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Investigation on Displacement Sensitivity of Cylindrical Dielectric Resonators for Sensor Applications
P. Boughedaoui, R. Barrère, M. Valentin
To cite this version:
P. Boughedaoui, R. Barrère, M. Valentin. Investigation on Displacement Sensitivity of Cylindrical Dielectric Resonators for Sensor Applications. Journal de Physique III, EDP Sciences, 1995, 5 (8), pp.1245-1253. �10.1051/jp3:1995189�. �jpa-00249376�
Classification Physics Abstracts
77.85 77.80 06.70M
Investigation on Displacement Sensitivity of Cylindrical Dielectric Resonators for Sensor Applications
P. Boughedaoui, R. BarrAre and M. Valentin
Equipe de Physique et Electronique des Capteurs, ENSMM, 16 route de Gray, 25030 Besangon Cedex, France
(Received 18 July 1994, revised 26 January 1995, accepted 3 May 1995)
Abstract. In order to design displacement resonant transducers in UHF-band, characteris- tics of shielded cylindrical dielectric resonators are investigated for the first higher order modes
by means of the transverse resonance method of cascaded waveguides. The effect of the geomet-
rical and electrical parameters of the structure on the displacement sensitivity and conductor Q-factor is examined for the TEo16 mode and for higher order modes.
1. Introduction
Dielectric resonators made with high permittivity, low loss and high temperature stable ce- ramics are commonly used in microwave filters and oscillators [1,2]. Recently several works have shown the capabilities of dielectric resonators in sensor applications. Thus quarter wave
dielectric resonators, partly metallized, have been used in the UHF range in order to realize displacement sensors with a very high sensitivity [3]. An active probe using a rectangular
dielectric resonator loaded by samples under test has been used as humidity sensor [4] and a passive dielectric probe has been realized to measure the complex electromagnetic parameters
If and ~) of materials [5).
In this paper, with displacement resonant sensor application at UHF in mind, a shielded cylindrical dielectric resonator placed in MIC configuration, is analyzed. At resonance, the
dielectric material confines the electromagnetic energy within the resonator where a real prop-
agation occurs internally along the dielectric rod. Fields decay exponentially outside the rod.
This exponential decay means that a finite amount of energy lies in the whole space outside and may be perturbed by any change of this space, in particular, by the displacement of a metal- lic plate in the vicinity of the dielectric. Consequently, a variation of the resonant frequency
occurs as function of the plate displacement.
A model based on the transverse resonance method [6-9] is briefly presented for the eval- uation of the resonant frequency of the structure. Influence of the structural parameters on
displacement sensitivity is studied taking the conductor Q-factor in consideration. Theoretical and experimental results are presented for the first higher modes.
© Les Editions de Physique 1995
1246 JOURNAL DE PHYSIQUE III N°8
2. Resonant Structure Analysis
The present analysis considers the structure shown in Figure I. A dielectric resonator of radius a, height h and permittivity end is placed coaxially in a cylindrical metallic cavity of radius b.
Lower and upper metallic planes denoted Pi and P2 are respectively distant of hi and h3 from the resonator faces. The structure may be divided into three regions, two cylindrical waveguides
filled with dielectric of relative permittivity err and er3 for region I and III, respectively, and a dielectric loaded waveguide with permittivity er2 around the rod for region II.
z b
a '
,
Pn Z=0
, '
' i
, i
i
Fig. I. Cross section of a cylindrical resonant structure and its equivalent circuit.
For practical use, a circular inhomogeneous waveguide with lossless propagation inside the dielectric rod and energy which must decay exponentially radially outside the rod, is considered.
The eigenfunctions (characteristic functions) are found by the process of matching tangential
components of the E and H fields on the boundary between adjacent regions or by radial
resonance transverse method. They give the conditions under which real propagation occurs
along the rod internally and fields decay exponentially outside the rod [10].
Usually, results are given for end * 36 [1,11]. In this paper, end
= 80 is chosen in order to realize UHF sensors of small size at frequencies around I GHz.
The normalized radial wavenumber ii a = [k(end p~j~~~ a is given in Figure 2 as function of the normalized wavenumber koa for the first modes of the dielectric loaded waveguide with
er2 " 1, bla
= 2 and for end
" 80. This mode chart is used to identify the series of mode
resonances which can be excited in the shielded dielectric rod.
3. Resonant Frequency Evaluation of a Multilayer Microwave Sensor
The studied sensor structure is represented by the three lines shown in Figure I and is solved
by means of a generalized axial transverse resonance relation where the mode- matching gen-
eral procedure is expressed in impedance or admittance terms of microwave networks. The transmission-line characteristic impedance Z~~ is related to the longitudinal propagation con-
stant -f~ in ith medium.
For lossless structures, -f~ can be either purely real (region I, III) or purely imaginary Ire- gion II) and the following relationships exist between the characteristic parameters for the
three layered structure.
1f2 " 3~ " 3 [h~~rd ~~j~~~ (~)
~ia
7
3
0.5 1.5 2
Fig. 2. Mode chart of a dielectric-loaded waveguide Erd
" 80, bla
= 2.
~ ~ l/2
'n °I (j) h~il (2)
~ l/2
~t3 = a3 = ((~) lfr3j (3)
where xm is the m~~ rooth of the Bessel function of the first kind and n~~ order, Jn(x) = o for TEnm or quasi TEnm and the m~~ rooth of J[(x) = 0 for Tmnm modes or quasi Tmnm
modes in the metallic guides I and III [12,13].
The resonance condition of the transmission-line system can be conveniently written from
the input-impedance looking both ways in a reference plane located at one step discontinuity
of the microwave structure. For a ground plane at the distance hi of the dielectric resonator and a perfectly-conducting plate located above the dielectric resonator, at the distance h3 from the resonator, the reference plane can be chosen according to the sensor application. For
displacement sensors, pressure sensors, position sensors, it is convenient to select the reference plane at the interface of the air medium (er3 " 1) and the dielectric resonator.
In the case of displacement sensors whose experimental results are given below, the resonance condition is given from ih~ the input impedance of the short-circuited line III, and from
~
Z h~+h the input impedance of line II loaded by the short-circuited line I. These impedances
are expressed by means of the well known equation of impedance transformation on lines and
lead to the resonance condition
Z~~
~~~~(~~ )
j~~l'~~) + Z~~th-f3h3 " 0 (4)
c~+ cit -fi it -f2
1248 JOURNAL DE PHYSIQUE III N°8
theory ~~3
exrenment £~d~ h
~# hi
mi~~
~
~l*0S5mm
i~1
~ $
~ =085mm
hj=o
2 4 6 8 IO 12 14 2 4 6 8
h3(mm) h3 (mm)
Fig. 3. Influence of hi and h3 on the resonant frequency and 8ensitivity of the TEo16 mode (Erd # 80,
a = 19 mm, h
= 17 mm).
From the characteristic equation [10] and for a given mode and parameters ~,end> the an- alytical expression of the eigenvalue (la as function of normalized frequency koaa is obtained
by polynomial regression. Then the resonant frequency is determinated by solving equation (4). These computations have been performed by using Mathematica built-in algorithms (least
squares and root finding) [14,15).
4. Results for the TEoi& mode
The results shown in Figure 3 are obtained in UHF-band, with the dielectric resonator param- eters end " 80, a =19 mm, h = 17 mm and with err
= er2 = er3 = 1 and b
= oo.
The theoretical values of the resonant frequency are also plotted for comparison.
The resonant frequency and sensitivity d f/dh3 of the TEoi& mode are higher since hi and
h3 are low. Therefore the choice of low spacing hi and h3 contributes to the improvement oi the displacement sensitivity so that it can reach the value of -50 kHz/~m. However, the
sensitivity is related to conductor Q-factor through the following relationship [16,17].
~ ~ ~
~~~
where 6s is the skin depth, Qc~ the Q-factor due to power loss in the upper metallic plate P2.
Then, the variation of Qc~ as function of h3 can be evaluated as shown in Figure 4 by substituting experimental frequencies and sensitivities into equation (5).
As can be seen from this Figure, when the spacing h3 decreases from 14 mm to 0, the Q- factor Qc~ decreases by about a factor of ten. Because of the structure symmetry, the same
property is verified by the lower plate Pi
Therefore, upper and lower plates must be far enough away from the resonator to minimize losses in the metallic plates.
The same dielectric resonator is now put on a substrate of thickness hi
" 1.27 mm dielectric
constant err " 10.5 and inserted in a metallic cavity as shown in Figure I. The resonant
0 2 4 6 S lo 12 14 h3 (mm)
Fig. 4. Influence of h3 on the conductor Q-factor Qc~ (Erd
" 80, a = 19 mm, h
= 17 mm, hi " 6
mm, Erl " I, b - W).
~
theory
~ a £r3
exrenment *~
h £~
hi £ri
5
l 13 IO 13
I
§ ( 15
~ /
20
~ I 97
~ '~~ ,'
3 2
2 4 6 8 lo 12 14 2 4 6 8 lo 12 14
h31mm) h3(mm)
Fig. 5. Influence of h3 and bla
on the resonant frequency and sensitivity of the TEo16 mode
(end " 80, a = 19 mm, h
= 17 mm, hi " 1.27 mm, Eri
" 10.5).
frequency f and sensitivity df/dh3 are evaluated and measured as function of h3 for different ratios bla. The theoretical and experimental results presented in Figure 5 for the TEoi& mode
are in good agreement and show that the sensitivity increases (up to 40 percent) with the metal shield radius b.
The ratio bla should be generally chosen greater than 1.5 to minimize the waveguide con-
ductor losses. Moreover, in order to reduce losses in the metallic planes Pi and P2 (Fig. I),
the wave must be propagating in the waveguide II and evanescent in the waveguides I and III.
Therefore, the normalized frequency must verify the following conditions at resonance,
(koa)~~ < koa < (koa)~ and koa < (koa)~ (6)
1250 JOURNAL DE PHYSIQUE III N°8
6
g
E 4
Gi 2
3
2
0 02 04 06 08
~ri~~rd
Fig. 6. (bla)rrax versus Er,/Erd Ii
= 1,3).
The cutoff normalized frequency (koa)c~ of the waveguide II is obtained as function of bla by solving the characteristic equation [10] at the limit of the cutoff straight line defined by
ii a = je~d i) koa (7)
In the case of waveguides I and III, the cutoff normalized frequency is given by
~~°~~~' blah ~~~
with I
= 1,3 and x = 3.832 for TEoi mode.
The geometrical parameters hi, h3, h, bla oi the resonant structure must be chosen so that the resonant frequency verifies conditions (6). In particular, bla should not exceed a certain value 16la)max given as a junction oi the ratio emlord for the TEoi mode in Figure 6.
It is found that the agreement between measured and computed values is quite good for the TEoi& mode. The accuracy oi the frequency calculation is about I percent with a single mode approximation. With the same model and for comparison, the resonant frequency and the
sensitivity d(af)/d(h3 /h) are theoretically evaluated as function of h3/h and shown in Figure
7 for a set of dielectric resonator parameters end " 20, 37, 80 and a/h
= 0.25, 0.5, 1, 2 in the
case oi hi - oo, b
- oo, eri " er2 " er3 = 1 I-e- dielectric resonator placed in tree space.
These results show that the transducer sensitivity at an upper metallic plate displacement, for a given radius a, is higher since the plate is nearer the dielectric resonator or that the res- onator height is low. The sensitivity increases also when the resonator permittivity decreases.
This agrees with the fact that the electromagnetic energy is not as confined as with a higher di- electric constant. For UHF transducers a compromise is needful between sensitivity, resonator
permittivity and a reasonable size.
5. Results for Higher Order Modes
The resonant frequency and the sensitivity of the structure are also evaluated for HEMII&, Tmoi& and HEM21& modes.
The agreement between measured and computed values (accuracy about 10%) is not as
good as for the TEoi& mode. This shows the limits of validity of the model with a single mode
approximation. However, in this investigation, the theoretical results allow the identification of
~h3
) h
af(mmGHz) ++ d(afJld(h31h)(mmGHz)
~
l I
~h =2 ~h= l14
~h = l12
~h
= I ~h =
~~_ ~j~
alh =2
0 05 J5 2
h~
37 h
af(mm GHz) ~/ d(aflld(h3lhJ (mm GHz)
0 5 5
h~ih
~~ ~ 5000 a/h = l14
~h = l12
~h =1 1°°°°
alh = I
~h = 2
20000
~h3
80 h
%+
af(mm GHz) a d(aflld(h3lh) (mm GHz)
os i is
~h=2 l14
J/~ ~h=I
IS 125
"2
o 5
Fig. 7. Influence of the parameters h3/h, a/h, Erd on the resonant frequency and sensitivity of the TEo16 mode.
higher modes. The measured sensitivities (Fig. 8) reach 700, 600 and 80 kHzll~m for HEMI IS, HEM21& and Tmoi& modes respectively. These high sensitivities are obtained for a low value of conductor Q-factor when the conducting plate is in close vicinity of the resonator (h3 < 0.5
mm).
6. Conclusion
A model using the transverse resonance method with a single mode approximation has been used for the evaluation of the resonant frequency of a cylindrical cavity loaded axially by a cylindrical dielectric resonator. It yields results with precision about I percent for the TEoi&
1252 JOURNAL DE PHYSIQUE III N°8
~~
~
a £~++
h £~
~~~21b
hi Eri
~
/
HEMII~
q
~~01b
2
h~(mm)
Fig. 8. Measured sensitivities of the four lowest modes as function of h3. (End " 80, a = 19 mm, h = 17 mm, hi
" 1.27 mm, Eri
= lo-S, bla = 1.97).
mode while for other modes, coupling to next higher modes must be included in the model.
The effect of the geometrical and electrical parameters of the structure on the resonant
frequency sensitivity to upper metallic plane displacement was investigated for the TEoi&
mode. The effect of the displacement of metallic plates on the conductor Q-factor is also taken into account in this investigation. A comparison between the first higher modes sensitivities is presented. Theses results are useful for understanding the general properties of shielded
cylindrical dielectric resonators as displacement resonant transducers. In the UHF range, it has been shown that compromises are needful between reasonable sizes, conductor Q-factor and displacement sensitivity.
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