# Recycler Upgrade Pre-Amplifier- Transition Module Design 

R. Webber, P. Prieto

Pre-Amp:

The pre-amp in the Recycler is connected to the BPM plate where it
converts the derivative of the beam current into a voltage. It uses a
shunting resistor network followed by a gain stage. A 25 MHz low-pass
filter follows the gain stage limiting the signal bandwidth in the
pre-amp. A differential amplifier drives the measured signal between the
tunnel and a transition module in the service. Provisions for BPM ion
clearing electrode connections and calibration circuitry are also
contained in the pre-amp.

Transition Module:

The transition module has a differential receiver stage and an impedance
matching network connected to the pre-amp in the tunnel. A gain stage
compensates for cable attenuation so the full dynamic range of the A/D
converter is used and anti-aliasing filter with a corner frequency of 8
MHz provides added filtering. In addition there is an adjustment to
match the signals from both BPM plates due to differences introduced by
the anti-aliasing filters.

Specifications:

The digital BPM has to resolve positions for the 4 bunch mode case,
separated at 396 ns and 12 bunch mode case separated at 132 ns with a
dynamic range of 2.0 to 50e10 injected particles. The

### 89.818 KHz de-bunched beam has intensities varying between 1e10 to 400e10 particles. The expected transfers into the recycler are 2.0e11 injected particles for protons and 5e10 injected particles for pbars. 

In the Recycler a horizontal BPM has a sensitivity factor of 0.273 dB/mm
and a vertical BPM has a sensitivity factor of 0.512 dB/mm. The position
in mm. is inversely proportional to the BPM sensitivity factor (dB/mm)
times the digital receiver sensitivity factor (V/dB), all multiplied the
measured signal with units of Volts.

$$p = \frac{1}{s_{x}B_{s}}F$$

The beam current is given by the expression:

$$i(t) = \frac{\text{eN}}{\sqrt{2\pi}\sigma}\begin{matrix}
e\frac{- t^{2}}{2\sigma^{2}} & 
\end{matrix}$$

The BPM plate current is the derivative the beam current:

> $$
>
> $$
> $\frac{\text{di}(t)}{\text{dt}} = - \frac{\text{eNt}}{\sqrt{2\pi}\sigma^{3}}e\frac{- t^{2}}{2\sigma^{2}}$
>
> $$$$

The expected $\sigma_{t_{1}}$(sigma t) for the different bunches are:

Table 1.

+-----------------------+-----------------------+-----------------------+
| Frequency             | $\sigma_{t_{1}}$      | > $\sigma_{t_{2}}$    |
|                       | (sec)                 | > (sec)               |
+-----------------------+-----------------------+-----------------------+
| 89e3                  | 1824e-9               | 11172e-9              |
+-----------------------+-----------------------+-----------------------+
| 2.5e6                 | 25e-9                 | 50e-9                 |
+-----------------------+-----------------------+-----------------------+
|                       |                       |                       |
+-----------------------+-----------------------+-----------------------+

Table 2 below lists the peak voltages at the output of the BPM where
there is a shunt circuit consisting of an 800 resistor network. The
values were obtained from SPICE simulations.

Table 2.

  -------------------- -------------- -------------- ---------------- ----------
  #Bunches             4              4              Unbunched        

  $$\sigma_{t_{1}}$$   25ns           50ns                            

  Tot. Inj. Part.      V peak         V peak         Tot. Inj Part.   V Peak.

  50e10                0.850          0.320          400e10 (1962ns)  0.40

  30e10                0.470          0.200          20e10 (2094ns)   0.015

  20e10                0.320          0.130          20e10 (8830ns)   0.003

  2e10                 0.032          0.012                           
  -------------------- -------------- -------------- ---------------- ----------

Factors Contributing to the Selection of the Pre-Amp Gain:

The pre-amp voltage gain is defined as the ratio of the voltages at the
shunt resistor and the output of the differential driver (V107/V103).

The output stage of the pre-amp (AD8131) can drive 7.75 Volts
peak-to-peak. The overall gain of the pre-amp is selected to produce the
maximum voltage possible at the output reducing signal loss due to cable
attenuation. The largest voltage in table 2 corresponds to the case
where the sigma_t is 25ns. for a total of injected particles of 50e10.

The pre-amp gain should be based on the 30e10 particle, 25ns case which
from table 2 shows the voltage at the shunt resistor to be 0.48
volts-peak. This value should be multiplied by 2 to account for the case
when the beam is nearest one of the plates, which is \~1.0 volts-peak.
Using this voltage the pre-amp gain becomes 3.8 V/V.

Table 3.

  -------------------- -------------- ---------- ----------------- -------------
  #Bunches             4              4          Unbunched Beam    

  $$\sigma_{t_{1}}$$   25ns           50ns                         

  Tot. Inj Part.       V peak         V peak     Tot. Inj Part.    V peak

  50e10                1.62           0.61       400e10            0.76

  30e10                0.90           0.38       20e10             0.029

  20e10                0.61           0.24       20e10             0.006

  2e10                 0.04           0.025                        
  -------------------- -------------- ---------- ----------------- -------------

The values in table 3 are the voltages listed in table 2 times G=3.8 and
divided by 2 due to the series cable matching output resistors. These
are the levels in volts-peak that would be obtained at the input to the
cable connected to the pre-amp.

Input Voltages at the transition Module due to Cable Losses:

The pre-amp drives a shielded twisted pair cable whose length varies
from 150ft. to 1300ft., with an attenuation factor approximating square
root of the frequency.

The cable attenuation factor is listed below in table 4 for the 4-bunch
case with 2 different different sigma_t's. Also the peak values for the
unbunched case are listed.

Table 4.

  ----------------- ----------------- ----------------- ------------------
  Cable Length      Bunch length      Bunch length      Bunch length

  feet              25ns              50ns              Unbunched

  100               0.804             0.859             0.891

  200               0.663             0.743             0.797

  300               0.553             0.639             0.712

  400               0.456             0.553             0.636

  500               0.373             0.48              0.57

  600               0.31              0.414             0.511

  700               0.261             0.362             0.459

  800               0.218             0.315             0.412

  900               0.18              0.274             0.37

  1000              0.154             0.24              0.334

  1100              0.13              0.209             0.301

  1200              0.109             0.184             0.272

  1300              0.094             0.161             0.245
  ----------------- ----------------- ----------------- ------------------

The resulting peak voltage at the input of the transition module due to
the cable length attenuations factorsand bunch sigma_t are listed in
table 5.

Table 5.

  -------- ------------- ----------- ----------- ------------- ----------- -----------
  Cable    Attn for      Total Inj   Total Inj   Attn for      Total Inj   Total Inj
  Length   2.5MHz        Particles   Particles   2.5MHz        Particles   Particles

  feet     Sigmat=25ns   3.00E+11    2.00E+10    Sigmat=50ns   3.00E+11    2.00E+10

  100      0.804         0.7236      0.03216     0.859         0.32642     0.021475

  200      0.663         0.5967      0.02652     0.743         0.28234     0.018575

  300      0.553         0.4977      0.02212     0.639         0.24282     0.015975

  400      0.456         0.4104      0.01824     0.553         0.21014     0.013825

  500      0.373         0.3357      0.01492     0.48          0.1824      0.012

  600      0.31          0.279       0.0124      0.414         0.15732     0.01035

  700      0.261         0.2349      0.01044     0.362         0.13756     0.00905

  800      0.218         0.1962      0.00872     0.315         0.1197      0.007875

  900      0.18          0.162       0.0072      0.274         0.10412     0.00685

  1000     0.154         0.1386      0.00616     0.24          0.0912      0.006

  1100     0.13          0.117       0.0052      0.209         0.07942     0.005225

  1200     0.109         0.0981      0.00436     0.184         0.06992     0.0046

  1300     0.094         0.0846      0.00376     0.161         0.06118     0.004025
  -------- ------------- ----------- ----------- ------------- ----------- -----------

Unbunched Beam voltage attenuation:

Table 6.

  ----------- ------------------------- ------------- ------------- ---------------
  Cable       Cable Attn. for           Total Inj     Total Inj     Total Inj
  Length                                Particles     Particles     Particles

  feet        [U]{.underline}nbunched   4.00E+12      2.00E+11      2.00E+11

  100         0.891                     0.67716       0.025839      0.005346

  200         0.797                     0.60572       0.023113      0.004782

  300         0.712                     0.54112       0.020648      0.004272

  400         0.636                     0.48336       0.018444      0.003816

  500         0.57                      0.4332        0.01653       0.00342

  600         0.511                     0.38836       0.014819      0.003066

  700         0.459                     0.34884       0.013311      0.002754

  800         0.412                     0.31312       0.011948      0.002472

  900         0.37                      0.2812        0.01073       0.00222

  1000        0.334                     0.25384       0.009686      0.002004

  1100        0.301                     0.22876       0.008729      0.001806

  1200        0.272                     0.20672       0.007888      0.001632

  1300        0.245                     0.1862        0.007105      0.00147
  ----------- ------------------------- ------------- ------------- ---------------

Transition Module:

The transition module input can be configured using a differential
receiver op-amp (AD8130) or using a passive element such as a
transformer. Both modes convert the differential signal to a single
ended one. The signal from the tunnel travels down 2 coupled
transmission lines (twisted pair) with equal magnitude but 180 degrees
out of phase. The termination placed at the end of the cable is two
resistors matching the cable common mode impedance (50 ) connected to
each terminal referenced to the center connection that is a virtual
ground.

The AD8130 has the limitation in that the maximum differential input
voltage is 2.5 Volts (1.25 Volts-peak). Any larger differential voltage
will clip the signal giving an erroneous position.

The main function of the transition module is compensate for cable
attenuation and normalize the input voltage to use the full dynamic
range of the A/D converter.

The gain is determined by the maximum signal amplitude at the input of
the transition module, the --6 dB attenuation at the anti-aliasing
filter and the --6dB lost in the input impedance matching at the input
of the A/D converter.

Table 7 below has a list of cable lengths and the input voltages at the
transition module after they have been attenuated. These voltages are
the largest peak voltages expected at the pre-amp output.

Table 7.

  --------------------- --------------------- ----------------------------
  Cable Length          Bunch length          

                        25ns                  Module Gain

  feet                                        

  100                   1.43916               1.52867

  200                   1.18677               1.853771

  300                   0.98987               2.222514

  400                   0.81624               2.695286

  500                   0.66767               3.295041

  600                   0.5549                3.964678

  700                   0.46719               4.709005

  800                   0.39022               5.637845

  900                   0.3222                6.828057

  1000                  0.27566               7.980846

  1100                  0.2327                9.454233

  1200                  0.19511               11.27569

  1300                  0.16826               13.075
  --------------------- --------------------- ----------------------------

![](media/image1.wmf)

This plots the range of gains vs. cable lengths the transition module
requires to normalize the output voltages to 1.1 Volts peak-to-peak.

The plots below show in graphical format the output voltages obtained
after applying the computed gains for the 30e10 total injected beam with
25ns and 50 ns sigma_t case and the 2e10 25ns and 50ns case. The
unbunched beam case data is plotted in two graphs, one for the 400e10
case where the signal intensities would be large. The second unbunched
beam case deals with the 20e10 case with two different head-tail time
separations 2094ns and 8830ns.

![](media/image2.wmf)

![](media/image3.wmf)

![](media/image4.wmf)

![](media/image5.wmf)

Transition Module Gain Selection:

The range of gains in the transition module can be lumped as follows:

-   150 to 300 ft use gain of 1.5

-   300 to 500 ft use gain of 2.2

-   501 to 800 ft use gain of 3.3

-   801 to 1100 ft use gain of 5.5

-   1101 to 1300 ft use gain of 9.5

> The plots below show the voltages obtained for the different injected
> intensities after the selected gain ranges have been applied.
>
> ![](media/image6.wmf)
>
> ![](media/image7.wmf)
>
> ![](media/image8.wmf)
>
> ![](media/image9.wmf)

Transition Chassis Requirements:

Table 8.

  --------- ---------- ------------- ------- ------- ------------ ----------------------
  Service   #Crates    #DDC          #DDC    #DDC    #Modules in  #Channels/Transition
  Bldg      per        Boards/Bldg   chan    Chan    Transition   Module
            service                  for     for     Crate        
            building                 Crate 1 Crate 2              

  MI-10     1          9             72              14           8

  MI-20     2          11            72      16      14           8

  MI-30     2          12            80      16      14           8

  MI-40     2          10            72      8       14           8

  MI-50     1          9             72              14           8

  MI-60     1          10            80              14           8
  --------- ---------- ------------- ------- ------- ------------ ----------------------

Table 8 columns 4 and 5 show the number of digital down-converter
channels required per service building. The maximum number of channels
the transition crate must handle is 96 based on the totals for MI-30
service building.

Connections:

The signals from each pre-amp come from the tunnel via a multi-conductor
twisted pair cable that has 3 pairs of #24 gauge wire for plate signals
and 1 pair of \# 18 gauge wire for pre-amp power and ground. In the
present BPM system the cable has a DB-9 connector that terminates on the
top plate in the service building.

The proposed design is to leave the DB-9 connector on this cable, mate
with cable extenders terminated with DB-9 at each end distributing the
signals within the house to the new location of the transition crate.

Module Functionality in the Transition Crate:

  ------------------------- --------------------- -----------------------
  Transition Crate                                

  Module function           \# Modules            #Channels/Module

  Signal Conditioning       13                    8

  Calibration Driver        1                     2 analog 3 digital

  DDC Clock Distribution    1                     12
  ------------------------- --------------------- -----------------------

Conditioning Channels:

The transition crate will have 10-12 signal conditioning modules and
each module will handle 8 plate signals. There are 2 plate signals per
BPM, therefore each module will be capable of handling 4 BPM's. Each
transition module channel output will then connect to an input of a
GC-814 board which has 8 channels.

Transition Module signal Conditioning Channel Topology:

Each transition module channel consists of a differential receiver
(AD8130) and a cable termination network at its input. It is then
followed by an adjustable stage that allows matching pairs of channels.
This matching is required due to small matching differences introduced
by the anti-aliasing filter between channels.

A non-inverting gain stage follows, it has the necessary gain to
compensate for cable attenuation between the pre-amp and the transition
channel and make up for the insertion loss (-6 dB) of the anti-aliasing
filter. This filter has a corner frequency of 8 MHz. Two buffers drive
the output signal, one to the front panel for monitoring and one to the
input of the GC-814 digital down-converter board. The output signals use
a BNC for the monitoring signal and an SMA female connector for the DDC
signal.

Board Layout Scheduling:

Layout Schedule:

  ----------------------------------- -----------------------------------
  Layer1                              Signal (each Channel trenched)

  Layer2                              Analog and Digital Ground (trench)

  Layer 3                             Analog and Digital Power (trench)

  Layer 4                             Analog and Pre-Amp Ground (trench)

  Layer 5                             Signal (each Channel Trenched)
  ----------------------------------- -----------------------------------

Board Power Requirements:

  ------------- ------------- ------------- ------------- ----------------
  Part \#       \# Parts      +5 VDC        -5 VDC        Total I
                Used/Chan                                 (amps)/Channel

  AD8130        1             11.6 mA       20 mA         32 mA

  LMH6655       4             7 mA          7 mA          56 mA

                                                          

                                                          
  ------------- ------------- ------------- ------------- ----------------

  ----------------------------------- -----------------------------------
  Total Board Current (8 Channels)    0.71 Amps

  Total Chassis Current (12 Modules)  8.5 Amps
  ----------------------------------- -----------------------------------

Chassis Power Distribution :

The transition module requires power for the different modules as well
as power for the pre-amps in the tunnel.

A power supply chassis all ready exists to provide power for the
pre-amps. It is proposed that four 4-pin Burndy's be placed on the rear
of the chassis frame where the connectors will distribute pre-amp power
through the back plane to each plug-in module. From there the transition
module will provide power through the DB-9 connectors on each board to
the pre-amps in the tunnel.

The power required by the modules themselves originates from a separate
power supply. This power supply could reside in the pre-amp chassis, but
this has to be analyzed.

Calibration Module Topology:

The calibration module distributes a 2.5 MHz waveform that can be a CW
signal or 4 bunch signal to each of the pre-amps in the tunnel for that
service building. The module receives the 4 bunch spaced at 398ns.
signal generated externally using a buffer through a front panel BNC,
distributes it to 2 channels driving the local houses' sectors through 2
rear-panel BNC's.

The board is capable of generating a 2.5 MHz CW signal that can be
activated using a jumper allowing it a stand-alone operation. Provision
for front-panel monitoring of the calibration signal exist through a
front panel BNC.

There are three calibration switches in the pre-amp that need to be
controlled. Presently there is an RG-58 cable daisy-chained between the
pre-amps that carries one of these control signals. It is intended to
use the spare twisted-pair in the cable between the pre-amp and the
tunnel to carry the control signal for the remaining two switches.

All of the switch control signals come into the module through
opto-couplers and then cable drivers (75451) send the signal to the
pre-amps or they are distributed through the back plane of the chassis
to the individual channels in each transition module for individual
distribution.

DDC Clock Distribution Module:

This module has 3 inputs from a clock-generating PMC module. The board
has 3 4-WAY-0 splitters mounted in it so the clock signal can be
distributed to each digital-down converter module in the VME64X crates.

### Pre-amp and Transition Module Noise Analysis : 

The following table lists the noise values computed for the resistor
elements around the two amplifiers used in the pre-amp.

Table 11.

  ----------------------- ------------------------- ------------------------
                          AD8001                    AD8131

  enw                     2nV/$\sqrt{\text{Hz}}$    5nV/$\sqrt{\text{Hz}}$

  inw                     2pA/$\sqrt{\text{Hz}}$.   2pA/$\sqrt{\text{Hz}}$
                          18pA/$\sqrt{\text{Hz}}$   

  Fh                      76 MHz                    40 MHz

  Fl                      1mHz                      1mHz

  Rs1                     14.71 Ω                   1.994 KΩ

  Rs2                     897 Ω                     1.994 KΩ

  et1                     13.67 μVrms               36.15 μVrms

  et2                     28.38 μVrms               36.15 μVrms

  et3                     4.2 μVrms                 

  In1                     17\. 44 nA rms            12.66 nArms

  In2                     156.90 nA rms             

  En                      17.44 μVrms               31.82 μVrms

  Esubtotal               38.13 μVrms               70.0 μVrms

  Etotal                  79.71 μVrms               
  ----------------------- ------------------------- ------------------------

Table 11 has the calculated noise for one pre-amp channel consisting of
two op-amps and their associated elements. The input of the GC-814 A/D
is a 14-bit converter that can resolve 61.04 μV. The total noise of a
single channel without any further electronics or resistors sets the
noise floor of the digital down-converter.

The output stage of the pre-amp (AD8131) could be optimized by making
the gain equal to one, and using lower values for the resistors.

Transition Module Noise analysis :

Table 12.

  ----------------------- ----------------------- -----------------------
                          AD8130                  LMH6655

  enw                     12.5 nV/                4.5 nV
                          $\sqrt{\text{Hz}}$      /$\sqrt{\text{Hz}}$

  inw                     1 pA /                  1.7pA
                          $\sqrt{\text{Hz}}$, 1.4 /$\sqrt{\text{Hz}}$
                          pA/ $\sqrt{\text{Hz}}$  

  Fh                      76 MHz                  8 MHz

  Fl                      1 mHz                   3e3

  Rs1                     200 Ω                   50 Ω

  Rs2                     500 Ω                   182 Ω

  et1                     15.78 μV rms            2.56 μV rms

  et2                     24.45 μV rms            4.89 μV rms

  En                      109 μV rms              12.85 μV rms

  In                      8.74 nA rms , 12.24 nA  4.85 nA rms
                          rms                     

  Ent1                    109.2 μV rms            2.71 μV rms

  Ent2                                            2.8 μV rms

  Ent                     114.7 μV rms            

                                                  
  ----------------------- ----------------------- -----------------------

Table 12 shows the calculated noise for the transition module. The noise
level of this module is \~2 times greater than the noise from the
pre-amp. It is equivalent to approximately two counts in the A/D
converter.

Pre-Amp and Transition Module Frequency Response Design:

The frequency response of the pre-amp is set by the capacitance of the
plate and cable connecting the pre-amp to the BPM together with the 800
Ω shunt resistor. This network produces a corner frequency at 10.1 MHz..
The coupling capacitor in the circuit (0.01 μF) together with the 800 Ω
resistor produce a corner frequency at 19.89 KHz .

Pre- Amp Frequency Measurement:

Plot 1.

![](media/image10.wmf)

Plot 1 shows the frequency response measured the input network and the
LPF output.

Plot 2.

![](media/image11.wmf)

Plot 2 is the frequency response measured between the input network and
the differential driver stage. The setup using a HP8751A network
analyzer and an 85047A S-parameter test set swept between 300 KHz and
100 MHz.

Plot 3.

![](media/image12.wmf)

Plot 3 is the frequency response of the pre-amp with the BPM connected
at both of its inputs. The system response up to 1 MHz is due to the BPM
response rising at 20 dB/decade. At 1.5 MHz the impedance corner
frequency flattens out the BPM response.

The measurements pass 20 MHz are not clearly understood but probably due
to standing waves in the BPM itself.

Plot 4.

![](media/image13.wmf)

Plot 4 has the measured response of the output of the gain stage and the
voltage at the end of a \~604 ft. twisted pair.

Plot 5.\
![](media/image14.wmf)

The cable attenuation is a function of frequency, increasing
approximately with the square root of the frequency. For the BPM system
it would be desirable to have a flat frequency response at least between
2.5MHz and 7.5MHz. By adding a network consisting of a resistor in
series with a capacitor in the feedback loop, the frequency response
seen in plot 8 was obtained.

Plot 6.

![](media/image15.wmf)

Plot 6 shows the frequency response of both channels in the transition
module prototype. The filter was designed with a slope with a minima
near or at 53 MHz to obtain the most attenuation possible at this
frequency.

The differences of the two channels at higher frequencies are attributed
to the prototype board and layout.

Dynamic Range Specifications:

1.  Bunched Beam: 2e10 to 30e10 Total Inj. Particles.

2.  Unbunched Beam : 20e10 to 400e10 particles.

Signal at A/D converter worst case considering cable attenuation and
transition and pre-amp gains

2e10 Bunched Total Injected Beam:

1.  0.02 Vpp for sigmat=50ns

> 0.017 Vpp for sigmat=25ns

20e10 Unbunched Beam:

2.  0.033 Vpp for 20e10 for 2094 ns bunch

> 0.007 Vpp for 20e10 for 8830 ns bunch
>
> Can the GC814 board resolve these signals and report a position with a
> resolution of 0.4 mm.
>
> Actual Bits Available:
>
> The GC-814 board has a 14 bit A/D converter whose signal to noise
> ratio and distortion at 15.5 MHz is 75 dB. This sets the effective
> number of bits to:
>
> $$\frac{\text{SINAD} - 1\text{.}\text{763}}{6\text{.}\text{021}}$$
>
> $$\frac{\text{75}\text{dB} - 1\text{.}\text{763}\text{dB}}{6\text{.}\text{021}\text{dB}} = \text{12}\text{.}1\text{bits}$$
>
> note: It can be stated the SINAD value would be different at 2.5 MHz
> but this needs to be measured.
>
> OverSampling:
>
> By sampling the input frequency at rates greater than nyquist, the
> system gains some resolution:
>
> $$\begin{matrix}
> R & & N
> \end{matrix}\begin{matrix}
>  + & \frac{1}{2} & \text{ln}_{2}M
> \end{matrix}$$
>
> N is the effective number of bits computed above
>
> $$\begin{matrix}
> M & & \frac{F_{s}}{\begin{matrix}
> 2 & F_{\text{in}}
> \end{matrix}}
> \end{matrix}$$
>
> for $F_{\text{in}}$= 2.5 MHz and $F_{s}$= 80 MHz M= 16
>
> $$\begin{matrix}
> R & & \begin{matrix}
> \text{12}\text{.}1 & + & \text{ln}_{2}\text{16}
> \end{matrix}
> \end{matrix}$$
>
> R = 14.
>
> The effective number of bits goes from the nominal 12 back to 14.
>
> The total Noise floor computed for the pre-amp and the transition
> module is $\begin{matrix}
> \text{200} & \mu\text{Vrms}
> \end{matrix}$.
>
> The measured non-linearity of the A/D converter on the GC-814 was 3
> counts for each channel.
>
> The A/D with the effective number of bits resolves:
>
> $\frac{\begin{matrix}
> 1\text{.}1 & \text{Vpp}
> \end{matrix}}{2^{\text{14}}}\begin{matrix}
>  & \text{64}\text{.}\text{17}\mu V
> \end{matrix}$ /count.
>
> It looses 3 counts of dynamic range due to the $\begin{matrix}
> \text{200} & \mu\text{Vrms}
> \end{matrix}$ noise in the signal.
>
> $\frac{\begin{matrix}
> 1\text{.}1 & \text{Vpp}
> \end{matrix}}{\text{16381}}\begin{matrix}
>  & \text{67}\text{.}\text{15}\mu V
> \end{matrix}$/count.
>
> For the unbunched case, the lowest predicted signal is 7 mVpp, the
> system needs 104 counts to resolve this input level.