Reentrant Klystron Cavity
as an Electromechanical Transducer
J. J. Barroso, P. J. Castro, L. A. Carneiro, and O. D. Aguiar
National Institute for Space Research/INPE
P. O. Box 515, S. José dos Campos 12245-670, SP, Brazil
Abstract. The resonance properties of reentrant cavities with
circularly cylindrical and conical inserts are examined to
quantify the resonant frequency dependence on the gap spacing
between the end of the insert and the cavity's top plate. An
experiment performed on a 1.0 GHz cavity fabricated from
aluminum shows that the resonant frequency downshifts when
the top plate made 1.0 mm thick is loaded at the center with
weights as light as 10 g. This translates into a tuning coefficient
of 3.0 MHz/µm, which can achieve a three fold increase through
optimization of the cavity dimensions looking at application of
the transducer in gravitational wave antennas.
Index Terms: reentrant cavity, electromechanical transducer
I. INTRODUCTION
Strong electric fields for accelerating or modulating an
electron beam find important microwave applications in
linear accelerators and RF power sources, where for a given
cavity stored energy the strongest field is desired [1]. Also
relying on intense fields to increase the energy sensitivity,
electromagnetic cavity-based transducers [2] must be
operated at high fields to maximize the electrical coupling to
an external mechanical transformer. A cavity of this sort is
usually accomplished by a reentrant klystron cavity (Fig. 1)
where intense electric fields develop across a short gap.
In this paper we examine both theoretically and
experimentally the resonance properties of azimuthally
symmetric reentrant cavities, namely the relationship between
the resonant frequency and the cavity dimensions with
emphasis on how the frequency varies when the top plate is
subjected to mechanical deformation due to an externally
applied force.
II. CAVITY ANALYSIS
As pictured in Fig. 1, for a small gap spacing the electric
field lines of the corresponding operation mode run in the gap
region as from one plate to the other of a parallel plate
capacitor, whereas in the rest of the cavity the field is
substantially as in a terminated coaxial line.
Joaquim J. Barroso, barroso@plasma.inpe.br, Pedro J. de Castro,
castro@plasma.inpe.br, Leandro A. Carneiro, leandro04@h8.ita.br, Odylio
D. Aguiar, odylio@das.inpe.br.
This work was supported by FAPESP, SP, Brazil.
Fig.1. Reentrant cavity schematic showing electric field lines
On condition that the gap spacing d is much shorter than
the resonant wavelength the concept of lumped circuit
elements becomes meaningful, whereby we treat the reentrant
cavity as a shorted coaxial line terminated by a capacitor
(Fig. 2).
Fig. 2. Approximate equivalent circuit of the cavity in Fig. 1
Thus for a line of length l, outer diameter 2r
2
, inner
diameter 2r
1,
and terminal capacitance C the resonance
condition requires that the loop impedance be zero, so that
0
1
tan
0
00
=+
Cj
jZ
ω
β l (1)
where
00000
/2 εµωλπβ == , )ln()2/1(
12000
rrZ εµ=
and in a first approximation the gap capacitance is expressed
as )./(
2
10
drC πε= Assuming l
0
β <<1, (1) simplifies to
1
000
=lβω ZC giving the resonant wavelength
1
2
2
10
ln2
r
r
d
r
l
πλ = (2)
We note that if d is small compared to l, as we are
assuming, then
0
λ is large compared to r
1
. If, as is usually
the case r
1
is of the same order of magnitude as r
2
and l ,
then this means that
0
λ is large compared to all the
dimensions of the cavity, justifying our assumption that the
cavity can be treated as a lumped constant problem. Then if
we wish to design a cavity for a given
0
λ , we see that the
smaller d is, the smaller the cavity dimensions become, so
that we can make in this way a conveniently small cavity
resonant at a long wavelength. Although enlightening, the
simple formula (2) does not provide an accurate estimate of
the resonant frequency (in some cases the error may be larger
than 40%) as its derivation lacks the cavity capacitance that
accounts for the fringing fields in the transition region
intermediate the coaxial and gap spaces. Calculated as [3],
d
rre
rC
2
)(
ln4
22
1
2
2
101
l+−
= ε (3)
the cavity capacitance C
1
when added to C
0
much improves
the accuracy of the equivalent circuit. Generalizing the
configuration shown in Fig. 1, a reentrant cavity with a
coaxial conical insert (Fig. 3) has been modeled by Fujisawa
[3] as a lumped LC circuit leading to the following
parameters:
)ln(ln
2
0
1
01
0
1
2
0
r
r
rr
r
r
er
L
−
−=
πµ
l
(4)
d
rC
2
0
0
0
π
ε
= (5)
=
0
1
ε
C
++
−
d
e
d
d
e
r
d
rr
MM
α
α
α
α
π
π
sin
ln
2
cot
sin
ln
2
)(
0
2
0
2
0
ll
(6)
where =
M
l
)2(3
)23()]2)((3)(2[
012
2
012
22
02112
2
01
rrr
rrrrrrrrrr
−−
−−+−+−+− l
(7)
01
1
tan
rr
d
−
−
=
−
l
α (8)
Fig.3. Definition of geometrical parameters for the reentrant cavity with
coaxial conical insert
for which the error incurred in estimating the resonant
frequency )(2/1
100
CCLf += π lies within a few percent
as has been verified by Fujisawa [3] upon comparison with
experiments. Accordingly, the accuracy of the formulas
becomes better for larger r
0
/l
M
and smaller l
M
/8
0
, indicating
the post radius and the resonant wavelength compared with
the relative size of the cavity.
To foresee the predictions of (4-8) we examine below how
the electrodynamical properties of the reentrant cavity relate
to the shape of the coaxial insert by considering two types of
posts: a truncated cone and a circular cylinder, the latter of
which the general expressions (6-8) apply when α=π/2 (r
1
=r
0
,
Fig. 3). Markedly different for each coaxial insert, the plots in
Fig. 4 show the dependence of resonant frequency f
0
on
radius r
1
for fixed major radius r
2
(=3.5 cm) and cavity length
l (=1.4 cm) with gap spacing d varying from 0.2 mm to 1.0
mm in steps of 0.2 mm. For the circularly cylindrical insert
(Fig. 4(a)), f
0
starts decreasing for increasing r
1
and after
reaching a flat region all the curves come nearer to the each
other at large values of r
1
, eventually merging to a single
curve in which the particular behavior entailed separately by
the gap d on each curve is lost. With most the
electromagnetic energy stored in the gap region and with the
electric-field lines running axially, this regime (r
1
→ r
2
)
closely resembles the TM
010
-mode operation in a circular
cavity. In fact we note that the frequency curves going
upward tend to an asymptotic value that is consistent with the
resonant frequency of a TM
010
-mode cavity with radius
r
2
=3.5 cm, i.e. f
TM010
=(15/π)(χ
01
/r
2
) =3.28 GHz, (where
χ
01=
2.4048 is the first zero of the Bessel function J
0
(χ)). By
contrast, for the cavity with the conical insert (Fig. 4(b)) all
the frequency curves slope upward and keep from
approaching to the each other as r
1
increases. Moreover, we
remark that the frequency separation given by the upper and
innermost curves, for instance, at r
1
=2.0 cm, for the uniform
cavity (0.57GHz) is nearly half that for the cavity with
tapered insert (0.97 GHz), which thus exhibits higher
sensitivity to variations in d. To verify the resonance
properties of the reentrant cavity with tapered insert an
experiment is carried out in the next section.
Fig. 4. Resonant frequencies as function of the inner radius r
1
for
reentrant
cavities with (a)
circularly cylindrical and (b) conical inserts. The curves are
parameterized at increments of 0.2 mm in the gap d.
III. EXPERIMENT
The resonance properties of a reentrant cavity with conical
insert is experimentally examined by looking at the effect on
the resonant frequency of reducing the gap spacing through
application of a bending force at the center of the circular top
plate with clamped edges. Fabricated from aluminum, the
cavity has dimensions to allow operation in the klystron
mode (with radial and axial electric field lines) around 1.0
GHz, a value well below the cutoff frequencies of potentially
competing modes, since the major radius r
2
(=3.2 cm) being
constrained to r
2
<λ
0
χ
11
/(2π) where χ
11
=1.8411, the first root
of J'
1
(χ)=0, bounds the lower frequencies for propagation of
either TM or TE modes on fc=(c/2π)(χ
11
/r
2
)=2.5 GHz.
The coaxial insert is a truncated cone of radii
0
r
=0.5 cm,
r
1
=1.00 cm and height that provides a gap of 0.2 mm
between the end of the post ad the upper plate made 1.0 mm
thick. Resonant frequencies are measured by using the
reflection-type circuit configuration in Fig. 5 where the cavity
fields are both excited and detected by means of a single
electric probe inserted through a 1.0-mm-diameter hole
drilled halfway across the cylindrical wall, as illustrated in
Fig. 6.
Fig. 5. Experimental setup to measure resonant frequencies
Fig. 6. Reentrant cavity under test
On applying a deflection force (using a set of calibrated
weights) we then measure the corresponding downshifted
frequencies, which are compared in Fig. 7 with calculated
values. In the calculation, the nominal gap spacing d (=0.2
mm) in (3)-(7) is reduced by the maximum deflection δ
max
at
the center (Fig. 8) determined from the following expression
that gives the deflections due to pure bending of a clamped
circular plate loaded at the center [4]:
)(
16
ln
8
Pr
),(
22
2
2
2
rr
D
P
r
r
D
Pr −+=
ππ
δ (9)
where P is the load applied, )1(12/
23
ν−= EhD denotes the
flexural rigidity of the plate of thickness h=1.0 mm, modulus
of elasticity E=69.0 GPa and Poisson's ratio ν=0.3. We see in
Fig. 7 that a weight of mass as low as 10 g loaded on the
plate is unambiguously ascertained, with the deflected plate
downshifting the free-loading 1.2003 GHz resonant
frequency to 1.1979 GHz, which lies within 5.6% above the
calculated value of 1.1309 GHz. Accordingly, since the
cavity parameters r
0
/l
M
=0.567 and l
M
/8
0
=0.065 are within the
applicability region (r
0
/l
M
>1/3) of the formulas, the
calculated values stay below those calculated within an error
of about 5.0 % in the observed range of frequencies. We note
in addition that the frequency calculation assumes a flat
spaced d-δ
max
apart form the top of the conical post, while in
the actual experiment the deflected plate takes on the shape
of a concave surface as illustrated in Fig. 8. And of course,
had we considered the gap d reduced by half the maximum
displacement, d-δ
max
/2, the resulting calculated curve would
have appeared closer to the experimental points, for the
klystron-mode resonant frequency increases with the gap
spacing.
Fig. 7. Measured and calculated resonant frequencies as function of the
loading force.
Fig. 8. Deflection of a clamped plate loaded at the center
IV. CONCLUSION
We have discussed the feasibility of a 1.0 GHz reentrant
cavity as a parametric transducer by demonstrating in an
exploratory experiment the transducer sensitivity to
deflections of the 7.0-cm-diameter, 1.0-mm thick aluminum
plate when loaded with weights as light as 10g. While
showing high energy sensitivity, the transducer tuning
coefficient ∆f/∆d=3.0 MHz/µm, which converts displacement
to electrical units. Through proper selection of the cavity
geometry by increasing r
1
(with r
2
and l fixed) and reducing
both
0
r
and the gap d, the tuning coefficient can achieve a
three fold increase aiming at the device application in a
resonant mass gravitational wave antenna under development
at INPE [5,6]. In this experiment the reentrant cavity actually
operates at 10.0 GHz, with its dimensions (in comparison
with the 1.0 GHz prototype described here) being scaled
down by a factor of 10, thus rendering the antenna’s cavity
100 times as sensitive.
REFERENCES
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[2] D. G. Blair, E. N. Ivanov, M. E. Tobar, P. J. Turner, F. van Kann, and I.
S. Heng, "High sensitivity gravitational wave antenna with parametric
transducer readout", Phys. Rev. Lett., vol. 74, pp. 1908-1911, March 1995.
[3] K. Fujisawa, "General treatment of klystron resonant cavities", IRE
Trans. Microwave Theory Tech., vol. 6, pp. 344-358, Oct. 1958.
[4] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells,
New York: McGraw-Hill, 1959, p. 69.
[5] O. D. Aguiar, et al. “The status of the Brazilian spherical detector”,
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o
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