Considerations for Ring and RTBT BPM Electronics

Prepared by: Martin Kesselman

07-09-02

Abstract:

The Spallation Neutron Source (SNS) Ring accumulates about 1060 pulses
of 38mA peak current 1GeV H-minus particles from the Linac thru the HEBT
line, then delivers this accumulated beam in a single pulse to a mercury
target via the RTBT line. Bunching frequency of beam in the HEBT line is
402.5MHz, and about 1MHz in the Ring and RTBT. Position monitor
electrodes in HEBT are of the shorted stripline type, with apertures of
12cm except in the dispersive bend, where the aperture is 21cm. Ring and
RTBT electrodes are open striplines, with apertures of 21, 26, 30, and
36cm. All pickups are dual plane. The electronics will be PC-based with
the Analog/Digital Front End passing data and receiving control and
timing thru a custom PCI interface developed by LANL[^1]. LabVIEW will
be used to direct the acquisition, process the data, and transfer
results via ethernet to the EPICS control system. To handle the dynamic
range required with well over 60dB variation in signal size, the Ring
and RTBT electronics will employ a fast gain switching technique that
will take advantage of the 300ns head-tail gap to provide position
measurement during the entire accumulation cycle. Beam-based alignment
will be utilized as part of the system calibration. This document will
discuss present plans, and initial design considerations leading to a
preliminary design configuration. We will also show that we can not meet
the requirement for turn-by-turn resolution of 0.15mm for a 105mm half
aperture BPM at study beam currents of 15ma.

**Basic Requirements -- Specifications:**

Ring-RTBT BPM

Quantity 44+17=61

Intensity 5E10 -- 2E14 ppp

Half Aperture +/- 105mm typical for Ring, +/- 130mm, +/- 150mm

Accuracy +/- 1% half aperture

Resolution +/- 1% of radius turn by turn, +/- 0.5% of radius averaged
for intensity \< 15E10ppp

Quantity:

Ring: (floated stripline)\
21 cm bpm: 28\
26 cm bpm: 8\
30 cm bpm: 8\
\
RTBT: (shorted stripline)\
21 cm bpm: 15\
36 cm bpm: 2

**Some associated Information:**

![](media/image1.png){width="9.583333333333334in"
height="5.629861111111111in"}

A paper that was generated a number of years ago, investigating the
output of the BPM system in the Ring is available as Technote 038, Under
SNS Reports, Technical Notes; "Simulations of SNS Accumulator Ring Beam
Position Monitor Signals", J. Beebe-Wang, A.U. Luccio, R. Witkover, at
the following website.

<http://server.ags.bnl.gov/bnlags/bnlsns/sns.html>

Minor changes to the system have been made since this investigation,
however, it is believed that the basic results will be representative of
expected signals. It should be noted that the RF structure of the
injected bunches is not simulated. The largest output signals are of the
order of +/- 20 Volts. Early turns are expected to have rise times near
the chopper edge of 10ns, while after many turns the edge is expected to
deteriorate to near 50ns.

The RF structure during the first turn dominates, and produces pulses of
high frequency information (402.5MHz plus harmonics). If these pulses
are filtered to base band with something near a 10MHz band pass, there
will be very little energy available at the output of this filter
(fundamental of 402.5MHz). Virtually nothing will be available for
processing (figure 1). After this single bunch has been allowed to
diffuse, by rotating around the Ring a number of times, the output
depicted by the above paper can be expected, and a crude PSpice
simulation showed that one might be able to obtain about 2.0mV peak from
a 5MHz low pass filter (figure 2). After the edge has deteriorated to
about 50ns (7.5MHz) the peak has been found to be about 1.5mV for a 15ma
signal (figure 3). However, this will take numerous times around the
Ring to slow the edge. The simulation circuit is shown below (figure 4).

![](media/image2.wmf){width="5.134722222222222in"
height="2.8868055555555556in"}Figure 1 -- Results of a simulation of
150ps RF bunched beam pulses chopped to 645ns with a 300ns gap. Shows
the low-pass filter response is poor due to the near zero low frequency
energy.

![](media/image3.wmf){width="4.958333333333333in"
height="3.9902777777777776in"}

Figure 2 -- BPM Simulation showing the response of five pole, 5MHz, Low
Pass Bessel and Gaussian Filters to a fast edge 0.015 Amp beam signal.
Note; the filtered response is shown ten times normal output to permit
it being viewed on the same scale as the input signal. Both filters
provide similar output signals. A slight undershoot for the Bessel
filter is observable.

![](media/image4.wmf){width="4.958333333333333in"
height="4.209722222222222in"}

> Figure 3 - BPM Simulation showing the response of 5MHz Low Pass Bessel
> and Gaussian Filters to a slow edge 0.015 Amp beam signal. Both
> filters provide similar output signals.

![](media/image5.wmf){width="5.996527777777778in"
height="2.3618055555555557in"}

Figure 4 -- Simulation Circuit

Challenges:

1.  Dynamic Range

2.  Resolution (noise) and system band-width selection

3.  Single turn (commissioning) vs. multi-turn operation (RF
    processing/base-band)

4.  Cable reflections due to out-of-band mismatch

5.  BPM linearity over a large range of aperture (+/- 100mm out of =/-
    105mm, ie: 95%)

6.  Method of position calculation

General Discussion:

The beam in the HEBT is primarily composed of short bunches. These give
rise to a large RF component at the bunching frequency and it's
harmonics. The character of the pick-up signals in this region of the
SNS is the same as that of the Linac section, and it is intended to use
identical electronics to that of the Linac in this region.

The Ring accumulates the injected bunches from the HEBT, and after a
number of turns the charge diffuses and the RF character is quickly
lost. With a millisecond of injection and about a 1 MHz revolution
frequency, the charge will grow by a factor of about 1000. Early
injected turns will have RF character, while turns that have been
accumulated for many turns will have the character of a 645ns pulse with
a repetition rate of 945ns. The pulses with little RF character will
provide a base-band signal, while those that have recently been injected
will provide an RF character. Therefore, to obtain position information
two types of electronics are required. For early turns something similar
to the Linac electronics is necessary, while for the later turns a
base-band approach is required.

The RTBT has a beam that displays primarily the character of a single
bunch of particles for 645ns. Therefore, this is similar to that of the
Ring base-band signal, and electronics based upon this design would be
most appropriate. During commissioning and studies, however, one could
expect to see a single mini-pulse transported through the entire system.
To observe this, electronics similar to that used in the Linac are
required.

A beam based alignment technique will be employed in the Ring to locate
the BPM/Quad center. A processing scheme employing normalization will
eliminate gain from the results, and the calculated position will become
dependent upon the basic sensitivity of the BPM.

History:

The original thought was to utilize a number of Linac BPM electronic
systems to get a general handle on beam position for early turns. In
addition, the base-band concept was pursued and a basic design scheme
was developed. This included a switched gain concept that follows the
beam current monitor system[^2]^,^[^3]^,^[^4]^,^[^5]^,^[^6]. The basic
concept provides different paths for the signal to follow, each with
different gain. The paths are summed in a sum stage. Each path is
capable of being switched, in or out, by a logic signal to a switched
amplifier (OPA3680). In so doing, the gain can be switched within 100ns,
sufficient for switching during the beam "gap" time in the Ring. A
detailed discussion of this approach is presented in the section
discussing amplifiers.

To permit a simpler more general solution to commissioning the Ring, an
approach that utilizes the electronics of the Linac has been considered,
with some modifications, to permit it's use with base-band signals. This
basic concept uses the RF sensitive electronics for early turns and also
during the commissioning process. Provision for switching the mixer off,
providing base-band filtering only, will permit the base-band signal to
propagate through the system to the ADC. Processing will continue in the
normal manner, and data analyzed for position. The Ring and RTBT do not
require the phase information, and I/Q demodulation need not be used. In
addition, some calibration-switching circuitry is suggested to be
replaced by local oscillator switches and front end filters to conserve
circuit board real estate.

Details:

The basic design is based upon the Bergoz BPM analog Front End (AFE ),
designed for the Linac. A simplified block diagram is shown in Figure
5a. This diagram shows an example of a 402.5MHz AFE. The pickup signals
are passed through a switching matrix that permits calibration
information to be placed on the inputs to the electronics or the BPM
pickup elements. Alternatively, the signals alone can be sent to the
electronics. These signals are band pass filtered to allow only the
402.5 MHz information to pass. They are amplified as necessary, and
mixed with a local oscillator of 352.5 MHz. This yields a difference
frequency of 50 MHz, which is selected by another band pass filter. The
result is digitized by about a 40MHz digitizer. This digitizer brings
the resulting signal to base-band, and the information of interest is a
5 MHz band after I/Q demodulation of the 10MHz-demodulated signal.

Figure 5a -- Block diagram of basic Linac BPM Electronics

![](media/image6.jpeg){width="5.997222222222222in"
height="3.921527777777778in"}

Figure 5b- Proposed Block Diagram showing the base-band Bergoz AFE

Bergoz has indicated that they can provide an AFE (figure 5b) designed
with the same format as that designed for the SNS Linac. It uses the
same I/Os (appendix D) and should plug into the present Digital Front
End (DFE) without a problem after the switch control pins are
designated. The product is BPM-AFE/1-10MHz.

It features:

0-70dB gain per channel, in steps of 10dB controlled by three (3) lines.

Gain precision better than 0.2dB (can be further calibrated by software)

Strict filtering of signal, with objective of fast settling time

50 Ohm signal path

Stable gains by current feedback op-amps.

Fast settling of filters after gain switching

Differential outputs matched to AD664x ADC inputs

Calibration by signals balanced to \<0.1dB into each channel

Calibration mode controlled by a single line

Calibration input with switchable attenuator controlled by one line

Implementing circuitry:

Four processing channels, each with:

Input attenuator 50 ohms

High-pass two pole filter

Low-pass two pole filter

402.5 MHz trap

Protective diodes

Input switch (high voltage) under common control line

Switchable gain stage 10dB under common control line

Switchable gain stage 20dB under common control line

Switchable gain stage 40dB under common control line

High-pass one pole filter

Low-pass one pole filter

Buffer and differential driver, in line with ADC recommendations

Calibration circuitry:

Switchable input attenuator

High precision four-way splitter

Calibration switches (4) (high voltage) under same control as inputs

Precision 50-ohm input/calibration summing circuit

![](media/image7.wmf){width="5.991666666666666in"
height="3.053472222222222in"}

Figure 5c -- Modified block diagram of AFE electronics

A modified design (Figure 5c) would eliminate or reduce the switching,
since the Ring will be calibrated by beam-based alignment. Some
switching will be required to establish filtering and channel gain. The
initial band-pass filtering will be switch selected to be either at RF
or a 5MHz low-pass. The local oscillator will be a 352.5 MHz oscillator
that will move the signal to 50MHz (for the 402.5MHz case), and the
mixer will be biased to avoid mixing for the base-band case. The 50 MHz
low-pass filter will be changed to a 5MHz filter for this application.

The available ADC bandwidth is 20MHz when clocked at 40MHz. Since I/Q
demodulation is not required (not interested in phase in the Ring and
RTBT) one could try to use the full ADC bandwidth, however, the full ADC
bandwidth is not easily utilized since the band-pass characteristics
required to restrict settling time to the 300ns "gap" time could result
in broad cut-off characteristics (see bandwidth section). As a result,
it is suggested that the digitizer be clocked at 64 times the revolution
frequency (about 68MHz). This will provide the separation between signal
and clock needed for anti-aliasing filter characteristics, and a clock
based upon the revolution frequency to provide better time stamping and
synchronization to the Ring signal.

In addition to the dual operating mode mentioned above, it would also be
desirable to examine the spectral lines of the tune system. This could
be accommodated with another modification, now under investigation, that
allowed a spectrum of 40MHz to 60MHz to be processed. If this was
delivered to the ADC, and sampled at 40MHz, the spectrum would be
brought back to base-band and occupy a 20MHz bandwidth. This could be
analyzed by an FFT algorithm to obtain the spectrum.

![](media/image8.wmf){width="5.992361111111111in"
height="4.0625in"}Figure 6 -- General system block diagram

The general system configuration follows that of figure 6. Signal is
processed by an analog front end (AFE) electronics block, fed to a
digitizer, and then interfaced to a personal computer. In this case,
Bergoz will provide the AFE, LANL/BNL will provide the digitizer, and
LANL/BNL will provide the digital PCI interface.

**Some base-band considerations:**

The details to follow reflect the thinking done prior to a decision to
use Bergoz to modify their present AFE for the Linac for use in the
Ring. Of particular interest is the processing, which will apply to any
hardware configuration. A summary is presented here, and more detailed
analysis is found in appendix C. Also of some interest is the bandwidth
chosen for processing, along with some of the additional work done
selecting switches, amplifiers, gains, and attenuators. Therefore, this
is presented in Appendix A of this report. Some of the details of the
BPM resolution, sensitivity and noise analysis, work are included as
appendicies B and C.

**Bandwidth:**

The bandwidth was determined by evaluating the transient response and
noise requirements. The system must settle, after a mini-pulse, within
the 300ns gap time to allow it to process the next turn. A 5 pole Bessel
or Gaussian filter exhibits minimal overshoot, and settles in about 6
time constants. This requires a filter of minimum bandwidth about 3.5MHz
(6/(6.28\*fc)=270ns). Higher order filters or filters with sharper
roll-off will ring much longer. A bandwidth of 5MHz will settle in about
190ns, and will pass a few more harmonics of the revolution frequency.

![](media/image9.wmf)

Figure 7 -- Estimated signal power in harmonics of the revolution
frequency at the output of a BPM element with a 38ma beam pulse of 645ns
duration with period 945ns. The power included in a "brick-wall" filter
BW is also shown, along with the resulting S/N ratio.

The chart above (figure 7) shows the expected frequency component power
in the signal from the BPM. One can observe the S/N ratio is fairly
constant. The added difficulties associated with anti-aliasing filtering
at wider bandwidths, and the lack of improvements in S/N indicate a
reasonable choice of bandwidth of about 1 MHz. It should be pointed out
that due to the pulse shape of the current, the third harmonic is near
zero and contributes little, as does the 6^th^ , 9^th^ & 13^th^ . A 1MHz
bandwidth is less than that required by transient response
considerations. Therefore, a compromise bandwidth near 5MHz is selected.

To achieve about 14 bits of anti-aliasing requires about 84dB rejection
at the Nyquist frequency (fs/2). Placing a sharp roll-off filter after a
Bessel or Gaussian filter will provide the good transient response of
the Bessel, with some overshoot introduced, with the good rejection of
the Chebyshev. Therefore a dual filter is suggested. A 5 pole 5MHz
Bessel or Gaussian filter provides about 40dB attenuation at 20MHz. A 5
Pole 10MHz, 0.01dB Chebyshev filter provides about another 34dB at 20MHz
for a total of 74dB at 20 MHz (Nyquist BW for a 40MHz clock). The design
can be improved with two Bessel filters, each with 7.07MHz BW (over-all
bandwidth of 5MHz, providing about 27dB attenuation each at 20MHz), for
a total of 88dB attenuation at 20MHz. The 7MHz filters have already been
design for the BCM system. A simulation is shown in Figure 8 & 9.

![](media/image10.png){width="5.759722222222222in"
height="4.322222222222222in"}

Figure 8 -- Simulated dual filter response 2 each 5-pole 7MHz Gaussian
followed by 5-pole 10MHz 0.01dB Chebyshev

![](media/image11.png){width="5.719444444444444in"
height="4.292361111111111in"}

Figure 9 -- Pulse response, showing the two Gaussian filters with good
transient response, and the over-all response corrupted by the Chebyshev
filter (amplitudes are arbitrary).

The transient response to a pulse shown in figure 9 has significant
ringing, barely recovering by the end of the "gap"time. This can be
reduced significantly by increasing the cut-off frequency of the
Chebyshev filter and requiring an increased sampling frequency. It was
considered to run the digitizer in the Ring electronics synchronously
with the Ring revolution frequency. A frequency of 64 times Frev was
selected (about 68MHz). For this case, an AD6645-80 (80MHz -- 14 bit
digitizer) would be used, and the Chebyshev filter, increased to a
bandwidth of 34/2=17MHz, would be selected. A comparison of using a
single 5-pole, 5MHz Gaussian filter and a 5-pole 0.01dB, 17MHz Chebyshev
filter vs. using two 5-pole 7MHz Gaussian filters and a 5-pole 0.01db,
10MHz Chebyshev filter is shown in figures 10 and 11. As pointed out in
the caption, the second 7MHz Gaussian filter provides a much needed 9dB
of attenuation at 20MHz. Clearly the transient response is greatly
improved by increasing the clock rate and it is suggested that the
digitizer be clocked at 64 times the revolution frequency.

![](media/image12.png){width="5.698611111111111in"
height="4.280555555555556in"}

Figure 10 -- Comparison of anti-aliasing capabilities of increasing
Sampling Frequency. The signal is down by about 92dB at 34MHz, while the
other provides about 79dB at 20MHz. If a single Gaussian filter was used
the signal would be down about 70dB at 20 MHz.

![](media/image13.png){width="5.698611111111111in"
height="4.240277777777778in"}

Figure 11 -- Comparison of pulse response for the two choices of
filtering and sampling frequency. The transient response is drastically
improved by using a 17MHz Chebyshev vs. using a 10MHz chebyshev.

**Processing:**

It is proposed to process the data using a mean square analysis coupled
with a log ratio calculation. This will provide position information
weighted more over the entire beam current pulse. Peak detection, on the
other hand, will favor the beam edges.

First, the log ratio technique offers a position output that is linear
over a larger range. Secondly, the squaring of the signals to compute
power will double the sensitivity ie:

Log(x^2^) = 2 \* log(x)

One would expect an output, then, similar to the standard log ratio
technique. The taking of the mean square for each PUE provides the
difference, and yields power in the signal, power in the noise, and a
cross-correlation term between noise and signal.

The mean square calculation basically generates a mean square power of
the signal plus the noise. This adds noise squared plus a
cross-correlation term that correlates the signal and the noise,
evaluated at zero. An excerpt from some MathCAD calculations is
presented below.

The general concept is to take the mean of the square signal of the
right PUE and divide by the mean of the square of the left PUE, and then
take a logorithm.

If we call the right pickup signal fr(t), and the left pickup signal
fl(t):

Then for small displacements of beam along the axis by a distance r:

![](media/image14.wmf){width="1.125in" height="0.17708333333333334in"}

![](media/image15.wmf){width="1.1041666666666667in"
height="0.17708333333333334in"}

F(t) is the signal on the PUE for zero displacement.

Let the noise on the right PUE be enr(t) and on the left PUE enl(t).

Then the signal out will be :

![](media/image16.wmf){width="2.1770833333333335in"
height="1.3229166666666667in"}

The output is approximately given by;

![](media/image17.wmf){width="3.9270833333333335in"
height="1.1354166666666667in"}

S is rms signal voltage (V), and σr^2^ and σl^2^ is the mean squared
noise voltages on right and left PUEs.

Assuming the terms in both the numerator and denominator are small
compared to 1 we get:

![](media/image18.wmf){width="5.996527777777778in"
height="0.4576388888888889in"}

This is approximately:

![](media/image19.wmf){width="5.09375in" height="0.5520833333333334in"}

The output is approximately (eliminating higher order terms):

![](media/image20.wmf){width="5.166666666666667in"
height="0.5520833333333334in"}

We observe that this requires 4\*a\*r to be equal to the other terms to
establish resolution requirements. The mean square noise terms for the
right and left PUE channels subtract from each other, and so do the
cross-correlation terms between the noise and the signal. There are
higher order terms that will add to noise but they can be neglected for
a\*r\<\<1.

Thus, mean square noise tends to cancel, as do cross-correlation terms.
However, the cross-correlation terms can be either positive or negative,
and will vary with a random character. Therefore, these terms could, in
fact, add. The result, then, assuming a good estimate of σr and σl are
achieved, and that these noise terms are near equal (both due to 50 Ohm
resistor noise), yields:

![](media/image21.wmf){width="2.5520833333333335in"
height="0.7291666666666666in"}

remember that a is:

![](media/image22.wmf){width="1.3645833333333333in" height="0.59375in"}

and an approximation for the minimum resolvable displacement is:

![](media/image23.wmf){width="4.5in" height="0.6979166666666666in"}

![](media/image24.wmf)

Figure 12 -- The comparison of different processing algorithms shown
above indicates that peak difference-over-sum is best, with mean-squared
difference-over-sum basically identical to log mean-squared ratio. The
indicated points reflect noise levels of 1,2,6,12,20 & 40 times the
noise of a 50 ohm resistor for reference. A S/N \> 33dB is necessary for
a 0.15mm resolution.

Processing methods were compared (see figure 12). Calculations were made
that approximated the BPM output signal, with noise added. Output
calculations were made for the case of difference over sum, using a near
peak signal, difference of mean squares over sum of mean squares (with
and without subtraction of mean square noise, although not plotted), and
log-ratio of mean squares. The results show that the standard
difference-over-the sum provides the best results when based upon signal
detection at time of peak. The log ratio of mean squares approach will
provide improved signal linearity with displacement, and is equivalent
to the difference over the sum of mean squares.

The plot of figure 12 shows the standard deviation (using 1000 values)
of the error in output signal vs. signal power to noise power ratio. In
each case the output signal was estimated with and without noise. The
difference between the output with noise and that without noise was
divided by the output without noise, and multiplied by 100 to obtain a
percent error. In this way, the standard difference-over-sum method,
using signals sampled near the expected peak time, was compared to a
difference of mean squares-over-sum of mean squares, and a log-ratio of
mean squares approach. The actual output errors could be as much as 3
sigma or more. Therefore, to estimate a single reading resolution one
should multiply the error on the plot by about 3 to 4. Shafer's estimate
of (1/(2\*sqrt2))\*sqrt(N/S) is also included for comparison to a
sinusoidal signal.

It is interesting to note that if the noise for the left and right PUE
are near equal, they tend to subtract as noise variance (power) squared.
However, the cross-correlation terms are functions of time and do not
subtract as nicely. It is these terms that dominate and limit the
resolution. Also interesting, is the attempt to subtract noise variance
values that are independent of time. The thought was to measure noise
power in each channel and then subtract it from the mean-squared values
of signal and noise measured (the output equation indicates that these
values subtract from each other). This was done and found to provide
virtually no difference to that of mean-squared difference over
mean-squared sum, and was not included in the plot. The fact that adding
and subtracting measured noise (basically of equal power) causes minor
differences in calculated resolution for the difference over sum of
mean-squares case, supports the conclusion that the noise variances tend
to cancel, and are not dominent. Furthermore, the added measurements
afford us no extra benefit, and this calculation need not be performed.

An estimate of the signal to noise ratio of 25dB was developed assuming
about a 3nV/rtHz amplifier and cable loss of 2dB, additional loss of 6dB
for switches and filters, and an intentional additional 6dB attenuator
loss for matching purposes. This yields an estimated best resolution
near 0.4mm for a 105mm half aperture. To achieve a better resolution
(near 0.15mm) one must average. One method of performing this averaging
would be to pass the digitized data through a narrow band digital filter
(about 22KHz at 2MHz). Selecting the second harmonic would provide a
near peak signal component, and would require averaging about 46 turns
to achieve this resolution.

APPENDIX A

The following information is included as documentation of work that was
performed before deciding to utilize Bergoz to design the AFE as a
modification of the LANL design. This effort includes investigations of
handling the large input signals, switch requirements, impedance
concerns, amplifiers and noise considerations, and calibration injection
methods.

**The protected amplifier:**

The system needs a method to deal with the large dynamic range and large
input signals that will become available during the later accumulated
turns in the Ring. The higher gain paths will require protected
amplifiers to assure that the input levels are not exceeded.

There are a number of considerations associated with the choice for
protection and components utilized. It is desired to maintain a linear
system. This requires an analysis of the distortion that will be
contributed to the system during circuit protection/non-linearities. It
is desired to keep the distortion to 0.1% if possible. One wants to
maintain a good input match to minimize reflections. If input diode
protection is employed, the current flowing in the protection diodes
must be low enough to protect the diodes and prevent sufficient
differential voltage build-up to cause amplifier failures. Some
wide-band amplifiers can only tolerate a differential input of +/-1.1
Volts. A method of injecting a calibration signal should be considered
as part of this effort, and noise considerations become a major concern.

A simulation assuming a 50ns bunch rise-time indicated about a 10 to 15
Volt peak signal for 100 Amps and a 10MHz bandwidth limited input
signal. The need for signal pre-filtering becomes evident when the input
signal peak voltage is estimated at high beam currents. The 100Amps is a
safety margin design point since the signal is expected to be 50Amps
with some cresting due to RF reshaping of the beam bunch. Therefore, it
may be possible to use a 15Volt op-amp at the input that exhibits large
differential and common mode input signal capability, and provide
limiting for the output of this stage to 4 to 5 volts for the remainder
of the system electronics. Using a 15 volt input stage will require an
additional power supply. A potential amplifier for this is a Texas
Instruments THS4021. This amplifier boasts a 1.5nV/rtHz noise voltage,
BW=350MHz at a gain of 10, a common mode voltage over the entire supply
range (+/- 15 Volts) and can tolerate as much as +/- 4 V differential
input voltage. It has a minimum stable gain of 10.

As pointed out earlier, the input noise must be kept low to achieve the
required resolution at low current study levels, near 10mA. Therefore it
is desirable to use low input resistor values. Using low values of
resistance will accentuate the distortion introduced by protection
components. A 15 Volt input stage will not require input protection
since the peak input will not exceed 15 volts for a pre-filtered input
scheme.

For an input scheme utilizing 5 volt amplifiers input protection is
required. The associated noise and distortion considerations for this
configuration must be dealt with. The distortion will require a large
input resistor to both minimize mismatch and limit diode currents due to
large input voltages. Diode protection requires about a 1K series
resistor to assure that the diode voltage and currents are kept low (for
a BAS70 it is 1V at 17 ma; for an HSMS2812 it is 1V at 35ma, a 1N4148 or
1N914/4448 can deal with larger currents but has more capacitance). This
assures the diodes would properly limit, and can support the signal
levels. In addition, to minimize the input loading/distortion, values
near 1K to 2K are also required.

*Ie: I= (V~pk~ -- V~diode~)/R and R~in~ = V~pk~ /\[(V~pk~ --
V~diode~)/R\]. The input resistance change when clamping signals to 0.5V
will change by 1+0.5/V~pk~ , yielding about a 3.33% change for a 15 volt
peak. To minimize this change, the resistor should be shunted by a low
value at the input. This will yield a parallel resistance change of
∆(1-R~2~/(R~1~+R~2~)); where ∆ is the fractional change in R~2~. From
this we see that for small changes in the parallel combination we
require a large value for R~2~ (R~2~ = 32.2\* R~1~ or 1.6K for R~1~=50
Ohms).*

Resistors of this value will generate significant noise and will degrade
performance considerably at low levels. It should be noted that if
attenuators are used before the amplifier the affect of the resistance
change is reduced on the input by about the power ratio ie: ½ for 3dB,
1/3 for 10dB.

Noise:

The equivalent rms input noise for the THS4021 is given by:

E~ni~=sqrt\[ (e~n~)\^2 + (I~n+~\*R~i+~)\^2 + (I~n-~\*R~i-~\|\|R~f~)\^2 +
4\*K\*T\* R~i+~ + 4\*K\*T\* (R~i-~\|\|R~f~) \] V/rtHz

Given: e~n~=1.5nV/rtHz

I~n~=2pA/rtHz

R~i+~=50

R~i-~=50

K=1.38E-23

T=300

R~f~=1000

Gain=20

E~ni~=sqrt\[ (1.5E-9)\^2 + (100E-12)\^2 + (95.2E-12)\^2 + 0.83E-18+
0.79E-18 \] V/rtHz

E~ni~=1.97nV/rtHz

A PSpice simulation using the manufacturer's model for this part yielded
2.1nV/rtHz at 10KHz.

Noise is amplified by the stages that follow, and there is ADC noise and
noise from the wide bandwidth ADC drive amplifier that is not filtered
that must be considered. The ADC noise is estimated from the data sheet
as 0.246mV. This adds in quadrature with noise from the 300MHz AD8138,
(5nV/rtHz, 2pA/rtHz, 500 Ohm unity gain) approximately 6.45nV/rtHz
(0.11mV over 300MHz). If we assume all of this folds into the pass band
when the ADC converts we have a digitizer with about 0.27mVrms noise. A
maximum gain of about 0.5/1.5E-3 = 333 is required yielding noise of
333\*2.1E-9\*sqrt(5E6)=1.56mVrms which dominates, providing about a 50dB
s/n ratio.

![](media/image25.wmf){width="5.997222222222222in"
height="1.5416666666666667in"}Figure A-1 -- Schematic of a low noise
amplifier providing 1.58nV/rtHz employing a virtual input impedance of
50 Ohms, and providing a Gain of 19 and Bandwidth of 60MHz.

The schematic shown in Figure A-1 is a simulation of a low noise
amplifier employing a THS4012, 15 volt amplifier operating at a gain of
19, and followed by a unity gain inverting OPA680 amplifier. The result
is a low noise amplifier with a controlled gain allowing the feedback
resistor to be divided by 20 to yield a virtual input impedance of 50
Ohms. This provides a match to a 50 Ohm transmission line, at no
additional noise penalty. Resistor values are selected that are low,
minimizing noise. The published noise voltage for the THS4012 is
1.5nV/rtHz, and a PSpice simulation yielded 1.58nV/rtHz at 10KHz.

![](media/image26.wmf){width="5.997222222222222in"
height="3.1083333333333334in"}

Figure A-2 -- A simulation of the Low Noise Amplifier of Figure 13
showing a solid state switch capable of dealing with +/- 15 Volt
signals.

The circuit of Figure A-2 can handle 15-volt signals, and protects the
amplifier by switching the signal to ground after a number of turns. The
problem with this is reliability of receiving the shutdown signal.
Should this signal fail to activate the switch the circuit can see the
high over-voltage at the input. Additional protection in the form of a
diode string can be used for safety. During normal operation the signal
is switched to ground long before the diodes clamp. During the switch
failure mode the diodes protect and cause distortion (however, the
circuit has already failed and needs repair).

The switch resistance may not be well matched from unit to unit, and the
amplifier noise is dependent upon the resistance at the input. An
estimate of equivalent input noise has been done to examine this affect.
The results are shown in figure A-3. Total input noise varies from 2nV
to 3.7nV per root Hz.

![](media/image27.wmf){width="5.136111111111111in"
height="3.0631944444444446in"}

Figure A-3 -- Noise estimates for the circuit of Figure 14

Calibration:

The system must allow for in-place calibration. Standard techniques
include a calibration source that is switched to each electrode, and
measurements made at all electrodes. These measurements are compared
with measurements made during bench calibration, and appropriate gain
adjustments are provided to each electrode channel. These gain changes
can be included in the digital processing part of the system if the
system is linear, requiring minimum impact on the analog front-end
electronics.

A "beam-based alignment" approach is intended to align the quad-center
and BPM. This is accomplished by applying a slowly varying current to
the "trim" winding in a Quad and observing the BPM signals. When the
beam is at the magnetic center of the quad under adjustment, there is no
force applied to the beam, and the BPM signal that is correlated to the
"trim" current frequency is zero. Reading position voltages on each PUE
channel should yield a zero position. Therefore, all voltages computed
by the electronics need to be stored for later subtraction.

The channel gains must be balanced to assure the proper sensitivity is
established for position. Thus, if the same signal was injected into all
channels, the resulting output signals should be the same for the same
gain through each channel. This must be done for different signal levels
to assure a gain calibration exists for each gain setting.

The implementation of the switching system and the required levels for
adequate coupling to the other electrodes could introduce problems with
component selection and availability. Therefore, it is desired to avoid
sending a signal to the PUE. Instead a signal will be placed on all
channels of the electronics. It is desired that calibration be
accomplished between macro-pulses. A fast switching scheme is preferred.
There is about 15ms between pulses for calibration purposes. In general,
relays do not respond quickly enough to permit calibration without
losing a mini-pulse/macro-pulse. Therefore, it is highly desired to find
a solid state switching solution. Wide band solid state switches have
limited voltage-handling capability, as do low noise variable gain
amplifiers. RF switches require AC coupling, and have typical 1dB
compression levels near 21 - 24dBm (2.5 - 3.5Vrms, 3.5 - 5Vpk). The
amplifiers under consideration (AD60x series) can handle a maximum input
of only +/- 2V continuously (+/- Vs for 10ms).

Approach 1 (not in priority order):

To accommodate switches and filtering with good cable matching, a scheme
of matching networks, attenuators, and switches has been devised.

![](media/image28.wmf)

Figure A-4 -- A scheme to maintain impedance match and switch signals
during calibration and gain changes.

Switches:

The switches will have to handle voltages that could be as high as 15
volts. The BPM signals could be higher but will be attenuated before
they reach the switches. Fast acting solid state switches characterized
in the RF frequency range of interest require AC coupling, and have
typical 1dB compression levels near 21 - 24dBm (2.5 - 3.5Vrms, 3.5 -
5Vpk). Therefore, a MAXIM switch (MAX4666) capable of handling +/- 15
volts is being considered for testing in this application. This switch
exhibits a large capacitance to ground, and the affect on frequency
response must be evaluated. In addition, this switch could see large
input signals with system power off. This could cause excessive currents
at the input protection diode and alternative switches like the MAX
4510/4520 and IH5341/5352 with fault protection are also being
considered. It is thought that the switch capacitance, if consistent one
switch to another, could be utilized as part of the output capacitance
required for a Diplexer or Bessel filter. A switch configuration is
under consideration to provide no reflections when switched "off". This
places an appropriate 50 Ohm termination on the line being switched.

Matching and attenuators:

To properly match the signal cable, two 100:50 Ohm minimum loss pads are
used to split the signal. These provide about a 7.7dB loss. To match a
switched attenuator, a technique has been chosen that will maintain
constant loading on the filter and attenuators when the switch is
positioned for attenuation or without attenuation. To accomplish this,
the switches place 50 Ohms across the attenuator when switched in the
"off" condition. Use is made of an attenuator made up of a series 22.36
Ohm resistor and a shunt 161.8 Ohm resistor. This combination provides a
50 Ohm impedance looking into the series end with the other end
terminated in 33.33 Ohms, and 50 Ohm looking into the shunt end with the
series end terminated in 50 Ohms. When loaded with 33.33 Ohms it
provides 5dB attenuation and a 50 Ohm input impedance (see figure A-4).
A similar attenuator placed at the output of a conventional 20dB - 50
Ohm pad will provide 20 dB additional attenuation with a match to 33.33
Ohms. The 33.33 Ohm termination is achieved by a parallel combination of
the100 Ohm load from the amplifier and the 50 Ohm impedance looking back
into the matching pad described above. Since this pad appears as a load
for both paths the system is matched with the switch in either position.

Approach 2 (not in priority order):

Using a splitter for the signal path, we lose 3dB in the signal path,
but decouple the calibrator from the signal path (Macom DS113/DSS113,
0.4-400MHz). Hybrid couplers capable of passing frequencies as low as
200KHz and with bandwidths of 35MHz (Macom HH-108) can be purchased for
this application as well. This could be used in coordination with a
protected amplifier gain selection approach, similar to that of the BCM
electronics. The system would use separate channels that are switched in
or out as gain is required. Signal processing must not rely upon DC, and
some other averaging calculation could be used on the full cycle signal.
An RMS approach will work after appropriate DC removal.

Amplifiers:

The AD602 variable gain amplifier is being considered for its low noise
(1.4nV/sqrt Hz) and controlled bandwidth with gain (35MHz) which matches
the ADC sampling frequency (68MHz) well. This low noise is only
attainable at full gain. This particular unit (AD602) can provide
attenuation as well as gain (-10dB to +30dB). The AD600 provides 0 to 40
dB gain with 35MHz bandwidth, and the AD603 provides --11 to +31 with a
90MHz bandwidth. The AD600/602 provides two amplifier sections. The
match between sections on a single chip is good, and this could be used
to help balance the characteristics between the two channels. Therefore,
it is recommended to use a single chip as the amplifier for both
channels, followed by a different dual amplifier chip that would provide
the second amplifier in both channels. The gain biasing will be selected
to provide the first stage with 3dB more gain than the second stage.
This will reduce the ripple in the gain as a function of control
voltage, and provide more gain in the earlier stage to reduce noise
contribution from the second stage.

Dealing with dynamic range:

One of the design challenges is to deal with a very large dynamic range
in the Ring and RTBT. The signal is expected to change by a factor of
about 1000 due to the storage of 1060 turns. In addition the target of
0.15mm resolution will necessitate an additional 1000 for a total of 6
orders of magnitude. It would be nice to adjust gain in such a way that
we lose no turn information. Input signal voltages could be quite large
directly out of the BPM (very wide-band, 20mV to 20V peak). When
filtered to 5MHz, signals will range from 1.8mV peak to 10 Volts peak,
for 15ma and 100 amp beams with no allowances for terminations and
matching circuits.

Of the various methods to deal with the required sensitivity during
studies with low currents, and large signals during normal operation, a
diplexer approach will yield the best signal with minimal reflections
and losses.

An alternate approach, a 6dB termination network placed at the BPM
output terminal to match the \>100 meter long transmission line for
reflections, will reduce signal. The result is a maximum signal near
0.9mV peak for a 15mA beam and a 6 Volt peak signal for 100 Amps. The
estimate is based upon a 10ns rise time at low currents and a 50ns rise
time at high currents. Therefore, it is desirable to filter the signal
as soon as practical. Placing a filter at the end of a long cable will
create a mismatch for out of band signal components. These components
will be reflected back to the source where they will encounter an open
circuited 50 Ohm stripline. They will again be reflected, and this will
create signal interference with "real" signals. To minimize this, a
specially designed 6dB pad is being considered to act as a moderate
matching element. This pad will look like 50 Ohms into each end, yet
will match an open circuit to terminate the long line at lower
frequencies (3400-25.5-51.1 see figure A-5).

![](media/image29.wmf){width="1.875in" height="1.4270833333333333in"}

Diplexer:

As an alternative to the resistive termination and matching scheme
mentioned above, a Diplexer circuit that will provide a broadband 50 Ohm
matching impedance for the 50 Ohm stripline, and band-limit the signal
for transmission on the long cable is presented below. The requirement
for a constant input impedance, independent of frequency is that L/C =
2\*R\^2 (for 50 Ohms L=5000\*C, cross-over frequency =
1/(2\*π\*sqrt(L\*C))).

![](media/image30.wmf){width="4.197916666666667in"
height="1.3229166666666667in"}Figure A-6 - Circuit of a 14.4MHz Diplexer

The circuit shown in Figure A-6 will separate the signal into a 14.1 MHz
low-pass band and a high--pass band above 14.1 MHz. This will provide a
good broadband match to the transmission line, and provide initial
filtering for the BPM system electronics to follow.

![](media/image31.wmf){width="2.2916666666666665in"
height="2.3645833333333335in"}

Figure A-7 -- Circuit of a 30MHz Diplexer

The circuit shown in Figure A-7 will separate the signal into a 30 MHz
low-pass band and a high--pass band above 30 MHz. This will provide a
good broadband match to the transmission line, and provide initial
filtering for the BPM system electronics to follow.

The circuits of figure A-6 and A-7 can be placed at the end or beginning
of the long transmission line connecting the BPM to the Ring BPM
electronics. If the circuit is placed at the beginning, near the BPM,
the stripline is terminated, the signal bandwidth is limited on the
transmission line and the peak signal at the electronics is smoothed and
reduced. This will limit signal bandwidth, and not permit RF processing.
Alternatively, placing this circuit at the receiving end, will split the
signal into a base-band signal and an RF signal.

Using a Diplexer will increase the available signal from that mentioned
earlier. It is estimated that the signal at the input to the electronics
will be about 8mV peak with the 30MHz filter. The 30 MHz Diplexer filter
will yield near unity gain at the 5MHz signal bandwidth expected for
this system, and add minimally to distortion. The electronics must
terminate the cable in 50 Ohms to minimize the reflections. After
bandwidth reduction to 5MHz the signal is expected to be about 1.76mV
peak.

Linearity:

In addition to the resolution, operation over a range of +/- 100mm for
narrow beam studies will be required. For a 210mm aperture, this
represents linear operation over 95% of the beam pipe aperture. Using
the linear approach (R-L)/(R+L) yields a linear range of about +/- 20mm.
An analysis of a 70 degree, stripline designed, indicates the BPM will
require non-linearity correction. An example of non-linear response for
the difference over the sum mode of operation is shown below derived
from equations developed by Shafer, AIP212, Accelerator Instrumentation
1989, "Beam Position Monitoring".

![](media/image32.wmf){width="4.0in" height="2.573611111111111in"}

Figure A-8- A plot of sensitivity along the axis of a pair of pick-up
elements for a 70 degree stripline designed BPM with a half aperture of
105mm. The sensitivity is shown to be 0.018V/V per mm. Linearity is
shown to be "reasonable" over a range of +/- 20mm.

![](media/image33.wmf){width="4.020833333333333in"
height="2.4895833333333335in"}

Figure A-9 - A plot of sensitivity along the axis of a pair of pick-up
elements for a 70 degree stripline designed BPM with a half aperture of
105mm. The sensitivity is shown to be 0.31dB per mm. Linearity is shown
to be "reasonable" over a range of +/- 65mm.

From the above plots (figures A-8 & A-9) it is clear that the log-ratio
approach provides a more linear range of measurement, along the axis of
the element pair, without corrections. For this reason the log-ratio is
a more advantageous processing approach. Problems associated with
balanced logarithmic amplifiers have made this approach challenging. We
propose doing the log-ratio digitally.

A hardware approach is included here for completeness. Recently (Feb.
2001,
http://content.analog.com/pressrelease/prdisplay/0,1622,301,00.html),
Analog Devices has developed an IC with two logarithmic amplifiers,
laser trimmed and matched for temperature, process, frequency and supply
voltage that will operate over a 60dB dynamic range (see AD8302 data
sheet, http://products.analog.com/products/info.asp?product=ad8302).
This device, although designed for RF applications with a demodulating
logarithmic amplifier system can be used at lower frequencies. Alternate
devices that provide a large dynamic range are the AD640 (350MHz
operation, provides two 50dB amplifiers in a single IC), and the AD8307
(500MHz operation, contains a single amplifier). The AD8302 (2.7GHz
operation, provides two 60 dB amplifiers and additional circuitry to
provide phase analysis) device can handle signals from 223uV to 223mV.
The maximum allowable input signal is 0.707V. Therefore, a method to
insure that this voltage is not exceeded must be employed for the Ring,
and RTBT. The automatic DC offset compensation control loop must be
adjusted to operate at frequencies below 1MHz to permit use in the Ring.
This can be achieved by using coupling and bypass capacitors that will
pass this frequency (using 0.02uF capacitors will extend the low
frequency response to near 100KHz. This approach has merit and should be
considered.

From manufacturer provided curves of magnitude ratio conformance, an
error of less than 0.25dB can be obtained over a +/- 25dB range. From
the above estimates of sensitivity, 1mm for a 105mm half aperture is
about 0.3dB. Therefore an accuracy of less than +/- 1mm can be expected
from this unit. Referencing the AD640 data sheet, a discussion of
"Effect of Waveform on Intercept" is presented. Since it would be
expected that both channels would see the same waveform, the intercept
would affect both channels equally, and the resulting output would not
exhibit any effect. This is also alluded to in the data sheet for the
AD8302 on page 15.

The full wave detector provided with each stage of the logarithmic
amplifier effectively provides an absolute magnitude function, and
doubles the frequency. This is a good feature for our purposes and we
can do an average of the absolute magnitude to develop a number
representing the signal over the entire mini-pulse. This number can be
used for the position information. Thus, the output from the log ratio
amplifier averaged over an entire mini-pulse can be used.

APPENDIX B

Taken from a MathCAD file.

**SENSITIVITY OF BPM**

This section will apply signals and develop plots and data of a
simulated beam. Noise is added at the end.

Shafer AIP Conference 212 -1989 \"Beam Position Monitoring\"

wall current density for a pencil beam Ib(t) at position r,θ inside a
grounded circular conducting beam pipe of radius b. The wall current
density iw at b,φw is then

iw(b,φw,t)=-(Ib(t)/2πb)\[1+2Σ(r/b)\^n (cos(n(φw-θ))\]

iw(b,φw,t)=(-Ib(t)/2πb)\[(b\^2-r\^2)/9b\^2+r\^2-2br cos(φw-θ)\]

Two forms are provided by Shafer. A sum and a closed form solution

![](media/image34.wmf){width="2.8229166666666665in"
height="0.5520833333333334in"}

Eq 2-1

Eq 2-2

![](media/image35.wmf){width="3.3333333333333335in"
height="0.6041666666666666in"}

Two electrodes with angular width φ placed at zero and 180 degrees the
resultant currents flowing parallel to the beam inside surface of these
electrodes are (assuming they are grounded and also at radius b):

Right pickup

let the electrode have an angle of 70 degrees:

![](media/image36.wmf){width="0.59375in" height="0.3854166666666667in"}

The beam is at point (r,θ):

Let θ=0 to examine along the axis of the pickup

![](media/image37.wmf){width="0.3333333333333333in" height="0.1875in"}

![](media/image38.wmf){width="0.4166666666666667in"
height="0.17708333333333334in"}

![](media/image39.wmf){width="0.4479166666666667in"
height="0.17708333333333334in"}

![](media/image40.wmf){width="0.19791666666666666in"
height="0.17708333333333334in"}

![](media/image41.wmf){width="0.7916666666666666in"
height="0.2604166666666667in"}

![](media/image42.wmf){width="1.53125in" height="0.2604166666666667in"}

![](media/image43.wmf){width="0.78125in" height="0.2604166666666667in"}

![](media/image44.wmf){width="0.7395833333333334in"
height="0.2604166666666667in"}

![](media/image45.wmf){width="2.9583333333333335in" height="0.46875in"}

![](media/image46.wmf){width="2.53125in" height="1.84375in"}

![](media/image47.wmf){width="3.4270833333333335in"
height="0.6041666666666666in"}

Left Pickup

![](media/image48.wmf){width="3.65625in" height="0.6041666666666666in"}

Consider n= only 1, for small displacements from center.

![](media/image49.wmf){width="2.6041666666666665in" height="0.40625in"}

![](media/image50.wmf){width="0.6979166666666666in" height="0.1875in"}

![](media/image51.wmf){width="1.0416666666666667in"
height="0.17708333333333334in"}

![](media/image52.wmf){width="1.09375in" height="0.40625in"}

![](media/image53.wmf){width="1.0833333333333333in"
height="0.3854166666666667in"}

![](media/image54.wmf){width="1.40625in" height="0.40625in"}

![](media/image55.wmf){width="1.25in" height="0.3854166666666667in"}

![](media/image56.wmf){width="1.1979166666666667in"
height="0.3854166666666667in"}

![](media/image57.wmf){width="0.90625in" height="0.17708333333333334in"}

![](media/image58.wmf){width="2.8229166666666665in"
height="1.7916666666666667in"}

![](media/image59.wmf){width="4.625in" height="0.4791666666666667in"}

![](media/image60.wmf){width="1.5416666666666667in"
height="0.3958333333333333in"}

![](media/image61.wmf){width="3.65625in" height="0.6041666666666666in"}

![](media/image62.wmf){width="3.1041666666666665in" height="0.40625in"}

![](media/image63.wmf){width="5.09375in" height="0.4791666666666667in"}

![](media/image64.wmf){width="2.2395833333333335in" height="0.40625in"}

![](media/image65.wmf){width="2.1875in" height="0.46875in"}

![](media/image66.wmf){width="2.1458333333333335in" height="0.46875in"}

Consider the closed form solution

![](media/image67.wmf){width="2.8229166666666665in"
height="0.5520833333333334in"}

Eq 2-3

Integrating from φw=-φ/2 to +φ/2

![](media/image68.wmf){width="3.6666666666666665in"
height="0.6041666666666666in"}

Eq 2-4

![](media/image69.wmf){width="7.802083333333333in"
height="1.1354166666666667in"}

Eq 2-5

![](media/image70.wmf){width="4.447916666666667in"
height="0.6041666666666666in"}

![](media/image71.wmf){width="4.145833333333333in"
height="2.5416666666666665in"}

![](media/image72.wmf){width="2.96875in" height="2.125in"}

Clearly , the log ratio technique is more linear over a larger range of
the beam pipe radius, and has about twice the sensitivity of the
diff/sum method.

Let the signals developed by an electrostatic pickup with 70 degree
angular width and 22cm length be the pickup element signals. Each signal
from the right and left is passed through a system filter with gain and
noise. We are interested in estimating the resolution.

From Shafer\'s sum equations, using only the linear term since we are
interested in displacements that are a small fraction of the radius, we
get the output of each element for an on axis beam at location (r,0).

![](media/image37.wmf){width="0.3333333333333333in" height="0.1875in"}

![](media/image73.wmf){width="2.71875in" height="0.40625in"}

![](media/image74.wmf){width="2.2708333333333335in"
height="0.6145833333333334in"}

assuming the pickup length is L, the charge deposited on the inside
surface of the electrode is:

![](media/image75.wmf){width="0.9791666666666666in"
height="0.3958333333333333in"}

The signal current flowing into the capacitance is the derivative of the
charge.

![](media/image76.wmf){width="0.9583333333333334in" height="0.40625in"}

![](media/image77.wmf){width="1.1979166666666667in" height="0.40625in"}

If this current is delivered to the characteristic resistance of a
matched transmission line the voltage developed is:

![](media/image78.wmf){width="0.9166666666666666in"
height="0.17708333333333334in"}

![](media/image79.wmf){width="1.3229166666666667in" height="0.40625in"}

![](media/image80.wmf){width="0.6041666666666666in"
height="0.2604166666666667in"}

![](media/image81.wmf){width="0.3854166666666667in"
height="0.17708333333333334in"}

![](media/image82.wmf){width="0.5520833333333334in" height="0.1875in"}

![](media/image83.wmf){width="0.40625in" height="0.17708333333333334in"}

![](media/image84.wmf){width="0.2916666666666667in"
height="0.17708333333333334in"}

![](media/image39.wmf){width="0.4479166666666667in"
height="0.17708333333333334in"}

Eq 2-6

![](media/image85.wmf){width="2.6145833333333335in"
height="0.6145833333333334in"}

Eq 2-7

![](media/image86.wmf){width="2.59375in" height="0.6145833333333334in"}

For :

![](media/image87.wmf){width="0.3541666666666667in"
height="0.17708333333333334in"}

![](media/image88.wmf){width="0.8229166666666666in"
height="0.17708333333333334in"}

![](media/image89.wmf){width="0.8125in" height="0.17708333333333334in"}

![](media/image90.wmf){width="2.6354166666666665in" height="1.84375in"}

Eq 2-8

![](media/image91.wmf){width="16.17222222222222in"
height="0.7291666666666666in"}

Filter response to an exponential is:

![](media/image92.wmf){width="0.6875in" height="0.2604166666666667in"}

use 2 times the fc for 1db point of a single order filter

![](media/image93.wmf){width="0.6666666666666666in"
height="0.3958333333333333in"}

![](media/image44.wmf){width="0.7395833333333334in"
height="0.2604166666666667in"}

Eq 2-9

![](media/image94.wmf){width="6.666666666666667in"
height="0.4895833333333333in"}

![](media/image95.wmf){width="4.03125in" height="2.9270833333333335in"}

![](media/image96.wmf){width="2.53125in" height="1.84375in"}

Define the bipolar response:

![](media/image97.wmf){width="1.8541666666666667in" height="0.1875in"}

![](media/image98.wmf){width="2.6354166666666665in" height="1.84375in"}

and when multiplied by the scaling:

define beam average current:

![](media/image99.wmf){width="0.5833333333333334in"
height="0.17708333333333334in"}

![](media/image100.wmf){width="3.8541666666666665in"
height="0.6145833333333334in"}

The output of the filter to the signal is:

![](media/image87.wmf){width="0.3541666666666667in"
height="0.17708333333333334in"}

![](media/image101.wmf){width="0.5104166666666666in"
height="0.17708333333333334in"}

For the right electrode;

Eq 2-10

![](media/image102.wmf){width="0.6458333333333334in"
height="0.17708333333333334in"}

![](media/image103.wmf){width="0.59375in" height="0.1875in"}

![](media/image104.wmf){width="0.9166666666666666in"
height="0.2604166666666667in"}

![](media/image105.wmf){width="2.7916666666666665in"
height="0.6145833333333334in"}

![](media/image106.wmf){width="1.6354166666666667in"
height="0.2604166666666667in"}

For the left electrode;

Eq 2-11

![](media/image107.wmf){width="2.7708333333333335in"
height="0.6145833333333334in"}

![](media/image108.wmf){width="1.625in" height="0.2604166666666667in"}

The difference is:

![](media/image109.wmf){width="2.4375in" height="0.2604166666666667in"}

For 15ma beam the difference is;

![](media/image110.wmf){width="2.90625in" height="0.3854166666666667in"}

This confirms a \"ball park\" number for the output of a 5MHz filter for
a 38mA input beam. If this is changed to a 15 mA beam the signal will
drop as shown: Earlier estimates showed an output peak at about 1.9mV
and this analysis shows about 2.1mV.

![](media/image111.wmf){width="4.114583333333333in"
height="2.5104166666666665in"}

Consider now the processing:

Consider taking the signal and squaring it and then taking the average
of the square over the time interval of an entire mini-pulse. We
therefore, obtain the power in the signal averaged over about 945ns.
This will automatically include power from the negative parts of the
signal, and will add the noise power to the signal power.

With the standard difference processing; we will have a signal at the
input of about 63uV per mm displacement (161\*15/38uV). The bandwidth
for this is 5MHz. For noise to equal signal, this reflects to a noise
voltage density of 63/2236 uV/rtHz.

![](media/image112.wmf){width="1.4791666666666667in"
height="0.5729166666666666in"}

this is 28nV/rtHz and is about 31 times the noise of a 50 Ohm resistor.

Using RMS analysis we get ;

Eq 2-12

![](media/image113.wmf){width="2.3020833333333335in"
height="0.8020833333333334in"}

Eq 2-13

![](media/image114.wmf){width="2.3020833333333335in"
height="0.8020833333333334in"}

The mean squared difference signal is:

![](media/image115.wmf){width="3.78125in" height="0.8020833333333334in"}

Eq 2-14

The rms difference signal is:

For :

![](media/image102.wmf){width="0.6458333333333334in"
height="0.17708333333333334in"}

![](media/image87.wmf){width="0.3541666666666667in"
height="0.17708333333333334in"}

![](media/image101.wmf){width="0.5104166666666666in"
height="0.17708333333333334in"}

Eq 2-15

![](media/image116.wmf){width="4.041666666666667in"
height="0.8229166666666666in"}

This difference mean square signal is the numerator with no noise, and
will be about 63uV as before, for a 15ma Beam current
(15/38)\^2\*417=65. This indicates little difference using mean-squared
analysis over peak signal analysis.

**EXAMINE THE SENSITIVITY**

To examine the sensitivity of the mean square approach consider the BPM
response of equation 2-4 and 2-5 again.

![](media/image117.wmf){width="3.6666666666666665in"
height="0.6041666666666666in"}

![](media/image118.wmf){width="4.447916666666667in"
height="0.6041666666666666in"}

![](media/image101.wmf){width="0.5104166666666666in"
height="0.17708333333333334in"}

![](media/image119.wmf){width="1.0416666666666667in"
height="0.17708333333333334in"}

![](media/image120.wmf){width="3.9270833333333335in"
height="2.4583333333333335in"}

![](media/image121.wmf){width="1.3541666666666667in"
height="0.5520833333333334in"}

![](media/image122.wmf){width="4.114583333333333in"
height="2.4583333333333335in"}

![](media/image123.wmf){width="1.3958333333333333in"
height="0.5520833333333334in"}

**ADD NOISE TO ANALYZE THE RESULTS**

Add noise:

![](media/image124.wmf){width="0.53125in" height="0.2604166666666667in"}

![](media/image125.wmf){width="0.3125in" height="0.17708333333333334in"}

k is the number of times the noise is a factor of a 50 Ohm resistor

![](media/image126.wmf){width="1.1041666666666667in"
height="0.2604166666666667in"}

Generate 4 independent noise sources (divide by Pi/2 to normalize to an
arbitrary bandwidth):

![](media/image127.wmf){width="2.90625in" height="0.59375in"}

Noise of right pickup

![](media/image128.wmf){width="2.90625in" height="0.59375in"}

Noise of left pickup

![](media/image129.wmf){width="2.90625in" height="0.59375in"}

Measured noise of right pickup

![](media/image130.wmf){width="2.90625in" height="0.59375in"}

Measured noise of left pickup

![](media/image131.wmf){width="1.4375in" height="0.3854166666666667in"}

15 ma beam with noise, Right pickup

15 ma beam with noise, Left pickup

![](media/image132.wmf){width="1.40625in" height="0.3854166666666667in"}

Mean Squared Noise:

![](media/image133.wmf){width="0.9895833333333334in"
height="0.2604166666666667in"}

![](media/image134.wmf){width="1.3854166666666667in" height="0.5625in"}

![](media/image135.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image136.wmf){width="0.8854166666666666in"
height="0.19791666666666666in"}

![](media/image137.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image138.wmf){width="1.3854166666666667in" height="0.5625in"}

![](media/image139.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image140.wmf){width="0.8854166666666666in"
height="0.19791666666666666in"}

![](media/image141.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image142.wmf){width="1.3854166666666667in" height="0.5625in"}

![](media/image143.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image144.wmf){width="0.8854166666666666in"
height="0.19791666666666666in"}

![](media/image145.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image146.wmf){width="1.3854166666666667in" height="0.5625in"}

![](media/image147.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image148.wmf){width="0.8854166666666666in"
height="0.19791666666666666in"}

![](media/image149.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image150.wmf){width="3.7604166666666665in"
height="2.4270833333333335in"}

![](media/image151.wmf){width="3.78125in" height="2.34375in"}

![](media/image152.wmf){width="3.78125in" height="2.34375in"}

For no noise:

![](media/image153.wmf){width="1.6770833333333333in" height="0.75in"}

![](media/image154.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image155.wmf){width="1.6458333333333333in" height="0.75in"}

![](media/image156.wmf){width="1.28125in" height="0.2604166666666667in"}

Mean squared signal is:

![](media/image157.wmf){width="1.375in" height="0.3854166666666667in"}

![](media/image158.wmf){width="1.3333333333333333in"
height="0.2604166666666667in"}

For no noise

![](media/image159.wmf){width="1.5520833333333333in"
height="0.3854166666666667in"}

![](media/image160.wmf){width="1.0625in" height="0.17708333333333334in"}

Including Noise:

![](media/image161.wmf){width="2.4375in" height="0.5625in"}

![](media/image135.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image162.wmf){width="1.8854166666666667in" height="0.5625in"}

![](media/image163.wmf){width="1.21875in" height="0.2604166666666667in"}

![](media/image164.wmf){width="1.7604166666666667in" height="0.5625in"}

![](media/image165.wmf){width="1.1875in" height="0.2604166666666667in"}

![](media/image166.wmf){width="1.3229166666666667in"
height="0.3854166666666667in"}

![](media/image167.wmf){width="1.3854166666666667in"
height="0.17708333333333334in"}

Using the noise subtraction we get:

![](media/image168.wmf){width="1.8125in" height="0.3854166666666667in"}

![](media/image169.wmf){width="0.9895833333333334in"
height="0.17708333333333334in"}

![](media/image170.wmf){width="1.6354166666666667in"
height="0.2604166666666667in"}

second sample point:

![](media/image171.wmf){width="1.6354166666666667in"
height="0.2604166666666667in"}

![](media/image172.wmf){width="1.6354166666666667in"
height="0.2604166666666667in"}

pk point

![](media/image173.wmf){width="1.5729166666666667in"
height="0.2604166666666667in"}

![](media/image174.wmf){width="1.625in" height="0.2604166666666667in"}

![](media/image175.wmf){width="1.6354166666666667in"
height="0.2604166666666667in"}

![](media/image176.wmf){width="2.3645833333333335in"
height="0.5520833333333334in"}

![](media/image177.wmf){width="2.0416666666666665in"
height="0.5520833333333334in"}

Output std Diff

notice exact timing not required, but same timing is:

![](media/image178.wmf){width="2.0416666666666665in"
height="0.5520833333333334in"}

With noise:

![](media/image179.wmf){width="4.520833333333333in"
height="0.5520833333333334in"}

![](media/image180.wmf){width="3.8958333333333335in"
height="0.5520833333333334in"}

Apply log ratio approach:

![](media/image133.wmf){width="0.9895833333333334in"
height="0.2604166666666667in"}

![](media/image181.wmf){width="2.1979166666666665in" height="1.15625in"}

![](media/image182.wmf){width="1.2291666666666667in"
height="0.17708333333333334in"}

Ratio of S/N is:

![](media/image183.wmf){width="1.2395833333333333in"
height="0.3854166666666667in"}

![](media/image184.wmf){width="1.46875in" height="0.3854166666666667in"}

![](media/image185.wmf){width="0.16666666666666666in"
height="0.17708333333333334in"}

![](media/image182.wmf){width="1.2291666666666667in"
height="0.17708333333333334in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image187.wmf){width="1.6041666666666667in"
height="0.3854166666666667in"}

![](media/image188.wmf){width="1.2395833333333333in"
height="0.17708333333333334in"}

![](media/image189.wmf){width="2.5in" height="0.3854166666666667in"}

![](media/image190.wmf){width="1.2395833333333333in"
height="0.17708333333333334in"}

![](media/image137.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image141.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image145.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image149.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image169.wmf){width="0.9895833333333334in"
height="0.17708333333333334in"}

![](media/image191.wmf){width="1.5in" height="0.17708333333333334in"}

![](media/image192.wmf){width="1.1979166666666667in"
height="0.17708333333333334in"}

![](media/image167.wmf){width="1.3854166666666667in"
height="0.17708333333333334in"}

![](media/image160.wmf){width="1.0625in" height="0.17708333333333334in"}

![](media/image193.wmf){width="2.96875in" height="0.3854166666666667in"}

![](media/image194.wmf){width="2.0in" height="0.2604166666666667in"}

![](media/image195.wmf){width="1.7916666666666667in"
height="0.3854166666666667in"}

![](media/image196.wmf){width="1.0208333333333333in"
height="0.17708333333333334in"}

with extra noise compensation

![](media/image197.wmf){width="1.6770833333333333in" height="0.59375in"}

![](media/image198.wmf){width="1.0833333333333333in"
height="0.17708333333333334in"}

no noise compensation

![](media/image199.wmf){width="1.8125in" height="0.3854166666666667in"}

![](media/image200.wmf){width="1.3541666666666667in"
height="0.3854166666666667in"}

Conclusion : The sensitivity seems to be improved by a factor of two
over the difference over sum calculation.

This calculation does not show a definitive advantage to subtract the
mean squared noise from earlier measurements. However, the mean squared
noise represents a smaller percentage of the signal power than the root
mean squared noise does from the peak signal.

![](media/image133.wmf){width="0.9895833333333334in"
height="0.2604166666666667in"}

Ratio of S/N is:

![](media/image183.wmf){width="1.2395833333333333in"
height="0.3854166666666667in"}

![](media/image201.wmf){width="1.21875in" height="0.3854166666666667in"}

Calculate cross-correlation:

![](media/image202.wmf){width="1.46875in" height="0.3854166666666667in"}

![](media/image185.wmf){width="0.16666666666666666in"
height="0.17708333333333334in"}

![](media/image203.wmf){width="1.7395833333333333in"
height="0.6666666666666666in"}

![](media/image204.wmf){width="1.1875in" height="0.2604166666666667in"}

![](media/image205.wmf){width="0.9166666666666666in"
height="0.3958333333333333in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image206.wmf){width="1.34375in" height="0.3958333333333333in"}

![](media/image207.wmf){width="1.34375in" height="0.3854166666666667in"}

![](media/image208.wmf){width="2.2291666666666665in"
height="0.2604166666666667in"}

![](media/image209.wmf){width="1.4479166666666667in"
height="0.6666666666666666in"}

![](media/image210.wmf){width="1.375in" height="0.2604166666666667in"}

![](media/image211.wmf){width="0.75in" height="0.3854166666666667in"}

![](media/image212.wmf){width="0.7083333333333334in"
height="0.17708333333333334in"}

![](media/image213.wmf){width="1.8333333333333333in"
height="0.6666666666666666in"}

![](media/image214.wmf){width="1.3854166666666667in"
height="0.2916666666666667in"}

![](media/image215.wmf){width="1.3854166666666667in"
height="0.2916666666666667in"}

![](media/image216.wmf){width="1.625in" height="0.6041666666666666in"}

![](media/image217.wmf){width="1.4583333333333333in"
height="0.2604166666666667in"}

![](media/image218.wmf){width="1.5625in" height="0.2604166666666667in"}

![](media/image154.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image219.wmf){width="1.4791666666666667in"
height="0.3958333333333333in"}

![](media/image220.wmf){width="0.9895833333333334in"
height="0.3958333333333333in"}

![](media/image135.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image221.wmf){width="1.6979166666666667in"
height="0.3854166666666667in"}

![](media/image135.wmf){width="1.3229166666666667in"
height="0.2604166666666667in"}

![](media/image222.wmf){width="3.0520833333333335in" height="1.84375in"}

![](media/image223.wmf){width="0.5416666666666666in"
height="0.17708333333333334in"}

![](media/image214.wmf){width="1.3854166666666667in"
height="0.2916666666666667in"}

![](media/image224.wmf){width="3.8229166666666665in" height="0.6875in"}

![](media/image215.wmf){width="1.3854166666666667in"
height="0.2916666666666667in"}

![](media/image225.wmf){width="1.9791666666666667in"
height="0.19791666666666666in"}

![](media/image226.wmf){width="1.6666666666666667in"
height="0.2604166666666667in"}

![](media/image217.wmf){width="1.4583333333333333in"
height="0.2604166666666667in"}

![](media/image227.wmf){width="1.4166666666666667in"
height="0.3854166666666667in"}

![](media/image228.wmf){width="1.3541666666666667in"
height="0.3854166666666667in"}

![](media/image229.wmf){width="0.8125in" height="0.17708333333333334in"}

![](media/image230.wmf){width="0.7604166666666666in"
height="0.2604166666666667in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image231.wmf){width="1.46875in" height="0.3854166666666667in"}

![](media/image185.wmf){width="0.16666666666666666in"
height="0.17708333333333334in"}

**SET UP A LOOP TO CALCULATE SOLUTIONS**

Add noise:

![](media/image124.wmf){width="0.53125in" height="0.2604166666666667in"}

![](media/image125.wmf){width="0.3125in" height="0.17708333333333334in"}

k is the number of times the noise is a factor of a 50 Ohm resistor

![](media/image126.wmf){width="1.1041666666666667in"
height="0.2604166666666667in"}

Generate independent noise sources:

![](media/image223.wmf){width="0.5416666666666666in"
height="0.17708333333333334in"}

![](media/image232.wmf){width="0.4895833333333333in"
height="0.17708333333333334in"}

![](media/image233.wmf){width="3.0in" height="0.6770833333333334in"}

Noise of right pickup

![](media/image234.wmf){width="3.0104166666666665in"
height="0.6770833333333334in"}

Noise on left pickup

![](media/image235.wmf){width="2.9479166666666665in"
height="2.0104166666666665in"}

![](media/image236.wmf){width="2.9479166666666665in"
height="2.0104166666666665in"}

Observe the function en2(t,2) seems to be recomputed each time the
function is called

Define signal period \"P\":

![](media/image43.wmf){width="0.78125in" height="0.2604166666666667in"}

![](media/image237.wmf){width="1.625in" height="0.3854166666666667in"}

15 ma beam with noise, Right pickup

![](media/image238.wmf){width="1.59375in" height="0.3854166666666667in"}

15 ma beam with noise, Left pickup

Plot of the signals

![](media/image239.wmf){width="3.4479166666666665in"
height="2.2604166666666665in"}

Output for log ratio calculation is:

![](media/image240.wmf){width="2.25in" height="1.15625in"}

![](media/image241.wmf){width="1.1354166666666667in"
height="0.17708333333333334in"}

![](media/image242.wmf){width="1.1354166666666667in"
height="0.17708333333333334in"}

note outputs changes with each call\....

change this to an array to solidify the calculation:

![](media/image243.wmf){width="0.8854166666666666in" height="0.25in"}

![](media/image244.wmf){width="2.53125in" height="1.84375in"}

![](media/image245.wmf){width="0.6875in" height="0.25in"}

![](media/image246.wmf){width="0.625in" height="0.25in"}

![](media/image245.wmf){width="0.6875in" height="0.25in"}

![](media/image246.wmf){width="0.625in" height="0.25in"}

now results do not change

display iterations N:

![](media/image247.wmf){width="0.7604166666666666in"
height="0.2604166666666667in"}

calculate mean of the squares:

![](media/image248.wmf){width="1.8854166666666667in"
height="0.6041666666666666in"}

calculate square of the mean:

![](media/image249.wmf){width="1.90625in" height="0.6875in"}

use following formula to compute the variance:

![](media/image250.wmf){width="3.2916666666666665in" height="0.6875in"}

![](media/image251.wmf){width="1.375in" height="0.19791666666666666in"}

![](media/image252.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

remember the ideal log ratio output is:

![](media/image188.wmf){width="1.2395833333333333in"
height="0.17708333333333334in"}

**compute the percent error of the log ratio output:**

![](media/image253.wmf){width="2.7395833333333335in"
height="0.3854166666666667in"}

check some values:

![](media/image254.wmf){width="1.1354166666666667in"
height="0.17708333333333334in"}

![](media/image255.wmf){width="1.0729166666666667in"
height="0.17708333333333334in"}

note outputs seem to be recomputed each time\....

![](media/image256.wmf){width="1.4791666666666667in"
height="0.17708333333333334in"}

![](media/image257.wmf){width="1.4791666666666667in"
height="0.17708333333333334in"}

![](media/image258.wmf){width="0.2604166666666667in"
height="0.17708333333333334in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image259.wmf){width="0.7083333333333334in"
height="0.2604166666666667in"}

**compute an array of output percent error values:**

![](media/image260.wmf){width="1.3541666666666667in"
height="0.4583333333333333in"}

![](media/image261.wmf){width="0.8020833333333334in" height="0.25in"}

![](media/image261.wmf){width="0.8020833333333334in" height="0.25in"}

values not changing now

as before find the variance of this error (which should have a mean of
zero:

![](media/image262.wmf){width="3.5104166666666665in" height="0.6875in"}

![](media/image263.wmf){width="1.9479166666666667in"
height="0.19791666666666666in"}

![](media/image264.wmf){width="1.1666666666666667in"
height="0.17708333333333334in"}

find the signal to noise ratio: (remember, RTms is the right signal mean
squared value)

![](media/image154.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

![](media/image154.wmf){width="1.3020833333333333in"
height="0.2604166666666667in"}

notice it does not change when recomputed.

![](media/image265.wmf){width="1.9583333333333333in"
height="0.7708333333333334in"}

![](media/image266.wmf){width="1.9583333333333333in"
height="0.7708333333333334in"}

![](media/image267.wmf){width="2.1875in" height="0.7708333333333334in"}

![](media/image268.wmf){width="2.1875in" height="0.7708333333333334in"}

However , the noise calculation causes the ratio to change when
recomputed.

This should not be a problem since we are trying to allow this ratio to
be influenced by noise anyway.

We do want to use an average S/N ratio for an eventual plot.

compute a mean squared power for the signals from each PUE. Form them
into an array.

![](media/image269.wmf){width="1.3958333333333333in" height="0.5625in"}

right PUE

left PUE

![](media/image270.wmf){width="1.4791666666666667in" height="0.5625in"}

some evaluations:

![](media/image271.wmf){width="1.2604166666666667in"
height="0.3333333333333333in"}

![](media/image271.wmf){width="1.2604166666666667in"
height="0.3333333333333333in"}

![](media/image272.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image272.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

notice these values do not get recalculated and are defined by the
subscript.

**compute a mean-squared difference over mean squared sum and form an
array with the answers**:

![](media/image273.wmf){width="1.2916666666666667in" height="0.53125in"}

sample calculated answers:

![](media/image274.wmf){width="0.9791666666666666in" height="0.25in"}

![](media/image274.wmf){width="0.9791666666666666in" height="0.25in"}

![](media/image275.wmf){width="0.9791666666666666in" height="0.25in"}

![](media/image276.wmf){width="1.1666666666666667in" height="0.25in"}

This is the equivalent calculation:

![](media/image277.wmf){width="3.2916666666666665in" height="1.15625in"}

Define an ideal output ratio with no noise corruption:

![](media/image278.wmf){width="1.2708333333333333in"
height="0.3854166666666667in"}

note the ideal mean squared output answer is, and is not influenced by
noise:

![](media/image279.wmf){width="0.78125in"
height="0.17708333333333334in"}

![](media/image279.wmf){width="0.78125in"
height="0.17708333333333334in"}

Define an array of percent error of the answers:

![](media/image280.wmf){width="2.6666666666666665in"
height="0.3854166666666667in"}

![](media/image281.wmf){width="2.6354166666666665in" height="1.84375in"}

or :

![](media/image282.wmf){width="1.6354166666666667in"
height="0.4583333333333333in"}

![](media/image283.wmf){width="0.7916666666666666in" height="0.25in"}

![](media/image284.wmf){width="0.7916666666666666in" height="0.25in"}

![](media/image285.wmf){width="0.7916666666666666in" height="0.25in"}

Compute the variance of the percent error:

![](media/image286.wmf){width="3.9583333333333335in" height="0.6875in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image287.wmf){width="2.1666666666666665in"
height="0.19791666666666666in"}

![](media/image288.wmf){width="1.1875in" height="0.17708333333333334in"}

**work on the difference over the sum using the peak signals**:

First find the ratio of the peak signal to the rms signal to compare
signal power calculations:

![](media/image289.wmf){width="1.4375in" height="0.4895833333333333in"}

Find the ideal difference over the sum, with no noise included:

![](media/image290.wmf){width="2.8020833333333335in"
height="0.8020833333333334in"}

some sample results:

![](media/image291.wmf){width="1.2604166666666667in"
height="0.17708333333333334in"}

![](media/image291.wmf){width="1.2604166666666667in"
height="0.17708333333333334in"}

![](media/image292.wmf){width="2.0416666666666665in"
height="0.5520833333333334in"}

Output std Diff

With noise:

![](media/image293.wmf){width="4.145833333333333in"
height="0.5520833333333334in"}

![](media/image294.wmf){width="1.8645833333333333in"
height="0.2604166666666667in"}

![](media/image295.wmf){width="1.875in" height="0.2604166666666667in"}

This is no good because it uses different numbers in the numerator and
the denominator

define a noise array:

![](media/image296.wmf){width="1.2708333333333333in"
height="0.3333333333333333in"}

![](media/image297.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image297.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image298.wmf){width="1.2708333333333333in"
height="0.3333333333333333in"}

![](media/image299.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image299.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image300.wmf){width="3.0625in" height="0.6979166666666666in"}

create an array of percent errors for this difference over sum (DOS)
calculation:

![](media/image301.wmf){width="2.0416666666666665in"
height="0.4583333333333333in"}

array of differences over sums percent error

find the variance:

![](media/image302.wmf){width="4.145833333333333in" height="0.6875in"}

![](media/image303.wmf){width="2.2916666666666665in"
height="0.19791666666666666in"}

![](media/image304.wmf){width="1.25in" height="0.17708333333333334in"}

compute the signal to noise ratio two ways. First as mean squared signal
to mean squared noise, then as peak squared signal to mean squared
noise.

![](media/image305.wmf){width="2.4375in" height="0.8125in"}

![](media/image306.wmf){width="1.4270833333333333in"
height="0.17708333333333334in"}

![](media/image185.wmf){width="0.16666666666666666in"
height="0.17708333333333334in"}

![](media/image307.wmf){width="2.9479166666666665in" height="1.84375in"}

number of iterations

![](media/image212.wmf){width="0.7083333333333334in"
height="0.17708333333333334in"}

![](media/image308.wmf){width="1.2083333333333333in"
height="0.6041666666666666in"}

![](media/image297.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image309.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image186.wmf){width="0.375in" height="0.17708333333333334in"}

![](media/image310.wmf){width="1.1770833333333333in"
height="0.2604166666666667in"}

![](media/image311.wmf){width="1.65625in" height="0.6041666666666666in"}

![](media/image312.wmf){width="1.28125in" height="0.2604166666666667in"}

![](media/image313.wmf){width="1.59375in"
height="0.17708333333333334in"}

![](media/image314.wmf){width="1.1458333333333333in"
height="0.17708333333333334in"}

Now using Peaks:

![](media/image309.wmf){width="1.2395833333333333in"
height="0.3333333333333333in"}

![](media/image315.wmf){width="2.21875in" height="1.1041666666666667in"}

![](media/image316.wmf){width="1.3020833333333333in"
height="0.3333333333333333in"}

![](media/image317.wmf){width="1.3020833333333333in"
height="0.3333333333333333in"}

![](media/image318.wmf){width="2.0208333333333335in"
height="0.6041666666666666in"}

![](media/image319.wmf){width="1.6979166666666667in"
height="0.2604166666666667in"}

![](media/image320.wmf){width="1.8645833333333333in"
height="0.17708333333333334in"}

compare answers in dB

![](media/image321.wmf){width="1.40625in" height="0.25in"}

![](media/image306.wmf){width="1.4270833333333333in"
height="0.17708333333333334in"}

![](media/image322.wmf){width="2.3958333333333335in" height="0.25in"}

compute variance of the signal to noise ratio calculation:

![](media/image323.wmf){width="4.53125in" height="0.6875in"}

![](media/image324.wmf){width="2.09375in"
height="0.19791666666666666in"}

![](media/image325.wmf){width="1.4270833333333333in"
height="0.2604166666666667in"}

Plotted data:

![](media/image326.wmf){width="5.5625in" height="5.520833333333333in"}

#### APPENDIX C

**APPROXIMATE ESTIMATION OF SENSITIVITY TO NOISE**

The general concept is to take the mean of the square signal of the
right PUE and divide by the mean of the square of the left PUE, and then
take a logarithm.

If we call the right pickup signal fr(t), and the left pickup signal
fl(t):

Then for small displacements of beam along the axis by a distance r:

F(t) is the signal on the PUE for zero displacement.

![](media/image327.wmf){width="1.125in" height="0.17708333333333334in"}

![](media/image328.wmf){width="1.1041666666666667in"
height="0.17708333333333334in"}

let the noise on the right PUE be enr(t)

and on the left PUE enl(t)

Then the signal out will be :

![](media/image329.wmf){width="2.1770833333333335in"
height="1.3229166666666667in"}

The numerator of the logarithm argument is:

![](media/image330.wmf){width="2.9895833333333335in"
height="0.6458333333333334in"}

![](media/image331.wmf){width="4.0625in" height="0.6458333333333334in"}

The denominator is:

![](media/image332.wmf){width="3.5625in" height="0.6458333333333334in"}

![](media/image333.wmf){width="4.03125in" height="0.6458333333333334in"}

With signal power squared =:

![](media/image334.wmf){width="1.1458333333333333in"
height="0.6458333333333334in"}

Dividing numerator and denominator by S squared, and σr^2^ and σl^2^ the
mean squared noise voltages of the right and left PUEs:

![](media/image17.wmf){width="3.9270833333333335in"
height="1.1354166666666667in"}

assuming the terms in both the numerator and denominator are small
compared to 1 we get:

![](media/image18.wmf){width="7.15625in" height="0.5520833333333334in"}

This is approximately:

![](media/image19.wmf){width="5.09375in" height="0.5520833333333334in"}

taking a natural log to provide an expansion

The log(w)=log(e)\*ln(w)

and ln(1+x)=x-(1/2)x\^2+(1/3)x\^3-\.....

The output is approximately (eliminating higher order terms):

![](media/image20.wmf){width="5.166666666666667in"
height="0.5520833333333334in"}

We observe that this requires 4\*a\*r to be equal to the other terms to
establish resolution requirements. The mean square noise terms for the
right and left PUE channels subtract from each other, and so do the
cross-correlation terms between the noise and the signal. There are
higher order terms that will add to noise but they can be neglected for
a\*r\<\<1.

Thus, mean square noise tends to cancel, as does cross-correlation
terms. However, the cross-correlation terms can be either positive or
negative, and will vary with a random character. Therefore, these terms
could, in fact add. The result, then, assuming a good estimate of σr and
σl are achieved, and that these noise terms are near equal (both due to
50 Ohm resistor noise), yields

![](media/image21.wmf){width="2.5520833333333335in"
height="0.7291666666666666in"}

remember that a is:

![](media/image335.wmf){width="1.3645833333333333in" height="0.59375in"}

and :

![](media/image336.wmf){width="4.5625in" height="0.6979166666666666in"}

APPENDIX D

**AFE card pins allocation --** Linac version -- provided by Bergoz

January 22, 2001

Revision 2, March 21, 2001

**Signal names structure:**

Signals that belong to a particular processing channel start with a
letter (A,B,C or D) referring to the channel.

Example: A.CAL-SW means Channel A, Calibration Switch control signal.

Each channel has 3 switches, every switch is individually controlled by
its own control line.

PICKUP-SW is the switch on the pickup side

RCVR-SW is the switch at the input of the analog processing chain

CAL-SW is the switch which sends the calibration signal to the common
point between the pickup and the processing chain.

The output signal gain for all 4 processing chains is controlled by a
single control line IF.HIGAIN

N.OUT+ and N.OUT- are the differential outputs of channel N (N=A,B,C,D)

**Control signals logic:**

TTL levels, active when low.

**Power supplies:**

The AFE requires 1 supply rail: +12 V for a nominal current of 500mA

**Connecting to AFE recommendations:**

ADCs implementation on DFE board should respect the provisions of AD6644
data sheet. Pls. note 25-ohm series resistors in each input line, close
to the input pins: AIN = pin 11 and AIN (inv.) = pin 12. AFE provides a
differential output signal for each

of the channels. The AFE driver output has impedance \< 200 ohms. It is
dc-isolated from

ground by a transformer. The output transformer center tap is connected
to AFE ground plane via a 100-nF capacitor. Signal connections to each
ADC should be routed as parallel lines of minimum line width (≤ 0.010" )
to minimize capacitive loading. They should be as short as possible and
be of same electrical length (signal delay) to ensure a common phase
reference. These analog signal lines are very sensitive to

noise pick up and should stay well away from all digital lines.
Considering this differential analog interface, we believe that a common
ground plane for AFE and DFE board is preferable and provide therefore
many ground pins over the whole width of the

interface connector. We recommend that all AFE ground pins be connected
to DFE ground plane. The PCI components side area under AFE should
preferably be a solid ground plane.

**Pin numbers**, per AFE pin numbering (Document: Connector location,
type, style and pins numbering dated Jan. 24, 2001 and John Power's
email dated 3/21/2001)

1 GND 2 +12V 1A

3 +5V analog \<0.1A 4 -12V 1A

5 -5V analog \<0.1A 6 A.PICKUP-SW

7 +5V digital 1A 8 A.RCVR-SW

9 +3,3V digital 1A 10 A.CAL-SW

11 GND 12 B.PICKUP-SW

13 GND 14 B.RCVR-SW

15 GND 16 B.CAL-SW

17 GND 18 C.PICKUP-SW

19 GND 20 C.RCVR-SW

21 GND 22 C.CAL-SW

23 GND 24 D.PICKUP-SW

25 GND 26 D.RCVR-SW

27 GND 28 D.CAL-SW

29 GND 30 IF.HIGAIN

31 GND 32 GND

33 GND 34 A.OUT-

35 GND 36 A.OUT+

37 GND 38 GND

39 GND 40 B.OUT-

41 GND 42 B.OUT+

43 GND 44 GND

45 GND 46 C.OUT-

47 GND 48 C.OUT+

49 GND 50 GND

51 GND 52 D.OUT-

53 GND 54 D.OUT+

55 GND 56 GND

57 GND 58 SPARE

59 GND 60 SPARE

Note: Even-numbered pins 6 to 30 may still be swapped around.

[^1]: J. Power et al, "Beam Position Monitors for the SNS Linac",
    PAC2001, NY.

[^2]: M. Kesselman, et-al, "SNS Project Wide Beam Current Monitors", BIW
    2000, Cambridge MA May 8-11, 2000

[^3]: M. Kesselman et-al, "SNS Project Wide Beam current Monitors", EPAC
    2000, Vienna Austria, June 26-30, 2000

[^4]: M. Kesselman, "Spallation Neutron Source Beam Current Monitor
    Electronics", PAC 2001, Chicago IL, June 18-22, 2001

[^5]: M. Kesselman, "Technote 104- Considerations for the Design of the
    Beam Current Monitor System for the Spallation Neutron Source
    (SNS)", SNS-BNL Technote, March 5, 2001,
    http://server.ags.bnl.gov/bnlags/bnlsns/sns.html

[^6]: M. Kesselman, et-al, Spallation Neutron Source Beam Current
    Monitor Electronics, to be published BIW 2002, New York NY, May 6-9,
    2002