Valeri Makeev

August 14, 2001

**NUMI BPM detectors**

# The goal.

1.  **Continuous monitoring of the beam centroid position.**

2.  **Beam intensity measurement.**

$$ **Specifications**

-   **12 detectors @ 0.2 mm resolution -- beam transport**

-   **4 detectors @ 0.05mm resolution -- beam targeting**

##### $$ The requirements for the BPM's functional accuracy 

1.  **BPM's along the transport line +/- 0.2mm.**

2.  **BPM's along the target line-up +/- 0.05mm.**

**3. Beam intensity measurement +/- 3%.**

$$ **What type BPM will be used?**

**Two different sizes, but the same in style of BPM's will bused. For
the first part transport line will be used standard BPM's (4 inches in
diameter) as in current**

**MI transfer lines. And for the target line-up will use the BPM's 2
.125 inches in diameter. For the last one no experimental data is
available.**

**Both BPM 's have diagonally cut circular cross-section pickup
electrode. That electrode shape was chosen to have a linear response to
beam displacement.**

To define the electronic specifications (which must be reliable and
affordable) a simple analytical calculation for was done output signal,
noise budget and position sensitivities for that type BPM's.

#######  **To be convinced of calculations the calculated values are compared to the measurements of standard BPM.**

########  $$ **Signal analysis and the main BPM's parameters** 

**We have made the approximation that the beam acts as a perfect current
source in generating image current on the inside surface of the
electrodes.**

**The analysis was done for Gaussian bunch shape:**

**I~b~ (*t*) = eN~p~ /**$\sigma$$\sqrt{2\pi}$$$**Exp(- *t*
^2^/2**$\sigma$**^2^)**

**where e is the electron charge in Coulombs, N~p~ is the number of the
protons,**$\sigma$ **is the RMS bunch temporal length (as bunch length
we take** $\pm$**2**$\sigma$**, that correspond to 95% protons in the
bunch).Further analysis will be done in the frequency domain.**

Assuming that the bunch is symmetric in time, centered at time *t* =0

and is in a pulse train with the bunch spacing T, the Gaussian bunch
shape can be expended in a cosine series with angular
frequency$\omega$~0~ = 2$\pi$/T.

I~b~ (t) = $$I~b~$$
$\left\lbrack 1 + 2\sum_{m}^{}{Am\text{Cos}(m\omega 0t)} \right\rbrack$

where A~m~ =Exp(-*m*^2^$\omega$~0~^2^$\sigma$^2^ /2) is the harmonic
amplitude factor and

$$I~b~$$= eN/T is the average (dc) bunch current.

**The image current flowing over the characteristic impedance**

**is equal to the time derivative of the induced charge Q( *t*** **) on
the electrode**

*i*~s~ (*t*) = *d*Q (*t*)/*dt* = *d*{I~b~ (*t*) /$\beta$~b~c $$ Z(*x,
m*$\omega$~0~)}/*dt*

or

*i*~s~ (*t*) = 2$\sum_{m}^{}{}$$$I~b~$$*m*$\omega$~0~
/$\beta$~b~c$$Exp(-*m*^2^$\omega$~0~^2^$\sigma$^2^ /2) Z(*x,
m*$\omega$~0~) $$sin(*m*$\omega$~0~ t)

where q(t)= I~b~ (*t*) /$\beta$~b~c is the beam charge density, Z(*x,
m*$\omega$~0~) is the "transfer function" from beam current to BPM
current and *x* is the offset center beam in horizontal plane. That is
true in some approximation when the inter -electrode coupling
capacitance is ignored .

The peak output BPM current for harmonic *m* of the bunch angular

frequency $\omega$~0~ occurs at *t* = +/- T/4*m* and is

*i* ~peak~ = 2$$I~b~$$ Z(*x, m*$\omega$~0~) *m*$\omega$~0~ /$\beta$~b~c
$$Exp(-*m*^2^$\omega$~0~^2^$\sigma$^2^ /2)

$$ What harmonic number is better to use?

To decrease the influence of the possible bunch length variation
(3$$`<!-- -->`{=html}8)ns on output signal it is preferable to use the
main harmonic 52.8MHz (*m*=1). In that case the signal variation will be
1.7dB. For *m*=2 the signal variation will be 6.6dB.

Remark, if we consider Microwave power sensors as the including many
harmonics, the peak output current which occurs at the time t = +/-
$\sigma$, in time domain representation, varies inversely as square of
bunch length RMS

i~peak~ = Z(*x*) $$ eN~p~
/$\beta$~b~c$\sigma$^2^$\sqrt{2\pi}$$$Exp(-1/2).

As a result the output signal would have a large variation (up to 17dB)
at the same condition as before.

$$The transfer function Z(*x,*$\omega$~0~).

To determine the diametrical part of the transfer function we used the
expression^/\ 1/^

$\frac{1}{\pi}$$\int_{0}^{\pi}\frac{L(\phi)(b^{2} - r^{2})}{b^{2} + r^{2} - 2\text{br}\text{cos}(\phi - \theta)}d\phi$

where *r* is the displacement from the BPM center, *b* is the inside

electrode radius and $\phi$ is the usual polar coordinate. $\theta$ is
the angle from the median electrode plane to the plane defined by the
beam and the BPM center.

The transfer function was calculated analytically for the real pickup
electrode design. Also taken into account was the electrodes axes
misalignment and inter-electrode coupling capacitance. The estimation
was done for the zero beam displacement on *y* =r$$sin$\theta$coordinate
($\theta$=0) from the center of the BPM.

For the diagonal type electrode L($\phi$)=L~0~ \[1+cos($\phi$)\]/2.

The transfer functions for hypothetical Right and Left electrodes are:

Z~R~(*x,*$\omega$~0~) = L~0~\[ 1+K~g~$$K~c~$$(*x +*$\Delta$*/*2) */b* \]
*/* 2K~g~

Z~L~(*x,*$\omega$~0~) = L~0~\[ 1 - K~g~$$K~c~$$(*x -*$\Delta$*/*2) */b*
\] */* 2K~g~

Where K~g~ =(1 + 2 *f* /L~0~ ) ^--1^ and

K~c~ = \[R~0~ (2 + 1/ R~0~$\omega$~0~ C~e~ ) $\omega$~0~ C~e~\]^--1^

(K~t~ = K~g~$$K~c~ )

are the geometrical and the inter-electrodes capacitance factors
accordingly.

The *f* and L~0~ are the lengths of the flat part and the diagonally
inside part

of the BPM electrodes, accordingly. The $\Delta$ is the electrodes axes
misalignment.

C~e~ is the inter-electrodes capacitance and R~0~ is the load impedance.

$$ The output signal voltage

(All calculations were done for beam intensity 1e10ppb)

> Table 1. Some BPM and Beam parameters

  -------------------------- --------------- --------------- -----------------
  Parameter                  Symbol          Standard BPM    Targeting BPM

  Inside BPM radius          *b*             49.149mm        25.349mm

  Diagonal electrode part    L~0~            98.298mm        103.947mm

  Distance between electrode *d*             4.49mm          5 .57mm

  Diagonally angle           $$\alpha$$      45^0^           26^0^

  Flat electrode part        *f*             14.351mm        6.968mm

  Offset electrodes          $$\Delta$$      0.4mm           0.2mm

  Required resolution        $\delta$*~x~*   0.2mm           0.05mm

  Bunch frequency            *f~0~*          52.8MHz         52.8MHz

  RMS bunch length           $$\sigma$$      2ns             2ns

  Measurable bandwidth       B               1MHz            1MHz

  Beam velocity              $$\beta$$       1               1

  Inter-electrodes           C~e~            2.97Pf          2.09pF
  capacitance                                                

  Geometrical factor         K~g~            0.774           0.882

  Inter-electrodes           K~c~            0.907           0.935
  capacitance factor                                         

  Total correction factor    K~t~            0.702           0.825

  Load impedance             R~0~            50$\Omega$      50$\Omega$
  -------------------------- --------------- --------------- -----------------

> If we use the previous formulas and parameters from Table 1 the peak
> signal in voltage is given by
>
> U~R(L)~ =9.362$$`<!-- -->`{=html}10 ^-3^Exp(-0.055$\sigma$^2^) $$
> L~0~\[1 +(-)$\frac{x + ( - )\frac{\Delta}{2}}{b}$ K~t~\]/2 K~g~
>
> The output signals amplitude U~R(L)~ for Standard and Targeting BPM's
> are presented in Table 2.

$$ The displacement sensitivity and Beam position computing

There are two formulae for displacement sensitivity:

(U~R~ - U~L~ )/( U~R~ + U~L~ ) = K~t~ (1+ K~t~ $$ $\Delta$/2*b* )^-1^
*x*/*b*

and

20 Log~10~ ( U~R~ / U~L~ ) = *x* S*~x~* .

The displacement sensitivity (S*~x~*) is measured in dB/mm.

These parameters for two size BPM's are presented in Table 2.

The wire measurements done by Jim Crisp and the calculated values S*~x~*
for standard BPM's are S*~x~* = 0.243 (0.277) dB/mm and S*~x~* = 0.249,
respectively.

$$ The position resolution

The position resolution can be derived as

$\delta$(*x*) = ( *b* /$\sqrt{2}$K~t~ ) $$$\delta$(U)/U

Where $\delta$(U)/U is the relative accuracy for amplitude voltage
measurement.

Assuming a required resolution $\delta$*~x~* and the parameters in Table
1 the value $\delta$(U)/U was specified. The results for both size BPM's
are presented in Table 2.

$$ The thermal noise and loss budget

The thermal peak voltage noise was calculated as

U ~noise~ =$\sqrt{2\text{ktBR}_{0}}$ = 0.64$\mu$V ( t=300 ^0^ K )

Translating the thermal voltage noise into a position resolution
$\delta$(*x*)

$\delta$(*x*)~noise~ = ( *b* /$\sqrt{2}$K~t~ ) $$ U ~noise~/ U~R(L)min~

For standard BPM's $\delta$(*x*)~noise~ = 0.16 $\mu$*m* and for
targeting BPM's

$\delta$(*x*)~noise~ = 0.06 $\mu$*m* for lowest beam intensity(6e9 ppb).

Assuming a required resolution $\delta$*~x~* (Table 1), we have a noise
and loss budget

62dB and 58dB accordingly.

$$Dynamic range in beam position and dynamic range

available for beam intensity.

*x*$$ \[ 2^N\ -1^$\sqrt{2}$$\delta$*~x~* (1+ K~t~$$$\Delta$/2*b*)
-$\varepsilon$ *b* / K~t~ - $\Delta$/2 \] /
($\varepsilon$+2^N\ -1^$\sqrt{2}$$\delta$ *~x~* K~t~/*b*)

where N is bit digitization number, $\varepsilon$ is the beam intensity
ratio(dynamic range).

Other values are specified in Table 1.

For $\varepsilon$ = 3.2 (Beam intensity range 9.5e10ppb to 3e10ppb) and
N=12, the dynamic range in beam position is +/- 6.0mm.

Table2.

+-------------------------+-------------------+-----------------------+
| BPM type                | Standard BPM      | Targeting BPM         |
+-------------------------+-------------------+-----------------------+
| Output signal amplitude | 0.47 (*x* = 0)    | 0.444 (*x* = 0)       |
| U~R(L)~ (Volts)         |                   |                       |
|                         | 61. (*x* =+20mm)  | 0.530 (*x* =+6mm)     |
|                         |                   |                       |
|                         | 0.34 (*x* =       | 0.358 (*x* = -6mm)    |
|                         | -20mm)            |                       |
+-------------------------+-------------------+-----------------------+
| Sensitivity conversion  | 0.249             | 0.58                  |
| factor S*~x~* (dB/mm)   |                   |                       |
+-------------------------+-------------------+-----------------------+
| Measurement accuracy    | 0.4               | 0.23                  |
| $\delta$(U)/U (%)       |                   |                       |
+-------------------------+-------------------+-----------------------+

**REFERENCES**

**1. R.Shafer, "Beam Position Monitoring", in *API Conf. Proc*. 212,
26-58(1990)**