Split-post dielectric resonators (SPDR) have been described in [I]-[3]. The purpose of this paper is to demonstrate that such resonators can perform accurate and ...
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Split Post Dielectric Resonator Technique for Precise Measurements of Laminar Dielectric Specimens - Measurement Uncertainties
J Krupka*,,R N Clarket,0 C Rochardt and A P Gregoryt
* Institute of Microelectronics and Optoelectronics Warsaw University of Technology, Koszykowa 75,OO-662Warsaw, Poland t Centre for Electromagnetic Metrology, National Physical Laboratory, Teddington T W l lOLW, UK Abstract- Split-post dielectric resonator was used to measure solid laminar dielectric specimens which had been previously measured by a number of other techniques. Detailed errors analysis of permittivity and dielectric loss tangent measurements has been performed. It was proved that using a 4 GHz split post resonator it is possible to measure permittivity with uncertainties down to 0.3% and dielectric loss tangent with resolution 2xlO-' for well machined laminar specimens. 1 Introduction.
Split-post dielectric resonators (SPDR)have been described in [I]-[3]. The purpose of this paper is to demonstrate that such resonators can perform accurate and convenient complex permittivity measurements upon low to medium loss solid iaminar dielectric specimens. This has been demonstrated by performing error analysis of SPDR technique and measurement upon materials which have also been measured accurately by other techniques. 2 The split-post dielectric resonator theory
The geometry of a split dielectric resonator fixture used in our measurements is shown in Fig. 1. ,coupling loop support D ? \ , I
dielectric resonators
L l-i
/ sample
'
I
dr
I
metal
enclosure
Fig. 1. Schematic diagram of a split dielectric resonator fixture. Such a resonator operates with the T&l6 mode which has only the azimuthal electric field component so the electric field remains continuous on the dielectric interfaces. We have employed rigorous RayleighRitz method to compute the T&l6 mode resonant frequencies, and the unloaded Q-factors which depends on conductor and dielectric lossess and all other related parameters. For low-loss materials the influence of losses on the resonant frequencies is negligible, so the real part of permittivity of the sample under test can be found on the basis of measurements of the resonant frequencies and physical dimensions only, as an iterative solution to the equation (1).
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where: h - thickness of the sample under test, 4 - resonant frequency of empty resonant fixture, f, resonant frequency of the resonant fixwre with dielectric sample, & fbnction of El and h
-
-
I& function was computed, employing the Rayleigh-Ritz metchd, for a number of El and h for a given resonant fixture and tabulated. Interpolation has been used to compute & for the current values of and h. The initial value of & in permittivity evaluation was taken to be the same as its corresponding value for given h and E, =l.Subsequent values of & were found for the subsequent dielectric constant values obtained in the iterative procedure. Because & is a slowly varying function and h so the iterations using formula (1) converge rapidly. of The dielectric loss tangent of the sample was determined using formula (2).
pa - electric energy filling factor of the sample defined as
Qc- Q-factor depending on metal enclosure losses for the resonant fixture with the sample
Qd - Q-fkctor depending on losses in a metal enclosure losses for empty resonant fixture
- electric energy filling factors for the dielectrics of the split post resonators, containing a sample and empty respectively, Q D R-~Q-factor depending on dielectric losses in dielectric resonators for empty fixture, Q - the unloaded Q-factor of the resonant fixture containing dielectric sample P ~ D R,P~DRO
Again the values of K1 and K2 vary slowly with h and E; and h so that they have been computed employing the Rayleigh-Ritz method and tabulated. Interpolation has been used to compute the values of K1 for the current values of h and
3. Uncertainties of measurements The main source of uncertainty of the real part of permittivity is yelate4 to uncertainty of the thickness of the sample under the test. Relative errors of dielectric constants due to thickness uncertainty are
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r A&' =Ah 1
T-
h
Er
where: I a < 2 ,
In most cases T is a number very close to unity except thick large permittivity samples for which T value increases but always remains less then two. Additional factors usually increase the overall uncertainty, e.g. differences between true dimensions of the resonant fixture and permittivity of dielectric resonators and the values assumed in computations. All those extra errors can be analysed indirectly via their influence on the computed values of I(E according to the equation
AK
Adr
K,
dr
Ahr Asd AD AL Ahg +Th +Td +TD+TL +Thg -
2= Tdr-
hr
sd
D
L
(7)
hs
Relative uncertainties for specific dimensions of the resonant structure depend on machining precision for particular parts of the resonator and their values for 4 GHz resonator used in our experiments were: Adr Ahr Aed AD AL = 0.2%, Ahg = 0.5% -= 0.1%, -= 0.5%,= 0.2%, -z z 0.1%,dr hr ed D L hg Computed values of error coefficients for two samples having thickness of 1.4 mm are presented in Table 1.
ET
2 10
TIU 0.853 0.803
TET 0.967 0.778
Tdr 0.3 12 0.432
TL -0.069 -0.058
TD -0.239 -0.257
Thg
0.097 0.050
As one can deduce from the data presented above the most important are coefficients T,, and T, related to properties of dielectric resonator. In practice it is possible to cancel out I(E errors related to those two coefficients by taking into account measured value of the resonant fiequency for empty split post resonator Assuming certain value for permittivity of the split post resonators and ail dimensions of the resonant structure except hr it is possible to choose hr such to get in computations the resonant frequency the same as measured value for empty resonator. Exact numerical analysis has shown that errors of K, due to uncertainty of hr and permittivity practically cancel out if computed and measured resonant frequencies for empty resonator are equal. Using such approach it is possible to compute I(E coefficients for specific resonant structure with uncertainties better then 0.15% so the total uncertainty assessment for real permittivity is A€', Ah I0.15%+T-
Er
h
(8)
Dielectric loss tangent uncertainty depends on many factors, mainly on of Q-factor measurement uncertainty and the value of the electric energy filling factor. For properly chosen sample thickness it is possible to resolve dielectric loss tangents to approximately 2xlO-' for Q-factor measurements with an accuracy of about 1%. 4. Measurement Results and Conclusions
In Table 2 results of complex permittivity measurements using 3.9 GHz split post dielectric resonator and other well established techniques. It is seen that all measurement data agree within specified measurement uncertainties. For very low loss materials, like sapphire or quartz, Q-factors of the
.
308 resonator with a sample is greater than the Q-factor for the empty resonator. Despite this the dielectric loss tangent value determined for such a case is greater than zero due to proper Qc and QDRcorrections. Table 2. Intercomparision of the complex permittivity measurements between 3.9 GHz split post
I
2.359M.3%11.4lE-O4f23% I 2.36H.5%(1.4E-04*10%
I
IHD
Polyethylene
I
The larger the thickness of the sample and its permittivity, the larger the influence of the sample on the electromagnetic field distribution in the fixture. In such cases Qc and QDR values can change significantly relative to their corresponding values for empty fixture. The split-post dielectric resonator has been shown to make a usefbl, accurate and convenient contribution to dielectric metrology. In this intercomparison, it has been shown that it can offer accurate measurements with quantifiable uncertainties for wide ranges of Permittivity and loss. The method plugs a gap in the frequency coverage of existing methods. 5. References.
[ l ] J. Krupka J. and Maj S., "Application of the TG16mode dielectric resonator for the complex permittivity measurements of semiconductors", CPEM '86 ,pp. 154-155, 1986. [2] Nishikawa T. et al, "Precise measurement method for complex permittivity of microwave substrate", CPEM '88, pp. 154-155, 1988, Figure captions [3] J. Krupka, R.G. Geyer, J. Baker-Jarvis, and J. Ceremuga, "Measurements of the complex permittivity of microwave circuit board substrates using split dielectric resonator and reentrant cavity techniques", pp.2 1-24, DMMA'96 Conference, Bath, U.K. 23-26 Sept. 1996. [4] J. Krupka, K.Derzakowski, M.E. Tobar, J. Hartnett, and R.G.Geyer, "Complex permittivity of some ultralow loss dielectric crystals at cryogenic temperatures", Measurement Science and T ~ C ~ ~ O vol. IO~ 10,Ypp.387-392, , Oct. 1999. [5] J. Krupka, K. Derzakowski, B. Riddle and J. Baker-Jarvis, "A dielectric resonator for measurements of complex permittivity of low loss dielectric materials as a function of temperature", Measurement Science and Technology, vo1.9, pp. 175 1-1756, Oct. 1998
