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

74 5. RADIOFREQUENCY MEASUREMENT
deduce the value of the quality factor; for brazed cavities, Q can be
measured with accurate methods, or can be extrapolated from numerical
calculations. Therefore, this procedure provides an indirect method
to determine the shunt impedance Rsh.
In the following a perturbation technique is shown [34] which involve
the measurement of the resonant frequency of the cavity as a
function of a certain perturbation objects.
Remembering the definition of quality factor <strong>and</strong> shunt impedance
energy stored ωU
Q = ω
=
energy lost per second W , Rsh
2 Edl
=
2W
where W is the power loss <strong>and</strong> the integral is usually calculated along
the axis where the particles pass through the cavity. If the electric field
E is constant along the path of integration, it is valid that
Rsh = d2 E 2
2W ,
where d is the length of the path, which should be the acceleration gap.
Combining the two definitions
Rsh/Q = d2 E 2
2ωU .
The quantity E 2 /U can be determined experimentally by placing a
perturbing object of volume ∆τ into the cavity where the electric field is
quite constant <strong>and</strong> measuring the resultant change in frequency. From
Slater Perturbation theorem [34, 36]
∆f
f
= −αε∆τ
4
E 2
U ,
where α is a constant which depends on the shape <strong>and</strong> on the material of
the perturbing object. It should be determined from the measurement
on a cavity with a known electric field distribution. Combining the
previous relations one obtains
Rsh/Q = −2d2 ∆f
, (5.1)
αωεf ∆τ
which links Rsh/Q <strong>and</strong> the change in frequency due to the perturbing
object.
It is worth noting that, in principle, it is not necessary to know
the constant α for the perturbing object. In fact, the equation (5.1)
depends on the derivative of the change in frequency <strong>and</strong> therefore, it
could be evaluated also for the unperturbed point (zero displacement
for the perturbing object), leading automatically to α = 1.
In order to have a more accurate measurement, it is possible to
measure also points when the plunger is displaced inside the conductor
plane, in order to have negative displacements. Finally, note that in