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

76 5. RADIOFREQUENCY MEASUREMENT
In principle, it is possible to measure the single resonant frequencies
also when the cavities are coupled together, it is sufficient to shortcircuit
the cavities near the one under measurement 6 .
Aim of this paragraph is to show a method which allows the measurement
of the own resonant frequencies <strong>and</strong> the true coupling coefficient
between two or three coupled cavities. The method is characterized
by an high degree of reliability.
3.1. Coupling measurements on two cells. In the case of two
coupled cavities, the system can be represented by an equivalent lumped
circuit, as it is explained in chapter 3, <strong>and</strong> by using the Kirchhoff equations
one can obtain [17]:
1 − f 2 1
f 2
1 − f 2 2
f 2
= k 2 , (5.2)
where f1 <strong>and</strong> f2 are the single cell frequencies (without any coupling),
k is the coupling coefficient <strong>and</strong> the equations are written for an infinite
quality factor. The two resonant frequencies of the coupled system are
the solutions of the above equation with respect to f:
⎡
⎤
f 2 ± = 1
1 − k 2
⎣ f1 2 − f2 2
2
±
f1 2 + f2 2
4
− f1 2 f2 2 (1 − k2 ) ⎦ (5.3)
In these relations f1, f2 <strong>and</strong> k are unknown quantities. From an experimental
point of view, one measures the two frequencies of the coupled
system f± (looking at the transmission coefficient of the system) keeping
one cavity unperturbed <strong>and</strong> varying the other one by introducing
a perturbation in the cell.
In this way, varying the perturbation we measure the corresponding
resonance frequencies f±, both modified, for each couple of unknown
values, f1 <strong>and</strong> f2 - the first one fix <strong>and</strong> the second variable.
Starting from equation (5.3) <strong>and</strong> looking for invariant expressions
we can find useful relations between the unknown quantities <strong>and</strong> the
measured ones. We used two methods, one more general <strong>and</strong> the second
6 The measurement on a single cavity stacked in a tank is possible if one shortcircuit
the cavities near the one under measurement. This technique takes into
account the coupling slots effect <strong>and</strong> the relative frequency change, but they do
not allow the measurement of coupling coefficient k. Furthermore this procedure,
still having a high degree of sensibility (depending on the instrument), it is not
sufficiently precise for our purpose, since it introduces a systematic error on the
frequency. This error depends on the particular realization of short-circuit <strong>and</strong>
cannot be always determined.
From the practical point of view, the short-circuited cavity becomes a coaxial cavity,
which resonates at a higher frequency respect to the unperturbed cavity (for
LIBO cavities the frequency goes from 3 GHz to around 5 GHz) but this frequency
displacement depends on the type of short-circuit.