# Beam position monitor handbook

## Cavity Beam position monitors

As a beam passes through a cylindrical cavity with position

, beam trajectory angle

and
tilt

, the cavity output voltage for the different contributions is given by

Generally we can write the output as (using complex oscillating functions)

So generally we can write the output of the cavity as

where is a complex amplitude which contains the information on the beam position, direction and angle of attack.
Alternatively we can consider the complex amplitude () consisting of in-phase and quadrature-phase components,
, giving

An example of a cavity signal is below

## Simple signal processing

- Calculate RMS signal in background region
- Calculate pedestal, background subtract
- Set to zero waveform upto end of background region

## Signal processing (mixing/filtering)

The analogue IF is digitized at some frequency

so that the IF can be written in sample time

, where

.

The digital signal is mixed with a digital complex local oscillator given by

Time domain gaussian filter with

## Reference frequency phase jitter problem

Consider two cavity signals which are mixed with a digital LO

If there is any timing jitter in the sample clock so then we find

Looking at just the phase of the oscillators

Simplifying we find

Subtracting the two phases gives

This is usally not a problem as the reference frequency is the same as the dipole frequency.
To use a reference at a different frequency then the phase must be corrected by
. Even if the digitizers both both reference and
dipole change together. It is worse for the S-band system as the signal and mixed either side
of the LO, so the term becomes something like

## Calibration

We can write I and Q as

Rotating I and Q so that position sensitive part

Fit line in I-Q plane for gradient so

The error on the rotation angle is

where is the gradient in the I-Q plane and its error.

The error on the rotated and values, and is
then given by

The BPM can be calibrated via

### Calibration Tone

## Noise sources

### Thermal noise

Thermal noise from cavity at temperature.

Thermal power (), temperature () and bandwidth ()

### Amplitude and phase noise

### Extrapolation (error)

The signal at sample time

,

can be extrapolated to

with the following equation :

The statistical error on the sample at the extrapolated time can be simply calculated via

The third term can be neglected as the error in the sample time is small

### Resolution

### Electronics design

So approximately

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