Notes on Dielectric Characterization in Waveguide

Transcription

1 Notes on Dieletri Charaterization in Waveguide R.Nesti, V. Natale IRA-INAF Aretri Astrophysial Observatory 1. Theory Let's suppose we have to haraterize the eletromagneti properties of a dieletri material, in terms of the relative dieletri onstant and the loss tangent tan. The loss tangent is given by ZZ b tan ~ (1) where is the angular frequeny and ~ À h F/m is the dieletri onstant of vauum. The ontribution of the bulk ondutivity ( ) have been here separated to any other sort of losses ( ZZ ). The problem of the eletromagneti sattering due to a dieletri slab is briefly introdued. To this aim the model shown in Fig. 1 is onsidered: a unit amplitude wave is inident on a dieletri slab of thikness inside a waveguide, whih is assumed air-filled in the remaining part. Fig. 1: Multiple refletion model of the eletromagneti sattering of a dieletri slab in a waveguide Due to the material disontinuity at the two interfaes between air and dieletri the eletromagneti field sattering an be modeled by multiple refletions and an be treated as an extension of the lassial problem of the inidene of a plane wave at the planar interfae between two semi-infinite dieletris.

2 We introdue the field impedane in the air and in the dieletri material, respetively, A and A whose expression is given by A~ where is the wave propagation fator and is given by the following general expression m ~ ~,, G (3) with the propagation onstant, the eigenvalue, the wave onstant and the attenuation onstant being given in Table I aording to material, domain and mode of propagation. As regards the magneti permeability, only standard materials (inluding vauum) having ~ ~ h H/m are here onsidered TABLE I l l tan Coaxial able Retangular waveguide Cirular waveguide (Free spae) (,, ) (Radius ~ 9) Mode TEM TE TE À 9 Air Dieletri $ ( m $ b tan 7 tan 4tan 5 m Air Dieletri tan $ l $ l h(hos 7 ] l $ 0 l sin tan $ h(h 7 ] l$ b tan $ $ (2) It is worth to spend a few words about Table I where medium eletromagneti properties are organised in a quite strange manner whih is however best suited to implement later formulas. Basially Table I has to be seen as a two olumn table, the first olumn giving an eletromagneti property and the seond one its expression in terms of known quantities. The vauum propagation onstant is given first than a relative propagation onstant is defined, both used to derive further parameters. A first ompliation omes out in the definition of the eigenvalue as it is different for the three domain we desire to haraterize: the oaxial able supporting TEM propagation, the retangular waveguide (TE mode propagation) and the

3 irular waveguide (TE mode propagation). It has to be notied that the oaxial able and free spae, as propagation domains, an be onsidered the same ase. A seond diversifiation is made as regards the medium of propagation and we distinguish between air (whih is assumed to have vauum features) and dieletri with the propagation onstant (that is the unbounded medium wave propagation fator) given for both media. It is also useful to define here $, ( and 7, mainly introdued to easily express further important parameters as they have no partiular physial meaning: in the air medium, $ represents the square of the wave onstant normalized to the vauum propagation onstant while in the dieletri medium, where losses are onsidered, it is used to ompute both a sort of normalized wave onstant amplitude (() and the wave onstant phase ( 7). The above definitions allow a ompat and simple form to express the wave propagation fator (Eq. (3)) and to analytially obtain its real and imaginary part, respetively ( ) and ( ); about these last items a useful approximation is also given in the ase of small losses tan $. Analogously, a simple and ompat form for the Fresnel oeffiient is defined at the seond interfae as follows: ~ A A A ba We an now try to desribe the multiple refletion proess in order to give, as elementary proesses ombined together, one single refletion and one single transmission oeffiient aounting for, respetively, the refleted and transmitted wave. As regards the refletion oeffiient, by using the Fresnel oeffiient (4) and the multiple paths shown in Fig 1, we have a summation of infinite terms ~ b b b b b ÀÀÀÀÀ (5) At eah rebound the refletion oeffiient gains a fator ontribution whih an be highlighted as follows B b ~ b 4 5 (6) Providing ** and arg, as in physial passive devies, the infinite summation in (6) onverges to, with ~, thus giving b 4 5 ~ b ~ (7) The same proedure holds analogously to the transmission oeffiient, giving ~ (4) 2 3 ~ b b b b ÀÀÀÀÀ ~ b 4 5 B ~ 0 (8) (9)

4 ~ b Note that the infinite summation in (9) onverges to 1. (10) 2. Dieletri material haraterization in retangular waveguide Although dieletri material haraterization an in priniple be done both in free spae and in waveguide, this last domain has the big advantage to allow very good auray with easy measurement setup and easy proedures for material preparation. On the ontrary, in free spae, setting-up testing failities requires very high attentions to prevent undesired effets, like spurious refletions and standing waves, that have sensitive effets on the measurement results. We now show how with only two different measurements giving the amplitude of the S 11 and S 21 parameters of a two port waveguide, we an obtain aurate predition of and tan. Let's suppose we have first, ase, a measurement of the empty waveguide of ²³ length. If the instrument alibration is well done we should read ideally S 11 ~ for the refletion oeffiient; in pratie, in the hypothesis of a perfetly fabriated waveguide, we read a very small value that an be interpreted as the zero of the ²³ measure. Ideally the amplitude of S should be equal to unity if no losses were present. In pratie we see, as a result of the measurement, a urve versus frequeny whih has a harateristi shape and whih is always down unity, highlighting some sort of losses; we an assoiate this to the ohmi losses of the metalli wall of the waveguide, a distributed effets obeying to an exponential law aording to the formula ²³ es e ~ (11) From this we an estimate the attenuation onstant. To evaluate aurately this parameter other effets, not only distributed but also loalized like for example radiation losses due to the leakage from the flanges, should be taken into aount. However the differential nature of the method here presented dramatially redues the effets of the non preise estimate of. In fat all the effets that are ommon to the two measurements are almost eliminated by differential omparison. As seond and last measurement, ase a dieletri material is inserted inside the waveguide and a new measurement is done. To get aurate information from the measurement it is important a very preise dieletri fabriation that should be in length and fill exatly the waveguide. In this ase the measurement of both sattering ²³ ²³ parameters, named S and S, is signifiant as they an be diretly related to, 11 1 respetively, and of Eq. (7) and (10). If we observe (7) we an predit minima and maxima of the measured refletion oeffiient. A first analytial approximation of suh extreme points is given by minimizing or maximizing the numerator in (7). Thus putting ~ (12)

5 we have the following ondition: ~ l $ h ~ l $ h ~ Á ~ ÁÁÀÀ minima (13) ~ l $ h ~ l $ h ~ b Á ~ ÁÁÀÀ maxima (14) where is the light veloity in vauum. Both (13) and (14) an be solved for. Considering for example a retangular waveguide with largest dimension the above equations lead to following extreme points: min ~ b o 8 9 l (15) max ~ o8b 9 b8 9 l (16) Fixed, depending on the length, obviously not all the points given above are meaningful sine the first 5²³ points an fall under the utoff region. However a numerial model based on (7), (10) and (11) an be implemented to predit the behavior of a waveguide filled in with dieletri, using for the propagation fator an expression of the form: ~, Á Á G (17) where and, as defined in Table I, analytially depends on geometrial and eletrial features and is obtained from the ( ) measurement aording to (11). Considering, tan and parameters (and thus, tan and, tan ) ²³ ²³ family of urves ( ) an be generated and ompared with S 11 and S 1 measurement data. To solve the problem of dieletri haraterization a distane 1 an be defined in the spae of real funtions of real variable and searh in the spae of the allowed value for ²³ ^ Á ~ + ²%³²%³ + (18) b and tan the one minimizing the distanes ^6* * Áe S 11 e7 and ^6* * Áe S 21 e7. The above onsiderations suggest to hoose a dieletri length to be few wavelengths so that two or more minima and maxima fall in the bandwidth of interest thus making easier to 'measure' the distane between the urves. In pratie, to ahieve about a few parts per thousand auray for and a few perent auray for tan, simply an eye inspetion to ompare the plots is suffiient if three minima (or maxima) or more fall inside the bandwidth. ²³ 1 An example of distane is the root mean square value + + l ²%³ ²%³ ~ % % %

6 3. Example results Some examples are here given to show appliations of the theory previously desribed. First we onsider a foam like material, that is Styrodur of BASF. A 50.3mm length sample have been mahined to fill in an aluminum WR42 standard waveguide and measurements have been done in the 18-26GHz band. First the empty waveguide has been measured in order to have an estimate of and the results are given in Fig. 2. In retangular waveguide the TEfundamental mode attenuation onstant due to wall ohmi losses may be modeled by the ondutivity through: 9 ~ 4 b 5 (19) where, are the waveguide setion dimensions ( ), and the surfae resistane is given by 9 ~ n (20) Fig. 2: Transmission of an empty 50.3mm WR42 aluminum waveguide. In the simulation model a finite ondutivity of ~ 5 h S/m is used. One has been estimated the dieletri parameters an be determined by omparing simulation urves with refletion and transmission measurements by inserting the dieletri in the waveguide. As final result in the Styrodur ase it was found a dieletri relative onstant ~À and tan ~h. As regards, the results have been determined by best mathing between the two plots of Fig. 3.

7 Fig. 3: Refletion oeffiient of a Styrodur filled 50.3mm WR42 aluminum waveguide. In the simulation model (Eq. 7) ~À, tan ~h ~ 5 h S/m are used. In the ase of tan the plots in Fig. 4 has been obtained as a result of best mathing simulations and measurements. Here the plots in Fig. 2 has been onsidered to remove ommon effets whih has not been taken into aount by using only wall ohmi losses to haraterize the attenuation onstant in (19). In fat, like ohmi losses, these not onsidered ommon effets produe the same offset in Fig. 2 and 4 (here mainly in the lower part of the band) between simulations and measurements. Fig. 4: Transmission oeffiient of a Styrodur filled 50.3mm WR42 aluminum waveguide. In the simulation model (Eq. 10) ~À, tan ~h, ~ 5 h S/m are used.

8 As a seond example we onsider high density Polyethylene (HDPE). A 98.6mm length sample material have been mahined to fill a opper WR28 standard waveguide and measurements have been done in the 26-40GHz band. In this ase we 4 obtained ~Àh S/m, ~ 2À 278, tan ~ 1 h as the plots in Figs. 5-7 show. Fig. 5: Transmission of an empty 98.6mm WR28 opper waveguide. In the simulation model a finite ondutivity of ~ 2.6 h 7 S/m is used. Fig. 6: Refletion oeffiient of a HDPE filled 98.6mm WR28 opper waveguide. In the simulation model (Eq. 7) ~ 2À278, tan ~ 1 h 4 ~ 2.6 h 7 S/m are used.

9 Fig. 7: Transmission oeffiient of a HDPE filled 98.6mm WR28 opper waveguide. In the simulation model (Eq. 10) ~ 2À278, tan ~ 1 h 4 ~ 2.6 h 7 S/m are used. Also if the seond material is quite different from the previous one the same results have been obtained in terms of and tan aurate estimation. In both ases it has to be notied the very good agreement between simulations and measurements. Sine in the HDPE ase the length of the dieletri is about 7-8 wavelengths a ²³ ²³ larger number of minima appears in the plots of S 11 and S 1 (Figs. 6, 7) with respet to the Styrodur ase (Figs. 3, 4), exhibiting about wavelengths dieletri length. As a third example we onsider Plexiglas. A 66.2mm length sample material have been mahined to fill a opper WR42 standard wave guide and measurements have been done in the 18-26GHz band. In this ase we obtained ~h 6 S/m, ~ 2À 54, tan ~ 4.35 h 3 as the plots in Figs show. One again it has to be notied the impressive agreement between simulations and measurements. In this ase the dieletri length is about 4 wavelengths.

10 Fig. 8: Transmission of an empty 66.2mm WR42 aluminum waveguide. In the simulation model a finite ondutivity of ~ 6.0 h 6 S/m is used. Fig. 9: Refletion oeffiient of a Plexiglas filled 66.2mm WR42 opper waveguide. In the simulation model (Eq. 7) ~ 2À54, 3 tan ~ 4.53 h ~ 6 h S/m are used.

11 Fig. 10: Transmission oeffiient of a Plexiglas filled 66.2mm WR42 opper waveguide. In the simulation model (Eq. 10) ~ 2À54, 3 tan ~ 4.3 h ~ 6 h S/m are used. As a general and pratial onsideration we an say that a larger helps for better haraterization of the dieletri onstant, sine it is assoiated essentially to the spaing between minima (or maxima) of the and urves. As regards the tan haraterization the hoie of is less ritial also if we an say that smaller lengths (2 or 3 wavelengths) are slightly preferable. For the sake of ompletness we olleted the above presented results in the following Table I together with some data already published. The data without referene are from this work. TABLE I Material f(ghz) tanx Ref Styrodur 4000CS (BASF) " * 4 " <1 8 " <1 8 " HDPE " " " Plexiglass " " " * in Np/m

12 Referenes [1] C.A. Balanis, Advaned Engineering Eletromagnetis, J.Wiley & Sons: New York (NY), [2] R.E. Collin, Foundations for Mirowave Engineering, MGraw-Hill: New York (NY), [3] F.E. Gardiol, Introdution to Mirowaves, Arteh House: Dedham (MA), [4] J.W. Lamb, Int J. Infrared and Millimeter Waves, 18, n.10, [5] Guozhong Zhao, et al., J.Opt. So.Am. B, 19, n.6, June [6] K. Seeger, IEEE Trans. MTT 39, n.2, Feb [7] Afsar and Button, Pro. IEEE 73, n.1, Jan [8] S. Mariotti, Nota Tenia IRA, Marzo 2005.