop in the kicker path to maintain constant
signal-to-noise.

C.  **PLL Signal Processing**

Commercial-Off-The-Shelf (COTS) components were widely used in
assembling the data acquisition and processing systems. Two exceptions
are the standard RHIC V108 'utility' module, and the Numerically
Controlled Oscillators (NCOs). The V108 module supports a variety of
utility functions, such as crate temperature monitoring and decoding
RHIC timing and global data signals. Its chief role in the present
system is to associate RHIC-specific timestamps to data. The NCO is a
standard RHIC module typically used in RF applications. While there may
be COTS equivalents, this module has a long and successful track record
at RHIC in similar applications, and was readily available. The
important characteristics for an NCO in this application include
appropriate frequency range, support for external clocking to
automatically track RF frequency changes during beam energy ramps, and
smooth transitions during frequency adjustments to maintain phase
continuity.

The software executes in the standard RHIC software environment. This
includes the VxWorks operating system, device drivers (typically written
in the "C" language), and a RHIC-specific abstraction layer called the
Accelerator Device Object (ADO) layer, developed using a C++-like
language and stored in files called 'rad files'.

  -------------- ---------------- ----------------------------------- -----
  Function       Model            Notes                               qty

  ADC            VMIVME 3123      16 channel, 16 bits, external       4
                                  clock, ping-pong buffers            

  DAC            VMIVME 4140,     8 channel, 16 bits                  3
                 4122                                                 

  Digital Output VMIVME 2170      32 channel                          2

  CPU            MVME2100         200MHz PowerPC 603                  5

  NCO            RHIC RF          External clock                      4

  Utility        V108             Provides timestamps                 1

  Shared memory  MM-6702          Non-volatile SRAM w/battery backup  1

  Reflective     VMIVME 5576      Broadcasts calculated magnet        1
  memory                          corrections                         
  -------------- ---------------- ----------------------------------- -----

Table 1. VME modules used in the tuned feedback system

Hardware modules used in the system are shown in Table 1. Device drivers
were already available for all hardware modules, and ADOs were available
for most of the needed functions, including basic analog and digital
control, utility crate monitoring, timestamp decoding, and even tune
feedback (from a previous generation RHIC tune feedback system). Two new
ADOs were developed for data acquisition and processing. One of these
ADOs supports closed-loop feedback on kicker strength and/or pickup
signal gain. The other ADO supports the data acquisition and processing
associated with tune and coupling measurement and will be described in
detail in the following paragraphs. Four of the 5 CPU modules in the
system are assigned to the four eigenmodes of the 2 RHIC rings. Each of
these CPUs runs a single instance of the data acquisition ADO and two
instances of the feedback ADO, one for kicker strength and one for
signal gain. The fifth CPU collects tune and coupling data from the
other CPUs, computes tune and coupling correction, and broadcasts those
values to magnet power supply controllers. The basic scheme for tune
feedback had been demonstrated in a previous generation RHIC tune
feedback system.

The ADC module is configured to continuously acquire data into one of
two 'ping-pong' buffers. When one buffer is full, the ADC switches to
the other buffer and interrupts the CPU, so that data from the first
buffer can be processed while the second buffer is being filled. The ADC
sampling clock is provided by the NCO, and is adjusted to be four times
the betatron frequency, so that the ADC samples contain alternately
in-phase and quadrature measurements of beam motion. The number of
samples in the ADC buffer is configured to be a multiple of 4 to
preserve the relative phase of samples within each of the two buffers.

The ADC interrupt initiates one pass of data processing. During this
pass, the ADC data is separated into an average of the in-phase and an
average of the quadrature values. Each of these values is inserted into
separate data streams. Each of these data streams is then passed through
one iteration of a low-pass (FIR) filter. The resulting low-passed
quadrature value is then used as the error signal to a PID loop which
adjusts the NCO frequency, realizing a PLL locked onto the betatron
frequency.

To aid in characterizing the performance of the PLL, certain parameters,
including the quadrature (error) signal and the NCO frequency are output
as voltages via a third, diagnostic DAC module. The DAC outputs are
available for analysis, for instance using a spectrum analyzer.

To measure coupling, the signal from the orthogonal plane is acquired
via a second ADC channel, using the same sampling clock. The resulting
signal is demodulated using the same technique as the primary channel.

Finally, after decimation several parameters, including the filtered I/Q
data from both ADC channels, are published via shared memory so that the
fifth (master) CPU can compute the necessary magnet correction strengths
to maintain programmed tune and minimize coupling.

D.  **Coupling Correction and Magnet Control**

For tune and coupling feedback in both of the RHIC rings, a total of 20
data streams (the 8 eigenmode projections for each of I and Q, plus the
4 tunes) are passed from the 4 eigenmode FECs to the Master FEC, where
amplitudes and phases are calculated from the I and Q data. For each
projection, the projection amplitude is given by sqrt(I^2^+Q^2^). The
projection phase is given by atan2(Q,I). If the eigentunes are well
tracked, the measure phases φ~1,x~ and φ~2,y~ are driven to zero by the
functioning of the PLL. The phase of the coupling coefficient is simply
given by φ~1,y~. With the eigentune split and the amplitude ratios r1
and r2, from Eq. (2) the amplitude of coupling coefficient is
determined. According to Eq.(3), the uncoupled set tunes are also
obtained.

For coupling correction, RHIC has 3 skew quadrupole families, F1, F2,
and F3. The coupling contributions from these 3 familes are about 120
degrees apart when they are powered with the same strength and proper
polarities. For coupling correction we combine F1 and F3 with the same
strength to produce a family F13 whose contribution to the total
coupling coefficient is approximately orthogonal to that only from F2
family. Knowing the coupling coefficient at the location of the PLL
pickup, based on the accelerator optics model the global decoupling
strengths for the two orthogonal families F2 and F13 can be calculated.
In reality, the coupling angles of F2 and F13 are slightly modified with
the measured phases of the eigenmode projections.

The feedback loop for tune control\[11\] has the measured tunes as
inputs. The differences between the measured and desired tunes are the
error signals, from which are calculated the focusing and defocusing
quadrupole strength outputs. Implicit in the operation of the tune
feedback loop is the assumption that the tunes are decoupled, so that
the measured eigentunes correspond to the correction planes of the
feedback loop. The two planes typically have different frequency
responses. The error signals from each plane are filtered separately,
permitting the overall loop responses of the two planes to be equalized.
A PID filter is utilized as a compensation filter. The magnet feedback
loop for coupling control is similar to the tune feedback loop.

# results 

Measurements of the coupling and tune parameters presented in equations
(1) through (3) are shown in Figure 9. This data was taken during a tune
scan made for working point optimization during RHIC Run 5. Vertical
tune was held constant while horizontal tune was scanned. The color
scheme used in this figure for set and measured tunes will be followed
throughout this paper. In this figure the set and measured tunes differ
as a result of the minimum tune split due to coupling, and are easily
distinguished. In several figures later in this paper the tunes are well
decoupled, and the measured tune data is partially or totally obscured
by the set tune data.

  ---------------------------------------------------- -- ------------------------------------------------------
  ![](media/image9.jpeg){width="2.852777777777778in"      ![](media/image10.jpeg){width="2.8520833333333333in"
  height="2.0854166666666667in"}                          height="2.061111111111111in"}

  **Figure 9.** (color) Parameters during a tune scan     **Figure 10.** (color) Parameters during a ramp
  ---------------------------------------------------- -- ------------------------------------------------------

The coupling parameters C^-^ and ∆ are shown in the upper pane of Figure
10, taken from an acceleration ramp during RHIC Run 5. The lower pane
shows measured and set tunes, and clearly shows crossing of the set
tunes during the ramp. One's understanding of the quality of tune
control in both Figure 9 and Figure 10 would be very different without
the availability of the coupling data. If tune feedback had been on in
these figures, the feedback would have been driven unstable by the
crossing of the set tunes.

Success of the tune feedback effort is predicated on decoupling.
Therefore, the first objectives when beam became available in Run 6 were
to validate the coupling measurement, then close the loop on coupling
feedback. Figure 11 shows data taken early in Run 6, during the week of
machine setup in preparation for ramp development. The upper pane shows
calculated correction strengths for the families F2 and F13, derived
from the measured coupling. The lower pane shows set and measured tunes
during decoupling. It can be seen that as the requested skew correction
strengths are adjusted towards zero, the set tunes converge on the
measured tunes.

The commonly accepted fiducial measure of coupling is accomplished by
moving the tunes as close as possible to determine the minimum tune
split. This method is not often used because it perturbs machine
operation, but is useful for comparing results with other measurement
methods. From the eigenmode-based coupling measurement, skew strengths
can be adjusted to zero, and it can be observed that set and measured
tunes agree in this case. But the real proof that the tunes are
decoupled is to measure the closest tune approach. Figure 12 shows data
from a tune scan after the tunes were decoupled via the method shown in
Figure 11. It can be seen both that the set and measured tunes were in
agreement, and that the closest tune approach was .001 or less.

  ----------------------------------------------------- -- -----------------------------------------------------
  ![](media/image11.jpeg){width="2.859027777777778in"      ![](media/image12.jpeg){width="2.847916666666667in"
  height="1.89375in"}                                      height="1.8520833333333333in"}

  **Figure 11.** (color) Manual de-coupling                **Figure 12.** (color) Tune scan
  ----------------------------------------------------- -- -----------------------------------------------------

Figure 13 shows set and measured tunes as the coupling feedback loop is
turned on, then off, and then back on again. With the loop off the
measured tunes are driven apart by the coupling. With the loop on the
set and measured tunes are essentially the same.

Figure 14 is a bit of a curiousity. During initial setup of the PLL, the
the impulse excitation of single kicks from the conventional tune meter
kicker were used to tune PID loop parameters. The resulting 'coupling
echoes' shown in the figure have two features, the response at twice the
synchrotron frequency and the slower response at \~2Hz. The beam physics
responsible for these features is not yet clear.

  ------------------------------------------------------ -- ------------------------------------------------------
  ![](media/image13.jpeg){width="2.9305555555555554in"      ![](media/image14.jpeg){width="2.8430555555555554in"
  height="2.3027777777777776in"}                            height="2.2708333333333335in"}

  **Figure 13.** (color) Coupling feedback on and off       **Figure 14.** (color) Coupling echoes
  ------------------------------------------------------ -- ------------------------------------------------------

Figure 15 shows data taken from 3 ramps during 100GeV ramp development.
It should be noted that PLL loop gains must be adjusted manually during
ramps with feedbacks on, to compensate the effect of chromaticity
variations. This will be discussed in more detail in the following
section. The uppermost pane in the figure shows blue ring set and
measured tunes, as well as beam currents in the blue and yellow rings,
during the first acceleration ramp with feedbacks enabled (in the blue
ring only). For the most part the correction from the coupling feedback
was so good that the measured tune data is obscured by the set tune
data. Loop gains in both the PLL tune tracker loop and the overall tune
and coupling feedback loops were set somewhat high to insure tracking of
fast changes. Additionally, PLL gains were manually adjusted through the
ramp to accommodate changing beam conditions.

![](media/image15.jpeg){width="5.797916666666667in"
height="4.802777777777778in"}

**Figure 15.** (color) Data from 100GeV development ramps in RHIC Run 6

The middle pane shows the eleventh and last of the 100GeV feedback ramps
in blue. Magnet correction strengths were fed forward from ramp to ramp,
so that by the end of this phase of ramp development the required
corrections were small, and loop gains could be lowered. The improved
quality of the data is clearly evident.

The lower pane shows data from the next ramp, with feedbacks off and
corrections from the previous ramp fed forward. This gives some idea of
the ramp-to-ramp variations in tune and coupling. There were no
deliberate orbit changes between the two ramps, and hence coupling was
not changed significantly.

# areas for further investigation

During the course of this work two additional obstacles to making the
system fully operational became clear. The first is the presence of
large amplitude direct excitation of the betatron resonances by high
harmonics (h\~350) of the 60Hz power frequency, and the second is the
need for precise chromaticity control (and possibly feedback) for proper
operation of the tune tracker.

A.  **Mains Harmonics**

The 3D AFE delivers significant improvement in position measurement
sensitivity, with estimated 10nm resolution at RHIC. This AFE has been
installed at the CERN PS and SPS, at the FNAL Tevatron, and at RHIC.
Mains harmonics have been observed in the betatron spectrum at every
installation \[12\]. Extensive studies have conclusively deomonstrated
that these harmonics result from direct excitation of the beam at the
betatron frequency by high harmonics of the power line frequency. At
RHIC the resulting spectral lines are as much as 40dB above the 3D AFE
noise floor at injection and store, and as much as 80dB above the noise
floor during ramping, when additional supplies are switched in to
provide the field time derivative needed for ramping. The source of
these harmonics is being actively investigated, and their possible
origin is discussed in detail elsewhere \[12,13\].

B.  **Chromaticity**

For reliable tune tracking the overall loop gain of the PLL must be
relatively constant. The beam-transfer-function portion of the loop gain
is dependent on chromaticity. With large chromaticity both the amplitude
and the phase slope of the betatron resonance response become small, and
loop gain can be diminished to the extent that the PLL cannot track fast
tune changes. Conversely, with small chromaticity the loop gain can
become so large that the loop becomes unstable.

The pickup for previous generations of the RHIC tune feedback system
utilized a stripline BPM resonated at 245MHz, above the coherent
spectrum. At that frequency the linewidth is dominated by broadening due
to the high harmonic number, and studies showed that PLL performance was
only weakly affected by chromaticity over the range from approximately
1-20 units. What effect was observed could be sufficiently compensated
by feeding back on the kicker excitation to keep the amplitude response
constant.

By virtue of its operating principle, the 3D AFE functions within the
coherent spectrum, where the linewidth is dominated by chromaticity. In
RHIC the 3dB point of the coherent spectrum is at about 60MHz when beam
is bunched in the 28MHz acceleration cavities. Operational experience
suggests that the tolerace of the 3D AFE PLL to chromaticity variations
is less than that of the accelerator itself, and that chromaticity must
be well controlled (in the range of 1 to 5 units) for reliable PLL
operation. In addition to the expected effects on overall loop gain, the
situation is aggravated by the fact the the 3D AFE functions as a peak
detector, and is sensitive to the 'micro-instabilities' that precede the
full blown head-tail instability caused by too small chromaticity. As
chromaticity becomes small the PLL senses not the kicker excitation, but
rather the self-excitations of beam, and loses lock. As previously
mentioned, this sensitivity to the effect of chromaticity on overall
loop gain required that loop gains be manually adjusted during Run 6
feedback ramps.

Past experience \[14\] with continuous chromaticity measurement using
the PLL has been good. Data taken under automated sequencer control from
three successive ramps during Run 5 is shown in Figure 16. This data was
taken with the 245MHz resonant pickup, and the quality of the data near
transition suffers due to excessive dynamic range as the bunch length
shortens and extends up to the pickup frequency. The 3D AFE functions in
the coherent spectrum, and does not have a problem with dynamic range
around transition. The measurements were made with radial modulation of
the beam with an amplitude of 200μ (this corresponds to a momentum
deviation of \~2x10^-4^) at a rate of 1 Hz. Excepting the time around
transition, the measurement appears to be of adequate quality to permit
feedback. It should also be mentioned that the algorithm for calculating
chromaticity suffers from a sign ambiguity as chromaticity becomes
small. This can be seen in the oscillations of the measured horizontal
chromaticity around zero near the end of all 3 ramps. The source of this
problem remains under investigation. Our expectation is that the quality
of chromaticity measurement from the 3D AFE PLL will be similar to or
better than that shown in the figure.

![](media/image16.jpeg){width="5.795138888888889in"
height="4.268055555555556in"}

**Figure 16**: (color) Automated chromaticity measurement during three
successive ramps

As a result of the observed sensitivity of the PLL to chromaticity, our
plan for RHIC Run 7 is to close the last loop, namely chromaticity, in
this multi-loop feedback system. We intend to implement simultanous
tune, coupling, and chromaticity feedback as early in the run as is
feasible.

# possible implications for lhc commissioning

The 'out of the box' success of simultaneous tune and coupling feedback
in RHIC greatly improves the possibility that robust feedbacks can be
implemented early in LHC commissioning. From the LHC perspective, the
most urgent motivation for feedbacks is management of the chromatic
effect of snapback at the beginning of the acceleration ramp \[15\]. The
present understanding is that compensation of this effect will require
real-time feedback control of chromaticity. This feedback is moving into
the foreground of our efforts, motivated both by LHC needs for snapback
control and the need to stabilize PLL loop gain in RHIC.

We emphasize that mains harmonics in RHIC are anomalously large, and
their origin is not understood. This has many implications for the LHC.
They impede needed system refinement, and obstruct development work on
chromaticity measurement and feedback. Their presence in the beam
spectrum introduces spurious phase any time the tune is such that they
are within the bandwidth of the PLL, an inevitable circumstance during
the radial modulation chromaticity measurement with feedbacks off. They
require careful selection of tunes when feedbacks are on. During normal
operations with feedbacks off the effects of their spurious excitation
diminish the confidence of non-specialist users of the system, impeding
its transition to full operational status. And finally, until their
origin is understood the possibility cannot be fully eliminated that a
similar difficulty might arise in the LHC. Understanding and resolving
this problem remains a high priority.

# acknowledgements

The results presented here would not exist without the early foresight
and guidance of Tom Shea, and the excellent foundation he provided to
support these efforts. We gratefully acknowledge his many contributions.

Hermann Schmickler initiated the collaborative effort between CERN and
BNL. He, Mike Harrison and Steve Peggs brought this collaboration into
the US LHC Accelerator Research Program (LARP), where it was expanded to
include FNAL and has prospered under the oversight of Alex Ratti. The
support of the LARP structure has been essential from a financial and
organizational perspective, and additionally has enhanced the scientific
and technical quality of the effort.

The authors acknowledge the support of the RHIC High Frequency
Instrumentation Group in earlier efforts and the C-AD Beam
Instrumentation Group in the present effort, and particularly Tom Russo,
Tony Curcio, Ron Schroeder, and Al Weston.

And finally, we acknowledge the tireless and efficient oversight of
Vadim Ptitsyn in his role as run co-ordinator of RHIC Run 6, as well as
the steadfast long-term support and encouragement of Thomas Roser from
the very beginnings of this effort.

This work is supported by the U.S. DOE under contract DE-AC02-98CH10886
and the US LHC Accelerator Research Program (LARP).

**\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_**

\[1\] The tune feedback effort at RHIC was initially a BNL effort, was
expanded into a bi-lateral collaboration with CERN in 2002, and became a
multi-lab LARP collaboration in 2003. For further information, see
<http://www.agsrhichome.bnl.gov/LARP/>

\[2\] P. Cameron et al., "RHIC Third Generation PLL Tune Measurement
System", PAC 2003, Portland.

> <http://accelconf.web.cern.ch/AccelConf/p03/PAPERS/ROAB009.PDF>

\[3\] P. Cameron et al., "Advances Toward the Measurement and Control of
LHC Tune and Chromaticity", DIPAC 2005, Lyon.
<http://dipac2005.web.cern.ch/dipac2005/default.htm>

\[4\] Y. Luo et al., "Possible Phase Loop for the Global Decoupling",
PAC 2005, Knoxville.

> <http://accelconf.web.cern.ch/AccelConf/p05/PAPERS/TPAP052.PDF>
>
> a much expanded version is available at
> <http://www.agsrhichome.bnl.gov/AP/ap_notes/ap_note_174.pdf>

\[5\]Y. Luo et al, "Continously measure global difference coupling using
a phase-locked-loop tune meter in the RHIC", this journal issue.

\[6\] M. Gasior and R. Jones, "The Principle and First Results of
Betatron Tune Measurement by Direct Diode Detection",
LHC-Project-Report-853.

<http://doc.cern.ch/archive/electronic/cern/preprints/lhc/lhc-project-report-853.pdf>

\[7\] M. Gasior and R. Jones, "High Sensitivity Tune Measurement by
Direct Diode Detection", DIPAC 2005, Lyon.
<http://dipac2005.web.cern.ch/dipac2005/default.htm>

\[8\] R. Jones et al., "Towards a Robust Phase Locked Loop Tune Feedback
System", ibid.

\[9\] G. Lambertson and D. Goldberg, "A Primer on Pickups and Kickers",
AIP 249, 1990.

\[10\] G. Decker, "The Design of Beam Pickups and Kickers", BIW 1994,
Vancouver.

\[11\] C. Schultheiss et al., "Real Time Betatron Tune Control in RHIC",
EPAC 2002, Paris.

> <http://accelconf.web.cern.ch/AccelConf/e02/PAPERS/THPRI041.pdf>

\[12\] P. Cameron et al., "The Effects and Possible Origins of Mains
Ripple in the Vicinity of the Betatron Spectrum", DIPAC 2005. Lyon.
<http://dipac2005.web.cern.ch/dipac2005/default.htm>

\[13\] P. Cameron et al., "Observations of Direct Excitation of the
Betatron Spectum by Mains Harmonics in RHIC", C-AD AP note xxx,
September 2006. This note is available at
http://www.agsrhichome.bnl.gov/AP/ap_notes/

\[14\] S. Tepikian et al., \"Measuring Chromaticity along the Ramp using
the PLL Tune-meter in RHIC\",\
EPAC 2002, Paris.\
[http://accelconf.web.cern.ch/AccelConf/e02/PAPERS/THPRI075.pdf](https://exchange2000.bnl.gov/exchweb/bin/redir.asp?URL=http://accelconf.web.cern.ch/AccelConf/e02/PAPERS/THPRI075.pdf)

\[15\] S. Fartoukh and J.P. Koutchouk., "On the Measurement of the
Tunes, Coupling, and Detunings with Momentum and Amplitude in LHC",
Functional Specification LHC-B-ES-004, 2002.
<http://www.rhichome.bnl.gov/LARP/papers/LHC_tune_spec.pdf>