# Difference: BeadPull (1 vs. 6)

#### Revision 619 Oct 2012 - convert_5facls_2esh

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#### Revision 517 Aug 2009 - StephenMolloy

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# Bead Pull Measurements of Microwave Cavities

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This frequency shift can be shown to be: %BEGINLATEX{label="freqshift" color="Black"}% \begin{displaymath*}
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\left(\frac{\Delta f}{f}\right) = \frac{k}{4U}\int\int{\left(\mu H\left(x,y\right)^{2} - \epsilon E\left(x,y\right)^{2}\right)dx~dy}
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\displaystyle\left(\frac{\Delta f}{f}\right) = \frac{k}{4U}\int\int{\left(\mu H\left(x,y\right)^{2} - \epsilon E\left(x,y\right)^{2}\right)dx~dy}
\end{displaymath*} %ENDLATEX%
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In the case of a circular bead of radius, r, and where the perturbation is small enough that k may be taken to be unity, %BEGINLATEX{label="freqshiftforbead" color="Black"}% \begin{displaymath*}
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\left(\frac{\Delta f}{f}\right) = -\left(\frac{\pi r^3}{U}\right)\left[\epsilon_0\left(\frac{\epsilon_r-1}{\epsilon_r+2}\right)E^2 + \mu_0\left(\frac{\mu_r-1}{\mu_r+2}\right)H^2\right]
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\displaystyle\left(\frac{\Delta f}{f}\right) = -\left(\frac{\pi r^3}{U}\right)\left[\epsilon_0\left(\frac{\epsilon_r-1}{\epsilon_r+2}\right)E^2 + \mu_0\left(\frac{\mu_r-1}{\mu_r+2}\right)H^2\right]
\end{displaymath*} %ENDLATEX%
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-- StephenMolloy - 17 Aug 2009

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#### Revision 408 Jul 2009 - StephenMolloy

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# Bead Pull Measurements of Microwave Cavities

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Where the fields are those before the perturbation, and the integral is over the volume of the probe. k is a geometrical factor to take account of the reorganisation of the fields around the perturbing object, and the electric and magnetic constants are those for the material of the probe.
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If a dielectric material (μ = 0) is used, it can be seen that this technique can be used to investigate the electric field only. In addition, since the integral is over the volume of the probe, different shapes (e.g. a bead or a needle) may be used to measure the absolute value and direction of the field vectors.
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In the case of a circular bead of radius, r, and where the perturbation is small enough that k may be taken to be unity,
 (1)

Where E and H are the average values of the fields (since the perturbation is small enough that the fields may be viewed as constant over the small volume of the bead).

If a dielectric material (μ_r = 1) is used, it can be seen that the frequency shift is related to the electric field only, thus allowing the measurements to be totally independent of the magnetic field. In addition, since the integral is over the volume of the probe, different shapes (e.g. a bead or a needle) may be used to measure the absolute value and direction of the field vectors.
Latex rendering error!! dvi file was not created.

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Thus, combinations of different materials (dielectric and magnetic) and shapes may be used to form a complete picture of the field structure within a cavity.
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Thus, combinations of different materials (dielectric and metallic) and shapes may be used to form a complete picture of the field structure within a cavity.

## References

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#### Revision 307 Jul 2009 - StephenMolloy

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# Bead Pull Measurements of Microwave Cavities

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Thus, combinations of different materials (dielectric and magnetic) and shapes may be used to form a complete picture of the field structure within a cavity.

## References

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1. J. Edighoffer, "Beam Breakup Considerations in the Design of Multiple Off-Axis gaps in an Induction Accelerator Cell for SLIA", Particle Accelerator Conference, 1991
2. J. Byrd, "Microwave Measurements Laboratory -- RF Cavity Bead Pull Measurements", USPAS/CCAST, Beijing, China, 1998
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#### Revision 207 Jul 2009 - StephenMolloy

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 META TOPICPARENT name="Public.StephenMolloy"

# Bead Pull Measurements of Microwave Cavities

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## Introduction

It is well known that rf cavities can support various modes of electromagnetic oscillations (so-called modes), and that, for simple geometries such as rectangular or cylindrical cavities, the shape of each of these modes in each of the appropriate degrees of freedom may be calculated analytically. The excitation of these modes by a beam of charged particles may also be calculated by integrating the electric field amplitude along the path of the beam (with a suitable transit time correction factor to take the oscillatory nature of the modes into account).

For more realistic (i.e. complicated) cavity structures such as might be found in a particle accelerator, it is normally not feasible to attempt an analytical calculation of these modes, and they must be determined using a computer simulation, or directly measured.

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## The Technique

The EM fields inside a resonant cavity may be perturbed using a small object whose permittivity and/or permeability is different to that of the contents of the cavity (normally a vacuum).

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In a standing wave, the energy is continually exchanged between the electric and magnetic fields, which means that the energies contained in each of these fields are identical. When one of these is perturbed by a probe, a frequency shift occurs in order to maintain this equality.

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## References

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This frequency shift can be shown to be:
 (2)

Latex rendering error!! dvi file was not created.

>
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Where the fields are those before the perturbation, and the integral is over the volume of the probe. k is a geometrical factor to take account of the reorganisation of the fields around the perturbing object, and the electric and magnetic constants are those for the material of the probe.

>
>
If a dielectric material (μ = 0) is used, it can be seen that this technique can be used to investigate the electric field only. In addition, since the integral is over the volume of the probe, different shapes (e.g. a bead or a needle) may be used to measure the absolute value and direction of the field vectors.

>
>
Thus, combinations of different materials (dielectric and magnetic) and shapes may be used to form a complete picture of the field structure within a cavity.

>
>

## References

1. J. Edighoffer, "Beam Breakup Considerations in the Design of Multiple Off-Axis gaps in an Induction Accelerator Cell for SLIA", Particle Accelerator Conference, 1991
2. J. Byrd, "Microwave Measurements Laboratory -- RF Cavity Bead Pull Measurements", USPAS/CCAST, Beijing, China, 1998

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#### Revision 107 Jul 2009 - StephenMolloy

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 META TOPICPARENT name="Public.StephenMolloy"

# Bead Pull Measurements of Microwave Cavities

## The technique

### Introduction

It is well known that rf cavities can support various modes of electromagnetic oscillations (so-called modes), and that, for simple geometries such as rectangular or cylindrical cavities, the shape of each of these modes in each of the appropriate degrees of freedom may be calculated analytically. The excitation of these modes by a beam of charged particles may also be calculated by integrating the electric field amplitude along the path of the beam (with a suitable transit time correction factor to take the oscillatory nature of the modes into account).

For more realistic (i.e. complicated) cavity structures such as might be found in a particle accelerator, it is normally not feasible to attempt an analytical calculation of these modes, and they must be determined using a computer simulation, or directly measured.

## References

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