Theory, Design and Tests on a Prototype Module of a Compact ...

82 5. RADIOFREQUENCY MEASUREMENT
All this means that it is required a RF measurement able to acquire
the relative electric field distribution inside the cavities <strong>and</strong> this is
possible using the bead pulling technique.
Bead pulling is a perturbation technique based on Slater’s theorem
[36, 37]. With the help of a perturbing object (the bead) which travels
along the axis of the multi-cell structure, the electric field distribution
is determined in terms of phase variation introduced by the bead. The
multi-cell structure is driven at a fixed frequency that is the accelerating
mode frequency.
The Slater perturbation theorem can be written as follows:
f 2 = f 2
∆τ
0 1 + α
(µH2 − εE2 ) dτ
V (µH2 + εE2
(5.12)
) dv
where α is a constant related to the particular perturbing object, the
numerator represents the integration over the volume of the perturbing
object <strong>and</strong> the denominator is equal to twice the average energy stored
in the cavity U. If perturbation is small the (5.12) can be simplified to
∆f
f0
= α
4U
∆τ
(µH 2 − εE 2 ) dτ. (5.13)
This important relation states that the change in frequency depends
upon the integral
∆τ (µH2 − εE2 ) dτ. And, if the perturbing object
acts in a zone where only the electric field is present in a significant
way, the previous relation simplifies again.
Then, a procedure to measure the relative level of electric field in a
multi-cell structure can be set up as follows:
(1) A nylon wire is stretched along the longitudinal axis of the
structure <strong>and</strong> some mechanical tools <strong>and</strong> a motor allow the
movement of the wire. Note that the wire itself modifies the
resonant frequency, since it is a dielectric 9 . Of course, a good
alignment is needed between the wire <strong>and</strong> the axis of the cavities
in order to have the same perturbation in all the cavities.
(2) A perturbing object (the bead) is put on the wire. The dimensions
should be little enough to perturb as little as possible
the field distribution, but sufficient to give a good signal
noise ratio. Using a cylindrical shape (a needle for example),
the perturbation acts on the axis <strong>and</strong> only the longitudinal
component of the electric field is perturbed.
(3) The network analyzer drives the structure at the resonant frequency.
The bead perturbs the electric field distribution along
the axis <strong>and</strong> this implies a change in frequency 10 <strong>and</strong> then,
in the phase of the signal detected by the network analyzer.
9 The nylon wire lowers the frequency since it acts in a zone where electric field
is strong <strong>and</strong>, as in a capacitor, it canalizes the field lines.
10 Strictly speaking, the bead grows the frequency.